INVESTIGATION OF DIFFERENT MODELING APPROACHES FOR COMPUTATIONAL FLUID DYNAMICS SIMULATION OF HIGH-PRESSURE ROCKET

The paper summarizes technical results and ¦rst highlights of the cooperation between DLR and Airbus Defence and Space (DS) within the work package ¤CFD Modeling of Combustion Chamber Processes¥ conducted in the frame of the Propulsion 2020 Project. Within the addressed work package, DLR Gottingen and Airbus DS Ottobrunn have identi¦ed several test cases where adequate test data are available and which can be used for proper validation of the computational §uid dynamics (CFD) tools. In this paper, the ¦rst test case, the Penn State chamber (RCM1), is discussed. Presenting the simulation results from three di ̈erent tools, it is shown that the test case can be computed properly with steady-state Reynolds-averaged NavierStokes (RANS) approaches. The achieved simulation results reproduce the measured wall heat §ux as an important validation parameter very well but also reveal some inconsistencies in the test data which are addressed in this paper.


INTRODUCTION
The main target of the Propulsion 2020 cooperation between DLR and Airbus DS is the strengthening of the knowledge and competence in the area of rocket propulsion combustion devices, particularly with regard to test, modeling, and simulation capabilities.One of the work packages is focused on CFD modeling of rocket combustion chambers whose ¦rst results are shown here.This work package is executed in close cooperation between DLR Institute of Aerodynamics and Flow Technology and Airbus DS.The objective is to develop numerical PROGRESS IN PROPULSION PHYSICS tools and methods which are applicable to real rocket thrust chamber hardware design and optimization.Hence, the tool packages must be designed to simulate multiinjector combustors in a reasonable time frame encompassing the driving mechanisms as propellant disintegration, combustion, and heat transfer.Conclusions on methods and models drawn from single-element combustor simulations as addressed in this paper are valuable but still insu©cient if not transferable to multiinjector simulations since only the latter can support engine layout and contribute to reduce development time and cost.
Airbus DS£s combustion modeling capabilities for H 2 /O 2 thrust chambers cover the complete range from the two-phase subcritical regime up to trans-and supercritical conditions (single phase).For this purpose, a commercial CFD tool (ANSYS CFX) is adapted and the implemented models are extended for the usage at the extreme boundary conditions of rocket combustors.The second approach taken at Airbus DS is working with in-house CFD tools for the simulation of §ow, combustion, and heat transfer in rocket thrust chambers.One of these in-house tools, Roc §am-II, has been operational for several years now and is a standard tool for design and analysis of rocket thrust chambers at Airbus DS.However, as working with in-house tools needs continuous development, there is a dedicated successor for Roc §am-II, consequently called Roc §am3, which has been applied for the work discussed in this paper.Additional computations with the DLR TAU Code have been performed to supplement the CFX and Roc §am3 investigations with data from alternative combustion and turbulence modeling approaches.The primary intention of the application of diverse numerical tools in the present work is to provide a broad spectrum of data for the assessment of di¨erent modeling methodologies.
In order to achieve high e©ciency and optimized mass relations, the thrust chambers used in modern rocket engines such as Vinci work at elevated pressure levels.At the same time, a very wide temperature range is covered inside the combustion chamber.The complexity of the dominant phenomena inside rocket combustors at these extreme thermodynamic conditions makes the modeling very di©cult.Especially, modeling the mixing and combustion of the propellants determining the integral performance characteristics and the wall heat loading of propulsion systems is a very challenging task.The well-known single element Penn State test case [1,2] provides test data for the validation of simulation tools in this ¦eld of application.In the present work, the three tools CFX, Roc §am3, and TAU are applied to this test case and the results are compared being aware that there are a variety of di¨erences regarding the numerical and physical models implemented in each tool.Therefore, it is not the objective of this work to compare individual model approaches but rather to compare the complete set of models used in each tool.The present investigations show that the Penn State single element combustor test case can be simulated with RANS approaches leading to a very good agreement with the measured wall heat §ux pro¦le.Former publications (see, for example, [3,4]) have promoted the neces-LIQUID AND ELECTRIC SPACE PROPULSION sity of time-accurate simulations for this purpose as they have not been able to achieve good results with steady-state RANS methods.The results documented in this paper show that the referred statements give a wrong impression of what is possible with RANS simulations in the ¦eld of rocket thrust chamber simulations and what is not.This is important since up to now, only RANS methods have been demonstrated to be e©ciently applicable to multiinjector combustor simulations.

