Numerical solution of incompressible Boussinesq equations using sixth-order combined compact scheme

In recent years, substantial amounts of research work have been devoted to using highly accurate numerical methods in the numerical solution of complex flow fields with multi-scale structures. The compact finite-difference methods are simple and powerful ways to attain the purpose of high accuracy and low computational costs. 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 the numerical simulation of the atmosphere–ocean dynamics problems.
This work is devoted to the application of the combined compact finite-difference method to the numerical solution of the gravity current problem. The two-dimensional incompressible Boussinesq equations constitute the governing equations that are used here for the numerical simulation of such flows. The focus of this work is on the application of the sixth-order combined compact finite difference method to spatial differencing of the vorticity-stream function-temperature formulation of the governing equations. First, we express formulation of the governing equations in dimensionless form. Then, we discretize the governing equations in time and space. For the spatial differencing of the governing equations, the sixth-order combined compact finite difference scheme is used and the classical fourth-order Runge–Kutta is used to advance the Boussinesq equations in time. Details of spatial differencing of the boundary conditions required to generate stable numerical solutions are presented. Furthermore, the details of development and implementation of appropriate no-slip boundary conditions, compatible with the sixth-order combined compact method, are presented. To assess the numerical accuracy, the Stommel ocean circulation model with known exact analytical solution is used as a linear prototype test problem. The performance of the sixth-order combined compact method is then compared with the conventional second-order centered and the fourth-order compact finite difference schemes. The global error estimations indicate the better performance of the sixth-order combined compact method over the conventional second-order centered and the fourth-order compact in term of accuracy.
The two-dimensional planar and cylindrical lock-exchange flow configurations are used to conduct the numerical experiments using the governing Boussinesq equations. In this work, we used the no-penetration boundary conditions for temperature and no-slip boundary conditions for vorticity at walls compatible with the sixth-order combined compact scheme.The results are then compared qualitatively with the results presented by other researchers. Quantitative and qualitative comparisons of the results of the present work with the other published results for the planar lock-exchange flow indicate the better performance of the sixth-order combined compact scheme for the numerical solution of the two-dimensional incompressible Boussinesq equations over the second-order centered and the fourth-order compact methods. Hence, such methods can be used in numerical modelling of large-scale flows in the atmosphere and ocean with higher resolutions.