A Finite Element Solver for Hypersonic Flows in Thermo-Chemical Non-Equilibrium Song Gao1, Jory Seguin1, Wagdi G. Habashi1, Dario Isola2 and Guido S. Baruzzi2 1
NSERC-Lockheed Martin-Bell Helicopter Industrial Research Chair for Multi-Physics Analysis and Design of Aerospace Systems, CFD Laboratory, Department of Mechanical Engineering, McGill University 680 Sherbrooke Street West, Montreal, Quebec, H3A 2S6, Canada 2 ANSYS Canada, 1000 Sherbrooke Street West, Montreal, Quebec, H3A 3G4, Canada
Email:
[email protected]
ABSTRACT A parallel finite element solver is developed for chemical and thermal non-equilibrium hypersonic flows. The finite element formulation is edge-based, with flow stabilization achieved via a Roe scheme. The flow, chemistry and thermal non-equilibrium solvers are loosely-coupled. A finite-rate chemistry model and a two-temperature thermal non-equilibrium model are used to account for the non-equilibrium processes. Numerical experiments are performed to assess the accuracy and efficiency of the proposed approach, and good agreement is found with solutions available in the literature.
1. INTRODUCTION The study of high-speed non-equilibrium flows is essential to the design of future hypersonic civil transport aircraft. The flow physics in this regime is dominated by complex phenomena such as strong shocks, thermo-chemical non-equilibrium and hightemperature effects [1]. In hypersonics, CFD has become the major tool to reliably evaluate aerothermodynamic quantities such as shear stresses, heat fluxes and temperatures on the surface of such vehicles. The numerical modelling of non-equilibrium effects requires solving the mass conservation equations for the species, and the mixture vibrational-electronic energy equation. The species mass conservation equations govern their production due to chemical reactions, while the vibrational-electronic energy equation accounts for relaxation between the energy modes and the vibrational energy production due to changes in gas composition. A two-temperature model [2] is used in which it is assumed that the translationalrotational modes and the vibrational-electronic modes
independently follow a Boltzmann distribution. This allows the definition of two temperatures; a translation-rotational one and a vibrational-electronic one. On the numerical side, Finite Element (FE)-based methods have been widely used for spatial discretization [3] due to their superior accuracy and robustness in using highly stretched grids in the boundary layer and across shocks, and in easily accounting for a variety of boundary conditions. To overcome the stability limitations of the FE method when addressing advection-dominated problems, an edge-based formulation has been proposed [4], facilitating the application of upwind stabilization schemes developed within the Finite Volume framework. The temporal discretization is based on a first-order backward Euler scheme for stability. This is especially important for stiff non-equilibrium flows that exhibit considerably different temporal scales [5]. After the discretization, the conservation equations of mixture quantities, mass of species and mixture vibrational-electronic energy, are solved in a looselycoupled fashion, reducing the implementation burdens, as well as minimizing the size of the linear system compared to a fully-coupled strategy [6]. This manuscript is organized as follows. Sections 2 and 3 describe the governing equations and numerical formulations. Results are presented in section 4 to show the accuracy and efficiency of the proposed methodology. Finally, conclusions and future developments are given in section 5.
