Kinetic Study with a 2-D Wedge Supersonic Flow

In the present section we conduct a detailed kinetic study for a two-dimensional supersonic nitrogen flow past a 25° finite wedge to investigate if previously defined breakdown criterion in Ref. [Wang and Boyd, 2003] is appropriate or not, especially near an isothermal solid wall, by employing the DSMC method. This test problem represents an idealistic flow for studying the continuum and thermal breakdown parameter since it includes a leading edge near the tip of the wedge surface, an oblique shock wave originating from the leading edge, a boundary layer along the wedge surface and an expanding fan starting at the end of wedge surface.

4.2.1 Flow and Simulation Conditions

A supersonic flow past a 2-D wedge, similar to the first benchmark test in Chapter 3 but

with an extending computational domain to the downstream region, is chosen as the test case for a detailed kinetic study. Free-stream conditions for this test case, which is the same as previous test case in Chapter 3, include: gaseous nitrogen as the flowing fluid, a Mach number (M_∞) of 4, a velocity (U_∞) of 1111.1m/s, a density (ρ_∞) of 6.545E-4kg/m³ and a temperature (T_∞) of 185.6K. The wedge has a wall temperature (Tw) of 293.3K and a length of 60.69mm.

Fig. 4.1 shows the surface mesh distribution, which is used for the current kinetic study.

Note only hexahedral mesh is used in this simulation. 103,520 DSMC cells and about 8 millions of simulation particles are used for this kinetic test. Fig. 4.2 illustrates the distribution of continuum breakdown parameter Knmax and the locations of random velocity sampling in this kinetic study. Totally 52 points in the regions, including near leading edge, oblique shock, boundary layer and expanding fan, are selected. Velocity distributions of three Cartesian directions at each selected point are sampled for particles up to at least 0.3 million, and then are compared with the corresponding local Maxwell-Boltzmann velocity distributions to understand the degree of continuum breakdown in these representative points.

In addition, the thermal non-equilibrium indicator P is also calculated at each selected _Tne^* point from the DSMC simulation. Fig. 4.3 shows the domain distributions of the maximum local Knudsen number based on the local gradient of the specific flow property according to eqns. (2.16) and (2.17). In general, the KnD and KnT dominates the most part of the

computational domain and across the oblique shock, respectively, while KnV dominates near the solid wall due to the high velocity gradient in the boundary layer and wake regions.

Results of the kinetic study in each specific region are described in the following in turn.

4.2.2 Region near the Leading Edge

Fig. 4.4a shows the locations of the sampling points 3-7, while Figs. 4.4b-f illustrate three corresponding Cartesian random velocity distributions at each point, respectively. In Figs.

4.4b-4.4f, the Maxwell-Boltzmann distribution represents the equilibrium state of flow in each translational degree of freedom, respectively, and temperature ratios in different degrees of freedom are also listed in these figures. Note the temperatures in each degree of freedom are normalized to the averaged temperature throughout the study unless otherwise specified.

Firstly, the flows are in continuum (Knmax<0.02 in Fig. 4.2) and in thermal equilibrium state in each degree of freedom and among all degrees of freedom at Point 3 (Fig. 4b) and 4 (Fig. 4c) in free stream due to very small property gradients. Secondly, the flow continuum breakdowns (Knmax>>0.02 in Fig. 4.2) and the flow deviates greatly from the equilibrium state at Point 5 (Fig. 4d), Point 6 (Fig. 4e) and Point 7 (Fig. 4f) very near the leading edge due to very large property gradients. For better understanding, more quantitative description all probing points (Figs. 4b-4f) will be stated in the following in turn.

Because the locations of Points 3 and 4 are close to the free stream region, the random

velocity distributions agree very well with the Maxwell-Boltzmann distribution in Figs. 4.4b and 4.4c. In addition, the maximum temperature deviation to average temperature is lower than 1% (normalized to T ) and general thermal non-equilibrium indicator _tot P , as defined _Tne^* in eqn. (4.1), is lower than 0.0021 at both Points 3 and 4. Thus, in the free stream region such as Points 3 and 4 the flows can be assumed to be in continuum and thermal equilibrium state, in which the NS equations are valid. As stated earlier, Fig. 4.2 shows the continuum

breakdown parameter Knmax at both Points 3 and 4 is lower than 0.02. Based on the threshold value Kn_max^Thr.=0.05 proposed previously by Wang and Boyd [2003], locations of Point 3 and 4

shall be assigned as the continuum domain, which is consistent with the above observation by the present kinetic study.

Figs. 4.4d-4.4f show the random velocity distributions deviate greatly from the Maxwell-Boltzmann distribution, in addition to the large discrepancy of temperatures existing among the various degrees of freedom. For example, the temperature at Point 6 (very near the leading edge as shown in Fig. 4e) in the x-direction deviates from the average temperature up to 87% more. Resulting P and Kn_Tne max is approximately 0.45 and 0.8, respectively. Thus, the use of DSMC method is necessary for such strong non-equilibrium and large deviation from Maxwell-Boltzmann distribution.

