Rowing Blade Design using CFD

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3 Rowing Blade Design using CFD Master of Science Thesis For obtaining the degree of Master of Science in Aerospace Engineering at Delft University of Technology M. Kamphorst B.Sc. June 30, 2009 Faculty of Aerospace Engineering Delft University of Technology Centre for Hypersonics University of Queensland

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5 Preface This report is a result of the research I did for my Master of Science Thesis as a conclusion of my studies at the faculty of Aerospace Engineering at Delft University of Technology. Last year has been a very eventful year in that I moved to Australia, got married to my wonderful wife Taryn and performed my graduation project. All of this would have been impossible without the help and support of many people, a number of whom I would like to mention. Firstly I thank Dr. Michael Macrossan from the Center for Hypersonics at the University of Queensland in Brisbane, Australia, for inviting me to perform this very interesting research for him. Under his supervision I was allowed to grow and develop in insight and analysis of the problem given to me. Secondly, I am Dr.ir. Bas van Oudheusden and Dr.ir. Leo Veldhuis grateful for their willingness to supervise this project from a different hemisphere. Their comments have been extremely valuable to me in improving the quality of my work. I would also like to thank Prof.dr.ir.drs. Hester Bijl for presiding my examination committee. Wietse en Niels, thank you very much for helping me with writing this report and giving me advise on any matter with respect to my graduation. Last but certainly not least I express my gratefulness and thanks to Taryn. Without her love, endurance and support I would not have been able do this. It is my hope that the reader of this report will enjoy studying it and that the results of my work will contribute to the understanding of the physics of rowing. Michiel Kamphorst t Harde, June 30, 2009 ii

6 Abstract How water and air flow around a rowing blade during a rowing stroke is poorly understood by the scientific community. Since important rowing races are won by time differences of only 0.4%, having a sound understanding of the flow will become more and more significant when it comes to rowing blade optimization. It is therefore the purpose of this Master Thesis project to investigate the flow around rowing blades using Computational Fluid Dynamics (CFD) in order to acquire enough knowledge and understanding of the flow to design a rowing blade. Being limited by the options provided by the CFD software ANSYS CFX v.11.0, it is decided to do steady state simulations of a flat plate at the catch and the square-off of a rowing stroke. This way typical phenomena at both sides of the rowing spectrum varying from fully attached boundary layer to a completely separated boundary layer flow can be investigated. Using the Shear Stress Transport k-ω model to model the turbulence and the Volume of Fluid method in combination with a mesh adaption technique to capture the water-air interface, a numerical model is set up with an approximate error of 3.7% at the square-off and an estimated error of 1.5% at the catch. The simulations of the flat plate at the square-off of the rowing stroke are compared to experiments performed by Caplan and Gardner (2007) [4][5] in order to validate the numerical model. It was found that when conducting scale model experiments to investigate rowing blade performance, it is important to establish what the minimum required towing tank dimensions are with respect to the dimensions of the scale model of the blade to be able to neglect blockage effects. When a flat plate is placed perpendicular to the flow, the depth of the towing tank is found to be required to be at least 4.5 times larger than the blade depth, whereas the required towing tank width is 4.3 times the blade width. To obtain dynamic similarity when performing scale model experiments, one has to obtain the same Froude number and Reynolds number as would be obtained in a full scale experiment. For every part of the rowing stroke but in particular at the square-off, the magnitude of the Froude number is of major importance. When the Froude number at the square-off is close to unity, the blade force coefficients are highest, after which they drop to a minimum at the Froude number where a perfect hydraulic jump appears. This is usually at a Froude number of At high supercritical Froude numbers, the force coefficients obtain an asymptotic value that will lie in between the two extrema described before. iii

7 For flow situations where the boundary layer is partially attached to the blade surface, the Reynolds number is important for the blade performance as well. Especially tangential forces are strongly influenced by the Reynolds number, whereas pressure dominated forces show only a very small increase when the Reynolds number is increased. When designing a rowing blade, postponing boundary layer separation will increase the fluid mechanical efficiency of the blade during the early parts of the stroke. Delaying boundary layer separation can be achieved by rounding off the leading edge of the blade and by introducing a camber that has a maximum that is not too close to either the leading or the trailing edge. Although introducing camber increases blade force coefficients, it directs the resultant blade forces away from the hull velocity vector and therefore decreases the fluid mechanical efficiency. When the higher blade force coefficients lead to the incorporation of a shorter shaft, the loss in efficiency can be balanced out. After camber is introduced, the blade geometry can be adjusted such as to give the blade more surface area on the part of the leeward side pointing outward of the boat. This way, the inward pointing tangential force is reduced thus increasing efficiency at low oar angles. Although a rectangular curved blade has a higher efficiency at the square-off, a compromise between a rectangular blade and a blade where the bottom corner at the trailing edge is removed will yield a high overall efficiency during the rowing stroke. If the proposed rowing blade designs are developed further, more optimization studies need to be performed with respect to the location and magnitude of maximum camber. Also the shape of the blade edges other than the leading edge as well as the blade planform shape need to be optimized. In addition to this, the shape of the part of the oar that connects the oar shaft to the rowing blade can be further designed as to improve efficiency at low angles of attack. iv

8 Contents Preface Abstract List of Figures List of Tables List of Symbols iii v vii ix xi 1 Introduction Background Problem statement Objectives Requirements Definitions and rowing terminology Report structure The efficiency of rowing Fluid mechanical efficiency Biomechanical efficiency Optimizing the fluid mechanical efficiency of the rowing blade Reference experiments Caplan and Gardner Coppel Current rowing blade designs Modern blade designs Design features v

9 5 Numerical aspects of modeling a free surface flow with turbulence Turbulence modeling Eddy viscosity modeling Large Eddy Simulation Selecting a turbulence model Interface capturing techniques Front tracking methods Volume tracking methods Volume of Fluid method in practise Flat plate at the square-off Computational method Mesh size study Influence of the domain dimensions on experimental results The influence of the Froude and Reynolds numbers on experimental results Froude number influence on the force coefficients Reynolds number influence on the force coefficients Verification of Reynolds number independence Flat plate at the catch Computational method Mesh size study Required domain dimensions Analysis of flow behaviour The influence of the Froude number on the force coefficients The influence of the Reynolds number on the force coefficients Blade design Design method Aspect ratio Leading edge shape Location of maximum camber Magnitude of maximum camber Geometry variation Quintagonal blade Triangular blade Comparison to rectangular blade Conclusions Recommendations 97 Bibliography 100 A Assignment description 101 B ANSYS CFX settings 103 C Engineering drawings 106 vi

10 List of Figures 1.1 Sign convention for rowing Sign convention for simulations [21] Blade velocity variation during rowing stroke Oar angle according to Kleshnev [13] Velocities relative to water [21] Blade velocities relative to water [21] Force profiles for seat 1 and 7 during rowing stroke at a rate of 40.8 min Mean efficiency, mean propulsion power, shape factor, center of force and maximum force displayed for different seats [23] Hull kinematics for stroke rate = 40.8 min Data for stroke rate = 40.8 min Setup of Caplan and Gardners experiments [4] Projected areas for the model oar blades tested [4] Projected areas for the model oar blades tested [5] Force coefficients compared for flat and curved rectangular blade [4] Caplan and Gardner results for rectangular flat plate Domain layout for simulations by Coppel (2008) [3] Drag coefficient C D vs. angle of attack α [3] Macon shape and dimensions of the tulip-shaped blade [8] Big Blade shape and dimensions [8] Smoothie2 Vortex Blade shape and dimensions [8] Fat Blade shape and dimensions [8] Slik Blade [26] Force coefficients compared for flat and curved Big Blade [4] Kinematics at initial phase of stroke [4] Force coefficients for different curvatures applied to rectangular blade [4] Comparison between curved Big Blade and curved rectangular blade [4] Vortex Edge [27] Fluid domain vii

11 6.2 Flat plate Plate and symmetry plane mesh after mesh refinement Force coefficient in flow direction C X for various mesh sizes Force coefficient perpendicular to flow direction C Y for various mesh sizes Vertical force coefficient C Z for various mesh sizes Volume fraction Φ and symmetry plane mesh after mesh refinement Domain length (x) vs. force coefficient in flow direction C X Domain length (x) vs. vertical force coefficient C Z Water surface for domain length = 500 mm Water surface for domain length = 800 mm Force coefficient in flow direction C X for varying domain half widths Vertical force coefficient C Z for varying domain half widths Flow development for various domain half widths at 30 mm depth. Top view of horizontal plane Force coefficient in flow direction C X for varying domain depth Vertical force coefficient C Z for varying domain depth Flow development for various domain depths in the symmetry plane Froude number F r vs. force coefficient in flow direction C X, constant water density and viscosity, quarter scale flat plate Froude number F r vs. force coefficient perpendicular to flow direction C Y, constant water density and viscosity, quarter scale flat plate Froude number F r vs. vertical force coefficient C Z, constant water density and viscosity, quarter scale flat plate Froude number F r vs force coefficient in flow direction C X, full scale flat plate Froude number F r vs. force coefficient perpendicular to flow direction C Y, full scale flat plate Froude number F r vs vertical force coefficient C Z, full scale flat plate Flow behaviour at various Froude numbers, side view Force coefficient in flow direction C X as a function of the inlet turbulence intensity, quarter scale flat plate Force coefficient in flow direction C X as a function of the inlet eddy length scale, quarter scale flat plate Reynolds number dependence for flat plate Force coefficient in flow direction C X dependence on Reynolds and Froude number for full scale flat plate Vertical force coefficient C Z dependence on the Reynolds and Froude number for full scale flat plate Flat plate dimensions Mesh close to flat plate Influence of number of mesh cells on flat plate forces Influence of maximum cell size on flat plate forces Influence of domain dimensions on normalized normal force on the blade Definitions of domain dimensions Water surface Velocity contour at a depth of 110 mm A close-up of the velocity contour at a depth of 110 mm Streamlines around the leading edge at a depth of 110 mm Separation bubble on windward side Streamlines around the windward side of the leading edge at a depth of 110 mm.. 61 viii

12 7.13 Pressure coefficient at a depth of 110 mm Streamlines around the trailing edge at a depth of 110 mm Locations of boundary layer profiles Boundary layer profiles for leeward side at 110 mm depth Boundary layer profiles for windward side at 110 mm depth Influence of the Froude number on flat plate forces Pressure coefficient C P distribution along the chord of the flat plate Water surface for various Froude numbers Contribution of pressure and viscous forces to tangential force coefficient C T Relation between ratio of tangential over normal forces and Froude number Separation bubble for various Froude numbers Separation bubble for various Froude numbers Influence of the Reynolds number on flat plate force coefficients Pressure coefficient for various Reynolds numbers Contribution of pressure and viscous forces to C T Wall shear stress for various Reynolds numbers Separation region for various Reynolds numbers Surface area exposed to air for various Re Boundary layer development for various Reynolds numbers Development of production term with increasing Reynolds number Development of pressure gradient with increasing Reynolds number Relation between viscous wall shear stress and Reynolds number Relation between viscous dissipation and Reynolds number Configuration of the rowing blade Configuration of the rowing blade Surface mesh of the oar configuration Surface mesh on the leading edge Outer mesh Outer mesh around the leading edge Blade performance as a function of aspect ratio various leading edge shapes Streamlines for rounded leading edge Streamlines for leading edge with lower corner tapered Blade performance as a function of location of maximum camber Streamlines for varying location of maximum camber Blade performance as a function of magnitude of maximum camber Layout of the quintagonal rowing blade Layout of the triangular rowing blade Water surface at angle of attack α = 90 o for rectangular blade Water surface at angle of attack α = 90 o for quintagonal blade Water surface at angle of attack α = 90 o for triangular blade Water surface at angle of attack α = 10 o for rectangular blade Water surface at angle of attack α = 10 o for quintagonal blade Water surface at angle of attack α = 10 o for triangular blade Geometry variation during next design iteration Fluid mechanical efficiency of rowing blade during Dr. Kleshnev s experiments [21]. 94 ix