TEST CASE DESCRIPTION
The Penn State combustion chamber con¦guration has been set up by NASA Marshall Space Flight Center and the Penn State University [1,2] in order to support the development of CFD tools for rocket combustor simulations with a test case for veri¦cation and validation.Therefore, wall heat §ux and temperature measurements have been performed in a rocket combustor with a single coaxial injector operating on gaseous oxygen (GO 2 ) and hydrogen (GH 2 ).Both propellants enter the main chamber in a preburnt state which corresponds to a staged combustion cycle.The operating point lies at a chamber pressure of p c = 5.42 MPa and an overall mixture ratio of O/F ≈ 6.6 which is a representative load point for rocket combustors.A schematic of this rocket chamber is shown in Fig. 1.
The main chamber has a circular cross section with a diameter of d c = 38.1 mm and a length of l c = 285.75mm.The throat diameter is d t = 8.2 mm.This results in a characteristic chamber length of l * = 6.46 m which is clearly longer than in common H 2 /O 2 rocket combustion chambers (l * < 1 m).While the cylindrical part of the chamber is cooled capacitively, the throat section is cooled actively with cold water §owing through the cooling channels in the chamber wall.Wall temperature and heat §ux measurements have been performed in the cylindrical part using coaxial thermocouples [1] with multiple measurement positions.While the wall temperature pro¦le serves as thermal boundary condition for the CFD simulations, the main target is to compute the correct heat §ux pro¦le.Measured temperatures, mass §ow rates, and mixture ratios are given for the two preburner §ows and serve as inlet boundary conditions for the main chamber.The test case description also provides fuel and oxidizer preburner gas compositions.These quantities are not measured but computed under the assumption of chemical equilibrium.All these values can be found in [2] and are repeated in Table 1.Unfortunately, it has not been possible to reproduce the measured chamber pressure p c = 5.42 MPa with the given in §ow conditions and the given chamber geometry neither in CEA2 [5] nor in the applied CFD tools indicating inconsistencies in the experimental data.This topic will be discussed in section 3.

INCONSISTENCY OF THE TEST DATA
During the evaluation and analysis of the Penn State test case, Airbus DS exposed some inconsistencies in the measured test data provided, for example, in [1] where the measured combustion chamber pressure is given as p c = 5.42 MPa.
With the given measured propellant mass §ow rate ' m and the throat area A t , the characteristic velocity is determined as follows: The theoretical or ideal characteristic velocity can be calculated with the chemical equilibrium tool CEA2 [5].The CEA2 calculation delivers an ideal characteristic velocity of c * ideal = 2041.4m/s for the in §ow conditions given in Table 1 and additionally taking into account the heat losses along the chamber body.Dividing the experimental value by the ideal value results in a combustion e©ciency of η c * ,Test = 112.5%.However, as values greater than 100% may not exist, something must be wrong.There must be some inaccuracies between speci¦ed and measured data.Apart from measurement errors, there are several possibilities to explain this gap.The most probable hypotheses are an incomplete preburner combustion or a geometric contraction of the throat area under thermal load.Furthermore, it is possible that inaccurate mass §ow measurements contribute to this inconsistency.
A contraction of the throat with constant propellant mass §ow rates directly results in a higher combustion chamber pressure.Assuming a realistic combustion e©ciency of 99%, a reduction of the throat diameter by 0.28 mm (3.4%) must have been given during the experiment.However, such a heavy distortion of the cooled nozzle is not realistic.This means the inconsistency cannot be completely explained only by this phenomenon.
An incomplete combustion in the two preburners can be a realistic explanation at least for the major part of this inconsistency.Both preburners run at very extreme mixture ratios meaning that the temperature of the burnt gases is rather low.Propellants which are not burnt in the stoichiometric §ame sheet where the temperatures are very high may also not be burnt further downstream in the post §ame region.This behavior is signi¦cantly di¨erent compared to the processes in the main combustion chamber where the temperatures also in the post §ame region are relatively high and where chemical equilibrium is a very good assumption.In the preburners, the temperatures in the post §ame region are very often below 900 K, i. e., the chemical kinetics are very slow and chemical nonequilibrium actually can exist for H 2 /O 2 .This means that even for extreme mixture ratios, it is possible that both propellants survive unburnt in the preburner.In this case, the ¦nal combustion occurs in the main combustion chamber generating higher heat release and combustion pressure.
To solve a rocket problem in CEA2, the user speci¦es the chamber pressure, the inlet enthalpies, and the mixture ratio.The propellant mass §ow rate is not a direct result from CEA2 but can be determined via the characteristic velocity computed by CEA2.In this case, the discussed inconsistency becomes visible as the computed propellant mass §ow rate is much higher than the given one.However, in CFD simulations, at least when the nozzle part is included, the chamber pressure is a simulation result and not a boundary condition.This is because cold propellant mass §ow rates are injected and the simulated e©ciency of the combustion process determines the resulting chamber pressure.Therefore, the explained inconsistency becomes evident as the computed chamber pressure is far below the measured value.This will be addressed again in the discussion of the simulation results in section 5.

