Case III: Simulation of Nitrogen Flow over the Double Cone Configuration

In this section, a laminar hypervelocity flow over a sharp 25/55 deg double cone, shown in Fig. 3.9, is considered and the predicted ES-BGK surface parameters are compared with DSMC simulations and an experiment conducted in the LENS I Tunnel. The detailed description of the experimental procedure and instrumentation is provided in Ref. [6]. In particular, we examine the run 7 test condition, a case that was studied extensively as a code validation exercise for both DSMC [108] and NS [6] methods. The free stream Mach and Reynolds numbers under this test conditions are 15.6 and 9.35×104_{, respectively for a nitrogen}

gas with a frozen vibrational temperature of 1986 K whereas the translational and rotational temperatures were in equilibrium at 42.6 K [108]. The detailed inflow conditions for this test case is presented in Table3.1 and numerical parameters are given in Table 3.2. For the ES-BGK simulation, the computational domain was divided into uniform 2,000 collision/background grids in the X and Y directions. The energy and velocity assignment process for the ES-BGK method as well as the calculation of the macro parameters were performed in these cells. After reaching steady state, it was found that the total number of particles in simulation domain was about 16 million which is comparable with the number of particles used in DSMC calculation performed by Mosset al.[108]. For the DSMC calculation, the number of background grids was the same as the ES-BGK method; however, each background collision cell could be divided up to 25 collision cells depending on the value of the density gradient. After each division, the number of particles per collision cell inside the shock region was found to be five and the ratio of the mean free path to the size of the collision cells was larger than unity throughout domain. At steady state, the total number of collision cells and total number of particles were found to be about 4.14 and 15.8 million, respectively. Additionally, a time step of 6.0×10−8_{was chosen such that the ratio of mean collision time to the time step is greater than unity.}

Figure3.10(a) and (b) show the translational temperature contours for the ES-BGK and DSMC methods, respectively. As can be seen, the temperature values are found to be in very good agreement especially downstream of the bow shock. It was observed that the highest values for ES-BGK and DSMC methods are 2086 and 2134 K, respectively. The vibrational temperature obtained by the ES-BGK method, shown in Fig.3.10(c), was also found to be in close agreement with the DSMC solution (not shown). The vibrational temperatures decrease downstream of the shock region, especially, in the vicinity of the separation region due to the high collision rate and the full thermal accommodated diffusive cold wall. It should be noted that the ES-BGK method predicts the vibrational temperature accurately although the free stream flow condition

Figure 3.9: CUBRC sharp double-cone model taken from Fig. 1 of Ref. [6]

is in thermal nonequilibrium. Figure 3.10(d) shows the pressure field with the highest value found to be about 500 times larger than the free stream value due to strong compression and the shock interactions on the second cone. It should be noted that the second cone is sharp enough to increase the adverse pressure gradients as can be seen in Fig.3.10(d) which in turn results in a separation region. As shown in Fig.3.10(a) and (b), there is no discernible difference in the location and size of the separation region predicted by both methods.

Moving to the surface parameters, Fig.3.11(a) shows the calculated and measured surface pressure values. As can be seen, the calculations are very close to the experiment and each other. However, discrepancies are observed at the aft part of second cone surface. Both methods underpredict the maximum measured values but the DSMC solution is relatively closer to experiment at the maximum. Similarly, the DSMC calculation performed by Mosset al.[108] is found to be in relatively better agreement in the prediction of the maximum values as compared with the ES-BGK solution. The skin friction coefficient is shown in Fig. 3.11(b) and compared with the DSMC simulations of Ref. [108]. Overall, all the results predict similar size and location of the separation region. Similar to the surface pressure profile, the ES-BGK solution overpredicts the DSMC values on the first cone.

Figure3.12(a) shows the measured and calculated heat transfer rates on the two cone surfaces. Both the ES-BGK and DSMC solutions are found to be in very good agreement with the experiment especially for the second cone. In particular, the ES-BGK result achieves better agreement with the experiment in the

(a) Translational Temperature, ES-BGK (b) Translational Temperature, DSMC

(c) Vibrational Temperature, ES-BGK (d) Pressure, ES-BGK

(a) Surface pressure (b) Skin friction

Figure 3.11: Calculated and measured surface parameters

prediction of the maximum value compared to the DSMC result. As seen both methods capture the decrease in heat flux due to separation in the region between locationsx= 0.08 and 0.1 m and the calculated values overlap with the data. However, there is a small overshoot on the first cone for the ES-BGK solution due to an insufficient number of particles per cell. The comparison between measured and calculated values using different number of particles for the ES-BGK case is shown in Fig. 3.12(b). The values are found to be insensitive to the number of particles per cell on the second cone. However, it is observed that increasing the number of particles results in closer values to the experiment on the first cone.

