保留原题，因要求不重复且内容独特

二维超声速空腔

空腔外形在内埋武器舱、起落架舱、超燃冲压发动机燃烧室等先进飞行器部件中有着广泛应用。本文以参考文献（Zhang and Rona, 1998, Journal of Sound and Vibration）提供的空腔外形为对象，采用ddes方法计算二维超声速空腔流动，检验SU2对于超声速非定常流场的模拟能力。

图片1.png

图片2.png

图 1 二维超声速空腔试验纹影结果

表 1 二维超声速空腔试验参数

参数名称	参数值
马赫数	1.5
空腔长度L	0.045 m
空腔深度D	0.015 m
空腔宽度W	0.114 m
雷诺数Re（参考长度为L）

2.网格生成

计算网格直接在pointwise软件中生成，网格包括空腔内部及平板上方两个网格块。空腔内部网格块为321×149个网格点，平板上方网格块为1011×359个网格点。平板上方边界层内第一层网格高度为4.5×10^-7^米。

图片3.png

a）二维空腔流动全局计算网格（i，j方向每8个网格点显示1个）

图片4.png

（b）二维空腔流动局部计算网格（i，j方向每4个网格点显示1个）

图 2 二维超声速空腔试验纹影结果

3.SU2求解器设置

3.1 流场求解cfg文件设置

下面介绍二维超声速空腔算例的参数设置。

+-----------------------------------------------------------------------+

| \% ------------- DIRECT, ADJOINT, AND LINEARIZED PROBLEM |

| DEFINITION ------------% |

| |

| \% |

| |

| \% Physical governing equations (EULER, NAVIER_STOKES, |

| |

| \% WAVE_EQUATION, HEAT_EQUATION, FEM_ELASTICITY, |

| |

| \% POISSON_EQUATION) |

| |

| PHYSICAL_PROBLEM= NAVIER_STOKES |

| |

| \% |

| |

| \% Specify turbulent model (NONE, SA, SA_NEG, SST) |

| |

| KIND_TURB_MODEL= SA |

| |

| HYBRID_RANSLES= SA_EDDES |

| |

| \% |

| |

| \% Mathematical problem (DIRECT, CONTINUOUS_ADJOINT) |

| |

| MATH_PROBLEM= DIRECT |

| |

| \% |

| |

| \% Restart solution (NO, YES) |

| |

| RESTART_SOL= NO |

| |

| \%initialization |

| |

| TESTCASE_TYPE= CAVITY |

| |

| \% -------------------- COMPRESSIBLE FREE-STREAM |

| DEFINITION --------------------% |

| |

| \% |

| |

| \% Mach number (non-dimensional, based on the free-stream values) |

| |

| MACH_NUMBER= 1.5 |

| |

| \% |

| |

| \% Angle of attack (degrees, only for compressible flows) |

| |

| AOA= 0.0 |

| |

| \% |

| |

| \% Init option to choose between Reynolds (default) or thermodynamics |

| quantities |

| |

| \% for initializing the solution (REYNOLDS, TD_CONDITIONS) |

| |

| INIT_OPTION= REYNOLDS |

| |

| \% |

| |

| \% Free-stream option to choose between density and temperature |

| (default) for |

| |

| \% initializing the solution (TEMPERATURE_FS, DENSITY_FS) |

| |

| FREESTREAM_OPTION= TEMPERATURE_FS |

| |

| \% |

| |

| \% Free-stream pressure (101325.0 N/m\^2, 2116.216 psf by default) |

| |

| FREESTREAM_PRESSURE= 5.38E4 |

| |

| \% |

| |

| \% Free-stream temperature (288.15 K by default) |

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| FREESTREAM_TEMPERATURE= 200 |

