CALIFORNIA STATE UNIVERSITY NORTHRIDGE. Evaluation of Ground Effect on the Drag on an HPV Fairing Using CFD

Transcription

1 CALIFORNIA STATE UNIVERSITY NORTHRIDGE Evaluation of Ground Effect on the Drag on an HPV Fairing Using CFD A thesis submitted in partial fulfillment of the requirements For the degree of Master of Science in Engineering, Mechanical Engineering By Dimitry Tsybulevsky May 2012

2 The Thesis of Dimitry Tsybulevsky is approved: Susan Beatty, Eng. Date Mike Kabo, Ph.D. Date Robert G Ryan, Ph.D., Chair Date California State University, Northridge ii

3 Acknowledgements I would like to thank Dr. Robert Ryan for being my graduate advisor and supporting me throughout this thesis. My thesis never would have been completed without his help. I would also like to thank Professor Susan Beatty for helping me during my time in California State University Northridge (CSUN) and being on my thesis committee. Additionally, special thanks goes to Dr Mike Kabo for assisting me with the application process for the graduate program in CSUN and being on my thesis committee. Lastly, I would like to thank the Department of Mechanical Engineering at CSUN for the encouragement and help to complete my Master s Degree in Mechanical Engineering. iii

4 Table of Contents Signature page.ii Acknowledgements....iii List of Tables...vii List of Figures...viii Abstract......xi Chapter 1: Introduction Problem Statement Purpose of the Thesis Background Information Definition of Drag Definition of Ground Effect Definition of CFD and CFD History Drag Measurement Techniques Using CFD Approach Theoretical Values of Drag on the Ellipsoid body Drag Values on Variation With Ground Clearance HPV Fairing Geometry Description Organization of the Thesis Chapter 2: Importation of Solid Model into ANSYS and Mesh Definition Meshing and Preprocessing Modeling of the HPV Fairing and Ellipsoid Geometries in SolidWorks Importing the Model into ANSYS WORKBENCH from SolidWorks Extracting A Fluid Volume for the Models iv

5 Opening the Models in ANSYS ICEM CFD Preparing the Geometry for Meshing Generating the Initial Mesh Using Octree Mesh Approach and Applying the Correct Mesh Size Generating the Tetra/Prism Mesh Using Delaunay Mesh Approach Smoothing the Mesh to Improve Quality Exporting the Mesh into ANSYS FLUENT Chapter 3: FLUENT Setup and Application of Spalart-Allmaras Turbulence Model Background Information in Computational Software and Methodology Turbulence Model Spalart Allmaras Turbulence Model Application of FLUENT Setup Initial Setup Boundary Condition Solution Setup and Mesh Adaption Solution to the Problem Graphical and Numerical solutions Drag Calculation Chapter 4: Baseline Solution and Calibration of FLUENT FLUENT Calibration Using Flat Plate FLUENT Calibration Using Oblate Ellipsoids Results for Oblate Ellipsoids Comparison between Hoerner s data and CFD data v

6 Chapter 5: HPV Fairing Results HPV CFD Test Results HPV Fairing Benchmark Results HPV Fairing at Different Ground Proximities Results Ground Clearance Effect on Pressure and Skin Frication Ground Clearance Effect on Drag and Lift Estimation of Discretization Error Discretization Error Calculation Tradeoff Study Between Ground Clearances Drag and Stability for a Typical HPV Chapter 6: Conclusion..133 References Appendix A Appendix B Appendix C vi

7 List of Tables Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table C Table C Table C Table C Table C Table C Table C Table C Table C Table C vii

8 List of Figures Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure viii

9 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure A Figure A Figure A ix

10 Figure A Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B Figure B x

11 Abstract Evaluation of Ground Effect on the Drag on an HPV Fairing Using CFD By Dimitry Tsybulevsky Master of Science in Mechanical Engineering The purpose of this study was to evaluate the ground effect on the Human Powered Vehicle (HPV) Fairing with different ground clearances, and its effect on drag using Computational Fluids Dynamics (CFD) software. The short term goal of this thesis was to use the CFD software package ANSYS FLUENT, to find how the ground clearance of the 2010 version of the HPV fairing affects the overall drag and to an optimal ground clearance for the vehicle. The long term goal was to create a guide to help future students use ANSYS FLUENT and other ANSYS software to create mesh and CFD studies to find external forces such as drag and lift coefficients on objects moving through a fluid. In order to create a good computational mesh for the HPV fairing flow field, the mesh was first created for standard geometries, i.e. flat plate and oblate ellipsoids. Drag values computed for various meshes were compared to known drag values for those geometries. The results for the flat plate matched within 3.5% of the theoretical results, and for the oblate ellipsoids the difference was less than 5.6% from experimental values. This process helped to optimize the final mesh settings for the HPV fairing and find acceptable results for the drag coefficient with the fairing at different ground clearances. xi

12 As mentioned previously, a long term goal for this thesis was to create a tutorial on how to use ANSYS and FLUENT to create good CFD studies. The tutorial can be used with future California State University, Northridge (CSUN) senior design teams to create body geometries and effectively to accurate results for drag and lift on various bodies. This tutorial can also help with regard to importing the geometry from CAD software and performing the correct model setup in ANSYS. The study for the HPV was conducted as a function of h/l, where h is the ground clearance and L is the length of the HPV fairing. (L= 99 inches and was constant). The ground clearance ranged from 3 to 18 inches including two baseline tests, at 30 and 297 inches away from the ground. All of the results are provided in terms of the streamlines, pressure and velocity magnitude fields, and vorticity contours. The goal was to see how high the body had to be off the ground to eliminate the drag ground effect. It was found that the fairing had to be at least 18 inches of from ground in order to see a significant reduction in ground effect. Additionally a trade off analysis was conducted on the HPV fairing to balance the speed benefit from high ground clearance with vehicle stability during cornering. However, the height required to minimize the ground effect was impractical for the HPV competition due the Center of Gravity (CG) considerations. xii

13 Chapter 1: Introduction 1.1. Problem statement The aerodynamics of human powered vehicles (HPV s) is greatly influenced by the shape of the body and the proximity of the ground to the surface of the HPV bodywork. In most cases the airflow between the ground and HPV bodywork results in a drag increase known as the ground effect. Approaches to lessen this effect fall into two categories: a) creating a specialized fairing skirt which helps to direct the airflow away from the underside of the vehicle; or b) increasing the height of the vehicle from the ground. Neither of those strategies is perfect; each strategy has its upside and its downside with respect to vehicle performance Scope of the Thesis The main goal of this thesis is to conduct a computational fluid dynamics (CFD) study on an HPV fairing by using ANSYS 12.1 and FLUENT in the Mechanical Engineering Design center at California State University Northridge (CSUN). This study analyzes airflow around a typical HPV fairing geometry and assesses the impact of the ground effect at typical HPV speeds. In addition, this study is designed to use the oblate ellipsoid and the flat plate as a calibration tool for the HPV s fairing mesh, boundary conditions and FLUENT setup. Then the experimental results found in Fluid Dynamics of Drag by Hoerner [11] are compared to the CFD results from FLUENT for the oblate ellipsoid to make sure that the software computationally precise. To accomplish these objectives the SolidWorks model created by 2010 CSUN s HPV design team was imported into ANSYS 12.1 and modified to be used within ANSYS-FLUENT. The geometry was cleaned within ANSYS 12.1 WORKBENCH 1

14 Geometry Design-Modeler; then the model was imported into ANSYS ICEM to create the mesh that was used by FLUENT. The mesh incorporates an estimation of boundary layer thickness to insure that sufficient points were used near the HPV s fairing surface to accurately predict velocity gradients in this region. Initially a study was performed on an ellipsoid geometry, which is somewhat similar to the shape of an HPV, and for which published drag data is available. In addition, velocities were chosen to match Reynolds numbers with available data. Using the ellipsoid geometry, a strategy was developed to optimize the program settings to get an effective convergence and solution accuracy in terms of drag force. This included running inviscid flow cases, using coarser mesh for the preliminary calculations, and then using FLUENT mesh refinement capabilities. In addition, different turbulence models such as the Spalart-Allmaras turbulence (SA) model and k-ε model within FLUENT were tried to assess the turbulence model s effect on solution convergence and drag calculations. This study was conducted using several different flow conditions and mesh configurations to determine their effect on the calculated drag values. The analysis was conducted on the 2010 HPV geometry at several different flow velocities with a maximum flow velocity of approximately 40 mph (58.67 ft ). These speeds corresponded to a Reynolds number range of approximately to That means the majority of the flow over the HPV fairing after the expected boundary layer transition point was in the turbulent region. Finally, a study was conducted to assess the impact of geometry changes on computed drag, i.e. changing the proximity of the HPV fairing to the ground surface. sec 2

15 Analyses were run for ground clearance of 3, 6, 9, 12, 15, 18, 30 inches and a freestream case of 297 inches above the ground Background Information Definition of Drag Drag refers to the forces that oppose the relative motion of an object through a fluid, either gas or liquid. Drag forces only act in the direction opposite to velocities not the oncoming flow velocity (or upstream velocity U). For a 3-D object moving through a fluid, the drag is the sum of forces due to pressure differences in the flow field (pressure drag) and shear forces on the object s surface (friction drag). Drag force has been found to be dependent on a fluid s density (ρ), object area (A), flow velocity (U) and a dimensionless drag coefficient (C D ), expressed by the following drag equation: D = 1 2 ρu2 AC D (1-1) The drag coefficient is a function of object shape and Reynolds number, and is usually determined experimentally or by CFD analysis. The area can either be the surface or wetted area, or the projected frontal area depending on the source of the drag coefficient values. Generally the wetted area is used if the total drag is dominated by friction drag. 3

16 Figure 1-1 a shows basic example of drag generated by a solid body moving through a fluid. Figure 1-1: Example of drag generated by solid object (Adapted from ) Definition of Ground Effect Ground effect is a term applied to a series of aerodynamic effects that are important in the automotive and aerospace industries. These effects usually cause an increase in drag force and a decrease in lift force (i.e. increase down force). Ground effects relevant to the automotive industry are due to the proximity of the underside of the moving vehicle to the stationary road surface. The ground effect is easily visualized by taking a canvas tarp out on a windy day and holding it close to the ground; when the canvas gets close enough to the ground it will suddenly be sucked downward due to the lowered pressure in the flow between the tarp and the ground. Some vehicle body components, such as a splitter and a diffuser, can be found under the vehicle s body to help increase the ground effect and improve the downforce of the vehicle. This helps it travel faster through the corners by increasing the vertical force on the tires. 4

17 Ground effects in aerospace applications are due to the proximity of the flying body to the ground. The most important of these effects is the wing in ground (WIG). This is due to the reduction in lift experienced by an aircraft as it approaches a height of roughly the aircraft s wingspan above the ground. Those effects increase as the aircraft approaches the surface, which can lead to loss of control and crashes Definition of CFD and CFD History Computational Fluid Dynamics (sometimes referred to as CFD) is a branch of fluid mechanics which uses complex algorithms in conjunction with numerical methods to solve the partial differential equations describing fluid flow. Advances in CFD software make it possible to perform complex calculations to simulate the interaction of gases and liquids with each other and geometric surfaces defined by Computer Aided Design (CAD) software. Yet even with modern high speed computers, only approximate solutions can be achieved in most cases, particularly for flows involving turbulence and flow separation around blunt bodies because CFD solution is a numerically based. CFD originated in the early part of the 20th century, marked by initial attempts to solve differential equations found in physics and engineering. The main equations governing fluid flow behavior are the Navier-Stokes equations, developed in the early part of the 19th century by George Stokes and Claude Navier. Although the Navier- Stokes equations were a significant development, the analytical mathematical solution of those equations proved untenable at that time period. This led to the development of a large number of simplified equations derived from the Navier-Stokes equation for special cases, which can be tackled analytically using pen and paper or a simple calculator. 5

18 However, these special cases were very limited in terms of describing practical applications. [36] The invention of digital computers led to many changes in solving the complicated Navier Stokes equations. In the late 1940 s, John von Neumann led a group of scientists and engineers to develop modern CFD. The digital computing machines have the analytical solutions of simplified flow equations with numerical solutions of full nonlinear flow equations for arbitrary geometries. Modern day CFD uses high-speed computers to achieve better solutions and improve accuracy of known exact and nonexact solutions to the Navier-Stokes equations such as nonlinear partial differential equations and turbulence analysis. [36] Common CFD codes have a specific structure that revolves around a numerical method or numerical algorithm able to undertake complex fluid flow studies. Most of the CFD codes currently on the market have only three basic elements, which divides the complete simulation to be performed on the specific domain or geometry. The basic three elements are the following: 1. Pre Processor, where the solution domain is defined and the mesh is generated; 2.Solver where the flow equations are solved for the previously defined mesh and domain; and 3. The Post-Processor, where the numerical results are displayed and analyzed Drag Estimation Techniques Using CFD Approach There are several approaches to calculate the drag on a 3-D geometry using the CFD approach. Perhaps the most common and widely used approach to finding drag using CFD is solving the Reynolds Averaged Navier-Stokes (RANS) equations, or the surface integration of stresses, i.e. near field methods. There are several problems with 6

19 this approach to solving CFD problems. For the near field method the problem is usually insufficiently accurate results, for example even if the flow solution is locally accurate in terms of pressure and velocity profile. As for RANS, the problem is mainly related to the numerical solution that generates the drag coefficient. A second problem for the RANS is near field drag computation; it only allows for distinction between pressure and friction drag. [26] Due to the mentioned problems above with the RANS methods, the following approach is used in this thesis to find the drag coefficient of the HPV fairing. This approach is to use the oblate ellipsoid to determine computational precision of FLUENT by finding the proper mesh parameters and turbulence model to provide accurate drag estimates. This approach establishes how fine the mesh should be in order to acquire proper results for drag forces over the HPV fairing. This mesh incorporates estimation of the boundary layer thickness to ensure that there are enough points used near the body surface to accurately predict the velocity gradient within the boundary layer, and the related friction drag. Using the ellipsoid body geometry, a strategy is developed to optimize the program settings within the FLUENT solver for effective convergence and solution accuracy Experimental Values of Drag on the Ellipsoid Bodies An oblate ellipsoid is a disk shaped spheroid where a=b>c, and prolate ellipsoid is a rugby ball shaped spheroid where a=b<c. Drag research on oblate ellipsoids and other similar shapes is very limited. There are only a few real-world examples of such types of bodies. The HPV fairing is being assumed as a streamlined geometry and the oblate ellipsoid is used to help with the initial 7

20 setup of the CFD approach. However, there is a lot of literature that discusses drag information on similar types of bodies, such as prolate ellipsoids and spheroids. This may be used as a baseline reference for the work being performed in this study. The information in Figure 1-2 comes from a well-known drag expert, Dr. S.F Hoerner. In his book Fluid-Dynamic Drag (1965), Hoerner presents the drag coefficient of numerous shapes such as oblate ellipsoids, prolate ellipsoids, and spheroids in both 2-D and 3-D flow fields. Figure 1-2 presents the wetted area drag coefficient of an oblate ellipsoid with different fineness ratios of body of revolution l d over a range of Reynolds number (R e ). The d is the diameter of the ellipsoid at its widest part, and l is the length of the ellipsoid. The points that are shown in Figure 1-2 are the experimental data that were found for those bodies, and the dashed lines represent the theoretical drag for fineness [6, 11, 12] ratio and is given with the following equation. C DWET = C f,lam 1 + d l d l 2 (1-2) 8

21 Figure 1-2: Drag Data on 3-D Bodies of Revolution Aligned Straight-and-Level (Adapted from Hoerner Fluid Dynamics of Drag, 1965, 6-16) Figure 1-2 represents the effect of Reynolds number on the drag of the ellipsoid with different fineness ratios of l d. In the laminar region where the Reynolds number is less than 10 5 the drag coefficients tend to be higher. When the Reynolds number reaches between 10 5 and 10 6, the boundary layer flow begins to transition from laminar to turbulent, and a significant drop is seen in the drag coefficient. After the drag reaches its minimum value, the drag begins to rise slightly as the boundary layer transition point continues to move forward. Finally, when the Reynolds number reaches 10 7, the flow is fully turbulent and the drag starts to decrease again. In reference to Figure 1-2 the higher the Reynolds numbers, the lower the drag at the fineness ratios. Additionally, the higher the fineness ratio the lower the drag coefficient will be. 9

22 To define the fineness ratio that is used in Figure 1-3 and its relationship to the friction drag coefficient C f the following equation is employed. C DWET C f = d l d l 3 (1-3) To find the correct ratio of wetted area to frontal area S wet S f for streamline bodies, the wetted area can be approximated as S wet = (0.7 to 0.8) l perimeter, where the perimeter is equal to π d, and the frontal area is equal to πd2. The ratio of wetted area to frontal area is equal to: 4 S wet S f = 0.75 l d π d 2π 4 = l d = 3 l d This expression is then substituted into equation 1-3 to find the C D C f for the frontal area coefficient and curve fit for Figure 1-3 as derived by Hoerner. C D = 3 l C f d d d + 21 l l 2 (1-4) Figure 1-3: Drag coefficient of streamlined bodies as a function of their thickness ratio (Adapted from Hoerner Fluid Dynamics of Drag, 1965, 6-19) 10

