Numerical Investigation Of Rarefaction Effect on wall flux in the vicinity of a sharp leading edge in a mach 6 argon flow
Shaowu Pan
National Lab for Computational Fluid Dynamics
Beihang University, Beijing, China
Advised by Professor Chun-hian Lee, (co-advised by Assistant Professor Zhenxun Gao)
Reference
Shaowu Pan, Zhenxun Gao, and Chunhian Lee. "Numerical investigation of rarefaction effects in the vicinity of a sharp leading edge." Proceedings of the 29th International Symposium on Rarefied Gas Dynamics. Vol. 1628. AIP Publishing, 2014
Shaowu Pan, Zhenxun Gao, and Chunhian Lee. "Numerical investigation of rarefaction effects in the vicinity of a sharp leading edge." Proceedings of the 29th International Symposium on Rarefied Gas Dynamics. Vol. 1628. AIP Publishing, 2014
Overview
A hypersonic rarefied flow about a sharp leading edge is studied numerically by solve compressible N-S-Newton-Fourier equations and kinetic (Direct Simulation Monte Carlo) approaches (which is identical with solution of Boltzmann equation). The DSMC tool is Multi-species Nonequilibrium Flow Simulator (MNFS) developed by Shaowu Pan, independently by FORTRAN 95 in part-time during undergraduate years. This paper is divided into three parts: 1) Code validation - a supersonic flat plate 2) analyzed rarefaction effect in detail by CFD, DSMC and comparing with theoretical results. 3) evaluated two continuum breakdown parameters proposed in current paper based on translational nonequilibrium. Mach number is 6 and pure argon is considered to exclude effect of nonequilibrium for internal energy. Freestream temperature is 300 K and wall temperature is 500 K. Four hypersonic inflow conditions: Kn = 0.05, 0.1, 0.2, 0.4 are considered here.
A hypersonic rarefied flow about a sharp leading edge is studied numerically by solve compressible N-S-Newton-Fourier equations and kinetic (Direct Simulation Monte Carlo) approaches (which is identical with solution of Boltzmann equation). The DSMC tool is Multi-species Nonequilibrium Flow Simulator (MNFS) developed by Shaowu Pan, independently by FORTRAN 95 in part-time during undergraduate years. This paper is divided into three parts: 1) Code validation - a supersonic flat plate 2) analyzed rarefaction effect in detail by CFD, DSMC and comparing with theoretical results. 3) evaluated two continuum breakdown parameters proposed in current paper based on translational nonequilibrium. Mach number is 6 and pure argon is considered to exclude effect of nonequilibrium for internal energy. Freestream temperature is 300 K and wall temperature is 500 K. Four hypersonic inflow conditions: Kn = 0.05, 0.1, 0.2, 0.4 are considered here.
|
For code validation, the classic example supersonic flat plate is considered here to check the validity of MNFS. It is shown in pressure contour that viscous interaction is significant. Nonequi- librium phenomena is obvious in rotati- onal temperature contour since the only way to transfer energy from translation to rotation is inelastic collision. Pressure coefficient compared with previous results are presented (see figure on the left). Drag and heat coefficient are also presented. The result shows current calculation has a good agreement with data in previous literature. The pressure coefficient reaches a peak after about five λ∞ because local Knudsen number exceeds the limit of continuum assumptions .From a kinetic perspective, the insufficient collisons due to rarefaction near the leading edge makes the peak of pressure coefficient a certain distance away from the leading edge. Under this circumstances, the tendency of drag coefficient is similar to that of classic Blasius solution as it is showed (see figure on the left)
|
Simulation of a Mach 6 argon flow over a sharp leading edge is conduct. It is clear that rarefaction would affect flowfield to a great extent. (see figure on the right) A comparison of temperature contour for two cases: Kn=0.1 and 0.4 is presented. It is clear that the inviscid region is larger in more rarefied cases. An important scale in compressible flow - shockwave stand-off distance - is much larger in Kn=0.4 case than that in Kn=0.1 case. Moreover, temperature is much higher in the more rarefied case, which may be resulted from the weakened heat transfer near cold wall due to insufficient collision in rarefied condition.
|
|
|
Surface properties, i.e., pressure, drag, heat transfer, wall influx coefficient are presented for different Kn numbers ranging from 0.05 to 0.4 (see figure on the left). As rarefaction effects increases from Kn=0.05 to 0.4, first a general trend is that the distribution becomes more and more uniform along the wall, which may be caused by less collisions in more rarefied condition that results in less interaction between wall and molecules. A step forward, it is coin- cident with Newton's sin-squared law that surface property has a direct relationship with local inclination. Second, as rarefaction increa- ses, pressure, drag, heat transfer coefficients becomes larger under current circumstances, which is also confirmed in previous literature.