PRESENTATION OF THE APPLIED TOOLS, MODELS, AND SETTINGS
In earlier times, the relevant processes inside rocket combustion chambers, especially the wall heat §ux distribution, were predicted by engineering tools based on one-dimensional (1D) §ow, chemical equilibrium, and Nusselt correlations.Further developments at Airbus DS led to the more sophisticated two-dimensional (2D) spray combustion tool Roc §am-II [6].However, especially in multiinjector chambers, three-dimensional (3D) phenonema are existent and not negligible in certain parts of the combustion chamber.Therefore, high e¨ort has been put in the in-house development of Roc §am3 (3D), which is the designated successor of Roc §am-II (2D), as well as in the adaptation of the commercial 3D-CFD solver CFX for the usage under these extreme thermodynamic conditions.Both Airbus DS CFD tools, i. e., CFX and Roc §am3, are presented in more detail (e. g., applied combustion models, mixing rules, etc.) together with the DLR TAU code in the following subsections.

ANSYS CFX (Including Modi¦cations Made by Airbus DS)
Especially for the rocket applications of the commercial CFD tool CFX, the main problem is that this solver is developed for a broad range of applications (e. g., turbomachinery) and not focused on rocket propulsion.The adaption process of CFX and the ¦rst validation studies have been shown in [7].Two di¨erent combustion models have been implemented and tested in CFX: the chemical equilibrium and the Flamelet approach.The equilibrium-based combustion model is not a CFX standard combustion model.It was developed and implemented via user-de¦ned functions into CFX by Airbus DS.The Flamelet model is a more sophisticated model because it ensures accurate chemistry kinetics and accounts for nonequilibrium e¨ects at low computational cost, because no species transport equations are solved but a mixture fraction equation.The in §uence of turbulence on combustion is treated by a presumed PDF (Beta-and clipped Gauss-pPDF) approach, where the main in §uencing factors for the PDF shape are the mixture fraction and its variance.The standard turbulence model applied in CFX for combustor simulation is the Menter shear stress transport (SST) model [8,9].The CFX simulation shown in this work applied the Flamelet combustion model and the SST model for turbulence [7,9] with a turbulent Lewis number unity approach (Le t = 1) and Pr t = Sc t = 0.85.
For the simulations with CFX, the so-called dense gas model is implemented in trans-and supercritical conditions.The dense oxygen jet is completely described within the Euler equations and appropriate material property data bases are essential for accurate modeling of the mixing of the cryogenic and dense oxygen with the surrounding hot gases.The §uid properties in CFX are extended with real gas data and sophisticated mixing rules for viscosity and heat conductivity are implemented also via user-de¦ned functions.For viscosity and thermal conductivity, the Brokaw and Wilke approaches are used [10].By this approach, it is ensured that the mixing and, also, the subsequent combustion process at much higher temperatures are covered properly by the §uid data base.Moreover, validation studies revealed that the mixing laws are important in order to get accurate results.Especially for a correct simulation of the wall heat transfer, a proper modeling of the gas mixture properties is essential.
The computational domain for the Airbus DS CFD simulations performed with CFX and Roc §am3 starts at the beginning of the main chamber.The grid is shown in Fig. 2. The inner parts of the injector as well as the recess are not resolved.However, the post-tip between oxidizer and fuel inlet is resolved by the grid.Most authors who have published on this test case have started their simulations further upstream thus resolving the inner part of the injection element.This has also been done with CFX in an alternative approach.The results will be shown in subsection 6.2.As CFX is a 3D solver, the grid is a segment of one degree with one grid cell used for resolving the circumferential direction.This means that all simulations presented here are axisymmetric.The grid shown in Fig. 2 consists of 91,000 grid cells and is a result of an intensive grid study.A ¦ne wall resolution ensures y + < 1 for the whole combustion chamber.Mass §ows and temperatures are used as boundary conditions for the inlets of fuel and oxidizer, i. e., the combustor pressure is a simulation result.All walls are treated as smooth no-slip walls with prescribed temperatures as thermal boundary condition.The wall temperatures of the post-tip area between the fuel and oxidizer inlets as well as the faceplate are set to the constant value of T w = 755 K.This value is not measured in the experiment but seems realistic and has been used by many other authors (see, for example, [11]).The measured wall temperature pro¦le is used as boundary condition for the chamber wall.Linear interpolation is applied to compute values between the measuring points.Downstream of the last measuring point, a constant wall temperature of T w = 511 K is prescribed.A supersonic outlet boundary condition is used in the outlet plane.

