Large-Eddy Simulation of Supercritical Mixing Layers

Transcription

1 Sonderforschungsbereich/Transregio 4 Annual Report Large-Eddy Simulation of Supercritical Mixing Layers By C.A. Niedermeier, M. Jarczyk, S. Hickel, N.A. Adams AND M. Pfitzner Institute of Aerodynamics and Fluid Mechanics, Technische Universität München Boltzmannstr. 1, 8748 Garching bei München Within the development of a reliable numerical tool for the simulation of a whole rocket combustion chamber, real gas thermodynamics have been implemented into two CFD codes, the in-house code INCA of the Institute of Aerodynamics and Fluid Mechanics at the Technische Universität München and OpenFOAM, used by the Institute for Thermodynamics at the Universität der Bundeswehr München. The present work is part of the validation process, where Large Eddy Simulations (LES) with different subgridscale (SGS) models are conducted with these codes. Predictions of the implicit LES method ALDM and two explicit one equation models dynamic and non-dynamic are compared with results for Direct Numerical Simulations (DNS) of a non-reacting, transitional, temporal mixing layer of counterflowing oxygen and hydrogen. All models are able to reproduce the DNS data very well. A future task will be the validation of the SGS models for real gas jet flows and subsequent simulations of the mixing and combustion process of coaxial injectors. 1. Introduction The technology of cryogenic rocket combustion engines has been used successfully for many years and is well known today. Demand for increasing rocket performance and reliability is challenging in particular if restricted budgets and shortened development cycles have to be considered. For this reason, the application of computational methods in the development process increases steadily and therefore also the demand for computational fluid dynamics (CFD) tools that are able to simulate the flow at rocket combustor conditions. The pressure in modern combustion chambers often exceeds MPa, causing many propellants, like oxygen and hydrogen, to become supercritical at injection. In this high pressure environment, molecular interactions significantly affect the fluid properties. Hence, a real gas equation of state and suitable relations for the transport properties have to be used for the numerical simulation. We apply the cubic Peng-Robinson (PR) [1] equation of state (EOS) combined with an empirical volume correction method to obtain the thermodynamic behaviour of the real gas [2]. The ultimate goal of our work is to analyze the propellant injection into a rocket combustion chamber through multiple coaxial injectors and the subsequent combustion process. As this process is strongly three-dimensional and unsteady, Large Eddy Simulation (LES) is the most suitable method. Institute for Thermodynamics, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 877 Neubiberg

2 136 C.A. Niedermeier, M. Jarczyk, S. Hickel, N.A. Adams & M. Pfitzner As a first step towards this goal, a non-reacting, transitional, temporal mixing layer of hydrogen and oxygen studied by Okong o et al. [3] is used in the present work as test case for validating the two CFD codes. The Institute for Thermodynamics of the Universität der Bundeswehr München uses the multi-component solver of the Open- FOAM package with an explicit one equation model (dynamic (DOEM) and non-dynamic (OEM)) as subgrid-scale (SGS) model, whereas the simulations carried out at the Institute of Aerodynamics and Fluid Mechanics at the Technische Universität München (TUM) are conducted by means of the in-house CFD code INCA using an implicit LES (ILES) method, namely the Adaptive Local Deconvolution Method (ALDM). This method gives very good results for compressible flows of ideal gases [4, ] and its usage is now extended to real gas flows. Additionally, Direct Numerical Simulations (DNS) are conducted with this code to have an own reference database for comparison with the LES results. The paper is organized as follows: In Sec. 2, the underlying conservation equations which are solved by our codes are briefly depicted. Secs. 3 and 4 describe the SGS and thermodynamic modeling approaches, while in Secs. and 6 the transport coefficients and the boundary and initial conditions of the simulations are given. The results consisting of flow visualizations of different variables at the transitional time are shown in Sec. 7. In Sec. 8, we draw conclusions from our results and give an outlook to future work. 2. Conservation Equations The general conservations equations for the H 2 /O 2 binary mixture under consideration are given by Okong o et al. [3] as ρe t t (ρu i ) t ρy O t ρ t + (ρu j) =, (2.1) + (ρu iu j +pδ ij ) + (ρy Ou j ) + [(ρe t +p)u j ] = τ ij, (2.2) = J Oj, (2.3) = q IKj + τ iju i. (2.4) Here x j, j {1,2,3} is a cartesian coordinate, t is the time, ρ is the density, u i is the velocity component in direction i {1,2,3}, e t = e + u i u i /2 is the total energy, p is the thermodynamic pressure and and Y O is the oxygen mass fraction. q IKj is the Irwing-Kirkwood (IK) form of the heat flux vector [6], J Oj is the oxygen mass flux in direction j and τ ij is the Newtonian viscous stress tensor. A detailed description of the flux calculation can be found in [3] and [7]. This flux formulation includes the Dufour and Soret effect as well as diffusion due to pressure gradients. In the investigated test cases, this very detailed diffusion modeling is needed in the DNS for the reproduction of the flow field to be as accurate as possible. The CFD codes follow two different approaches. INCA is a density-based finite volume method that directly solves Eqs. (2.1)-(2.4), whereas the OpenFOAM solver uses a pressure based PISO (Pressure Implicit with Splitting of Operators) algorithm proposed by Issa et al. [8, 9].

