Development and Assessment of Altitude Adjustable Convergent Divergent Nozzles Using Passive Flow Control

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2 Development and Assessment of Altitude Adjustable Convergent Divergent Nozzles Using Passive Flow Control A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Aerospace Systems of the College of Engineering by Mohamed Mandour Eldeeb October, 2014 B.S. Military Technical College, Egypt 1994 M.S. Military Technical College, Egypt 2001 Committee Chair: Shaaban Abdallah, Ph.D.

3 Abstract The backward facing steps nozzle (BFSN) is a new developed ow adjustable exit area nozzle for large rocket engines. It consists of two parts, the rst is a base nozzle with small area ratio and the second part is a nozzle extension with surface consists of backward facing steps. The number of steps and their heights are carefully chosen to produce controlled ow separation at steps edges that adjust the nozzle exit area at all altitudes (pressure ratios). The BFSN performance parameters are assessed numerically in terms of thrust and side loads against the dual-bell nozzle with the same pressure ratios and cross sectional areas. Numerical method is validated by solving two- and three-dimensional turbulent ow through a planar dual-bell nozzle at dierent pressure ratios and comparing the numerical results with available experimental data and numerical results. The numerical results for the pressure distributions over the planar dual-bell nozzle walls show a good agreement with the experimental data. Cold ow inside the planar BFSN and planar DBN are simulated using three-dimensional turbulent Navier-Stoke equations solver at dierent pressure ratios. The pressure distribution over the upper and the lower nozzles walls show a symmetrical ow separation location inside the BFSN and an asymmetrical ow separation location inside the DBN at same vertical plane. The side loads are calculated by integrate the pressure over the nozzles walls at dierent pressure ratios for both nozzles. Time dependent solution for the DBN and the BFSN are obtained by solving two-dimensional turbulent ow. The side loads over the upper and lower nozzles walls are plotted against the ow time. The BFSN side loads history shows a small values of uctuated side loads compared with the DBN which shows a high values with high uctuations. Hot ow three-dimensional numerical solutions inside the axi-symmetric BFSN and DBN are obtained at dierent pressure ratios and compared to assess the BFSN performance against the DBN. Pressure distributions over the nozzles walls at dierent circumferential angels are plotted for both nozzles. The results show that the ow separation location is axi-symmetric inside iii

4 the BFSN with symmetrical pressure distributions over the nozzle circumference at dierent pressure ratios. While the DBN results show an asymmetrical ow separation locations over the nozzle circumference at all pressure ratios. For further conrmation of the axi-symmetric nature of the ow in the BFSN, two-dimensional axi-symmetric solutions are obtained at same pressure ratios and boundary conditions. The results show that the side loads in the BFSN is 0.01%-0.6% of its value in the DBN for same pressure ratio. The ow separation position from the 3-D simulations for each PR shows a good agreement with its position in the 2-D axi-symmetric simulation for same PR. The ow parameters at the nozzle exit are calculated the 3-D and the 2-D solutions and compared to each other. The maximum dierence between the 3-D and the 2-D solutions is less than 1%. All the numerical results conrmed that the ow inside the BFSN is axi-symmetric. That is a very important nding which has the following implications: 1) two dimensional solution can be used to analyze the BFSN, calculate the nozzle thrust, the ow exit velocity, etc, 2) unsteady ow solution are now possible because of the major reduction of the CPU time for 2D solutions compared to 3D solution, and 3) the axi-symmetric solution is suitable for design practices of unsteady ow. To study the eect of the number of the backward facing steps on the nozzle performance, the number of the backward facing steps varied from two to forty. Since the most of the rocket mission is taking place at high altitude, a high PR of 1500 is taken as design point for the parametric study. The thrust values for all BFSNs are calculated at the design point and compared to the thrust of the DBN at same PR. The results show that as the number of backward facing steps increase, the nozzle performance in terms of thrust becomes closer to the DBN performance. The BFSN with two and six steps are simulated for pressure ratios range from 148 to 1500 and compared with the DBN and a conventional parabolic contour bell nozzle. Expandable BFSN study is carried out on the BFSN with two steps where the nozzle operation is divided into three modes related to the operating altitude (PR). At sea-level, the BFSN operates in mode-1 with expansion area limited to the base nozzle exit area. As altitude increase, the rst transition takes place by expanding the rst step results in increasing the expansion area to the rst step exit area. Second transition takes place at higher altitude by expanding the second step results in increasing the expansion area to the total nozzle exit area. Backward facing steps concept is applied to a full scale parabolic contour nozzle by adding two iv

5 backward facing steps at the end of the parabolic nozzle increasing its expansion area results in 1.8% increasing in its performance in terms of thrust coecient at high altitudes. v

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7 Acknowledgment All thanks are due to my Lord. It was only through ALLAH's help and support that this work was accomplished. I would like to express my special appreciation and thanks to my advisor Professor Dr. Shaaban Abdallah, He has been a tremendous mentor for me. I would like to thank him for encouraging my research and for allowing me to grow as a research scientist. He oers unconditional time to me in his oce and over the phone to discuss problems that I have encounter. I would also like to thank my committee members, professor Mark Turner, professor Jong Guen Lee, professor Milind Jog for serving as my committee members and for giving me advices about my work. My deepest appreciation to Dr. Mark Turner for his guidance and help. I am also very thankful to the people at School of Aerospace Systems. Special thanks to Mr. Rob Ogden for their assistance. A special thanks to my family. Words cannot express how grateful I am to my father for all of the sacrices that they made on my behalf, my motherin-law, and brothers. Their prayer for me was what sustained me thus far. I would also like to thank all of my friends who supported me in writing, and incented me to strive towards my goal. At the end I would like express appreciation to my beloved wife Asmaa Gabr who spent sleepless nights with and was always my support in the moments when there was no one to answer my queries. vii

8 Contents 1 Introduction Flow separation in rocket nozzles Side loads Flow separation control Trip rings Vented nozzle Active uid injection Nozzles with temporary insert A new nozzle concept Dual-bell nozzle Dissertation Overview Outline Validation of the Numerical Procedure Introduction Numerical method Unsteady Reynolds Averaged Navier-Stokes equations (URANS) SST k ω turbulent model Transport equations for the SST k ω model Time-step calculation Computational domain and boundary conditions Results and discussion Planar Backward Facing Steps Nozzle 33 viii

9 3.1 Introduction Numerical method Computational domain and boundary conditions Results and discussion Side loads D Results for Axisymmetric Backward Facing Steps Nozzle and Dual-bell Nozzle Introduction Numerical method Computational domain and boundary conditions Results and Discussion Side loads calculation D Axisymmetric Parametric Studies on Backward Facing Steps Nozzle Introduction Numerical method Computational domain and boundary conditions Results and Discussion Equal length backward facing steps Equal length steps Vs. variable length steps Temperature distributions Extendable Backward Facing Steps Nozzle Introduction Numerical methods Computational domain and boundary conditions Results and Discussion BFSN Two-modes 7-BFSN Backward Facing Steps Application for Full-Scale Nozzle Introduction ix

10 7.2 Numerical method Computational domain and boundary conditions Numerical results Flight performance Conclusions and Future Work Conclusions Future work Bibliography 100 x

11 List of Figures 1.1 Rocket nozzle FFS in over-expanded nozzle RSS on over-expanded nozzle Nozzles contours and as-designed wall pressure Relative side loads of TIC and PAR nozzles Typical side load magnitude for LE-7 and LE-7A nozzles during start up operation Jump of the separation point at LE-7A Side loads generated during separation point jump Mach number contours from SSME simulation comparing free shock separation (upper) and restricted shock separation (lower) Trip rings nozzle Vented nozzle Nozzle with ow injection Nozzle with ejectable insert Nozzle with consumable insert New nozzle prole Computed pressure contours for PR=37 without secondary ow Computed pressure contours for PR=37 with secondary ow Wall Pressure prole for PR=37 without secondary ow Wall Pressure prole for PR=37 with secondary ow Sketch of the dual-bell nozzle Operatioal modes of the dual-bell nozzle Specic impulse of the DBN in comparison with base and extension nozzles Schlieren observation of shock system within a dual-bell nozzle ow transition. 18 xi

12 1.24 Side load generation during a typical test run Normalized side loads measured during experimental work D schematic diagrams of the nozzles physical models Computational domain boundaries Computational domain at the mid-plane section Cross section (A) zoomed at nozzle geometry Y+ at the mid-plane upper wall for DBN, PR= Wall pressure distribution at PR= Wall pressure distribution at PR= Dual-bell nozzle geometry DBN upper wall pressure distribution at dierent planes PR= DBN upper wall pressure distribution at dierent planes for PR= Side loads variation with time D schematic diagrams of the nozzles physical models Computational domain at the mid-plane section Cross section (A) zoomed at BFSN geometry Y+ at the mid-plane upper wall for BFSN Upper wall pressure distribution at dierent planes Upper and lower wall pressure distribution at BFSN plane of symmetry for PR= Upper and lower wall pressure distribution at BFSN plane C for PR= BFSN upper wall pressure distribution at dierent planes for PR= Upper and lower wall pressure distribution at BFSN plane of symmetry for PR= Upper and lower wall pressure distribution at BFSN plane C for PR= Mach contours at BFSN mid-plane for PR= Mach contours at BFSN plane C for PR= Mach contours at BFSN mid-plane for PR= Mach contours at BFSN plane C for PR= Velocity vectors at plane of symmetry zoomed near second step edge (PR=30.83) 42 xii

13 3.16 Velocity vectors at plane of symmetry zoomed near second step edge (PR=45.5) Side loads over the BFSN walls as function of the ow time Side loads comparison between the BFSN and the DBN D schematic diagrams of the nozzles physical models Cross-section shows the computational domain and the boundary conditions Computational grid inside the BFSN at mid-plane section Computational grid inside the DBN at mid-plane section Wall pressure distribution at dierent azimuth angels for PR= Wall pressure distribution at dierent azimuth angels for PR= Wall pressure distribution at dierent azimuth angles for PR= Wall pressure distribution at dierent azimuth angles for PR= Isosurface for wall xial shear stress equal to or less that zero for DBN at PR=400_xy view Isosurface for wall xial shear stress equal to or less that zero for DBN at PR=600, x-y view Isosurface for wall axial shear stress equal to or less that zero for BFSN at PR=600, x-y view Isosurface for wall axial shear stress equal to or less that zero for BFSN at PR=800, x-y view Mach contours at x-y plane of symmetry for PR= Mach contours at x-y plane of symmetry for PR= Mach contours at x-y plane of symmetry for PR=800 zoomed at 1st step Mach contours at x-y plane of symmetry for PR= Mach contours at x-y plane of symmetry for PR= Axial-velocity contours for PR= Axial-velocity contours for PR= Axial-velocity contours at eective exit area for DBN at PR= BFSN with 2, 4, and 6 backward facing steps BFSN with 20 and 40 backward facing steps BFSN adjusted step cross-sectional area xiii

14 5.4 Mach contours for 2-BFSN at PR= Mach contours for 4-BFSN at PR= Mach contours for 6-BFSN at PR= Mach contours for 20-BFSN at PR= Mach contours for 40-BFSN at PR= Thrust Curves for the BFSN with two steps and the DBN with respect to PR Mach contours of 2-BFSN at dierent PRs Mach contours of 2-BFSN at dierent PRs Thrust curve for 6-BFSN with equal steps length Vs. 6-BFSN with adjusted length Static temperature contours in the 2-BFSN at dierent pressure ratios Static temperature contours in the 6-BFSN at dierent PRs Dierent operating modes of 2-BFSN computational grid of the 2-BFSN in mode Mach contour for 2-BFSN transition from mode-1 to mode Mach contour for 2-BFSN transition from mode-2 to mode Thrust curve for 2-BFSN xed steps Vs. 2-BFSN extendible steps Mach contour for 7-BFSN transition from mode-1 to mode Thrust comparison between 6-BFSN and 2-modes_7-BFSN Conventional bell shape nozzle contour Designed parabolic conventional nozzle AR= BFSN with AR= Computational domain for conventional bell nozzle and BFSN Mach contours for classical bell nozzle at dierent altitudes Mach contours for BFSN at dierent altitudes Thrust coecient for classical bell nozzle and BFSN as a function of altitude Thrust coecient Vs. nozzle area ratio Nozzles mounted in rocket base Flight performance comparison between conventional nozzle and BFSN xiv