Roc §am3
The Airbus DS in-house tool Roc §am3 is a ¦nite-volume solver for nonorthogonal, boundary-¦tted and block-structured grids that applies the SIMPLE (semiimplicit method for pressure-linked equations) algorithm to solve the compressible RANS equations in three spatial dimensions.Roc §am3 features di¨erent kε and kω turbulence models and an equilibrium-based pPDF-chemistry model which takes into account e¨ects of turbulent combustion as well as real gas effects.The simulation of the Penn State test case shall contribute to the validation process of Roc §am3 which is still under development.For the present simulations, Roc §am3 uses an equilibrium-based pPDF-combustion model and the LaunderSharma turbulence model [12] with Pr t = 0.9 and Sc t = 0.6 resulting in a turbulent Lewis number of Le t = 0.67.Furthermore, the Durbin realizability constraint [13] and the Yap correction [14] are implemented in order to enhance modeling of the turbulent heat transfer.
Regarding the numerical mesh and the prescribed boundary conditions, the Roc §am3 simulations use values and settings which are identical to the ones used in the presented CFX simulations.

DLR TAU Code
The TAU code is a ¦nite-volume §ow solver for the Euler and NavierStokes equations including a wide range of turbulence modeling capabilities [15].The present investigation is based on steady-state RANS computations using the SpalartAllmaras [16] and Menter SST [8] eddy viscosity models.The AUSMDV §ux vector splitting scheme was applied together with MUSCL (monotonic upwind scheme for conservation laws) gradient reconstruction to achieve secondorder spatial accuracy [17].Combustion was modeled using a detailed-chemistry and a Flamelet approach.
The detailed-chemistry combustion model in the DLR Tau code is an extension of the models for chemical and thermal nonequilibrium §ows in high enthalpy (re-)entry aerothermodynamics.The §ow is considered to be a reacting mixture of thermally perfect gases.A transport equation is solved for each individual species.The chemical source term in this set of transport equations is computed from the law of mass action.The forward and backward reaction rates are computed using the modi¦ed Arrhenius law and equilibrium constants.Turbulencechemistry interactions are modeled using an assumed probability density function (PDF) approach [18] which considers local §uctuation of temperature and species concentrations.Two sets of reaction mechanisms for hydrogen combustion were considered: a 19-step [18] and a reduced 7-step [19] mechanism based on the Jachimowski scheme.A detailed model description and performance assessment for steady and unsteady combustion problems is given in [20,21].
The applied compressible Flamelet model for nonpremixed combustion [22] is a conserved scalar method.The §ow ¦eld is assumed to consist of fuel and oxidizer at an arbitrary state of mixing which is characterized by a nonreactive scalar, the mixture fraction.Two additional transport equations for the mixture fraction and its variance are solved and the evaluation of the local chemical composition is performed on the basis of a precomputed lookup-table (Flamelet library).The §ame properties and the species composition inside the §ame zone are governed by the thermodynamic state and composition of the fuel and oxidizer stream and the scalar dissipation rate of the mixture fraction which is related to the variance of the mixture fraction and the speci¦c turbulent dissipation rate.The Flamelet library was computed with the FlameMaster package [23] and was based on a 19-step hydrogen combustion mechanism described by Pitsch [24].
The thermodynamic properties (energy, entropy, and speci¦c heat) are calculated using the partition functions for each individual species in the reacting gas mixture.Knowing the mixture composition and the thermodynamic state of the individual species, the properties of the reacting gas mixture are computed using suitable mixture rules [20].The species di¨usion §uxes are modeled using Fick£s law applying an averaged di¨usion coe©cient for all species This approximate di¨usion coe©cient is computed using the laminar and eddy viscosities and constant Schmidt numbers of Sc = Sc t = 0.6.The turbulent heat conductivity was computed assuming a constant ratio of laminar to turbulent Prandtl numbers of 0.85.shown in Fig. 2 was also used for the 2D axisymmetric TAU simulations.A small section of the coaxial injector was added at the in §ow boundary (visible in Fig. 3).The computational plane was meshed with a hybrid grid.The near-wall regions were discretized with structured quadrilateral sublayers.

