Continuum breakdown in hypersonic viscous flows

This paper presents a study of the breakdown of the Navier-Stokes equations in hypersonic viscous flows over a sharp cone tip and a hollow cylinder/flare ge- ometry. Investigations are performed through detailed comparisons of the numerical results obtained with continuum and particle techniques. The objective of the study is to predict conditions under which the con- tinuum approach may be expected to fail. A modified breakdown parameter is proposed that can predict the failure of the continuum approach accurately for the simple cone flow and fairly well for the more complex cylinder/flare flow. The study of continuum break- down is the first step toward development of a hybrid numerical code. Introduction Numerical simulation of hypersonic viscous flows over complex geometries is of great importance be- cause of its application in trans-atmospheric vehicle design. In the flight of such a vehicle, the hypersonic free-stream undergoes large variation in properties due to interactions with shock waves from a wing or con- trol surface and with the boundary layer from the wall surface. The large variation in properties results in some regions where the flow is described as a contin- uum and can be modeled by the Navier-Stokes (NS) equations and solved numerically by Computational Fluid Dynamics (CFD) approaches. The wide varia- tion in flow properties may also lead to some regions where the flow is rarefied and the NS equations break down because of physical limitations. A particle simu- lation technique such as direct simulation Monte Carlo (DSMC) is commonly employed in this region. The DSMC method cannot be used for the full sys- tem, as it demands huge amounts of computational capacity in regions where the flow is dense and in the continuum regime. For instance, in Refs. 1 and 2, com- putations of the same geometry and free-stream con- ditions (Run 28) were performed by CFD and DSMC methods, respectively. The CFD method consumed * Graduate Student Research Assistant, AIAA Student Mem- ber (aerowwlQengin.umich.edu) t Associate Professor, AIAA Senior Member (iainboydQengin. umich. edu) Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. about 20 hours of 32 processors of an IBM-SP machine and captured all the flow details in good agreement with experiment whereas the DSMC method spent more than twice the computational time and a lot more memory but satisfactory results still were not reached. With the well known fact that the NS equa- tions will fail in rarefied flows, it is necessary to have an approach that is physically accurate and numeri- cally efficient. One way to achieve this objective is to combine the DSMC and CFD methods. There are two primary issues associated with the combination of the two numerical methods. First of all, we need to determine when to switch between the methods. As the NS equations are not valid under rar- efied conditions, it is general to use a continuum break- down parameter as the criterion for switching between methods. In his pioneering work, Bird first advo- cated a semi-empirical parameter for expanding flows. Another empirical parameter based on local flow gra- dients was later developed specifically for hypersonic flows. More recently, a new breakdown parameter based on the Chapman-Enskog perturbation expan- sion of the Boltzmann equation has been developed again for expanding flows. In the present investiga- tion, the determination of an applicable parameter for hypersonic flows is addressed. The second issue concerned in a hybrid DSMC-CFD approach is to deal with the information exchange at the interface of the two methods. Several approaches have been considered, such as the Marshak condi- tion, the KFVS scheme' and AMAR embedding a particle method. Unfortunately, none of these schemes was designed for nonequilibrium, hypersonic compressed flows. Since this issue is beyond the scope of the present research, it will be left for the future study. The layout of the paper is as follows. A description of two types of breakdown parameter is provided in the next section. We will show the relations between these parameters and propose a modified parameter that is believed to be more conservative and adequate for complex flows. In the section of Numerical Examples, a hypersonic flow over a simple axisymmetric cone tip is first considered, followed by a hypersonic flow over a relatively complex hollow cylinder/flare geometry. In the last section, conclusions and suggestions for future