Development and critical assessment of a new flic method for base flows

Abstract

The motivation for the present work stems from two complex base flow problems of relevance to satellite launch vehicles, namely the flow about the base of a rocket body containing an exhaust jet and the aerodynamics of stage separation during the flight of a multistage rocket. With the rapid increase in computer power and advances in numerical techniques, numerical solution of Navier-Stokes equations has become possible. The present work formulates and adopts a new Fluid-in-cell (NFLIC) method for obtaining steady-state Navier-Stokes solution of laminar base flows, which may include flow past a backward-facing step, base flow with a single jet exhaust, and stage separation aerodynamics. Before doing this, we first investigate the effectiveness of the classical Fluid-in-cell method (termed FLIC), which consists of two phases. Phase I neglects convection effects and considers change in the solution due to pressure and viscous terms alone, while Phase II considers convection effects neglected in Phase I. The finite difference operator in Phase I employs central differencing for pressure and viscous terms and is second-order accurate in space but first-order accurate in time. The operator in Phase II treats the convection terms through donor-cell differencing, which is first-order accurate both in space and time. Because of this, it is found that results on the above-mentioned problems, though qualitatively correct, are quantitatively not sufficiently accurate; for example, there is unacceptable smearing of shocks. It is therefore essential to modify the existing FLIC to develop a new FLIC method devoid of the above disadvantages and still retaining its attractive features like physical appeal, relatively less operation count per mesh point, and economy in storage. One of the major contributions of the present thesis is precisely the development of a new FLIC method which is second-order accurate in space but first-order accurate in time. The enhanced accuracy of the present FLIC has been obtained by treating the convection phase through a second-order accurate flux-splitting procedure and employing conventional sequence of operators. The eigenvalues of Jacobians of flux vectors in the convection phase are found to be of the same sign—they are either all positive or all negative depending on the sign of the velocity components. The flux splitting of convection phase equations, which are Euler equations without pressure terms, therefore turns out to be very simple compared to the flux splitting in the case of full Euler equations where the eigenvalues of the Jacobians of flux vectors can be of mixed sign. These split fluxes are then used in a second-order accurate upwind differencing expression to provide second-order accuracy in space in the convection phase. Considerably enhanced accuracy of NFLIC could thus be achieved without increasing significantly the operation count per mesh point, which is slightly less than half of that required for explicit methods based on flux splitting of full Euler equations. Another interesting fact discovered during this work is that FLIC is equivalent to a first-order accurate upwind differencing procedure and can be obtained as a special case of the more general upwind differencing procedure adopted in the present work. Thus, a link between FLIC and modern upwind differencing procedure is established. Finally, the efficiency of FLIC is further improved by a factor of six through the use of convergence acceleration devices which include grid sequencing and local time stepping. The performance of this NFLIC method is critically assessed with reference to the supersonic laminar base flow problem of a backward-facing step. These results from the NFLIC on the above problem are also compared with the numerical results obtained by using MacCormack's explicit scheme employing operator splitting with and without flux vector splitting and an unfactored upwind implicit scheme. It is concluded that the NFLIC is a robust second-order accurate scheme that can be used to obtain steady-state Navier-Stokes solution of base flow. It has a certain edge in terms of computation time and memory requirement over other explicit upwind methods based on the splitting of flux vectors for the full Euler equations.