23 Figure 1-3 illustrates the drag coefficients based on frontal area of streamline bodies as a function of their fineness ratio, the points in Figure 1-3 are the experimental data and the dashed lines are evaluated from equation 1-4. When the fineness ratio increases, the drag coefficient also increases expect for low fineness ratios. The drag coefficient for the HPV fairing based on its frontal fineness ratio of 3.53 is between 0.02 and for Reynolds numbers This was found using Figure 1-3 and equation 1-4. It is difficult to isolate the critical Reynolds number on the oblate ellipsoid where the transition will occur from laminar to turbulent flow with estimated Reynolds numbers from 500 to 600 thousand for that geometry. Figure 1-4 [8] shows the wetted area drag coefficient for the x l = 0. 5 prolate spheroid for several different surface roughnesses. The roughness has an enormous effect on the drag coefficient in the low Reynolds numbers. This is because the flow is not fully developed and this adds to the total skin friction coefficient as illustrated in figure 1-4. During Dr. Dress s study the critical Reynolds number reached about 800 thousand where the transition from laminar to turbulent region occurs, and the minimum drag coefficient happened at a Reynolds number of almost 1.2 million for a fine grit of 80. The different types of runs show the effect of skin roughness from laminar to turbulent flow, and the effect on the wetted drag. 11

24 Figure 1-4: Drag Data from a x = 0. 5 Prolate Spheroid Aligned Straight-and-Level free transition l is the base run, 80 is the fine grit, and 40 is the rough grit (Adapted from Dress, NASA Technical Paper , 29) Drag Values on Variation With Ground Clearance Once the potential of using aerodynamic downforce in automotive racing applications was realized, many teams started to experiment with other methods to increase aerodynamic downforce other than simply attaching inverted wings. It was found that with a larger underbody area of the vehicle, significant levels of downforce could be generated. This kind of effect was first seen in 1935 in the racing circuit with early wing prototypes used in ground effect models. [13] 12

25 Figure 1-5 illustrates a basic principle of ground effect on typical car shapes represented by an oblate ellipsoid and half streamlined body. However, to understand ground effect the nature of the flow under the vehicle must be considered. The top part of the Figure, shows an oblate ellipsoid that is approaching the ground. The flow under the oblate ellipsoid and the downforce ( C L ) are increasing as distance to the ground reduces and creates low pressure. If one looks at the bottom part of the Figure and closely examines the half streamlined body, the drag coefficient is seen to be nearly the same as the oblate ellipsoid. The lift force is opposite due to the reduced flow under the body, with the result of increased lift due the reduced ground clearance. In both Figures the transition to significant ground effect starts to occur at h l < However, this only applies to these specific geometries. The transition point can shift to either left or right depending on the fineness ration and overall shape of the geometry. There are several options for the car body shape to generate lower pressure under the body. Option one is to streamline the underbody to create low pressure. Option two is to create a seal between the underbody of the car and the ground and only leave the rear portion of the car open. Then the low pressure behind the car would dictate the pressure [14, 15] under the car. 13

26 Figure 1-5: Effect of ground Proximity on the lift and drag of two streamline bodies (Adapted from Race Car Aerodynamics by Joseph Katz 1995) 14

27 1.4. HPV Fairing Geometry Description Figures 1-2 and 1-3 are used as a reference to estimate the expected drag coefficient for the HPV fairing. If one assumes the HPV fairing is a body of revolution then the oblate ellipsoid can be used as a computational precision tool for the mesh setup, turbulent model selection, and optimize FLUENT parameters. To apply Figures 1-2 and 1-3 one needs to estimate an equivalent fineness ratio for the HPV fairing, and a range of drag values can then be estimated for the HPV fairing in freestream flow. This is used as a benchmark for the HPV fairing analysis. Figure 1-6 shows the dimensions of the HPV fairing; this data can then be used to find the fineness ratio based on the height of the HPV fairing which is equal to 3.53 for half of a body of revolution. However, because the HPV fairing is assumed to be a body of revolution the height needs to be doubled to get the correct fineness ratio l = the h resulting wetted area drag coefficient value for Re = is approximately C D,Wetted =0.009 and C D,surface area = Figure 1-6: Dimensions of the HPV fairing from SolidWorks 2010 where l= inches, h=d= inches 15

28 1.5. Organization of the Thesis The remainder of this thesis will be organized as follows. Chapter 2 describes the model design and importation of the model into ANSYS WORKBENCH and fluid volume extraction. It also explains how to import the model in to ANSYS ICEM and the mesh setup and creation. Lastly, it will be explained how to import the mesh from ANSYS ICEM to ANSYS FLUENT. Chapter 3 explains how to operate FLUENT using ANSYS WORKBENCH and apply FLUENT setups as an initial setup, materials for the fluid and geometry, dynamic mesh, and boundary conditions. It will demonstrate how to use FLUENT to generate numerical and graphical solutions for the HPV fairing geometry with different ground clearances ranging from 3 inches to 18 inches away from the ground. Chapter 4 presents the results of the baseline solution of the oblate ellipsoid with l d = 2&4 and results for the flat plate. This chapter also compares the CFD results of the baseline solution to the results found in Chapter 6 in Fluid Dynamics of Drag by Hoerner. [11] Chapter 5 presents the results of the HPV fairing with different ground clearances ranging from 3 inches to 18 inches away from the ground. Then the results from the HPV fairing CFD analysis are compared to the benchmark results (freestream and 30 inch ground clearance). In addition, the results for drag and lift are discussed, and calculations of discretization error are presented. Then the final part of Chapter 5 will include the trade-off study regarding the optimum vehicle height while considering both vehicle stability and aerodynamic drag. 16

29 Chapter 6 is the conclusion and the summarization of the study. It is based on the results shown in Chapters 4 and 5. References and an appendix follow the conclusion. 17

30 Chapter 2: Importation of Solid Model into ANSYS and Mesh Definition 2.1.Meshing and Preprocessing The pre-processing of a CFD procedure consists of several inputs for the flow problem that are done by the user in CFD software. For this study the pre-processing software is ANSYS ICEM CFD, and the solver software is ANSYS FLUENT. The inputs are then transferred into a form made suitable for use by the solver. The pre-processor is the main connection between the CFD solver and the user. The user has to complete several significant steps in the pre-processing stage of the CFD problem. A schematic of the process is shown in Figure 2-1.The following definition, gives a brief explanation of these steps. 1. Define the geometry of interest: This step uses ANSYS DesignModeler CAD software within ANSYS WORKBENCH to help design and model the topology of the fluid flow domain inside or outside the geometry. This domain is defined and optimized for the best CFD results. 2. When the geometry preparation is defined within the pre-processor software, the fluid domain and every surface affected by the fluid is then also defined. Each fluid and surface has its own distinct property; those properties are used in the CFD process and must be defined at this stage. The output of the DesignModeler software is a xxxx.agdb file. 3. Meshing is the third step. Because the CFD process uses a finite volume method, the domain of interest has to be divided into structured and unstructured elements. All the elements are connected to each other through nodes to and from the flow domain. For this study ANSYS ICEM CFD software is used to create the mesh in the form of a 18

31 xxxx.mesh file. The quality of the mesh contributes to the accuracy of the final results. 4. Definition of boundary conditions is the final step at the pre-processing stage. Each CFD domain needs an initial condition to begin calculations, which is defined by the user s input. In addition, the CFD code implements the boundary conditions at a specific locations. The following few sections will explain these four steps in complete detail and explain how to use ANSYS 12.1 for external flow problems. Lastly, Figure 2-1 illustrates how the files from the different software packages move through the overall solution process..agdb ANSYS ICEM CFD.mesh SolidWorks.SLDPRT ANSYS DesignModeler ANSYS FLUENT.wbpj ANSYS WORKBENCH Figure 2-1: Block Diagram illustrates where each file type goes to 19

32 2.2. Modeling of the HPV Fairing and the Ellipsoid Geometries in SolidWorks All of the solid models that were used in this study were designed and drafted using SolidWorks Computer-Aided Design (CAD) software, using inches for dimensions. The fairing was originally designed and modeled by the California State University Northridge (CSUN) Human Powered Vehicle (HPV) Team for their competition in April An ellipsoid model was also designed to represent a simpler geometry and was used as the baseline for this thesis. The ellipsoid model establishes the mesh fineness requirements to acquire good results for the drag force, based on comparison with published results from Fluid Dynamics Drag by Hoerner data. [11] The modeling of the ellipsoid geometry in SolidWorks was a little challenging, because the ellipsoid had to represent the fairing shape as closely as possible. The ellipsoid was created using the lofted boss/base tool in SolidWorks. However, before that could be done, several planes were created so that a 2-D ellipse could be drawn on each plane with different chord lengths A and B. This is illustrated in Figure 2-2. A B Figure 2-2: Representation of an ellipse geometry B=99in and A=49.5in After all of the 2-D schematic geometries were drawn, the lofted boss/base tool was used to create the 3-D ellipsoid body that can be seen in Figure 2-3. The ellipsoid 20

33 model dimensions are: the chord length (l) is 99 inches; height(x) is 49.5 inches, and the diameter (d) of the ellipsoid is inches. The fineness ratio of l d is then can be found as 99 = 2. This ratio is then used to find the drag of a non-oblate ellipsoid body Additionally, another ellipsoid was created in SolidWorks with a fineness ratio of l d = 4, and was used as a baseline test in FLUENT. Additional comparisons were made with a flat plate geometry which is useful because the drag force on a flat plate is completely due to surface stresses. There are a few reasons why two oblate ellipsoids are used to calibrate FLUENT and set correct mesh parameters for the HPV fairing. The first reason is to match the results from FLUENT runs to the known results from Fluid Dynamics Drag by Hoerner. The second reason is to find the limitation of FLUENT on predicting drag on similar geometries with different fineness ratios, as the flow behaves differently for a Falter shape. Generally, a smaller l d ratio will have a larger contribution of pressure forces to the overall drag, especially if the boundary layer separates on the rear portion of the body. Figure 2-3: 3-d Ellipsoid body from SolidWorks 21

34 After the models were created and saved in SolidWorks, one needed to import those models into ANSYS 12.1 for geometry calibration and model clean up before the models were meshed and used within ANSYS FLUENT Importing Model into ANSYS WORKBENCH from SolidWorks ANSYS WORKBENCH is a Computer Aided Engineering (CAE) software package that is used in engineering simulation and analysis. It is an innovative project organizer that ties together the entire simulation process. It helps the user go through several complex studies at once with drag and drop menus. It also has powerful user controls, automated meshing abilities, project level update mechanisms, and integrated [40, 37] optimization tools, which enable complex simulation and product optimization. The next few Figures show a step by step explanation process to import any SolidWorks model into ANSYS WORKBENCH, and clean up the geometry so it can be properly meshed. Figure 2-4 shows how to load the geometry in ANSYS WORKBENCH. In order to load the SolidWorks model in ANSYS WORKBENCH, the user first has to open ANSYS WORKBENCH, then go to the component systems and select Geometry (A). Then the geometry tab is placed on the main WORKBENCH screen, and it then becomes a cell. In order to load the geometry, the user must right-click the Geometry..? tab, and then scroll down until import geometry has been reached. After left-clicking on this item, a new window will open. Then user must left-click browse tab and load the specific geometry (B) to be modified. After the geometry is loaded into the WORKBENCH, the user must double click with the left mouse button on the geometry cell number 2, and ANSYS DesignModeler 22

35 will load. The user then is able to clean, modify, edit and fix the geometry so a better mesh can be created for future analysis of the model. This is explained in Section A B Figure 2-4: ANSYS WORKBENCH front screen; A- geometry is selected first; B-geometry cell where geometry is going to be imported 23

36 Extracting a Fluid Volume for the Models The next few Figures will show step by step how to extract the fluid volume around the imported geometry. The fluid volume must be extracted because one must correctly define the volume that is being occupied by the fluid around a specific solid model. Figure 2-5: Ellipsoid model with in ANSYS DesignModeler and the selection of the external flow. Figure 2-5 illustrates how once the geometry is loaded into ANSYS DesignModeler the user can then begin to select what kind of fluid volume to apply to the specific model, such as internal or external fluid volume. For this study an external fluid volume is being used. This is because the imported geometry represents a solid body and the air flow is external to the body surface. 24

37 Figure 2-6: Selection of shape and cushion type Figure 2-6 illustrates the shape and the cushion size of the fluid volume enclosure. The cushion size is also known as the domain size. For this study the shape of the fluid volume is the box shape, since it is convenient for generating the mesh around the solid body. Since the CFD process is a numerical approximation approach that uses the finite volume method to solve the Navier Stokes equations, the fluid volume domain is going to be composed of an Octree Mesh, sometimes referred to as an unstructured mesh. In order to create the fluid volume domain, the user must set the cushion size and select either uniform or non uniform size. For this thesis the non-uniform cushion size will be used on all the models. This is done to make a more efficient study that does not require a large quantity of computing power. The ellipsoid model was run in freestream condition without any ground plane representation. The HPV fairing simulation consisted of eight different cases. The first two cases are set as benchmarks, where one is in freestream condition and the other one simulation a ground clearance of 30 inches. The other six cases will simulate the HPV fairing with ground clearances ranging from 3 to

38 Figure 2-7: Generated enclosure for the oblate ellipsoid in freestream Figure 2-8: Editing of the enclosure based on symmetry Figures 2-7 and 2-8 illustrate the generated fluid volume enclosure for the solid model, and the editing process for the fluid enclose based on model symmetry about the XY plane. This makes the computation more efficient because it only has to analyze half 26

39 of the model to achieve the same results. In order to create the symmetric model, the user must right click on the Enclosure tab in the tree outline, and then select the edit selection tab. After the user has selected the previous command, the model enclosure can then be edited to the user s specifications and the correct symmetry plane. The user can then select up to three planes of symmetry. As mentioned earlier this model is only symmetric to one plane, the XY plane. In order to select the symmetry plane, the user must left click on the not selected tab and then the user must select the corresponding plane from the tree outline, then press apply. In order to generate the new model, the user must press the Generate tab to create the symmetric model about the XY plane. This is illustrated in Figure 2-9 where one can see the selection of the total number of planes that can be used at the same time, and the symmetry plane selection. Figure 2-9: Selection of symmetry planes. For this study it is the XY plane. 27

40 Y + =3x X + =3X X - =6x Z=3x Y - =3x Figure 2-10: Fluid volume for the ellipsoid model Figures 2-10 and 2-11 show the final view of the oblate ellipsoid s model and fluid enclosure, and the HPV fairing within the non-uniform fluid volume box. The oblate ellipsoid fluid volume box is X + =Y + =Y - =Z=3 times chord length, and X - =6 times chord length. The fairing fluid volume box is X + =Y + =Z=3 times chord length, X - =6 times chord length, Y - =3 to 18 inches for the test cases, and for the benchmarks it is 30 inches and 297 inches. The domain size was selected to help decrease the total computing power while maintaining accuracy. The optimal domain size for a wing was found by Amir Mohammadi in his thesis and this data is being used as a reference for the domain size used here. [21] Before the mesh can be created, the model needs to be exported as an xxxxxx.agdb file. In order to save the ANSYS DesignModeler file, the user must do the following steps; File>Export > xxxxxx.agdb> then Save. Once the file is saved, it then can be opened by ANSYS ICEM CFD, and a proper mesh can be applied to the solid model and the fluid volume box. 28

41 In addition to creating the fluid volume, naming the surfaces that represent the boundary conditions will help later with ANSYS FLUENT setup and the meshing process in ANSYS ICEM CFD. In order to name the different surfaces, the user must right click on the surface and then click edit to name the surface. For the oblate ellipsoid and the HPV fairing model, the surfaces that are created are the inlet velocity, outlet, boundary volume box, and symmetry plane. The boundary volume box for the oblate ellipsoid is made out of three surfaces that surround the geometry. However, for the fairing the bottom surface is named ground plane and the volume box is made only of two adjacent surfaces. This is illustrated in Figure 2-12 and Table 2-1. Y + =3X X + =3X X - =6X Z=3X Y - =3 to 18 inches Figure 2-11: Fluid volume for the Fairing model 29

42 Plane Name Surface Name For Ellipsoid Surface Name For HPV Right (YZ X+) Velocity Inlet Velocity Inlet Left (YZ X-) Outflow Outflow Top (XZ Y+) Bottom (XZ Far side (XY Z+) Symmetry (XY Fluid Volume Box Fluid Volume Box Fluid Volume Box Symmetry Plane Table 2-1: Surface names for ellipsoid and HPV Fairing Fluid Volume Box Ground plane Fluid Volume Box Symmetry Plane Fluid Volume Box Outflow Velocity Inlet Ground plane Symmetry Figure 2-12: Plane location and names 30

43 Opening the Models in ANSYS ICEM CFD Before the meshing procedure can begin, the file that was saved by DesignModeler must be opened in ANSYS ICEM CFD. In order to do that, the user must do the following steps; File>WORKBENCH Reader>select xxxxxx.agdb file> then Open. Prior to the file being completely loaded into ANSYS ICEM CFD, the user has to go to the scroll down menu below and select the options that are illustrated in Figure Then the user must press apply. A B Figure 2-13: Importing an xxxxxx.agdb file into ANSYS ICEM CFD CFD (A). Opening the xxxxxx.agdb in ANSYS ICEM CFD CFD (B) 31

44 2.3.3.Preparing the Geometry for Meshing Figure 2-14: Extracting the feature curve from the symmetry plane Select those locations for the fluid volume area Figure 2-15: Demonstration the correct location 32