15 List of Tables 2.1 Grid dependence study Grid dependence study Grid dependency study Grid dependency study Flow parameters comparison between 3-D and 2-D simulations Side load values for DBN and BFSN Thrust values for BFSNs and DBN at PR= Basic rocket parameters xv

16 Chapter 1 Introduction The main system which is used for space propulsion is the rocket a device that stores its own propellant mass and induce thrust. This thrust is produced by the rocket engine, by accelerating the combustion gases to the desired velocity and direction, and the nozzle is that part of the rocket engine extending beyond the combustion chamber, see Figure 1. Typically, the combustion chamber is a constant diameter ductwhere the propellants are injected, mixed and burned. Its length is sucient to allow a complete combustion of the propellants before the nozzle accelerates the gas products. The nozzle is said to begin at the point where the chamber diameter begins to decrease. The ow area is rst reduced giving a subsonic (Mach number < 1) acceleration of the gas. The area decreases until the minimum or throat area is reached. Here the gas velocity corresponds to a Mach number of one. Then the nozzle accelerates the ow supersonically (Mach number > 1) by providing a path of increasing ow area. The nozzle exit velocity that can be achieved is governed by the nozzle area ratio which is commonly called the expansion ratio, ε. 1

17 Figure 1.1: Rocket nozzle 1.1 Flow separation in rocket nozzles In today's launch vehicles, the main engine usually operates from take o at sea level up to high altitudes with variable ambient pressures. To get an optimum performance over the whole trajectory, the nozzle is usually designed for an intermediate operating PR, at which the exhaust ow is adapted to the ambient pressure [1]. That leads to over-expansion conditions when the nozzle operates at low altitudes. When the supersonic ow exposed to an adverse pressure gradient, it adapted to the higher pressure by means of a shock wave system. Basically, the separation occurs when the turbulent boundary layer cannot withstand the adverse pressure gradient imposed upon it. Thus, the ow separation in any supersonic ow is a process involving complex shock wave boundary layer interactions [2]. This condition occurs when a nozzle is operating under strongly over-expanded conditions. As soon as the exit nozzle wall pressure is slightly lower than the ambient pressure, an oblique shock system is formed from the trailing edge of the nozzle wall due to induced adverse pressure gradient [2]. When the 2

18 nozzle wall pressure is further reduced, the viscous layer cannot sustain the adverse pressure gradient imposed upon it by the inviscid ow, and the boundary layer separates from the wall. Summereld criterion [3] predicts that the ow in a nozzle separates when the ratio of the wall pressure to the ambient pressure is less than or equal to 0.4. Many experimental studies carried out on sub-scale nozzle [4, 5, 6] or full scale nozzle [4] and dierent numerical studies [7, 8, 9, 10, 11] demonstrated the existence of two dierent types of ow separation. The rst type is called Free Shock Separation (FSS) where the ow is separated from the nozzle wall and never reattaches as shown in Figure (1.2). The second type is called Restricted Shock Separation (RSS) which is characterized by a closed recirculation bubble downstream of the separation point with reattachment on the wall as shown in Figure (1.3). The RSS mainly appears in the parabolic contour nozzle [2] because of presence of internal shock due to due to non ideal nozzle wall curvature just downstream of the throat [12]. If the internal shock is strong, its interaction with the Mach disk can cause the ow to deect outward and reattach the nozzle wall which results in a recirculation bubble [12]. Figure 1.2: FFS in over-expanded nozzle 3

19 Figure 1.3: RSS on over-expanded nozzle 1.2 Side loads Many experiments have been carried out to study the side loads generated in rocket nozzles for full scale and sub-scale nozzles, see reference [13, 14, 15, 16, 17, 18, 19, 12]. Flow separation in rocket nozzles is considered undesirable because of the asymmetry in the ow separation locations which lead to asymmetrical pressure distribution over the nozzle wall that can cause a high side loads, which may damage the nozzle structure [20]. The structural damage caused by the transient nozzle side loads during testing at sea level has been found for almost all rocket engines during their initial development [21]. Many examples for the nozzle failure caused by side loads are mentioned in references [22, 23, 14, 24]. As a result, whether during sea-level testing or in ight, transient nozzle side loads have the potential of causing real systems failure [21]. In 2004, marshal Space Flight Center (MSFS) began an experimental study to quantify the relative magnitude of the side loads on two dierent types of nozzle contours, ideal contour and parabolic contour, [12]. They found that, the maximum side loads measured in the truncated ideal contour nozzle is 45% of that measured in the parabolic contour nozzle. They gured out the reason of that is because of the ow transition between FSS and RSS in the parabolic nozzle [12]. Figure (1.4) shows the the nozzles contours and the as-designed wall pressure distribution, 4

20 while Figure (1.5) shows the side loads measured in both nozzles. Further support to the analysis of the ow separation behavior has been provided by means of numerical simulations, see references [25, 26, 9, 27, 28, 21, 29]. Figure 1.4: Nozzles contours and as-designed wall pressure Figure 1.5: Relative side loads of TIC and PAR nozzles Many examples for the nozzle failure caused by side loads are mentioned in references [22, 23, 14, 24, 12]. Y. Watanabe et al. in 2002 [23] presented a study on the LE-7A engine (the rst stage main engine of the Japanese H-IIA launch vehicle) separation problem which caused a large side loads and a failure in the cooling tubes. The LE-7 A nozzle consists of two parts, the upper part is assembled from cooling tubes and the lower part is made from forging material that needs to be cooled by injecting a lm cooling, a contour discontinuity exists between the upper and the lower parts of the nozzle. They found that the there are two kinds of side loads that have dierent origins. The rst side load is due to the transition of the ow separation from FSS to RSS during start up and shut down the engine. The second side 5

21 load is due to the sudden movement/jump of the ow separation point. Figure (1.6) shows the side loads magnitude level in LE-7 and the modied LE-7A nozzles. Figure 1.6: Typical side load magnitude for LE-7 and LE-7A nozzles during start up operation During the start up, the ow separation is located at the contour discontinuity until the PR is increased high enough. The stagnation separation point jumps suddenly to the nozzle exit lib. The separation point motion is asymmetric around the nozzle circumference which caused a very large side loads that is responsible of the structural failure of the nozzle[23]. Figures (1.7) to (1.8) show the asymmetrical jump of the separation point from the step to the nozzle end and the side loads generated. Figure 1.7: Jump of the separation point at LE-7A 6

22 Figure 1.8: Side loads generated during separation point jump The Space Shuttle Main Engine (SSME) also suered from nozzle side loads in terms of low cycle fatigue crack which caused a damages in the nozzle [14]. SSME has performed reliably and safely for more than 25 space shuttle mission. However, until 1987, four failures have occurred during launches or launch attempts[22]. In 2012, Eric L. Blades et al. [30] carried out a numerical study to investigate the side loads in the SSME. Figure (1.9) demonstrates both modes in which the ow can separate in the SSME nozzle. The upper half of the gure illustrates a free shock separation (FSS) structure, and the lower half illustrates a restricted shock separation (RSS) structure. In the FSS mode, the ow separates from the wall and continues as a free stream. Since the SSME is a thrust-optimized/parabolic nozzle, an internal shock forms due to the curvature discontinuity where the wall contour transitions from a circular arc contour to a parabolic contour. As the chamber pressure increases, the internal shock interacts with the Mach disk (the normal shock identied in Figure 1), causing the annular supersonic plume to deect outward and reattach to the nozzle wall. This reattachment creates a local recirculation region that is restricted in location between the separation location and the reattachment location. The transition from FSS to RSS creates a signicant side load. As the chamber pressure increases further, the RSS reattachment location moves further downstream, closer to the end of the nozzle. As it approaches the nozzle exit plane, the plume oscillates between FSS and RSS, creating additional side loads [30]. 7

23 Figure 1.9: Mach number contours from SSME simulation comparing free shock separation (upper) and restricted shock separation (lower) High side loads also resulted in the failure of the gimbal block retaining bolts for the J-2 engine [14]. The J-2X engine, which is currently under development as an upper-stage engine for NASA's next heavy-lift launch vehicle, is a derivative of the J-2 engine and will likely experience side loads similar to its predecessors, the J-2 and J-2S [30]. Thus, accurate prediction of nozzle side loads is of great interest for current and future nozzle designs. Recent rocket nozzle failure due to the side loads is occurred with the Ariane 5 ECA launcher [24]. An intensive engineering and testing program are carried out by Volvo Aero Corporation to reinforce the Vulcain 2 Nozzle extension. It was formed that the ight loads had been much higher than expected which led to a failure of the nozzle [24]. One possible solution to avoid side loads generation is to adapt the nozzle contour during ight to changes of ambient pressure. Several methods to adapt the nozzle during ight are exist either mechanically or non-mechanically. For the mechanically methods, the weight and mechanical complexities of such devices are a big issue [1]. For the non-mechanically methods, many techniques have been used to control the ow separation and reduce the side loads generated due to asymmetrical ow separation. 1.3 Flow separation control A nozzle capable of varying eective expansion area ratio can optimize delivered impulse over the entire ight trajectory, resulting in enhanced performance gain. The ideal altitude com- 8

24 pensating nozzle would continuously vary nozzle exit area ratio such that the nozzle is always pressure matched. Theoretically, if the nozzle is available to change its exit area continuously with altitude, the over-expanded operating condition can be avoided. That means the ow will not separates from the nozzle wall and no side loads would be generated. In real life, the ideal altitude compensating nozzle doe not exist. Instead, a one- or multi-step altitude adapting nozzles are used to adapt the nozzle exit area by forcing the ow to separate at certain locations related to the operating altitude. Many studies [2, 31, 32, 33, 34, 35, 36, 37, 38, 39] have been carried out with a main objective of controlling the ow separation and/or the unsteady motion of the separation front to reduce the side loads aecting the nozzle structure. The main ideas used to control the ow separations are presented as follows: Trip rings The main concept of using a trip rings inside a bell nozzle, Figure (1.10), is to disturb the turbulent boundary layer and force the ow to separate at certain locations (ring location) in symmetrically way to decrease the side loads aecting the nozzle wall. At high altitudes, the ow reattaches to the nozzle wall behind the trip ring, and a full owing nozzle is achieved. Using several rings mounted after each others, several altitude adaptation can be achieved. However, this will results in performance loss at high altitudes [32]. The main problem with this concept is the performance losses and the ring resistance in high temperature boundary layers and also the uncertainty in transition between low altitude mode to high altitude mode [33, 34]. Figure 1.10: Trip rings nozzle 9

25 1.3.2 Vented nozzle In the vented nozzle concept [35], a section of the nozzle wall has slots or holes opened to the atmospheric surrounding, Figure (1.11). At low altitudes, these slots or holes allow adequate passive inow to sustain a symmetrical and stable ow separation. By closing the holes at high altitude, the full owing nozzle is achieved. The main disadvantages of this nozzle concept is that the number and position of the holes limit the altitude range of the nozzle operation because the concept is built on the condition that the pressure within the nozzle must be lower than the ambient pressure. The second disadvantage is that the mechanism required to close the slots or holes at high altitude will increase the rocket engine mass and reduce the reliability. Figure 1.11: Vented nozzle Active uid injection The concept is to force the ow to separate by injecting a second ow into the nozzle, normal or at angle from the nozzle wall, at certain locations, Figure (1.12). Experience on this concept [36]shows that a large amount of the secondary ow injection is required to induce signicant ow separation. Figure 1.12: Nozzle with ow injection 10

