Two-Dimensional Flows

Flow and Simulation Conditions

Flow conditions are the same as those of Koura and Takahira [78] and represent the

experimental conditions of Bütefisch [79]. This flow problem is chosen to demonstrate the capability of resolving the expected high density in the stagnation region and the high-density gradient across the detached bow shock around the cylinder. For completeness, they are briefly described here as follows: Variable Hard Sphere nitrogen gas, free-stream M ach number M_∞=20, free-stream number density n_∞=5.1775E19 particles/m³, free-stream temperature T_∞=20 K, fully thermal accommodated and diffusive cylinder wall with T_w/T₀=0.18, where T_w and T₀ are the wall and stagnation temperatures, respectively. Temperature dependent rotational energy exchange model of Parker [80] is used to model the diatomic nitrogen gas with the following parametric setting: limiting rotational collision number Zr_∞=21, potential well-depth temperature T*=79.8 K. Resulting Knudsen number based on free-stream condition is 0.025, based on the free-stream mean free path and diameter of the cylinder. The diameter of the cylinder is 1m. An unstructured triangular mesh is used for the simulation. Sketch of computational domain and complete listing of physical and VHS parameters are shown in Fig. 3.8 and Table 3.1. The variable time-step (VTS) and constant time-step (CTS) schemes with adaptive mesh are used in this simulation.

M esh Adaptation Concerns

Corresponding adaptation criteria for mesh adaptation is Kn_cc=1.1 with maximum number of adaptation levels equal to 4. Additional constraint, free-stream parameter, φ₀

=1.05, which reduces greatly the final refined total cell numbers, is used not to refine those cells with normalized density ratio close to unity (within 5% in this case). The side effect of this constraint might increase the skew of the interfacial cells between un-adaptive free-stream cells and adaptive cell; however, the reduction of computational cost is appreciable and up to approximately 20-30%. Final free-stream cell size is expected to be much longer than the local mean free path; however, the solution is not expected to deteriorate since nearly uniform flow properties prevail in the free-stream region. Thus, initial 7,025 triangular cells are used for the simulation.

Evolution of adaptive mesh at each level (only 0,2,4 shown) with cell quality control is presented in Fig. 3.9 and corresponding results are summarized in Table 3.2.

In Table 3.2, the cell numbers increase from 7,025 to 75,099 after four levels of mesh adaptation, the number of cells is much less than that used by Koura and Takahira [78], which had 200,000 cells, but the positioning of the cells in the present study may be superior to theirs due to the mesh adaptation scheme applied. As illustrated in Figs. 3.9,

the mesh is refined across the strong bow shock around the cylinder as well as the stagnation region in front of the cylinder. It is clearly that the proposed mesh adaptation method captures the important flow features such as the bow shock in this case. We would expect the results in these mesh-refined regions to be better than those without mesh adaptation.

Comparisons of the Results Using Different M eshes

Figure 3.10 is a normalized density contour with different meshes. The upper and the lower figures are the density contours with the original and the level-4 adaptive meshes, respectively. There are some results that we need to note. First, a rather strong bow shock stands off at some distance away from the cylinder. The flow is highly compressed across the nearly normal shock to the stagnation point, where density increases tremendously. Second, the flow is slightly compressed across the oblique shock away from the cylinder and then is slightly expanded further downstream. A relatively rarefied region (as compared with free-stream) with the size of cylinder diameter is formed with density ratio less than 0.5 behind the cylinder since most gas particles are directed away from the cylinder across the oblique shock as discussed earlier. Third, maximum values of normalized density are 14.4 and 39.3 at the stagnation point due to the highly refined mesh in this region, while minimum values of 0.279 and 0.283 are observed just behind the cylinder with the initial and adaptive mesh, respectively. Figure 3.11 presents the comparisons of the contours of normalized temperatures with different resolution meshes. The distributions of translational and rotational temperatures can be demonstrated clearly in this figure, where T_tr and T_rot represent the translational and rotational temperature, respectively. In general, the trends of the results with different meshes are similar, but the distributions of value are different. These figures show that these temperatures are increased according to the bow shock and reached to a maximum value at the stagnation point. And then it decreases away from the bow shock and the cylinder by expansion effect. Clearly, strong temperature non-equilibrium exists in the bow shock especially for the regions near the stagnation line.

