Modified Entropy Correction Formula for the Roe Scheme

1To avoid un-realistic solutions like expansion shocks from appearing as a part of a solution it is necessary to satisfy the entropy condition for the Roe scheme. A vari-ety of entropy fix formulae for the Roe scheme have been addressed in the literature. Three of the most famous are due to Harten-Hyman and Hoffmann-Chiang. These for-mulations have been assessed in this paper by applying them to the inviscid Burgers ’ equation and shock tube problem. These entropy fix formulations are unable to totally diminish the expansion shock in the vicinity of sonic expansions. Moreover, they are not universal, i.e. a single formulation is not adequate for the scalar Burg-ers ’ equation, shock tube problem and multi-dimensional cases and different formulations were suggested for each case. A simple modification to the Harten formulation is presented in this paper. This modification basically enlarges the band over which the entropy fix condition is enforced. The resulting formulation is able to totally remove the non-physical expansion shocks from the re-gion of sonic expansion without affecting the rest of the computational domain. Comparison among the exact solution, and the entropy correction formulae of Harten-Hyman and Hoffmann-Chiang and the currently mod-ified formula are shown here. The modified entropy fix formulation can totally diffuse the expansion shock. Moreover, the current formula does not affect the solu-tion in the rest of the computational domain. Besides the modified formula, as a single formulation, can uni-versally be applied over a wide range of applications from scalar equations to the governing equation of fluid mo-tion. Finally in this paper, the following test cases are performed to assess the accuracy of the modified entropy formulation: inviscid shear flow, transonic flow over a bump, and transonic flow in a Laval Nozzle. Very accu-rate results are obtained. In the current study a second order upwind scheme of Roe with minmod flux limiter is applied. ∗Ph.D. Candidate, AIAA student member