There is a large variety of numerical methods that are used to solve the many flow problems to which boundary-layer theory is applied Two particular methods, the Crank-Nicolson scheme and the Box scheme, seem to dominate in most practical applications Of course these methods are not uniquely defined and there are many variations in their formulation as well as in the procedures used to solve the resulting (nonlinear) algebraic equations They are both implicit with respect to variation normal to the boundary layer, they can both be second-order accurate in all variables,
and both can be improved to yield even higher-order accuracy However, I prefer and stress the Box scheme as it is very easy to adapt to new classes of problems It also allows a more rapid net variation and ease in obtaining higher-order accuracy
This scheme was devised in Keller (1971), for solving diffusion problems, but it has subsequently been applied to a broad class of problems A recent survey by Blottner (1975a) has stressed the Crank-Nicolson scheme I suggest that article as well for a
brief presentation of various other numerical methods that have been used A sketch of the applications of the Box scheme to a variety of boundary-layer flow problems is given in Keller (1975b)

Many of the problems and techniques presented in this survey have been worked out with T Cebeci Details of some of these techniques and even listings of a few of the codes appear in a text by Cebeci & Bradshaw (1977)

Numerical methods in boundary-layer theory

A quasi-simultan eous method is described to calculate laminar, incompressible boundary layers interacting with an inviscid outer flow. The essential feature of this method is an interactive boundary condition prescribing a linear combination of pressure and displacement thickness which models the behavior of the outer flow. This way the quasi-simultaneous method avoids difficulties incurred when either direct or inverse methods are used, resulting in fast convergence of the iterative procedure involved. The method is consistent with the asymptotic triple-deck theory. Results will be presented for two problems which exhibit strong interaction between the viscous and inviscid regions: a boundary layer with a separation bubble, and the flow near the trailing edge of a flat plate. fj Nomenclature cf = skin friction coefficient, cf-Re h = mesh size in x direction H = thickness of computational boundary-layer domain L = characteristic length M,N = number of grid points in y and x direction, respectively p = pressure RJ = remainder term in discrete interaction law Re = Reynolds number , Re = U0L I v u = component of flow velocity parallel to surface Ue>ue0 = value of u at edge of boundary layer with and

New, quasi-simultaneous method to calculate interacting boundary layers

For high-Reynolds-number flow over bodies or in confined channels the effects of viscosity are generally limited to a thin layer, the boundary layer, adjacent to the bounding surface. When the imposed pressure gradient is adverse, however, the thickness of the viscous layer increases as momentum is consumed by both wall shear and pressure gradient, and at some point the viscous layer breaks away from the bounding surface. Downstream of this point (or line) of breakaway the original boundary-layer fluid passes over a region of recirculating flow. The point at which the thin boundary layer breaks away from the surface and which divides the region of downstream-directed flow, in which the viscous effects are quite limited in extent, from the region of recirculating flow is known as the separation point.! Two different types of post-separation behavior are known to exist. In some cases the original boundary layer passes over the region or ' recirculating fluid and reattaches to the body at some point downstream, trapping a bubble of recirculating fluid beneath it. The characteristic length of this separation bubble may be of the same order as the upstream boundary-layer thickness or ma�y times the boundary-layer thickness. In other cases, the original boundary-layer fluid never reattaches to the body but passes downstream, mixing with recirculating fluid, to form a wake. For this wake-type of separation the characteristic dimension of the recirculating region is generally of the same order as the characteristic body dimension. In either case, the recirculating flow alters the effective body shape and hence the inviscid flow about the body.

Incompressible Boundary-Layer Separation

: A study is presented of the characteristics of transitional and turbulent layers, and regions of shock wave-turbulent boundary layer interaction in high Reynolds number hypersonic flow. An examination and correlation of skin friction, heat transfer and pressure measurements in laminar, transitional and turbulent boundary layers on sharp flat plates and cones are presented for the Mach range from 3 to 13 at wall-to-free stream temperature ratios from 0.1 to 0. 4.