3 LES of Supercritical Mixing Layers 137 The PISO algorithm for compressible flows used in OpenFOAM is an extension of the incompressible approach. As in the incompressible limit the continuity equation becomes an auxiliary condition more than a conservation equation, it is no longer reasonable to solve the system of equations above. Therefore, pressure based algorithms have been developed, where a conservation equation for the pressure (developed from the discretized momentum and continuity equation) is solved instead of Eq. (2.1). A special issue here is the nonlinearity of the equation of state needed in high pressure environments, which has to be accounted for in the solution approach. Energy conservation in OpenFOAM is realized by solving the enthalpy equation (ρh) t + (ρhu j) = q IKj + Dp Dt with h = e t + p ρ (2.) instead of the total energy equation (2.4). The viscous heating term τ ij u i / is neglected here as it is insignificant in the subsonic flows studied in this work and known to negatively affect stability in pressure based solution approaches. 3. Turbulence Modeling and Numerics As an intermediate approach between Reynolds-averaged Navier-Stokes (RANS) and DNS, LES resolves the large scales of turbulence while the small scales below the grid width have to be modeled. In this paper, we apply two distinct modeling approaches: Implicit LES (ILES) with the Adaptive Local Deconvolution Method (ALDM) and explicit LES with a one-equation model in dynamic (DOEM) and non-dynamic form (OEM). The basic idea of ILES is to directly use the truncation error of the unmodified conservation equations (2.1)-(2.4). ALDM incorporates free parameters in the discretization scheme which can be used to control the truncation error. A physically motivated implicit SGS model that is consistent with turbulence theory is obtained through parameter calibration, see Ref. [4]. ALDM is implemented in INCA for Cartesian collocated grids and used to discretize the convective terms of the Navier-Stokes equations (see Ref. [] for a detailed description). The diffusive terms are discretized by 2 nd order centered differences and a 3 rd order explicit Runge-Kutta method is used for time integration. For the DNS conducted with INCA in the present work, the convective terms are discretized by 6 th order centered differences with all other properties of the computational framework being the same as for the ILES. For explicit LES, a compressible version of the OEM and DOEM is used in OpenFOAM []. An additional transport equation for the turbulent kinetic energy is solved in this model, which is then used to calculate the eddy viscosity. With OpenFOAM, 2 nd order centered differences are applied for spatial discretization and a 2 nd order backward differencing scheme is used for time integration. 4. Thermodynamic Modeling To account for real gas effects, all thermodynamic properties are calculated as the sum of an ideal reference value and a departure function that accounts for real gas effects. For closing the system of conservation equations, enthalpy h and constantpressure specific heat c p must be provided in OpenFOAM, while internal energy e and

