Numerical computation of spatially developing flows by full- multigrid technique

Abstract

The instability of spatially developing laminar flows, such as that through converging/diverging channels, is often fundamentally different from flows that do not vary downstream. Another class of flows whose stability and transition behaviour is not well understood is pulsatile flows. In most laminar flows which fall under these categories, it is not possible to obtain the basic flow profiles analytically. The aim of this thesis is to develop codes which will compute the basic flow for two dimensional and axisymmetric geometries. A long-term objective is to understand the transition to turbulence in such flows. The Navier-Stokes equations in the vorticity and streamfunction formulation have been solved for computing the mean flow. Two types of spatially developing flows have been considered, namely, flow in a divergent channel and axisymmetric flow in a divergent pipe. The code can handle unsteady problems, but has been used up to now to solve a pseudo-unsteady problem to obtain steady state solutions. The Gauss-Seidel iteration method was found to be alarmingly slow in solving the elliptic streamfunction and vorticity equation with vorticity as a source term. To increase the rate of convergence, a multigrid technique has been implemented. Algorithms like Jacobi or Gauss-Seidel are local because the new value for the solution at any lattice site depends only on the value of the previous iterate at neighbouring points. The basic idea behind multigrid technique is to reduce long wavelength error components rapidly by updating blocks of grid points. We used a simple V-cycle for the present algorithm. For Poisson equation with 128x128 number of grids, using six multigrid levels, it was found that multigrid technique is about a hundred times faster then the Gauss-Seidel method. The present code has been tested with a number of experimental and theoretical bench-mark results for the developing flow in a channel and a flow in a backward-facing step. The multigrid algorithm has been compared with the Gauss-Seidel iteration method for the Poisson equation. With the present code, we were able to simulate the separated flow with reattachment for a divergent channel and pipe, with straight exit portions, for different angles of divergence and Reynolds number. As an analytical solution is not possible for such kind of flows, with large angle of divergence, solution of full Navier-Stokes equation is required. The code in the present form can be used for this purpose. In summary, we now have the capability of studying the stability of a wide class of spatially developing and time-periodic flows.