2. GOVERNING EQUATIONS The governing equations for hypersonic flows in thermo-chemical non-equilibrium can be written as [7]
𝜕𝜌 + 𝛻 ⋅ (𝜌𝑽) = 0 𝜕𝑡
𝜕𝜌𝑌𝑠 + 𝛻 ⋅ (𝜌𝑌𝑠 𝑽) = −𝛻 ⋅ 𝑱𝒔 + 𝑆𝑠𝑐 𝜕𝑡 𝜕𝜌𝑽 + 𝛻 ⋅ (𝜌𝑽𝑽) = −𝛻𝑃 + 𝛻 ⋅ 𝝉 𝜕𝑡 𝜕𝜌𝐸 + 𝛻 ⋅ (𝜌𝐻𝑽) = −𝛻 ⋅ 𝒒 + ∇ ⋅ (𝝉𝑽) 𝜕𝑡 𝜕𝜌𝑒𝑣𝑒 + 𝛻 ⋅ (𝜌𝑒𝑣𝑒 𝑽) = −𝛻 ⋅ 𝒒𝑣𝑒 + 𝑆 𝑣𝑒 𝜕𝑡
(1) (2) (3) (4) (5)
where 𝜌 is the total density, P is the pressure, 𝑽 is the velocity vector, 𝐸 is the total energy per unit mass, 𝐻 is the total enthalpy per unit mass, 𝑌𝑠 is the mass fraction of the sth species and 𝑒𝑣𝑒 is the vibrationalelectronic energy as defined by the two-temperature model. The conservative variables vector is thus given by 𝑇 (6) 𝑸 = {𝜌, 𝜌𝑌1 , … , 𝜌𝑌𝑁𝑠 , 𝜌𝑽, 𝜌𝐸, 𝜌𝑒𝑣𝑒 }
∫ 𝑊𝑖 Ω
Ω
+ ∫ 𝑊𝑖 𝑛⃗ ∙ (𝑭 𝐴 − 𝑭𝑉 − 𝑭𝑇 ) = ∫ 𝑊𝑖 𝑺 ∂Ω
where 𝑭 𝐴 , 𝑭𝑉 and 𝑭𝑇 are the inviscid, viscous and thermal fluxes, respectively, and 𝑺 is the source term vector. Introducing edge coefficients 𝝌 and 𝜼 and using Roe’s scheme to provide a stabilized numerical formulation of the inviscid fluxes, the above equation can be assembled in a hybrid edge-based fashion as [4] 𝑳𝑖
𝑭𝑗 − 𝑭𝑖 𝑑𝑸𝑖 + ∑ 𝜼𝑖𝑗 ⋅ 𝑭𝑅𝑜𝑒 − 𝝌𝑖𝑗 ⋅ 𝑑𝑡 2 𝑗∈𝐾𝑖
+ ∑ ∫ 𝑾𝑖 ⋅ (𝑭𝑉 + 𝑭𝑇 )𝑑𝑉 𝑒∈𝐸𝑖 𝑉𝑒
𝑁𝑟 𝑓 ′′ ′ = 𝑀𝑠 ∑(𝜈𝑠,𝑟 − 𝜈𝑠,𝑟 ) (𝑅𝑟 − 𝑅𝑟𝑏 )
(7)
𝑟=1 ′ ′′ where 𝜈𝑠,𝑟 and 𝜈𝑠,𝑟 are the forward and backward stoichiometric coefficients and 𝑅𝑟𝑓 and 𝑅𝑟𝑏 are the forward and backward rates of chemical reactions. The source term governing the relaxation of the vibrational-electronic energy includes two components, as
𝑆 𝑣𝑒 = 𝑆𝑐−𝑣 + 𝑆𝑡−𝑣
(8)
where 𝑆𝑐−𝑣 is the relaxation due to chemical reactions, modelled using non-preferential dissociation, which assumes that molecules are created and destroyed at the average vibrational energy level, and 𝑆𝑡−𝑣 is the relaxation between the translational and vibrational energy modes, modelled using Landau-Teller theory [7]. The equation of state is 𝑁𝑠
𝑃 = ∑ 𝜌𝑌𝑠 𝑅𝑇
(10)
𝛺
The chemical source terms are expressed as 𝑆𝑠𝑐
𝜕𝑸 − ∫ 𝛻𝑊𝑖 ∙ (𝑭 𝐴 − 𝑭𝑉 − 𝑭𝑇 ) 𝜕𝑡
(9)
𝑠=1
where each species is assumed to obey the ideal gas law.
3. NUMERICAL ALGORITHM A weak-Galerkin method is used to obtain a FE representation of Eq. (1-5)
+ ∑ ∫ 𝑾𝑖 𝒏 ∙ (𝑭 𝐴 − 𝑭𝑉 − 𝑭𝑇 )𝑑𝐴 = 𝑳𝑖 𝑺𝑖
(11)
𝑒∈𝐹𝑖 𝐴
where 𝑭𝑅𝑜𝑒 are the numerical fluxes obtained from the Roe scheme. In the present work a segregated methodology is adopted, in which at each time level the gasdynamic, chemical and thermal nonequilibrium systems are solved sequentially. The Roe flux of each system is obtained from solving an approximate Riemann problem for each system. The scheme is made second-order in space by utilizing a MUSCL reconstruction and a van Albada slope limiter [8]. The linear system is solved using the GMRES method with an ILU preconditioner implemented within the PETSc framework [9].