13 List of Tables 7.1 Domain size Parameter values for fluid mechanical efficiency Blade performance at catch for various leading edge shapes at α = 10 o Blade performance for various blade geometries α = 90 o Blade performance for various blade geometries at α = 10 o Ranking of the three different blade geometries at catch and square-off Comparison of rectangular and quintagonal blade to Big Blade B.1 simulation settings (1/2) B.2 simulation settings (2/2) x

14 List of Symbols English Symbols B n normal blade force [N] B t tangential blade force [-] C N normal force coefficient [-] C P pressure coefficient [-] C T tangential force coefficient [-] C X, C Y, C Z force coefficients in Cartesian directions [-] D blade depth [mm] Ė blade flux of blade energy loss to water [J/s] Ė in energy flux put into oar by rower [J/s] F r Froude number [-] G source term [m 2 ] H height [m] L length [m] M order of series expansion [-] N number of mesh cells [-] P h handle power [J/s] P p propulsive power [J/s] P wb waste power [J/s] P ws waste power due to hull fluctuations [J/s] Re Reynolds number [-] S blade surface area [-] S shape factor [-] S ij strain rate tensor [s 1 ] T torque [Nm] V velocity [m/s] W width [m] X n ordinate normal to plate surface [m] xi

15 c blade chord length [m] c 1, C 2 empirical constants [-] e b Kleshnev blade efficiency [-] e N percentage error in normal force [-] e res percentage error in resultant force [-] e T percentage error in tangential force [-] es effectiveness of boat hull propulsion [-] f variable for Richardson Extrapolation [-] f s propulsive effectiveness for hull propulsion [-] g k independent scalar [-] h mesh cell edge length [m] l m mixing length [m] k kinetic energy of turbulence [J] r location vector of rowing blade [m] t i inflation layer thickness [m] x, y, z Cartesian coordinates for position [m] x CoP Location of center of pressure along chord line [m] u i,j,k velocity in Cartesian directions [m/s] u i,j,k turbulent velocity fluctuation in Cartesian directions [m/s] x/c blade chord fraction [-] Greek Symbols α blade angle of attack [deg] δ ij Kronecker delta [-] ɛ turbulent dissipation [J/s] η fluid mechanical efficiency [-] µ dynamic viscosity [Pa s] ν kinematic viscosity [m 2 s 1 ] ν t eddy viscosity [m 2 s 1 ] ρ density [kg/m 3 ] Π tangential force to normal force ratio [-] τ ij viscous stresses [N s m 1] τij R Reynolds stresses [N s m 1] Φ volume fraction [-] φ oar angle [deg] ψ oar angle in horizontal plane [deg] ψ cof weighted average oar angle [deg] ω oar angular velocity, [deg/s] specific dissipation rate of turbulent kinetic energy [s 1 ] xii

16 CHAPTER 1 Introduction 1.1 Background Rowing is a form of propulsion on water that has been used throughout the course of history. From the impressive Phoenician triremes in ancient times to the 2008 Olympic games in Beijing, rowing is a form of propulsion that survived the ages into modern times. But although human kind has known the concept of rowing for several millenniums, the fundamental flow behaviour around a rowing blade has never been fully understood. In recent history, only a few scientists are known to have attempted to perform experiments with rowing blades. Dr. V.I. Kleshnev from the Australian Institute of Sport has measured oar bending during a rowing stroke, whereas Dr. N. Caplan and Dr. T.N. Gardner have measured rowing blade forces in a water flume. Although these experiments give an idea of the performance of rowing blades throughout a rowing stroke, the behaviour of the water and air flow and the dependence of the blade performance on various factors such as water depth, Froude number and Reynolds number are are still ill understood. The development of Computational Fluid Dynamics (CFD) software and fast computersystems has now made it possible to compute and visualize fluid flows. Although this has been applied in many area s of engineering and science, the combination of moving meshes and interface capturing has made the accurate simulating of a rowing blade during the rowing stroke a challenging project. Considering the narrow margins with which rowing races are won nowadays, CFD will become increasingly important in designing new rowing blades. During the Men s eight rowing final in the 2008 Beijing Olympic games, the difference between the number one and number two on the finish line of the 2000 m was 1.22 seconds. Assuming the hull speed at the finish was 7 m/s, this means that the Canadian and the Great Britain teams finished with a 8.5 m difference. On a 2000 m race, this is a difference of 0.4%. Narrow margins like this signify the importance of capabilities of optimizing rowing blade design. It is the purpose of this thesis to investigate the capabilities of a commercial CFD package to 1

17 model the flow around a rowing blade, investigate the dependence of the blade performance on several parameters and use this knowledge to design a rowing blade. 1.2 Problem statement The goal of this Master Thesis research can be defined as: Use a commercial CFD package to investigate the flow around a rowing blade, in particular the effects of the Froude number and the Reynolds number on the rowing blade forces. Use the results to design a rowing blade To accomplish this goal, the commercial CFD package ANSYS CFX v.11.0 is used to simulate the fluid flow around a rowing blade at several stages of the rowing stroke. Because the current version of ANSYS CFX is not capable of using mesh refinement in transient simulations and because its capabilities for mesh deformation are limited, the choice has been made to do steady state simulations. 1.3 Objectives As stated above, it is the purpose of this Master Thesis project to investigate the water and air flow around a rowing blade. This is done using CFD. As with any computational method, the CFD model that is used will have to be tested against experiments. The experiments that will be used as reference material are the tests performed by N. Caplan and T.N. Gardner [4] [5], a more detailed description of which will follow in chapter 3. After the numerical model is validated, it can be used to investigate water and air flow around the blade. Often when tests are performed on rowing blades, the size of the blades is limited by the dimensions of the towing tank that is available. Since this results in the use of scale models of rowing blades, it is important to obtain dynamic similarity with the full scale flow. As chapter 3 will show, not all scientists are fully aware of this. Therefore it is the purpose of a part of this report to investigate the effects of the two similarity parameters the Froude number and the Reynolds number on the rowing blade performance. Once the effects of the Froude number and the Reynolds number on the blade forces are established, the rowing blade performance can be investigated for various typical flow situations. Using a rectangular flat plate to model a rowing blade, knowledge is accrued about various flow features that appear. This knowledge can then be used to start the conceptual design of a rowing blade. Summarizing, the objectives of this Master Thesis project can be stated as follows: 1. use existing experimental results to validate the results of the numerical model 2. investigate the influence of the Froude number on the blade forces 3. investigate the influence of the Reynolds number on the blade forces 4. analyze the water and air flow around a flat rectangular blade at different stages during the rowing stroke 5. use the acquired knowledge to design a rowing blade 2

18 1.4 Requirements For each rowing team it will depend upon the configuration of the boat what sort of rowing blade is chosen. Children teams for example will use rowing blades that are easier and lighter to handle than adult teams. For this Master Thesis research it is decided to design a blade for a Men s Eight rowing team that uses sweeping blades, i.e. each rower operates only one oar. Sweeping blades generally have a frontal surface area of m 2 Rowing Australia, the rowing authority in Australia, has posed requirements on the geometry of rowing boats and rowing oars in Rowing Australia rules of racing and related by-laws [29]. With respect to sweeping blades, Rowing Australia s safety requirements state that each blade must have a minimum thickness of 5 mm at a distance of 3 mm from the edge. This is the only requirement that Rowing Australia poses on the geometry of the rowing blade. In addition to this, a visit to the local rowing club resulted in the conclusions that oar shafts generally have a thickness of 45 mm and are placed at an angle of 160 o with respect to the blade s top edge. To be able to assume a rigid rowing blade, sufficient stiffness has to be provided. Most rowing blades used in competitive rowing are made from laminated timber or a composite material based on laminated timber and carbon fibre. The combination of blade curvature, the chosen material and the connection part between the oar shaft and the blade make the blade sufficiently rigid when its thickness is equal to the required minimum thickness of 5 mm. Since it is the aim of this Master Thesis to design a rowing blade that could be used in competition, the rowing blade that will be designed has to have an efficiency that is equal to or better then the efficiency of existing rowing blades. As a reference the performance of a Big Blade as used in experiments performed by Dr. V.I. Kleshev, will be used. The requirements posed to the design can be summarized as follows: Geometry 1. minimum blade thickness: 5 mm at 3 mm distance from blade edge 2. blade frontal surface area S = m 2 3. oar shaft diameter: 45 mm 4. angle between oar shaft and blade top edge: 160 o Structural 1. provide blade with place for shaft attachement 2. provide sufficient blade thickness for structural stiffness Performance 1. the designed rowing blade should be at least as efficient as existing rowing blades 1.5 Definitions and rowing terminology Since the technicalities of rowing may be new to most of the readers of this report, a list of rowing terms used in this report is given below. 3

19 sweeping - type of rowing where the rowers use both hands to move one oar. Average blade size is big. sculling - type of rowing where each rower operates two oars, one with each hand. Average blade size is small rowing blade - the large surface area part at the end of the oar that moves through the water, thus propelling the boat shaft - the timber, carbon fiber or fiber glass boom connecting the handle and the blade handle - the part of the oar the rower exerts force on through his hands hull - the outer shell of the rowing boat that is in contact with the water catch - phase of stroke when blade enters the water and a torque is applied to the handle square-off - phase of stroke when oar is at an angle of 90 o with respect to the hull centerline in the horizontal plane release - phase of stroke when oar is pulled out of the water gate - pivot point of the oar Figure 1.1 shows the angles and velocities that will be used throughout this report. The oar angle ψ is the angle between the symmetry axis of the rowing boat hull and the oar in the horizontal plane. The angular velocity of the oar is defined as ω = ψ. Adding the velocity of the hull V hull to the angular velocity of the oar at the position of the rowing blade r results in the blade velocity V blade relative to the water (equation 1.1). Angle of attack α is the angle between V blade or V b and the oar axis. V blade = V hull + ω r (1.1) Figure 1.1: Sign convention for rowing Taking the rowing blade as reference, figure 1.2 shows the relative velocities and angles for the simulations. In this figure, B n and B t are the normal and tangential force respectively. The tangential force direction is defined away from the boat. 4

20 Figure 1.2: Sign convention for simulations [21] How α relates to ψ is shown in figure 1.3 (a). Figure 1.3 (b) shows the resulting blade velocity for the angle of attack. The data for these figures has been provided by Dr. V.I. Kleshev, who measured oar angles, handle forces and hull speeds for a men s eight rowing at a stroke rate of 40 strokes/min. From these figures it can be deduced that at the catch, where ψ = 37 o, α = 10 o with a corresponding V b = 4.7 m/s, whereas at the square-off, both α and ψ are 90 o and V b = 1.4 m/s. The forces on a rowing blade depend on three categories of properties that are listed below: fluid properties - density, viscosity flow properties - freestream velocity, plate depth geometrical properties - plate size and orientation, water depth and distance from hull and bank To be able to compare the performance of different blades, these properties need to be nondimensionalized in a logical manner. Assuming constant geometrical properties, the blade forces will depend on both the fluid and flow properties. The four parameters belonging to these categories can be captured by two non-dimensional parameters: the Reynolds number Re and the Froude number F r. In this report, the effect of these two non-dimensional parameters on the normal and tangential force coefficients (C N and C T ) will be investigated. These non-dimensionalized parameters are defined as: Re = ρv D µ F r = (1.2) V gd (1.3) C N = B n 1/2ρV 2 S (1.4) C T = B t 1/2ρV 2 S Where ρ is the water density (usually with a value of 997 kg m 3 ), g the gravitational acceleration (9.81 m s 2 ), S the blade surface area (0.125 m 2 ). In case of a curved blade, this surface area is defined as the projection of the rowing blade upon a plane that touches the four corners of the windward side of the rowing blade. µ is the water dynamic viscosity ( Pa s), D is the depth of the blade and V is the inlet velocity. (1.5) 5