26 1.3.4 Nozzles with temporary insert This concept is based on controlling the ow separation using a temporary inserts inside the nozzle and remove them at high altitude, Figure (1.13). These inserts can be either consumable or ejectable. Previous studied on this nozzle concept [37, 38]showed that the nozzle operation at low altitudes resulting in a slightly performance loss compared to the equivalent bell shape nozzle. It should be stressed that the ejectable concept is highly depends on the ejection mechanism which have to provide a sudden and symmetrical detachment of the insert. Another issue to concern about is that during the ejection process, the insert acts as an obstacle in a supersonic ow, which will induce shocks that will interact with the nozzle wall and increase the pressure loads [2]. The nonsymmetrical ejection will lead to high side loads. furthermore, the danger of collision with the nozzle wall arises because the inserts might experience a transversal momentum toward the nozzle wall [2]. Figure 1.13: Nozzle with ejectable insert Another method to remove the insert is to use an ablative or combustible insert [38, 39], Figure (1.14). During the ascent of the nozzle through the atmosphere, the size of the insert is continuously decreased until its complete consumption. This will results in a full owing bell nozzle at high altitudes. The main uncertainties of this concept are the stability and the surface regression rate of the consumable insert. Furthermore, a symmetrical and homogenous consumption must be guaranteed to avoid the side loads generation [2]. 11

27 Figure 1.14: Nozzle with consumable insert A new nozzle concept Another method to overcome the side loads that found in the literature is presented by Luca et al. [31] in 2008 by introducing a new nozzle concept that eliminates the physical cause of the ow separation. The new concept is based on generating a second annular ow at the nozzle lip to act as an aerodynamic barrier to prevent the ambient air from going inside the nozzle Figure (1.15). That will decrease the local pressure in the recirculation region and will remove the adverse pressure gradient. Figure 1.15: New nozzle prole Luca at al. carried out a series of numerical and experimental tests to conrm his new nozzle concept in preventing the ow separation. 2-D and 3-D with 18 degree annular sector numerical solutions are obtained at dierent PRs. The numerical results showed that the ow separation did not occured when the secondary ow is injected. Figures (1.17) and (1.16) show the numerical results of the Mach contours with and without the secondary ow respectively. Experimental results for pressure distribution over the nozzle wall for PR 37 with and without the secondary ow showed that the ow separation is prevented with the presence of the secondary ow and the pressure keep decreasing as going towards the nozzle exit as shown in Figure (1.19) and Figure (1.18). 12

28 Figure 1.16: Computed pressure contours for PR=37 without secondary ow Figure 1.17: Computed pressure contours for PR=37 with secondary ow Figure 1.18: Wall Pressure prole for PR=37 without secondary ow 13

29 Figure 1.19: Wall Pressure prole for PR=37 with secondary ow In my point of view, although this new concept prevented the ow separation which lead to decrease the side loads, the pressure distribution over the inner and outer walls of the spike (secondary jet nozzle) showed a non symmetrical values as appeared in the pressure distribution curve in Figure (1.19). This can lead to a side loads over the spike nozzle Dual-bell nozzle One of the most promising non-mechanical altitude compensating nozzles is the DBN [40, 41, 42], Figure (1.20). It was rst studied at the Jet Propulsion Laboratory by Foster and Cowles in 1949 [43]. This nozzle concept is patented in late of 1960s by Rocketdyne. The main advantage of the DBN is the one step altitude adaptation achieved only with presence of a wall inection and therefore the absence of any movable parts which leads to a high reliability [44]. 14

30 Figure 1.20: Sketch of the dual-bell nozzle Dual-bell nozzle is a combination of two bell nozzles with dierent geometric area ratios. Compared with single-bell nozzles, it has advantages of providing a stable separated ow at low altitudes and high specic impulses at high altitudes [45]. Figure (1.21) shows schematics of ow patterns in a dual-bell nozzle at low- and high-altitude operation modes. At the lowaltitude operation mode, the wall inection yields stable and symmetrical ow separation. This results in a smaller eective area ratio and in small side loads. At the high-altitude operation mode, high specic impulse is obtained with the larger area ratio of the nozzle throat and the nozzle exit. As shown in Figure (1.22), however, there are three instances of performance loss during the low-altitude operation mode, the operation mode transition, and the high-altitude operation mode. During low altitude operation, the pressure in the separated ow region of the extension part becomes lower than the ambient pressure, and the drag force called aspiration drag acts on the wall of the extension part [45]. At the transition between the low- and the highaltitude operation modes, the specic impulse decreases because the transition occurs at a lower altitude than the optimum [45, 46]. The dual-bell contour is designed just as a combination of two bell nozzles; therefore, the combined contour is not optimized, and the specic impulse becomes smaller at the high-altitude operation mode. Although the mentioned disadvantages should be improved, the dual-bell nozzle has a simpler nozzle structure without any moving parts and is more lightweight in comparison with other altitude compensation concepts, such as an extendible nozzle, an expansion deection nozzle, or an aerospike nozzle. 15

31 Figure 1.21: Operatioal modes of the dual-bell nozzle Figure 1.22: Specic impulse of the DBN in comparison with base and extension nozzles In recent years, several experimental and computational studies on dual-bell nozzles have provided insight [47, 48, 49, 50, 51, 52, 53, 54] into the ow phenomenon under dierent operating conditions, varying from sea level to high altitude. Of prime interest are transient studies, especially those that aim to address the parameters that inuence the 1) transition nozzle pressure ratio (PR) and transition duration, 2) transition stability, and 3) the associated side-load activity. At sea level, the contour inection in the dual-bell nozzle wall forces the ow to separate controlled and symmetrically. The base nozzle ows full and the extension is separated, the dual bell is operating in sea level mode. Because of a smaller eective area ratio the sea level, 16

32 the specic impulse increases compared with a conventional nozzle. At the designed altitude the ow attaches abruptly to the wall of the extension down to the exit plane. This transition to high-altitude mode results in a short time specic impulse loss but later on in a higher vacuum performance. The transition from one operating mode to the other is particularly of interest as the ow potentially separates asymmetrically within the extension as shown in Figure (1.23), resulting in a strong side load peak[46]. Hagemann et al. [41] presented an experimental work including cold as well as hot ow studies with respect to side load generation. He found that depending on the type of nozzle contour used in the dual-bell nozzle extension, a sudden transition from sea level to altitude mode can be achieved. One remarkable fact is that the side load peak during re-transition (while the nozzle is shut down) was shown to be signicantly higher than during transition [41] as shown in Figure (1.25). An opposite result is given in studies performed by Hieu et al. [55], where the transition to high altitude mode generates higher side loads. The experimental cold ow results were recalculated at DLR, German Aerospace Center by Karl and Hannemann [56] using the in-house code TAU. The transient simulations showed that the calculated side load peak during transition mainly depended on the nozzle pressure ratio gradient. The side load generation was studied in a numerical parametric study by Martelli et al. [49] as a function of the wall pressure gradient, the base length, the Reynolds number, and the contour inection angle.the eect of a cooling lm applied on full scale hot ow dual bell nozzles is presented in [57]. The dual bell nozzle principle was found to improve the lm cooling eciency compared with conventional nozzles. The side loads generated during the transition increase with increasing lm mass ow. Chloe Genin at al. [46] introduced the phenomenology of side load generation in dual bell nozzles and gave a detailed experimental parametric study. The cold ow experimental study has shown the dierent phases of operation of dual bell nozzles and the side load generation corresponding to each phase. In its operating modes, a dual bell nozzle produces smaller side loads than a conventional TIC nozzle. However, the transition goes ahead with a short time side load peak that must be taken into account to avoid harm to the engine structure, Figure (1.24). 17

33 Figure 1.23: Schlieren observation of shock system within a dual-bell nozzle ow transition Figure 1.24: Side load generation during a typical test run Figure 1.25: Normalized side loads measured during experimental work 18

34 1.4 Dissertation Overview The goal of this dissertation is development and assessment a new altitude adjustable area nozzle using backward facing steps to force the ow to separate at certain symmetrical locations, with specic aims of: Stationary adaptive nozzle geometry Dynamically adaptive nozzle geometry Hybrid adaptive nozzle geometry These goals are achieved by controlling the ow separation passively using a backward facing steps nozzle geometry. The nozzle contour is replaced by a number of backward facing steps to force the ow to separate at certain locations (steps edges). A symmetrical ow separation can be guaranteed and side loads can be avoided which will increase the nozzle life time and safety margin. The assessment of the BFSN performance is done numerically using uent code. The numerical procedure is validated rst by simulating a planar dual-bell nozzle and the results are compared with experimental data. Dual-bell nozzle is chosen as a reference nozzle to assess the performance of the new developed BFSN. Backward facing steps approach is applied on both planar and axisymmetric nozzles geometries with cold ow in the planar nozzle and hot ow in the axisymmetric nozzle. 3-D solutions for both BFSN and the DBN at dierent PRs show that the ow is axisymmetric in the BFSN at all PRs with symmetrical ow separation locations. While the ow in the DBN is naturally asymmetric with dierent separation locations over the nozzle circumference. Further conrmation for the ow symertical behavior in the BFSN are carried out by the 2-D axisymmetrical simulation of the BFSN at derent PRs and comparing the results of the ow parameters with the results of the 3-D solutions. The comparison showed that the maximum dierence in the ow parameters at the nozzle exit section is less than 1%. The conrmation of the axisymmetrical nature of the ow in the BFSN is a very important nding which has the following implications: 1) two dimensional solution can be used to analyze the BFSN, nozzle thrust, ow exit velocity, etc, 2) unsteady ow solution are now possible because of the major reduction of the CPU time for 2D solutions compared to 3D solutions, and 3) the axi-symmetric solution is suitable for design practices. 19

35 Parametric studies on the BFSN are carried out in two ways. First, changing the number of backward facing steps with keeping their length to be equal. Second, choosing a certain number of backward facing steps and changing their length based on the ideal exit areas (area corresponding to fully expands) suitable for dierent altitudes. For the rst parameter, the BFSN is simulated with 2, 4, 6, 20, 40 backward facing steps. A high PR is chosen as a design point and all BFSNs thrust values are calculated and compared to each other and to the DBN. The BFSNs with two and six backward facing steps are chosen (as a lower and medium number of backward facing steps) and their performance in terms of thrust are obtained for a wide range of PR and compared to the DBN. For the second parameter, the BFSN with six backward facing steps is chosen and the steps are redesigned so that each step local cross-sectional area present an ideal area suitable for specic PR (altitude). The BFSN with ideal backward facing steps cross-sectional area is simulated for the all PRs and compared to the BFSN with equal six backward facing steps. Expandable capability of the BFSN is studied in two dierent approaches. First, the BFSN can be contracted at sea level and the steps starts to expand one after other as the rocket ascent through the atmosphere. This approach is studied using the BFSN with two backward facing steps. At sea level, the two steps are contracted and the BFSN operates with its base section only limiting the nozzle expansion area to the base nozzle exit area (mode-1). As the PR increases, the rst step expands and BFSN expansion area increased to the rst step crosssectional area (mode-2). At high altitude, the second step expands and the BFSN operates with its nal expansion area which is the second step cross-sectional area (mode-3). Second, the BFSN can operates with certain number of xed and contracted backward facing steps to sustain its length to be equal to the DBN (as a comparable reference nozzle) until it reach a designed high altitude. For a higher altitudes, this nozzle can be expanded to a higher expansion area ratio and operates eciently using a contracted backward facing steps. 1.5 Outline This thesis consists of eight chapters. The rat chapter is the introduction. The second chapter is numerical procedure validation. The third chapter is the planar backward facing steps nozzle. The fourth chapter is 3-D numerical simulations for the axisymmetric backward facing steps 20

36 nozzle and the dual-bell nozzle. The fth chapter is parametric studied for the BFSN. The sixth chapter is the expandable BFSN. The seventh chapter is a full sclae nozzle application. Finally, the eighth chapter is the conclusions. 21

37 Chapter 2 Validation of the Numerical Procedure 2.1 Introduction This chapter presents three-dimensional unsteady solutions and two-dimensional time dependent solutions of turbulent ow in a planar DBN as shown in Figure (2.1). The two- and the three-dimensional numerical results of the DBN are validated by comparing with an available numerical results and experimental data respectively in ref [58]. The three-dimensional unsteady numerical results showed a high side load value for the dual-bell nozzle. The twodimensional time dependent results showed a high uctuated side load values in the dual-bell nozzle during the transition from low to high altitude modes. (a) DBN Figure 2.1: 3-D schematic diagrams of the nozzles physical models 22