|
A detailed comparison for CFD and DSMC approaches is conduct for Kn = 0.01, 0.05, 0.1, 0.4,.(see figures on the left) It is revealed in pressure, temperature contour that the fully diffusive wall will affect a larger area when the flows comes more rarefied as it is also observed in previous literature. The shockwave is much thicker in higher Kn as it is displayed in the figures on the right. Moreover, CFD underestimates temperature, pressure near the leading edge, which is also confirmed by observation in previous studies. A step forward, the property along the stagnation line for hypersonic flow past a sharp leading edge is not studied in previous literature according to author's limited knowledge. The variation of shockwave stand-off distance can be clearly seen in figures below. It should be noted that CFD underestimates again in pressure and temperature along the stagnation line with a small lag off in onsite of shockwave. It is different in the cases of 2-D cylinder since in that case the region near the stagnation point is near-continuum while in sharp leading edge, it is still in non-equilibrium due to the infinite small radius.
|
Two novel parameters for predicting continuum breakdown based on nonequilibrium in translational temperature are proposed. These two parameters are intrinsically consistent. It is seen that these two parameters are non-zero before discrepancy happens between CFD and DSMC. It can be explained by the strong sensitivity of translational temperature in nature. Based on current numerical results, 0.2 seems to be a safe cut-off number for parameter 1. Also, it is interesting to point out that, the results obtained from CFD in the case of high Kn (0.4) is better than what might be predicted from the cut-off parameter. Despite of the strong sensitivity, it can be seen that this method might not work very good under high Kn. But it might help for giving a safe, simple criteria. Further study is still needed to give an accurate criteria for high Kn flows.
|
It is clear that as the flow becomes more and more rarefied, the entropy layer is thicker and grows much faster than boundary layer for the three cases (S/L=22%, 56% ,90%). Also, the entropy is larger at most region for more rarefied condition. Moreover, two maximum points are found in relative low Kn cases, but disappear as the flow becomes more rarefied or closer at the front leading edge. However, there is tendency for returning to two maximum points for high Kn case at downstream (S/L=90%). This phenomena could be explained by the overlap of thick shockwave and boundary layer.
|
The leading edge area might affect the flow downstream even though rarefaction effects near the leading edge are insignificant. The flow passing through the near-normal bow shock detached from the blunted leading edge causes formation of an inviscid entropy layer near the body. Further downstream, the entropy layer is swallowed by the growing viscous boundary layer. Entropy/boundary layer interaction causes significant changes in boundary layer charact- eristics. In particular, the entropy layer greatly affects the boundary layer stability. At small bluntness, the laminar/turbulent transition point is displaced downstream. Greater bluntness produces a boundary layer transition reversal, i.e. additional blunting produces forward displacement of the transition point. The swallowing distance of the entropy layer is an important correlation parameter for the transition reversal in previous literature. In current paper, a infinite small bluntness leading edge is considered for a first step to study this problem. Further study should be focused on a comparison between CFD and DSMC on a blunt leading edge.
Due to the nature of this study is comparison on entropy not quantifying. So a simplified form of entropy is taken here. After numerical simulation, it is found that CFD underestimates the entropy generally in this case (see figures above). But the error is relatively negligible probably due to the small entropy increase in oblique shockwave in the flow past a 14° wedge. It is also important to note that CFD underestimates the entropy layer thickness by 20% in a certain surface normal to the freestream, which may lead to a inaccurate process of entropy swallowing. |
The most important calculation for aircraft design is wall flux, i.e.,pressure, drag, heat transfer. At a relative small Kn = 0.01, CFD, DSMC and analytic results in free molecule flow limit predictions for the above parameters after dimensionless process are shown. (see figure on the right). It is found that CFD has a good agreement with DSMC in drag and heat transfer prediction for all four cases except for region near leading edge. Moreover, it is shown in current cases that the analytic results in free molecule flow limit have a good agreement with DSMC at the leading edge for Cf and Ch, which can be explained by the relative high Kn near leading edge. However, it is not the same rule for Cp. But the value at the leading edge seems to be nearly the same in all the cases considered, with a bit higher that what free molecular theory predicts. Moreover, it can be seen that the analytic methods in free molecule flow, underestimates pressure but overpredicts heat transfer and drag. Future studies on the verification of this kind of approach on wedge-like configuration is needed since it would be beneficial to aircraft design of future hypersonic vehicle.
|
|