Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-Dimensional Viscous Compressible Flow in a Shock Tube Using Entropy and Kinetic Energy Preserving Schemes

Abstract

This paper follows up on the author's recent paper "The Construction of Discretely Conservative Finite Volume Schemes that also Globally Conserve Energy or Enthalpy".In the case of the gas dynamics equations the previous formulation leads to an entropy preserving (EP) scheme. It is shown in the present paper that it is also possible to construct the flux of a conservative finite volume scheme to produce a kinetic energy preserving (KEP) scheme which exactly satisfies the global conservation law for kinetic energy. A proof is presented for three dimensional discretization on arbitrary grids.Both the EP and KEP schemes have been applied to the direct numerical simulation of one-dimensional viscous flow in a shock tube. The computations verify that both schemes can be used to simulate flows with shock waves and contact discontinuities without the introduction of any artificial diffusion. The KEP scheme performed better in the tests.2

Analysis of One-Dimensional Inviscid and Two-Dimensional Viscous Flows Using Entropy Preserving Method

“…It should be noted that governing equations were written in references [1] and [2]. In this section, derivation of the available equations is shown in more details comparing to the cited references.…”

“…The rate of energy changes which is predicted by discrete equation will be the same as the exact solution only if the amount of flux is calculated using the following equations [1] f j+…”

“…Weak solutions of conservative laws (solutions with discontinuity like compression wave) with applied initial conditions may do not have unique solution [1]. To eliminate the irrelevant physical solutions, additional conditions must be applied.…”

Aerodynamics, computers and the environment

“…The emissions targets in terms of CO 2 , NO x and noise are defined. The paper has a positive message on the future role of CFD and its particular importance for simulating coupled airframe and engine interactions where rig tests will be especially expensive.…”

“…Hence, now the ability to directly predict turbulence, for complex engineering systems, without recourse to accuracy reducing assumptions is close at hand. Computer-processing speeds have increased by a factor of around one million in the past 25 years (see Jameson [2]), and continued improvements are expected.…”