38 2.2 Numerical method The commercial CFD software, Ansys Fluent, is used to simulate the turbulent ow of the sub-scale BFSN and the DBN shown in Figures (4.1-a) and (4.1-b). The SST k ω is used to predict the turbulence quantities of the ow eld behavior. It was chosen due to its accuracy in computing the ow separation from smooth surface and predicting the details of the wall layer characteristics [59]. Second order accuracy upwind scheme is used which accurately predicts the interaction between the oblique shock and the turbulent boundary layer. Ideal air is modeled as the driving gas at constant inlet pressure and temperature. Varied ambient pressure is specied at the far downstream boundary condition. The three-dimensional computations are done using parallel processing on eighteen node cluster at OSC (Ohio Super Computer). Each node is a 2.5 GHz processor Unsteady Reynolds Averaged Navier-Stokes equations (URANS) In turbulent ows, the randomly changing ow variables of the conventional Reynolds decomposition are replaced by two parts: 1) a steady quantity time average, 2) its uctuation quantity. More details about turbulent decomposition, dierent forms for RANS and boundary layer equations are available in references [59, 60]. The system of turbulent ow governing equations for a single-component uid, written to describe the mean ow properties, is cast in integral Cartesian form for an arbitrary control volume V with dierent surface area da as follows: ˆ W.dV + t V ˆ [F G].dA = V H.dV (2.1) where the vectors W,F,and G are dened as ρ ρv 0 ρu ρvu + p i τ xi W = { ρv }, F = { ρvv + p j }, G = { τ yi } (2.2) ρw ρvw + p k τ zi ρe ρve + pv τ ij v j + q and the vector H contains source terms such as body forces and energy sources. 23

39 2.2.2 SST k ω turbulent model The shear-stress transport (SST) k ω model was developed by Menter [61] to eectively blend the robust and accurate formulation of k ω model in the near-wall region with the free-stream independence of the k ε model in the far eld. To achieve this, the k ε model is converted into k ω formulation. More details of the standard k ωturbulent model can be found in [62]. The SST k ω model is similar to the standard k ω model, but includes the following renements: The standard k ω model and the transformed k ε model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the near-wall region, which activates the standard k ω model, and zero away from the surface, which activates the transformed k ε model. The SST model incorporates a damped cross-diusion derivative term in the ω equation. The denition of the turbulent viscosity is modied to account for the transport of the turbulent shear stress. The modeling constants are dierent. These features make the SST k ω model more accurate and reliable for a wider class of ows (for example, adverse pressure gradient ows, airfoils, transonic shock waves) than the standard k ω model Transport equations for the SST k ω model The SST k ω model has a similar form to to the standard k ω model: and t (ρk) + (ρku i ) = x i k (Γ k ) + x j x j G k Y k + S k (2.3) t (ρω) + x j (ρωu j ) = ω (Γ ω ) + G ω Y ω + D ω + S ω (2.4) x j x j In these equations, Gk represents the generation of turbulence kinetic energy due to mean velocity gradients, calculated from G k and dened in as: 24

40 G k = min(g k, 10ρβ kω) (2.5) and G k is the generation of ω ans calculated in same manner as in the standard k ω model. Γ k and Γ ω represent the eective diusivity of k and ω respectively. Y k and Y ω represent the dissipation of k and ω due to turbulence. D ω represents the cross difussion term and S k and S ω are user-dened source term. More details for the methods of calculation of these terms are described in [62] Time-step calculation Time step size, Δt must be small enough to resolve time-dependent features; make sure convergence is reached within the number of Max Iterations per Time Step. The time step size can be estimated as: t = T ypical cell size Characteristic f low velocity (2.6) Where the typical cell size depends on the grid generated for each case and the characteristic ow velocity (C ) can be calculated base on the following relation: C = P t A t ṁ (2.7) where P t...total pressure at nozzle inlet A t...throat area ṁ...mass ow rate 2.3 Computational domain and boundary conditions The grid of the DBN consists of ve blocks for each, two of which located inside the nozzle and the other two are placed outside as shown in Figure (2.2). The computational domains are designed as described in ref [58] with wind tunnel walls. To model the ow accurately, the mesh is concentrated in the turbulent boundary layer and across the geometrical corners to 25

41 capture the ow separation and the high ow eld gradient. The nozzles walls are assumed to be smooth and adiabatic during the simulations with no-slip and no-ux velocity condition imposed at the walls. A grid dependence studies are conducted here using the 3-D grids at an operating PR of Four levels of grid are generated and the ow parameters at the nozzle exit cross-sectional area are compared to each others. Table (2.1) shows the grid levels and the ow parameters values with the CPU time required for each grid level. Taking into account the dierence in the ow parameters and the CPU time required for each grid level, grid level 3 is chosen (with maximum dierence less than 0.2% corresponding to the ne grid) to carry out all the studies on the planar DBN. The quality of the ow solution depends upon the ability to capture the ow phenomena inside the boundary layers that are developing along the solid walls. Fluent recommends locating the nearest grid point along the wall at a distance that corresponds to non-dimensional coordinate y+ near one for the SST k ω turbulent model [59]. The rst cell distance from the wall is 1.5 micrometer and the numerical results showed that y+ average is less than 0.9 as shown in Figures (2.5). These results conrm the requirement that y+should be near one. Figure 2.2: Computational domain boundaries 26

42 Figure 2.3: Computational domain at the mid-plane section Figure 2.4: Cross section (A) zoomed at nozzle geometry Grid level Cells number Mass ow rate [kg/s] Flow axial velocity [m/s] Flow static pressure [psi] CPU time [days] 1 778, ,547, ,220, ,934, Table 2.1: Grid dependence study 27

43 Figure 2.5: Y+ at the mid-plane upper wall for DBN, PR= Results and discussion The solution convergence criterion is sited up to satisfy three conditions: (1) the residuals for all cases were decreased by more than three orders of magnitude for all simulations, (2) the average axial velocity and static pressure become constant with for 1000 iterations at least, and (3) the mass ow rate dierence was less than 0.2 %.The experimental data in ref [58] were used for validation of the numerical procedure. In this study, the numerical results of the pressure distributions at the upper DBN wall mid-plane (plane of symmetry) are compared to the experimental data and numerical results from Ref [58]at PRs 30.83, 37.86, and The comparison of static pressure distribution over the upper wall at nozzle mid-plane for dierent nozzle pressure ratios are shown in Figures (2.6), (??), and (2.7). The Fluent computed results of the pressure distribution are compared to the experimental data and numerical results from ref [58]. No signicant dierences can be seen. The main ow features, especially the pressure values and the separation location, compare well. 28

44 Figure 2.6: Wall pressure distribution at PR=30.83 Figure 2.7: Wall pressure distribution at PR=45.5 Further results analysis are carried out for the planar DBN by generating three planes in the distance between the plane of symmetry and the side wall (in each half of the nozzle) with equal spaces from each other. the planes are named A, B, and C as shown in gure (2.8). the pressure distribution over the upper and lower nozzle walls at each plane for PR=45.5 (as sample PR) are plotted and compared to each other to show the ow separation location change from upper to lower wall as the source of the side loads and the eect of the side walls. 29

45 Figure 2.8: Dual-bell nozzle geometry The wall pressure distributions are plotted for the nozzle upper wall at plane of symmetry, plane A, plane B, and plane C. The nozzle side walls aect the ow separation location for dierent planes as shown in Figure (2.10). The separation locations in planes A, B, and C are located more upstream than the plane of symmetry as going near to the side wall. Plane C has the most eect of the nozzle side wall as it is the nearest plane to it. However the ow separation location is strongly aected by the nozzle side walls in z-direction (nozzle depth), the wall pressure distribution over the upper and the lower nozzle wall for same plane show also an asymmetrical behavior for PR=45.5. Figure 2.9: DBN upper wall pressure distribution at dierent planes PR=

46 Figure 2.10: DBN upper wall pressure distribution at dierent planes for PR=45.5 Time dependent two-dimensional numerical simulations are used to decrease the CPU time. Two-dimensional time dependent numerical simulations are obtained for the planar DBN with computational domain similar to the mid-plane of the three-dimensional domain. A grid dependence studies are conducted here using the 2-D grids at an operating PR of 45. The maximum dierence in the ow parameters between the ne grid points (150,487 grid points) and the coarse grid points (48,478 grid points) did not exceed 1.3 %. However a grid size of 74,647 grid points is used with an error less than 0.6% corresponding to the ne grid. The rst cell distance from the wall is 1.5 micrometer and the numerical results showed that Y + average is less than Time step is calculated to be 50 Micro-seconds. The two-dimensional time dependent numerical simulation is carried out for time; t =0 9 seconds corresponding to nozzle pressure ratio; PR= In the numerical simulation, the total pressure at nozzle inlet is kept constant while the ambient pressure is varied with altitude as follows: P = e ( H/7000) Assuming that the rocket is ying with constant velocity equal 500 m/s, the ambient pressure related to the time (t) can be written as 31

47 P = e (( 500 t)/7000) As the ambient pressure equivalent to PR=24.24 (t=0 seconds), the pressure equation as a function of time is modied by curve tting to get the ambient pressure-time relationship used in this study: P = e ( t) The calculations were done using implicit scheme with second order accuracy in space and time. the SST k-ω turbulent model was used through calculations with rst cell distance near wall equal to 1.5 micrometer to keep y+ value near 1.During the calculations, the pressure integral over the upper and lower walls are calculated every second to calculate the net side force aecting the nozzle structure during nozzle ascending through the atmosphere. Figures (2.11) shows the side load values as a function with ow time. High uctuated side load values are generated on the planar DBN wall during the transition duration from low to high altitude modes. These high side loads values present a source of nozzle structure failure. Figure 2.11: Side loads variation with time 32

48 Chapter 3 Planar Backward Facing Steps Nozzle 3.1 Introduction This chapter presents three-dimensional unsteady solutions and two-dimensional time dependent solutions of turbulent ow in a planar BFSN as shown in Figure (3.1). The threedimensional unsteady numerical results showed a strong reduction in the side load value for the backward facing steps nozzle compared to the reference DBN. The two-dimensional time dependent results showed a high uctuated side load values in the dual-bell nozzle during the transition from low to high altitude modes while the results showed a small values of side load in the backward facing steps nozzle. (a) BFSN Figure 3.1: 3-D schematic diagrams of the nozzles physical models 33

49 3.2 Numerical method The commercial CFD software, Ansys Fluent, is used to simulate the turbulent ow of the sub-scale BFSN shown in Figure (4.1-a). The SST k ω is used to predict the turbulence quantities of the ow eld behavior. It was chosen due to its accuracy in computing the ow separation from smooth surface and predicting the details of the wall layer characteristics [59]. Second order accuracy upwind scheme is used which accurately predicts the interaction between the oblique shock and the turbulent boundary layer. Ideal air is modeled as the driving gas at constant inlet pressure and temperature. Varied ambient pressure is specied at the far downstream boundary condition. The three-dimensional computations are done using parallel processing on eighteen node cluster at OSC (Ohio Super Computer). Each node is a 2.5 GHz processor. 3.3 Computational domain and boundary conditions The computational domains are designed as described in ref [58] with wind tunnel walls. Similar computational domain with similar boundary conditions is used for the BFSN as used in chapter (2) for the DBN. To model the ow accurately, the mesh is concentrated in the turbulent boundary layer and across the geometrical corners to capture the ow separation and the high ow eld gradient. The nozzles walls are assumed to be smooth and adiabatic during the simulations with no-slip and no-ux velocity condition imposed at the walls. A grid dependence studies are conducted here using the 3-D grids at an operating PR of Four levels of grid are generated and the ow parameters at the nozzle exit cross-sectional area are compared to each others. Table (3.1) shows the grid levels and the ow parameters values with the CPU time required for each grid level. Taking into account the dierence in the ow parameters and the CPU time required for each grid level, grid level 3 is chosen (with maximum dierence less than 0.3% corresponding to the ne grid) to carry out all the studies on the planar BFSN. The quality of the ow solution depends upon the ability to capture the ow phenomena inside the boundary layers that are developing along the solid walls. Fluent recommends locating the nearest grid point along the wall at a distance that corresponds to non-dimensional coordinate y+ near one for the SST k ω turbulent model [59]. The rst cell distance from 34