21 (a) Oar angle vs. angle of attack (b) Angle of attack vs. blade speed Figure 1.3: Blade velocity variation during rowing stroke 1.6 Report structure Because rowing is such an old and widespread activity, there exist numerous views on how to row as efficient as possible. Chapter 2 will therefore give an overview of scientific approaches towards defining rowing efficiency. Then in chapter 3, the work of a number of scientists on the analysis of the performance of rowing blades will be discussed, followed by an overview of modern rowing blade designs in chapter 4. To be able to set up a computational method, in chapter 5 the methods to model the turbulence and to capture the water-air interface are discussed. Chapter 6 then starts with the analysis of the flow around a flat plate at square-off, followed by an analysis of flat plate performance at the catch of the rowing stroke (chapter 7). The accrued knowledge will then be used to design a rowing blade in chapter 8, after which this report is concluded with chapters 9 and 10. 6

22 CHAPTER 2 The efficiency of rowing During the last few years a lot of effort has been put into finding a solid theoretical basis for rowing. Rowing myths have been revealed and created and a unified scientific model seems to be a mere dream. This chapter is dedicated to the ideas and work with respect to rowing efficiency of two players in this field: Dr. V.I. Kleshnev and Dr. M.N. Macrossan. In the first section the focus will be on fluid mechanical efficiency, while the second section will take into account the broader perspective of rower and boat. Section 2.3 will conclude with a discussion on how to proceed in the course of the current research. 2.1 Fluid mechanical efficiency Looking purely at the fluid mechanical efficiency of rowing blades, there are several interpretations available throughout the scientific community of what that exactly means. In two of his reports, which basically cover the same material but to different extents, Dr. V.I. Kleshnev [13] [14] uses the following definition for oar blade efficiency, e b : e b P p P h (2.1) where P h is the handle power, which is the power exhibited at the handle of the rowing oar. The propulsive power P p is defined as the difference between the handle power and the waste power due to blade movement through the water, P wb. This waste power is the product of the normal force on the oar blade (F b ) and the velocity of the blade projected unto the direction of the hull centerline, or written in mathematical form: P wb = F b v b cos φ, with φ the angle between the normal to the hull velocity direction and the oar shaft longitudinal axis, as shown in figure 2.1 as taken from [22], where φ = θ c In the performed experiments to which both reports refer, forces and speeds were measured for several seating configurations, boat speeds and rowing rates. The blade efficiencies for the 7

23 Figure 2.1: Oar angle according to Kleshnev [13] different situations are all within a range of 72% to 96% with an average of 80 83%. Kleshnev concludes that improvement of the blade efficiency can increase the overall rowing efficiency with 3 5%. Based on the work of Dr. M.N. Macrossan, a different approach towards oar blade efficiency can be taken. Firstly, Macrossan shows how the myth of back-splash acts against both intuition and physics [22]. In professional rowing, rowers are often instructed by their coaches to lower their blade into the water just before the blade is at its furthest rearward point. What happens then is that water is splashed in hull velocity, thus exerting a force on the blade opposite hull velocity, which in turn helps the rower to start the rowing stroke with a higher initial angular momentum. It is easily understood that by exerting a force on the rowing blade opposite to the direction of hull speed, the hull velocity is reduced, thus reducing the overall rowing efficiency. Figure 2.2: Velocities relative to water [21] In addition to this, one can look at the energy efficiency of the rowing blade whilst in the water during a propulsive rowing stroke between catch and square-off. Using the definitions provided in figures 2.2 and 2.3, the energy the blade exchanges with the water Ėblade can be defined as follows: 8

24 Figure 2.3: Blade velocities relative to water [21] Ė blade = B n v s B t v p (2.2) with v s the slipping velocity component and v p the sliding, parallel velocity component. Both velocity components are defined as: v s = V hull sinψ + ωl (2.3) v p = V hull cosψ (2.4) In these equations, V hull is the hull velocity and l the distance between the gate and the rowing blade center of pressure. The energy the rower puts into the oar Ėin is defined as: Ė in = T ω (2.5) with T the torque the rower applies to the oar and ω the angular velocity of the oar. Knowing this, the fluid mechanical efficiency of the rowing blade can be calculated as: η = Ėin Ėblade (2.6) Ė in Combining equations 2.2 through to 2.6, the fluid mechanical efficiency of the blade is: η = V hull ωl [ sinψ + B ] t cosψ B n During the first part of the rowing stroke, B t will be negative. Therefore, a tangential to normal force ratio Π can be defined as Π = B t B n (2.8) Π is an indication of the direction of the resultant force with respect to the hull velocity. At oar angles smaller than 90 o a low Π means that the angle between the hull velocity vector and the resultant blade force vector is smaller thus increasing fluid mechanical effiency. Substituting equation 8.2 into equation 2.7, the fluid mechanical efficiency of the rowing blade is calculated as: η = V hull ωl (2.7) [sinψ Πcosψ] (2.9) 9

25 2.2 Biomechanical efficiency Increasing the efficiency of rowing depends on more than just increasing blade efficiency. In one of his earliest reports on the subject, Kleshnev takes into account the efficiency of the hull (which he calls shell). In Kleshnev (1998) [13], the overall efficiency is defined as the sum of the blade efficiency and the shell velocity efficiency. The shell velocity efficiency results from velocity fluctuations of the hull and the resulting loss of energy. Kleshnev concludes that the overall energy loss of 22.4% is due to water movement around the blade (17.8%) and shell velocity fluctuations (5.6%) It appears that improving the blade efficiency can have a large positive influence on the overall rowing efficiency. This is underlined by [14], where Kleshnev states that the (undefined) rowing effectiveness can be increased by 8.2%, of which 6.4% comes from blade efficiency improvements and 1.8% from hull velocity fluctuations. In his 1999 report, Kleshnev [14] defines a large series of parameters. First of all, there is waste energy due to shell velocity fluctuations, P ws, which is calculated as: E ws = kvi 3 kv3, with v the average shell velocity during the stroke. Then there is the propulsive effectiveness for shell propulsion, f s = V real V max = ( espp/k P p/k ) 1/3 = es 1/3 = effectiveness of boat shell propulsion. Kleshnev concludes in his report that reducing the drive time to stroke time ratio will increase the effectiveness of boat shell propulsion. He also states that an increased stroke rate will increase the shell velocity fluctuations, thus decreasing es. In addition to this there are two more reports from Kleshnev, one [15] of which shows the results of measuring the power produced by different parts of the human body. The legs prove to provide 45% of the rowing power, the trunk 32% and the arms account for the remaining 23%. This results in the fact that 53% of the power is provided at the oar handle, while 47% is applied at the footstretcher. Secondly, [16] states that the best results in rowing races will be obtained when a flat force profile is created by accelerating the body as fast as possible. These conclusions are elaborated upon by Macrossan in M.N Macrossan and N.W. Macrossan (2008) [23]. In his report, Macrossan uses Kleshnev s data to investigate the influences of blade efficiency, force profile and force magnitude on the rowing performance of an elite eight rowing at the Australian Institute of Sport. There is a number of parameters Macrossan incorporates in his research. Firstly there is the force profile, that is described by the shape factor (S). The shape factor is defined as S B B max and varies from 0 to 1. A shape factor close to 1 indicates a flat force profile. When at the catch the maximum force is reached sooner and kept constant throughout the stroke, a rectangular force profile is developed, resulting in a shape factor close to one. Two examples of a force profile are shown in figure 2.4. As seen before, Kleshnev seems to be in favour of a shape factor close to unity. To be able to say something about how the forces are applied over the stroke, Macrossan defines the: weighted average oar angle where the weighting is proportional to the force developed. This variable ψ cof can be calculated as ψ cof = Bnψ. It indicates at which oar angle half of the B n integrated force is applied, therefore it is a measure of when most of the force is applied: near catch 10

26 Figure 2.4: Force profiles for seat 1 and 7 during rowing stroke at a rate of 40.8 min 1 or near square-off. Finally there is the average propulsion power, Ė p Ėprop = B n sinψv and the mean efficiency η. For a mean hull speed V = 5.19 m/s and a stroke rate of 21.1 min 1, values for the different variables are given in Figure 2.5, taken from [23]. Figure 2.5: Mean efficiency, mean propulsion power, shape factor, center of force and maximum force displayed for different seats [23] Because of the many factors involved, no clear conclusion can be drawn on what to do to win a rowing race. Based on his findings, Macrossan suggests that applying a greater force near square-off might increase blade efficiency. This is based on the idea that although efficiency is high at the beginning of the stroke, the blade force is very low and angled away from the hull velocity vector. Because of the increased directional efficiency near square-off, it can be worthwhile investigating the idea of shorter strokes at higher rates. This opposes Kleshnev s claim of better rowing efficiency by applying higher forces close to the catch of the stroke. 11

27 (a) Hull velocity vs. oar angle ψ (b) Hull acceleration vs. oar angle ψ Figure 2.6: Hull kinematics for stroke rate = 40.8 min Optimizing the fluid mechanical efficiency of the rowing blade Estimating and optimizing rowing performance is a process that involves a large number of variables. The fluid mechanical efficiency of the blade, the stroke rate, the shape factor and the magnitude of the shell velocity fluctuations each individually have to be optimized according to the wishes of the rowing team. For a rowing blade designer it is important to design a blade such that its fluid mechanical efficiency is as high as possible. Looking back at the equation for the fluid mechanical efficiency of the blade (8.1), every variable has to be taken into account to achieve a high efficiency. Firstly, the hull velocity, V hull, is to be kept as high as possible. Figures 2.6 (a) and (b) are obtained using measurements performed with a Men s Eight rowing team. The oar angle ψ depicted in these figures is the mean value of the oar angles for the eight seat positions at a particular moment in time. These figures show how the fluid mechanical efficiency has potential for increase just after the catch and for 55 o ψ 80 o as well as just before release. Secondly, l and ω need to be kept as small as possible. Whilst l will be constant for a particular oar shaft, ω depends on the stroke rate of the rowing team. Figure 2.7 (a) shows, low ω requires a small handle force, i.e. fluid mechanical efficiency will be increased when the blade forces are small. Figure 2.7 (b) indicates that a low ω corresponds to the phase of the stroke just after the catch and before release. Thirdly and finally, the parameter that will be of main interest for this research is force ratio Π. Π will strongly depend on the rowing blade design and has to be kept as small as possible to achieve optimal fluid mechanical efficiency. In other words, the inward pointing tangential force has to be as small as possible with respect to the normal force to increase fluid mechanical efficiency. In chapter 8, the influence of blade design on this factor Π will be demonstrated. 12

28 (a) Normalized handle force vs. oar angular velocity ω (b) oar angular velocity ω vs. oar angle ψ Figure 2.7: Data for stroke rate = 40.8 min 1 13

