Direct simulation Monte Carlo (DSMC) of shock interaction in hypersonic low density flow is developed. Three collision molecular models, including hard sphere (HS), variable hard sphere (VHS), and variable soft sphere (VSS), are employed in the DSMC study. The simulations of double-cone and Edney’s type IV hypersonic shock interactions in low density flow are performed. Comparisons between DSMC and experimental data are conducted. Investigation of the double-cone hypersonic flow shows that three collision molecular models can predict the trend of pressure coefficient and the Stanton number. HS model shows the best agreement between DSMC simulation and experiment among three collision molecular models. Also, it shows that the agreement between DSMC and experiment is generally good for HS and VHS models in Edney’s type IV shock interaction. However, it fails in the VSS model. Both double-cone and Edney’s type IV shock interaction simulations show that the DSMC errors depend on the Knudsen number and the models employed for intermolecular interaction. With the increase in the Knudsen number, the DSMC error is decreased. The error is the smallest in HS compared with those in the VHS and VSS models. When the Knudsen number is in the level of 10−4, the DSMC errors, for pressure coefficient, the Stanton number, and the scale of interaction region, are controlled within 10%.
Shock interactions play an important role in hypersonic flows. In the past decade, significant efforts in computational fluid dynamics (CFD) have been exerted to develop prediction techniques for simulating these complex flow structures [
Flows with
Unfortunately, hypersonic shock is not an entirely understood domain. Flow property
In low-density hypersonic flows, both CFD and DSMC are used in the prediction of pressure and heat characteristics. However, conclusions, drawn from the simulation and experiment, are different. Unfortunately, many CFD and DSMC simulations are incompatible with precision direction [
Some double-cone flow simulation studies, including
Test case in [
Run |
|
|
|
Ma | Re/m ( |
Nose |
---|---|---|---|---|---|---|
26 | 2581 | 1.093 | 48.7 | 10.34 | 1.704 | Sharp |
28 | 2520 | 0.727 | 30.0 | 10.50 | 1.106 | Sharp |
35 | 2575 | 0.608 | 18.5 | 12.49 | 0.945 | Sharp |
32 | 2572 | 0.758 | 32.1 | 10.56 | 1.177 | R0.250 |
33 | 2636 | 1.288 | 59.8 | 10.35 | 2.050 | R0.250 |
36 | 2584 | 1.108 | 49.6 | 10.34 | 1.729 | R0.250 |
38 | 2453 | 0.498 | 13.9 | 12.47 | 0.738 | R0.288 |
For studying the influence of the DSMC molecular model on hypersonic shock interaction, DSMC simulations are conducted by three molecular models, namely, HS [
In DSMC, gas is represented by the velocity components and positions of a large number of simulated molecules. The position coordinates, velocity components, and internal states of the simulated molecule data are stored. These parameters are updated with time because the molecules are concurrently followed by representative collisions and boundary interactions in the simulated domain [
DSMC flowchart.
The core of the DSMC algorithm consists of four primary processes: moving the particles, indexing and cross-referencing the particles, simulating collisions, and sampling the flow field [
In the first process, all the simulated molecules are moved through distances, which are proportionate to their velocity components and discrete time step. Appropriate action is taken if the molecule crosses boundaries, such as solid surfaces and symmetry boundary. Collisions with surfaces can be treated as being fully secular, fully diffuse, or any combination of the two. Secular collisions involve a simple reversal of the molecular velocity component normal to the incident surface. Diffused collisions cause a random reorientation of the reflected molecule, where the postcollision velocity is based on surface temperature.
The second DSMC process involves indexing and tracking the particles. This molecular referencing scheme is needed as prerequisite for the next two steps: modeling the collisions and sampling the flow field.
The third step, simulating collisions, is an important process. And it is different from the MD (molecular dynamics) method. In DSMC simulation, elastic collision model is used in which no interchange of translational and internal energy is considered. And also, in the collision, linear momentum and energy are conserved.
In homogenous gas, the probability of two molecules collision is proportional to their relative speed and total collision cross section. In DSMC, cross section
Illustration of the impact parameters [
Then, the total collision cross section can be written as
Collision geometry of HS molecules [
In the VHS model, the closest approach
In the VSS model, the closest approach
Diffuse reflection [
The sampling of the macroscopic flow properties is the final process in DSMC. The spatial coordinates and velocity components of the molecules in a particular cell are used to calculate the macroscopic quantities at the geometric center of the cell. Consider
The parameters selected in the present paper are pressure coefficient and the Stanton number:
To validate the DSMC method, three 25°/55° hypersonic double-cone configurations are selected under five conditions. The experimental data, with four configurations as shown in Figures
Number 1 configuration (in) (sharp nose 25°/55° double cone).
Number 2 configuration (in) (R0.25 blunt nose 25°/55° double cone).
Number 3 configuration (in) (R0.288 blunt nose 25°/55° double cone).
Number 4 configuration (in) (R0.6 blunt nose 25°/55° double cone).