45 Figure 2-14 illustrates how to prepare the geometry that was loaded into ANSYS ICEM CFD so that the correct mesh and grid can be generated. In order to extract the curves from the surface, the user must do the following steps: Geometry tab > Create/ Modify Curve icon> Extract Curves from Surfaces icon, then select the surface on the screen. The user has to click on the glass icon to select all appropriate visible objects, or use the following shortcut key v. The plane that is selected for this study is the symmetry plane. After all the correct surfaces are selected, the user must click apply or OK. Following the Extract Curves procedure, the body for the fluid has to be created. In order to do that, the user must start with the Geometry tab again, and then the user must click the Create Body icon. Following that, name the part as the fluid name; any name can be used to name the region. For this study the name that is used is Fluid Volume. In order to name the fluid region, the user must select Centroid of 2 points for the location and the Material Point icon to select the location of the fluid volume. Then the user must click the two screen locations to select the fluid body region as demonstrated in Figure Following that, the user must click OK to finish creating the fluid volume area and proceed to the meshing setup. In addition, the user must create parts from the Subsets by selecting the inlet velocity, outlet, and the fluid volume boundary, and then right click on the Subsets to create parts. These names, are used when meshing in ANSYS ICEM CFD, and setting the boundary conditions and parameters in ANSYS FLUENT. 33

46 Generating the Initial Mesh Using the Octree Mesh Approach and Applying the Correct Mesh Size The strategy that is used for the mesh process is to have a prismatic or structured mesh around the solid model and then transition to an unstructured mesh. The prismatic mesh represents the boundary layer and is defined as a stair step mesh to decrease the required computing power. The height and the mesh density of the prismatic layer was set to represent the estimated boundary layer thickness around the solid models, i.e. oblate ellipsoids, flat plate and HPV fairing. Then the prismatic mesh transitions to an unstructured mesh to create a hybrid mesh around the solid model and inside the fluid region. Assigning the correct mesh for each model was a trial and error method. The reason behind this is that each model used slightly different mesh parameters, and it also varied from robust to fine mesh. It also depended on the size and shape of the geometry. The Scale Factor multiplies other mesh parameters to globally scale the model, for example if a Max Element Size of a given entity is 64 units and the Scale Factor is 0.3 units, then the actual maximum element size will be = 19.2 units. After countless tries, the correct scale factor was found to be approximately 0.3 for all the models. For that reason, all the models used a proper mesh for balancing accuracy with computed memory requirements. The maximum element size that was selected ranged from This value was selected due to the fact that an Octree Mesh scales by a power of two, and the Octree algorithm is limited to datasets of resolution of power of two. For that reason our values range from (or ). This is very important because all other values that will be input into the maximum scale factor will be rounded off to the closest power of two. In 34

47 order to set the parameters, the user must select Mesh tab> Global Mesh Setup icon > Global Mesh Size. After the correct input is input the user must click apply/ok. This is demonstrated in Figure Lastly, the general grid topology will be talked in chapter 3. Figure 2-16: Meshing sizing with ellipsoid of ratio l/d=2 After the meshing sizing is completed, the user must select the Part Mesh Setup icon. This icon is selected in the Mesh tab area to specify the mesh parameters. In order to create the prism mesh, the user must first select the prism option in the mesh parameter area, only for the solid model and the symmetry plane. The prism height is set to , depending on the model, so it can build the correct boundary layer as learned in ME692. For the ellipsoid and fairing geometry surfaces the maximum size is set in the range of 2.5-3; this creates a proper surface mesh for the solid geometry. Also the user needs to input at least 90 for number of prism layers of to be created, and a height ratio of for the growth factor. This corresponds to the maximum thickness (δ) in the turbulent boundary layer, which is approximately 2 inches. This number was found using the 35

48 calculations that can be seen in appendix A for the boundary layer thickness for the laminar, turbulent, and transition layers on a flat plate with a length equal to that of the ellipsoid and the fairing models. For the fluid volume box (inlet velocity, outlet, symmetry and open domain) the maximum size is set to 64 to allow create an appropriate volume mesh. After the mesh parameter setting are complete, the user must press apply. This is shown in Figure 2-17 for the ellipsoid and HPV fairing models. Figure 2-17: Mesh parameters step for the ellipsoid A B Figure 2-18: Mesh density box setup (a). Shifting of mesh density box to refine wake region (b) The density box is created to represent the wake region of recirculation flow immediately behind the model. The wake region is chaotic due to boundary layer separation on the rear portion of the body. The density box allows local control over the mesh density in the wake region to correctly represent the flow. 36

49 In order to create the density box that represents the wake region, the user must first select the Mesh tab> Create Mesh Density icon, then select the size of the density box. For this study the size was selected at 32, and the ratio and width were left at zero. The user then must select the density location as an entity. After the density box is selected, the user must click OK to generate it. Note that at this point the box surrounds the solid body. In order to shift the density box to the expected wake region location, the user must click Geometry tab> Transform Geometry icon > Translate Geometry icon, then select the density box and keep the translation method as explicit. Before the density can be shifted the model needs to be measured by the Measure Distance feature. Following that the density box is shifted by half of the model length. In this study the model was 99 inches long so the density box was shifted 44.5 inches in the negative X direction to represent the True Wake region. This is illustrated in Figure Following the completion of the creation and shifting of the density box, to generate the mesh, the user must first click the Mesh Tab> Compute Mesh icon, then the user must select the Create a Prism Layers and click Compute, as Figure 2-19 illustrates. Following that another mesh has to be defined to refine the present mesh of the model that can be correctly analyzed within ANSYS FLUENT. This is the Delaunay mesh step, and it will be discussed later in the chapter. The reason why an Octree Mesh was used as opposed to a Delaunay Mesh is to minimize the numerical error as much as possible. This also helps to minimize the total computing power needed to create a solid mesh. [39] 37

50 Figure 2-19: Computing the initial mesh Figure 2-20 illustrates the cut plane that allows the examination of the prism layers in the mesh around the solid model. Please note that the prism height floats, as the height was initially set to These numbers illustrate that the first few prism layers start growing very slowly and there after grow exponentially. The variation in layer thickness (float) is not significant for the model because the surface mesh size is relatively uniform. The mesh density near the solid body does not vary with axial position as defined in ANSYS ICEM CFD, note that the mesh is adjusted during the analysis in ANSYS FLUENT with mesh adaption. 38

51 A B C Figure 2-20: Mesh analysis using a cut plane in the XY plane (A); YZ plane (B); XZ plane (C) The mesh process is completed by performing a check done on the mesh to find any errors that may cause problems during the analysis in FLUENT. In order to check the mesh, the user must do the following steps; Edit Mesh tab > Check Mesh tab. The user needs to keep the default settings and then click OK. 39

52 Generating the Tetra/Prism Mesh Using the Delaunay Mesh Approach Once an Octree Mesh has been checked and no errors have been found, the Delaunay Mesh can be generated. The Delaunay Mesh more efficiently fills the volume, and it has a smoother volume transition. This kind of mesh works a lot better with FLUENT to help calculate better results for drag for all the models according to the ANSYS ICEM CFD user manual. [39] Figure 2-21 displays the steps to generate the Delaunay Mesh within ANSYS ICEM CFD. In order to generate the Delaunay Mesh, the user must do the following steps: click on Mesh tab> Global Mesh Setup > Volume Meshing Parameters, and select the Delaunay option from the drop down menu. The user must enter a scale factor of 1.2, memory scaling factor of 1 and the Delaunay Scheme must be T-Grid according to the ANSYS ICEM CFD user manual. [39] After all the correct options have been selected, the user must click Apply. In order to start the computing process, the user must click on the Compute Mesh icon and select the Delaunay method from the drop-down menu, and then disable the Create Prism Layers option. The user must make sure that the Existing Mesh option is selected from the drop-down menu because the mesh is generated based on the Octree Mesh; then finally click Compute. 40

53 Figure 2-21: Delaunay Mesh Setup Smoothing the Mesh to Improve Quality The smoothing of the mesh is done to improve its quality. The smoothing approach involves the initial smoothing of the interior elements without adjusting the prism elements. After the initial smoothing is complete, the prism elements then will be smoothed by themselves. In order to smooth the mesh, the user must click the Edit Mesh tab > Smooth Mesh Globally tab. To smooth the mesh that was generated using the Delaunay Approach the user first has to smooth the interior elements without touching the prism elements. This is done by opening the Smooth Elements Globally control panel. The first step that the user must do in this process is to set the number of smoothing iterations; this number was set to 25. The second step is to enter the Up to Value ; this value 41

54 specifies the quality level up to which the program will attempt to smooth the mesh. It was set to 0.5 this was based on ANSYS ICEM CFD settings. [39] Then for the criterion the user must select the quality option from the drop down menu. Lastly the user must set all the elements to get smooth except for PENTA_6 which was set to freeze. The reason why PENTA_6 was set to freeze is because it is a five sided element with six nodes as a prism element. These elements are usually perfect, but they may be damaged by the smoother as it adjusts to optimize the nearby tetra elements. By selecting the freeze option in the Smooth Mesh type for the PENTA_6 elements, it protects them from being damaged. When smoothing those kinds of elements the values for the Up to Value should be reduced to 0.01 so only the worst of the PENTA_6 elements are adjusted, and the number of smoothing iterations should be dropped to 2. Figure 2-22 illustrates how the smoothing step is setup and the quality Histogram for the mesh elements. [39] Figure 2-22 Mesh smoothing setup and quality histogram. 42

55 2.4. Exporting the Mesh into ANSYS FLUENT When the mesh process is finally completed, checked, and smoothed, the user has to then save the project and transfer the mesh into ANSYS FLUENT. This procedure applies to all the models for this thesis and can be used as a general guideline for future CFD projects. There are several steps in this procedure of transferring the mesh file from ANSYS ICEM CFD to FLUENT. The first step is to save the ICEM project by clicking on File > Save Project As, which creates a xxxx.uns file. The second step is to go to the Output tab and select the red tool box (Select Solver). After the Select Solver is clicked a menu will appear on the screen with two drop down lists. The first list is Output Solver; the user must select the FLUENT_V6 option in order to produce a mesh file that is compatible with FLUENT. The second drop down list is the Common Structural Solver; the user must select ANSYS option, and then click Apply ; as illustrated in Figure Figure 2-23: Output step and solver selection 43

56 Following the Output Solver and the Common Structual Solver selection, the user then can apply the boundary conditions to mesh. The boundary conditions are located in the Output tab, where the user can apply the boundary conditions and check that all the surfaces are defined and represented correctly. This is illustrated in Figure Figure 2-24: Boundary condition step. After all the the above steps are completed the user can then write the input file for ANSYS FLUENT. This is done in the Output tab once again. In order to write the mesh as a FLUENT compatable file, the user must select the Write Input tab. First the correct ANSYS.uns file for the project, (that was saved in the first step) must be opened. Then the FLUENT_V6 window will appear. Following the window s appearance a name for the file must be entered in the Output File line. All other options can remain as the defult values; the step is completed by clicking Done. This is all illustreted in Figures 2-25 and 2-26 below. 44

57 Figure 2-25: Opening of the ANSYS.uns File Figure 2-26: Fluent_V6 window that appears after the ANSYS.uns file is selected. After the mesh is saved as a FLUENT file (.msh file) the user then can close ANSYS ICEM CFD, and open ANSYS WORKBENCH. In order to load the mesh into 45

58 FLUENT, the user must select the mesh option from the component systems list and drag it to the WORKBENCH. The same thing is then done for the FLUENT option. After the two boxes appear on the WORKBENCH, the user must right click on the mesh cell in the mesh box and load the FLUENT mesh, as illustrated in Figure After the mesh has been loaded in the ANSYS WORKBENCH, it then can be loaded in FLUENT. This is done by dragging the Mesh cell from the Mesh box to Setup cell in FLUENT Box. Figure 2-27: Loading of the.msh file in ANSYS WORKBENCH 46

59 Chapter 3: FLUENT Setup and Application of Spalart-Allmaras Turbulence Model 3.1. Background Information in Computational Software and Methodology The major reason behind the growth of CFD usage in various industries is due to its accuracy, reliability, and replacement for running experimental tests. There is also much more advanced computing technology available today for much less cost than running a physical experiment, which may require major equipment such as a wind tunnel. This kind of software is capable of solving large two and three dimensional problems numerically in a short period of time. The accuracy and reliability of a CFD simulation depends on the numerical algorithms employed by the software. This means selecting the appropriate options such as a turbulence model, appropriate spatial and temporal discretization scheme, and correct computational grid topology. The grid topology can have significant weight on the final results of the CFD simulation. Each one of the options mentioned earlier can have either a positive or negative effect on the simulation. With reference to the grid topology, structured grids are more common, preferred, and efficient in the boundary layer region along the model surface for the simulation of the flat-plate, oblate ellipsoid and HPV fairing. In addition, structured grids allow more efficient computations and parallelization. However, an unstructured grid requires less grid points outside the boundary layer region. Considering the oblate ellipsoid geometry, the unstructured grid was a lot easier to generate; it also adapted to the flow gradients more easily. However, the structured grid was much harder to generate around the model within the boundary layer. The reason why all the models use hybrid grids is to simplify the mesh creation and provide accurate and reliable results. 47

60 FLUENT is a finite-volume solver that is based on the full Navier-Strokes equations with a Blasius assumption for turbulence. FLUENT works on structured and unstructured grids. As noted above, the mesh for each model is composed of both kinds of grids. Various grids were examined in order to find the optimum size grid to use for this study. In the thesis Computation of Flow Over a High Performance by Amir Mohammadi, [21] grid optimization was considered, and some of those findings, have been used here. The following section discuss the way FLUENT solves the grid and provides the user with the proper results. FLUENT uses cell faces to integrate for a solution, since the software must handle both structured and hybrid meshes. The hybrid mesh contains many different types of cells such as TETRA_4 (Tetrahedral), TRI_3 (Triangles), PENTA_6 (Prisms), QUAD_4 (Quadrilateral) and PYRA_5 (Pyramids) cells. The structured mesh is a uniform mesh, composed entirely of QUAD-4 cell Turbulence Model Turbulence modeling is the construction and use of a model such as Spalart- Allmaras (SA), k-epsilon (k-ε), or k-omega (k-ω) to predict the effects of turbulence around or inside blunt objects. [33] Averaging is used to simplify the solution of the governing equations of turbulence; hence the models are required to represent different scales of the flow that are not resolved. Consideration of turbulent flows phenomena includes transport properties, boundary layer separation, and other major phenomena; because of this, the most recent work focuses on different types of turbulent models that consist of one or two equation 48

61 models. For instance, examples of two-equation models are the k-ε and k-ω models, and the most popular one-equation model is the SA model. For this thesis, the SA model is being used because of its strong performance in the baseline studies versus the k-ε and k- ω turbulent models Spalart-Allmaras Turbulence Model The SA model was developed in 1992 by Dr. Steven R. Allmaras and Dr. P.R Spalart. The SA model is an approach for modeling different types of turbulent flows, specifically aerodynamics flows with a high Reynolds number. This model is basically a transport equation for the eddy viscosity ν, or a parameter that is proportional to the turbulent viscosity. The main idea that Spalart and Allmaras used to develop this model was very similar to the Nee & Kovasznay (NK) model, which was developed in 1969, and more recently the Baldwin & Barth (BB) in However, all one-equation models have been based on the turbulent kinetic energy equation.[42] It was discovered during the preliminary and baseline tests on the flat and ellipsoid models that the SA model provided better results for drag forces and prediction of flow separation, compared to other options such as the k-ε and k-ω models. Due to its performance during these tests for different flow conditions, the SA model was selected as the main turbulent model for this thesis. As noted above the SA model employs only one-equation, which is a partial differential equation for the modified eddy viscosity. The basic equation is setup as: DF = F + (u )F = Production + Diffusion Destruction (3-1) Dt t 49

62 This can be written as: ν + u ν t j = c x b1 (1 f t2 )S ν c w1 f w c b1 f j k t2 ν 2 d (ν + ν ) ν + σ x j x j c b2 ν x j ν x i (3-2) Equation 3-2 can be simplified and the term by term explanation will be given over the next few paragraphs. ν + u ν t j = c x b1 S ν + 1 (ν + ν ) ν + c ν ν j σ x j x b2 j x j x i Produciton term [c w1 f w ] ν d 2 Destruction term (3-3) Or Diffusion term ν + u ν t j = c x b1 S ν + 1 [ (ν ν ) + c j σ b2 ( ν )2 ] [c w1 f w ] ν d 2 Produciton term Diffusion term Destruction term (3-4) Or in words: Rate of change of viscosity parameter ν Transport + of ν by = production + by turbulent - convection Rate of of ν Transport of ν diffusion Rate of dissipation of ν The production, diffusion, and destruction terms that were defined in the SA model were based on the NK model. The production term defined by NK was based on a statement that was made by Nee & Kovasznay about what defines eddy viscosity and turbulent flow. The eddy viscosity can be regarded as the ability of turbulent flow to transport momentum. The ability must be directly related to the general level of activity, and therefore, to the turbulent energy". [28] 50

63 Based on the above argument, NK defines the production term analogous to the production of turbulent energy. Based on this assumption, NK then assumed that the production term must increase monotonically with magnitude of the mean vorticity U y and the increase of the total viscosity. The SA model is slightly different in defining the production term in terms of its consideration of the appropriate form of mean vorticity. Since the NK model focuses on the simulation of the turbulent shear flow, then the mean vorticity form of U was the best choice. However the SA model s emphasis is on high Reynolds number aerodynamic flow in which turbulence is found only where the vorticity is located. Consequently the SA model uses only magnitude of the vorticity. The diffusion term that was defined in NK used a general definition of diffusion of a scalar F based on the general diffusion equation: y Diffusion = φf (3-5) Here φf is the flux of F due to diffusion and it can be rewritten as φf = D F F, where D F is the coefficient of diffusion. In addition to the diffusion assumption by NK, they also considered the total viscosity n = ε + ν as a portable quantity, where ν is the molecular viscosity and ε is the eddy viscosity. NK also assumed that turbulent motion diffuses by itself; for that reason the coefficient of diffusion is assumed to be D n =n, and henceforth the turbulent Prandtl and Schmidt numbers are approximately one and D n n 1. Based on all the above NK assumptions for the diffusion term, the equation is given as: Diffusion = (n n) (3-6) 51