29 CHAPTER 3 Reference experiments When a numerical model is developed, it is important to test it and compare the results to experiments performed by others. These experiments can be both empirical and numerical. This chapter gives an overview of relevant experiments that have been done in the area of the current research and that can be used to validate the numerical model that is to be developed during the course of this project. Section 3.1 will describe the empirical experiments, while section 3.2 reports the results of numerical experiments. 3.1 Caplan and Gardner In Caplan and Gardner (2007) [4][5] Nicholas Caplan and Trevor Gardner measured forces on a flat plate and several rowing blades in a water flume experiment. Because of the dimensional limitations of their flume, they used quarter scale blades. Not realizing that the Froude number could be of any importance, they tried to realize dynamic similarity by investigating the effects of Reynolds number, but took no account of the Froude number variation between model and experiments. The flume was limited to a water velocity of 0.85 m/s, but they concluded that for velocities higher than 0.7 m/s the Reynolds number had no influence on the experimental results. Therefore they performed the experiments at 0.75 m/s. Figure 4.7 shows experimental setup and the forces and definitions used in the papers. The projected areas of the used blades are depicted in Figures 3.2 and 3.3. Figure 3.4 shows their results for a flat and curved rectangular blade. To get an estimation of the quality of Caplan and Gardners experiments, their force coefficients for the rectangular flat plate are transformed from lift and drag components into normal and tangential components. This is depicted in figures 3.5 (a) and (b). The tangential force component can be expected to change sign when the angle of attack increases from α 90 o to α 90 o. Caplan and Gardners results show a positive tangential force component for every angle of attack. This C T has an average value of and a standard deviation of It can be concluded therefore that the experimental setup Caplan and Gardner 14

30 Figure 3.1: Setup of Caplan and Gardners experiments [4] used is not able to detect tangential forces, i.e. forces parallel to the oar shaft axis. This is an interesting conclusion, since they do use a strain gauge for this direction, namely strain gauge V in figure 4.7. Thus, assuming all their strain gauges have the same characteristics, it can be concluded that the magnitude of tangential forces is smaller than the error in the force measurements. This means that the error in the calculated force coefficient is larger than The measured C N for 15

31 Figure 3.2: Projected areas for the model oar blades tested [4] Figure 3.3: Projected areas for the model oar blades tested [5] Figure 3.4: Force coefficients compared for flat and curved rectangular blade [4] α = 90 o is This value can be used as a reference value when investigating the flow around a flat plate at square-off in chapter Coppel In an attempt to numerically investigate the flow around a rowing blade, Coppel et al. (2008) [3] used Fluent to model this flow. Instead of using a free surface, the air-water interface was modeled using a symmetry plane as is shown in figure 3.6. Two turbulence models (k-ɛ and SST k-ω) are tested together with a first and a second order upwind scheme. The calculated drag coefficients are 16

32 (a) Normal force coefficient C N vs. angle of attack α (b) Tangential force coefficient C T vs. angle of attack α Figure 3.5: Caplan and Gardner results for rectangular flat plate compared to the empirical results obtained by Caplan and Gardner (2007) [4]. The resulting graph is depicted in figure 3.7. This graph implies that SST k-ω performs better in these type of experiments than the k-ɛ model. Care needs to be taken though since the free surface has not been taken into account while its influence is expected to be significant. Coppel et al. conclude that the Reynolds number assumption of Caplan and Gardner [4] is incorrect. For a Macon blade at α = 90 o, they measure a C D of 1.97 at an inlet velocity of 0.75 m/s. For inlet velocities of 2.5 m/s and 5 m/s, the measured C D is This is interpreted as a Reynolds number effect whilst it may well be a Froude number effect in as far as the latter is allowed for by a numerical model that does not include air flows and models the water-air interface with a symmetry plane. Figure 3.6: Domain layout for simulations by Coppel (2008) [3] 17

33 Figure 3.7: Drag coefficient C D vs. angle of attack α [3] 18

34 CHAPTER 4 Current rowing blade designs Designing a rowing oar blade is not a unique phenomenon in history; many have gone before us and many will follow. Section 4.1 gives an overview of the mainstream blade designs. The designs described in this section are produced by Concept2, a company specialized in rowing materials and equipment. An introduction into the various features used in modern rowing blade design is given in section Modern blade designs A traditional rowing blade used for over thirty years is the tulip-shaped Macon blade. Figure 4.1 shows the main features of this blade. Figure 4.1: Macon shape and dimensions of the tulip-shaped blade [8] During the last decade a lot of improvement has been made in the area of rowing blade design. Firstly, there is the Big Blade. The Big Blade is designed to displace a large amount of water during the square-off and the last part of the stroke. Shape and dimensions of this blade are shown in Figure 4.2. The wide outboard edge of this blade produces an increased amount of leverage at 19

35 square-off, thus providing the rower with the option of using a shorter shaft. Figure 4.2: Big Blade shape and dimensions [8] Secondly, there is the Smoothie2 Vortex Blade. The philosophy behind this blade is that a turbulent flow at the low pressure side of the blade, created by the so called vortex edge, during the initial part of the stroke just after catch, provides more water grip. For rowers who prefer to use most of their power during the first part of the stroke, this blade will prove very convenient. The shape and dimensions of this blade are depicted in Figure 4.3. Figure 4.3: Smoothie2 Vortex Blade shape and dimensions [8] Concept2 s most recent design is the Fat2 Blade. This blade provides the rower with an even heavier load at the first phase of the stroke. After the last phase of the stroke the blade is designed to create less energy dissipation. An overview of the blade s characteristics is given in Figure

36 Figure 4.4: Fat Blade shape and dimensions [8] A different approach to blade design has been taken by Croker Oars with their Slik Blade (Figure 4.5). The Slik Blade relies on as little splashing at catch and release as possible and is visibly not designed to create extra turbulence. Figure 4.5: Slik Blade [26] 4.2 Design features Other than defining the cross-sectional shape, one of the main features to optimize the design is to change the curvature of the blade. As the static experiments of Figures 4.6 and 3.4 show, adding curvature to a flat blade increases the C D near square-off and the C L at the initial stage of the stroke, where lift and drag are defined according to Figure 4.7. There does seem to be an optimimum though, since after increasing the curvature depth more in Figure 4.8, the hydrodynamic characteristics of the blade deteriorate. Interesting to see is that a well defined cross-sectional shape does not always improve blade performance in steady state conditions (Figure 4.9). 21

37 Figure 4.6: Force coefficients compared for flat and curved Big Blade [4] Figure 4.7: Kinematics at initial phase of stroke [4] 22

38 Figure 4.8: Force coefficients for different curvatures applied to rectangular blade [4] Figure 4.9: Comparison between curved Big Blade and curved rectangular blade [4] A rather modern design feature is the Vortex Edge [27] (see Figure 4.10). The Vortex Edge is a strip of thin triangles attached to the leading edge at the low-pressure side of the blade. The principle of this design is that the increased turbulence at the low pressure side increases the lift of the blade at the initial stage of the stroke. Although Concept2 claims to have had no negative response in terms of speed or handling, the actual benefits of this feature still need to be proven in public. If the flow is partially separated near the leading edge, the attached flow part at the lowpressure side of the blade will produce a force component in the direction of the hull velocity, which 23

39 is positive. What happens with the Vortex Edge though is that this attached region is transitioned into a separated region, increasing the size of the wake, therefore possibly reducing the advantageous effect of an attached flow near the leading edge. If on the other hand the velocity is not sufficiently low, such that the wake is fully developed behind the blade, i.e. there is no attached region left at the low-pressure side, the Vortex Edge will have no effect on the flow at all. Figure 4.10: Vortex Edge [27] 24

40 CHAPTER 5 Numerical aspects of modeling a free surface flow with turbulence During a rowing stroke, the Reynolds number dependent on the blade depth is has values between 350,000 and 1,200,000. This indicates that the flow will be turbulent. Section 5.1 describes a range of turbulence models in order to conclude which available turbulence model would perform best for rowing simulations. In addition to this, the numerical capturing of the water-air interface is important to obtain a sufficiently accurate solution. In section 5.2 an overview is given of numerical models that have been developed in order to capture a fluid-fluid interface. 5.1 Turbulence modeling In order to reduce computational effort in the calculation of turbulent flows, turbulence closure models are used to solve problems with turbulent flows. There exists a large range of models, each with its own limitations are applications. This section will treat a selection of these models starting with eddy viscosity models and followed by Large Eddy Simulations (LES) and a conclusion on which method to use to model the turbulence Eddy viscosity modeling The larger part of the turbulence models described below are based on a turbulent viscosity model. The principle behind turbulent viscosity modeling is the following. Starting out with the momentum equation: ρ u i t + ρ (ū ) u i = p u i + τ ij x j (5.1) where [ ui τ ij = ρν + u ] j x j x i (5.2) 25

41 the Reynolds averaged momentum equation results in: ρ (ū ) u i = p + [ τij + τ R ] ij u i x j (5.3) with τij R as the Reynolds stresses. Boussinesq came up with a shear-stress strain-rate relationship for time-averaged flows with a one-dimensional nature that eventually resulted in the following equation for the Reynolds stresses: [ τij R = ρu i u j = ρν ui t + u ] j ρ x j x i 3 u k u k δ ij (5.4) The applicability of the kinetic theory of gases for predicting the macroscopic property of viscosity leads to the prediction that viscosity ν t is proportional to a length scale and the rms speed of the molecules. Prandtl brought this theory from a molecular level to a level where lumps of fluid exchange momentum between different layers. This leads to Prandtl s mixing-length model to calculate eddy viscosity: ν t = lm 2 ū i x j (5.5) Prandtl s mixing-length model has great impact on all the turbulence models that are based on modeling eddy viscosity. This choice of eddy viscosity suggests that the length scales of the vortices in all three Cartesian directions are roughly the same, leading to the limitation that these turbulence models are all limited to flows with isotropic turbulence. Accurately modeling the eddy viscosity is the main topic of turbulence methods that are discussed in this section. The k-ɛ model With the k-ɛ model, an answer is sought for the description of the eddy viscosity, ν t. Firstly it is assumed that ν t k l, where l is of the order of the integral scale and k the kinetic energy of the turbulence. The k-ɛ model then assumes that the rate of turbulent energy destruction ɛ u3 l, leading to ν t k2 ɛ. This is then put into the form ν t = c µ k2 ɛ. The coefficient c µ is usually given a value of After this two empirical transport equations have to be solved: [( k t + u (k) = ν + ν ) ] t k + G ɛ (5.6) σ k [( ɛ t + u (ɛ) = ν + ν ) ] t Gɛ ɛ + c 1 σ ɛ k c ɛ 2 2 k (5.7) ( with G = τ R ij ρ ) S ij. 26