The computational domain
The wallnormal size of the wall-adjacent cell ranging between 1 and 0.5 µm was chosen to ensure a nondimensional wall distance of y + < 1 for the ¦rst cell (as needed by the applied low-Reynolds turbulence models) and a su©cient resolution of the temperature gradients close to the wall.The inner §ow ¦eld was discretized with an unstructured triangular grid with an embedded structured block for better resolution of the shear layer and §ame zone.A detail of the applied grid (medium resolution) in the vicinity of the injector is shown in Fig. 3).
The combustor walls were treated using a noncatalytic, no-slip boundary condition with a prescribed temperature pro¦le taken from the experimental test case description [2].The in §ow at the coaxial injector was modeled using a mass §ow boundary condition with prescribed mass §ow rates, species concentrations, and in §ow temperatures according to Table 1.

EVALUATION AND COMPARISON OF THE FINAL SIMULATION RESULTS
This section shows the ¦nal simulation results obtained with CFX, Roc §am3, and TAU.Extensive parameter studies have been conducted with each tool in order to develop the ¦nal settings.In this process, three di¨erent model packages have been designed to perform the same task.The modeling approaches are not identical for the three tools.Therefore, as it can be seen in Fig. 4, there are, of course, di¨erences between the heat §ux pro¦les from the three tools but they all correspond very well to the measured heat §ux data.Figure 5 shows the temperature ¦elds computed with the three tools and reveals strong di¨erences especially between CFX and the two other codes which may be attributed to the in §uence of the chemistry model- Wall heat §ux distribution for the three applied tools (1 ¡ TAU; 2 ¡ CFX Flamelet; and 3 ¡ Roc §am3); 4 ¡ test data ing.The CFX Flamelet solution shows in general lower temperatures than the equilibrium solution of Roc §am3 or the ¦nite rate solution of TAU.Flame quenching processes which reduce the temperature are improbable at these locations behind the §ame, because the turbulence intensity is rather small there.This di¨erence in the thermal ¦eld is under investigation.
The combustion chamber pressure is a result of the present simulations, i. e., there is no pressure boundary condition at the inlet or outlet.Numerical simulations prescribing the chamber pressure as inlet boundary conditions as often found in the literature are set up in an inconsistent manner.The simulations performed at Airbus DS with CFX and Roc §am3 compute much lower combustion chamber pressures (p c ≈ 5.0 MPa) than the measured pressure (p c ≈ 5.4 MPa).TAU computes even a slightly lower combustion chamber pressure than the Airbus DS simulations of about 4.9 MPa.Due to that, the combustion e©ciencies of the simulations are much lower than η c * ,Test computed from CEA2 and the experimental data.The simulations use the measured mass §ow rates, inlet temperatures, and the species compositions from a complete combustion process in the preburner as boundary conditions.The composition of the complete preburner combustion is given in Table 1 and in [5] but as discussed in section 3 of this paper, the complete combustion in the preburners is questionable.However, all simulations performed so far are based on the assumption of complete combustion in both preburners.Considering incomplete combustion would result in a greater heat release inside the combustion chamber and would, thereby, increase the chamber pressure.

PARAMETER STUDIES CONDUCTED DURING THE DEVELOPMENT OF THE TOOL SETTINGS
In order to develop the settings for the simulations presented in the previous section, a number of parameter studies have been conducted with each of the applied tools.This section provides an insight into some of these studies.Due to the fact that the Penn State combustor is a single element con¦guration and the characteristic length is very high, the outer recirculation zones (see e. g., Figs. 9 and 14) are much more intensive than in ordinary multiinjector con-¦gurations which usually possess much smaller distances between injector and chamber wall.The correct simulation of the very intensive recirculation zones, i. e., also the back-transport of the hot gases towards the face plate, is an essential feature which must be captured very accurately by the CFD tool in order to reproduce the measured data like the wall heat §ux correctly.This section shows some of the parameter and sensitivity analyses on turbulent Prandtl and Schmidt numbers (Roc §am3), on grid convergence, turbulence and combustion modeling (TAU), and, ¦nally, on the in §uence of the detailed numerical resolution of the inner injector §ow (CFX).

