Large Eddy Simulation of Supercritical Nitrogen Jets

Transcription

1 Sonderforschungsbereich/Transregio 40 Annual Report Large Eddy Simulation of Supercritical Nitrogen Jets By M. Jarczyk, M. Pfitzner, C. A. Niedermeier, S. Hickel AND N. A. Adams Institut für Thermodynamik, Universität der Bundeswehr München Werner-Heisenberg-Weg 39, Neubiberg 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 Technische Universität München and OpenFOAM, used by the Institute for Thermodynamics at Universität der Bundeswehr München. The present work, where Large Eddy Simulations (LES) are conducted for the transcritical injection of nitrogen, is part of the validation process. The comparison against experimental test data underlines that the results of both codes are in excellent agreement with each other as well as with the experiments. As a future task, further validation will be done by simulating the multicomponent mixing and combustion processes of coaxial injectors typically applied in rocket combustions engines. 1. Introduction The technology of cryogenic rocket combustion engines has been used successfully for many years and is well known today. The 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 need for reliable computational fluid dynamics (CFD) tools that are able to simulate the flow at rocket combustor conditions. The pressure in modern combustion chambers often exceeds more than 100 bar, 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. In recent years, intensive research efforts have been rendered to gain a deeper insight into the characteristic phenomena of trans- and supercritical flows. Shear layer flows in high pressure environments have been investigated in great detail by the group of Bellan [1 3] using Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS). LES simulations of reacting and non-reacting supercritical jet flows have been performed by Zong and Yang [4], Oefelein [5], Schmitt et al. [6 8] and Matsuyama et al. [9]. The final objective of this work is to analyze the propellant injection into a rocket Lehrstuhl für Aerodynamik und Strömungsmechanik, Technische Universität München, Boltzmannstr. 15, Garching b. München

2 146 M. Jarczyk, M. Pfitzner, C. A. Niedermeier, S. Hickel & N. A. Adams combustion chamber through multiple coaxial injectors and the subsequent combustion process. As this process is strongly three-dimensional and unsteady, Large Eddy Simulation is the most suitable method. At the Institute for Thermodynamics at Universität der Bundeswehr München the OpenFOAM package is used applying a pressure based PISO (Pressure Implicit with Splitting of Operators) algorithm proposed by Issa et al. [10, 11], whereas at the Institute of Aerodynamics and Fluid Mechanics at Technische Universität München the in-house code INCA has been developed as a densitybased finite-volume method using the Adaptive Local Deconvolution Method (ALDM) as an implicit LES (ILES) method. Both codes have been extended by the cubic Peng- Robinson [12] (PR) equation of state (EOS) combined with an empirical volume correction method developed by Harstad [13] as well as a transport property modeling accounting for high pressure effects. As a first validation of both real-gas LES solvers, DNS and LES simulations of a non-reacting, transitional, temporal mixing layer of hydrogen and oxygen have been performed [14, 15] and compared to the detailed investigations done by Okong o et al. [16]. The work presented here briefly summarizes the numerical methods developed for real gases within the two codes OpenFOAM and INCA in Sec. 2. Special emphasis is placed on a further validation of the solvers by simulating pure component transcritical nitrogen jet flows. The experimental and numerical setup as well as the comparison against experimental results obtained by Mayer et al. [17] are presented in Sec. 3. A detailed description and validation of the pressure-based numerical method applied along with OpenFOAM have also been published recently by Jarczyk and Pfitzner [18]. Finally, conclusions and an outlook to future work are given in Sec Numerical and Physical Modeling 2.1. Conservation Equations The general conservation equations for a compressible single component flow are: (ρe t ) (ρu i ) ρ + (ρu j) =0, (2.1) + (ρu iu j +pδ ij ) + ([ρe t+p]u j ) = τ ij, (2.2) = q j + (τ iju i ). (2.3) Here, x j are cartesian coordinates, t is the time, ρ is the density, u i is the velocity component in direction i, p is the thermodynamic pressure, e t = e+u i u i /2 is the total energy and τ ij is the Newtonian viscous stress tensor. In OpenFOAM, the energy conservation is realized by solving an equation for the enthalpy h instead of Eq. (2.3): ρh + (ρhu j) = q j + τ iju i + Dp Dt. (2.4) The viscous heating term τ ij u i / is omitted as it is negligible for the low velocity flows considered in this study. For both codes, q j is the heat flux vector modeled by Fourier s law with the thermal