42 For general engineering purposes, the k-ɛ model seems to do rather well although it has some major drawbacks. The first one is its incapability of distinguishing between homogeneous isotropic turbulence and homogeneous anisotropic turbulence when u freestream = 0. This is because the k-ɛ model always assumes freely decaying turbulence to be isotropic. It is known that it is usually very difficult to get rid of anisotropies in a turbulent flow. The k-ɛ model gives acceptable solutions for one-dimensional, homogeneous shear flows because it predicts that the flow develops towards a self-similar solution. This makes the model very well suited for simple shear flows. But when the flow situation becomes more complex, such as stagnation point flows, boundary layers with strong adverse pressure gradients, rapid strain rate flows or flows with anisotropic turbulence such as buoyancy or strong swirl flows, the k-ɛ model starts to show major shortcomings. [9] To solve the shortcomings of the k-ɛ model, two variations to this model have been developed: the RNG k-ɛ and the Kato-Launder k-ɛ model. Re-normalization group k-ɛ Because the standard k-ɛ model assumes that large scales dominate the flow, the re-normalization group (RNG) k-ɛ has been developed in order to account for the influence of smaller scales. This is done by adjusting the ɛ equation. [6] Because the main problems of the standard k-ɛ model are related to basically any flow that deviates from the idealized one-dimensional homogeneous shear flow, it is not expected that the RNG k-ɛ model solves many of the problems that are encountered by the standard k-ɛ model. This is supported by experiments. With little exceptions (Lasher and Sonnenmeier (2008) is one of these [19]) RNG k-ɛ often performs equally or less than the standard k-epsilon model ([1] [28] [31]). Kato-Launder k-ɛ A better modification to the standard k-ɛ is the Kato-Launder k-ɛ model. The Kato-Launder k-ɛ is a modification that improves the prediction of turbulence in stagnation flows, which were a problem for the standard k-ɛ. In many researches where this stagnation flow is important, this modification is proven to be a success [10]. In computing the flow past a partially enclosed square cylinder, Saha et al (1999) [1] proved the Kato-Launder k-ɛ to be superior when it comes to lift and drag coefficients and the velocity profile. Also Lakehal and Rodi (1997) [18] showed that Kato-Launder k-ɛ eliminates the excessive kinetic energy production in the separation region in the flow over a surface-mounted cube. Bosch and Rodi (1996) [2] confirm that Kato-Launder k-ɛ is superior to the standard k-ɛ model in predicting vortex-shedding past a square cylinder. The k-ω model A minor conceptual difference to the k-ɛ model is the k-ω model. Instead of a second equation for ɛ, a second equation for ω, which is the specific dissipation rate of the turbulent kinetic energy [10]. Since ω is defined as ω = ɛ / k, the ω equation becomes: t (ρω) + ρpω (ρu j ω) = c ω1 x j k c ω 2 ρω 2 + [( µ + µ ) ] t ω (5.8) x j σ ω x j Since the k-ω model is based on the same principles as the k-ɛ model, the same applications and limitations can be expected with an exception for adverse pressure gradient boundary layers and separation for which the 1998 Wilcox model is said to work well [19]. This statement is supported by the following scientific data. 27

43 Firstly Kuznik et al. (2007)[31] investigated the fluid flow in a full scale ventilated enclosure using the k-ɛ realizable model, RNG k-ɛ, k-ω and SST k-ω. For the hot and isothermal jet case, none of the models were able to provide the researchers with a satisfactory solution. Although in the cold jet case k-ω performs better, it still does not match experimental data. The k-ω model did give the best solutions for the jet-expansion rate, but since jet-expansion is not part of the problem areas of the k-ɛ model, k-ω does not solve the problems inherent to this range of closure models. Another example is Liu et al. (2006)[28]. In their numerical simulation of UV disinfection reactors none of the used models (k-ɛ, RNG k-ɛ, k-ω ( 88) and k-ω ( 98)) stood out with respect to the rest. It proved that although both the k-ω models seemed to perform better than the k-ɛ models, the results of omitting the cross-diffusion term still largely limited their performance. In addition to this, Lasher and Sonnenmeiers spinnaker sail computations [19] prove that both the standard k-ɛ and k-ω models have the least accuracy when compared to the RNG k-ɛ, realizable k-ɛ and SST k-ω. Finally, Smith and Pongjet (2008) [32] show that k-ɛ and RNG k-ɛ have superior results with respect to k-ω when it comes to the heat transfer of turbulent channel flow over periodic grooves. Shear Stress Transport k-ω In order to be able to avoid the weaknesses of the k-ɛ and the k-ω models and combine their strengths, the Shear Stress Transport (SST) k-ω model has been developed. This model uses the k-ω model formulation in the viscous sublayer, while basing its free stream behaviour on the k-ɛ method. Although the model does produce too much turbulence in large normal strain regions, the overproduction is smaller than what the k-ɛ model would do. In regions of adverse pressure gradients and separation, the SST k-ω model is expected to perform well [7]. One of the main problems of SST k-ω though is its dependence on grid refinement. Rumsey (2007) [30] shows that with increasing grid density on the nose of an airfoil, its separation point moves further backwards. In addition to this, Lasher and Sonnenmeier (2008) [19] mention that SST k-ω together with RNG k-ɛ experiences robustness problems. Result wise though, the SST k-ω here is an improvement of the standard k-ω method, whereas in Kuznik et al. (2007) [31] the SST k-ω results for velocity profiles are generally worse than the standard k-ω s. Also for the heat transfer in [32] the SST k-ω performs less than its k-ɛ counterparts. Spalart-Allmaras A slightly different approach to turbulence modeling is that of Spalart and Allmaras (1992) [10]. They devised a one-equation model that solves a transport equation for the kinematic eddy viscosity, which is specifically designed for aerodynamic purposes and has been used and trusted widely for that [30]. One of the problems though with this model is its transition prediction as shown by Rumsey (2007) [30]. As well as SST k-ω, the Spalart-Allmaras model s prediction of the transition point is dependent on the grid refinement. But in the intermediate region between the viscous sublayer and the logarithmic layer in a recirculating flow, the Spalart- Allmaras model produces satisfactory results [11] Large Eddy Simulation If Direct Numerical Simulation (DNS) is too expensive and the implementation of a turbulence model to crude, a solution can be sought in Large Eddy Simulation (LES). LES uses the fact that turbulence energy cascades from larger length scales to smaller length scales. This means that 28

44 whatever happens in the larger length scales influences the smaller length scales but not vice versa. For the engineering application for which the larger length scales are of a major importance, the smaller length scales can be modeled by some heuristic viscosity model, while the larger length scales are accurately calculated. This is the general principle of LES [9]. A demonstration of the higher accuracy of LES with respect to Reynolds Averaged Navier Stokes (RANS) is Min and Gao (2006) [24]. In this paper, the curve of tracer response, power demand and mixing time for LES and RANS are compared to experimental data. In these three cases, LES proved to provide more accurate results than RANS. Difficulty with the comparison here is that in this paper, the k-ɛ model is used for the RANS comparison, while in section it was shown that the k-ɛ has severe difficulties in solving for flows with rapid strain rate flows, which is exactly the type of flow that can be expected with impeller speeds up to 300 revolutions per minute in a viscous fluid (µ is calculated to be kg/(ms)). Although a different choice of RANS model may have been better, the result is what has to be expected: LES is more accurate than RANS Selecting a turbulence model In order to save a large amount of time by not performing DNS during the current research, it will be important to select a turbulence method. To do so a choice has to be made either for RANS or for LES. Since ANSYS CFX v.11.0 does not have a LES module, a RANS model has to be chosen. Since one of the main features of the flow around a rowing blade is the fact that a large part of the stroke is a stagnation flow, it is important to select a turbulence model that is able to cope with this type of flow. The standard k-ɛ and k-ω both seem to have trouble with stagnation flows. Both the RNG and SST modifications are not designed to improve the performances of the standard models in this area. About the Spalart-Allmaras model behaviour in stagnation flows there is not enough data to get a clear view on this. Therefore if a turbulence closure model has to be incorporated, the choice should be for Kato-Launder k-ɛ, since its capability to cope with stagnation flows has been proven in section Because ANSYS CFX v.11.0 does not have a Kato-Launder k-ɛ model, a choice is made from the remaining turbulence models. The SST k-ω model is a good improvement of the k-ω model and performs better than the k-ɛ model. Based on its incorporation of good characteristics of both the k-ω and the k-ɛ model, it is decided to use the SST k-ω model for the simulations. 5.2 Interface capturing techniques A main difficulty with the flow simulations around rowing blades is the presence of a free surface. This free surface, the water-air interface, has a major influence on the behaviour of a rowing blade during the different phases of the stroke. Effectively simulating and tracing the location of this interface is a very recent development in CFD. In this chapter, a summary of several available methods will be given, beginning with front tracking methods in section 5.2.1, followed by volume tracking methods in section 5.2.2, and after the specific treatment of the Volume of Fluid (VOF) method in section 5.2.2, the chapter is concluded with a discussion of the findings in section Unless otherwise indicated, the information provided sections and is based on van der Pijl (2004) [33] 29

45 5.2.1 Front tracking methods One of the Eulerian free surface methodologies is front tracking. With front tracking, the interface is represented by a large number of nodes that are connected by lines or surfaces (depending on whether the problem is 2D or 3D). These nodes are followed using mass and momentum equations, thus keeping track of the location of the interface or front. The main problem that is encountered with front tracking methods is the inability to cope with numerical coalescence, although there exist methods that deal with this problem. Several methods exist that make use of the front tracking principle. Hybrid front-tracking front-capturing The hybrid front-tracking front-capturing method solves for the flow on a stationary grid. Using the resulting information, the advected location of the interface can be determined. Due to the fact that the interface is not divergence-free, as opposed to the flow, this hybrid method encounters interpolation errors. Immersed boundary This method is designed for biological fluid dynamics. The interface is represented by an elastic membrane that interacts with the incompressible viscous fluid. It basically has the same characteristics as the hybrid front-tracking front-capturing method, although the volume conservation is improved by equating the stationary velocity field divergence to the interpolated interface velocity field divergence. The resulting smoothing of the interface over several cell widths can cause parasitic currents. This method is inappropriate for solid-fluid calculations. Cut-cell As opposed to the previously described methods that experience smoothing of the interface, the cut-cell method is a sharp front-tracking method, which works for both solid-fluid and fluid-fluid calculations. With the finite volume method the sharpness is obtained by identifying the material the node of the finite volume belongs to. The rest of the finite volume that does not consist of the same material is added to the control volume of the neighbouring node. The finite-difference version of this method is used for solidification problems. Fictitious domain The fictitious domain method works similar to the hybrid front-tracking front-capturing method, with a difference that it is developed for solid-fluid interactions. The total domain consists of two domains: the flow domain and the fictitious solid-domain. Again, the flow equations are solved for the fluid domain, after which they are enforced on the fictitious domain, thus initializing the movement of the solid particles Volume tracking methods As opposed to front tracking methods where extra nodes are added to the system, one can also choose tho add an extra variable to the problem. This will result in what is called a volume tracking method. Most of the volume tracking methods are based on a marker function that works similar to tracing particles in a windtunnel. Generally the advantage of volume tracking methods is that there are no problems with respect to numerical coalescence. A list of volume tracking methods that are of interest for the current research is given below. Marker and cell With the marker and cell method, the interface is marked by a marker function, i.e. no extra elements are introduced but an extra variable. The cells contain mass-less markers that advect with the flow. Their velocities are interpolated from the calculated flow field. All the cells with fluid contain these particles whereas the rest of the cells are voids. This method only works for void-fluid flows. It experiences trouble with surface cells. Also no viscosity is allowed at the fluid surface. 30

46 Surface marker and micro cell (SMMC) This method works similar to the marker and cell method with this difference that the SMMC method only has markers in the surface cells thus following the progression of the surface. Macro cells bordering voids are subdivided into micro cells in order to make the method more accurate and sharp than the conventional marker and cell method. Volume of fluid (VOF) VOF defines the interface by a fraction of fluid in the interface cells. The value of this fraction ranges from 0 (no fluid) to 1 (completely filled with fluid). It works like the marker method but then with an infinitely large amount of markers. VOF is a simple straight-forward method but it does experience problems with mass conservation. Simple Line Interface Calculations (VOF/SLIC) SLIC reconstructs the interface with lines either parallel or perpendicular to the coordinate direction, based on VOF information. When the shape of the interface becomes somewhat complex, a significant amount of flotsam and jetsam can be created, which compromises the solution. When surface tension is modeled, a lot of flotsam and jetsam is created which makes it inappropriate to use the volume of fluid method in combination with the modeling of surface tension. Especially where the modeling of surface tension is required, this problem becomes important. Hirt-Nichols VOF Hirt-Nichols VOF, also known as the donor-acceptor method, approximates the flux for the flux advection equation: Φ t + u jφ = 0 (5.9) x j The donor-acceptor method works as a flux approximation for the flux-advection equation. It sometimes induced volume errors, but they may be caused by other issues such as grid generation. Piecewise Linear Interface Calculation (VOF/PLIC) Instead of recreating the interface in a cell with a horizontal and perpendicular direction, PLIC constructs a linear interface for both the interface conditions and the calculation of fluxes and is therefore a refinement of SLIC and Hirt-Nichols VOF. Its characteristics are superior to those of SLIC and Hirt-Nichols VOF. Flux-corrected-transport (VOF/FCT) A middle way between VOF/PLIC and VOF/SLIC is the flux-corrected transport method. VOF/FCT solves the problems SLIC and PLIC have with mass conservation due to interface smoothing, by maintaining extremely sharp interfaces. It is inferior to VOF/PLIC, but is still more accurate than VOF/SLIC. Constrained Interpolation Profile (CIP) The Constrained Interpolation Profile works similarly to the Volume of Fluid method and is suitable for large density ratios in compressible and incompressible flows. CIP also uses the flux advection equation, but uses a different approach to discretize this. Level-set The Level-set function is analogous to a VOF marker method with the difference that the sign changes over a material boundary. That means that at zero level set value for the marker function Φ a material boundary is encountered. Level-set deals with the interface time-evolution problem by advecting Φ as a material constant according to: 31