4.2.3 Region near the Oblique Shock

Fig. 4.5a shows the locations of the sampling points 14-19, while Figs. 4.5b-f illustrate three corresponding Cartesian random velocity distributions at each point, respectively. The random velocity distributions in each direction at Points 14 (pre-shock, Fig. 5b) and 19 (post-shock, Fig. 5c) agree very well with the Maxwell-Boltzmann distribution, respectively, because the property gradients are small at Points 14 and 19 which are distant from the oblique shock. The maximum deviation of the temperature from the average temperature is less than 3%, while P and Kn_Tne max (Fig. 4.2) is lower than 0.03 and 0.04, respectively. As the locations are close to the oblique shock region, such as Points 15-18, either the random velocity begins to deviate from the Maxwell-Boltzmann distribution (Point 15), the temperature in some degree of freedom begins to deviate from the average temperature (Point 18) or show very large discrepancies of both velocity distribution and thermal non-equilibrium among various degrees of freedom (Points 16 and 17).

For example, temperature in the y-direction at Point 17 can deviate greatly from the average temperature up to 27% more, which results in the P and Kn_Tne max approximately as 0.223 and 0.424, respectively. This indicates the regions at Points 16 and 17 near the oblique shock are in continuum breakdown and strong non-equilibrium among various degrees of freedom and should be treated using the DSMC method. In addition, even the random velocity distributions agrees very well to the Maxwell-Boltzmann distribution at Point 18 in

Fig. 4.5f, strong thermal non-equilibrium among various degrees of freedom exists (P =0.101 and Kn_Tne max=0.155).

From the above observation of the kinetic study, continuum breakdown parameter Knmax , as defined in eqn. (2.16), with a threshold value K_max^Thr.=0.05 and previously proposed thermal non-equilibrium indicator, as defined in eqn. (2.18), with a threshold value 0.03 can successfully predict the breakdown of the flow in the leading edge and oblique shock regions.

In the next section, the continuum breakdown will be reinvestigated in the regions near the boundary layer and near the expanding fan.

4.2.4 Region near the Boundary Layer

Figs. 4.6b-4.6f and Fig. 4.7b-4.7f show the random velocity distributions at Points 26-30 and Points 31-35 near the boundary layer, respectively, along with the local Maxwell-Boltzmann distribution. Note that Points 26-30, as compared to Points 31-35, are at locations closer to the leading edge, which are expected to have larger property gradients. As shown in Fig. 4.2 in both regions in the boundary layer, very large breakdown parameter

Knmax occurs due to the large velocity gradient (Fig. 4.3), especially near the solid wall (Knmax>0.4). Normally these two regions in the boundary layer would be considered as continuum breakdown domains based on previously proposed criterion of Knmax.

Astonishingly at first, at Points 31-35 the velocity distributions are in very good

agreement with the local Maxwell-Boltzmann distribution and the temperature variation among different degrees of freedom is very small, even with very large value of Knmax (all higher than 0.05 as shown in Fig. 4.2). At Points 26-30 that are closer to the leading edge, the velocity distributions are also in excellent agreement with the local Maxwell-Boltzmann distribution, although the temperature in both x- and y-direction begins to deviate from the average temperature. Even at Point 30, which is very near the solid wall, the maximum temperature deviation to the average temperature is less than 5-6% (P =0.034). In addition, _Tne^* at Points 31-35, which is further downstream in the boundary layer, not only the velocity distribution agrees very well with the local Maxwell-Boltzmann distribution, but also the temperature deviation among the various degrees of freedom is very small. Even at Point 35 that is very close to the solid wall, the maximum temperature deviation is less than 3%

(P =0.018). The reason of being capable of maintaining the continuum condition and _Tne^* thermal equilibrium among various degrees of freedom lies in the fact that the particles collide with the isothermal solid wall and are thermalized to the wall temperature before emitting into the region near the wall.

In Fig. 4.2, the continuum breakdown parameter Knmax in the boundary layer region is higher than K_max^Thr.=0.05 recommended by Wang and Boyd [2003]. That means the boundary layer regions would be assigned as the breakdown regions. However, it can be found the random velocity distributions in the x-, y-, z-direction agree excellently with the

Maxwell-Boltzmann distribution, respectively, in Figs. 4.7. Furthermore, the value of P _Tne are lower than 0.0185 for all of the Point 31-35. Thus, we can conclude that the degree of the continuum breakdown in the locations, such as Point 31-35, is overestimated based on the previous criterion of Knmax. The above kinetic studies indicate that it is not necessary to utilize the DSMC method in the whole boundary-layer region, even the continuum breakdown parameter Knmax is very large. This observation is critical in improving the efficiency of a coupled DSMC-NS scheme presented in Chapter 3.

4.2.5 Region near the Expanding Fan

Fig. 4.8 shows the locations of the sampling points and random velocity distributions of Point 41-46. Along the direction of Point 41 to Point 46, the value of Knmax increase from 0.021 to the maximum value 1.182 as shown in Fig. 4.2. Most of the Points 41-46 are located in the breakdown domain based on the previous recommended criterion of Knmax, except Point 41. In Figs. 4.8b-4.8d, random velocity distribution in each direction agrees well with Maxwell-Boltzmann distribution at Point 41 to 43, while the values ofP are higher than _Tne 0.03 except at Point 41 (<0.001). In addition, velocity distributions at Points 44 to 46 disagree appreciably with the local Maxwell-Boltzmann distribution with very high values of Knmax

(>0.2 as shown in Fig. 4.2). Thus, previously proposed criterion of Knmax can correctly predict the breakdown domain in the expanding fan region.