| |

| \% |

| |

| \% Reynolds number (non-dimensional, based on the free-stream values) |

| |

| REYNOLDS_NUMBER= 1.35E6 |

| |

| \% |

| |

| \% Reynolds length (1 m by default) |

| |

| REYNOLDS_LENGTH= 0.045 |

| |

| \% |

|

| \% Compressible flow non-dimensionalization (DIMENSIONAL, |

| FREESTREAM_PRESS_EQ_ONE,

| |

| \% FREESTREAM_VEL_EQ_MACH, FREESTREAM_VEL_EQ_ONE) |

| |

| REF_DIMENSIONALIZATION= FREESTREAM_VEL_EQ_ONE |

| |

| \% ---------------------- REFERENCE VALUE |

| DEFINITION ---------------------------% |

| |

| \% |

| |

| \% Reference origin for moment computation |

| |

| REF_ORIGIN_MOMENT_X = 0.25 |

| |

| REF_ORIGIN_MOMENT_Y = 0.00 |

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| REF_ORIGIN_MOMENT_Z = 0.00 |

| |

| \% |

| |

| \% Reference length for pitching, rolling, and yawing non-dimensional |

| moment |

| |

| REF_LENGTH= 1.0 |

| |

| \% |

| |

| \% Reference area for force coefficients (0 implies automatic |

| calculation) |

| |

| REF_AREA= 0.045 |

| |

| \% ------------------------- UNSTEADY |

| SIMULATION |

| -------------------------------% |

| |

| \% |

| |

| \% Unsteady simulation (NO, TIME_STEPPING, |

| DUAL_TIME_STEPPING-1ST_ORDER, |

| |

| \% DUAL_TIME_STEPPING-2ND_ORDER, TIME_SPECTRAL) |

| |

| \%UNSTEADY_SIMULATION= NO |

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| UNSTEADY_SIMULATION= DUAL_TIME_STEPPING-2ND_ORDER |

| |

| \% |

| |

| \% Time Step for dual time stepping simulations (s) |

| |

| \% U_inf = 425 - L=4.5E-2 dt=0.002 - |

| |

| UNST_TIMESTEP= 2.12E-7 |

| |

| \% |

| |

| \% Total Physical Time for dual time stepping simulations (s) |

| |

| UNST_TIME= 1.06E-2 |

| |

| \% |

| |

| \% Unsteady Courant-Friedrichs-Lewy number of the finest grid |

| |

| UNST_CFL_NUMBER= 0.0 |

| |

| \% |

| |

| \% Number of internal iterations (dual time method) |

| |

| UNST_INT_ITER= 30 |

| |

| \% |

| |

| \% Iteration number to begin unsteady restarts |

| |

| UNST_RESTART_ITER= 2 |

| |

| \% -------------------- BOUNDARY CONDITION |

| DEFINITION --------------------------% |

| |

| \% |

| |

| \% Navier-Stokes wall boundary marker(s) (NONE = no marker) |

| |

| MARKER_HEATFLUX= ( wall-down, 0.0 ) |

| |

| \% Format: (inlet marker, temperature, static pressure, velocity_x, |

| |

| \% velocity_y, velocity_z, ... ), i.e. primitive variables |

| specified. |

| |

| MARKER_SUPERSONIC_INLET= ( inlet, 200,5.38E4,425.22,0,0 ) |

| |

| MARKER_SUPERSONIC_OUTLET= ( outlet ) |

| |

| MARKER_SYM= ( wall-up ) |

| |

| \% |

| |

| \% Farfield boundary marker(s) (NONE = no marker) |

| |

| \%MARKER_FAR= ( FARFIELD ) |

| |

| \% |

| |

| \% Marker(s) of the surface to be plotted or designed |

| |

| MARKER_PLOTTING= ( wall-down ) |

| |

| \% |

| |

| \% Marker(s) of the surface where the functional (Cd, Cl, etc.) will |