3 Large Eddy Simulation of Supercritical Nitrogen Jets 147 conductivity λ: q j = λ T. (2.5) 2.2. Turbulence Modeling and Numerical Methods 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, two distinctly different modeling approaches are applied: An explicit LES with the Smagorinsky model in OpenFOAM and an implicit LES (ILES) with the Adaptive Local Deconvolution Method (ALDM) in INCA. OpenFOAM For the investigations presented here, the subgrid-scale (SGS) contribution is calculated by a Smagorinsky model where C S = 0.17 is taken for the Smagorinsky constant and Pr t =1.0 for the turbulent Prandtl number. Furthermore, a pressure-based OpenFOAM solver is used where a conservation equation for the pressure is solved instead of the continutiy equation (2.1). This approach has been extended by Issa et al. for compressible flows [10,11] and is known as PISO (Pressure Implicit with Splitting of Operators) algorithm. The derivation of the pressure equation depends on the particular equation of state to be considered. For an ideal gas, Eq. (2.6) has to be solved whereas for a real gas, the nonlinearity of the equation of state requires a special treatment, which finally leads to Eq. (2.7) to be used in the solution process: ψp (ρ 0 ψ 0 p 0 ) + (ρ( H p A p )) ( ρ A p p )=0, (2.6) + ψ 0p + (ρ( H p A p )) ( ρ A p p )=0. (2.7) Similarly, also the solution algorithm has to be adapted for real gases as presented in Fig. 1. Generally, the compressibility ψ is the only thermodynamic quantity that appears within the PISO loop. For ideal gases ψ can be expressed by 1/(RT) and is thus pressure independent. Consequently, all thermodynamic properties only have to be calculated once before the PISO loop is excecuted. For real gases, however, ψ is a function of temperature and pressure. In order to avoid thermodynamic inconsistencies in this case, the thermodynamic properties have to be updated with every excecution of the PISO loop. This procedure increases the computational effort, but provides a stable solver behavior. A further improvement of the solver stability is achieved by solving the enthalpy equation also within the PISO loop. A detailed description and validation of the pressure based numerical method applied along with OpenFOAM can be found in Ref. [18]. INCA The basic idea of ILES is to directly use the truncation error of the unmodified conservation equations (2.1) - (2.3). 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. [19]. ALDM is implemented in INCA for Cartesian collocated grids and used to discretize

4 Time Step Time Step 148 M. Jarczyk, M. Pfitzner, C. A. Niedermeier, S. Hickel & N. A. Adams Continuity Eqn. Continuity Eqn. Momentum Pre. Momentum Pre. PISO Loop Enthalpy Eqn. Thermodynamics Pressure Eqn. Turbulence PISO Loop Enthalpy Eqn. Thermodynamics Pressure Eqn. Turbulence FIGURE 1. PISO algorithm for ideal (left) and real gases (right) adapted from Ref. [18]. the convective terms of the Navier-Stokes equations (see Ref. [20] for a detailed description). The diffusive terms are discretized by 2nd order centered differences and a 3rd order explicit Runge-Kutta method is used for time integration. As INCA is a density-based code, pressure and temperature have to be calculated at each timestep from internal energy and density. For ideal gases, this can be done in a straightforward manner by using the ideal gas law. For real gases, however, an iterative procedure is necessary to find the best-suited values for pressure and temperature to minimize the following measure: 2 χ= ρ CFD ρ EOS ( ) ρ CFD +( e 2 CFD e EOS ). (2.8) e CFD Here, ρ is the density and e is the internal energy while the subscript CFD refers to the target values at the current timestep and the subscript EOS denotes the values corresponding to the current pair of pressure and temperature being under trial. It becomes obvious from Eq. (2.8) and the nonlinear form of the EOS that a nonlinear least squares optimization method is best suited to efficiently solve the problem. A trust region method from the Intel Math Kernel Library has therefore been chosen for the final implementation of the iterative calculation Thermodynamic Modeling and Transport Properties 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, the enthalpy h and the constant-pressure specific heat c p are calculated in OpenFOAM, while the internal energy e and the constant-volume specific heat c V must be provided in INCA. These are defined as p h(t,p)=h 0 (T)+ (V m T( V m T ) ) dp, (2.9) p p 0 e(t,v)=e 0 (T)+ V V (p T( p T ) ) dv, (2.10) V