4 138 C.A. Niedermeier, M. Jarczyk, S. Hickel, N.A. Adams & M. Pfitzner constant-volume specific heat c v must be provided in INCA. These are defined as p ( ( ) ) Vm h(t,p) = h (T)+ V m T dp, (4.1) T p p e(t,v) = e (T)+ V V ( p T T c p (T,p) = c v (T,p) c v (T,V) = ( e T ( ) ) p dv, (4.2) T V ( ) 2 p T ( ) V p V ) V T, (4.3). (4.4) Here the subscript refers to the ideal reference state at low pressure. In the present work, the departure functions on the right hand side are determined by the Peng- Robinson (PR) EOS [1] p = RT V b a(t) V 2 +2Vb b 2. (4.) V is the molar volume and R is the universal gas constant with a value of R = J/molK. The constants a(t) and b are calculated from empirical relations. a(t) accounts for attractive forces between the molecules in the fluid and is calculated from the empirical equation ( ( a(t) =.4723 R2 T 2 c p c 1 κ )) 2 T 1, (4.6) where κ = ω.26992ω 2 is a function of the acentric factor ω. The effects of the reduction of free volume by the particular volume of the molecules are taken into account via b =.77796RT c /p c. T c and p c are the critical temperatures and pressures of the modeled species (H 2 : 33.1 K / 1.3 MPa, O 2 : 14.6 K /.4 MPa). For multi-component mixtures, the extended Corresponding States Principle is applied, where a mixture of a fixed composition is assumed to behave like a pure fluid. The mixture properties are also calculated using the PR equation of state with parameters a and b calculated from real gas mixing rules a = x i x j a ij, (4.7) i j T c b = i x i b i. (4.8) Here a i and b i are the pure component parameters, and a ij is calculated from a ij = ai a j (1 k ij ) with the binary interaction coefficient k ij, which is set to zero for the investigations presented in this work. As the Peng-Robinson equation of state is known to be not very accurate in predicting the density in transcritical regions, an empirical correction method established by Harstad et al. [2] is chosen for the final implementation.

5 LES of Supercritical Mixing Layers 139 FIGURE 1. Mixing layer configuration: fluid 1 = oxygen, fluid 2 = hydrogen.. Transport Coefficients The transport coefficients are calculated with empirical correlations given by Okong o et al. [3] as µ = µ R (T/[.(T 1 +T 2 )]).7, (.1) Sc = µ/(ρα D D) = ( Y O.186YO 2.268YO 6 ) [ 1+(88.6/T) 1.], (.2) Pr = µc p /(mλ) = 1.33/T.1. (.3) The Reynolds number Re of the mixing layer is adjusted using the reference viscosity µ R, which can be calculated from Re =.(ρ 1 +ρ 2 ) U δ ω, µ R. (.4) The investigations presented in this work are carried out at Re = 7, which is sufficient for the mixing layer s transition [3]. ρ 1 and ρ 2 are the densities of the two different layers of the shear flow, U is the initial velocity difference across the layer ( U = U 1 U 2 ) and δ ω, is the initial vorticity thickness (δ ω, = 6.89 mm). 6. Initial and boundary conditions We chose the configuration of a temporal oxygen/hydrogen mixing layer investigated by Okong o et al. [3] as a validation test case. The domain is periodic in x 1 - and - direction and of outflow type in the -direction. The layer is not symmetric in as the layer growth is expected to be significantly larger on the hydrogen side. Table 1 summarizes the the mixing layer s mean flow properties. The free stream densities (ρ 1 and ρ 2 ) are calculated at the free stream temperatures (T 1 and T 2 ) and the initial uniform pressure p using the specified EOS. The initial mean profiles for streamwise velocity, temperature and species mass fractions are prescribed using an error function profile erf( π ). To initiate the layer roll up, vorticity distur-