64 The SA model is still slightly different than the NK model for the diffusion term. SA considers the general diffusion operator as ([ν σ] ν ), where ν is the eddy viscosity and σ is the Prandtl number. In the SA model, the molecular viscosity does not play a major role, and the Prandtl number is still about one. The main difference between the SA and NK models comes in the conservation of the ν integral. Spalart and Allmaras pointed out that manipulation of two-equation models such as the k-ε model often brings out diffusion terms that are not conserved. For example, if a cross product of k ϵ is [28, 42] calculated, a non conservative diffusion term will then be allowed in the equation. Lastly, the destruction term in the SA model is very similar to the NK approach. NK again uses the same assumptions as the production term for the eddy viscosity to construct the destruction term. NK states that the rate of decay of the energy of highintensity uniform turbulence is a very rough approximation, and it is inversely proportional to the square of the energy: du 2 = dt β(u2 2 ) (3-7) Separating the terms and then integrating both sides will then get the decay law: u 2 1 β t (3-8) Since Equation (3-1) considers the quantity F to be the total viscosity n, if the production and the diffusion terms are removed from Equation (3-1), it will then reduce to: n t = Destruction (3-9) 52

65 Based on the NK assumption of similar behavior of total turbulent energy and viscosity, it can be assumed that Destruction βn 2. Finally, based on dimensional analysis, the final form of the destruction term is given as: Destruction = B L2 n(n ν) (3-10) The term B is a universal constant for the turbulence production and L is the characteristic length. The L term was introduced in order to make B a non-dimensional term. Usually L is a function of y, but in this area of the outer edge of the turbulent flow, L is assumed to be equal to the boundary layer thickness (δ). However, when L is analyzed closer to the wall, it can be assumed that L=y. In addition, the destruction term depends on the distance from the wall. This accounts for the high rate of dissipation in nearness of solid boundaries. It is very important to note that the maximum dimension of the dissipating eddies in the direction perpendicular to the flow must be equal to the distance from the wall. As mentioned earlier, the SA model for the destruction term is very similar to the NK approach. The major difference in the derivation of the destruction term is the way SA defines the non-dimensional function beside the constant in the term. The SA model assumes that the blocking effect on the wall in the boundary layer is felt at a distance through the pressure term. The pressure term acts as the main destruction term in the Reynolds shear stress. For that reason the first term of the destruction term can be written as; c w1 (v t /d ) 2, where c w1 is constant and d is the distance to the wall. To overcome the problem with slow decay in the outer region, SA multiplied the destruction term by a 53

66 non-dimensional function f w which is equal to 1 in the log layer near the wall. Consequently the new destruction term then becomes: Destruction = c w1 f w (v t d) 2 (3-11) Now that the production, diffusion, and destruction terms are defined, the rest of the SA model will be explained. The relationship between all the working terms in the equation and the turbulent kinematic eddy viscosity is ν t = ν f ν1 = μ t ρ and the wall function f v1 is defined as: f ν1 = X3 X 3 +c ν1 3 and X = ν ν (3-12) The S term is defined as the modified Vorticity magnitude that is maintained in the buffer layer with log behavior. This is defined as: S Ω + ν f k 2 d 2 ν2, where Ω = 2W ij W ij, W ij = 1 2 u i u j and f x j x ν2 = 1 i (3-13) X 1+Xf ν1 The destruction term function f w is: 1 f w = g 1+c 6 w3 6 g 6 +c6 w3, where g = r + c w2 (r 6 r), and r = min ν S k 2 d 2, 10 (3-14) f t2 = c t3 exp ( c t4 X 2 ) (3-15) Now that the SA model is completely defined as a one-equation model. SA suggests the following constants to be used with its equation to do numerical simulation. The suggested values for the constants are: 54

67 c b1 = ; c b2 = 0.622; c w1 = c b1 k c b2 ; c σ w2 = 0.3; c w4 = 2 ; c ν1 = 7.1 c t3 = 1.2 ; c t4 = 0.5 ; σ = 2 3 ; k = Appling FLUENT Setup All of the modeling and analysis was done using ANSYS FLUENT. Before all this could be done the software had to be calibrated and initial parameters had to be applied to the model within FLUENT. The following section will explain the setup, application of boundary conditions to each model, and application of the mesh refinement Initial Setup In order to open FLUENT and start the CFD analysis, the user first has to open ANSYS WORKBENCH as illustrated in Figure 2-27 and load the mesh from ANSYS ICEM CFD. In order to do that, the user has to drag the correct cells in ANSYS WORKBENCH to the workbench window. The cells are the geometry block and the FLUENT block. The user must right click on the mesh cell to load the ANSYS ICEM CFD mesh to the workbench. Then to load the mesh into FLUENT, the user must drag the mesh cell to the FLUENT block and then double click on the setup cell to open FLUENT. However, the FLUENT Launcher window will open first, and the user must select the following options that are illustrated in Figure 3-1. Then press OK to start FLUENT. 55

68 Figure 3-1: Fluent Launcher option selection As soon as FLUENT opens the user needs to set up the problem. Almost all the steps are the same for each simulation except for the boundary condition setup that will be discussed later. Figure 3-2 shows how to apply the problem setup within FLUENT to get the best results. The first thing that is done in the Problem Setup is the General setup. This is where the mesh is checked; after that is completed the correct scale and the units are then selected for the model. The reason why the correct scale and the units are selected is because the model is in SI units and it needs to be converted to British units and scaled to the correct size. In order to convert the units from SI to British units, the user must click on the General tab and then select the Units menu. After that task is completed, the user must scale the model to the correct size. In order to scale the model, the user must click on the Scale menu and select ft for the View Length Unit In and then in the 56

69 scaling region the user must select the Convert Units option and units of inches. This is illustrated in Figure 3-2. Then press Scale and close the dialog box by clicking Close. Figure 3-2: General setup step in Fluent After the General setup is completed, the user must select the following steps to complete Problem Setup. The steps are: Models, Materials, Cell Zone Conditions, Boundary Conditions and Reference Values setups. The Model setup allows the user to set various flow model options, e.g. phase change, mass transfer, etc. For this study, the Viscous model is the only one selected. In the Viscous model option the SA turbulent model is selected and the SA model uses the constants that are explained in Section 3.2. This is illustrated in Figure

70 Figure 3-3: The Viscous Model Dialog box displaying the SA model setup The next step is the Materials setup where the user must select the fluid and solid materials. As mentioned earlier, the outside fluid (fluid box) for this thesis is going to be air, and the solid will be set as aluminum in the Materials setup. The reason why aluminum was set as the solid material, and not carbon fiber, is due to two reasons. First FLUENT does not have carbon fiber in its data base. The second reason is because the wall is assumed to be smooth and an arbitrary material is used. The next steps in the Problem Setup are the Cell Zone Conditions and the Reference Values setup. The Cell Zone Condition task allows setting the type of cell zone condition parameters for each zone i.e. fluid domain is set as fluid. The Reference Value Task page allows setting the reference quantities that are used for computing different variables after the solution process has finished. Figure 3-4 illustrates the reference values that are being used for the HPV model; however, the reference values for the other 58

71 models such as the oblate ellipsoid and the flat plate are all the same except for the area and the velocity values that change with each simulation. Figure 3-4: Reference Values task for the HPV model Boundary Conditions The following discussion summarizes the Boundary Conditions task in FLUENT, and Boundary Conditions for each simulation, are shown in Tables 3-1 and 3.2. Recall that while creating the mesh in ICEM, the boundary types were then set for each face in the domain. The right boundary plane (YZ plane in the positive X direction) is the inflow of the flow field (u = V ), and the left boundary plane is the outflow. The top and bottom planes, and the XY plane in the positive Z direction are set as Symmetry planes, as well 59

72 the Symmetry plane for the fluid boundary. The exception is the simulation of the moving ground plane as shown in Table 3-2, the bottom plane is defined as a wall for these cases. Symmetry boundary conditions are used when the physical geometry and the expected pattern of flow solution have mirror symmetry in order to reduce the total computational time and power needed for the simulation. In addition, symmetries are also used to model zero-shear slip walls in viscous flow. In the Problem Setup a Boundary Conditions Task can be opened and this where the boundaries are specified for each region, this is done according to Tables 3-1 and 3-2 for the flat plate, oblate ellipsoid and the HPV fairing simulations. plane Position Name Type Right (YZ X+) Inflow Velocity Inlet Left (YZ X-) Outflow Outflow Top (XZ Y+) Top of the outer volume Symmetry Bottom (XZ Bottom of the outer volume Symmetry Far side (XY Z+) Far side of outer volume Symmetry Symmetry (XY Symmetry Symmetry Model Model Wall Table 3-1: Boundary type for the Flat Plate, Oblate Ellipsoid, and HPV fairing run without a ground plane 60

73 plane Position Name Type Velocity Right (YZ X+) Inflow Inlet Left (YZ X-) Outflow Outflow Top (XZ Y+) Top of the outer volume Symmetry Bottom (XZ Ground Plane Wall Far side (XY Z+) Far side of outer volume Symmetry Symmetry (XY Symmetry Symmetry Model Model Wall Table 3-2: Boundary type for the HPV fairing model with moving simulation ground plane Figures 3-5 and 3-6 illustrate the velocity inlet and the outflow setup. This step is performed on all the models. A Velocity Inlet boundary condition is used to define the velocity and the scalar properties of the flow at the inlet. By clicking on Velocity Inlet and setting the momentum parameter. In the momentum parameter the user must select the following options; the Velocity Specification Method is set to the Magnitude, Normal to Boundary, the Reference Frame setting is set as Absolute, and the Velocity/Magnitude setting is set to the freestream velocity. As the Velocity Magnitude varies the Modified Turbulent Viscosity varies with it. The following equations are used to find the values that are illustrated in Table 3-3. u 1 I = 0.16 Re u D avg H 8 (3-16) l = 0.07L (3-17) ν = 3 2 (u avgil) (3-18) 61

74 In these equations I is the turbulence intensity, u is the root-mean-square of the turbulent (defined as; u = 1 u 3 x 2 + u 2 y + uz 2, and u avg is the mean flow velocity. The turbulence intensity can be also found using the Reynolds number. The turbulence length l is a physical quantity related to the size of the large eddies that contain the energy in turbulent flow, and L is the length of the model. In order to find the modified turbulent viscosity ν, the user must use Equations 3-16 and 3-17 to find I and l then plug these values into Equation 3-18 to get the value for ν as illustrated in Table 3-3 u avg (f/sec) I l (ft) Re modified turbulent viscosity [ν ],(ft 2 /sec) modified turbulent viscosity [ν ],(m 2 /sec) 6.562E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-01 Table 3-3: Change in modified turbulent viscosity (ν ) with velocity. 62

75 Figure 3-5: Boundary condition setup for the velocity inlet The outflow boundary condition is used to define the flow that exits the region. At this plane, the details of the pressure and the flow velocity are unknown before the solution has been generated by FLUENT. The pressure outlet is set to outflow and the flow rate weighting parameter is set to one; this specifies that 100% of the outflow is leaving the bounded area. These steps are illustrated in Figure 3-6. [38] 63

76 Figure 3-6: Boundary condition setup for the outflow Solution Setup and Mesh Adaption The next step in the problem definition process is the Solution Setup. This setup is the same for all the models. Once the Solution Setup is completed the user can proceed to the mesh adaption process. Mesh adaption is performed on the model to ensure that the mesh is sufficiently fine to obtain accurate form the simulation. The first step in the Solution Setup is defining the Solution Method, and specifying the various parameters associated with the calculation of the final result. FLUENT provides implicit and explicit solver formlations from the drop down lists. The Pressure-Velocity Coupling option contains the settings for this scheme, and the SIMPLE algorithm (Semi-Implicit Method for the Pressure-Linked Equation) is used for all the models. This algorithm is selected due to its relationship between velocity and pressure corrections to help enforce mass conservation and to accurately obtain the pressure field. In addition, the SIMPLE algorithm substitutes the flux correction equations into a 64

77 separate continuity equation in order to obtain a separate equation for the pressure correction term in each cell. The next step is selecting the Spatial Discretization of the convection terms in the solution equation. Choices for these options were based on ME692 course notes. The SA model has only four terms in the Spatial Discretization menu: Gradient, Pressure, Momentum, and Modified Turbulent Viscosity term, in Table 3-4. Spatial Discretization Option Gradient Green-Gauss Node-Based Pressure Second Order Momentum Second Order Upwind Modified Turbulent Viscosity Second Order Upwind Table 3-4: Spatial Discretization Setup Following the Solution Method setup, the user must then apply the solution controls. The user must set solution parameters for the under relaxation factors of pressure, density, body forces, momentum, modified turbulent viscosity and turbulent viscosity. Those factors were initially set at default values, and then if the residuals continued to increase after four or five iterations, the under relaxation factors were then reduced. The default under relaxation factors are shown in Table 3-5. [38, 41] 65

78 Under Relaxation Factors Pressure 0.3 Density 1 Body Forces 1 Momentum 0.7 Modified Turbulent Viscosity 0.8 Turbulent Viscosity 1 Table 3-5: Default under Relaxation Factors After all the preliminary steps are completed, the user then defines which flow parameters will be monitored by FLUENT, and select where the solution will be initialized from. The main parameters that need to be monitored are the residual, drag, lift and moment solutions. The residual monitor allows us to set the absolute convergence criteria for all the residuals that apply to the SA model. The residual monitor setup is illustrated in Figure 3-7. The Convergence Absolute Criteria for all the residuals was set to 1e-05. This was done to achieve the best results, the solution process continue until the difference between successive values is less then 1e-05. The drag, lift, and moment monitors all have the same setup. In order to setup the drag monitor, the user must double click on the drag option and select the wall zone as the geometry itself, i.e. HPV fairing or oblate ellipsoid. The user has to make sure that the forces are monitored in the correct direction by setting the vector values. For these models the vector value was 1 for X and 0 for Y and Z; this means that the forces are monitored in the X direction or the flow direction. 66

79 Figure 3-7: Setup of the Residual Monitors To initialize the solution, the user must select where it will be computed from. The program then references the frame and checks the initial values to see if they are correct. In order to do that, the user has to follow a few steps. The first step is selecting where the solution will be computed from, and what direction and zone the flow is coming from and heading toward. This step is the same for all the simulations; the flow is coming from the velocity inlet zone and going to the pressure outlet zone. For that reason the user must select the Compute From as the Velocity Inlet zone. After the zone is selected the initial values that are applicable variables to that zone are displayed. The reference frame specifies whether the initial velocities are absolute velocities, by selecting the Absolute option, or the velocity is relative to the motion of each cell zone by selecting the Relative to Cell Zone option. For this study Relative to Cell Zone option is used, due to the assumption that was made earlier that the model is stationary and the air is moving around it at some speed. After the user has selected the correct options, the user must then click Initialize to initialize the flow field. 67

80 In order to start the calculation and do mesh adaption, the user first has to start the calculation by going to the Run Calculation task page and then checking the mesh to make sure there no problems with the mesh. The user then needs to set the number of iterations to 200 and click Calculate as illustrated in Figure 3-8. After these preliminary results are produced they are used for the mesh adaption process. In order to do mesh adaption, the user must click on the Adapt drop down menu, and the select the Gradient Adaption option, as illustrated in Figure 3-9. The gradient adaption option is a function that allows the user to mark specific cells and adapt the mesh based on the gradient. For this thesis the method that is used to adapt the mesh is gradient based with respect to velocity magnitude; all the adaptions were done to refine the mesh, and to make sure the solution is grid independent. The gradient method was selected because the momentum equation is the only governing equation that is used for this simulation to find drag. Velocity gradient in the boundary layer determine the drag on the surface, which is why velocity gradient were chosen to control the adaption. Mesh adaption was done until the mesh reached independency and the solution converged. That means the solution for drag was no longer dependant on the mesh fineness. This process will be explained in depth in chapter 4. 68

81 Figure 3-8: Run Calculation Setup Figure 3-9: Gradient Adaption / Mesh Adaption Setup 69

82 The mesh adaption process is illustrated in Figure 3-10, as the mesh changes from the original one (A) to the mesh is used to set the final solution (D). The mesh adaption is done every 200 iterations and data for viscous and pressure drag is then recorded for each mesh as well. Figure 3-10: Mash adaption, A the original model, B D is the adaption every 200 iteration Solution to the Problem The solutions to each model are provided in the results task page, where the user can set up and display the results of the CFD simulation using FLUENT. The graphical results that let the uses visually inspect the results are also generated using FLUENT. Those graphical results include contours, vectors, pathlines, particle tracks, animations and plots. In addition the user also has the ability to get the numerical solutions for the drag and lift forces from the Reports task page. 70

