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Numerical Investigations on the Impact of Turbulent Prandtl Number and Schmidt Number on Supersonic Combustion

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Numerical Investigations on the Impact of Turbulent Prandtl Number and Schmidt Number on Supersonic Combustion Fluid Dynamics & Materials Processing Tech Science Press DOI:10.32604/fdmp.2020.09694 Article Numerical Investigations on the Impact of Turbulent Prandtl Number and Schmidt Number on Supersonic Combustion Yongkang Zheng1,2,* and Chao Yan1 1National Key Laboratory of Computational Fluid Dynamics, Beihang University, Beijing, 100191, China 2State Key Laboratory of Aerodynamics, Mianyang, 621000, China *Corresponding Author: Yongkang Zheng. Email: [email protected] Received: 15 January 2020; Accepted: 08 April 2020 Abstract: The flow field inside the combustor of a scramjet is highly complicated and the related turbulent Prandtl and Schmidt numbers have a significant impact on the effective numerical prediction of such dynamics. As in many cases researchers set these parameters on the basis of purely empirical laws, assessing their impact (via parametric numerical simulations) is a subject of great impor- tance. In the present work, in particular, two test cases with different characteris- tics are selected for further evaluation of the role played by these non-dimensional numbers: Burrows-Kurkov case and DLR case. The numerical results indicate that these parameters influence ignition location. Moreover, the temperature dis- tribution is more sensitive to them than to H2O mass fraction and velocity distributions. Keywords: Turbulent Prandtl number; turbulent Schmidt number; ignition position; supersonic combustion 1 Introduction For accurate predictions on scramjet engines with turbulence models, calculation requires the turbulent Prandtl number (Prt) and the Schmidt number (Sct) for the purpose of accounting for turbulence/chemistry interactions [1]. Eklund demonstrates that predicting Prt and Sct for turbulence models should be as part of the solution, with the assumption that low Schmidt number can lead to unstart phenomena in combustor while high Schmidt number can result in blowout phenomena for flame [2]. However, traditional turbulence models, which are applied mostly to scramjet engine simulations, only take velocity fluctuations into consideration and these two parameters as constants. Since the variance of temperature and the mixing between fuel and oxidizer are inversely proportional to Prt and Sct, respectively, these two dominant parameters have remarkable effect on the conduction and diffusion processes. Among the researches, both variable and constant Prt and Sct are adopted as the critical factors for scramjet flow field prediction. Baurle chooses the Prt and the Sct as 0.9 and 0.5 respectively to control the turbulent mass and energy transport within the simulation of cavity flameholder using a hybrid Reynolds-averaged/large-eddy scheme [3]. Edwards sets both two parameters as 0.7 while simulating a This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 638 FDMP, 2020, vol.16, no.3 three-dimensional flame/shock wave interaction problem [4]. Gao assumes these two parameters as 0.9 in supersonic turbulent combustion flows with the flamelet model [5]. As in many cases researchers set these parameters on the basis of purely empirical laws, assessing their impact is of great importance. On the other hand, Xiao proposes a strategy about variable Prt and Sct and applies it to model scramjet flows [1]. Two additional equations are introduced into the governing system to calculate Prt and Sct as a part of the solution, which avoids involving the assumed probability density function into supersonic combustion flows. Variable Prt is implemented in Xiao’s research on heat transfer prediction for separated flows [6] and variable turbulent Schmidt Number is implemented in Large-eddy simulations by Ingenito [7]. Jiang et al. investigates some of the eddy dissipation combustion results obtained with the Prt and Sct varying from 0.25 to 0.85 and the calculated results are in good agreement with the experimental database [8]. Therefore, it can be concluded that the numerical solution is affected greatly by the selection of the Prt and Sct. For the express purpose of providing reliable evidence for further research, it is inevitable and significant to evaluate the affection of these tow parameters on the flow field prediction of scramjet configuration. In this paper, the sensitivity analysis of turbulent Prandtl number Prt and Schmidt number Sct for supersonic combustion is performed, and ignition location, temperature and combustion efficiency are focused as well. Two representative experimental cases are selected and the numerical results with different combinations of parameters are compared in detail. Some statistical analysis on the affection of the two critical factors are also performed. 