5 Large Eddy Simulation of Supercritical Nitrogen Jets PR + volume Correction NIST ρ [kg/m 3 ] T [K] FIGURE 2. Density of nitrogen at 40 bar, PR EOS with volume correction vs. NIST data. c p (T,p)=c V (T,p) T( p T )2 V ( p V ) T, (2.11) c V (T,V)=( e T ) V. (2.12) Here, the subscript 0 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 [12] p= RT V b a(t) V 2 +2Vb b. (2.13) 2 V is the molar volume and R = J/(mol K) is the universal gas constant. 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)= R2 T 2 c p c 2 T 1 κ 1, (2.14) T c whereκ= ω ω 2 is a function of the acentric factorω. The effects of the reduction of the free volume by the particular volume of the molecules are taken into account via b= rt c /p c. T c and p c are the critical temperature and pressure of the modeled species (N 2 : T c = K / p c =33.96 bar). 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. [13] was chosen for the final implementation. Figure 2 exemplifies the excellent agreement between the PR EOS with volume correction and experimental data from the National Institute of Standards and Technology (NIST) for the density of nitrogen under supercritical pressure. Viscosity and thermal conductivity are calculated according to an approach developed by Chung et al. using empirical correlations for dense fluids. A detailed description of this method can be found in Ref. [21].

6 150 M. Jarczyk, M. Pfitzner, C. A. Niedermeier, S. Hickel & N. A. Adams [kg/m 3 ] Case 3 Cp [J/(kg*K)] Case Case 4 Chamber conditions Case 4 Chamber conditions T [K] T [K] FIGURE 3. Thermodynamic properties of LN2 and GN2 for Cases 3 and 4. D = 122 mm d = 2.2 mm L = 250 mm FIGURE 4. Modeled chamber geometry [18]. 3. Results 3.1. Experimental Setup In order to validate the solvers for jet flows in trans- and supercritical regimes, a test case injecting cold into gaseous nitrogen (GN2) at ambient temperature and supercritical pressure (39.7 bar) has been chosen. This configuration has also been investigated experimentally by Mayer et al. [17] using a square duct mixing chamber of 60x60 mm which is about one meter long. The injector has a diameter of 2.2 mm and a sufficient length to assure a fully turbulent pipe flow at the injector exit. For the investigations presented in this work, particularly the two test conditions illustrated in Fig. 3 by density and specific heat capacity are considered. Following Mayer s classification, the injection temperature is below the pseudo boiling temperature (T bp = K) in Case 3, whereas it is significantly above this value in Case Numerical Setup In OpenFOAM, the numerical setup is similar to the one Schmitt et al. [6] already used for their jet flow investigations. It is schematically shown in Fig. 4. Here, the injector dimensions are identical to the experiment, while the mixing chamber is modeled rotationally symmetric with a diameter of 122 mm and a total length of 250 mm. A suitable mesh refinement was obtained for an O-grid with a total number of about 1.7 million cells. A detailed description of the blocking can be found in Ref. [18]. In INCA, the full square duct of the chamber with a cross section of 60x60 mm is modeled, while the length of the simulation domain is 80 mm. The grid is refined near the centerline of the jet, leading to a total number of about 3.9 million cells.