Boyd’s DSMC for Run number 35 [
Figures
Moss’s DSMC for Run number 28 [
Boyd’s DSMC for Run number 28 [
In [
The computational grid, employed for Run numbers 26, 32, 33, 36, and 38, consists of 2,048 cells along the body by 512 cells normal to the body. This grid is the same as that of Boyd’s DSMC simulation [
The properties of nitrogen molecule, used in the current study, are presented in Table
Properties of nitrogen molecule in DSMC.
Degree of freedom | 5 |
Molecular mass × |
46.5 |
Viscosity coefficient | 1.656 |
Viscosity index | 0.74 |
|
1.36 |
Molecular diameter (VHS) × |
4.17 |
Molecular diameter (VSS) × |
4.11 |
Molecular diameter (HS) × |
3.784 |
In this section, the DSMC simulation result of Run numbers 26 and 36 is presented. As shown in Table
Flow conditions for Run Numbers 26 and 36.
Ma | Re/m | Kn | |
---|---|---|---|
Run 26 | 10.34 |
|
|
Run 36 | 10.34 |
|
|
The pressure coefficient and the Stanton number, obtained with HS, VHS, and VSS, are shown in Figures
DSMC simulation of Run number 26 (
DSMC simulation of Run number 36 (
DSMC simulation of Run number 26 (
DSMC simulation of Run number 36 (
In this section, the DSMC simulation result of Run number 33 is presented. As shown in Table
Flow conditions for Run Numbers 33 and Numbers 36.
Ma | Re/m | Kn | |
---|---|---|---|
Run 33 | 10.35 |
|
|
Run 36 | 10.34 |
|
|
The pressure coefficient and the Stanton number, obtained from HS, VHS, and VSS using the DSMC method, are shown in Figures
DSMC simulation of Run number 33 (
DSMC simulation of Run number 33 (
In this section, the DSMC simulation results for Run numbers 32 and 38, as shown in Table
Flow conditions for Run Numbers 32 and 38.
Ma | Re/m | Kn | |
---|---|---|---|
Run 32 | 10.56 |
|
|
Run 38 | 12.47 |
|
|
The pressure coefficient and the Stanton number obtained by HS, VHS, and VSS using the DSMC method are shown in Figures
DSMC simulation of Run number 32 (
DSMC simulation of Run number 38 (
DSMC simulation of Run number 32 (
DSMC simulation of Run number 38 (
Table
Comparisons of calculations
Run | Kn |
The scale of interaction region |
Peak value of the Stanton number | ||||
---|---|---|---|---|---|---|---|
HS | VHS | VSS | HS | VHS | VSS | ||
33 |
|
|
73.4 | 53.4 |
|
209 | 245 |
36 |
|
|
70 | 43 |
|
92.8 | 78 |
26 |
|
|
66 | 38 |
|
85.7 | 48.5 |
32 |
|
|
59 | 31 |
|
57.8 | 47.4 |
38 |
|
|
72.4 | 31.04 |
|
68.4 | 42.1 |
In the present study, results of the numerical simulations for Mach 10 air flow, as shown in Table
Flow conditions of Edney’s type IV shock interaction.
Ma | Re/m | Kn |
---|---|---|
10 |
|
|
Configuration of Edney’s Type IV shock interaction.
The results of the grid sensitivity investigation are presented in [
The DSMC data of the pressure coefficient and the Stanton number are shown in Figures
DSMC of type IV shock interaction (
DSMC of type IV shock interaction (
The result from the DSMC calculation for Edney’s type IV shock interaction also confirmed the same conclusion. The error is the smallest in the HS model compared with those in the VHS and VSS models. In the region of large Kn number, DSMC using the HS model could be used in pressure and thermal prediction of hypersonic complex flow.
In the present study, the double-cone and Edney’s type IV shock interactions were studied. Molecular models were employed with DSMC including the HS, VHS, and VSS models. Comparisons between DSMC and the experiment were also conducted.
Drawing firm conclusions from this study prior to full comparisons of these results is somewhat difficult. However, some issues are immediately clear from our attempt to compute these flows. The DSMC method appears to predict qualitatively the structure of flow separation found in previous CFD calculations. Investigation of the double-cone hypersonic flow reveals that the three collisions models can predict the pressure coefficient and the Stanton number trends. The agreement between the DSMC simulation and the experiment is the best in the HS collisions model. Investigation of type IV shock interaction shows that the agreement between the DSMC simulation and the experiment is generally good in the HS and VHS collision models. However, it fails in the VSS model. Both double-cone flow and type IV shock interaction simulations show that the DSMC error depends on the Kn number and the models employed for intermolecular interaction. Increasing the Kn number decreases the DSMC error. The error is the smallest for the HS model compared with those for the VHS and VSS models. When the Kn number is in the level of
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported in part by the Chinese Aerospace Innovation and in part by the Basic Research Fund of Northwestern Polytechnical University.