Solutions to three laminar viscous flow problems by an implicit finite-difference method

Abstract

This paper deals with three problems, (1) laminar incompressible viscous flow past a cylinder and a sphere, (2) laminar incompressible viscous flow past a finite flat plate (second-order solutions), and (3) laminar viscous past a sphere at a high Mach number. These problems are solved by using an implicit finite-difference method.

The first problem (flow past a sphere and a cylinder) involves the classical boundary-layer equations which are the first approximation to the Navier-Stokes equations in a region near to the body surface for high Reynolds number. The computational results were obtained for the distribution of velocity components in the boundary-layer, and the variation of skin-friction and displacement-thickness along the body.

The second problem (second-order flow past a finite flat plate) involves the second-order boundary-layer equations which introduce only the effect of the displacement-thickness in the case of flow past a flat plate. An assumption is made that the displacement-thickness is constant in the wake behind the flat plate. The adequacy of this assumption is checked from solutions based on the calculated displacement-thickness in the wake. The wake behind the finite flat plate is assumed laminar, and its displacement-thickness and the velocity distribution are computed downstream, by using the implicit finite-difference method.

In the third problem (high Mach number flow past a sphere), constant density is assumed in the shock layer. This is nearly true in the stagnation-point region, especially if the flow is hypersonic and the temperature of the sphere is nearly the same as the stagnation-temperature. It is also assumed that the shock is nearly spherical, even though it is not spherical as it is in the inviscid case. The numerical results will show that the assumption of a spherical shock will, however, nearly be true. This problem involves the solution of the complete Navier-Stokes equations. These equations are solved for various Reynolds numbers by two methods; namely the truncated series method and the implicit finite-difference method.

The solutions by the implicit finite-difference method are in excellent accord with those obtained by the series solutions in the stagnation-point region. As the computation by the finite-difference method proceeds downstream, the deviation of the finite-difference solution from the series solution increases. This is due to the fact that the series is valid only around the stagnation-point, and is thus expected to give inaccurate solutions downstream. The finite-difference method has no such restrictions, however, and gives accurate results in the whole flow field. In conclusion, solutions by the implicit finite-difference method have proven not only to be accurate but also to be stable in all examples computed.