6 14 C.A. Niedermeier, M. Jarczyk, S. Hickel, N.A. Adams & M. Pfitzner Mean quantity = (O 2) = (H 2) ū 1, m/s ρ, kg/m p, atm T, K 4 6 Y O 1 TABLE 1. Mixing layer mean flow properties. bances are superimposed on the mean velocity profile. Their wavelength corresponds to the most unstable incompressible mode [11]. These streamwise and spanwise disturbances are specified as ω 1 (, ) = F 3D λ 1 U Γ 1 f 2 ( )f 3 ( ), (6.1) λ 3 U ω 3 (x 1, ) = F 2D f 1 (x 1 )f 2 ( ). (6.2) Γ 3 The functions f 1, f 2 and f 3 are are given by 2 ( ) ( πx1 πx1 f 1 (x 1 ) = A i sin 2 i +A 3 sin π ), (6.3) λ 1 8λ 1 2 i= f 2 ( ) = exp [ λ 3 π ( x2 δ ω, ) 2 ], (6.4) ( ) ( ) 2πx3 πx3 f 3 ( ) = B sin +B 1 sin. (6.) The relative amplitudes of the vorticity disturbances are chosen asf 2D =.1 andf 3D =.. The coefficients A i and B i have the values A = 1, A 1 =., A 2 = A 3 =.3, B = 1 and B 1 =.2. The perturbation wave length is λ 1 = 7.29δ ω, and λ 3 =.6λ 1, corresponding to the domain size which is chosen to cover four wavelengths each in streamwise and spanwise direction. The circulations Γ 1 and Γ 3 are calculated by integratingf 2 f 3 andf 1 f 2 over the respective planes and dividing by twice the number of wavelengths in x 1 and once the number of wavelengths in, respectively. The initial velocity fluctuations can then finally be determined by solving the corresponding Poisson equations for the vorticities. Table 2 summarizes the domain sizes and grid resolutions which are used for the DNS and LES. L 3 7. Results In this section we compare the results of our reference DNS with ALDM, OEM and DOEM. Of course, an LES cannot recover all the features of a much higher resolved

7 LES of Supercritical Mixing Layers 141 Cases Domain size Grid resolution L 1 L 2 L 3,m 3 N 1 N 2 N 3 DNS LES TABLE 2. Domain size and grid resolutions for the H 2/O 2 mixing layer. DNS by definition. Therefore, the DNS results must be filtered and coarsened to the LES grid resolution for comparison with the LES results. We obtain the filtered DNS (FDNS) as reference solution by top-hat filtering over cubes of 4x4x4 cells. Figure 2 (left) shows the oxygen mass fraction in the - -plane at x 1 =.176 m. All SGS models are able to reproduce the mushroom-like structures in the O 2 mass fraction, only the fact that the leftmost structure already crosses the domain boundary in the FDNS is not covered to the full extent by the different LES. The tilt of the structures is best reproduced with ALDM and OEM, while the penetration of hydrogen into the oxygen side of the layer is a bit overestimated by all SGS models. The amount of unmixed oxygen remaining between the mushroom-like structures fits the FDNS very well in all LES, especially with ALDM, where these oxygen streaks are most prominent. The streamwise vorticity (Fig. 2 (right)) gives similar results. Shape, magnitude and position of the upper vortices obtained with all models are very close to each other and agree well with the FDNS results. While the minimum values are best captured with DOEM, ALDM gives the best result regarding the shape of the areas of positive streamwise vorticity. Only the fact that the positive peak values differ from each other in the FDNS is captured to the full extent by neither of the SGS models, but besides that the overall agreement is very good. The results for the non-dimensional spanwise vorticity (Fig. 3 (right)) in the x plane at =.6 m show comparable behaviour. The LES are able to reproduce many of the geometrical features of the FDNS contour as well as the overall position. The magnitude is generally in the right range as well. For example, ALDM is able to capture the maximum values in the middle and near the right boundary of the domain, only the area of maximum vorticity near the left boundary of the domain is overestimated. For OEM and DOEM the situation is contrariwise. While the vorticity magnitude near the left boundary matches the FDNS reference better, the maxima in the middle and near the right boundary are missing. That the results obtained with OpenFOAM do not match the shape of the high vorticity region in the lower part of the flow field is probably caused by boundary interactions.