4. RESULTS In this section, two test cases are presented. The first test case is an inviscid nitrogen flow past a 2D cylinder in thermo-chemical non-equilibrium. The second test case is a Mach 20 viscous nitrogen flow past a 2D cylinder, in which two flow conditions are considered. The first is thermal non-equilibrium with frozen chemistry, and the second is thermo-chemical nonequilibrium.
4.1 Inviscid flow past a cylinder An inviscid nitrogen flow past a cylinder in thermochemical non-equilibrium is studied. The flow conditions are listed in Table 1. The mass fractions and temperatures along the stagnation line are shown in Figure 1. In the figure, solid lines are results from HALO3D, while dots are solutions from references. Good agreement is found with Kessler [10] and the
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experimental shock position from Hornung [11]. The translational-rotational (T-R) temperature reaches its maximum value just behind the shock, whereas the vibrational-electronic (V-E) temperature lags due to the finite amount of time available for the exchange between translational and vibrational energies. Chemical reactions are initiated within the shock layer due to the large post-shock temperature, leading to dissociation of molecular nitrogen, and a decrease of both the translational-rotational and vibrationalelectronic temperatures in the shock layer.
number based on cylinder radius is 5,913. A 2D structured mesh is shown in Figure 2, consisting of 78,400 nodes and 78,204 elements. The height of the first layer of elements near the wall is 2 μm. The initial conditions can be found in Table 2. An isothermal wall is used, with both temperatures set to 1,000 K, and a far-field Kn of 5.1 × 10−3 . Two configurations are considered: the first one is thermal non-equilibrium with frozen chemistry, and the second is chemical and thermal non-equilibrium. Electronic energy modes are not considered.
Table 1. Flow conditions for inviscid flow past cylinder in chemical-thermal non-equilibrium
Table 2. Flow conditions for viscous flow past cylinder
𝑀∞ 𝑇∞ , 𝑇𝑣𝑒,∞ 𝑃∞ 𝑢∞
6.13 1833 K 2831 Pa 5590 m/s
𝑌𝑁2 𝑌𝑁 𝑅𝑐𝑦𝑙
0.927 0.073 0.045 m
Figure 1. Mach 6.31 flow past a cylinder: nitrogen and atomic nitrogen mass fractions (top), translation-rotational and vibrational-electronic temperatures (bottom) along the stagnation line.
4.2 Viscous flow past a Cylinder This test case is a Mach 20 laminar flow of nitrogen past a cylinder with a radius of 1 m. The Reynolds
𝑀∞ 𝑇∞ , 𝑇𝑣𝑒,∞ 𝑃∞ 𝑢∞
20 220 K 0.89 Pa 6047 m/s
𝑌𝑁2 𝑌𝑁 𝑅𝑐𝑦𝑙
1.0 0.0 1m
Figure 2. Mach 20 flow past a cylinder: mesh (top) and mesh near the wall (bottom) 4.2.1 Non-reacting Results Figure 3 plots the T-R and V-E temperature contours at the top, and pressure contours below. Figure 4 (top) plots the convergence curves for the flow and thermal non-equilibrium solvers. The flow solver achieves roughly three orders of magnitude reduction in the residual norm, with both solvers leveling off after
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5,000 Newton iterations. Figure 4 (bottom) plots the convergence of the integrated heat flux until 8,000 Newton iterations. Figure 5 plots the skin friction coefficient, pressure coefficient and heat fluxes on the wall. Figure 6 plots the T-R temperature, V-E temperature, density and Mach number profiles, along the stagnation line. The numerical results with the noslip boundary condition (labelled as HALO3D) are compared with Casseau et al. results [12] (labelled as Casseau 2015) where the Smoluchowski temperature jump condition and the Maxwell velocity slip jump [13] boundary conditions are employed. Since the farfield Knudsen number is still in the continuum regime, minor differences are expected between the profiles obtained from the two boundary conditions. With the exception of the friction coefficient on the wall, which is under-predicted, the comparisons of the various quantities are in good agreement.