50 the wall is 1.5 micrometer and the numerical results showed that y+ average is less than 1.5 as shown in Figure (3.4). These results conrm the requirement that y+should be near one. Nozzles inlet total pressure of 135 psi and total temperature of 503 R are used with varied back pressure to simulate the dierent operating PRs Figure 3.2: Computational domain at the mid-plane section Figure 3.3: Cross section (A) zoomed at BFSN geometry Grid level Cells number Mass ow rate [kg/s] Flow axial velocity [m/s] Flow static pressure [psi] CPU time [days] 1 812, ,499, ,811, ,122, Table 3.1: Grid dependence study 35

51 Figure 3.4: Y+ at the mid-plane upper wall for BFSN 3.4 Results and discussion Numerical solutions are obtained for the planar BFSN at dierent operating PRs. The pressure distribution over the nozzles wall are plotted for each operating PR at dierent vertical plane parallel to the side walls. Studying the eect of the 3-D geometry is obtained by detecting the ow separation location at each plane. Same planes locations (plane of symmetry, plane A, plane B, and plane C) used in the DBN (chapter 2) are used in the BFSN. the pressure distribution over the upper and lower nozzle walls at each plane are plotted and compared to each other to show the ow separation location change from upper to lower wall as the source of the side loads and the eect of the side walls on the ow separation location. The pressure distributions over the BFSN upper wall for plane of symmetry and planes A, B, and C are plotted and compared to each other for PRs 30.83, and The results show that the pressure distributions at all planes are identical in the region where the ow is attached to the nozzle wall. However, the pressure at plane C shows a dierence over the other three planes at the separated ow region as shown in Figure (3.5). Figures (3.6) and (3.7) show a pressure distribution over the upper and lower nozzle wall at the plane of symmetry and plane C for PR The pressure distribution over the upper and lower nozzle wall show a symmetrical behavior for the same plane. However the pressure distribution in plane C is dierent than the 36

52 other planes, the pressure over the upper and the lower walls in plane C show a symmetrical distribution too, as shown in Figure (3.7). Figure 3.5: Upper wall pressure distribution at dierent planes Figure 3.6: PR=30.83 Upper and lower wall pressure distribution at BFSN plane of symmetry for 37

53 Figure 3.7: Upper and lower wall pressure distribution at BFSN plane C for PR=30.83 Figure 3.8: BFSN upper wall pressure distribution at dierent planes for PR=

54 Figure 3.9: Upper and lower wall pressure distribution at BFSN plane of symmetry for PR=45.5 Figure 3.10: Upper and lower wall pressure distribution at BFSN plane C for PR=45.5 The pressure drops occurred at the end of the base nozzle and step edges are due to the expansion occurred at these locations due to sudden change in cross-sectional area, which cause a strong negative pressure gradient. Flow separation occurred at the edge of the rst step for the PR 30.8, while it occurred at the edge of the third step for PR Figures (3.11) and (3.12) show the Mach contours at the plane of symmetry and plane C respectively for PR=30.83, 39

55 while Figures (3.13) and (3.14) show the Mach contours at plane of symmetry and plane C respectively for PR=45.5. The Mach contours show a symmetrical behavior between the upper and the lower half for all operating PRs. Figure 3.11: Mach contours at BFSN mid-plane for PR=30.83 Figure 3.12: Mach contours at BFSN plane C for PR=

56 Dual-bell nozzle Figure 3.13: Mach contours at BFSN mid-plane for PR=45.5 Figure 3.14: Mach contours at BFSN plane C for PR=45.5 Figures (3.15) and (3.16) show the velocity vectors at the plane of symmetry for PR and 45.5 respectively zoomed near the second step edges. Figure (3.15) shows that the ow is separated at the rst step edge for PR and didn't attach to the nozzle walls again. The 41

57 rst step exit cross-section is then considered as the nozzle eective area. Figure (3.16) shows that the ow separated at the second step edge cross section which considered as the nozzle eective exit area for PR=45.5. Figure 3.15: Velocity vectors at plane of symmetry zoomed near second step edge (PR=30.83) Dual-bell nozzlethe governing equations for mass-weighted variables are given as: Figure 3.16: Velocity vectors at plane of symmetry zoomed near second step edge (PR=45.5) Side loads For purpose of studying the eect of the backward facing steps on the side loads, a time dependent two-dimensional numerical simulations are used to decrease the CPU time. Twodimensional time dependent numerical simulations are obtained for planar BFSN with compu- 42

58 tational domain similar to the mid-plane of the three-dimensional domain. A grid dependence studies are conducted here using the 2-D grids at an operating PR of 45 for the BFSN. The dierence in the average velocity and average static pressure values at the nozzle exit between the ne grid (230,548 grid points) and the coarse grid (64,477 grid points) did not exceed 1.5 %. However a grid size of 117,355 grid points is used which has an error less than 0.7 % corresponding to the ne grid. The rst cell distance from the wall is 1.5 micrometer and the numerical results showed that Y + average is less than 0.8. The two-dimensional time dependent numerical simulation is carried out for time; t =0 9 seconds corresponding to nozzle pressure ratio; PR= In the numerical simulation, the total pressure at nozzle inlet is kept constant while the ambient pressure is varied with altitude as follows: P = e ( H/7000) Assuming that the rocket is ying with constant velocity equal 500 m/s, the ambient pressure related to the time (t) can be written as P = e (( 500 t)/7000) As the ambient pressure equivalent to PR=24.24 (t=0 seconds), the pressure equation as a function of time is modied by curve tting to get the ambient pressure-time relationship used in this study: P = e ( t) The calculations were done using implicit scheme with second order accuracy in space and time. the SST k-ω turbulent model was used through calculations with rst cell distance near wall equal to 1.5 micrometer to keep y+ value near 1.During the calculations, the pressure integral over the upper and lower walls are calculated every second to calculate the net side force aecting the nozzle structure during nozzle ascending through the atmosphere. Figures (3.17) shows the side load values as a function with ow time. The side loads values 43

59 in the planar BFSN showed a small values values compared to the DBN. The average side loads value in BFSN decreased by 95% compared to the average value of the DBN side load as shown in Figure (3.18). Figure 3.17: Side loads over the BFSN walls as function of the ow time Figure 3.18: Side loads comparison between the BFSN and the DBN 44

60 Chapter 4 3-D Results for Axisymmetric Backward Facing Steps Nozzle and Dual-bell Nozzle 4.1 Introduction In this chapter, a complete three-dimensional numerical study for the BFSN and the DBN; with same area ratio, throat diameter, and length; are obtained at several operating PRs. Figure (4.1) shows the geometry of the BFSNs and equivalent dual bell nozzle. The wall pressure distributions over the nozzles walls and the axial velocity contours at the eective exit area have been obtained and used to assess the ow symmetrical distribution for both nozzles at all operating PRs. The ow parameters at the nozzles exits are obtained and the thrust forces are calculated for all operating PRs. Finally, the three-dimensional numerical results for the BFSN are compared to the two-dimensional axi-symmetric numerical results. (a) BFSN (b) DBN Figure 4.1: 3-D schematic diagrams of the nozzles physical models 45

61 4.2 Numerical method The commercial CFD software, Ansys Fluent, is used to simulate the turbulent ow of the subscale BFS nozzle and the DBN shown in Figures (4.1-a) and (4.1-b). Both nozzles have throat diameters of 0.53 inches and area ratios of 89. The length of both nozzles is 5.6 inches. The SST k ω was utilized to predict the turbulence quantities of the ow eld behavior. It was chosen because of its accuracy in computing the ow separation from smooth surface, and predicting the details of the wall layer characteristics [59]. Second order accuracy upwind scheme is used which accurately predicts the interaction between the oblique shock and the turbulent boundary layer. Ideal air is modeled as the driving gas at constant inlet pressure and temperature. Varied ambient pressure is specied at the far downstream boundary. The computations are done using parallel processing on eighteen node cluster at OSC (Ohio Super Computer). Each node is a 2.5 GHz processor. Time step size is calculated as the ratio between the typical cell value and the characteristic ow velocity. For all axi-symmetric nozzles simulations, the time step size is 10 Micro-second. 4.3 Computational domain and boundary conditions The grid of the BFSN and the DBN consist of ve blocks for each, two of which located inside the nozzle and the other three are placed outside as shown in Figure (4.2). A preliminary study was done to determine the downstream distance in the axial and the radial directions for both nozzles at dierent PRs. It is concluded from that study that the computational domain should be extended to 115 D in the axial direction and 28 D in the radial direction, where D is diameter of the nozzle throat. Similar computational domain with similar boundary conditions is used for the BFSN and the DBN as shown in Figure (4.2). To model the ow accurately, the mesh is denser in the turbulent boundary layer and across the geometrical corners to capture the ow separation and the concentrated high ow eld gradient. The nozzles walls are assumed to be smooth and adiabatic during the simulations with no-slip and no-ux velocity conditions imposed at the walls. A grid dependence studies are conducted here using the 3-D grids at an operating PR of 800 for both the BFSN and the DBN respectively. Four levels of grid are generated for each nozzle and the ow parameters at the nozzles exit 46

62 cross-sectional areas are compared to each others. Table (4.1) and table (4.2) show the grid levels and the ow parameters values for the BFSN and the DBN respectively with the CPU time required for each grid level. Taking into account the dierence in the ow parameters and the CPU time required for each grid level, grid level 3 is chosen for both the BFSN and the DBN (with maximum dierence less than 0.2% and 0.3% for the BFSN and the DBN respectively, corresponding to the ne grid) to carry out all the studies on both nozzles. Figures (4.3) and (4.4) show the grid in the mid-plane (x-y plane of symmetry) inside the BFSN and the DBN respectively. The quality of the ow solution depends upon the ability to capture the ow phenomena inside the boundary layers that are developing along the solid walls. Fluent recommends locating the nearest grid point along the wall at a distance that corresponds to non-dimensional coordinate y+ near one for the SST k ω turbulent model [59]. The rst cell distance from the wall is 1.5 micrometer for both the BFSN and the DBN. From the numerical results, the calculated y+average is less than 0.8, and 0.75 for the BFSN and the DBN respectively, that conrm the requirement that y+should be near one. Figure 4.2: Cross-section shows the computational domain and the boundary conditions 47

63 Figure 4.3: Computational grid inside the BFSN at mid-plane section Figure 4.4: Computational grid inside the DBN at mid-plane section 48

64 Grid level Cells number Mass ow rate [kg/s] Flow axial velocity [m/s] Flow static pressure [Pa] CPU time [days] 1 457, ,532, ,812, ,624, Table 4.1: Grid dependency study Grid level Cells number Mass ow rate [kg/s] Flow axial velocity [m/s] Flow static pressure [Pa] CPU time [days] 1 447, ,499, ,498, ,321, Table 4.2: Grid dependency study 4.4 Results and Discussion Numerical solutions are obtained for both the BFSN and the DBN at dierent operating PRs for assessment of the BFSN performance against the DBN. Nozzles inlet total pressure of 15 MPa and total temperature of 2000 K are used with varied back pressure to simulate the dierent operating PRs. The pressure distribution over the nozzles walls are plotted for each PR at dierent azimuth angles. Studying the eect of the 3-D geometry is obtained by detecting the ow separation location around the nozzles circumferences. The pressure distributions over the BFSN and the DBN walls at azimuth angel φ equal 0, 90, 180, and 270 degree are plotted and compared to each other for PRs of 400 and 600. The results show that the pressure distributions are identical at all azimuth angels for the BFSN which lead to a symmetrical ow behavior and ow separation location over the nozzle circumferential as shown in Figure (4.5) and Figure (4.6). In these Figures, sudden drops in the pressure occur at the discontinuity points of the nozzle surface which is caused by expansion of the ow due to sudden change in the nozzles cross-sectional area. 49

