Verification of Virtual Mesh Refinement Method

In Fig. 3.2, it is computational domain of benchmark. Each cell size is 1/4 mean free path based on free stream. Its total cells number is 48000. The solution of benchmark is showing in Fig. 3.3 (a)-(d). We know that hypersonic flow over a block to cause result of shock and density to increase four times. Cell size of other cases besides benchmark is one mean free path, in which it is too big for resolution of the shock. We know that it will be not resolution enough in here and near the block. Therefore, we will be to verify that the same cells using virtual mesh refinement scheme can be to obtain better resolution than without VMR.

3.2.2.2 Compare Contours of Different Properties 3.2.2.2.1 Density, Temperature and Velocity

In this section, we compared some properties of these four cases including contours of density, temperature and velocity to verify VMR scheme. It is able to obtain better resolution than other at the same mesh. Fig. 3.4 show computational domain with VMR, TAS and None.

The total cells number and cell size of each case is 3000 and one mean free path. The simulation conditions of these cases are show in Table II. Fig. 3.5 is show contour of density for simulation results of these four cases. Fig. 3.6, Fig 3.7 and Fig. 3.8 show contours of temperature and velocity in x and y direction, respectively. We can be found that using VMR scheme have better resolution than TAS and None at location of high density.

3.2.2.2.2 Collision Quality

Fig. 3.9 (a)-(d) show contour of collision quality with benchmark, VMR, TAS and None.

that can be to obtain better collision quality. These results obviously show that the same grid number using VMR scheme have better collision quality than TAS and None.

3.2.2.3 Properties along Different Profile

In this section, we show some properties along different profile. Fig. 3.10 (a)-(d) show density, temperature and velocity along x=0.01 m, respectively. Fig. 3.11 (a)-(d) show density, temperature and velocity along x=0.005 m, respectively. Fig. 3.12 (a)-(d) show density, temperature and velocity along x=0.0005 m, respectively. Fig. 3.13 (a)-(d) show density, temperature and velocity along y=0.02 m, respectively. We observe that result of VMR have better resolution near the block.

3.2.2.4 Local Coefficient on Surface of the Block

Fig. 3.14 show compare of local pressure coefficient along x=0 m on the surface of block. Fig. 3.15 show compare of local friction coefficient along x=0 m on the surface of block. Fig. 3.16 show compare of local pressure coefficient along y=0.01 m on the surface of block. Fig. 3.17 show compare of local friction coefficient along y=0.01 m on the surface of block. Similarly, the local coefficients at location of edge obviously show that the results of using VMR scheme have better resolution with the same mesh number. Next section, we will to simulate other flow problem about two dimension hypersonic flow over a cylinder to verify VMR module.

3.3 2-D Hypersonic Flow over a Cylinder

3.3.1 Problem Description and Simulated Condition

The condition for this simulation with Mach number 10 Argon is shown in Fig. 3.18 and Table III. The upstream velocity, temperature and number density is equal to 2634.1 m/s,

200 K and 4.274E+20 particles/ m , respectively. Similarly, this problem is a ³ two-dimensional case, will be simulated several cases with the different cell size and geometric girds. The simulation conditions of these cases show in Table IV, Table V and Table VI. The grid spacing of benchmark was chosen to be 1/5~1/2 the mean free path based on condition of free stream. The grid spacing of other cases was chosen to be 1~3 the mean free path. In these cases, will used different module to simulate in PDSC. Therefore, to be compared and verified the effects of using virtual mesh refinement module in PDSC.

3.3.2 Verification of Virtual Mesh Refinement Method 3.3.2.1 Results of the Benchmark

Fig. 3.19 is showing computational domain of benchmark. Each cell size is equal to 1/5~1/2 mean free path of free stream. Its total cells number is 195000. The solution of benchmark is showing in Fig. 3.20 (a)-(d), including contours of density, temperature, velocity in x- and y-direction. We know that hypersonic flow over a block to result shock and density to increase. Cell size of other cases besides benchmark is bigger than local mean free path in here and near the cylinder. Therefore, we will verify that the same cells using virtual mesh refinement scheme to obtain better resolution than without VMR.

3.3.2.2 Using Different Geometric Grids

In this section, we had simulated several cases using different module with geometric grids, including quadrilateral, triangular and mixed quadrilateral-triangular mesh. From these test cases, we can verify that PDSC simulation using VMR method has better resolution than TAS-case and None-case on unstructured grids. The detailed discussions are as follows sections.

3.3.2.3 Results of simulation with quadrilateral mesh

In this section, we have simulated several cases with VMR, TAS and None to verify that it has better resolution with VMR scheme. Fig. 3.21 show computational domain with VMR, TAS and None. The total cells number and cell size of each case are 7650 and 1~3 mean free path based on condition of free stream. The simulation conditions of these cases are show in Table IV. Simulation of these cases with VMR, TAS and None use quadrilateral mesh.

The detail of simulation results are as follows sections.

3.3.2.3.1 Compare Contours of Different Properties 3.3.2.3.1.1 Density, Temperature and Velocity

In this section, we compared some properties of these cases including contours of density, temperature and velocity to verify VMR scheme. It is able to obtain better resolution than other at the same mesh. Fig. 3.22-Fig. 3.25 show contours of density, temperature and velocity of these cases, respectively. From these results, we can be found that the results with VMR and TAS module have good resolution before cylinder, but they are bad after cylinder.

However, it can also obviously show that results of VMR have better resolution after cylinder with less grid number.

3.3.2.3.1.2 Collision Quality

Fig. 3.26 (a)-(d) show contour of collision quality with benchmark, VMR, TAS and None. The mcs/mfp of None is greater than eight at stagnation point near cylinder. The result of TAS is to improve the collision quality in here. However, the result of VMR obviously show that the same grid number using VMR scheme have better collision quality than TAS and None.

