Chapter 3 Evaluation of a Particle Ellipsoidal Statistical BGK Approach for Simulation
3.2 Case II: Simulation of Nitrogen Flow over Spherical Blunt Body – a Strong Shock Case
Body – a Strong Shock Case
The second simulation is performed to show the characteristics of the ES-BGK method under strong shock conditions at a free stream Mach Number of 15.6 flow over a blunt body. For this test case, the freestream flow density is ten times larger than the previous case, the sphere diameter is 0.1 m and the corresponding Knudsen number is 1.285×10−3. The simulated gas is nitrogen and due to the existence of the internal
modes, this case involves the full procedure presented in the previous section. Constant rotational and vibrational collision numbers were used in the Borgnakke-Larsen approach [88] with corresponding values of
Zr and Zv of 10 and 100, respectively. It was found from zero-dimensional calculations that at least three iterations are necessary for the ES-BGK calculation to predict the correct relaxation rate given in Eq.2.62. Figure 3.4 shows the translational, rotational and vibrational temperature contours in the forebody region of the spherical blunt body obtained by the ES-BGK method and the percentage differences of this solution with respect to DSMC method as well. It can be seen that overall the results are in good agreement throughout the simulation domain especially inside the shock. However, the largest differences, up to 20%, is observed just upstream of the shock region. In order to better analyze this finding, data was extracted
(a) Effect of number of particles on temperature without scaling
(b) with/out scaling
(c) Effect of number of particles on temperature with scal- ing
Figure 3.3: Comparison of translational temperature (K) at y=0.013 m for flow over a cone (Case I), using both DSMC and ES-BGK methods.
along the stagnation line and was presented in Fig. 3.5. Note that the temperature values are normalized using the freestream equilibrium and post-normal shock temperature values, T1= 200 and T2 = 9660 K,
respectively. Similarly, the distance is normalized with freestream mean free path value,λ1= 1.29×10−4m.
The difference in the region as the shock starts to form is caused by a phenomenon similar to the discussion about Fig. 3.3(c) where the ES-BGK method might require more particles in that region to have better agreement with the DSMC solution.
Moving to the rotational temperature comparison, both methods seem to overlap for the majority of the stagnation streamline. However, it is seen that there is a small shift in the ES-BGK solution. This shift is caused by the translational temperature differences at the location where the shock is formed, which in turn, changes the energy transfer rate from the translational to rotational mode. However, the vibrational temperature obtained by ES-BGK is found to be in excellent agreement with the DSMC solution. This is more likely due to the fact that the T-V rate is much slower than T-R relaxation. In general, the ES-BGK solution captures all the temperature profiles predicted by DSMC from the peak temperature locations to the temperature values close to the wall. Note that a diffusive wall model was used in all simulations.
Figure3.6compares the variation of the vibrational temperature with respect to the number of particles along the stagnation line. Vibrational temperature is chosen due to its more sensitive dependence on the number of particles. To test the dependence of the ES-BGK method to the number of computational particles, the number of particles was gradually decreased. Note that results presented in Figs.3.4 and3.5 employed N number of particles, where N is the number of particles given in Table3.2. It was observed that the result was essentially the same (see the “N/4” curve) but a further decrease resulted in temperature fluctuations, especially in the upstream region of the shock and underprediction of the highest temperature value around at x/λ1 = −20 where the number of particles was about three (“N/16” case). The same computational particle dependence was performed using DSMC method (not shown) and it was observed that the amount of underprediction at this location was almost the same as the ES-BGK method. This demonstrates that the ES-BGK method is able to predict flowfield parameters accurately with using the same number of particles as DSMC under this flow condition.
Table3.3presents the elapsed time for both methods for the case that employs about 9 million particles, designated as N in Fig.3.6. It can be seen that the elapsed time for the ES-BGK simulation is found to be about two times faster than that for the DSMC method. As compared to Case I, we can observe a speed-up due to fact that the number of collisions becomes a major part of the total computation which is indicated by the increase in the collision rate for this case, as given in Table3.3for Case II.
(a) Translational Temperature (b) Rotational Temperature
(c) Vibrational Temperature
Figure 3.4: Temperature (K) contours obtained by ES-BGK at top and their percentage differences with DSMC method at the bottom for flow about spherical blunt body (Case II), using both DSMC and ES-BGK methods.
Figure 3.5: Translational, rotational and vibrational temperatures (K) along the stagnation line.
Figure 3.6: Variation of vibrational temperature with the number of particles along stagnation line for ES-BGK method
Figure3.7shows a comparison of the velocity distributions obtained from the ES-BGK and the DSMC simulations at two locations, one at the start of the shock (denoted as point A in Fig.3.4(a)) and and the other close to the surface (denoted as point B in Fig.3.4 (a)). At the upstream location, the peak velocity values in the x and y directions for both DSMC and ES-BGK, as seen in Fig.3.7(a), reflects the different contributions of the free stream bulk velocity in the direction of the shock and the cross wise direction. It should be noted that in this strong shock case, the distribution is bimodal and in fact, is the summation of the subsonic and supersonic streams as pointed out by Mott-Smith [106]. The low temperature peak corresponds to the free stream condition (point A) whereas the broader distribution (higher temperature at point B shown in Fig.3.4(b)) corresponds to those particles that have undergone collisions inside the shock region and collisions with the wall. At the downstream location, point B, the distribution is Maxwellian with higher temperature since particles have had enough time to collide and equilibrate. The flow is found to be almost stationary in bothxandydirections and the agreement between DSMC and ES-BGK is good.
Table 3.4: VHS parameters for Nitrogen collisions
Model w dref[˚A] Tref[K]
Bird [69] 0.745 4.11 273
MD/QCT VHS Fit [107] 0.725 3.768 1000
(a) at x = -0.00645 m (shown in Fig.3.4(a)as point A) (b) at x = -0.0011 m (shown in Fig.3.4(a)as point B)
Figure 3.7: X and Y component velocity PDF inside the shock along the stagnation line, Case II Finally, in order to investigate the effects of viscosity on the ES-BGK and DSMC solutions, two different Variable Hard Sphere (VHS) parameters given in Table3.4are selected. The reference viscosity used in the
calculation of the characteristic relaxation frequency, given in Eq.2.16, is calculated by µref= 30(mkTref/π) 0.5 4(5−2w)(7−2w)d2 ref (3.1)
where the viscosity power constant, w, is given in Refs. [69] and [107]. The corresponding translational temperatures along the stagnation line is presented in Fig. 3.8(a). When the data given by MD/QCT VHS Fit [107] was used, as opposed to Ref. [69], a better agreement was surprisingly observed between the ES-BGK and DSMC solutions at the shock starting location. More specifically, in this location, the corresponding percentage differences when MD-QCT and Bird models were used in ES-BGK and DSMC methods are found to be approximately 5% and 10%, respectively as shown in Fig. 3.8(b). This is caused by the fact that MD/QCT derived fit of Ref. [107] results in higher total cross section values which in turn increases the total number of assignments in the ES-BGK method, consistent with the previous observation. Additionally, the ES-BGK results were found to be more sensitive to viscosity in the shock on-set region, −80< x/λ1<−50 where the largest difference was observed as compared with the DSMC results.
(a) Translational Temperature (b) Percentage Difference