83 Graphical and Numerical Solutions The main kinds of graphical solution used for this thesis are the contours and the vectors solutions. The vectors solutions let us inspect specific scalar fields as a vector field or as a contour plot showing variations of velocity, vorticity, pressure, etc. In order to find the drag and lift forces, the user must go to the report task page and select the Forces option and then select the appropriate geometry for the wall zone and appropriate vector directions to compute the forces. For the drag force the user must set the vectors to X=1 and Y=Z=0, and for the lift force the user must set the vectors to Y=1 and X=Z= Drag Calculation The main aerodynamic force that is important in this thesis is the drag force. Drag force is the net horizontal force with respect to direction of motion. The flow domain is illustrated in Figure This Figure shows domain and the coordinate system around the HPV fairing; all other models use the same domain and coordinate system. The fluid domain provides a basis for the calculation of the drag force by FLUENT. The velocity inlet Y-Z plane (S 1 ) is assumed to be located in front of the model, and far enough ahead of the model so that the incoming flow will satisfy the undisturbed free stream conditions. The outflow Y-Z plane (S 2 ) is the downstream condition, located sufficiently downstream of the model. All the walls of the virtual tunnel form a uniform cross section that is parallel to the ongoing flow, and blowing and suction is not allowed on any of these surfaces. Lastly, the flow in the domain is steady, subsonic and incompressible. The fluid domain defined in FLUENT can be used to 71

84 describe a control volume which allow, the formulation of the drag force, as discussed below. S2 S1 U Figure 3-11: Control volume for derivation of drag equation using HPV model The total drag on the body can be determined by calculating the change of momentum equation in the direction of flow which in our case is the X-direction. F D = (P + ρ U 2 )ds S 1 (P + ρu 2 )ds S 2 (3-19) Equation 3-19 represents the pressure forces driving the flow through the control volume and the flux of momentum across the faces of the control volume. Where U is the free stream velocity, u is the axial velocity, ρ is the free stream density, ρ local density, and P is the static pressure. The conservation of mass for steady flow through the control volume is given as: 72

85 ρ u S 1 = S ρ( u )ds n = 0 (3-20) The total pressure equation is given as: P = P + ρ 2 (U2 + V 2 + W 2 ) (3-21) Where U, V and W are the velocity magnitude in the x, y and z direction. As assumed earlier, flow at the S 1 plane corresponds to the undisturbed free steam conditions. By substituting Equation (3-21) into Equation (3-19), the total drag equation can be obtained as and over all drag comes : F D = Wake (P i + P )ds + ρ 2 (U 2 U 2 )ds S 1 + ρ 2 (V 2 + W 2 )ds S 2 (3-22) Equation 3-22 is based on Kusunose 1997 and FLUENT user guide [44, 38], where P is the total pressure from equation 15, P i is the free steam total pressure, and U is the free stream velocity. If it is assumed that the upstream (velocity inlet) and the downstream (pressure outflow) planes have the same area, the drag equation can be rewritten (3-22) as: F D = ρu(u + u) + (P i P)ds (3-23) s 1 Equation 3-23 can be normalized by the dynamic pressure in order to produce the non dimensional coefficient of drag. C D = D q S where q = 0.5ρ U 2 (3-24) 73

86 C D is the coefficient of drag, F D is the drag force, q is the freestream dynamic pressure, and S is the reference area. The non-dimensional quantity is a function of the Reynolds number. The Reynolds number influences boundary layer behavior, especially the transition to turbulence and the location of the separation point. Both of these factors affect the total drag. FLUENT provides the results of the total drag as a sum of two major components where C D = C Df + C Dp. The first component is the pressure drag coefficient, and the second component is the viscous (or friction) drag. [24] For this thesis three major models i.e. flat plate, HPV fairing, and oblate ellipsoid were used. For the flat plate the drag is totally due to the viscous drag.. For the oblate ellipsoid and the HPV fairing the drag is composed of pressure and viscous contributions. The reason why pressure drag plays such a major role is due to the flow separation around the rear portion of these blunt objects. 74

87 Chapter 4: Baseline Solutions and Calibration of FLUENT Before using and trusting CFD analysis techniques on a new configuration, one should benchmark or validate the technique against a known test case similar to the new configuration. If no appropriate test case exists, then the CFD analysis results would be compared with another analysis or an experimental technique such as a wind tunnel. The benchmark (baseline) test is a process of numerical analysis performed on a case which is a replica of the previous results of numerical simulations or real time testing. In this study, testing is performed by comparing test case results against published values in Fluid Dynamics of Drag by Hoerner and standard fluids texts. For CFD, the baseline process should result in guidelines for a specific class of problems. The guidelines should describe what kind of turbulence model, preferred boundary conditions, and meshing strategy (i.e. growth rate, cell size and clustering) are required to achieve a desired level of precision and confidence in the results. Following the procedure outlined in Chapter 3, results were generated for baseline models, i.e. the oblate ellipsoid and the flat plate. The oblate ellipsoid and the flat plate of very small thickness were modeled using SolidWorks CAD software to match the models in Hoerner [11] in Chapter 1, Figure 1-2. Hoerner presented selective results on the drag of rotationally symmetric bodies with different ratios of l d based on wind tunnel tests. Hoerner s results are compared to the results of the numerical simulation using ANSYS FLUENT. Flat plate results are compared to values found in standard fluid text. 75

88 4.1. FLUENT Calibration using Flat Plate A CAD and mesh model of a flat plate with thickness of almost zero and the same length as the HPV fairing was created. This flow case is simple but useful since the drag is entirely due to wall shear stress, and this depends solely on the wall velocity gradient in the boundary layer. The drag coefficient found for the flat plate was compared to the drag coefficient found in Fundamentals of Fluid Mechanics by Munson. [24] This also helped find the correct FLUENT parameters for the oblate ellipsoid such as initial setup, materials for the fluid, geometry, boundary conditions, and dynamic mesh setup. The type of mesh that was used for the flat plate is a combination of structured and unstructured mesh. The structured mesh is used around the flat plate, i.e. for boundary layer mesh, and the unstructured mesh is used everywhere else. The mesh was created usingansys ICEM CFD, and this was also done for all other models such as the oblate ellipsoid and the HPV fairing. The mesh for the flat plate is displayed in Figure

89 A B Figure 4-1: The computational mesh for the flat plate, overall view (A), and structured mesh (boundary layer) (B) The drag coefficient for the flat plate is found at various Reynolds numbers, and then compared to the calculated drag for those Reynolds numbers based on correlation analytical and empirical results. As noted above, the drag for the flat plate is composed of 100% viscous drag (and no pressure drag) because the flat plate had no flow separation and thickness equal to zero. Four Reynolds numbers are selected to be compared, two from the laminar region and two from the turbulent region. The laminar Reynolds numbers are 1X10 5 and 2X10 5, and the turbulent Reynolds numbers are 1.5X10 6 and 2X10 6. Following the CFD process, the numerical values were then compared to the calculated values to identify differences. 77

90 Table 4-1 shows the results for the drag coefficients found using FLUENT and compared with the calculated drag and empirical drag. Table 4-2 compares the FLUENT values with empirical values from Munson [24]. The calculated drag assumed to be fully turbulent for the turbulent Reynolds number cases, i.e. there was no assumption made for laminar to turbulent transition. The drag that was calculated used the following equations: C Df = Re0.5, Laminar flow (4-1) C Df = (log Re) 2.58, Turbulent flow, smooth plate (4-2) Percent error and percent difference were computed for each case using the following equations: % Error = Experimental(CFD) empirical empirical 100 (4-3) % Diff = x 1 x 2 x1+x2 100 (4-4) 2 Reynolds number CFD values for C D Percent error for CFD vs. calc Calculated values for C D 1.0E E E % 2.38% 2.0E E E % 2.20% 1.5E E E % 3.54% 2.0E E E % 4.95% Percent Diff CFD vs. calc Table 4-1: CFD drag data vs. calculated data for flat plate using integral method 78

91 Reynolds number CFD values for C D Empirical values for C D from Munson Percent error for CFD vs. Munson Percent diff CFD vs. Munson 1.0E E E % 2.28% 2.0E E E % 2.10% 1.5E E E % 2.32% 2.0E E E % 3.45% Table 4-2: CFD drag data vs. Empirical data found from Fundamentals of Fluid Mechanics 5 th edition by Munson for flat plate. 4.45E E E E-03 coefficient of drag 3.85E E E E-03 CFD Data calc data Munson Data 3.25E E E E E E E+06 Reynolds number 2.5E+06 Figure 4-2: Comparison of three coefficients of drags for a flat plate Figure 4-2 illustrates graphically what is displayed in Tables 4-1 and 4-2 for CFD data that was found using ANSYS FLUENT, compared to the calculated and empirical values. 79

92 Figure 4-3: Empirical data for drag coefficient of a flat plate adopted from Fundamentals of Fluid Mechanics 5 th edition by Munson Figure 4-3 illustrates the empirical data that is shown in Table 4-2. This figure indicates how the drag coefficient behaves in turbulent flow with respect to surface roughness, and Reynolds number FLUENT Calibration Using Oblate Ellipsoids The oblate ellipsoid was used to calibrate ANSYS FLUENT and ANSYS ICEM CFD, in order to find the best mesh density and ANSYS FLUENT parameters. Two oblate ellipsoids geometries of l ratios 2 and 4 were modeled in SolidWorks to match the d tested solid geometries from Hoerner s books and Figure 1-2. The mesh for each model was generated, and the mesh sensitivity was found. That data from the oblate ellipsoid was then used as a reference for the HPV fairing mesh. 80

93 The boundary condition types that were set in ANSYS ICEM CFD were also used as the reference for the boundary condition setup for the HPV fairing. The Right boundary (YZ X+) was set as the inflow of the flow field (u = V ). The left boundary plane is the outflow. The rest of the fluid boundaries, including the top plane, bottom plane, the XY plane in the positive Z direction, and the Symmetry plane are set as symmetry. The type of each boundary condition is specified in Table 3-1 in Chapter 3. The boundary conditions that were created in ANSYS ICEM CFD in order to be used in FLUENT are set as follows. Inflow boundary condition is set as the velocity inlet to define the velocity and scalar properties of the flow at the inlet boundaries. Outflow boundary condition is used to model the exit pressure and the flow velocity that is not known prior to solution of the flow problem. The wall boundary condition is used to model the actual physical geometry that is stationary, and the moving wall for ground effect cases. Symmetry boundary conditions are used to model zero-shear slip walls in the viscous flow. They are also used when the physical geometry and the expected flow solution have mirror symmetry. The following global mesh characteristics used for the oblate ellipsoid mesh generation are: global element scale factor of 0.4 of an inch, and global element seed size of 128 inches. Also during the Part Setup the oblate ellipsoid surface is the only surface that will have a prismatic layer of maximum size of 3 inches, and inch first cell transverse dimension. The first cell transverse dimension depends on the total size of the model, and the growth factor. For this thesis the growth factor is set to 1.08 for the structured cells and 1.2 for the unstructured cells. 81

94 Figure 4-4: The computational mesh for the oblate ellipsoids of ratio l/d=4 and 2 Figure 4-4 illustrates the computational mesh for the two oblate ellipsoids with different l ratios, where a structured mesh is shown around the ellipsoid geometry and d the unstructured mesh is used everywhere else. 82

95 4.2.1 Results for the Oblate Ellipsoids The flow field about the oblate ellipsoids ( l ratios of 2 and 4) and the associated d drag forces are presented in this section. The Reynolds number was varied from 230 thousand to 3 million, and the freestream velocities were based on the Reynolds number, and range from 1.98 to ft/sec. These Reynolds numbers were selected because they match the known values of drag that were done experimentally by Hoerner. The flow field data are presented graphically in terms of velocity magnitude, pressure field, and streamlines. The drag force is normalized by the freestream dynamic pressure and the chord length of the ellipsoid and then compared with values in Hoerner. The pressure field (dynamic and static) around both oblate ellipsoids is shown in Figures 4-5 and 4-6 at Reynolds number and V = ft/sec. A low pressure region (shown in blue) is present over and under the oblate ellipsoid. This is because the angle of attack is 0 and it is symmetric about the X axis. There is also a stagnation region as expected near the leading edge for both oblate ellipsoids that is indicated by dark red in the static pressure plots. However, if one looks at the dynamic pressure plots one can see the stagnation region is indicated by dark blue and the low dynamic pressure regions are indicated by dark red. The velocity magnitude data for the same conditions as Figure 4-5 are presented in Figure 4-7; the flow accelerates from the stagnation region near the leading edge to the top of the oblate ellipsoid, and then the boundary layer is visible on the rear portion of the body. It is clear that the fatter ellipsoid ( l = 2) produces d a larger wake region. 83

96 A B C D Figure 4-5: Pressure field (Static) about the oblate ellipsoids of ratios of l/d=2 and 4. Dark red indicates stagnation regions and dark blue low pressure regions. Filled contour and contour lines of oblate ellipsoid (A) Filled contour l/d=2; (B) contour lines l/d=2; (C) Filled contour l/d=4 ;(D) contour lines l/d=4. 84

97 A B C D Figure 4-6: Pressure field (Dynamic) about the oblate ellipsoids of ratios of l/d=2 and 4. Dark red indicates low pressure regions and dark blue stagnation regions. Filled contour and contour lines of oblate ellipsoid (A) Filled contour l/d=2; (B) Contour lines l/d=2; (C) Filled contour l/d=4; (D) Contour lines l/d=4. 85

98 A B C D Figure 4-7: Velocity magnitude about the oblate ellipsoids of ratios of l/d=2 and 4. Red indicates high velocities and the light blue the stagnation region near the leading and trial edges. Filled contour and contour lines of oblate ellipsoid (A) Filled contour l/d=2; (B) Contour lines l/d=2; (C) Filled contour l/d=4; (D) Contour lines l/d=4. 86

99 However, when turbulent flow is analyzed in FLUENT, the program has a problem in predicting the correct separation point, which then produces slightly higher values for drag coefficient in the turbulent region compared to the experimental values. This will be explained in more detail in Section 4.3. To verify how the flow behaves around the ellipsoid, streamline plots are used as illustrated in Figure 4-8, which shows no reverse flow on the surface. The boundary layer can be absorbed more clearly by examining the vorticity contours as plotted in Figure 4-9. The boundary layer gets thicker as it moves farther away from the leading edge of the ellipsoid. In addition, the boundary layer at the trailing edge is about two to three times thicker than at the leading edge and the flow is fully turbulent by that time. A B Figure 4-8: Streamlines about oblate ellipsoids (A) l/d=2; (B) l/d=4. 87

100 A B Figure 4-9: Vorticity about oblate ellipsoids (A) l/d=2 and (B) l/d=4, showing the boundary layer thickness on each model Comparison Between Hoerner s data and CFD data Tables 4-3, 4-4, 4-5, and 4-6 demonstrate how the optimum mesh was found for the oblate ellipsoid cases and how each drag component changes after each level of mesh adaptation. One can see that the viscous and pressure C D values change, however when the values start to converge and the difference between them is less than 1% the mesh adaptation process is then stopped. The final value is used as the CFD C D value to compare with the empirical value for each Reynolds number and l value. d 88

101 2.000E+06 l/d=2 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E+06 Table 4-3: Pressure C D, Viscous C D and mesh density as separate entities for R E =2E6 for oblate ellipsoids of fineness ratio E+06 l/d=4 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E+06 Table 4-4: Pressure C D, Viscous C D and mesh density as separate entities for R E =2E6 for oblate ellipsoids of fineness ratio E+06 l/d=2 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E+06 Table 4-5: Pressure C D, Viscous C D and mesh density as separate entities for R E =3E6 for oblate ellipsoids of fineness ratio 2 89

102 3.000E+06 l/d=4 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E+06 Table 4-6: Pressure C D, Viscous C D and mesh density as separate entities for R E =3E6 for oblate ellipsoids of fineness ratio 4 The oblate ellipsoid s drag coefficient (C D ) values are shown in Figure 4-10 and Table 4-7 as a function of Reynolds number for both l ratios and compared to the d experimental data from Hoerner. The agreement in the laminar range is within 3% and in the turbulent range is within 8.5%. However, the CFD results for drag coefficient are higher in the turbulent region. This is illustrated in Figure 4-10 and Table 4-7, for both l d ratios. 90

103 9.90E E-03 Coefficient of Drag based on surface area (C D ) 8.70E E E E E E E-03 l/d=4 Hoerner l/d=4 CFD l/d=2 Hoerner l/d=2 CFD 4.50E E E+05 Reynolds number on a log10 scale 1.00E+06 Figure 4-10: Drag Coefficient of the oblate ellipsoids vs. experimental data by Dr. Hoerner. l d When looking at Figure 4-10; it is clear that there is a bigger deviation in the = 2 plots. This is probably due to the greater importance of pressure drag to the overall drag for lower l d ratios. Therefore, the drag results found from FLUENT are more sensitive to the prediction of the separation point in the back side of the body. Table 4-7 tabulates the CFD and experimental C D values, including percent differences. It should be noted that the Hoerner data was read from Figure 1-2. It was 91

104 hard to read the numbers and there is a small uncertainty of about 2-3% for the empirical data that may add to the total percent difference. Re # l/d=2 CFD l/d=2 Hoerner % Difference for l/d=2 Re # l/d=4 CFD C D C D C D C D l/d=4 % Hoerner Difference for l/d=4 2.5E E E % 2.3E E E % 3.7E E E % 2.6E E E % 5.0E E E % 3.0E E E % 6.0E E E % 1.5E E E % 2.0E E E % 2.0E E E % 3.0E E E % 3.0E E E % Table 4-7: CFD drag data for both oblate ellipsoids CFD and experimental data from Hoerner 92