47 Φ t + u Φ j = 0 (5.10) x j The level-set method has rather poor mass conservation characteristics. Variational Level-set Variational Level-set uses different level-set methods for different fluids and phases. It is especially designed for surface angles between drops of fluid and solid walls. Therefore it is available for steady state and viscosity dominated unsteady flows. Coupled Level-Set Volume of Fluid There are several methods to couple Level-set to VOF methods. Mass conservation characteristics are similar to VOF methods. It is quite elaborate to start from a Level-set method. When the basis for the method is Volume of Fluid though, a rather straightforward method gives accurate surface curvatures and surface tension can be measured. This is underlined by van der Pijl et al. (2008) [34] Volume of Fluid method in practise The Volume of Fluid method has proven itself for a large range of computations. Kvinsky et al (1999) [17] validated VOF for the case of a jet flow onto an inclined flat plate. A three-dimensional unstructured mesh was created using hexahedric elements, to capture the flow of the water on the plate. After the numerical simulation was run and compared with experiments, it was shown that the VOF method produced results that were very close to the experimental data. In addition to this, Kleefsman et al (2004) [35] performed two numerical experiments using VOF. Firstly they simulated the very violent dam brake flow. Using a fixed Cartesian grid, VOF proved to give satisfactory results in simulating this highly complex flow. Also the water entry and exit of wedges and cylinders gave results which were in good agreement with performed experiments. A weakness though of VOF proved to be the capturing of spray. Thirdly, Yang et al (2006) [20] simulated the interaction between a freely floating ship at sea and the sea itself. Using an adaptive unstructured grid it was proven that the Volume of Fluid method is capable of simulating this flow, which incorporated a LNG carrier with a sloshing cargo on a rough sea. Since VOF has been satisfactorily used to simulate complex water-air interface flows with objects entering the water and low frequency behaviour of freely floating objects, it is expected that it will perform robust simulations for the movement of rowing blades entering and exiting the water, where unstructured moving grids might be needed. 32

48 CHAPTER 6 Flat plate at the square-off Using the Computation Fluid Dynamics software ANSYS CFX v.11.0, a model is set up for a flat plate at an angle of attack of 90 o. The purpose of investigating this particular flow is to compare the performance of ANSYS CFX v.11.0 to the experimental results of Caplan and Gardner [4] for a flat plate at 90 o angle of attack and to investigate the influence of the Froude number and Reynolds number on the force coefficients of the flat plate. In addition to this, the influence of the fluid domain dimensions on the flat plate forces are investigated. 6.1 Computational method The fluid domain is shown in figure 6.1 with a symmetry plane in the x-z plane cutting the flat plate in half and a model is set up using the Volume of Fluid method (see section 5.2) to capture the water-air interface and the SST k-ω method to model the turbulence. Using the dimensions Caplan and Gardner provided in their work, the half width of the domain of the quarter scale flat plate is set to be 320 mm with a domain depth of 150 mm. The plate (shown in figure 6.2) thickness is given to be 1.8 mm, this is the thickness Caplan and Gardner needed to provide their aluminum flat plate with enough stiffness to assume a rigid plate. Since no length dimensions are given in their paper, the influence of the length of the domain on the simulation results is investigated in section 6.3. Important to note is that the domain length upstream of the flat plate is kept constant at 250 mm. The aspect ratio of the flat plate is deduced from a figure provided in the publication of Caplan and Gardner and is measured to be a half-width of 70 mm and a depth of 55 mm. Since the aspect ratio of the flat plate has a large influence on the plate forces as is shown in chapter 8, deviations of the force coefficients obtained from the simulations with respect to Caplan and Gardners experimental results can be caused by having drawn a wrong conclusion about the aspect ratio of the flat plate they used. To be able to investigate the influence of the mesh size on the simulation results, a mesh size study is performed in the next section using the flume dimensions provided by Caplan and Gardner. Then in section 6.3, the influence of the domain dimensions on the experimental results is determined. These results are then used in section to make the domain just big enough to 33

49 be able to assume far field boundary conditions for the quarter scale and the full scale domain. Using Caplan and Gardners domain sizes, in section 6.4 the influence of the Froude number and the Reynolds number on the flat plate forces are investigated for both the quarter scale and the full scale models. Figure 6.1: Fluid domain Figure 6.2: Flat plate 6.2 Mesh size study Defining the mesh size in ANSYS CFX is done by setting the maximum and minimum element size in the tetrahedral mesh that is to be created. For tetrahedral mesh elements is chosen because the ANSYS CFX module called CFX-mesh is specifically designed for CFD applications and this module uses only tetrahedral elements. After setting the minimum and maximum element sizes the transition factor can be set to prescribe the growth rate of the cell size from the minimum to 34

50 maximum element size. The purpose of this section is to show the results of a mesh size study that is performed to determine what mesh size would be appropriate to use. For this research various meshes were created by changing the values of the maximum and minimum element sizes. Per mesh a simulation was performed for the quarter scale flat plate at 90 o angle of attack, with an inlet velocity of 0.75 m/s. Figure 6.3: Plate and symmetry plane mesh after mesh refinement The finest mesh was chosen at a maximum element size of 15 mm, because at smaller maximum element sizes the used computer did not have enough RAM memory to support the calculations. The minimum element sizes were set as 1.8 mm (equal to the plate thickness), 1.2 mm and 0.6 mm. To capture the water-air interface with a better accuracy, a mesh adaption method is used. With this method extra nodes are added in regions where the volume fraction gradient, Φ (see section 5.2), is high. Using two adaption steps, in each adaption step the number of nodes added to the mesh is such that the number of nodes in the mesh doubles. The mesh adaption steps start when the RMS difference values of the air and water mass and momentum reach a value of After the second mesh adaption step the solution stops when a convergence level of is reached. Figure 6.3 shows the mesh after two mesh adaption steps. After the simulation finished, the force coefficients in x-, y- and z-direction were calculated by summing pressure and viscous forces over the six plate sides and then dividing this total force over the product of the dynamic pressure and the frontal surface area of the plate. The results of these 35

51 calculations are found in figures 6.4, 6.5 and 6.6. Figure 6.4: Force coefficient in flow direction C X for various mesh sizes Figure 6.5: Force coefficient perpendicular to flow direction C Y for various mesh sizes For C X, the values vary from to with an average value of If the results for the series with h max = 30 mm are omitted, the average C X rises to 1.888, a value fairly close to 36

52 Figure 6.6: Vertical force coefficient C Z for various mesh sizes the required Still there is a difference of 3%, a value that can mean the difference between winning or losing a rowing race as stated by Macrossan [21]. A typical grid size needs to be selected to work with for the remainder of this part of the research. Looking at figure 6.4, the three size families with h max = 15, 17 and 20 mm, are both closest to the required value, a choice has to be made between these three. The typical calculation time for the 15 mm series is 45 minutes, whereas the 20 mm series reduces that to 20 minutes. Therefore, for the remainder of this part of the research, the maximum edge length will be set to 20 mm.for the minimum edge length 1.2 mm is chosen. The resulting distribution of the volume fraction on the symmetry plane is shown in figure 6.7. Although the interface is thin on top of the flat plate due to the already existing fine mesh there, in the wake behind the flat plate their appears a large area where the water volume fraction slowly decreases from 1 to Influence of the domain dimensions on experimental results Although the main focus of this part of the research is to validate the ANSYS CFX software by comparing its results to the Caplan and Gardner experiments [4], also the effect of Caplan and Gardner s water flume dimensions on their experimental results is investigated. It appeared that there is a very strong influence of the Froude number of the flow on the non-dimensionalized forces. To be able to further investigate this effect, blockage influences had to be such that they could be neglected. The purpose of this section is to find out at which domain dimensions the blockage can be neglected. 37

53 Figure 6.7: Volume fraction Φ and symmetry plane mesh after mesh refinement Domain length Firstly the length of the domain is considered. The length of the domain has to be decided upon without knowledge of the flume length Caplan and Gardner used. The results are depicted in figures 6.8 and 6.9. Figure 6.8: Domain length (x) vs. force coefficient in flow direction C X Firstly a behaviour is noted that looks like a scatter of data. Although the first half gives rise 38

54 Figure 6.9: Domain length (x) vs. vertical force coefficient C Z to a hope of a smooth graph, increasing the number of data points around a domain length of 1400 mm diminishes that hope by introducing a large amount of sharp peaks. Neglecting the value for a domain length of 420 mm, the C X values vary between and 1.913, that is a bandwidth of 3.7%. The average value equals Based on these it is tempting to take a domain length of 500 mm as sufficiently large. But as figure 6.10 shows, the flow downstream of the flat plate has hardly had any time to recover from the wake, before it leaves the simulation domain. To get a better picture of the complete behaviour of the flow, the 800 mm domain will be chosen. Figure 6.11 shows that the flow has had more time to recover from the wake, whereas the calculation time is still reasonably short (25 minutes). Figure 6.9 shows a possibly more chaotic scatter of results for C Z, with a bandwidth of 8.1% and a mean C Z of Domain width Restricted by the width of the available water flume, Caplan and Gardner choose to use a plate width that is a fifth of the width of the flume. The question arises how much the horizontal blockage effect influences their results. The results for the non-dimensionalized force coefficients for various domain widths is shown in figures 6.12 and To half the calculation speed, the simulation domain has been cut in half by inserting a symmetry plane on the XZ-axis. The domain depth is the same as Caplan and Gardner s: 150 mm Figure 6.12 shows that, using an inlet velocity of 0.75 m/s, C X quickly drops from at 200 mm to at 300 mm. From then onwards the value of C X remains relatively constant. It still fluctuates, but the bandwidth of these fluctuations is only 1.7% of This is amplified in figure 6.14 where the blockage effect has a strong influence on the flow velocity around the side of the flat plate at a domain half width of 200 mm. From 300 mm onwards, velocity of the flow that is accelerated around the sides of the plate remains constant. It appears that Caplan and Gardner were in the safe zone with respect to their flume width. 39

55 Figure 6.10: Water surface for domain length = 500 mm Figure 6.11: Water surface for domain length = 800 mm C Z remains relatively constant, with small fluctuations. For the future calculations where far field boundary conditions are assumed, a half-width of 300 mm will be used. Domain depth 40

56 Figure 6.12: Force coefficient in flow direction C X for varying domain half widths Figure 6.13: Vertical force coefficient C Z for varying domain half widths Another interesting investigation is to determine the influence of the domain depth on the results. For this investigation, the width is kept the same as Caplan and Gardners (2 x 320 mm) and the depth is varied from 150 mm to 500 mm. The results are depicted in figures 6.15 and The observed relationship between force coefficients and flume depth is very similar to what was 41

