Computation of hypersonic transitional flows by solving three-dimensional Burnett equations

Abstract

This paper describes the recent research progress in computation of hypersonic transitional flows by solving the three-dimensional Burnett equations. The derivation of the three-dimensional Burnett equations is presented. A three-dimensional computer code, Burnett-3D to solve the three-dimensional Burnett equations was developed. The Burnett-3D code adapts the solution methodology in the NPARC^ flow simulator. The Beam and Warming approximate factorization algorithm is used to discretize the connective terms of the Burnett equations. Higher-order Burnett viscous stress and heat flux terms are discretized using central-differencing scheme and axe treated as source terms explicitly. The three-dimensional hypersonic transitional flow over an ellipsoid body is investigated and the results are presented in this paper. The free stream Mach number is 8.0, the reference free stream Reynolds number ranges from 80 to 10,000. Studies of the transitional flow characteristics in terms of free stream Reynolds number are presented. Comparisons between the Burnett and Navier-Stokes solutions show that as the Reynolds number decreases to 80, the transitional flow characteristics can be captured by the Burnett solutions. This study represents the first effort to numerically investigate the transitional aerodynamics based on solutions of the three-dimensional Burnett equations. INTRODUCTION For hypersonic flight vehicles maneuvering at altitude above 75 km, the flow field around the vehicle is in the transitional to rarefied regimes. In this regime, the Knudsen number Kn, which is based on the scale length of the gradients of the macroscopic flow properties, is in the range between 0.1 to 10.0. In general, the flow in this regime is characterized by a low Reynolds number and a high Mach number. Because of the extreme difficulties in obtaining flight and ground testing data, the transitional aerodynamics above 75 km are poorly understood. Even with the current X-33 advanced demonstrator it would involve speeds only up to Mach 15 and altitudes only up to approximately 75 km. It is imperative to investigate the transitional flow characteristics through numerical simulation. As the altitude is below 75 km, the flow field around the hypersonic flight vehicle is generally in the continuum regime. The conventional Navier-Stokes equations are valid. As the altitude increases from 75 to 100 km, the ambient air becomes gradually rarefied. As shown in Figure 1, the flow is primarily in the transitional regime. The conventional constitutive equations that relate the shear stress and heat fluxes to these gradients breakdown. The Navier-Stokes equations fail to provide accurate solutions in this transitional regime. Figure 2 shows the literature reviews on the accuracy of the numerical algorithm and its limitations. It concluded that in order to take advantage of the continuum modeling and to improve the solution accuracy, the Burnett equations are more accurate than the Navier-Stokes equations in solving the transitional aerodynamics. Mathematically, the Navier-Stokes and Burnett equations were derived from the Hilbert-ChapmanEnskog expansion of the Boltzmann equation. Based on this expansion, the velocity distribution function was expressed in the series expansion of Knudsen number, f Assistant Professor, Member AIAA J Professor, Member AIAA Copyright ©American Institute of Aeronautics and Astronautics, Inc., 1998. All rights reserved. which is the perturbation expansion of the velocity distribution function about the Maxwellian distribution, where /(°) is the Maxwellian distribution function, aj, i = 1,2, 3, ... are functions of density, molecular velocity, and temperature. Substituting this expansion into Boltzmann equation, taking moments of the Boltzmann equation yields the continuum equations of fluid mechanics, a set of conservation equations describing global conservation of density, momentum, and energy. To close this system of equations requires constitutive equations which express viscous stress and heat flux in terms of the distribution function rather than macroscopic gradients. Closure of these equations can be achieved by using any of the nth order approximations to the distribution function described in the expansion. The constitutive conditions relating viscous stress tensor and heat flux to