105 Chapter 5: HPV Fairing Results Following the procedures outlined in Chapter 3, and baseline solutions for the oblate ellipsoids and the flat plate discussed in Chapter 4, FLUENT program setting to obtain accurate drag values were established. This chapter provides a detailed overview of the results for the HPV fairing simulations for various ground clearance and the resulting drag and lift coefficients. The results are presented with flow profiles, graphs, and tabulated values. The discussion sheds light on the total drag on the HPV fairing at various ground clearances. The plots and graphs for the HPV fairing simulation at Re = and a ground clearance of 15 inches are presented in detail in this chapter. Results for other ground clearances can be found in Appendices B and C HPV CFD Test Results The CFD simulation was conducted on the HPV fairing geometry with various ground clearance ranging from 3 to 18 inches using the S-A turbulent model, for these Re = ( i.e. with inlet velocity of 40 mph, or ft/sec). In addition to these simulations, two benchmark tests were also conducted at ground clearances of 30 inches and 297 inches. This was done to find the drag coefficient corresponding to zero ground effect. Since the process of CFD meshing is an iterative process, a number of meshes were applied and tested. However, due to the limited time and computing power, only two major benchmarking simulations were done (297 and 30 inches) to check for mesh convergence. 93

106 HPV Fairing Benchmark Results The benchmark CFD simulations for the HPV fairing were done for freestream conditions (297 inches) and 30 inches away from the ground surface. This was done to find the drag on the HPV fairing with zero or minimum ground interference. The data could then be compared to the HPV fairing at different ground clearances corresponding to realistic design condition. The freestream ground clearance of 297 inches was selected because it was assumed that at a distance of 3 HPV length away from the ground, there will not be any ground interference on the fairing drag. In addition, another benchmark case was selected at 30 inches away from the ground surface. This was estimated as the minimum ground clearance where the ground interference would be negligible. Figures 5-1 and 5-2 show the velocity vectors, and Figures 5-3 and 5-4 show the velocity profiles for the two benchmark simulations with a freestream velocity of V = ft ft. The magnitude of the velocity vectors in is indicated by the color scale, and sec sec the length of each vector depends on the direction of the freestream velocity around the HPV fairing. The velocity profile reveals that the highest velocities (in red) are encountered in the bottom leading edge and on the upper surface as the flow accelerates around the front portion of the fairing geometry. As the flow along the rear of the fairing slows down a thicker boundary layer appears at the trailing edge. There is also a high velocity region below the HPV fairing in both benchmark cases. At the trailing edge there is a low velocity region that is indicated by a dark blue color, which indicates a stagnation region. Essentially, this means that the fluid stops moving at the trailing edge of the HPV fairing. 94

107 In the 30 inch benchmark case one can see the effects of ground proximity on the velocity distribution underneath the HPV fairing. When looking at Figures 5-2 and 5-4 one can see how the velocity shape changes under the fairing, as the fairing boundary layer starts to interact with the ground boundary layer. This interference would be expected to affect the total drag on the fairing. When comparing the two benchmark simulations (30 inches and freestream), a 2.6% difference is observed for the drag values, and a 3.7% difference is seen for the lift values. Figure 5-1: Velocity Vectors about the HPV fairing in freestream 95

108 Figure 5-2: Velocity Vectors about the HPV fairing with ground clearance of 30 inches Figure 5-3: Velocity magnitude of HPV fairing in freestream, red indicates high velocity regions and dark blue indicates stagnation region at the trailing edge of the HPV fairing. 96

109 Figure 5-4: Velocity magnitude of HPV fairing with 30inches ground clearance, red indicates high velocity regions and dark blue indicates stagnation region at the trailing edge of the HPV fairing. To confirm the accuracy of these drag values, mesh adaption was used as discussed previously in Chapter 4. Tables 5-1 and 5-2 demonstrate how the optimum mesh was found for each benchmark simulation by noting the change in each drag coefficient component after each level of mesh adaption. The stopping criteria for the simulation is when there is no longer a significant change (i.e. less than 1%) in between the drag coefficients for each mesh. Also the major contributor to the total drag coefficient is the pressure drag; for these cases the ratio of pressure drag to viscous drag is roughly 5:1. Ground Clearance 30 Inches Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E E E E+06 Table 5-1: Drag coefficient components for HPV fairing with ground clearance of 30 inches 97

110 Ground Clearance 297 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E E E E+06 Table 5-2: Drag coefficient components for HPV fairing at freestream The static, dynamic, and total pressure fields around the HPV fairing for the freestream case are displayed in Figures 5-5 to 5-7. Figure 5-5 shows the static pressure contour around the HPV fairing at the symmetry plane. There is a high pressure region that is indicated by the red color directly in front of the fairing; this represents the frontal pressure, and it is caused by the air attempting to flow around the front of the fairing nose. The low pressure regions are indicated in green and blue on top of the fairing s surface. This represents the flow that is accelerating above and below the fairing and it is at a lower pressure compared to the front portion of the surface. This wake region plays a major role in the total drag coefficient. A B Figure 5-5: Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 297 inches using contour lines(a) and filled contour (B). Red indicates stagnation pressure and green-blue low pressure regions 98

111 Figure 5-6 shows the dynamic pressure contour; the low pressure region is indicated using blue and green colors, and the high pressure regions are indicated using red-orange. The low dynamic pressure regions are found in front of, behind, and on the surface of the HPV fairing, i.e. in areas where the velocity is low. A B Figure 5-6: Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 297 inches using contour lines(a) and filled contour (B). The low pressure region is indicated using blue and the high pressure regions are indicated using red-orange Figure 5-7 shows the total pressure contour; is indicated using red, the low pressure wake region is indicated using yellow, light blue, and light green, and the low pressure regions are indicated using color blue. As mentioned above, the low pressure wake region is created when the boundary layer separates from the fairing. The size of the wake region plays a very significant role on the total drag coefficient. If the size of the wake region is reduced with increasing ground clearance, or the ground clearance stays the same but aerodynamic enhancements are added, then the low pressure in the wake region will then increase. This will cause the pressure differences on the fairing in the flow direction to decrease, and this eventually helps reduce the pressure drag. 99

112 A B Figure 5-7: Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 297 inches contour lines(a), and filled contour (B). The high pressure around the HPV fairing on the surrounding air is indicated using red; the low pressure wake region is indicated using yellow, light green and light blue; and low pressure is indicated using blue. The static, dynamic, and total pressure fields around the HPV fairing with a ground clearance of 30 inches and at a freestream velocity of V = ft sec are displayed in Figures 5-8 to In those figures, one can see the pressure distribution and ground effect on the fairing using frontal and isometric views. The pressure fields for this case have a very similar pressure distribution as the freestream case. However, when one examines the pressure field below the fairing, it appears to be influenced by the ground surface boundary. 100

113 A B C D Figure 5-8: Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 30 inches Isometric view of contour lines (A), filled contours (B), front view of contour lines (C), filled contours (D).Red indicates stagnation pressure and green-blue low pressure regions A B C D Figure 5-9: Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 30 inches. Isometric view of contour lines (A), filled contours (B), front view of contour lines (C), filled contours (D).The low pressure region is indicated using blue and the high pressure regions are indicated using red-orange 101

114 A B C D Figure 5-10: Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 30 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).The high pressure around the HPV fairing on the surrounding air is indicated using red, the low pressure wake region is indicated using yellow and light green, and low pressure is indicated using blue. The boundary layer formation on the HPV fairing can be observed from the vorticity contours, shown in Figures 5-11 and 5-12 for the two benchmark simulations. One can see how the boundary layer develops and thickens on the upper surface of the HPV fairing for each benchmark simulation. In association with the velocity magnitudes shown in Figure 5-3 and 5-4, the vorticity values represent the boundary layer as a gradient of velocity. Figure 5-11 shows the in freestream, case and Figure 5-12 shows the 30 inch case. Comparing the two figures, the proximity of the ground appears to have a definite effect on the size of the wake region downstream of the fairing. 102

115 Figure 5-11: Vorticity contours of HPV fairing in freestream flow Figure 5-12: Vorticity contours of HPV fairing with ground clearance of 30 inches The pressure coefficients for the top and bottom surfaces of the HPV fairing for both benchmark simulations are shown in Figures 5-13 and Note that the flow direction in these figures is right to left. Figure 5-13 show the pressure underneath the HPV fairing. When examining the graph one can see that the ground effect is already present in the 30 inch benchmark, there is a significant deviation between the two curves. In addition, a separation bubble is present on the bottom of the HPV fairing that crates a negative value for the pressure 103

116 coefficient near the leading edge (point at 8.5 ft) before it then recovers to freestream pressure toward the trailing edge of the fairing. Figure 5-14 shows the pressure coefficient on the upper surface of the HPV fairing. This pressure coefficient reaches a value nearly equal to 1 at the leading edge, then it drops below 0 over the length of the fairing, and then starts recovering toward the trailing. In addition, one can also see that the shape of the curves are nearly the same. It may be that the ground clearance affects the location of the stagnation point on the nose of the fairing, which then causes small changes down stream of this point C p (Coefficient of Pressure) Flow Direction freestream 30 inches Length of HPV fairing (ft) Figure 5-13: Pressure Coefficient the beneath the HPV fairing, 30 inch benchmark (blue) and freestream benchmark (red) 104

117 freestream 30 inches Cp (Coefficient of Pressure) Flow Direction Length of HPV fairing (ft) Figure 5-14: Pressure Coefficient above the HPV fairing at 30 inch (blue), and freestream (red) The computed skin (viscous) friction coefficient of the fairing for both benchmark simulations is shown in Figure The skin friction coefficient for the 30 inches ground clearance simulation is clearly higher than the freestream case. This is a clear indication of the ground effect, which results in a higher viscous drag coefficient (refer to Tables 5-1 and 5-2). The highest values of the skin friction coefficient are found near the leading edge of the fairing. This is because of the initial impact of the airflow on the fairing and the development of the boundary layer. Following the initial impact, the skin friction coefficient starts decreasing gradually over the majority of the fairing before it spikes at the trailing edge. 105

118 C f (skin Friction Coefficient) inches freestream Flow Direction E Length of HPV fairing (ft) Figure 5-15: Skin Friction Coefficient on the bottom of the HPV fairing at 30 inch (black), and freestream (red) HPV Fairing at Different Ground Proximities Results The results of the HPV fairing with ground clearances ranging from 3 inches to 18 inches, at a constant freestream velocity of V = ft, are discussed in this section. However, only the figures for the 15 inch ground clearance are shown in this section, and detailed results for other ground clearances can be found in Appendix B. Figures 5-16 and 5-17 show the velocity profile and the velocity vectors around the fairing with 15 inch ground clearance. The velocity magnitude data reveals that the highest velocities are on the top surface of the fairing and the leading edge below the 106 sec

119 fairing as the flow accelerates. This is shown in red and dark orange. One can also notice that the ground boundary layer interacts with the fairing s boundary layer at the trailing edge and this causes a reduction in velocity in this region. There is also a high velocity region just after the leading edge of the fairing, and this is due to the boundary layer not being fully developed yet. In addition, there is also a high velocity near the trailing region above the fairing. This seems to be caused by the shape of the top surface of the fairing, where the flow picks up some speed and then slows down again. At the trailing edge of the fairing there is a low velocity region that is indicated by light blue, which indicates a stagnation region. This means that the fluid stops moving directly behind the HPV fairing, and a wake region is created in a very similar manner to the benchmark simulations. Figure 5-16: Velocity Vectors about the HPV fairing with 15 inches ground clearance 107

120 Figure 5-17: Velocity magnitude of HPV fairing with 15 inches ground clearance, red indicates high velocity regions and blue indicates stagnation region at the trailing edge of the HPV fairing. Table 5-3 shows the pressure and viscous drag coefficient components of total drag of the fairing with a ground clearance of 15 inches, at various points in the simulation. When looking at Table 5-3 one can see the changes in viscous and pressure drag coefficients at different mesh adaption levels. In order to stop the simulation, the change between the drag components should be theoretically zero or very close to it. The criteria is used as the stopping guideline for the simulation is the percent change between the runs is less than 1%. Also it should be noted the major contributor to the total drag coefficient is the pressure drag. Ground Clearance 15 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E E E E+06 Table 5-3: Drag coefficient components for HPV fairing with ground clearance of 15 inches 108

121 The static, dynamic, and total pressure fields around the HPV fairing with ground clearance of 15 inches and freestream velocity of V = ft sec are displayed in Figures 5-18 to All pressure figures show the fairing in isometric and front view. The front view shows how the pressure is distributed around the fairing, and the isometric view shows the pressure distribution on the ground under the fairing. Figure 5-18 shows the static pressure contours around the HPV fairing. There is a high pressure region in front of the fairing, which is indicated by the red. It is caused by the air attempting to flow around the front of the faring. As the air molecules approach the front of the fairing they begin to compress, and in doing so raise the air pressure. The low pressure regions are indicated with green and are located above and below the HPV fairing. The main low pressure regions are located in the lower leading edge and directly above the fairing, corresponding to the high velocity region discussed previously. The pressure then starts to increase toward the trailing edge under the fairing. This is because the pressure is trying to equalize around the fairing. A B C D Figure 5-18: Static pressure contours of 15 inches ground clearance simulation at symmetry plane. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).Red indicates stagnation pressure (frontal pressure) and green low pressure regions 109

122 Figure 5-19 shows the distrbution of dynamic pressure around the HPV fairing with a ground cleanance of 15 inches. The low pressure regions are indicated using blue and green and high presure regions are indicated using red and orange. Low dynamic pressure regions are corrspond to the low velocity regions in the flow,and vice versa. A B C D Figure 5-19: Dynamic pressure contours of 15 inches ground clearance simulation at symmetry plane. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).The low pressure region is indicated using blue and the high pressure regions are indicated using red-orange Figure 5-20 shows the total pressure around the HPV fairing with a ground clearance of 15 inches. The low pressure regions are indicated using green, high presure regions are indicated using red and orange, and the low pressure wake region is indicated using light green and several shades of yellow. The wake region plays a significant role in the total drag of the HPV fairing. The size of the low pressure region behind the HPV fairing is directly related to the wake region. If the wake region gets bigger the low pressure region will decrease. This is because the ground proximity also decreased. This then causes the pressure difference of the HPV fairing in the flow direction to increase, and total drag will also increase. 110

123 A B C D Figure 5-20: Total pressure contours of 15 inches ground clearance simulation at symmetry plane. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D).The high pressure around the HPV fairing on the surrounding air is indicated using red and orange; the low pressure wake region is indicated using yellow, and light green; and low pressure is indicated using green. The velocity contours on the top and bottom surfaces of the HPV fairing with a ground clearance of 15 inches are shown are Figure This can be associated with the velocity magnitude found in Figure Common knowledge dictates that vorticity is a gradient of velocity. Thus the vorticity contours can be used to represent the boundary layer around, above, and directly under, the fairing. One can see how the boundary layer starts to develop on the ground directly under the HPV fairing, as result of decreasing the height between the fairing and the ground. The boundary layer of the HPV fairing has some interaction with the ground s boundary layer, and the separating boundary core starts to disappear toward the rear of the fairing. The separating boundary core disappears where the boundary layers of the fairing and the ground are completely developed. The compression between all ground clearances is found in Figure

124 Figure 5-21: Vorticity contours of HPV fairing with ground clearance of 15 inches A B C D E F G H Figure 22: Vorticity contours compression of HPV fairing, 3 inches (A), 6 inches (B), 9 inches (C), 12 inches (D), 15 inches (E), 18 inches (F), 30 inches (G), freestream (H) Ground Clearance Effect on Pressure and Skin Friction 112

125 Figures 5-23 to 5-25 displayed below illustrate the comparison of the pressure coefficient above and beneath the HPV fairing, and the skin friction coefficient data for the fairing with different ground clearance ranging from 3 to 18 inches, and a constant freestream velocity of V = ft sec. Figures 5-23 and 5-24 show a comparison of the pressure coefficient beneath and above the HPV fairing, as a function of ground clearance. The reduced ground clearance causes reduced pressure beneath the fairing. In addition, the rear of the fairing also acts as a diffuser; this is due to the low pressure region behind the fairing and the pressure starts to rise at X<2.4 in Figure 5-23 coincide with the tapering of the cross-section. This assists with the increased speed of the airflow between the ground and the HPV fairing. Pressure drag is the major contributor to the total drag of the HPV fairing. There is also a separation bubble beneath the fairing similar to the benchmark simulation of the 30 inch and freestream cases. This creates a negative value for the pressure coefficient near the leading edge, before the pressure then recovers to the freestream pressure towards the trailing edge of the HPV fairing. [13, 14] Figure 5-24 shows that the pressure coefficient behaves very similarly to the benchmark simulation at the top surface of the fairing, but the curves are shifted as ground clearance changes. It is believed that the shift is caused by the effect of the ground clearance on the location of the stagnation point on the nose of the fairing. 113

126 C p (Coefficient of Pressure) inches 6 inches 9 inches 12 inches 15 inches 18 inches Length of HPV fairing (ft) Flow Direction Figure 5-23: Pressure Coefficient beneath the HPV fairing with different ground clearance ranging from 3 to 18 inches 114

127 C p (Coefficient of Pressure) inches inches inches inches Flow Direction inches inches Length of HPV fairing (ft) Figure 5-24: Pressure Coefficient the above the HPV fairing with different ground clearance ranging from 3 to 18 inches Figure 5-24 shows a comparison of the skin friction coefficient on the center of the underside of the HPV fairing at different ground clearances. As expected, the skin friction coefficient increases as the ground clearance decreases. The skin friction coefficient increases and the pressure decrease as a result of decreasing the distance between the bottom of the HPV fairing and ground plane. The highest value of skin friction coefficient is found in the leading edge, and this is due to the initial impact of the airflow with the HPV fairing. Following the initial impact the skin friction coefficient then gradually decreases before it spikes at the trailing edge as a function of ground clearance. 115