N2 + N → 2N + N The Arrhenius constants in the chemistry model are given as Table 3. Coefficients for the chemistry model Reaction N 2 + N2 → 2N + N2 N2 + N → 2N + N
𝐴𝑓𝑟
Arrhenius constants 𝐸𝑎 𝜂𝑟𝑓
7.0 × 1021
−1.6
113,200
3.0 × 1022
−1.6
113,200
Figure 4. Mach 20 flow past a cylinder (nonreacting): convergence curves of solvers (top) and integrated heat flux (bottom)
Figure 3. Mach 20 flow past a cylinder (nonreacting): Translational-rotational and vibrational-electronic temperature contours (top) and pressure contour (bottom). 4.2.2 Reacting Results For the reacting case, two irreversible reactions are considered: N2 + N2 → 2N + N2
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Figure 5. Mach 20 flow past a cylinder (nonreacting): skin friction coefficient, pressure coefficient and surface heat flux on the wall.
Figure 6. Mach 20 flow past a cylinder (nonreacting): translational-rotational temperature, vibrational-electronic temperature, density and Mach number along the stagnation line. Figure 7 (top) plots the convergence curves for the flow, chemistry, and thermal non-equilibrium solvers. The various solvers achieve roughly two to three orders of magnitude reduction in the residual norm, leveling off after 10,000 Newton iterations. Figure 7 (bottom) shows that physical quantities may require more Newton iterations to reach convergence. Figure 8 and Figure 9 compare temperature contours and profiles between non-reacting and reacting cases. The reacting case has a smaller shock standoff distance and a lower V-E temperature compared to the non-reacting case, indicating that internal energy is consumed in the chemical dissociations. Figure 10 plots the skin
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friction coefficient, pressure coefficient and heat fluxes on the wall. Figure 11 plots the T-R temperature, V-E temperature, species density and Mach number profiles along the stagnation line. The numerical results (labelled as HALO3D) are compared with Casseau et al. results [14] (labelled as Casseau 2016). Both simulations use a Dirichlet boundary condition on the wall. The comparisons of stagnation line quantities are in good agreement, while the friction coefficient on the wall is slightly underpredicted. For the heat flux on the wall, both CFD and DSMC results from [14] are plotted. The results from HALO3D are in better agreement with the DSMC results. For the pressure coefficient on the wall, good agreement is achieved.
Figure 8. Mach 20 flow past a cylinder (compare non-reacting and reacting): T-R temperature contour (top) and V-E temperature contour (bottom), the upper half is non-reacting contours and the lower half is reacting contours.
Figure 7. Mach 20 flow past a cylinder (reacting): convergence curves of solvers (top) and integrated heat flux (bottom)
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Figure 9. Mach 20 flow past a cylinder (compare non-reacting and reacting): comparison between non-reacting (red) and reacting (black) profiles of T-R temperature (top) and V-E temperature (bottom) along stagnation line
Figure 10. Mach 20 flow past a cylinder (reacting): skin friction coefficient, pressure coefficient and surface heat flux on the wall.
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Figure 11. Mach 20 flow past a cylinder (reacting): translational-rotational temperature, vibrational-electronic temperature, 𝐍𝟐 and 𝐍 density and Mach number along the stagnation line.
5. CONCLUSIONS Chemical and thermal non-equilibrium flow physics have been addressed in a loosely-coupled numerical approach that combines the best of FV and FE methods. The code is validated by considering both inviscid and viscous simulations using 2D structured meshes. Additional validations will be performed for viscous flows on canonical 3D geometries such as spheres and cones. These will provide good points of comparison to other codes in terms of accuracy and performance.
ACKNOWLEDGEMENTS The authors acknowledge financial support of the NSERC-Bell Helicopter-Lockheed Martin Industrial Research Chair at the McGill CFD Lab, and through Lockheed Martin grant G238199 to ANSYS. We are grateful to Compute Canada and CLUMEQ for the access to their computational resources that facilitated this work.
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