Investigation of the In §uence of Di¨erent Turbulent Prandtl and Schmidt Numbers in Roc §am3
The turbulent Prandtl and Schmidt numbers are inherent in RANS modeling (also, in unsteady RANS (URANS) and large-eddy simulation (LES)) and govern turbulent mixing and heat transfer.Thereby, the turbulent Prandtl number controls the wall heat §ux process via the turbulent thermal conductivity and the turbulent Schmidt number con-Figure 6 Wall heat §ux distribution com- puted with Roc §am3; in §uence of a variation of the turbulent Prandtl number (Sct = 0.6): 1 ¡ Prt = 0.6; 2 ¡ 0.7; 3 ¡ 0.8; 4 ¡ 0.9; 5 ¡ 1.0; and 6 ¡ Prt = 1.1.Signs refer to experiment trols the mixing process via turbulent di¨usion.Since there are no de¦nite ¦gures for both quantities, they have to be stipulated by each model package individually.Therefore, it is appropriate to check their sensitivity range ¦rst.In Roc §am3, both parameters can be adjusted independently from each other.Figure 6 shows the wall heat §ux pro¦les of Roc §am3 results for various turbulent Prandtl numbers.
As it is well known, the choice of the turbulent Prandtl number has a strong in §uence on the wall heat §ux.By reducing the turbulent Prandtl number, the wall heat §ux increases mainly because of the increase of the e¨ective thermal conductivity.In the Roc §am3 results shown in Fig. 6, the turbulent Schmidt number is always kept constant at Sc t = 0.6.The shape of the heat §ux distribution does not change in principle by varying the turbulent Prandtl number.This means that the axial position of the maximum wall heat §ux is constant for all turbulent Prandtl numbers while there is only a shift to higher wall heat §ux by reducing the turbulent Prandtl number.No in §uence on the aerodynamic ¦eld (streamlines) is visible when changing the turbulent Prandtl number, but the temperature ¦eld is slightly changing as it is shown in Fig. 7.
A higher turbulent Prandtl number leads to a broader and slightly more intensive high temperature region in the cylindrical part of the chamber, especially close to the walls.This is due to the fact that more heat is transferred from the §uid into the chamber wall.
Apart from the pure in §uence of the turbulent Prandtl number, also, the turbulent Schmidt number is very important.A separate variation of the turbulent Schmidt number has, therefore, been performed with Roc §am3. Figure 8 displays the wall heat §ux distributions for di¨erent turbulent Schmidt numbers.Thereby, the turbulent Prandtl number is kept constant at 0.9.
A strong dependency of the wall heat §ux on the turbulent Schmidt number is visible, but di¨erent to the turbulent Prandtl number: the turbulent Schmidt number mainly a¨ects the shape of the heat §ux pro¦le.This means that the axial position of the maximum wall heat §ux moves downstream by increasing the turbulent Schmidt number.As a consequence of this, it is clear that the  turbulent Schmidt number has not only an e¨ect on the wall heat §ux, it also in §uences the aerodynamic §ow ¦eld, the mixing process, and due to this, also the combustion process.This e¨ect is visualized in Fig. 9.
In Fig. 9, one can see that for higher turbulent Schmidt numbers, the outer recirculation zone expands.The hot gas stagnation point of the outer recirculation zone on the combustor wall is located further downstream for the higher turbulent Schmidt number (indicated by the blue line).By reducing the turbulent Schmidt number, the turbulent di¨usivity is enhanced.This means that the mixing and combustion processes are also intensi¦ed.One can also see that the cold injection jet is a bit shorter for the lower turbulent Schmidt number (Sc t = 0.6) which is a direct consequence of the enhanced mixing by turbulent di¨usivity.As a secondary e¨ect of this, a faster combustion occurs for the lower turbulent Schmidt numbers.Such global in §uence is not observed for the turbu-

PROGRESS IN PROPULSION PHYSICS
lent Prandtl number where the in §uence is limited only to the energy transport which, however, is essential for the wall heat §ux.
From these individual parameter variations, one could conclude that a tool package which assumes Le t = 1, would predict the heat §ux pro¦le also fairly well with Pr t = Sc t = 0.85 since the reduction of the turbulent Prandtl number to 0.85 could be compensated by an increase of the turbulent Schmidt number to 0.85.At this place, it is recalled that the latter is the chosen setting for CFX.