128 C f (skin Friction Coefficient) inches 15 inches 12 inches 9 inches 6 inches 3 inches Flow Direction Length of HPV fairing (ft) Figure 5-25: Skin Friction Coefficient beneath the HPV fairing with different ground clearance ranging from 3 to 18 inches 116

129 C p (Coefficient of Pressure) inches 6 inches 9 inches 12 inches 15 inches 18 inches 30 inches Freestream Length of HPV fairing (ft) Flow Direction Figure 5-26: Pressure Coefficient the beneath the HPV fairing for all the simulations 117

130 Cp (Coefficient of Pressure) inches 6 inches 9 inches 12 inches 15 inches 18 inches freestream 30 inches Length of HPV fairing (ft) Flow Direction Figure 5-27: Pressure Coefficient the above the HPV fairing for all the simulations 118

131 C f (skin Friction Coefficient) inches 15 inches 12 inches 9 inches 6 inches 3 inches 30 inches freestream Flow Direction Length of HPV fairing (ft) Figure 5-28: Skin Friction Coefficient for HPV fairing for all the simulations Ground Clearance Effect on Drag and Lift The drag and lift coefficients that result from the HPV fairing simulations of different ground clearances are examined in the section in order to explain the results that were achieved using ANSYS FLUENT. The following description of ground effect is based on literature by J.B. Barlow. [3] The interaction of boundary layer between a moving object and a ground surface depends strongly on the ground clearances. 119

132 B A C D Figure 5-29: Effect of ground proximity (h) on the boundary layer of the body; (A) Large ground clearance i.e. freestream; (B) Medium ground clearance i.e. normal road vehicles; (C) Small ground clearance i.e. racecars; (D) Very small ground clearance when body boundary layer starts to touch the ground. (Adapted from Low Speed Wind Tunnel Testing 3 rd edition by J.B. Barlow) Figure 5-29 shows the effect of the ground on the body boundary layer for various ground clearances. Figure 5-29A illustrates a case where the ground clearance is very large and there is no interaction between the flow around the body and the ground. This description would correspond to the benchmark simulations of freestream flow around the fairing. Figure 5-29B illustrates a case where the ground clearance is at medium height, which is the case for a normal road vehicle. For this thesis HPV fairing ground clearance ranges from 30 to 18 inches. Figure 5-29B shows the flow accelerating under the body, then the flow starts to interact with the moving ground and a deformation occurs in the body s boundary layer. In addition to the deformation of the body s boundary layer, there is also a boundary layer generated along the moving ground. The speed at the edge of ground boundary layer is not equal to zero, but rather it is equal to the velocity of the body. This is because the local speed at the edge of the ground s boundary layer is equal 120

133 to the speed of the flow separated flow in the wake region behind the body. The shape of the body s boundary layer starts to change due to the presence of the ground s boundary layer, with decreased ground clearance (h). Figure 5-29C illustrates a case where the ground clearance is at small height, which is the case for a racecar. For this thesis HPV fairing ground clearance ranges from 15 to 6 inches. Here one has a strong interaction between the body s boundary and ground s layer, due to the fact that there is almost no core separating them. For a smaller ground clearance as illustrated in Figure 5-29D, the underbody will be hindered by the combined ground and body boundary layer, and this will cause blockage and lower the maximum speed beneath the underbody; this causes lower underbody downforce and therefore effect of ground clearance on the max speed beneath the underbody increases lift and drag on the fairing. This is observed in dynamic pressure plots in Figure

134 A B C D E F G H Figure 5-30: Dynamic Pressure contours compression of HPV fairing, 3 inches (A), 6 inches (B), 9 inches (C), 12 inches (D), 15 inches (E), 18 inches (F), 30 inches (G), freestream (H) The computed lift and drag coefficients are plotted in Figure 5-31 for a freestream velocity of V = ft sec versus all HPV fairing ground clearances normalized by the fairing length. In addition, Table 5-4 shows the values for pressure, viscous, total drag coefficients, while Table 5-5 shows the values for pressure, viscous and total lift coefficients. The results for the force coefficients with various ground clearances follow similar trends to the theoretical data found in Figure 1-6 in Chapter one. Significant drag increases are seen as the normalized ground clearance (h/l) decreases below

135 As expected, the lift and drag coefficients are the highest when the ground clearance is the smallest, and have the highest percentage difference from the freestream values. Having a streamlined HPV fairing shape this causes a blockage of air beneath the fairing. This causes a lot of the flow near the ground to create a higher drag on the bottom surface of the HPV fairing. However, due to the shape and the ground clearance, the flow is no longer symmetric between the top and bottom surfaces. Because of that, there are larger speeds and a low pressure region near the top and bottom surfaces of the HPV fairing. This results in the positive lift being higher than the negative lift (downforce) and higher drag. 1.50E-01 C D and C L vs normalized ground clearance 1.40E E-01 Drag Coefficient Lift Coefficient 1.20E-01 C D and C L 1.10E E E E E E Ground Clearness (h/l) on a Log 10 scale Figure 5-31: Effect of ground proximity on the lift coefficient (C L ), and drag coefficient (C D ) of HPV fairing. Where h is the ground clearance and L is the length of the HPV fairing (99 inches). 123

136 Ground Clearance C D pressure C D viscous Total C D % difference 297 (freestream) 7.413E E E % E E E % E E E % E E E % E E E % E E E % E E E % E E E % Table 5-4: Effect of ground proximity on the drag coefficient (C D ) of HPV fairing data Ground Clearance C L pressure C L viscous Total C L % difference 297 (freestream) 6.614E E E % E E E % E E E % E E E % E E E % E E E % E E E % E E E % Table 5-5: Effect of ground proximity on the lift coefficient (C L ) of HPV fairing data 5.2. Estimation of Discretization Error According to the American Institute of Aeronautics and Astronautics (AIAA) guidelines for the definition of an error and uncertainty are as follows. Uncertainty is defined as: "A potential deficiency in any phase or activity of the modeling process that is due to the lack of knowledge." (AIAA G ) Error is defined as: 124

137 A recognizable deficiency in any phase or activity of modeling and simulation that is not due to lack of knowledge. (AIAA G ) There are three situations in this study that could possibly leave room for error. The cases are as follows: 1. Computer round-off error, 2. Iterative convergence error, and 3. Spatial discretization error. The following section will discuss these possible errors and how they could develop. Computer round-off error: This is due to the accuracy at which numbers are stored on a computer. However, with modern computers and it s their ability to store data with 32 or 64 bits, the round-off error is considered to be least significant compared to the other errors. Iterative convergence error: This occurs during the iterative process that is used during the CFD simulation. Ultimately, it must have a stopping point. The error varies with the solution variation at the end of the simulation. Tables 5-1, 5-2 and 5-3 demonstrate how stopping criteria is used to get the final values. The criteria that is used as the stopping guideline for the simulation is the percent change between the runs is less than 1%. Spatial Discretization errors: These are also known as numerical errors. They happen from representation of governing partial differential flow equations and other physical models with algebraic expressions on a discrete spatial domain, also known as a grid or mesh. As the mesh is refined the solution should no longer depend on the grid size, and a solution should converge. This is called grid convergence, and it is very important in finding levels of numerical errors in CFD solutions. 125

138 Discretization Error Calculation According to the Fluids Engineering Division of ASME the following steps are used to find the Discretization error. [4] Step one: Define the representative cell size h, where V i is the volume of the i th cell, and N is the total number of cells used for this computation. The value of h is only used for integral quantities such as drag or lift coefficients. h = 1 N ( V N i=1 i) 1 3 ( 5-1) Step two: Select three significantly different sets of meshes that are demonstrated by subscript 1, 2 and 3; this is done to find the values of main variables. Three sets of meshes from all the HPV fairing simulations were selected; however, only the data for the ground clearance of 15 inches will be shown here. The rest can be found in Appendix C. Step three: Let h 1 < h 2 < h 3 and r 21 = h 2 /h 1, r 32 = h 3 /h 2, and then calculate the apparent order, p, by using Equations 5-2, 5-3 and 5-4. p = 1 ln r 21 ln ε 32 ε 21 + q(p) (5-2) q(p) = ln r p 21 s p r 32 s (5-3) s = sign ε 32 ε 21 (5-4) where ε 32 = f 3 f 2 and ε 21 = f 2 f 1, and f k denotes the solution on the k th mesh. Also it should be noted that q(p)=0 for the first guess for r=constant in Equation 5-2, and 126

139 this equation can be solved using fixed point iteration with initial guess equal to the first term. done for f 32 est. Step four: Calculate the extrapolated values from Equation 5-5. This also can be f 21 est = r p 21 f1 f 2 r 21 p 1 (5-5) Step five: Calculate the estimated errors using Equations 5-6 and 5-7, along with the apparent order p relative approximate errors using Equation 5-8. e a 21 = (f 1 f 2 ) f 1 (5-6) e a 32 = (f 2 f 3 ) f 2 (5-7) 21 = (f 21 ext 2 f 3 ) (5-8) f 2 e ext Step six: The fine grid convergence index is found by 21 GCI fine = 1.25e a 21 r 21 p 1 (5-9) Table 5-6 illustrates this calculation procedure for three selected grids. As mentioned previously the rest of the values are shown in Appendix C. The GCI value represents the resolution level and how much the solution approaches the asymptotic values. [43] Table 5-6 shows the successive grid refinement results and a reduced GCI less than 5%. Therefore, it can be said that the solution on the finest grid resolution is nearly grid independent. 127

140 C D 15 inches N 1, N 2, N , , , , h 1, h 2,h , , , , r r f f f p f ext f ext e a % % 32 e a % % 21 e ext % % 32 e ext % % 21 GCI fine % % Table 5-6: Calculation of Discretization error results for a ground clearance of 15 inches C L 5.3. Tradeoff Study Between Ground Clearances Drag and Stability for a Typical HPV This section will discuss what happens when the HPV fairing ground clearance changes, and its effect on the drag, lift, and handling. Tables 5-4 and 5-5 illustrate the percentage difference in drag and lift coefficients for the HPV fairing with various ground clearances, compared to the freestream cases. 128

141 According to Race Cars Aerodynamics, by Katz, the handling of the vehicle with similar body shape increases with a lower center of gravity and closer ground [14, 15] proximity. However, the drag and lift coefficient increase as well. In order improve the stability of the HPV, it needs to be closer to the ground; however, it will consequently have higher aerodynamic drag which limits top speed. Ultimately, an HPV design has to balance these effects. In order to estimate what is the best height for the HPV a few trade-off studies were done. The first trade-off study estimates the power required as a function of velocity for each ground clearance and compares it to the average human output of 221 ft lb used by the 2010 HPV team. In order to find the required power the following equations are used: P = FV = F D V = 1 2 C DAρV 2 V = 1 2 C DAρV 3 (5-10) sec Here A = 2.84ft 2 and ρ = 2.37E 3 slug ft 3 are constants and C D is used from the Table 5-4 for different ground clearances.required power values are calculated as a function of velocity for each ground clearance, and compared with the average human output, as shown in Figure

142 250 Power as a function of velocity 225 Power [(lb-ft)/sec] Clearance=3" Clearance=6" Clearance=9" Clearance=12" Clearance=15" Clearance=18" Clearance=30" Clearance=free" Average human power Velocity (ft/sec) Figure 5-32: Theoretical power as a function of velocity for various ground clearances 130

143 Power [(lb-ft)/sec] Clearance=3" Clearance=6" Clearance=9" Clearance=12" Clearance=15" Clearance=18" Clearance=30" Clearance=free" Average human power Velocity (ft/sec) Figure 5-33: Zoomed values from Figure 5-30 that shows the behavior of each line The second trade-off study is to find the rollover speed for a corner with a 15ft radius as a function of ground clearance. It is assumed that the ground clearance and center of gravity location differ by a constant. Using an Excel spreadsheet for rollover calculation provided by Dr. Robert Ryan and the CSUN HPV design team the rollover speeds for ground clearances of 3, 6, 9, 12, 15, 18, and 30 inches were found and shown in Figure 5-33 and Table

144 31 Ground Clearance vs. rollover speed Rollover Velocity (ft/ssec) inches 6 inches 9 inches 12 inches 15 inches 18 inches 30 inches Groound Clearance (inches) Figure 5-34: Rollover speed for various ground clearances of the HPV Ground clearance to the bottom of the HPV fairing in inches ground clearance and seat height total in inches Rollover velocity in ft/sec V in ft/sec for max power Table 5-7: Rollover speed and maximum velocity to achieve average human power of for various ground clearances of the HPV 132

145 Chapter 6: Conclusion The flow fields around the HPV fairing at different ground proximities and oblate ellipsoid baseline models have been computed using the ANSYS FLUENT CFD package. This software solves the Reynolds Averaged Navier Stokes equations for 3-D flow with incompressible flow assumption using the one equation Spalart-Allmaras (SA) turbulence model. The surfaces of the HPV fairing and the oblate ellipsoid were assumed to be smooth. The computational domain was selected to be large enough, and the mesh was fine enough, so that the flow calculated field is independent of these parameters, which results in solution convergence. The drag and lift coefficients for the HPV fairing at different ground clearances and the drag coefficient of oblate ellipsoids were calculated from the CFD computations. The computed values for the oblate ellipsoids were compared to experimental values from Fluid Dynamics of Drag by Hoerner to verify that FLUENT was providing correct values and that the mesh was sufficiently fine. Then the computed values for the HPV fairing with different ground clearances ranging from 3 to 18 inches were compared to the HPV fairing in freestream and 30 inches away from the ground. Based on the results for the HPV fairing with different ground clearance, the following can be concluded. 1. The streamlined shape of the HPV fairing does in fact help to reduce drag coefficient; however, due to its shape when the ground proximity increases, a lot of the flow is blocked near the ground and it creates a higher coefficient of drag is created. Also, in addition to the increase in drag there is also an increase in lift coefficient. This is because due to the ground proximity, the flow is no longer 133

146 symmetric and it will have an increased speed near the top of the HPV fairing and low pressure region beneath the HPV fairing. This results in a more positive lift coefficient. 2. The pressure coefficient (C P ) on the fairing bottom surface decreases as ground clearance decreases. This is because as air tries to flow around the HPV fairing as the ground becomes closer, less airflow is permitted to flow beneath the vehicle. This then causes a low pressure region. Additionally, the pressure above the HPV fairing also decreased due to the effect on velocity on this region. 3. As discussed in Chapter 5, the HPV fairing s drag coefficient (C D ) and lift coefficient (C L ) increase as ground clearance decreases. For example, the HPV fairing with a ground clearance of 15 inches has a remarkable increase in C D from to This value represents a nearly 34.13% increase in drag from the benchmark simulation. In addition, the C L increases from to This value is nearly % higher than the C L in freestream. 4. To take advantage of the drag decrease afforded by greater ground clearance the vehicle height is such that cornering stability is severely compromised. It may be possible that changes to fairing geometry can be used as an alternative approach to minimizing ground effect. This is recommended as a topic for further study. 134

147 References 1. Anderson, J.D. Jr., Computational Fluid Dynamics the Basics with Applications, McGraw-Hill, Inc, 1 st edition, February Anderson, J.D. Jr., Fundamentals of Aerodynamics, McGraw-Hill, Inc, 3 rd edition, February Barlow, J.B., Rae, W.H. Jr., and Pope, A., Low-Speed Wind Tunnel Testing, New York, NY: John Wiley& Sons Inc, 3 rd edition, Celik, I.B., Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications, ASME Journal of Fluids Engineering, Vol. 130, July Date, A.W., Introduction to Computational Fluid Dynamics, Cambridge University Press, DeMoss, J.A., Drag Measurements on an Ellipsoidal Body, Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA, August Diasinos, S., Barber, T.J., Leonardi, E., and Hall. S.D., A Two-Dimensional Analysis of the Effect of a Rotating Cylinder on an Inverted Aerofoil in Ground Effect, 15 th Australasian Fluid Mechanics Conference, The University of Sydney, Sydney, Australia, December 13-17, Dress, D.A., Drag Measurements on a Laminar-Flow Body of Revolution in the 13- inch Magnetic Suspension and Balance System, NASA Technical Paper 2895, Fox, R.W., McDonald, A.T, and Pritchard, P.J., Introduction to Fluid Mechanics, New York, NY: John Wiley& Sons Inc, 6 th edition, July Hamamoto, N., Nagayoshi, T., and Koike, M., Research on Aerodynamics Drag Reduction by Vortex Generators, Mitsubishi Motors Technical Review, No.16:11-16, January Hoerner, S., Fluid Dynamic Drag, Published by the Author, Hoerner, S., and Borst, H.V., Fluid Dynamic Lift, Mrs Liselotte A. Hoerner, Hunco, WH., Aerodynamics of Road Vehicles, Warrendale, PA: SAE Int, 4 th edition, Katz, J., Aerodynamics of Race Cars, Annual Reviews Fluid Mechanics Journal, 38:27-63, January Katz, J., Race-Car Aerodynamics, Cambridge, MA, Bentley Publishing,

