Finite-difference 4th-order compact scheme for the direct numerical simulation of instabilities of shear layers

An algorithm based on the 4th-order finite-difference compact scheme is developed and applied in the direct numerical simulations of instabilities of channel flow. The algorithm is illustrated in the context of stream function formulation that leads to field equation involving 4th-order spatial derivatives. The finite-difference discretization in the wall-normal direction uses five arbitrarily spaced points. The discretization coefficients are determined numerically, providing a large degree of flexibility for grid selection. The Fourier expansions are used in the streamwise direction. A hybrid Runge-Kutta/ Crank-Nicholson low-storage scheme is applied for the time discretization. Accuracy tests demonstrate that the algorithm does deliver the 4th-order accuracy. The algorithm has been used to simulate the natural instability processes in channel flow as well as processes occurring when the flow is spatially modulated using wall transpiration. Extensions to three-dimensional situations are suggested.