7 Large Eddy Simulation of Supercritical Nitrogen Jets 151 Investigated Cases Case 3 Case 4 Injection velocity [m/s] Injection temperature [K] Chamber pressure [bar] Chamber temperature [K] TABLE 1. Initial and boundary conditions for the trans- and supercritical jet flows. The inital and boundary conditions applied in the simulations are listed in Tab. 1. At the inlet, a constant temperature is prescribed along with a time varying fully turbulent velocity profile extracted from a turbulent pipe flow. The chamber front wall is adiabatic and the outer walls have a constant temperature of 298 K. A wave transmissive boundary condition has been applied at the outlet Jet Simulations The results are presented in Fig. 5 for the transcritical Case 3 (top) and the supercritical Case 4 (bottom) as density gradient in the symmetry plane (left) and a comparison of the axial density distribution predicted by both solvers against the experimental data measured by Mayer et al. [17] (right). Especially the density plots enable a detailed investigation of the transition and breakup behavior of the dense jet. As the density of the injected nitrogen is much higher for Case 3 than for Case 4 (Fig. 3), the dense core of the jet is much longer here. Also the pseudo-boiling process taking place delays the heating of the core region. A very sharp, non-disturbed jet boundary can be detected right after the injector exit which starts to roll up and becomes fully turbulent further downstream. That means that the transcritical jet in Case 3 resists transition to turbulence for quite a long time. The coarse grids used for these studies might be identified as a reason here. This behaviour is expected to change when future investigations will be carried out with higher resolution. The quantitative comparison of the axial density distribution in Fig. 5 (right) underlines that an excellent agreement can be found for the supercritical Case 4 for both LES solvers. For the very demanding transcritical Case 3, both solvers still reproduce the experimental results very well. 4. Conclusion The transcritical injection of nitrogen has been simulated by means of an LES in order to validate the implementation of real-gas thermodynamics in two different CFD codes, the in-house code INCA of the Institute of Aerodynamics and Fluid Mechanics at Technische Universität München and OpenFOAM, used by the Institute for Thermodynamics at Universität der Bundeswehr München. The simulations were compared against the experimental results of Mayer et al. [17], where cold nitrogen is injected into gaseous nitrogen at supercritical pressure and ambient temperature. The reproduction of experimental data is excellent for the supercritical Case 4 for both solvers while the results obtained for the transcritical Case 3 are still in very good agreement with the experiments.

8 152 M. Jarczyk, M. Pfitzner, C. A. Niedermeier, S. Hickel & N. A. Adams 1.e06 1.e04 1.e02 ρ [kg/m 3 ] Experiment OpenFOAM INCA x/d 1.e Experiment OpenFOAM INCA 1.e04 1.e02 1 ρ [kg/m 3 ] x/d FIGURE 5. Magnitude of density gradient (left) for Case 3 (top) and Case 4 (bottom) computed with OpenFOAM (inserted pictures show a magnification of the inlet region) and comparison of axial density distribution (right) with experimental data (Case 3 (top), Case 4 (bottom)) [18]. With the presented investigations, both solvers qualify as promising tools for the final goal of simulating reacting flows in a rocket combustion chamber. Acknowledgments Financial support has been provided by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) in the framework of the Sonderforschungsbereich Transregio 40. Computational resources have been provided by the Leibniz-Rechenzentrum München (LRZ) under grant h0983. References [1] MILLER, R., HARSTAD, K. AND BELLAN, J. (2001). Direct numerical simulations of supercritical fluid mixing layers applied to heptane-nitrogen. J. Fluid Mech., 436,1 39. [2] OKONG O, N. AND BELLAN, J. (2003). Perturbation and Initial Reynolds Number Effects on Transition Attainment of Supercritical, Binary, Temporal Mixing Layers. Comp. & Fluids, 33, [3] TASKINOGLU, E. AND BELLAN, J. (2010). A posteriori study using a DNS database describing fluid disintegration and binary-species mixing under supercritical pressure: heptane and nitrogen. J. Fluid Mech., 645,

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