57 Figure 6.14: Flow development for various domain half widths at 30 mm depth. Top view of horizontal plane seen at the domain width investigation. The difference though is that this time, the water flume dimensions Caplan and Gardner used is located during the large decrease in C X. Only when the domain depth is more than 250 mm, the influence of the domain depth can be neglected. This is amplified with figure This figure shows how the flow varies for different domain depths. The acceleration of the flow around the bottom of the flat plate is increased by the presence of a significant blockage effect for a domain depth of 150 mm. From a depth of 250 mm onwards, the increase in flow velocity in this accelerated flow area due to the presence of the bottom wall becomes negligible. 42

58 Figure 6.15: Force coefficient in flow direction C X for varying domain depth Figure 6.16: Vertical force coefficient C Z for varying domain depth 6.4 The influence of the Froude and Reynolds numbers on experimental results While performing the simulations it was found that the inlet velocity has a major influence on the experimental results. Whether this influence of the inlet velocity is driven by viscous effects 43

59 Figure 6.17: Flow development for various domain depths in the symmetry plane (Reynolds number) or by free surface effects (Froude number) is determined in this section Froude number influence on the force coefficients In this subsection the influence of the Froude number upon the force coefficients for the quarter scale and the full scale flat plates at α = 90 o is investigated for a Froude number range of about 0.5 to 6.5 by keeping the Reynolds number constant by varying the fluid densities whilst varying the inlet velocity. The results for the quarter scale flat plate are figures 6.18, 6.19, and To check the quality of the solution, simulations are performed without symmetry plane where C Y should be zero. Figure 6.19 shows that the values of C Y are in between and The solution seems to produce an accuracy of order O(10 4 ), which is satisfactory. The vertical force coefficient C Z is mainly driven by the buoyancy force as shown previously in figure A surprising result is the strong dependence of C X on F r. At subcritical Froude numbers, C X shows a wavy behaviour along a value of 1.9 to a maximum at the critical Froude number, after which it quickly drops to a value of 1.15 at F r = 1.7. Then C X asymptotically recovers to a value slightly higher than 1.5. The behaviour of the flow close to the plate at various Froude numbers with far field boundary conditions is shown in figure At low Froude numbers a large counterclockwise circulation of flow appears downstream of the flat plate. As the dynamic pressure is increased, in between this circulation and the flat plate, a strong low pressure area is formed with such a strength that at the critical Froude number it pulls the water surface down to the bottom of the flat plate. This strong suction creates the high C X visible in figure At F r = 1.25 a hydraulic jump appears downstream of the flat plate, whilst at the low pressure side of the plate a square-shaped vortex is created that appears like two counter rotating vortices in the symmetry plane. As the Froude number increases into the high supercritical regime, the square vortex behind the flat plate increases and gives rise to the appearance of a large wave that grows further downstream as the Froude number is increased. 44

60 Figure 6.18: Froude number F r vs. force coefficient in flow direction C X, constant water density and viscosity, quarter scale flat plate Figure 6.19: Froude number F r vs. force coefficient perpendicular to flow direction C Y, constant water density and viscosity, quarter scale flat plate The same calculations have been done for the full size flat plate: D W t = 220mm 560mm 7.2mm. These results are depicted in figures 6.21, 6.22 and Although the flow behaviour of 45

61 Figure 6.20: Froude number F r vs. vertical force coefficient C Z, constant water density and viscosity, quarter scale flat plate the full scale model is similar to the flow behaviour of the quarter scale model, as the figures show, the values for the force coefficients are slightly lower for the quarter scale model. This difference may be partially caused by the fact that the inlet turbulence conditions were not scaled up when the transition was made from quarter scale to full scale simulations. Although the inlet turbulence intensity is relative, the inlet turbulence length scale was kept at 55 mm, i.e. the flat plate depth for the quarter scale model Reynolds number influence on the force coefficients By changing the density of the water and the air, the Reynolds number that relates to the depth of the plate was varied while the Froude number remained the same. The results of this process are shown in figure The Reynolds number for the quarter scale flat plate at 0.75 m/s at square-off is 43,000. Figure 6.27 shows that the flow is virtually independent of the Reynolds number for Reynolds numbers larger than 430. There is some fluctuation visible, but it is not sure if that is due to a physical phenomenon or a numerical error. To verify that the force coefficients are indeed independent of the Reynolds number section is written Verification of Reynolds number independence Using the model for the full scale flat plate and changing either the velocity or the density of the water and air, a plot can be created wherein for various Froude numbers and Reynolds numbers the force coefficients are plotted. The results are shown in figures 6.28 and From the plot it can be concluded that the 46

62 Figure 6.21: Froude number F r vs force coefficient in flow direction C X, full scale flat plate Figure 6.22: Froude number F r vs. force coefficient perpendicular to flow direction C Y, full scale flat plate force coefficients for a flat plate at an angle of attack of 90 o are independent on the Reynolds number. The domain is dimensioned according to the results of the domain dimension study. The quarter scale domain is: L W 2 D = mm and the full scale domain is: L W 2 D = mm. 47

63 Figure 6.23: Froude number F r vs vertical force coefficient C Z, full scale flat plate Figure 6.24: Flow behaviour at various Froude numbers, side view 48

64 Figure 6.25: Force coefficient in flow direction C X as a function of the inlet turbulence intensity, quarter scale flat plate Figure 6.26: Force coefficient in flow direction C X as a function of the inlet eddy length scale, quarter scale flat plate 49

65 Figure 6.27: Reynolds number dependence for flat plate Figure 6.28: Force coefficient in flow direction C X dependence on Reynolds and Froude number for full scale flat plate 50

66 Figure 6.29: Vertical force coefficient C Z dependence on the Reynolds and Froude number for full scale flat plate 51

67 CHAPTER 7 Flat plate at the catch In addition to the investigation of the flow around a flat plate at the square-off of a rowing stroke as performed in chapter 6, this chapter focuses on the flat plate at the catch. The catch is the phase of the rowing stroke where a force is applied on the rowing blade just after blade entry into the water. Assuming the rowing blade is fully submerged at ψ = 37 o, figure 1.3 is used to derive that the blade velocity is 4.7 m/s and the angle of attack α = 10 o. 7.1 Computational method Using the computational fluid dynamics program ANSYS CFX v.11.0, a simulation is set up to solve the Reynolds Averaged Navier-Stokes (RANS) equations for a flat plate at α = 10 o. To model the turbulence, the Shear Stress Transport (SST) k-ω model was used. This model combines the widely used k-ɛ model in the outer flow with the k-ω model close to the wall. Because the k-ɛ model lacks accuracy close to the wall [12] and also fails to predict separation properly [9], the SST k-ω model uses the k-ω model close to the wall. To accurately keep track of the air-water interfase, the volume of fluid (VOF) method is combined with mesh adaption as is described in section 6.2. The rowing blade is modeled using a flat plate with a width of 560 mm, a depth of 220 mm and a thickness of 5 mm (see figure 7.1). This length and width are chosen such that the surface area is similar to that of sweeping blades used in rowing. Rowing Australia dictates a minimum sweeping blade thickness of 5 mm at a distance of 3 mm from the blade edge [29]. The water density ρ = 997 kg/m 3, but varies depending on the Reynolds number when the Reynolds number is kept constant. The dynamic viscosity µ = Pa s. Appendix B gives more details about the simulation settings and boundary conditions. 7.2 Mesh size study Tetrahedrons are used to build up the mesh, where a maximum and minimum element size are the parameters that determine the mesh size. The boundary layer on the plate is captured using 52

68 Figure 7.1: Flat plate dimensions (a) Leading edge (b) Trailing edge Figure 7.2: Mesh close to flat plate an inflated layer technique. This technique builds up a layered mesh close to the wall where a boundary layer is expected. Using the mesh that exists on the surface of the flat plate, 13 layers of thin mesh cells are placed on top of each other. The thickness t of each layer depends on the layer number i according to equation 7.1 where t is in millimeters and 1 i 13. t i = i 1 (7.1) Figures 7.2 (a) and (b) show how the inflation layers are built up around the leading and trailing edge of the flat plate. Using an expansion factor, the mesh layers around the blade are thickened until the outer layer has a thickness that is similar to the average cell size in the freestream mesh. Important to note is that this technique has not been used in the domain size study. The boundary layer on the plate is not important in this case because the domain dimension influence is governed by blockage effects. This means that the overall size and orientation of the object in the flow is of more importance than the actual shape and detail on this object. For the mesh size study and Froude and Reynolds number investigations the inflation layer technique has been used. 53

69 (a) Normal force (b) Tangential force Figure 7.3: Influence of number of mesh cells on flat plate forces The influence of the mesh size on the normalized normal force on the flat plate has been investigated in two different ways. Firstly, the maximum cell size is kept constant at 40 mm, whilst the maximum number of mesh cells after two mesh adaption steps is increased. The result of this is shown in figures 7.3 (a) and (b). The y-axis of this figure is defined as the normalized force using the limit value for the force calculated with Richardson Extrapolation [25]. With Richardson Extrapolation, variable f is expressed using a series expansion: f = f h=0 + g 1 h + g 2 h g M h M (7.2) where M is the order of the series expansion, g k are independent scalars, f is the normal force on the blade and h is the mesh spacing, in this case defined as h = 1 N (7.3) with N the number of mesh cells. Assuming a second order solution with results for f calculated on two meshes of spacing h 1 and h 2 with the former being the finer spacing, two equations of the form of equation 7.2 can be written as follows, neglecting higher order terms and forming an estimation of the value of f for N infinitely large: lim f N = f1 + f 1 f 2 ( N1 N 2 ) 2 1 Using the maximum number of mesh cells as allowed by the capacity of the computer that is used (details of which are given in Appendix B), it is found that the normal force has an error of 0.8% with respect to the value of the normal force that would have been achieved if the number of mesh cells was infinitely large, i.e. if the mesh cells in the water-air interface would be refined to an infinitesimal size. The error for the tangential force is significantly larger; it has a value of 7%. (7.4) 54

70 (a) Normal force (b) Tangential force Figure 7.4: Influence of maximum cell size on flat plate forces Secondly, the maximum number of cells after mesh adaption is kept constant and the maximum cell size in the mesh is varied. This means that for smaller cell sizes, the water-air interface will be less refined. The results of this study are shown in figures 7.4 (a) and (b). This time, the forces are not normalized using the Richardson Extrapolation but using the mean value of all the forces in the dataset belonging to this mesh refinement study. The reason for this is the fact that the forces are not converging towards a limit but show a fluctuating distribution. Richardson Extrapolation can only be used under the assumption of a smoothly converging trend. This figure shows that for larger h, the solution for the tangential force is more accurate. A larger h results in less nodes in the outer flow, thus leaving more nodes to be distributed at the water-air interface. Figure 7.4 (b) indicates that refining the interface better improves the accuracy of the solution of the tangential force. Based on these figures, it is decided to use a maximum cell size of 40 mm. For the normal force this results in an deviation of 0.5% from the mean value, whilst for the tangential force this deviation is 2.8%. Although the error of the tangential force is significant, since the tangential force is generally ten times smaller than the normal force, the effect of this error becomes negligble with respect to the resultant force on the blade. The combined errors of normal and tangential force coefficients (e N and e T ) on the resultant force coefficient can be calculated as e res = CN (e N ) 2 + (C T e T ) 2 C 2 N + C 2 T = ( )2 + ( ) ( ) 2 = (7.5) 7.3 Required domain dimensions To make the simulations as time efficient as possible, the influence of the domain size is investigated. This way the domain can be kept as small as possible such that the mesh can be made finer. The resulting graphs are depicted in figure 7.5. In these graphs, the domain length, width, depth and height are normalized. The domain length is divided by W plate cos(α), the domain width by W plate sin(α) and the domain depth and height by the plate depth. The definitions of the domain dimensions are shown in figure 7.6 From these graphs it is concluded that a domain with the dimensions as shown in table 7.1 is 55