| be evaluated |

| |

| MARKER_MONITORING= ( wall-down ) |

| |

| \% ------------- COMMON PARAMETERS DEFINING THE NUMERICAL |

| METHOD ---------------% |

| |

| \% |

| |

| \% Numerical method for spatial gradients (GREEN_GAUSS, |

| WEIGHTED_LEAST_SQUARES) |

| |

| NUM_METHOD_GRAD= GREEN_GAUSS |

| |

| \% |

| |

| \% Courant-Friedrichs-Lewy condition of the finest grid |

| |

| CFL_NUMBER= 10 |

| |

| \% |

| |

| \% Adaptive CFL number (NO, YES) |

| |

| CFL_ADAPT= NO |

| |

| \% |

| |

| \% Parameters of the adaptive CFL number (factor down, factor up, CFL |

| min value, |

| |

| \% CFL max value ) |

| |

| CFL_ADAPT_PARAM= ( 1.5, 0.5, 1.0, 100.0 ) |

| |

| \% |

| |

| \% Number of total iterations |

| |

| EXT_ITER= 50000 |

| |

| \% |

| |

| \% Linear solver for the implicit formulation (BCGSTAB, FGMRES) |

| |

| LINEAR_SOLVER= BCGSTAB |

| |

| \% |

| |

| \% Min error of the linear solver for the implicit formulation |

| |

| LINEAR_SOLVER_ERROR= 1E-6 |

| |

| \% |

| |

| \% Max number of iterations of the linear solver for the implicit |

| formulation |

| |

| LINEAR_SOLVER_ITER= 20 |

| |

| \% -------------------------- MULTIGRID |

| PARAMETERS -----------------------------% |

| |

| \% |

| |

| \% Multi-Grid Levels (0 = no multi-grid) |

| |

| MGLEVEL= 0 |

| |

| \% |

| |

| \% Multi-grid cycle (V_CYCLE, W_CYCLE, FULLMG_CYCLE) |

| |

| MGCYCLE= W_CYCLE |

| |

| \% |

| |

| \% Multi-grid pre-smoothing level |

| |

| MG_PRE_SMOOTH= ( 1, 2, 3, 3 ) |

| |

| \% |

| |

| \% Multi-grid post-smoothing level |

| |

| MG_POST_SMOOTH= ( 0, 0, 0, 0 ) |

| |

| \% |

| |

| \% Jacobi implicit smoothing of the correction |

| |

| MG_CORRECTION_SMOOTH= ( 0, 0, 0, 0 ) |

| |

| \% |

| |

| \% Damping factor for the residual restriction |

| |

| MG_DAMP_RESTRICTION= 0.95 |

| |

| \% |

| |

| \% Damping factor for the correction prolongation |

| |

| MG_DAMP_PROLONGATION= 0.95 |

| |

| \% -------------------- FLOW NUMERICAL METHOD |

| DEFINITION -----------------------% |

| |

| \% |

| |

| \% Convective numerical method (JST, LAX-FRIEDRICH, CUSP, ROE, AUSM, |

| HLLC, |

| |

| \% TURKEL_PREC, MSW) |

| |

| CONV_NUM_METHOD_FLOW= JST |

| |

| \% |

| |

| \% Monotonic Upwind Scheme for Conservation Laws (TVD) in the flow |

| equations. |

| |

| \% Required for 2nd order upwind schemes (NO, YES) |

| |

| MUSCL_FLOW= YES |

| |

| \% |

| |

| \% Slope limiter (VENKATAKRISHNAN, MINMOD) |

| |

| SLOPE_LIMITER_FLOW= VENKATAKRISHNAN |

| |

| \% |

| |

| \% Coefficient for the limiter (smooth regions) |

| |

| VENKAT_LIMITER_COEFF= 0.03 |

| |

| \% |

| |

| \% 2nd and 4th order artificial dissipation coefficients |

| |

| JST_SENSOR_COEFF= ( 0.5, 0.02 ) |

| |

| \% |

| |

| \% Time discretization (RUNGE-KUTTA_EXPLICIT, EULER_IMPLICIT, |