Centerline Properties Distribution

Results of normalized number density (n/n_∞), and normalized translational and rotational temperatures ((T-T_∞)/(T_o-T_∞)) along the stagnation line are presented in Fig.

3.12. Previous experimental data of Bütefisch [79] and DSM C data of Koura and Takahira [78] are also included in these figures for comparison. In these figures,

normalized density and temperature are nearly the same and the agreements are remarkable, except for the data in front of the cylinder. Also appreciable thermal non-equilibrium occurs in the wake and shear layer around the cylinder. The simulated results with initial mesh are under predicted due to the fact that the cell is too coarse for solving the flow field. However, thermal non-equilibrium due to complicated flow field is well resolved using the adaptive mesh.

Comparisons of Adaptive M esh With or Without Cell Quality Control

Distribution of the adaptive mesh with or without cell quality control after fourth level adaptation is presented in Fig. 3.13. As this figure illustrated, the refined cells with high aspect ratio are removed by isotropic adaptation with cell quality control. Figure 3.14 illustrates normalized density and temperatures along the stagnation line. These results are approximately the same and agree with the previous experimental and simulated data. Although the discrepancy of the present results is not obvious, it is intuitive that the results with good cell quality should be more accurate.

Comparisons of the Results Using CTS and VTS Schemes

The constant time-step scheme (CTS) and variable time-step scheme (VTS) are both used with the same adaptive mesh (level-4) in this simulation. Both the simulation time-steps are set as 20,000. Variation of particle distribution is reduced greatly as illustrated in Fig. 3.15, which shows the comparison of distribution of averaged number of particles per cell on level-4 adaptive mesh using constant time-step (CTS) and variable time-step (VTS) methods. In this figure, the upper part is the result using constant time-step scheme. It is clear that the particle distribution is extremely non-uniform. The free-stream and wake regions have larger particle numbers, while fewer particle numbers at the bow shock region, due to the cell dimension and density effect as mentioned in Section 3.1. The particle distribution seems more uniform and desirable which prevents the waste of computational time. In addition, using VTS method, the reduction of averaged number of particles per cell can be as large as 10-fold and 30-fold in the regions of oblique bow shock and wake region behind the cylinder, respectively. This is achieved by keeping the particle weight the same in the reference cell (minimum cell, near the stagnation point in this case) for both VTS and CTS methods. In the other words, the statistical uncertainty in the minimum cell is kept the same for comparing both methods. Resulting simulated particles at steady state are reduced from 0.85 million, using constant time-step (CTS), to 0.16 million, using variable time-step (VTS) in this case (Fig. 3.16). The number of simulated particles

using variable time-step scheme can be reduced about five times of the constant variable time-step scheme. In addition, another benefit of applying VTS method to unstructured adaptive mesh is that it can reduce dramatically the number of iterations of transient period towards steady state, as illustrated in Fig. 3.16. In addition, the required number of iterations for transient period, when VTS is applied, is only about 25% of that if CTS is applied. In the current case, if we expect roughly the same statistical uncertainty in the minimum cell to obtain macroscopic properties for both CTS and VTS methods, the combination of VTS and unstructured adaptive mesh could save the computational time up to one order of magnitude.

Density Distributions

Figures 3.17 and 3.18 illustrate comparisons of normalized density and temperature contours with different time schemes, respectively. The upper and lower regions are the results using CTS and VTS methods, respectively. All the properties distributions are almost the same except for the density distribution at the wake region due to a relative fewer particles are applied by variable time-step scheme.

Centerline Properties Distribution

Normalized density and temperatures along the stagnation line are shown in Fig.

3.19. The results of variable time-step method seem more agreeable with previous experimental by Bütefisch [79] and simulated results by Koura and Takahira [78] than constant time-step scheme. Therefore, the variable time-step scheme not only can reduce the computational time but also obtains a more accuracy results in this case.