148 16. Fidkowski, F.K.., and Darmofal, L.D., Review of Output-Based Estimation and Mesh Adaptation in Computation Fluid dynamics, AIAA Journal, Vol. 49, No. 4, April Krishnami, P.N., CFD Study of Drag reduction of a Generic Sport Utility Vehicle, Master Thesis, California State University, Sacramento, CA, Fall Little, R.P., Flight Simulator Database Population from Wind Tunnel and CFD Analysis of a Homebuilt Aircraft, Master Thesis, California Polytechnic State University, San Luis Obispo, CA, May Mafi, M., Investigation of Turbulence Created by Formula One Cars with the Aid of Numerical Fluid Dynamics and Optimization of Overtaking Potential ANSYS Conference & 25 th CADFEM Users Meeting 2007, Congress Center Dresden, Germany, November 21-23, Milliken, W.F., and Milliken, D.L., Race Car Vehicle Dynamics, Warrendale, PA: SAE Inc, August Mohammadi, A., Computation of Flow Over a High Performance, Master Thesis, California State University, Northridge, CA, May Monsch, S.C., A Study of Induced Drag and Spanwise Lift Distribution for Threedimensional Inviscid Flow Over a Wing, Master Thesis, Clemson University, SC, May Munk, M.M., Fundamentals of Fluid Dynamics for Aircraft Designers, New York, NY, The Ronald Press Company, Munson, B.R., Young, D.F., and Okiishi T.H., Fundamentals of Fluid mechanics, New York, NY: John Wiley& Sons Inc, 5 th edition, Panton,R.L., Incompressible Flow, New York, NY: John Wiley& Sons Inc, 3 rd edition, Paparone, L., Tognaccini, R., A Method for Drag Decomposition from CFD Calculations, ICAS 2002 Congress, pp , Roy, C.J., Raju, A., and Hopkins, M.M., Estimation of Discretization Errors Using the Method of Nearby Problems, AIAA Journal, Vol. 45, No. 6, June Spalart, P.R., and Allmaras, S.R., A One-equation Turbulence Model for Aerodynamic Flows, AIAA paper No , January

149 29. Scibor-Rylski, A.J., Road Vehicle Aerodynamics, London, UK, Pentech Press, Schlichting, H., Boundary-Layer Theory, McGraw-Hill, Inc, 7 th edition, Steenbergen, C.K., Vortices in favorable pressure gradients, Master Thesis, Delft University of Technology, Delft, South Holland, Netherlands, July Thompson, J.F., Soni, B.K.., and Weatherill, N.P., Handbook of Grid Generation, CRC-Press, 1 st edition, September Versteeg, H.K.., and Malalasekera, W., An Introduction to Computation Fluid Dynamics The Finite Volume Method, Pearson Education Ltd, 2 nd edition, February Tutorial on CFD verification and Validation, NPARC Alliance CFD Verification and Validation Web Site, 20 December 2010, The Spalart Allmaras Turbulence Model, Langley Research Center, 15 April 2011, History of Theoretical fluid Dynamics, Centrum Wiskunde & Informatica, 17 September 2010, ANSYS Tutorials: 12 August2011, ANSYS FLUENT 12.0 User Guide, ANSYS ICEM CFD 12.1 User Guide, ANSYS WORKBENCH 12.1 User Guide, California State University, Northridge, ME692 Course notes 42. Javaherchi, T., Review of Spalart-Allmaras Turbulence Model and its Modifications, University of Washington, ME Department, March Roache, P. J., Quantification of Uncertainty in Computational Fluid Dynamics, Annual Reviews Fluid Mechanics Journal, 29:123-60, Kusunose, K.., Development of a Universal Wake Survey Data Analysis Code. AIAA , pp ,

150 Appendix A Boundary layer calculation using the integral approach: The integral Equations: Figure A-1: The infinitesimal controlled volume of thickness dx for the boundary layer (adopted from Schaum s Outline of Fluid Mechanics ) Figure A illustrates an infinitesimal controlled volume of a thickness dx. The continuity equation supplies the mass Flux ṁ top that crosses into the controlled volume from the top. ṁ top = ṁ out ṁ in = x The X-component momentum equation is: F x = mom out mom ın Then it will become the following equation: δ ρu dy dx (1) 0 mom top (2) 138

151 τ w dx δdp = δ x ρu2 dy dx U(x) 0 x δ ρu dy dx (3) 0 In the above equation the term pdδ and dpdδ are neglected because they are a smaller order then the entire term because pdδ is very small since the δ assumed is to be very small and dδ is then an order smaller ; also the term for the momentum mom top = U(x)m top. In addition divide the entire equation (3) by (-dx) and the new equation becomes the von Karman integral equation: τ w + δdp dx = ρu(x) d δ u dy ρ d dx 0 δ dx u2 dy (4) 0 After using ordinary derivatives on equation (4), where the density is assumed to be constant over the entire boundary layer and the δ is a function of x. as a result for a flow over a flat plate with a zero pressure gradient, such as U(x)=U and p = 0 this can be simplified and put into the following form: τ w = ρ d δ u(u dx u) dy (5) 0 For the velocity profile of u(x, y) for the specific flow, equation (5) along with τ w = μ u lets both δ(x) and τ 0 (x) be determined. Where δ=y and u=0.99u y y=0 x δ δ d = (1 u ) dy = 0 (6) 0 U δ θ = u (1 u ) dy = 0 (7) 0 U U δ d is the displacement thickness and it is the distance the streamline outside the boundary layer is displaced due the slow moving fluid inside the boundary layer. Θ is the momentum thickness and it is the thickness of the fluid layer with the velocity U that 139

152 possesses the momentum lost because of the viscous effect. It is frequently used as the characteristic length for the boundary layer. τ w = ρu 2 dθ dx (8) Laminar and Turbulent Boundary Layer: The main boundary conditions that need to be meet for the velocity profile in the laminar boundary layer for the flat plate with a zero pressure gradient are: u=0 at y=0 u=0.99u at y=δ u y = 0 at y= δ As figure B illustrates a general flow over a flat plate with uniform velocity and the development of the boundary layers along the flat plate. Figure A-2: Boundary layer development along a flat plate. (Adopted from Fluid Mechanics for Engineering a Graduate Textbook by Meinhard T. Schobeiri) Laminar boundary layers: 140

153 According Prandtl/Blasius boundary layer solution that can be solved for by the governing Navier-Stokes equations with negligible gravitational effects. They become the following two equations: u u u + v = 1 x y ρ u v v + v = 1 x y ρ u + ν( 2 u x v + ν( 2 v y x u x v y 2 ) (9) y 2 ) (10) Figure A-3: Typical characteristics of boundary layer thickness (δ) and wall shear wall stress (τ w ) for laminar and turbulent boundary layer. (Adopted from Fundamentals of Fluid Mechanics 5 th edition by Munson) Using conservation of mass equation for incompressible flow becomes. In addition Where v<<u and x y u + v = 0 (11) x y u u u + v = v 2 u (12) x y y 2 141

154 Although both the boundary equations (11) and (12) and Navier-Stokes equations (9) and (10) are non-linear partial differential equations, there a considerable difference between them. For one, the y momentum equation has been eliminated and only leaves the x momentum equation. The pressure variable has been eliminated and only leaving the x and y velocity components as the only unknowns. In addition for the boundary layer flow over the flat plate the pressure is assumed to be constant and the flow represents a balance between viscous and inertial effects with the pressure playing no role. For the laminar boundary layer, a parabolic velocity profile is assumed due to the fact that the boundary layer is very thin. u = A + By + Cy 2 (13) U The above boundary conditions are used to find the values for A, B and C 0=A 1=A+Bδ+Cδ 2 0=B+2Cδ The solution of the problem then: A = 0 B = 2 δ C = 1 δ 2 This results in the laminar flow velocity profile: u U = 2 y δ y2 δ 2 (14) Then substitute equation (14) in to equation (5) and the result of it is: δ 2 τ w = ρu d 2 y y2 dx δ δ y y2 0 δ δ 2 dy = 2 ρu 15 2 dδ dx (15) 142

155 The wall shear stress is given by the following expression: τ w = μ u 2 = μu y y=0 δ (16) Then equate the equation (15) and (16) to obtain δdδ = 15μ dx (17) U ρ Then integrate equation (17) with δ=0 at x=0 to find the expression for δ(x) in the laminar boundary layer, where μ ρ = ν: δ(x) = 5.48x νx U (18) To find the shear stress (τ w (x)), local skin friction coefficient (c f ) and dimensionless drag force that is the skin friction coefficient (C f ) substitute equation (18) into equation (16). τ w (x) = 0.365ρU 2 ν c f (x) = = 0.365ρU 2 xu R e x τ w = ρU2 ν = 0.73 xu R e x (19) (20) C f = F d = L 0 τ wdx 0.5 ρu 2 L 0.5 ρu 2 L = ν LU = 1.46 R el (21) Buffer boundary Layer: Buffer boundary layer is the layer in between the laminar and turbulent boundary layers; it is sometimes call the transition layer this is shown in figure B. The boundary layer thickness increases in proportion to x where x denotes the distance from the leading edge according to Boundary-Layer theory by Schlichting. Near the leading edge of the flat plate the flow is always laminar and it is becoming turbulent further 143

156 downstream on the plate. The transition takes place at distances x from the leading edge and it determined by: R x,crit = U x = 5 ν 105 to (22) And δ crit δ x=xcrit determined by: 1 δ crit = 5. 48x ν 2 x U crit (23) In addition the process of transition also involves a large decrease in shape factor H 12 =δ 1 /δ 2 where δ 2 =θ and δ 2 = δ d given by: by: Figure A-4: Changes in the shape factor (adopted from Boundary-Layer Theory 7 th edition by H. Schlichting) Using Prandtl-Schlichting local skin friction equation for a Smooth Plates that is c f = A (log 10 R e x )2.58 R e x Where the A is the function of only the transition Reynolds number and it is given (24) A = log 10 R e x 2.58 R e x R e x (25) 144

157 Where local shear stress is a function of the local skin friction and is given by: c f = 2τ w ρu 2 ; τ w = log 10 Rex A Rex ρu 2 (26) Turbulent boundary layer: In the turbulent boundary layer it is often assume a power law velocity profile for this study the maximum R e number was found to be 2.95x10 6 so the n=7 1 u = y n U δ (27) Then substitute this velocity profile equation (27) into equation (5) and integrate to get the following expression for τ w : τ w = 7 ρu 2 dδ 72 dx (28) This τ w from the velocity profiles yields a τ w = u = at y = 0 so this expression cannot be used at the wall. A new expression is needed to be derived for τ w ; the Blasius formula is selected for local skin friction coefficient for turbulent model and given by: y 1 c f = ν 4 U δ (29) In addition to the local skin friction coefficient an experimentally determined formula for shear stress is given by: 1 τ w = ρU 2 ν 4 U δ (30) Combine equation (28) with equation (30) and we get: δ 1 1 4dδ = ν 4 dx (31) U 145

158 Equation (31) can be integrated from δ=0 and x=0 to obtain δ δ 1 x 4 dδ = ν 4 dx; δ = 0.37x ν 0 0 U 1 U x 1 5 R ex < 10 7 (32) Equation (29) can substitute into equation (32) to get the local skin friction coefficient to be: 1 c f (x) = ν 5 U x R ex < 10 7 (33) Equation (33) can be substituted into equation 30 and divided by 2 and right side multiplied by ρu 2 to get the shear stress 1 τ w (x) = ρU 2 ν 5 U x R ex < 10 7 (34) becomes: In addition the dimensionless drag force that is the skin friction coefficient (C f ) 1 c f (x) = ν 5 U L R el < 10 7 (35) Law of the Wall: Law of the wall is the average velocity of the flow at a specific point. It is proportional to the logarithm of the distance a point to the wall (flat plate surface). The law was first published in the early 1930 s by von Karman; it only applies to the flow that is close to the wall. The general logarithmic equations are: u + = u u t with y + = yu t υ and u t = τ w ρ (36) Where: 146

159 y + is the wall coordinate, the dimensionless distance y to the wall u + is the dimensionless velocity τ w is the wall shear stress ρ is the fluid density u t is the friction velocity For laminar study the y + is assumed to be 5, for the buffer layer the y + =11 and, turbulent the y + =30 147

160 Appendix B In Appendix B we can find all of the figures for velocity, pressure and vectors contours from ANSYS FLUENT for HPV fairing at ground clearance of 3,6,9,12,18 inches. Figure B-1: Velocity Vectors about the HPV fairing with ground clearance of 3 inches Figure B-2: Velocity contours about the HPV fairing with ground clearance of 3 inches 148

161 A B C D Figure B-3: Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 3 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). A B C D Figure B-4: Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 3 inches. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). 149

162 A B C D Figure B-5: Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 3 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). Figure B-6: Vorticity contours of HPV fairing with ground clearance of 3 inches 150

163 Figure B-7: Velocity Vectors about the HPV fairing with ground clearance of 6 inches Figure B-8: Velocity contours about the HPV fairing with ground clearance of 6 inches 151

164 A B C D Figure B-9: Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 6 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). A B C D Figure B-10: Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 6 inches. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). 152

165 A B C D Figure B-11: Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 6 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). Figure B-12: Vorticity contours of HPV fairing with ground clearance of 6 inches 153

166 Figure B-13: Velocity Vectors about the HPV fairing with ground clearance of 9 inches Figure B-14: Velocity contours about the HPV fairing with ground clearance of 9 inches 154

167 A B C D Figure B-15: Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 9 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). A B C D Figure B-16: Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 9 inches. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). 155

168 A B C D Figure B-17: Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 9 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). Figure B-18: Vorticity contours of HPV fairing with ground clearance of 9 inches 156

169 Figure B-19: Velocity Vectors about the HPV fairing with ground clearance of 12 inches Figure B-20: Velocity contours about the HPV fairing with ground clearance of 12 inches 157

170 A B C D Figure B-21: Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 12 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). A B C D Figure B-22: Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 12 inches. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). 158

171 A B C D Figure B-23: Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 12 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). Figure B-24: Vorticity contours of HPV fairing with ground clearance of 12 inches 159

172 Figure B-25: Velocity Vectors about the HPV fairing with ground clearance of 18 inches Figure B-26: Velocity contours about the HPV fairing with ground clearance of 18 inches 160

173 A B C D Figure B-27: Static pressure contours of the benchmark simulation at symmetry plane with ground clearance of 18 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). A B C D Figure B-28: Dynamic pressure contours of the benchmark simulation at symmetry plane with ground clearance of 18 inches. Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). 161

174 A B C D Figure B-29: Total pressure contours of the benchmark simulation at symmetry plane with ground clearance of 18 inches Isometric view of contour lines (A); filled contours (B); front view of contour lines (C); filled contours (D). Figure B-30: Vorticity contours of HPV fairing with ground clearance of 18 inches 162

175 Appendix C 2.500E+05 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E+05 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E+05 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E+06 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E+06 Tables C-1: Pressure C D and Viscous C D as separate entities for oblate ellipsoids of finesse ratio 4 163

176 2.500E+05 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E+05 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E+05 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E+05 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E+06 Tables C-2: Pressure C D and Viscous C D as separate entities for oblate ellipsoids of finesse ratio 2 164

177 Ground Clearance 3 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E E E E+06 Ground Clearance 6 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E E E E+06 Ground Clearance 9 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E E E E+06 Ground Clearance 12 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E E E E+06 Ground Clearance 18 Iterations C D pressure C D viscous Total C D mesh density (cells) E E E E E E E E E E E E E E E E E E E E+06 Tables C-3: Pressure C D and Viscous C D as separate entities for HPV fairing 165

178 C D 3 inches N 1, N 2, N , , , , h 1, h 2,h , , , , r r C L f f f p f ext 21 f ext 32 e a 21 e a 32 e ext 21 e ext 32 GCI fine % % % % % % % % % % Table C-4: Calculation of Discretization error for HPV fairing with ground clearance of 3 inches 166

179 C D 6 inches N 1, N 2, N , , , , h 1, h 2,h , , , , r r f f f p f ext 21 f ext 32 e a 21 e a 32 e ext 21 e ext 32 GCI fine 21 C L % % % % % % % % % % Table C-5: Calculation of Discretization error for HPV fairing with ground clearance of 6 inches 167

180 C D 9 inches N 1, N 2, N , , , , h 1, h 2,h , , , , r r f f f p f ext 21 f ext 32 e a 21 e a 32 e ext 21 e ext 32 GCI fine 21 C L % % % % % % % % % % Table C-6: Calculation of Discretization error for HPV fairing with ground clearance of 9 inches 168

181 C D 12 inches N 1, N 2, N , , , , h 1, h 2,h , , , , r r f f f p f ext 21 f ext 32 e a 21 e a 32 e ext 21 e ext 32 GCI fine 21 C L % % % % % % % % % % Table C-7: Calculation of Discretization error for HPV fairing with ground clearance of 12 inches 169

182 C D 18 inches N 1, N 2, N , , , , h 1, h 2,h , , , , r r f f f p f ext 21 f ext 32 e a 21 e a 32 e ext 21 e ext 32 GCI fine 21 C L % % % % % % % % % % Table C-8: Calculation of Discretization error for HPV fairing with ground clearance of 18 inches 170

183 C D 30 inches N 1, N 2, N , , , , h 1, h 2,h , , , , r r f f f p f ext 21 C L f ext 21 e a 32 e a 21 e ext 32 e ext 21 GCI fine % % % % % % % % % % Table C-9: Calculation of Discretization error for HPV fairing with ground clearance of 30 inches 171

184 C D Freestream N 1, N 2, N , , , , h 1, h 2,h , , , , r r C L f f f p f ext f ext 21 e a 32 e a 21 e ext 32 e ext 21 GCI fine % % % % % % % % % % Table C-10: Calculation of Discretization error for HPV fairing in freestream 172