3.3.2.3.2 Properties along Different Profile

In this section, we show some properties along different profile. Fig. 3.27 (a)-(d) show density, temperature and velocity along x=0.005 m, respectively. Fig. 3.28 (a)-(d) show density, temperature and velocity along x=0.4 m, respectively. Fig. 3.29 (a)-(d) show density, temperature and velocity along x=0. 5 m, respectively. Fig. 3.30 (a)-(d) show density, temperature and velocity along y=0. 2 m, respectively. In Fig. 3.27 (a), we observe that result of VMR have better resolution near the cylinder.

3.3.2.3.3 Surface Property on the Cylinder

Fig. 3.31 (a)-(c) show compare of different local coefficient along surface of cylinder, including local coefficient of pressure, friction and drag. From Fig. 3.31, we can not obviously observe variation of these results with different module. However, Fig. 3.31 (b) shows that VMR and TAS have improve results of local friction coefficient. Next section, we will to simulate the same flow problem with triangular mesh.

3.3.2.4 Results of simulation with triangular mesh

In this section, we have simulated several cases with VMR, TAS and None to verify that it has better resolution with VMR scheme. Fig. 3.32 show computational domain with VMR, TAS and None. The total cells number and cell size of each case are 9802 and 1~3 mean free path based on condition of free stream. Simulation conditions of these cases are show in Table V.

3.3.2.4.1 Compare Contours of Different Properties

3.3.2.4.1.1 Density, Temperature and Velocity

Fig. 3.33-Fig. 3.36 show compare of contours of density, temperature and velocity of these cases, respectively. From these results, we can also be found that the results with VMR and TAS module have good resolution before cylinder, but they are bad after cylinder.

However, it can also obviously show that results of VMR have better resolution after cylinder with less grid number.

3.3.2.4.1.2 Collision Quality

Fig. 3.37 (a)-(d) show contour of collision quality with benchmark, VMR, TAS and None. The mcs/mfp of the None is equal to four at location of stagnation. The result of TAS is to improve the collision quality in here. Similarly, the result of VMR obviously show that the same grid number using VMR scheme have better collision quality than TAS and None.

3.3.2.4.2 Properties along Different Profile

In this section, we show some properties along different profile. Fig. 3.38 (a)-(d) show density, temperature and velocity along x=0.005 m, respectively. Fig. 3.39 (a)-(d) show density, temperature and velocity along x=0.4 m, respectively. Fig. 3.40 (a)-(d) show density, temperature and velocity along x=0. 5 m, respectively. Fig. 3.41 (a)-(d) show density, temperature and velocity along y=0. 2 m, respectively.

3.3.2.4.3 Surface Property on the Cylinder

Fig. 3.42 (a)-(c) show compare of different local coefficient along surface of cylinder, including local coefficient of pressure, friction and drag. From Fig. 3.42, we can not obviously observe variation of these results with different module. However, Fig. 3.42 (b) shows that VMR and TAS have improve results of local friction coefficient. Next section, we will to simulate the same flow problem with mixed quadrilateral-triangular mesh.

3.3.2.5 Results of simulation with mixed quadrilateral-triangular mesh

In this section, we have simulated several cases with VMR, TAS and None to verify that it has better resolution with VMR scheme. Fig. 3.43 show computational domain with VMR, TAS and None. Total cells number and cell size of each case are 12825 and 1~2 mean free path based on condition of free stream. Simulation conditions of these cases are show in Table VI.

3.3.2.5.1 Compare Contours of Different Properties 3.3.2.5.1.1 Density, Temperature and Velocity

Fig. 3.44-Fig. 3.47 show compare of contours of density, temperature and velocity of these cases, respectively. From these results, we can also be found that the results with VMR and TAS module have good resolution before cylinder, but they are bad after cylinder.

However, it can also obviously show that results of VMR have better resolution after cylinder with less grid number.

3.3.2.5.1.2 Collision Quality

Fig. 3.48 (a)-(d) show contour of collision quality with benchmark, VMR, TAS and None. The mcs/mfp of the None is equal to six at location of stagnation. The result of TAS is to improve the collision quality in here. Similarly, the result of VMR obviously show that the same grid number using VMR scheme have better collision quality than TAS and None.

3.3.2.5.2 Properties along Different Profile

In this section, we show some properties along different profile. Fig. 3.49 (a)-(d) show

density, temperature and velocity along x=0.005 m, respectively. Fig. 3.50 (a)-(d) show density, temperature and velocity along x=0.4 m, respectively. Fig. 3.51 (a)-(d) show density, temperature and velocity along x=0. 5 m, respectively. Fig. 3.52 (a)-(d) show density, temperature and velocity along y=0. 2 m, respectively.

3.3.2.5.3 Surface Property on the Cylinder

Fig. 3.53 (a)-(c) show compare of different local coefficient along surface of cylinder, including local coefficient of pressure, friction and drag. From Fig. 3.53, we can not obviously observe variation of these results with different module. However, Fig. 3.53 (b) shows that VMR and TAS have improve results of local friction coefficient.

3.3.2.6 Comparison of Surface Properties with VMR on Different Geometric Grids

Fig. 3.54 (a)-(c) show compared of different local coefficient using VMR scheme along surface cylinder with quadrilateral, triangular and mixed quadrilateral-triangular grids. From these results, they obviously show that results of PDSC simulation using VMR scheme with different geometric grids don’t have different. Thus we know that PDSC simulation using VMR scheme on different geometric grids all have resolution enough.