Ramesh Balakrishnan

Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime

In hypersonic flows about space vehicles in low earth orbits or flows in microchannels of microelectromechanical devices, the local Knudsen number lies in the continuum–transition regime. Navier–Stokes equations are not adequate to model these flows since they are based on small deviation from local thermodynamic equilibrium. To model these flows, a number of extended hydrodynamics or generalized hydrodynamics models have been proposed over the past fifty years, along with the direct simulation Monte Carlo (DSMC) approach. One of these models is the Burnett equations which are obtained from the Chapman–Enskog expansion of the Boltzmann equation [with Knudsen number (Kn) as a small parameter] to O(Kn2). With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with the Burnett model. As a result, several improvements to the conventio...

Numerical Simulation of Bhatnagar -Gross-Krook— Burnett Equations for Hypersonic Flows

A kinetic-theory-based upwind algorithm for the Bhatnagar - Gross - Krook (BGK) - Burnett equations is presented. The Boltzmann equation, with the BGK approximation for the collision integral, describes the spatial and temporal variations of the second-order distribution function that forms the basis of this formulation. The second-order distribution function is derived by considering the first three terms in the Chapman-Enskog expansion and using the Navier-Stokes equations to express the material derivatives, present in the second-order terms, in terms of the spatial derivatives. The BGK-Burnett equations are derived by taking moments of the BGK-Boltzmann equation with the collision invariant vector. A kinetic wave/particle split scheme for the BGK - Burnett equations is derived by taking moments of the upwind discretized BGK - Boltzmann equation. This algorithm is applied to a hypersonic shock structure problem. This is the first time that a kinetic-theory-based method has been developed for solving the BGK-Burnett equations.

BGK-Burnett Equations for Flows in the Continuum-Transition Regime

To extend the range of applicability of continuum formulations into the continuum-transition regime, an extended set of fluid dynamic equations has been derived. These equations, termed as the Bhatnagar-Gross-Krook (BGK)-Burnett equations, have been derived by taking moments of the Boltzmann equation by using the BGK model for the collision integral, The second-order distribution function that forms the basis of this derivation is formulated by considering the first three terms of the Chapman-Enskog expansion. It is shown that the BGK-Burnett equations have been used to compute the hypersonic shock structure and the hypersonic flow past a blunt body. The results of these computations are compared with the augmented Burnett and Navier-Stokes solutions. The second-order distribution function does not violate Boltzmanns H-theorem; as a consequence the BGK-Burnett equations are entropy consistent for the range of Knudsen numbers for which computations have been performed

A comparative study of several higher-order kinetic formulations beyond Navier-Stokes for computing the shock structure

This paper presents a comparitive assessment of the BGKBurnett, conventionul Burnett and Woods equations with regard to the linearized stability analysis and hypersonic shock structure computations. A detailed stability analysis of these equations is carried out and the role of the internal energy term on the stability of the BGK-Burnett equations is explored. The results of the computations are presented. Nomenclature et : (0) f(l) = fwp f(2) G :, = (f’O’lI) ?= I/IO Pr 4i Q 72 = (-cqoo) R+ = [O, Co) R= (-CqO] T

A kinetic theory based scheme for the numerical solution of the BGK-Burnett equations for hypersonic flows in the continuum-transition regime

A kinetic theory based upwind algorithm for the BGK-Burnett equations is presented. The Boltzmann equation, with BGK approximation for the collision integral, describes the spatial and temporal variations of the second-order distribution function which forms the basis of this formulation. The second order distribution function is derived by considering the first three terms in the Chapman-Enskog expansion and using the Navier-Stokes equations to express the material derivatives, present in the second-order terms, in terms of the spatial derivatives. The Burnett equations are derived by taking moments of the BGK-Boltzmann equation with the collision invariant vector. A Kinetic Wave/Particle Split scheme for the BGK-Burnett equations is derived by taking moments of the upwind discretized Boltzmann equation. This algorithm is applied to a 1D shock tube problem and a hypersonic shock structure problem. This is the first time that a kinetic-theory based method has been developed for solving the Burnett equations. (Author)

On the performance of SPAI and ADI-like preconditioners for core collapse supernova simulations in one spatial dimension

The simulation of core collapse supernovae calls for the time accurate solution of the (Euler) equations for inviscid hydrodynamics coupled with the equations for neutrino transport. The time evolution is carried out by evolving the Euler equations explicitly and the neutrino transport equations implicitly. Neutrino transport is modeled by the multi-group Boltzmann transport (MGBT) and the multi-group flux limited diffusion (MGFLD) equations. An implicit time stepping scheme for the MGBT and MGFLD equations yields Jacobian systems that necessitate scaling and preconditioning. Two types of preconditioners, namely, a sparse approximate inverse (SPAI) preconditioner and a preconditioner based on the alternating direction implicit iteration (ADI-like) have been found to be effective for the MGFLD and MGBT formulations. This paper compares these two preconditioners. The ADI-like preconditioner performs well with both MGBT and MGFLD systems. For the MGBT system tested, the SPAI preconditioner did not give competitive results. However, since the MGBT system in our experiments had a high condition number before scaling and since we used a sequential platform, care must be taken in evaluating these results.