High-order FV-WENO Scheme Based Numerical Simulation Of A Gravity Current By Parabolized Navier-Stokes Equation

The finite-volume weighted essentially non-oscillatory schemes (FV-WENO) can be verified to have the properties such as robustness,the genuine high order accuracy in smooth regions,and the non-oscillatory property of general solutions with discontinuities.A gravity current is a flow of relative motion which are driven by a density difference.Gravity currents are widely applied in natural environment. Great effect is had on the transport of the airborne pollution, the dam break,the spread of the oil slick etc.The fourth-and fifth-order FV-WENO schemes in this paper are presented and employed to simulate by parabolized Navier-Stokes equations a typical gravity current characteristic of Rayleigh-Taylor instability with wide applications in envi-ronmental hydrodynamics.The third-order TVD Runge-Kutta method is used to discretize the partial derivative term of the time. The Burgers equation is adopted to demonstrate the numerical orders of accuracy of the methods.The numerical research is done for the gravity current as follows:(1)The convergence of the numerical results is discussed for the fourth-and fifth-order FV-WENO schemes.(2)The conversation of the schemes is analyzed in this paper, and the study shows that the conservation of mass is well reserved for the fourth-and fifth-order FV-WENO schemes and the fifth-order finite-difference weighted essentially non-oscillatory(FD-WENO) scheme.(3)In order to find more economical and efficient scheme,we compare the mesh numbers and CPU time needed by the fourth-and fifth-order FV-WENO schemes and the fifth-order FD-WENO scheme to reach the same resolution.(4) The effect of the boundary condition on the numerical resolutions is dis-cussed.In the condition of the same mesh size,for example, h=1/(480),and the same numerical method adopted, for example,the fifth-order FV-WENO scheme, the results at different time are obtained under the solid boundary condition and the reflective boundary condition, respectively. The y coordinations of the top of the spikes are captured.