| EULER_EXPLICIT) |

| |

| TIME_DISCRE_FLOW= EULER_IMPLICIT |

| |

| \% -------------------- TURBULENT NUMERICAL METHOD |

| DEFINITION ------------------% |

| |

| \% |

| |

| \% Convective numerical method (SCALAR_UPWIND) |

| |

| CONV_NUM_METHOD_TURB= SCALAR_UPWIND |

| |

| \% |

| |

| \% Monotonic Upwind Scheme for Conservation Laws (TVD) in the |

| turbulence equations. |

| |

| \% Required for 2nd order upwind schemes (NO, YES) |

| |

| MUSCL_TURB= NO |

| |

| \% |

| |

| \% Time discretization (EULER_IMPLICIT) |

| |

| TIME_DISCRE_TURB= EULER_IMPLICIT |

| |

| \% ---------------- ADJOINT-FLOW NUMERICAL METHOD |

| DEFINITION -------------------% |

| |

| \% Adjoint problem boundary condition (DRAG, LIFT, SIDEFORCE, |

| MOMENT_X, |

| |

| \% MOMENT_Y, MOMENT_Z, EFFICIENCY, |

| |

| \% EQUIVALENT_AREA, NEARFIELD_PRESSURE, |

| |

| \% FORCE_X, FORCE_Y, FORCE_Z, THRUST, |

| |

| \% TORQUE, FREE_SURFACE, TOTAL_HEAT, |

| |

| \% MAXIMUM_HEATFLUX, INVERSE_DESIGN_PRESSURE, |

| |

| \% INVERSE_DESIGN_HEATFLUX) |

| |

| OBJECTIVE_FUNCTION= DRAG |

| |

| \% |

| |

| \% Convective numerical method (JST, LAX-FRIEDRICH, ROE) |

| |

| CONV_NUM_METHOD_ADJFLOW= JST |

| |

| \% |

| |

| \% Monotonic Upwind Scheme for Conservation Laws (TVD) in the adjoint |

| flow equations. |

| |

| \% Required for 2nd order upwind schemes (NO, YES) |

| |

| MUSCL_ADJFLOW= YES |

| |

| \% |

| |

| \% Slope limiter (NONE, VENKATAKRISHNAN, BARTH_JESPERSEN, |

| VAN_ALBADA_EDGE, |

| |

| \% SHARP_EDGES, WALL_DISTANCE) |

| |

| SLOPE_LIMITER_ADJFLOW= NONE |

| |

| \% |

| |

| \% Coefficient for the sharp edges limiter |

| |

| ADJ_SHARP_LIMITER_COEFF= 3.0 |

| |

| \% |

| |

| \% 2nd, and 4th order artificial dissipation coefficients |

| |

| ADJ_JST_SENSOR_COEFF= ( 0.0, 0.01 ) |

| |

| \% |

| |

| \% Reduction factor of the CFL coefficient in the adjoint problem |

| |

| CFL_REDUCTION_ADJFLOW= 0.75 |

| |

| \% |

| |

| \% Time discretization (RUNGE-KUTTA_EXPLICIT, EULER_IMPLICIT) |

| |

| TIME_DISCRE_ADJFLOW= EULER_IMPLICIT |

| |

| \% |

| |

| \% Adjoint frozen viscosity (NO, YES) |

| |

| FROZEN_VISC_CONT= YES |

| |

| \% --------------------------- CONVERGENCE |

| PARAMETERS --------------------------% |

| |

| \% |

| |

| \% Convergence criteria (CAUCHY, RESIDUAL) |

| |

| \% |

| |

| CONV_CRITERIA= RESIDUAL |

| |

| \% |

| |

| \% Residual reduction (order of magnitude with respect to the initial |

| value) |

| |

| RESIDUAL_REDUCTION= 3 |

| |

| \% |

| |

| \% Min value of the residual (log10 of the residual) |

| |

| RESIDUAL_MINVAL= -15 |

| |

| \% |

| |

| \% Start convergence criteria at iteration number |

| |

| STARTCONV_ITER= 10 |

| |

| \% |

| |

| \% Number of elements to apply the criteria |

| |

| CAUCHY_ELEMS= 100 |

| |

| \% |