65 (a) Wall pressure distribution at angles 0, 90, 180, and 270 degrees (b) Section A Figure 4.5: Wall pressure distribution at dierent azimuth angels for PR=400 50

66 (a) Wall pressure distribution at angles 0, 90, 180, and 270 degrees (b) Section A Figure 4.6: Wall pressure distribution at dierent azimuth angels for PR=600 On the other hand, the pressure distribution over the DBN wall showed an asymmetrical behavior leads to a dierence in the ow separation locations over the nozzle circumference as shown in Figure (4.7) and Figure (4.8). Similar sudden pressure drops occurs in the DBN at the wall inection point. The maximum dierence in the ow separation position is found to be about 4% and 9% for PR 400 and 600 respectively. 51

67 (a) Wall pressure distribution at angles 0, 90, 180, and 270 degrees (b) Section A Figure 4.7: Wall pressure distribution at dierent azimuth angles for PR=400 52

68 (a) Wall pressure distribution at angles 0, 90, 180, and 270 degrees (b) Section A Figure 4.8: Wall pressure distribution at dierent azimuth angles for PR=600 The ow separation location can be dened as the point at where the axial wall shear stress become zero. Figures (4.9) and (4.10) show the isosurface of the DBN wall where the wall axial shear stress is equal to or less that zero for PR=400 and PR=600 respectively. The isosurface of the zero or negative wall axial shear stress shows the asymmetrical behavior of the separation process inside the nozzle. The positive values of wall axial shear stress near the nozzle exit is 53

69 due to the ow recirculation where the entrained air from the downstream have a positive axial velocity near the wall at that location. Figures (4.11) and (4.12) show the same isosurface in case of the BFSN at PR=600 and PR=800 respectively, where the ow separation is occurs at symmetrical way. The ow separated from the second step and the third step edge edge at PR=600 and PR=800 respectively, where a small positive wall shear stress region appears just before the those step edges is indication that the ow is attached to the wall in this region and separates after than and did not attach to the nozzle wall again. The positive wall shear stress regions appear after the separation location are due to the ow recirculation which leads the ow to have a positive axial velocity near the wall at some locations. Figure 4.9: Isosurface for wall xial shear stress equal to or less that zero for DBN at PR=400_xy view Figure 4.10: Isosurface for wall xial shear stress equal to or less that zero for DBN at PR=600, x-y view 54

70 Figure 4.11: Isosurface for wall axial shear stress equal to or less that zero for BFSN at PR=600, x-y view Figure 4.12: Isosurface for wall axial shear stress equal to or less that zero for BFSN at PR=800, x-y view Mach contours in the plane of symmetry for PR 400 and 600 are plotted for both the BFSN and the DBN as shown from Figures (4.13) to (4.17). The symmetrical nature of the ow inside the BFSN can be noticed from the Mach contours for both PRs of Figures (4.13) and (4.14). The BFSN Mach contours show that the ow separation occurred at the second step step for PR 400 and 600. Figure (4.15) shows the subsonic region in the ow circulation zone near the rst steps corner for PR 600. Similar ow behavior and Mach contours distribution appears at all closed subsonic recirculation zones at all operating PRs. The nozzle eective exit area is the last nozzle cross-sectional area where the ow is separated for a given PR. The ow 55

71 parameters at the eective exit area are used to calculate the nozzle thrust at dierent PRs. The asymmetric ow separation inside the DBN can also be noticed from the Mach contours as shown in Figures (4.16) and (4.17). The Mach contours show an asymmetric distribution behavior around the nozzle axis of symmetry which shows that the ow inside the DBN is asymmetric. Figure 4.13: Mach contours at x-y plane of symmetry for PR=400 Figure 4.14: Mach contours at x-y plane of symmetry for PR=600 56

72 Figure 4.15: Mach contours at x-y plane of symmetry for PR=800 zoomed at 1st step Figure 4.16: Mach contours at x-y plane of symmetry for PR=400 57

73 Figure 4.17: Mach contours at x-y plane of symmetry for PR=600 Further conrmation of the ow behavior inside the BFSN and the DBN is obtained by plotting the axial velocity contours at the nozzle eective exit cross-sectional area shown in Figure (4.18) and Figure (4.19). Figure (4.18) shows the cross-sectional axial-velocity contours, for PR 600, at the second step where the ow is still attached to the nozzle wall (with minimum positive axial-velocity value near the wall) and the third step where the ow is separated from the nozzle wall (with minimum negative axial-velocity value near the wall). Figure (4.19) shows the cross-sectional axial-velocity contours, for PR 800, at the third step where the ow is still attached to the nozzle wall (with minimum positive axial-velocity value near the wall) and the fourth step where the ow is separated from the nozzle wall (with minimum negative axialvelocity value near the wall. The results show a smooth and axi-symmetric velocity distribution for both attached and separated ow cross-sections. The axial velocity contours at the DBN eective exit cross-sectional area show an asymmetrical distribution for PR 600 as shown in Figure (4.20). That indicates the ow is axi-symmetric. 58

74 (a) Second step exit cross-section (b) Third step exit cross-section Figure 4.18: Axial-velocity contours for PR=600 59

75 (a) Third step exit cross-section (b) Fourth steps exit cross-section Figure 4.19: Axial-velocity contours for PR=800 60

76 (a) (b) Section A (c) Section B Figure 4.20: Axial-velocity contours at eective exit area for DBN at PR=600 61

77 For further conrmation of the axi-symmetric nature of the ow in the BFSN, two-dimensional axi-symmetric solutions are obtained computational grid identical to the 3-D grid plane of symmetry. Comparison between the 3-D and the 2-D axisymmetric results are shown in table (4.3). The ow separation position from the 3-D simulations for each PR shows a good agreement with its position in the 2-D axi-symmetric simulation for same PR (at same step edge). The ow parameters at the ow separation location cross-sectional area are calculated the 3-D and the 2-D solutions and compared to each other in table (4.3). The maximum dierence between the 3-D and the 2-D solutions is less than 1%. From theses results, we conclude that the ow inside the BFSN is axi-symmetric. That is a very important nding which has the following implications: 1) two dimensional solution can be used to analyze the BFSN, calculate the nozzle thrust, the ow exit velocity, etc, 2) unsteady ow solution are now possible because of the major reduction of the CPU time for 2D solutions compared to 3D solution, and 3) the axisymmetric solution is suitable for design practices of unsteady ow. PR Parameter 3-D 2-D dierence % Eective exit area location 2nd step 2nd step 400 Pressure [Pa] Velocity [m/s] Eective exit area 2nd step 2nd step 600 Pressure [Pa] Velocity [m/s] Eective exit area 3rd step 3rd step 800 Pressure [Pa] Velocity [m/s] Eective exit area 4th step 4th step 1100 Pressure [Pa] Velocity [m/s] Table 4.3: Flow parameters comparison between 3-D and 2-D simulations Side loads calculation The side loads are calculated for the BFSN and the DBN at dierent PRs from the threedimensional solutions by integrating the pressure over the nozzles walls at the operating PRs 400, 600, and 800. It has been found that the maximum side loads in the BFSN is less than 0.7% of the side loads in the DBN at PR 400, 2.35% at PR 600, and 2.45% at PR 800. Note 62

78 that these results are obtained for scaled nozzles. We expect the values of the side loads in actual nozzles to be much higher. Table (4.4) shows the side load values in y- and z- directions for dierent PRs for the DBN and the BFSN. PR Direction DBN [N] BFSN [N] % y z total y z total y z total Table 4.4: Side load values for DBN and BFSN 63

79 Chapter 5 2-D Axisymmetric Parametric Studies on Backward Facing Steps Nozzle 5.1 Introduction In this chapter, a parametric studies are carried out to study the eect of the number and height of the backward facing steps on the performance (in terms of thrust) of the BFSN. Two approaches are carried out in this chapter. First, equal length backward facing steps are considered, with steps number of two, four, six, twenty, and forty as shown in Figure (5.1). 64

80 (a) 2-BFSN (b) 4-BFSN (c) 6-BFSN Figure 5.1: BFSN with 2, 4, and 6 backward facing steps 65

81 (a) 20-BFSN (b) 40-BFSN Figure 5.2: BFSN with 20 and 40 backward facing steps Second, varied steps length and height with step edge cross-sectional area suitable for the related operating altitude, with number of steps equal six as shown in Figure (5.3). Since the most of the rocket mission takes place at high altitude, a high PR of 1500 (altitude) is chosen as a design point to compare the performance (in terms of thrust) of the dierent BFSNs and the DBN. 66

82 Figure 5.3: 6-BFSN adjusted step cross-sectional area It is important to note here that the thrust curve for the DBN calculated from the 2-D axi-symmetric solutions following many studies have been done using the 2-D axi-symmetric solutions [63], [64], [65], [66], and [45]. The thrust curve is calculated from the ow parameters (pressure and axial velocity) based on the base nozzle exit area before the ow separation point leaves the end of the base nozzle following Refs [40] and [67]. As the ow separation moves downstream toward the nozzle exit, the thrust is calculated based on the nozzle total exit area which takes into account the eect of the ow in the separation region. All BFSNs geometry have the same DBN geometrical parameters of throat area, exit area, and length. 5.2 Numerical method The SST k ω was used to predict the turbulence quantities of the ow eld behavior. Second order accuracy upwind scheme is used which accurately predicts the interaction between the oblique shock and the turbulent boundary layer. Ideal air is modeled as the driving gas at constant inlet pressure and temperature. Varied ambient pressure is specied at the far downstream boundary. 5.3 Computational domain and boundary conditions Similar computational domain with similar boundary conditions from chapter (4) is used for all BFSNs. To model the ow accurately, the mesh is concentrated in the turbulent boundary layer and across the geometrical corners to capture the ow separation and the high ow eld 67

83 gradient. The nozzles walls are assumed to be smooth and adiabatic during the simulations with no-slip and no-ux velocity conditions imposed at the walls. The rst cell distance from the wall is 1.5 micrometer for all BFSNs and the numerical results showed that the calculated Y + average is less than 0.8, and 0.75 for the BFSN and the DBN respectively, that conrm the requirement that Y + should be near one. 5.4 Results and Discussion Equal length backward facing steps Five dierent geometries of the BFSN with two, four, six, twenty, and forty backward facing steps are generated as shown in Figure (5.1). All BFSNs are simulated for the design point PR of 1500 and compared with each others and with the DBN in terms of thrust value. Mach contours for all BFSNs geometries at PR=1500 are shown from Figure (5.4) to Figure (5.8). The ow parameters at nozzles exits (pressure and axial velocity) are used to calculate the thrust for all BFSNs. Table (5.1) shows the thrust value comparison for all BFSNs with respect to the DBN. 68

84 Figure 5.4: Mach contours for 2-BFSN at PR=1500 Figure 5.5: Mach contours for 4-BFSN at PR=

85 Figure 5.6: Mach contours for 6-BFSN at PR=1500 Figure 5.7: Mach contours for 20-BFSN at PR=

86 Figure 5.8: Mach contours for 40-BFSN at PR=1500 Nozzle DBN 2-BFSN 4-BFSN 6-BFSN 20-BFSN 40-BFSN Thrust [N] Dierence from DBN % % -0.15% -0.08% -0.04% % Table 5.1: Thrust values for BFSNs and DBN at PR=1500 The results show that the BFSN with two steps has the lowest thrust value compared to the DBN. As the number of steps increase, the thrust of the BFSN become close to the thrust of the DBN. Theoretically, if the number of steps increased innitely, the wall of the BFSN will become like a smooth wall of the DBN. In this study, BFSNs with two and six backward facing steps are chosen to be simulated for PR varies from 50 to 1500 and compared to the DBN. The thrust curves for the DBN and the BFSNs are plotted and compared to each others as shown in Figure (5.9). The DBN thrust curve shows the traditional schematic thrust curve behavior of the DBN. The DBN thrust increase with PR until the PR reaches the transition value, then the transition from low altitude mode to high altitude mode takes place resulting in a thrust drop as shown in Figure (5.9). The thrust curve for the 2-BFSN experience less thrust drop as the transition is 71

