A consistent scheme of the high-speed source term in probability density function methods for supersonic flows

To improve the particle energy accuracy of the probability density function (PDF) method in supersonic flows, a consistent numerical scheme for the high-speed source term in the particle energy equation is put forward. The proposed scheme is designed and computed based on the characteristic format, which shares the same right and left eigenvectors for the Jacobian of the convection term of the system. Therefore, this scheme is fully consistent with the spatial discretization of the inviscid term in the finite-difference solver of compressible flows. To show the advantages of the proposed scheme, a redundant energy equation is solved along with the Euler equation. The consistent scheme and the quasi scheme are numerically tested and compared in several canonical flows. The results show that the high-speed source term calculated by the consistent scheme generates much less numerical oscillation than the quasi scheme around discontinuities. Due to the accumulation of the numerical errors of the high-speed source term in every single iteration, the energy redundantly computed by the consistent scheme agrees better with the Euler results than the other scheme. Following this new high-speed source computation scheme, the large eddy simulation-PDF method is further developed and tested in a shock tube problem interacting with an isotropic turbulent flow and a supersonic temporally developing mixing layer. The results show that PDF with this consistent high-speed source scheme can improve the energy accuracy as well as turbulent combustion in supersonic flows.

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