| \% Epsilon to control the series convergence |

| |

| CAUCHY_EPS= 1E-6 |

| |

| \% |

| |

| \% Function to apply the criteria (LIFT, DRAG, NEARFIELD_PRESS, |

| SENS_GEOMETRY, |

| |

| \% SENS_MACH, DELTA_LIFT, DELTA_DRAG) |

| |

| CAUCHY_FUNC_FLOW= DRAG |

| |

| \% ------------------------- INPUT/OUTPUT |

| INFORMATION --------------------------% |

| |

| \% |

| |

| \% Mesh input file |

| |

| MESH_FILENAME= cavity-LD3-fine.su2 |

| \% |

| |

| \% Mesh input file format (SU2, CGNS, NETCDF_ASCII) |

| |

| MESH_FORMAT= SU2 |

| |

| \% |

| |

| \% Mesh output file |

| |

| MESH_OUT_FILENAME= mesh_out.su2 |

| |

| \% |

| |

| \% Restart flow input file |

| |

| SOLUTION_FLOW_FILENAME= restart_flow.dat |

| |

| \% |

| |

| \% Restart adjoint input file |

| |

| SOLUTION_ADJ_FILENAME= solution_adj.dat |

| |

| \% |

| |

| \% Output file format (PARAVIEW, TECPLOT, STL) |

| |

| OUTPUT_FORMAT= TECPLOT_BINARY |

| |

| \% |

| |

| \% Output file convergence history (w/o extension) |

| |

| CONV_FILENAME= history |

| |

| \% |

| |

| \% Output file restart flow |

| |

| RESTART_FLOW_FILENAME= restart_flow.dat |

| |

| \% |

| |

| \% Output file restart adjoint |

| |

| RESTART_ADJ_FILENAME= restart_adj.dat |

| |

| \% |

| |

| \% Output file flow (w/o extension) variables |

| |

| VOLUME_FLOW_FILENAME= flow |

| |

| \% |

| |

| \% Output file adjoint (w/o extension) variables |

| |

| VOLUME_ADJ_FILENAME= adjoint |

| |

| \% |

| |

| \% Output objective function gradient (using continuous adjoint) |

| |

| GRAD_OBJFUNC_FILENAME= of_grad.dat |

| |

| \% |

| |

| \% Output file surface flow coefficient (w/o extension) |

| |

| SURFACE_FLOW_FILENAME= surface_flow |

| |

| \% |

| |

| \% Output file surface adjoint coefficient (w/o extension) |

| |

| SURFACE_ADJ_FILENAME= surface_adjoint |

| |

| \% |

| |

| \% Writing solution file frequency |

| |

| WRT_SOL_FREQ= 100 |

| |

| \% |

| |

| \% Writing convergence history frequency |

| |

| WRT_CON_FREQ= 1 |

| |

| \% Writing solution file frequency for physical time steps (dual |

| time) |

| |

| WRT_SOL_FREQ_DUALTIME= 30 |

| |

| WRT_SURF_FREQ_DUALTIME= 1 |

+-----------------------------------------------------------------------+

3.2 并行运算脚本sh文件设置

在算例cfg文件所在目录，创建如下内容的sh文件，采用sbatch命令提交即可。

+----------------------------------------------------------------------+

| #!/bin/bash |

| |

| #SBATCH -N 7 #并行节点数 |

| |

| #SBATCH -n 168 #并行cpu数，=24*节点数 |

| |

| #SBATCH --job-name=cavity #job的名称 |

| |

| #SBATCH --ntasks-per-node=24 #每个节点用到的cpu数，无需修改 |

| |

| #SBATCH --output=%j.out #算例运行过程中在屏幕上显示的信息 |

| |

| #SBATCH --error=%j.err #报错信息 |

| |

| mpirun SU2_CFD ddes_2d_cavity_unsteady.cfg #流场求解 |

| |

| mpirun SU2_SOL ddes_2d_cavity_unsteady.cfg #输出tecplot结果文件 |

+----------------------------------------------------------------------+