87 taking place at higher PR (ow separation stagnate at the base nozzle exit for a higher range of PRs). In this case the transition PR is closer to the optimum value resulting in a smaller thrust drop. For 6-BFSN, the transition takes place at PR higher than the DBN and lower than the 2-BFSN. That is because the expanded ow from the base nozzle exit attach the rst step edge in case of 6-BFSN much earlier than in case of 2-BFSN (rst step exit cross-sectional area in the 6-BFSN is much lower than the rst step exit cross-sectional area in the 2-BFSN). As the PR increases, the 2-BFSN, 6-BFSN and the DBN produces approximately equal thrust with low dierences as mentioned before in table (5.1). Figure 5.9: Thrust Curves for the BFSN with two steps and the DBN with respect to PR Figure (5.10) and Figure (5.11) show the Mach contours for the 2-BFSN at all PRs. Figure (5.10-a) shows the Mach contours at PR=50 where the ow separation occurs just before the base nozzle exit. The ow structure in this case takes a cap shock pattern where a trapped vortex occurred after the small normal shock at the nozzle center. As PR increase, the ow is completely attached to the base nozzle wall and expands gradually until the ow attach the rst step edge at PR=650 as shown in Figure (5.10-f). Further increasing of the PR makes the ow expands after the rst step edge to attach the nozzle exit lib at PR=950 as shown in Figure (5.11-a). Further increasing of the PR makes the nozzle operates with under-expansion 72

88 conditions. It is very important to mention here that the ow separations occurred at base nozzle exit and the rst step edge at dierent PRs is due to expansion waves at those locations instead of oblique separation shock in conventional nozzles. (a) PR=50 (b) PR=148 (c) PR=200 (d) PR=400 (e) PR=600 (f) PR=650 Figure 5.10: Mach contours of 2-BFSN at dierent PRs 73

89 (a) PR=900 (b) PR=950 (c) PR=1100 (d) PR=1500 Figure 5.11: Mach contours of 2-BFSN at dierent PRs Equal length steps Vs. variable length steps The 6-BFSN with variable back steps length is designed based on the suitable area required for PRs. The required area are calculated using the isotropic ow relation for ideal nozzle from [68]. Comparing the thrust values for the 6-BFSN with variable steps length with the thrust of the 6-BFSN with equal step length, it has been found that both are quit similar to each other with maximum thrust dierence of 0.9% at PR=200 where the ow separation exits at the end of the base nozzle for both BFSNs. At high altitudes, both BFSNs produce equal value of thrust. Figure (5.12) shows the thrust curve of both nozzles as a function of PR. 74

90 Figure 5.12: Thrust curve for 6-BFSN with equal steps length Vs. 6-BFSN with adjusted length Temperature distributions One of the most important issues to display is the temperature distribution over the nozzle wall. Figure (5.13) and Figure (5.14) show the static temperature contours in the 2-BFSN and 6-BFSN respectively.the static temperature in the backward facing steps regions reaches high values when they become a closed recirculation regions. That is a result of a low velocity of the recirculated ow with high total temperature. Figure (5.13) explain that very well. For PR=600 where the oe separation located at the end of the base nozzle, there is no closed recirculation zones. At PR=650, the ow separation point jump to the rst step edge resulting in generating a closed recirculation zone at the rst step corner with high static temperature. After the second transition at PR=950, the second step corner transfer to a closed recirculation zone with high static temperature too. Figure (5.14) shows the same phenomenon occurs in the 6-BFSN with dierent PRs. One of the methods to overcome the danger of high temperature eect on the nozzle walls is the thermal barrier coating which can be used to isolate the nozzle walls form the high temperature ow. Studying the high temperature issue and the suitable cooling methods are required in the future work. 75

91 (a) PR=600 (b) PR=650 (c) PR=900 (d) PR=950 Figure 5.13: Static temperature contours in the 2-BFSN at dierent pressure ratios 76

92 (a) PR=400 (b) PR=800 (c) PR=1100 (d) PR=1500 Figure 5.14: Static temperature contours in the 6-BFSN at dierent PRs 77

93 Chapter 6 Extendable Backward Facing Steps Nozzle 6.1 Introduction One of the most important advantages of the BFSN is its capability to be contracted and expanded. In this chapter, studyning of the expanding concept is carried out on the 2-BFSN. The 2-BFSN in this study has three operation modes. First mode is taking place at sea-level, with the two steps are in contracted form as shown in Figure (6.1-a). The 2-BFSN in this mode acts as a small area ratio conventional bell nozzle (base nozzle). The second operation mode is taking place when the rst step is expanded while the second step is still contracted as shown in Figure (6.1-b). While the third operation mode is taking place when the two steps are expanded as solved before in chapter (5), Figure (6.1-c). There are two dierent scenarios available in this study. The rst is that the 2-BFSN is operating as the main engine nozzle where its operation starts from the sea-level up to the high required altitude. In this case the nozzle passes through its three operation modes as the altitude changes. While the second is that the 2-BFSN is operating as the second stage engine nozzle. In this case, the nozzle will be stored in its contracted form until the rocket reaches the altitude where the second stage engine start to provide thrust, and the nozzle will then expand suddenly to operates at mode 3 directly. 78

94 (a) Mode-1 (b) Mode-2 (c) Mode-3 Figure 6.1: Dierent operating modes of 2-BFSN 79

95 6.2 Numerical methods Same numerical methods used in simulating all BFSNs in chapters (4 and 5) is used in this study. In order to simulate the motion of the steps while the nozzles transition between dierent modes, dynamic grid with layering technique is used to obtain a time dependent solution with changing the ambient pressure during nozzle transition. The transition times in these studies are chosen to be 0.1 second. Using the unsteady solutions obtained for both 2-BFSN and 6-BFSN in Chapter (5), the rst transition for the 2-BFSN is calculated to take place at PR= and nish at PR=750 with duration time of 0.1 second. The second transition for the 2-BFSN is calculated to take place at PR= and nish at PR= with duration time of 0.1 second. While for the 6-BFSN, the transition is calculated to take place at PR=4000 and nish at PR= with duration time of 0.1 second. All of PRs calculations are carried out with assuming the rocket velocity of 500 m/s. 6.3 Computational domain and boundary conditions The computational domains of the 2-BFSN mode 1 and mode 2 have the same dimensions, boundary types, and node distributions of the computational domain used before in simulating the mode-3,see chapter (5). Figure () and Figure () show the computational grid inside the nozzle for mode-1 and mode-2 respectively. The only dierence in this computational domain is that it consists of structured grid inside the nozzles and the two steps zones as shown in Figure (), while an unstructured grid is used in the rest of the domain. Figure 6.2: computational grid of the 2-BFSN in mode 1 80

96 6.4 Results and Discussion BFSN The rst transition of the 2-BFSN from mode-1 to mode 2 is numerically simulated with time dependent solution and dynamic grid. First, the unsteady solution is obtained for the 2-BFSN on its mode-1 shape at PR= Second, the time dependent solution with dynamic grid is obtained taking the previous unsteady solution as initial values. The simulation duration is 0.1 second with solution time step of 0.05 ms. Figure (6.3) shows the Mach contours at dierent ow time starting from t=0 s (PR= ) to t=0.1 s (PR=750). The ow separates at base nozzle exit at t=0 second as the 2-BFSN operates in mode-1. As the geometry transition takes place with expanding the rst step, the ow attached the rst step edge and hence the 2-BFSN operates at mode-2 with expansion area ratio base on the rst step exit area. (a) t=0 s (b) t=0.05 s (c) t=0.097 s (d) t=0.1 s Figure 6.3: Mach contour for 2-BFSN transition from mode-1 to mode-2 As the rst transition (from modee-1 to mode-2) occurred, the 2-BFSN operates in mode- 81

97 2 for PR range brom PR=750 to PR= 945. At PR=945, the second transition takes place The second transition of the 2-BFSN from mode-2 to mode-3 is then simulated starting from unsteady solution of the 2-BFSN on its mode-2 shape as initial values. The second transition time is 0.1 s starting from t=0s (PR=945 ) to t=0.1s (PR=950). Figure (6.4) shows the Mach contours at dierent ow time starting from t=0 s (PR=) to t=0.1 s (PR=950). At t=0 s, the 2-BFSN operates with mode-2 shape where the ow is attach to the rst step exit edge, Figure (6.4-a). As the second step starts to expand, the transition from mode-2 to mode-3 takes place and the ow attaches to the second step edge. Finally, the 2-BFSN takes its mode-3 shape, Figure (6.4-d), nd operates with full owing condition where the ow is attached to the nozzle exit lib and total expansion area is in use. (a) t=0 s (b) t=0.5 s (c) t=0.09 s (d) t=0.1 s Figure 6.4: Mach contour for 2-BFSN transition from mode-2 to mode-3 The main advantage of nozzle contraction and expansion is to limit the nozzle exit area ratio to base nozzle exit, rst-step edge, and nozzle exit through mode-1, mode-2 and mode-3 respectively. This will increase the nozzle thrust compared to the thrust of the 2-BFSN with expanded steps due to elimination of the aspiration drag of the separated region inside the 82

98 nozzle. Figure (6.5) shows a comparison of thrust values of the 2-BFSN through its three modes and the thrust of the 2-BFSN if it operates at mode-3 only at all altitudes. the 2-BFSNmode-1 has little higher thrust at low PR as the expansion area ratio limited to the base nozzle exit area with absence of aspiration drag. At PR=600, the rst transition takes place in the 2-BFSN with xed steps causing a thrust drop while in the 2-BFSN-mode-1 the thrust keep increasing until the rst step expands and the transition takes place at higher PR causing a smaller thrust drop. The second step expands at PR=944 where the second thrust drop occurs in the expandable 2-BFSN. At high PR, both nozzle have the same thrust values as they have the same geometry. Figure 6.5: Thrust curve for 2-BFSN xed steps Vs. 2-BFSN extendible steps Two-modes 7-BFSN Seven steps BFSN is designed to operates with two modes. The rst mode takes place from sealevel up to certain PR (altitude) with BFSN geometry consists of six xed backward facing steps and seventh step is contracted. Second mode takes place when the seventh step is expanded enlarging the nozzle expansion area ratio hence generating higher thrust at higher altitudes. The transition from mode-1 to mode-2 is numerically simulated with time dependent solution 83

99 and dynamic grid. First, the unsteady solution is obtained for mode-1 at PR=2000. Second, the time dependent solution with dynamic grid is obtained taking the previous unsteady solution as initial values. The simulation duration is 0.1 second with solution time step of 0.05 ms. Figure (6.6) shows the Mach contours at dierent ow time starting from t=0 s (PR=2000) to t=0.1 s (PR=2014). 84

100 (a) t=0 s (b) t=0.05 s (c) t=0.1 s Figure 6.6: Mach contour for 7-BFSN transition from mode-1 to mode-2 85

101 Figure 6.7: Thrust comparison between 6-BFSN and 2-modes_7-BFSN 86

102 Chapter 7 Backward Facing Steps Application for Full-Scale Nozzle 7.1 Introduction In this chapter, the backward facing steps approach is applied on a full-scale nozzle having a dimensions and operating conditions as a typical values for todays rocket nozzles. The backward facing steps approach is applied on a cionventional bell shape nozzle with parabolic contour. The conventional bell shape nozzle is designed based on the values form [68] with area ratio of 25 and inlet nozzle angle 30 degree and exit angle of 8.5 degree as shown in Figure (7.1). Figure 7.1: Conventional bell shape nozzle contour The conventional bell shape nozzle has a parabolic contour with 80% length of equivalent 87

103 15 degree conical nozzle. The parabolic conventional nozzle has expansion area ratio of 25 and length of m. The throat radius is chosen to be m which is equal to the throat radius of Vulcain 2 engine nozzle [69]. Two backward facing steps are added at the end of the parabolic nozzle increasing its expansion area ratio to 43 and its length to m. Figures (7.1) and (7.3) show the parabolic conventional nozzle and the BFSN respectively. Figure 7.2: Designed parabolic conventional nozzle AR=25 Figure 7.3: BFSN with AR= Numerical method Two-dimensional axi-symmetric solutions are used to simulate the turbulent ow of the fullscale conventional bell nozzle and BFSN. The SST k ω is used to predict the turbulence quantities of the ow eld behavior. It was chosen because of its accuracy in computing the 88