Grid Convergence and Variations of Turbulence and Combustion Modeling (TAU-Code)
Di¨erent grid resolutions with a total number of points of 25,000 (¤coarse¥), 50,000 (¤standard¥), and 80,000 (¤¦ne¥) were tested using the SpalartAllmaras turbulence model in conjunction with the detailed chemistry combustion model (7-step reaction mechanism, neglect of turbulencechemistry interaction).The grid topology is shown in Fig. 3 and described in section 4. The results in Fig. 10 show good agreement of the surface heat §ux distributions for the di¨erent grid densities.The ¤standard¥ grid density was used for all further investigations.The heat §ux level of the SpalartAllmaras results (used for the initial grid convergence study) in Fig. 10 is signi¦cantly too low and position of the heat §ux peak is too far upstream.Due to these observed de¦ciencies, further parametric studies were carried out.Exemplary results from the application of a Menter SST turbulence model and di¨erent combustion models as described in section 4 are shown in Fig. 11.The main conclusions that can be drawn from the results of Figs. 10 and 11 are that the turbulence model is of primary importance for the prediction of the wall heat §ux and that the in §uence of the applied combustion model is comparatively small.The application of the Menter SST model without realizability limiter led to a qualitatively wrong prediction of the surface heat §ux distribution including two distinct peaks.This is due to a large sensitivity of the structure of the recirculation zone in the upstream part of the combustor to the turbulence model.The di¨erence between the 7-step and 19-step reaction mechanisms is negligible and the in §uence of the assumed-PDF modeling of the turbulencechemistry interaction is small.This is consistent with previous numerical results for this test case [11].The largest deviation is observed to occur between the detailed-chemistry and the Flamelet results.However, this deviation is small compared to the in §uence of the applied turbulence model.
All results obtained with the SpalartAllmaras and Menter SST turbulence models do not reproduce the available experimental data for the surface heat §ux distribution.Further investigations were carried out and a dramatic improvement of the computational result for the heat §ux was observed by introducing the Durbin realizability constraint [13] to the Menter SST model.The resulting heat §ux distribution is shown in Fig. 4 in section 5.The applied combustion model for this case was the 7-step laminar chemistry scheme.Both, the location of the heat §ux peak and the total heat §ux level, are well reproduced by this numerical model setup.

Simulations with Resolved Injector in ANSYS CFX
All CFX simulations shown up to now in this paper do not resolve the inner injection §ow.In these simulations, the propellant in §ow started at the injection plane (plane of the faceplate, see, e. g., Fig. 2) of the combustion chamber which is a simpli¦cation often done in CFD of combustion chambers.This results, of course, in block pro¦les for the velocity at the inlet.Resolving the inner injector §ow, the velocity inlet pro¦les change towards parabolic pro¦les.Especially for certain parametric studies, it is usually too complex and time-consuming to resolve for every simulation the inner injector aerodynamics.Would the resolution, however, be necessary, it had to be transferred to multiinjector con¦gurations too; otherwise, the conclusion drawn here is not conclusive for a design tool.But the in §uence of this inner injector §ow has then to be investigated in order to justify correctly such simpli¦cations.In this subsection, the inner injector §ow is analyzed in detail and the in §uence on the processes and on the results inside the combustion chamber is evaluated in order to legitimate this simpli¦cation.Figure 12 shows the numerical mesh with the resolution of the injection element.
The numerical mesh which resolves the injection element is, again like the one without injection element, 2D (axially symmetric), and has in total 79,000 cells.Hereby, 16,000 cells (20%) are used for the resolution of the inner §ow of the injection element.As it can be seen in Fig. 12, the post-tip and the recess of the injection element are highly resolved with 52 cells in radial direction.This cell number is enough to resolve the vortex system which exists behind the post-tip and which is shown later in this paper.Figure 13 compares the wall heat §ux distribution for simulations of CFX with and without the resolution of the inner injector §ow.
The CFX simulation uses the Flamelet combustion model with a turbulent Prandtl number of Pr t = 0.8, i. e., slightly lower than the number used for Fig. 4 result.In Fig. 13, one can see that there is a di¨erence when resolving the inner injector §ow compared to the solution without injector.But the di¨erence is not very pronounced for the CFX solution.Especially further downstream, the results with and without resolution of injector are very similar.More important deviations are visible in the region where the wall heat §ux has its maximum values, around the axial position of X = 0.07 m.But here also the di¨erences are not very pronounced.CFX shows here a deviation of around 7% in the maximum value when resolving the inner injector §ow.CFX shows that in the ¦rst part of the combustion chamber, the wall heat §ux is lower when resolving the inner injector §ow.The highest di¨erence is exactly at the position with the maximum wall heat §ux which is close to the reattachment point.But further downstream, this picture changes.Here, the wall heat §ux by resolving the inner injector §ow is slightly higher.The comparison to the test data is, in general, very satisfying for all simulations.A more qualitative evaluation of the in §uence of the inner injector §ow is shown in Fig. 14.Here, the temperature contour ¦eld is shown with overlaid §ow streamlines.The top part of Fig. 14 shows the solution with resolving the inner injector §ow and the bottom part without.The aerodynamics inside the combustion chamber are relatively similar, but one di¨erence is obvious.The reattachment or stagnation point of the outer recirculation zone, indicated by the red line, is a bit further downstream in the case with resolution of the inner injector §ow.As it was shown in Fig. 13, the maximum wall heat §ux with resolution of the inner injector §ow is slightly further downstream than without the inner injector §ow.This is in-line with the results of Fig. 14 where the position of the stagnation point is shown.
Also, for the evaluation of the mixing process, the detailed §ow ¦eld directly behind the post-tip is essential.The resolution of the inner injector §ow has a contribution, but the dominant factor is the §ow §ied and the turbulence production in the region downstream of the post-tip.Figure 15 shows the aerodynamic ¦eld at the post-tip region with the §ame anchoring at the tip for both CFX results.
The CFX solution shows very good and stable convergence for the case shown in Fig. 15.The vortex system directly behind the liquid oxygen (LOx) tip is resolved well and the turbulence production by this vortex system is essential for the mixing behavior and due to that also for the combustion process further downstream in the reactive shear layer.The combustion intensity in the shear layer determines the wall heat §ux but obviously, the main features can be reproduced also not by resolving the details of the inner injector §ow as long as the aerodynamics behind the LOx tip are simulated accurately.

