Numerical solution of incompressible Boussinesq equations using fourth-order compact scheme: Lock exchange flow

In recent years, the number of research works devoted to applying the highly accurate numerical schemes, in particular compact finite difference schemes, to numerical simulation of complex flow fields with multi-scale structures, is increasing. The use of compact finite-difference schemes are the simple and powerful ways to reach the objectives of high accuracy and low computational cost. Compact schemes, compared with the traditional explicit finite difference schemes of the same order, have proved to be significantly more accurate along with the benefit of using smaller stencil sizes, which can be essential in treating non-periodic boundary conditions. Applications of some families of the compact schemes to spatial differencing of some idealized models of the atmosphere and oceans, show that the compact finite difference schemes are promising methods for numerical simulation of the atmosphere–ocean dynamics.
This work is devoted to the application of a fourth-order compact finite difference scheme to numerical solution of gravity current. The governing equations used to perform the numerical simulation are the two dimensional incompressible Boussinesq equations. The two-dimensional lock-exchange flow configuration is used to conduct the numerical simulation of the Boussinesq equations. The lock-exchange flow is a prototype problem which has been studied numerically and experimentally by many researchers.
For the spatial differencing of the governing equations the second-order central and the fourth-order compact finite difference schemes are used. The predictor-corrector leapfrog scheme is used to advance the Boussinesq equations in time. The boundary condition formulation required to generate stable numerical solutions without degrading the global accuracy of the computations, is also presented.
The fourth-order compact scheme is compared in detail with the conventional second-order central finite difference method. Qualitative comparison of the results of the present work with published results for the planar lock-exchange flow indicates the validity and accuracy of the fourth-order compact scheme for numerical solution of the two-dimensional incompressible Boussinesq equations.