Numerical Solutions of Supersonic and Hypersonic Laminar Compression Corner Flows

C.M. Hung* and R.W. MacCormacki"NASA Ames Research Center, Moffett Field, Calif.An efficient time-splitting, second-order accurate, numerical scheme is used to solve the complete Navier-Stokes equations for supersonic and hypersonic laminar flow over a two-dimensional compression corner. Afine, exponentially stretched mesh spacing is used in the region near the wall for resolving the viscous layer.Good agreement is obtained between the present computed results and experimental measurement for a Machnumber of 14.1 and a Reynolds number of 1.04x l0 s with wedge angles of 15", 18", and 24". The details of thepressure variation across the boundary layer are given, and a correlation between the leading edge shock and thepeaks in surface pressure and heat transfer is observed.!. IntroductionONTINUING advances in numerical methods and com-uter capabilities have now made feasible many flowfieldcalculations which were formerly intractable. One suchproblem which has renewed interest is supersonic or hyper-sonic flow over a two-dimensional compression corner. Thisproblem has received considerable attention within the pastdecade because of its importance to the design engineer inpredicting the pressure and heat loads at a wing-flap junctionon re-entry vehicles. When flow separation occurs, reducedflap effectiveness results, and in some regions, the surfaceheating could become severe on a maneuverable re-entryvehicle.The problem to be considered is illustrated schematically inFig. 1. The pressure rise generated by the wedge extends up-stream along the flat plate, thickening the boundary layer,and results in a complicated interaction between the viscousflow near the body surface and the outer inviscid stream. Sin-ce the inner part of the boundary layer may not have suf-ficient momentum to overcome the combined effects of skinfriction and adverse pressure gradient, the interaction canlead to flow separation for certain ranges of Mach number,Reynolds number, and wedge angle. The separated boundarylayer will then become a free shear layer external to a recir-culating inner flow near the corner. Reattachment occursbecause of the interaction between free shear flow and theouter flow. The surface pressure continues to rise through theseparated and reattached regions, until the boundary layerreaches a minimum thickness or "neck." Downstream of theneck, the boundary layer returns to a normal state of weak in-teraction with the outer inviscid stream at a new Mach num-ber. Although, in most practical situations the region of shockwave and boundary-layer interaction is turbulent, at highaltitude flight, fully laminar flows can exist and are importantfor design considerations.Previous theoretical treatment of such a problem hasusually been made with the boundary-layer equations togetherwith a "coupling" equation relating the development of theinner viscous flow to the outer flow. The governing partialdifferential equations can then be solved by finite differencetechniques L2 or can be expressed as integral relations andPresented as Paper 75-2 at the AIAA 13th Aerospace SciencesMeeting, Pasadena, California, January 20-22, 1975; submittedFebruary 27, 1975; revision received July 7, 1975.Index categories: Supersonic and Hypersonic Flow; BoundaryLayers and Convective Heat Transfer-Laminar; CompulerTechnology and Computer Simulation Techniques.*NRC Research Associate. Member AIAA.[-Assistant Chief, Computational Fluid Dynamics Branch. MemberAIAA.solved as ordinary differential equations. 3-7This treatment, ingeneral, involves the question of uniqueness because certainportions of the flowfield contain substantial upstream in-fluence, and the initial and downstream boundary conditionscannot be completely specified. In some integral tech-niques, 4-7 there is also the question of so-called jump con-ditions for supercritical to subcritical types of boundarylayers.Consideration of the Navier-Stokes equations avoids thesequestions and removes some restrictive assumptions, viz., thatthe static pressure is constant across the boundary layer, andthat the viscous and inviscid flows interact only along a line ator near the edge of the boundary layer, which can be difficultto define in hypersonic flow. Also, as mentioned by VanDyke, s a solution of the Navier-Stokes equations is necessaryin the immediate vicinity of a sharp corner.Cart,:r 9 has obtained numerical solutions of the Navier-Stokes equations for laminar flow past a compression cornerat low Mach numbers. He used the Brailovskaya differencescheme, which is first-order accurate in time and second-orderaccurate in space. In the present study a more efficientnumerical method, _0.H which is second-order accurate in bothtime and space, is used to calculate supersonic and hypersonic