2 Numerical Methods A finite volume in-house code developed by the authors is utilized to simulate all the numerical cases. The compressible governing equations, including the mass, momentum, energy and species transport equations, are solved using Reynolds-averaged Navier-Stokes simulations (RANS). The finite-rate chemistry model is adopted as combustion model in this study. Main algorithms of this code are presented follows: 2.1 Governing Equations The Favre-averaged conservation equations in the Cartesian coordinate are interpreted as: Continuity equation: @q @qu þ j ¼ 0 (1) @t @xj Momentum equation: @qu @qu u @p @s i þ i j ¼ þ ij (2) @t @xj @xi @xj Energy equation: ! @q @q @ l @ @ Xns l @ @s E þ Huj ¼ þ tCp T þ q þ t Ys þ ijui @ @ @ @ @ Ds hs @ @ (3) t xj xj Prt xj xj s¼1 Sct xj xj Species continuity equation: @q @q @ l @ Ys Ysuj t Ys þ ¼ qDs þ þ x_ s (4) @t @xj @xj Sct @xj FDMP, 2020, vol.16, no.3 639 th where q, uj, p, T are the density, velocity component in j Cartesian, pressure and temperature, respectively. Ys, Ds, hs are the mass fraction, diffusion coefficient and enthalpy per unit mass of species s, respectively. , E and H are the thermal conductivity coefficient, total energy per unit mass of mixture and total enthalpy per unit mass of mixture, respectively. s is the viscous stress tensor defined as: ij @ui @uj 2 @uk sij ¼ l þ À l dij (5) @xj @xi 3 @xk where dij is the Kronecker symbol. The source term x_ s is given with finite rate chemical model as follows: Xnr ÀÁÀÁ x_ ¼ b À a À s Ms js js Rj RÀj (6) j¼1 a b fi where Ms is the molecular weight of the species s, js and js are the stoichiometric coef cients of the related th th species in the j reaction. Rj and RÀj the forward and backward net rate of the j reaction respectively: a b Yns q js Yns q js s s Rj ¼ Kj ; RÀj ¼ KÀj (7) s¼1 Ms s¼1 Ms th th where Kj is the forward rate coefficient of the j reaction, and KÀj is the backward rate coefficient of the j reaction. The state equation of mixture gas is provide in the form: Xns qYs p ¼ RuT (8) s¼1 Ms where Ru is the universal gas constant, Ms is the molecular weight of species s. The static enthalpy and the specific heat at constant pressure are considered with temperature and species dependent thermodynamic properties as follow: Xns Xns h ¼ hsYs; Cp ¼ Cp;sYs (9) s¼1 s¼1 where hs, Cp;s are the enthalpy and specific heat at constant pressure of species s, modeled with a polynomial function of static temperature. For each species, thermal conductivity and molecular viscosity are calculated with Sutherlands formula. Wilkes formula [9] is then applied to calculate the mixture thermal conductivity and the mixture molecular viscosity. Besides, mass diffusivity is calculated according to the kinetic theory as follows [10]: 1 À Ys l qDs ¼ (10) 1 À Xs Sc where Xs is the mole fraction of species s. The total energy per unit mass of mixture E and total enthalpy per unit mass of mixture H are defined as: 1 p H ¼ h þ u u ; E ¼ H À (11) 2 j j q 640 FDMP, 2020, vol.16, no.3 2.2 Turbulence Model As shown in previous investigations, turbulence model plays very important roles in prediction of the flow field variables and the interaction between turbulence and combustion for supersonic combustion simulations. According to its good performance for less computational cost and more numerical robustness [11], Menter’s shear stress transport (SST) model [12] has extensive applications in engineering and thus employed in this paper. The SST turbulence model introduces the original k À x model inside boundary layer region and switches into the standard k À E model outside boundary layer region as well as in free shear flows [12]. The transport equations used for SST turbulence model take the form: @q @q @ @ k þ ujk ¼ ðÞl þ r l k þ À bÃqx k t Pk k (12) @t @xj @xj @xj @qx @q x @ @x qr @ @x þ uj ¼ ðÞl þ r l þ À bqx2 þ ðÞÀ x2 k x t Px 21 f1 (13) @t @xj @xj @xj x @xj @xj where k and x are the turbulent kinetic energy and the specific dissipation rate of turbulence, and Pk and Px denote the corresponding production term respectively. Pk and Px are defined as follows: ¼ l 2; ¼ q 2 Pk t Px Cx (14) l where is the magnitude of vorticity and t is the eddy viscosity, q l ¼ a1 k t (15) maxðÞa1x; f2kk The constants rk; rx; b and Cx are calculated via ÀÁ f ¼ f þ ðÞÀ f ; ¼ À4 f1 1 "#1 f1 2 f1 pffiffiffitanh!1 500l k 4qrx2k À1 ¼ min max ; ; qx 2 : x x 2 (16) d 0 09 d CDk d qr @ @x x2 k À20 CDkx ¼ max 2 ; 1 Â 10 x @xj @xj where d is the nearest distance to the wall.
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