Boundary layer equations and methods of solution

Abstract

The objective of computational fluid dynamics is to calculate an entire flow fieldeither around an arbitrary obstacle or through a channel of any shape. The flow maybe unsteady, three dimensional, compressible and turbulent. At hypersonic speeds,regions of reacting flow (dissociation, ionization, etc.) might also be considered.The equations to describe this task, as derived in Chap. 2, are the Navier-Stokesequation, the energy equation, the global and partial continuity equations and otherclosure model equations describing turbulence and reacting gas effects. It can easilybe shown that, at present, no computer could provide either the capacity or thenecessary calculation speed to fulfil this task.Thereby, coming back to the reality of today, the governing equations have to besimplified such that the properties of the remaining set of equations still describe theflow to be considered. For instance, when the viscous terms in the Navier-Stokesequations are neglected, one arrives at the Euler equations. They can be used to determinefar-field flows where the interaction with the viscous layer is not dominant.However, a separation bubble on the surface of a wing cannot be detected withoutproviding viscous flow calculations near the body surface; separation is a matter ofviscous effects.As indicated already, different types of flows can be treated by examining theirphysical characteristics in detail and in this way establishing the appropriate governingequations by the correct reduction of the general set of fluid mechanicalequations. This is what Prandtl [1] did in 1904 concerning a thin layer near wallswhere the influence of viscosity normal to the wall is dominant. He called this layera 'boundary layer'. The important detail of the physical meaning of this kind of flowis that the main flow velocity tends to zero approaching the wall. The gradient ofthis velocity component in the direction normal to the surface is large compared tothe gradient of this component in the downstream direction.This observation leads to an important change in the character of the governingequations from the elliptic to the parabolic type which makes a numericaldownstream marching procedure applicable. Reverse flow therefore cannot be calculatedwith a simple boundary layer method, but the flow field very near to theseparation point, where the reverse flow starts, can be detected very well. Referenceto the parabolic nature of differential equation is already given in Sect. 4.3.The boundary layer theory will be the subject of this chapter. Prandtl's idea willbe described in detail as an introduction. The hierarchy of the boundary layer equationswill be discussed; that is, the relationship of the different types of boundarylayer equations to the Navier-Stokes equations will be demonstrated. Furthermore,it will be pointed out that there are transformation techniques to reduce the problemsof solution. A generalized discretization scheme will be applied to a set of laminarcompressible boundary layer equations and a numerical solution scheme for calculatingthe remaining tridiagonal linear difference equations will be shown. A samplecalculation of a laminar boundary layer along the symmetry line of a highly inclinedellipsoid will conclude the discussion.This chapter deals only with laminar boundary layer theory. The description ofturbulence needs additional effort, especially in seeking suitable turbulence modelsfor specific purposes, but it does not affect the principal solution procedure. As thesenotes are meant to give an introduction to the boundary layer theory and its methodsof solution, turbulent boundary layers will not be considered, but overviews onrecent turbulence models are given in Refs. [2-5]. © Springer-Verlag Berlin Heidelberg 2009.