High Resolution Entropy Consistent Schemes For Compressible Navier-stokes Equations

With the rapid development of computer technology and numerical methods, numerical simulation has become an important technique of computational fluid dynamics research. For nonlinear hyperbolic conservation laws, even if the initial conditions are very smooth, the discontinuous solution may also occur at finite time, i.e., the shock wave formation. The design principle of many numerical methods is to be able to have good shock capturing ability. Based on the physical background of the Second Law of Thermodynamics, this thesis introduces the construction methods and theories of the entropy conservation / entropy stable / entropy consistent scheme, develops a new high-resolution entropy consistent scheme, and finally extends it to solve the Navier-Stokes equation. The main works are as follows:1) At first, the related background knowledge of the hyperbolic conservation laws and some numerical methods were introduced to pave the way for the following study.2) The constructing theories and ideas of entropy conservation / entropy stable / entropy consistent schemes were introduced in detail. Based on the second order entropy conservation schemes which Tadmor had put forward, the viscous term was added to them and the concept of entropy production was introduced. When the entropy production is less than zero, the right dissipation direction of the numerical solution is ensured and oscillation is eliminated. At the same time, the above scheme becomes the entropy stable scheme with first order accuracy. When entropy production is less than zero and can reach the cube order of the shock strength, the scheme becomes the entropy consistent with first order accuracy.3) Based on the constructing idea of TVD schemes and with the flux limiters, a kind of high-resolution entropy consistent schemes were developed. Numerical experiments for Burgers equation and Euler equations shows that the new schemes have features of high accuracy, non-oscillation, and high-resolution for shocks.4) One of the high resolution entropy consistent scheme was extended to the Navier-Stokes equations. Navier-Stokes equations are described as the Newtonian viscous fluid for aero-dynamics, which does not belong to hyperbolic conservation laws; but when its viscous terms approaches zero, Navier-Stokes equations becomes Euler equations. Numerical results showed that the high resolution entropy consistent schemes were feasible to extend to solving Navier-Stokes equations, and the schemes have strong stability and non-oscillatory.