Viscous Transonic Flow Research Articles

Solution of viscous transonic flow over wings

Knowledge of the viscous flow about wings is very important in 3-D wing design. In transonic flow about a typical supercritical wing, the viscous effect results in a sizable reduction of the lift-to-drag ratio. The Reynolds number dependence of the flow is not clearly defined, and no known similitude exists that can be used to scale the experimental data for a particular design. Recent advances in computer technology and numerical technique have relieved the difficulty of obtaining a theoretical solution somewhat, but the lack of a proper reliable method of treating the turbulence in a time-averaged Navier-Stokes solution remains the major stumbling block. For this paper, a “zonal” approach has been used for a viscid-inviscid interaction analysis to yield an iterative solution for the viscous flow about wings in the transonic flow regime. The chord Reynolds number considered was of the order of 10 6 and above so that the flow was predominantly turbulent. The inviscid flow field was obtained by solving the 3-D potential flow equation. A parabolic coordinate mapping was used in the computation, in conjunction with a finite volume formulation. A new approximate factorization scheme has been developed for the iterative solution of the inviscid flow. A special far field asymptotic boundary condition that improves the accuracy and convergence of the method was derived. For the 3-D boundary layer calculation, the integral method of Myring-Smith-Stock was extensively modified to make it suitable for the interaction calculation. The effect of wing thickness was taken into account and the 3-D viscous wake was computed. The interaction calculation was formulated with a set of coupling conditions that includes the source flux distribution due to the surface boundary layer on the wing, the flux jump distribution due to the viscous wake, and the effect of the viscous wake curvature. The transpiration boundary conditions have been used for the inviscid flow in the coupled calculation. In addition, a method was devised so that the results of an analysis of the trailing edge strong interaction solution for a 2-D viscous airfoil could be adapted for the normal pressure correction near the trailing edge. The theory has been applied to supercritical wing geometries of practical interest. The converged viscous flow results compare favorably with experimental pressure data.

Analysis of circulation-controlled airfoils in transonic flow

A method developed for the analysis of the transonic viscous flow over circulation-controlled airfoils is described. A finite-differe nce method is used to solve the inviscid portion of the flow, and a combination of integral and finite-difference methods is used to calculate the development of the compressible viscous layers. An iterative procedure is employed to give solutions which satisfy the trailing-edge Kutta condition and incorporate the interaction between the viscous and potential regions of the flow. Comparisons between calculated and experimental results show good agreement for surface pressure distributions and lift coefficients over a range of blowing momentum coefficients and Mach numbers. a B c CE C = Nomenclature speed of sound

Embedded mesh multigrid treatment of two-dimensional transonic flows

We report the development of a highly flexible computer code which uses a hierarchy of embedded meshes, communicating via multigrid, to solve the axisymmetric or planar viscous transonic flow problem. The purpose of the code is to demonstrate the practicality of techniques sufficiently general to be applicable to complex three-dimensional geometries, and to serve as a test vehicle for mesh communication and relaxation procedures. The code is also of interest in its own right for the fast solution of planar and axisymmetric problems in a very large domain. It consists of a multigrid solver for the full-potential equation (which incorporates the embedded meshes), with a provision for later coupling mass injection to a separate boundary layer program. The principal feature of the code is its treatment of the geometry by means of several levels of locally defined body-fitted meshes embedded in a system of coarser global grids which do not conform to the body. The user may define meshes as large and coarse as desired, in order to solve the problem in a large domain, and also as fine as desired, in order to resolve body geometry and flow field details. The grids need not be coextensive, so that mesh lines placed close to detail features need not extend throughout the field. The intent throughout has been to do nothing whose extension to three dimensions is not straightforward; in particular, global mappings are avoided. We have also emphasized flexible data structures and modular code, with a view toward the development of a highly vectorized three-dimensional version.

Computation of Viscous Transonic Flow about a Lifting Airfoil

The Reynolds averaged Navier-Stokes equations are solved numerically for the viscous transonic flow about a stationary NACA 64A010 airfoil in free air. This paper presents descriptions of the numerical method, turbulence models employed, and boundary conditions appropriate to simulation of free-air flight. Computed results are presented for the airfoil at a free-stream Mach number of 0.8, angles of attack of 0 and 2 deg, and a Reynolds number based on a chord of 4 x 10 to the 6th. For the lifting case, unsteady periodic motion was calculated along the aft portion of the airfoil and in its wake. Recent experimental results obtained by Johnson indicate periodicity aft of the shock closely approximates the computed frequency, but the amplitude of the disturbances was significantly less than the calculated amplitude.

A coordinate system for a viscous transonic cascade analysis

A coordinate system suitable for the numerical computation of viscous transonic cascade flows is constructed. The system consists of coordinate loops surrounding the airfoil and radial coordinate lines normal to the airfoil surface. The outermost loop is constructed so that the cascade periodicity conditions can be applied without interpolation between grid points. The coordinates are orthogonal on the airfoil surface but gradually become nonorthogonal away from the airfoil. The coordinate distribution of mesh points is simple and direct; this is a useful property for the resolution of large solution gradients. In addition to the above, the coordinates are generated from discrete input data, little restriction is placed on airfoil camber or spacing, and the entire analysis is easily extended to three dimensions. Moreover, the method of coordinate generation can be readily applied to a wide variety of other problems.

Numerical solution for viscous transonic flow

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Transonic Navier-Stokes flow computations over wing-fuselage geometries

The steady-state transonic viscous flows over two wing-fuselage configurations are solved numerically. The 3D compressible thin-layer Navier-Stokes equations are solved by means of multistage timestepping similar to the Runge-Kutta method. The timestepping technique is applied with a multigrid scheme to make the transonic Navier-Stokes computations using the Baldwin-Lomax turbulence for closure and converging the solution on a grid of 1.52 million grid points with a C-O topology. Both generic and fighter wing-fuselage geometries are solved, and the results are compared to experimental studies. Solution convergence is found to be fast, and the fine grid generates accurate results for both configurations by means of the Baldwin-Lomax turbulence model. Results from the Johnson-King model calculations produce results that are more consistent with experimental results for the fighter configuration.