104 ow separation from smooth surface, and predicting the details of the wall layer characteristics [59]. Second order accuracy upwind scheme is used which accurately predicts the interaction between the oblique shock and the turbulent boundary layer. Ideal air is modeled as the driving gas at constant inlet pressure and temperature. Varied ambient pressure is specied at the far downstream boundary. 7.3 Computational domain and boundary conditions The computational domain is extended for thirty-ve times nozzle exit radius in downstream and eight times nozzle exit radius in vertical direction as shown in Figure (7.4). To model the ow accurately, the mesh is concentrated in the turbulent boundary layer and across the geometrical corners to capture the ow separation and the high ow eld gradient. The nozzles walls are assumed to be smooth and adiabatic during the simulations with no-slip and no-ux velocity conditions imposed at the walls. The rst cell distance from the wall is 20 micrometer for both nozzles and the numerical results showed that the calculated Y + average is less than 1 for both nozzles, that conrm the requirement that Y + should be near one. Constant total pressure equal to 10 MPa is used in the simulation with total temperature of 2000 K at the nozzle inlet. The freestream pressure is varied to simulate the nozzle operating conditions form sea-level to vacuum. Both nozzle are simulated at altitudes 0, 10, 20, 80, 160 Km and the thrust coecient for both nozzle are compared to each other. Figure 7.4: Computational domain for conventional bell nozzle and BFSN 89

105 7.4 Numerical results Figure (7.5) shows the Mach contours in the classical bell nozzle at dierent altitudes. The classical bell nozzle operates at slightly over-expansion conditions at sea level appears in the weak oblique shock at the nozzle exit to adapt the ow pressure to the ambient pressure as shown in Figure (7.5-a). As the altitude increase, the nozzle become under-expanded the the ow pressure at nozzle exit is higher than the ambient pressure as shown in Figures (7.5-b), (7.5-c), and (7.5-d). (a) H=0 km (b) H=20 Km (c) H=80 Km (d) H=160 Km Figure 7.5: Mach contours for classical bell nozzle at dierent altitudes Figures from (7.6) show the Mach contours for the BFSN at dierent altitudes. At sea level, the expansion area ratio is limited to the base nozzle exit area with ow structure similar to the conventional bell nozzle shown in Figure (7.6-a) while the ow is attached to the second step exit at high altitude as shown in Figures (7.6-b) and (7.6-c) increasing the nozzle expansion area and results in a higher thrust. 90

106 (a) H=0 Km (b) H=20 Km (c) H=80 Km (d) H=160 Km Figure 7.6: Mach contours for BFSN at dierent altitudes The thrust coecients for both nozzle at dierent altitudes are calculated base on the ow pressure at the nozzle exit using the equation from [68] as following: where: P 1...total pressure at nozzle inlet P 2...pressure at nozzle exit A 2...Nozzle exit area k...specic heat ratio Figure (7.7) shows the comparison between the conventional bell nozzle and the BFSN thrust coecient at dierent altitudes. The results show that the BFSN and the conventional 91

107 bell nozzle have same thrust coecient at sea level as both of them have same expansion area ratio. At high altitudes, the BFSN has higher thrust coecient than the conventional bell nozzle because of the higher expansion area ratio with presence of the backward facing steps. The numerical result in terms of the thrust coecients are in a good agreement with the thrust coecient curve as function of nozzle expansion area ratio from [68] as shown in Figure (7.8) Figure 7.7: Thrust coecient for classical bell nozzle and BFSN as a function of altitude Figure 7.8: Thrust coecient Vs. nozzle area ratio 92

108 7.5 Flight performance To demonstrate the main benets of the BFSN, two simplied ight scenarios are obtained for a rocket with three engines from sea-level up to altitude of 200 km. The rst scenario is carried out using the conventional parabolic nozzle (Rocket 1) while the second is obtained using the BFSN (Rocket 2). For demonstration purposes, gravity-free Drag-free space ight simplication assumptions are assumed. (a) Parabolic nozzle (b) Parabolic nozzle (c) BFSN (d) BFSN Figure 7.9: Nozzles mounted in rocket base Using basic motion equations, the ight performance parameters of both rockets are ob- 93

109 tained from sea-level up to altitude of 160 km. The thrust is assumed to be change linearly between the calculated altitudes (0, 10, 20, 80, 160 km). The following table shows the main ight performance parameters for both rockets. The nozzles material is assumed to be Aluminum alloy Al 6061-T6 with density equal to 2700 kg/m^3. By calculating the nozzles volume, The mass of both nozzles (conventional nozzle and BFSN) are obtained. The dierence in the nozzles mass are calculated and added to the rocket-2 initial mass as shown in table (7.1). Parameter Rocket 1 Rocket 2 Total initial mass [kg] 550, ,729 No. of engines 9 9 Mass ow rate [kg/s] Max. altitude [km] Propulsion time (t p ) [s] Final mass (m f ) [kg] Nozzles mass [kg] Table 7.1: Basic rocket parameters Figure (7.10) shows the comparison the thrust-altitude curves, Figure (7.10-a) and the altitude-time curve, Figure (7.10-b) of both rockets. Both rockets have approximately same thrust from sea level up to 10 km, then the rocket-2 produce higher thrust due to it's higher nozzles expansion areas (BFSN). Figure (7.10-b) shows that, the time required to reach any altitude from the rocket-2 is always less than the time required from rocket-1. 94

110 (a) Thrust-altitude curve (b) Altitude-time curve Figure 7.10: Flight performance comparison between conventional nozzle and BFSN From the calculations, the nal mass m f for rocket-1 and rocket-2 are and kg respectively, and can be expressed as follows: m f 1 = m P L 1 + m R 1 + m N 1 and 95

111 m f 2 = m P L 2 + m R 2 + m N 2 where m f 1 and m f 2 are the nal mass of the rocket-1 and rocket-2 respectively m P L 1 and m P L 2 are the payload mass for rocket-1 and rocket-2 respectively m R 1 and m R 2 are the mass of thr rocket-1 and rocket-2 structures respectively m N 1 and m N 2 are the rocket-1 nozzles (conventional nozzles) and rocket-2 nozzles (BFSNs) respectively Since the both rockets have same structure (m R 1 = m R 2 ), m f 2 m f 1 = m P L 2 m P L 1 + m N 2 m N 1 m f = m P L (7.1) m P L = = kg The percentage of the payload increasing is calculated related to the m f 1 and it has been found that, using of the BFSNs instead of the conventional nozzles results in increasing the payload by 12.8%. 96

112 Chapter 8 Conclusions and Future Work 8.1 Conclusions A new backward facing steps nozzle that is an altitude adjustable exit area is developed. The nozzle geometry consists of two parts, the rst is a base part of the dual-bell nozzle and the second part is a nozzle extension with surface consists of backward facing steps. The performance of the nozzle is assessed numerically by solving Unsteady Reynolds Averaged Navier-Stokes equations (URANS) equations using Fluent code against the classical bell and the dual-bell nozzles. The assessment parameters are thrust and side loads. The following conclusions are drawn: The numerical procedure is validated by comparing the numerical results of a planar Dual-Bell Nozzle (DBN) with experimental data. The comparison of the results validate the procedure; Figure (2.6) and Figure (2.7). Assessment of the side loads generated by the Backward Facing Steps Nozzle (BFSN) compared to the side loads generated by the DBN shows a 0.6% of the DBN side loads is generated by the BFSN; Figure (2.11), Figure (3.17) and table (4.4). 3-D numerical solutions for the BFSN and the DBN show that the ow in the BFSN is found to be axisymmetric while the ow in the DBN is three dimensional. That makes numerical solution for the BFSN ow can be computed eciently. The parametric study for the number and geometry of the backward facing steps that varied the number of the steps from two to forty indicated that the thrust of the BFSN 97

113 approaches the thrust of the DBN at high altitude as the number of steps increases which was expected; table (5.1). At lower altitudes, the BFSN thrust is higher than the DBN and experience smaller drop at the transition from the base nozzle to the extension part of the nozzle; Figures (5.9) Improved thrust in the BFSN is achieved by using a dynamic exit area adjustment with altitude. That is achieved by having the nozzle in a contracted form then the steps open with exit areas appropriate for each altitude, Figure (6.5). A hybrid xed area and a contracted last step nozzle is design to allow the rocket to reach altitudes higher than the designed altitude. An improvement over the classical bell nozzle for a given pressure ratio by adding backward facing steps extension allows the nozzle to produce higher thrust at high altitudes; Figure (7.7) Comparison of the advantages and the disadvantages of the BFSN, DBN, and the conventional bell nozzle can be concluded as follows: BFSN Advantages: Multi-altitudes adjustable exit area nozzle with low thrust drop at transition between altitude modes Generates axi-symmetric ow at all pressure ratios which has advantages for time dependent solutions and for design practices No side loads generated at all pressure ratios due to the guaranteed axi-symmetric separation at the steps edges Can be stored in combat form for multi-stage rockets, and can be expanded at a given altitude by mechanical or other means Generating higher thrust than conventional bell and DBN at lower altitude by controlling the ow separation at the steps edges 98

114 Can operate in hybrid mode of conventional and backward facing step modes Can improve thrust by using dynamic backward facing steps Disadvantages: High static temperature at closed recirculation regions which can be treated using thermal barrier coating and/or vented steps Low thrust drop during transition between the dierent modes which can be reduced by using more steps Questions about manufacturing and additional weight due to the geometry of the steps DBN Advantages: Two altitudes adjustable exit area nozzle that improve thrust at high altitude Disadvantages: High side load peak during transition High thrust dope at transition between two modes Transition takes place earlier than the optimum pressure ratio Conventional bell nozzle Advantages: Highest thrust at high altitudes (at full owing condition) because of the optimum contour Disadvantages: High side loads due to non-symmetrical separation. That can be reduced by using expensive active or passive ow separation control 99

115 8.2 Future work While this dissertation provides promising results for using the high-order Discontinuous Galerkin method, especially combined with the Chimera scheme, some potential future work directions are outlined blow: 1. Combustion simulation : simulating the inlet ow at the nozzle as a ow mixture resulting from the fuel combustion in the combustion chamber with high temperature. 2. Conjugate heat transfer: studying the eect of the high ow temperature (heat conduction) the free stream temperature (heat convection) on the nozzle wall. 3. Nozzle cooling system: searching for alternative cooling mechanism to avoid the high temperature eect on the nozzle wall, specially at the closed recirculation regions (steps corners). 4. Contracting and expanding of the nozzle: studying the suitable systems that can be used with the BFSN to provide a reliable expanding mechanism for changing the nozzle area dynamically with altitude or for expanding all steps suddenly at high altitude (for working in multi-stage rockets) 5. Fluid Solid Interaction (FSI): using a FSI to produce expanding mechanism to expand the BFSN suddenly at high altitude (for multi-stage rockets) using a diaphragm attached at the last step exit. 6. Practical application: designing a BFSN with full scale dimensions to operates with a real rocket to satisfy a certain mission goal taking into account the real parameters (mass, payload, required thrust, required altitude, mission time,...etc) and assess the BFSN performance against a conventional rocket nozzle with the same operating conditions. 7. Vibration and stability of the nozzle (dynamic performance): obtaining a complete time history of nozzle performance and side loads aecting the full scale BFSN through its mission to assess its dynamic performance. 100

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123 Appendix A Thrust Calculations Thrust was calculated for each nozzle PR based on the ow parameters at the nozzle eective exit cross-sectional area using the following relation : ˆ F = (ρ V 2 e ) + (P e P a ) da (8.1) Where ρ...the ow density at nozzle exit V e...flow exit velocity at nozzle exit P e...flow static pressure at nozzle exit P a...ambient static pressure As long as the ow separation located at the base nozzle exit area, The thrust is calculated base on the base nozzle exit cross-sectional area. Otherwise, the thrust is calculated base on the ow parameters at the total nozzle exit cross-sectional area. 108