CONCLUDING REMARKS
This paper presents the simulation results for the Penn State test case obtained with three di¨erent CFD tools: the Airbus DS in-house solver Roc §am3, the commercial solver CFX adapted for rocket applications by Airbus DS, and the DLR code TAU.All three tools are able to compute the wall heat §ux pro¦le achieving good agreement with the available test data.
The axial position of the heat §ux peak is well captured.An overestimation of the wall heat §ux simulated by TAU of about 20% occurs in the vicinity of the heat §ux peak and of about 10% in the downstream section of the combustion chamber.A large in §uence of the applied turbulence model is observed.In particular, the application of the Durbin realizability limiter to the Menter SST model leads to a signi¦cantly improved prediction of the surface heat §ux distribution.
From the conducted investigations, it can be concluded that the choice of the turbulence model has major impact on the solution just as the selection of the chemistry model and the choice of the turbulent transport characteristics Pr t and Sc t for which no speci¦c numbers exist.This is not surprising since the huge outer recirculation zone triggered by the rocket thrust chamber untypically large contraction ratio of the combustor governs the §ow ¦eld evolution.This is also a reason why this test case is very challenging.Converged solutions are obtained with CFX using the SST with and without resolution of the inner injector §ow.The investigation discussed in subsection 6.3 reveals that it is not necessary to resolve the inner injector §ow as long as the aerodynamics behind the LOx tip and the processes inside the reactive shear layer are resolved accurately.This means that a lot of computation time can be saved by neglecting the inner in-jector §ow.The correct resolution of the vortex system behind the LOx tip is essential because this determines the turbulence production in the shear layer which is the dominant process driving the mixing and combustion in the reactive shear layer.The main outcome is that these dominant parameters in the reactive shear layer can also be captured without detailed resolution of the inner injector §ow but adequate resolution of the LOx-post and the reactive shear layer.
Moreover, there are still some uncertainties which prohibit the full understanding of the test case.The test data show some inconsistencies, i. e., the measured mass §ow rates do not ¦t to the measured combustion pressure.An incomplete combustion in the preburners could be one possible contribution to this mismatch.However, there may also be other reasons that cannot be quanti¦ed.
Finally, the results shown in this paper clearly justify the application of RANS models for this test case.This means that it is not necessarily essential to apply more sophisticated model approaches like URANS or LES, as it is stated in other publications, e. g., in [3] or [4].This is crucial for all tool developers designing a tool for multiinjector combustor layout and optimization work.

Figure 1
Figure 1 Schematic of the Penn State single element combustor with speci¦ed pro- pellant inlet conditions [1].Dimensions are in millimeters

Figure 2
Figure 2 Computational grid for the CFX and Roc §am3 simulations

Figure 3
Figure 3 Computational grid (TAU, standard grid density) in the vicinity of the injector

Figure 12 Figure 13
Figure 12 Numerical mesh with resolution of the injection element

Figure 14
Figure 14 Temperature ¦eld (CFX-solution) with (upper frame) and without (bot- tom frame) resolution of injection element

Figure 15
Figure 15 Vortex system behind the post-tip with (a) and without (b) resolution of inner injector §ow.Combustion model: CFX-Flamelet; turbulence model: CFX-SST

Table 1
[1]ector and main chamber quantities: measured values and calculated exhaust compositions of the preburners as speci¦ed in[1]