8 142 C.A. Niedermeier, M. Jarczyk, S. Hickel, N.A. Adams & M. Pfitzner (a) (b) (c) (d) (e) (f) (g) (h) FIGURE 2. Y O for (a) FDNS, (c) ALDM, (e) OEM and (g) DOEM and ω 1δ ω,/ U for (b) FDNS, (d) ALDM, (f) OEM and (h) DOEM at t = 1 in the --plane at x 1 =.176 m.

9 LES of Supercritical Mixing Layers (a) (b) (c) x /δ 1 ω, (d) x /δ 1 ω, (e) x /δ 1 ω, (f) x /δ 1 ω, (g) x /δ 1 ω, (h) x /δ 1 ω, (i) x /δ 1 ω, (j) x /δ 1 ω, (k) x /δ 1 ω, (l) x x x 1 FIGURE 3. Y O for (a) FDNS, (d) ALDM, (g) OEM and (j) DOEM, ρ in kg/m 4 for (b) FDNS, (e) ALDM, (h) OEM and (k) DOEM and ω 3δ ω,/ U for (c) FDNS, (f) ALDM, (i) OEM and (l) DOEM at t = 1 in the x 1--plane at =.6 m.

10 144 C.A. Niedermeier, M. Jarczyk, S. Hickel, N.A. Adams & M. Pfitzner Regions with so-called high density gradient magnitude (HDGM) are of special interest among the geometrical features of the mixing layer. They redistribute turbulent energy from the normal to the tangential direction [12], therefore being crucial for the correct simulation of mixing and combustion. As can be seen in Fig. 3 (middle), again all models are able to reproduce the most important geometrical features of the FDNS solution s density gradient. The pressure based nature of the OpenFOAM approach (OEM and DOEM) obviously leads to higher density fluctuations, leading to more distorted results than with INCA (FDNS and ALDM). Besides that, the LES results are quite similar to each other with the strong density gradient in the lower left part of the domain being best captured with ALDM. Also, the area of the maximum density gradient in the middle is the smallest for ALDM, leading to the best match with the FDNS reference. The results for the O 2 mass fraction in the x 1 - -plane at =.6 m (Fig. 3 (left)) are similar to the previous findings. The general flow characteristics are met well with all SGS models. The substantial amount of hydrogen being present in the core of the vortex is well captured especially by ALDM and DOEM. Again, the results obtained with OpenFOAM look more distorted, but besides that both codes give comparable, good results. 8. Conclusions and outlook We simulated a non-reacting, transitional, temporal mixing layer of counterflowing hydrogen and oxygen by means of LES with different SGS models. The ILES method ALDM was used as well as the explicit SGS models OEM and DOEM. Additionally, a DNS of the configuration was simulated as a reference for comparison with the LES results. All SGS models generally show a good agreement with the FDNS results in the instantaneous flow visualizations. They are able to capture the most important geometrical flow features with their current parameterization. For a few distinct features, an advantage of ALDM and DOEM over OEM can be seen, but for a final assessment additional fully turbulent test cases will be used. There we expect the differences between the SGS models to become more prominent than in the transitional test case presented in this work. The TUM in-house code INCA showed its capability of performing high quality realgas simulations (DNS and LES). These yield the basis for the usage of ALDM for realgas flows and thereby for the very detailed investigation of supercritical flow phenomena. Additionally, OpenFOAM qualified as a promising industry-oriented tool for the final goal of simulating reacting flows in a rocket combustion chamber. Future work will focus on further validation of the SGS modeling for real-gas flows and later on the simulation of supercritical jet injection. The next step will be the investigation of fully turbulent oxygen/hydrogen and nitrogen/nitrogen mixing layers. Additionally, test cases like coaxial jet flow will be used to support this validation and optimization process. Acknowledgments Financial support has been provided by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) in the framework of the Sonderforschungsbereich Transregio 4. Computational resources have been provided by the Leibniz-Rechenzentrum München (LRZ).

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