Hypersonic Flow Over a 15^o-Compression Ramp Flow and Simulation Conditions

A hypersonic flow over a flat plate with a 15°-compression ramp is investigated by Holden and M oselle [81] and Robinson [24]. The corresponding boundary settings for simulation and general features of flow field are depicted in Fig. 3.20. The flow conditions are briefly described in the following; Variable Hard Sphere nitrogen gas, free-stream M ach number M_∞=14.36, free-stream density ρ_∞ and temperature T_∞ are 5.221E-4 kg/m³ and 84.83 K, the length of the cylinder, X_c, and ramp, X_r, are 43.891 cm and 36.86 cm, respectively, fully thermally accommodated and diffusive flat and ramp wall with T_w =294.4 K. Resulting Knudsen numbers and Reynolds numbers based on X_c are 0.0002 and 1.04E5, respectively. Constant rotational energy exchange model is used with the rotational collision number Zr=5. Vibration energy transfer is neglected due to the low temperature involved. The complete listing of physical and VHS parameters are

summarized in Table 3.3 M esh Adaptation Concerns

In this simulation, free-stream parameter φ₀ and adaptation criteria Kn_cc for mesh adaptation are set to 1.05 and 1.1, respectively. The cell number of the initial triangular mesh is 15,063 and the maximum number of adaptation levels equal to 2. The simulation procedure is the same as the cylinder flow.

Figure 3.21 is the evolution of adaptive mesh at each level with cell quality control and corresponding results are summarized in Table 3.4. The number of cells increases from 15,063 to 83,776 after two levels mesh adaptation. The number of cells is larger than that used by Robinson [24], which had 66,482 cells. The numbers of simulated particles are about 350,000 and 640,000 of the present and Robinson, respectively. And the simulation time-steps are 24,000 and 160,000 of the present and Robinson, respectively. The mesh is refined across a weak leading-edge shock stands off at several mean free path from the tip of the plate. At the same time, a viscous boundary layer grows downstream along the flat plate due to the low Reynolds number laminar flow (Re_L=1x10⁵). As mentioned earlier, the layer is thickened greatly by the pressure rise caused by the compression ramp. It is clearly captured in this figure. We would expect the results in these mesh-refined regions are better than those without mesh adaptation.

Comparisons of the Results Using Different M eshes

Results of normalized density (ρ/ρ_∞) contour with initial and level-2 adaptive meshes are presented in Fig. 3.22. Both the contours have the similar trend but the distributions are much different especially at the weak leading-edge shock and the portion of the ramp. The maximum density ratio occurs at the mid portion of the ramp, where the leading-edge shock and the compression ramp shock interact with each other (Type VI interaction in Fig. 3.20). These ratios of initial and level-2 adaptive meshes are 6.39 and 8.79, respectively. The minimum value of density ratio above the plate is lower than ambient value in the early portion of the plate due to the leading-edge shock, and the values are 0.28 and 0.25 with respect to initial and level-2 adaptive meshes.

Figures 3.23 illustrates the pressure (

2

and heat transfer (

3

12 _∞

= u C_h q

ρ ) coefficient distributions, along the solid wall, with initial and level-2 adaptive meshes. Previous experimental data of Holden and M oselle

[81] and the DSM C data of Robinson [24] are also included in these figures for comparison. In these figures, the circle hollow symbols and the solid line represent the experimental by Holden et al. and simulated data by Robinson, respectively. The lines with hollow triangle, hollow quadrilateral are the present data with respect to use initial and adaptive meshes. As shown in Fig. 3.23(a), there are something needed to note.

First, the pressure distribution generally increases with the distance from the leading edge, reaches a maximum value at approximately the position of x=0.76 m, and then decreases to some value before the ramp corner. Second, the result of level-2 adaptive mesh is more agreeable to experimental data of Holden and simulation of Robinson due to the reasonable mesh is obtained after refinement. But the present results are worse than Robinson’s because the numbers of simulated particles and sampling are not enough. In addition, shear stress and heat transfer coefficient distributions are shown in this figure, although there is no experimental and simulated result available. This represents that there have a large improvement after mesh adaptation.

Comparisons of Adaptive M esh With or Without Cell Quality Control

M esh refinements with or without cell quality control are also discussed in this simulation again. Figures 3.24 and 3.25 are the level-2 adaptive mesh distribution and normalized density contour with or without cell quality control, respectively. From Fig.

3.24(b), there exist some cells with high aspect ratio in font of the leading edge and the region after the leading edge shock. These high aspect ratio cells are removed by cell quality control (in Fig. 3.24(c)). The local normalized density contour with a value equal to 0.66 presents more scatter distribution than the data with quality control in Fig.

3.25. Hence, a suitable adaptive mesh with good cell quality can obtain accurate simulated results.

3.3.2 Three-Dimensional Flows