71 (a) Height (b) Width (c) Depth (d) Upstream length (e) Downstream length Figure 7.5: Influence of domain dimensions on normalized normal force on the blade 56

72 Figure 7.6: Definitions of domain dimensions large enough to neglect the influence of the walls. Upstream length Downstream length Height Depth Width 700 mm 1500 mm 375 mm 850 mm 1200 mm Table 7.1: Domain size When the results of table 7.1 are compared to the original domain size as described in chapter 6, it is apparent that the required size is considerably smaller than required for the plate perpendicular to the flow. This has two reasons. Firstly, because of the plate s orientation to the flow, the blockage effect is much smaller than for the plate that is placed perpendicular to the flow direction. This results in a reduction of required domain width, depth and height. Secondly, the flow in the case of the low angle of attack plate has a high supercritical Froude number. This means that the flow behaves like a supersonic flow in air: pressure disturbances will not travel upstream. Because of this reason, the locations of both the inlet and the outlet boundaries have a negligible influence on the flow behaviour at the location of the flat plate, thus reducing the required domain length. Although a supercritical Froude number flow has similarities to a supersonic airflow, this is not the case with the boundary layer development. It is the Reynolds number that will strongly influence this phenomenon. 57

73 7.4 Analysis of flow behaviour Shortly after the catch of a rowing stroke, the blade velocity is 4.7 m/s. For this situation with a water density of 997 kg m 3, corresponding to Re = 1,158,442, the flow behaviour is analyzed in this chapter. Firstly the water surface in figure 7.7 is investigated. This water surface is formed using an iso-surface for a value of the volume fraction of 0.5. The high inlet velocity causes the water that flows from the high pressure side to the low pressure side of the plate to form a vortex that lays exposes a part of the low pressure leeward side of the plate to air. Figure 7.7: Water surface Then consider the velocity contour of the x-y plane at a depth of 110 mm in figure 7.8. In this contour the rapid development of a turbulent boundary layer on the leeward side of the plate is distinguished. Using a close-up of the velocity contour near the plate (figure 7.9), the windward side of the plate shows there is also a turbulent boundary layer developing, but on this side the boundary layer is thinner than on the leeward side of the flat plate. In addition to this, a boundary layer separation bubble appears on the part of the leeward side close to the leading edge. This turbulent boundary layer separation is caused by the sharp edge of this flat plate model. Along the largest part of the leeward side the boundary layer grows, but it reduces in thickness when it approaches the trailing edge of the plate. To examine the flow behaviour near the leading edge a closer look is taken to the streamlines around the leading edge in figure There is a stagnation point on the leading edge of the plate. The particles that flow around the top corner of the leading edge form the earlier discussed 58

74 Figure 7.8: Velocity contour at a depth of 110 mm Figure 7.9: A close-up of the velocity contour at a depth of 110 mm separated turbulent boundary layer. Then this boundary layer is reattaching downstream of the separation bubble. The shape of the separation bubble is depicted in figure In this figure, the wall shear stress in x-direction is shown. The light area is where the wall shear stress is positive (caused by an attached boundary layer). In the dark area, the wall shear stress is negative, indicating recirculation. The separation bubble has an asymmetrical conical shape with a top just below the middle of the flat plate. The streamlines around the bottom corner of the leading edge also suggest the formation of a separation. This is further examined in figure There indeed is a region that would give rise to a separation bubble, but it is suppressed by the high pressure in this region. 59

75 Figure 7.10: Streamlines around the leading edge at a depth of 110 mm Figure 7.11: Separation bubble on windward side This region of suppressed separation can be recognized with a little bump in the pressure coefficient development along the windward side close to 0% chord. The pressure coefficient is defined using the hydrostatic pressure at a depth d of the point of measuring (in this case 110 mm) 60

76 Figure 7.12: Streamlines around the windward side of the leading edge at a depth of 110 mm C P = p p atm ρgd 1 2 ρv 2 (7.6) On the leeward side, the magnitude of C P ( C P ) rapidly increases from 0.5 to 2 over the length of the first mesh cell at the corner. From then on it shows a steep decrease throughout the separation bubble and a gradual decrease towards a constant value in the reattached boundary layer. Near the trailing edge it quickly increases again. On the windward side C P decreases gradually throughout the attached turbulent boundary layer. Close to the trailing edge it becomes negative and shows a steep decline. The water flow at the trailing edge is visualized using streamlines in figure At the trailing edge, a small vortex is formed that is rapidly dissipated when the streamlines from the windward and the leeward side come together. Figure 7.15 shows the locations of the boundary layer profiles depicted in figures 7.16 and On the y-axes the variable X n represents the distance from the flat plate along the lines of figure On the horizontal axis V t is the velocity tangential to the blade chord, but with a positive component in positive x-direction; i.e. the direction of V t is opposite to that of C T. V t is normalized using the freestream velocity of 4.7 m/s. In figures 7.16 (a) and (e) the thickness of the recirculation area is shown to be two times 2.5 mm. Figures 7.16 (b) and (c) show how the boundary layer grows to a thickness of almost 40 mm. The boundary layer reduces in thickness when it approaches the trailing edge (figure 7.16 (d)). The windward side of the plate shows a different behaviour. Due to the high pressure on this side, the boundary layer thickness stays below a value of 20 mm. It is only closer to the trailing 61

77 Figure 7.13: Pressure coefficient at a depth of 110 mm Figure 7.14: Streamlines around the trailing edge at a depth of 110 mm edge that the boundary layer is allowed to grow to a value close to 30 mm. Many of the flow features discussed in this chapter are caused by the sharp corners of the flat plate model. On rowing blades, these corners can be expected to be rounded to some extend. This might lead to a fully attached flow at the catch. The effect of edge shape will be a topic of research during the blade design process that will be described in chapter 8. 62

78 Figure 7.15: Locations of boundary layer profiles 7.5 The influence of the Froude number on the force coefficients To investigate the effect of the Froude number on the blade forces, the Reynolds number is kept constant at Re = 1,000,000. This is done by varying the density as to counteract the effect of changing velocity with varying Froude number. The effect of the Froude number on the flat plate force coefficients is shown in figures 7.18 (a) and (b). Figure 7.19 shows how the C P varies significantly for the different Froude numbers. For F r = 0.9, C P on the leeward side grows downstream of the separation bubble, whilst C P for F r = 0.5 decreases in this region. The value of C P for F r = 4.0 is generally much smaller than for the subcritical Froude numbers. The normal force coefficient grows in the subcritical Froude number range to a maximum of 0.47 at F r = 0.8. After this maximum, C N decreases to a value of 0.22 at F r = 4.0. This behaviour is explained in figure Figure 7.20 shows the isosurface for a volume fraction of 0.5. the colours of the isosurface indicate the height of the water surface with respect to the undisturbed watersurface. The flow is from left to right. In the subcritical Froude number range, the growth of an air filled wake on the leeward side and the growth of a wave on the windward side of the flat plate is visible. At F r = 0.5, the wake and wave form close to the leading edge of the plate. Both the wave and the wake grow until their sizes equal the flat plate chord length. This coincides with maximum C N. After this, the wake extends to downstream of the trailing edge of the flat plate, causing C N to decrease. 63

79 (a) 4% chord (b) 14% chord (c) 50% chord (d) 90% chord (e) Recirculation details at 4% chord Figure 7.16: Boundary layer profiles for leeward side at 110 mm depth 64

80 (a) 4% chord (b) 14% chord (c) 50% chord (d) 90% chord Figure 7.17: Boundary layer profiles for windward side at 110 mm depth (a) Normal force coefficient (b) Tangential force coefficient Figure 7.18: Influence of the Froude number on flat plate forces 65

81 Figure 7.19: Pressure coefficient C P distribution along the chord of the flat plate A similar behaviour is visible for C T in figure 7.18, but the location of maximum C T does not coincide with that of C N ; the location of maximum C T is at F r = 0.9. Dividing C T up in pressure and viscous components results in figure C T viscous has a maximum at F r = 0.85 and C T pressure at F r = 0.9. The location of the maximum C T coincides with the decrease of plate surface area exposed to air as F r increases in the subcritical Froude number range and the increase of plate surface area exposed to air as F r increases into the supercritical Froude number range (figure 7.23). A practical application of the knowledge of the Froude number effect is that it gives the possibility to optimize a rowing blade. Figure 7.22 indicates that the ratio of tangential over normal forces Π is strongly dependent on the Froude number. If at the catch a Froude number close to unity can be achieved, the fluid mechanical efficiency of the blade will increase. The separation bubble reduces in size as the Froude number increases (figure 7.24). 7.6 The influence of the Reynolds number on the force coefficients Keeping the freestream velocity constant at 4.7 m/s, the Reynolds number is varied by changing the density of the water and the air. This way, the influence of the Froude number is ignored and the dependence of the blade force coefficients on the Reynolds number can be investigated. The Reynolds number in this research varies from 100,000 to 2,000,000; this is the Reynolds number range experienced during a rowing stroke. The results are shown in figure Firstly looking at the normal force, a small increase of C N with increasing Re is detected (0.7%). Since C N is generated by pressure forces, this negligible influence of Re on C N is confirmed by investigating the C P for a Re = 100,000 and Re = 2,000,000 (figure 7.26). The pressure coefficient distribution shows no difference for the two situations. 66

82 Figure 7.20: Water surface for various Froude numbers 67

83 Figure 7.21: Contribution of pressure and viscous forces to tangential force coefficient C T Figure 7.22: Relation between ratio of tangential over normal forces and Froude number 68

84 Figure 7.23: Separation bubble for various Froude numbers The influence of Re on C T is significant: The magnitude of C T, C T, decreases from at Re = 100,000 to at Re = 2,000,000. This is a decrease of 17.6%. As figure 7.27 shows, this decrease is carried by the decrease of the viscous component of C T. There are several phenomena that influence the tangential force coefficient on the blade. The separation bubble on the leeward side of the blade results in a viscous force that is pointed in the positive t-direction. An increase in separation bubble size would therefore suggest a change in C T towards the positive range. But, figures 7.28 and 7.29 indicate that the size of the separation bubble decreases with increasing Re. The effect of the separation bubble size decrease is that with increasing Re, C T would grow. In addition to this, figure 7.30 indicates that the amount of surface area of the leeward side of the blade that is covered with water increases with increasing Re. In this figure, the dark grey area is air and the light grey area is water. Since the C T is defined using the density of water, whilst part of the blade is exposed to air, this phenomenon should increase C T. What then causes this decrease of C T? To answer this question the Reynolds Averaged momentum equation as given by Davidson [9] is written in a form that accounts for the Reynolds stress: ρ(u )u i = p + [ τ ij + τ R ] ij x i x j In this equation, τ ij is the viscous stress of the mean flow, defined as (7.7) 69

85 Figure 7.24: Separation bubble for various Froude numbers (a) Normal force coefficient (b) Tangential force coefficient Figure 7.25: Influence of the Reynolds number on flat plate force coefficients 70

86 Figure 7.26: Pressure coefficient for various Reynolds numbers Figure 7.27: Contribution of pressure and viscous forces to C T 71

87 (a) Re = 100,000 (b) Re = 2,000,000 Figure 7.28: Wall shear stress for various Reynolds numbers Figure 7.29: Separation region for various Reynolds numbers 72