Similarly, in order to investigate the dependence of solutions obtained by both methods on the spatial and temporal resolution, the number of cells and time step are changed gradually. The heat flux has been shown to be highly sensitive to the DSMC numerical parameters. Figure 3.13 shows that the maximum heat flux value (atx= 0.105 m) changes significantly with the number of cells for the DSMC calculation. It should be noted that 2,000 background cells without grid adaptation are insufficient to resolve the simulation domain up to the order of the local mean free path and non-physical diffusion effects become sufficiently large to change the surface heating. In fact, the DSMC solution with 2,000 background collision cells and each background cell being allowed to be subdivided into divided 25 cells was found to be in good agreement with experiment, as shown in Fig. 3.13(a). Therefore, further grid resolution would not change the surface heating values. Similarly, 1,500 cells in thex-direction were not sufficient to resolve the flow gradient for the ES-BGK method causing the overshoot in the maximum heat transfer rate, shown in Fig.3.13(b). However, it was observed that 2,000 cells in thexdirection are sufficient for the ES-BGK method in contrast to the

(a) Comparison of the ES-BGK and DSMC results (b) Variation of values with the number of particles

Figure 3.12: Computed and measured heat transfer rates

DSMC solution. It should be noted that since the total number of particles in the simulation domain is the same for the 1,500 and 2,000 cell ES-BGK cases, the 1,500 cell case has more particles per cell causing it to be in better agreement, compared to the 2,000 cell case, with experiment on the first cone surface. The number of particles for the 2,000 cells with subdivision up to 25 case is increased by a factor of two for both methods to ensure sufficient number of particles for the relaxation process. Therefore, perfect agreement with experiment is achieved at the cone surfaces even at the first cone. More specifically, the calculated heat flux values become closer to the measurements betweenx= 0.12 and 0.15 m. This shows that the cell size is sufficiently small to resolve the simulation domain to any characteristic length scale for gradients in any macroscopic flow quantities.

Figure 3.14 shows the variation of the calculated heat flux values with time step for DSMC and ES- BGK. As shown in Fig. 3.14(a), using larger time step results in an overprediction in heat flux values at

x = 0.105 m for the DSMC calculation due to the fact that time step is larger than the mean collision time. However, Fig. 3.14(b) shows that this variation for the ES-BGK calculation is less. Note that the number of particles in the simulation domain is the same and the only variable here is the time step for this test case. In contrast to DSMC, using too small a time step results in an overprediction along the first cone surface due to an insufficient number of particles selected for relaxation in Eq. 2.17. On the other hand, if the time step is sufficiently large, all particles in a cell are selected for relaxation and the overall distribution becomes Maxwellian in the cell. Thus, the time step should be small enough to capture any non-equilibrium phenomena in the simulation domain but large enough to avoid computational overhead

(a) DSMC (b) ES-BGK

Figure 3.13: Variation of heat flux values with the number of cells caused by the requirement to have a large number of particles per cell for a small time step.

Lastly, in the flow over the double cone configuration the characterization of the effect of the size of the separation region provides an opportunity to test the fidelity of all particle approaches for modeling the key flow physics as well as establish numerical convergence in near-continuum flow conditions. Reference [108] in fact used the slip velocity as a convergence criterion for DSMC simulations due to its very sensitive nature to the DSMC numerical parameters. The slip velocity (Vs) is calculated using Eq. 2.9. Figure 3.15shows the comparison of the slip velocity calculated by the ES-BGK and DSMC methods and with the DSMC simulations of Ref. [108]. All simulations show that the slip velocities are found to be high near the cone apex due to the rarefaction effect and decrease in the downstream direction, especially in the separation region. Interestingly, the ES-BGK solution achieves perfect agreement with the DSMC solutions and is able to model the gas surface interaction in an accurate manner. However, at the top side of the second cone (where there are no heat flux measurements), the ES-BGK solution overpredicts the velocity slip. This is more likely due to the lack of the sufficient number of particles arising from the suddenexpansionof the flow or the rarefaction effect.

The flow over the double cone involving multiple length scales and large density gradient provides an opportunity to test further the accuracy of the ES-BGK model in terms of the probability distribution function. Previously, Gallis et al. [109] compared the velocity distribution functions obtained by the ES- BGK and Boltzmann models in terms of Sonine polynomials for one-dimensional argon gas flows between parallel walls. It was observed that the ES-BGK model failed to reproduce precisely the higher order moments

(a) DSMC (b) ES-BGK

Figure 3.14: Variation of heat flux values with time step

of the velocity distribution function, especially when the hard-sphere interaction model was employed. To investigate this effect, the velocity distribution functions corresponding to the locations shown in Fig. 3.9 are presented in Fig. 3.16. It can be seen that the ES-BGK method is well able to capture the classical progression of the bi-modal nature of the velocity distribution function and its change as we move to different locations in the flow. As expected, the velocity distribution function differs slightly at the leading edge of the cone (i.e. location A) due to the rarefaction effect. However, for the other locations the ES-BGK method predicts accurate distribution functions especially in the locations where Knudsen number is less than 0.01. It should be noted that location C (i.e. in recirculation region) and D (the point where hot and cold streams interact) are selected to see the effect of the viscous heating effect on flow-field which was studied extensively in Ref. [110]. In contrast to the observation of Menget al.[110], the distribution functions are found to be in perfect agreement with the benchmark DSMC. This is due to the fact that Knudsen number at these locations is about two orders of magnitude less than that given in Ref. [110]. Finally, the very slight overprediction of the velocity slip shown in Fig.3.15can be attributed to the slight differences in the velocity distribution at location E.

To compare the efficiency of ES-BGK compared to DSMC, the elapsed time values are tabulated in the Case III row of Table3.3, employing 32 CPUs. It is found that the ES-BGK baseline case is only 3% faster compared to the DSMC method. In fact, similar to the test case I, there is no discernible difference observed in the running time of both methods due to low collision frequency. If the number of particles is increased or decreased by a factor of two with respect the baseline case (keeping all other numerical parameters the same), the ES-BGK simulation time was found to increase by 80.87% and decrease by 95.33%, respectively. Therefore, similar to the DSMC method, the ES-BGK method demonstrates a roughly linear scaling in terms of the relationship between the number of particles and simulation time.