Shock Wave Turbulent Boundary Layer Interaction in Hypersonic Flow

Viscous-inviscid interactions in external aerodynamics

Numerical solutions are presented for the laminar and turbulent boundary-layer equations for incompressible flows with separation and reattachment. The separation angularity is avoided by using an inverse technique in which the displacement thickness is prescribed and the pressure is deduced from the resulting solution. The turbulent results appear qualitatively correct despite the use of a two-layer eddy-viscosity model which is generally assumed appropriate only for mild-pressure-gradient flows. A new viscous-inviscid interaction technique is presented in which the inviscid flow is solved inversely by prescribing the pressure from the boundary-layer solution and deducing the new displacement thickness from the solution of a Cauchy integral. Calculations are presented using this interaction procedure for a laminar flow in which separation and reattachment occur on a solid surface.

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Solutions for incompressible separated boundary layers including viscous-inviscid interaction

Viscous effects can have a substantial impact on the aerodynamic performance of internal and external flow configurations. For a significant number of flows of practical interest, the Reynolds number is sufficiently large for the flow field to be represented by interacting boundary layer theory (IBLT), whereby the flow is divided into viscous and in viscid flow regions with the two regions coupled through the viscous displacement thickness. The formal justification for the use of IBLT in the prediction of high Reynolds number strong interaction laminar flows is provided by triple deck theory [316], which is obtained from an asymptotic expansion of the Navier—Stokes equations. In IBLT the viscous region is represented by the Prandtl boundary layer equations; the inviscid flow can be represented in a number of different ways which depend on the flow configuration and the Mach number. It is clear from previous work (see [340] for a review) that many flows with separation can be solved accurately (neglecting the problems of modeling the turbulence and transition processes) through the use of IBLT. A critical step in the numerical solution of the governing equations in IBLT is the coupling mechanism which connects the viscous and inviscid flow equations. Several coupling algorithms currently exist for strongly interacting flows; however, the efficiency of these procedures, which are discussed below, generally decreases, along with the possibility of unstable behavior, as the size of the separation region increases.

A Quasi-simultaneous Finite Difference Approach for Strongly Interacting Flow

Numerical solutions of Navier-Stokes equations are presented for the supersonic laminar flow over a two-dimensional compression corner. A well-known time-dependent method has been used wherein the asymptotic steady solutions of the unsteady Navier-Stokes equations are obtained with the Brailovskaia (1965) finite-difference scheme.

Numerical solutions of the supersonic, laminar flow over a two-dimensional compression corner

A forward-marching procedure for separated boundary-layer flows which permits the rapid and accurate solution of flows of limited extent is presented. The streamwise convection of vorticity in the reversed flow region is neglected, and this approximation is incorporated into a previously developed (Carter, 1974) inverse boundary-layer procedure. The equations are solved by the Crank-Nicolson finite-difference scheme in which column iteration is carried out at each streamwise station. Instabilities encountered in the column iterations are removed by introducing timelike terms in the finite-difference equations. This provides both unconditional diagonal dominance and a column iterative scheme, found to be stable using the von Neumann stability analysis.

Forward marching procedure for separated boundary-layer flows

An analysis based on the boundary-layer equations is presented for the p rediction of threedimensional separated flow. In this analysis, t he boundary-layer equations are solved with finite difference techniques in four i nverse (pressure is unknown) modes. One inverse mode, in which the component of vorticity normal to the surface is specified at the boundary-layer e dge, is shown to result in an elliptic system of boundary-layer equations which has departure solutions when solved with a f orward marching technique. The accuracy o f the p resent analysis in the other t hree inverse modes using forward marching techniques is demonstrated for: (1) separated flow over an infinite swept wing which has been rendered three dimensional with a rotated coordinate system; and (2) a three-dimensiona l separated flow problem analyzed by Smith. The principal conclusion o f this paper is that the methodology is now in place to solve the boundary-layer equations for three-dimensiona l separated flow and hence work should now be focused on the development of three-dimensiona l viscousinviscid interaction procedures.

James E. Carter

Papers

Solutions for incompressible separated boundary layers including viscous-inviscid interaction

A Quasi-simultaneous Finite Difference Approach for Strongly Interacting Flow

Numerical solutions of the supersonic, laminar flow over a two-dimensional compression corner

Forward marching procedure for separated boundary-layer flows

Analysis of three-dimensional separated flow with the boundary-layerequations