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NASA NTRS1975

Supercritical wing sections 2, volume 108

A mathematical theory for the design and analysis of supercritical wing sections was previously presented. Examples and computer programs showing how this method works were included. The work on transonics is presented in a more definitive form. For design, a better model of the trailing edge is introduced which should eliminate a loss of fifteen or twenty percent in lift experienced with previous heavily aft loaded models, which is attributed to boundary layer separation. How drag creep can be reduced at off-design conditions is indicated. A rotated finite difference scheme is presented that enables the application of Murman's method of analysis in more or less arbitrary curvilinear coordinate systems. This allows the use of supersonic as well as subsonic free stream Mach numbers and to capture shock waves as far back on an airfoil as desired. Moreover, it leads to an effective three dimensional program for the computation of transonic flow past an oblique wing. In the case of two dimensional flow, the method is extended to take into account the displacement thickness computed by a semi-empirical turbulent boundary layer correction.

Bauer, F.↗
NASA NTRS1975

A theoretical method for the analysis and design of axisymmetric bodies

A theoretical method is presented for the computation of the flow field about an axisymmetric body operating in a viscous, incompressible fluid. A potential flow method was used to determine the inviscid flow field and to yield the boundary conditions for the boundary layer solutions. Boundary layer effects in the forces of displacement thickness and empirically modeled separation streamlines are accounted for in subsequent potential flow solutions. This procedure is repeated until the solutions converge. An empirical method was used to determine base drag allowing configuration drag to be computed.

Beatty, T. D.↗
NASA NTRS1975

Supercritical wing sections II: A handbook

The numerical aspect of theoretical work on transonics and supercritical wing sections are compiled. A model of the trailing edge is introduced which eliminates the loss of 15 to 20 percent experienced with heavily aft-loaded models, and it is indicated how drag creep can be reduced at off-design conditions. A rotated finite difference scheme is presented which can handle supersonic as well as subsonic free stream Mach numbers and leads to an effective three-dimensional program for the computation of transonic flow past an oblique wing. In the case of two-dimensional flow, the method is extended to take into account the displacement thickness computed by a semiempirical turbulent boundary layer correction. A series of supercritical wing sections is discussed together with comparisons between experimental and theoretical data. Computer programs and a brief manual for their operation are listed. It is shown that the programs furnish a physically adequate computer simulation of the compressible flows that arise in problems of transonic aerodynamics.

Bauer, F.↗
NASA NTRS1977

Swept wing aerodynamics

Technique analyzes viscosity-dependent aerodynamic characteristics of multielement infinite swept wings in incompressible flow. Use of source distributions rather than displacement thickness to represent boundary layer effect on potential flow and of iterative technique for matrix inversion reduces computer time for overall analysis.

Dvorak, F. A.↗
NASA NTRS1975

Aerodynamic heating to the gaps and surfaces of simulated reusable-surface-insulation tile arrays in turbulent flow at Mach 6.6

An experimental investigation was made on a simulated reusable-surface-insulation tile array in a turbulent boundary layer to determine aerodynamic-heating distributions representative of those expected on the surface of the shuttle orbiter during earth entry due to the presence of longitudinal and transverse surface gaps. The tests were conducted in an 8-foot high-temperature structures tunnel in a test medium of methane-air combustion products at a nominal Mach number of 6.6 and over a free-stream Reynolds number range from 2,000,000 to 4,900,000 per meter (600,000 to 1,500,000 per foot). The results were used to assess the aerodynamic heating effects produced by parameters that include gap width, boundary-layer displacement thickness, in-line and staggered tile arrangement, and tile protrusion.

Weinstein, I.↗
NASA NTRS1975

Inverse solutions for laminar boundary-layer flows with separation and reattachment

Numerical solutions of the laminar, incompressible boundary layer equations are presented for flows involving separation and reattachment. Regular solutions are obtained with an inverse approach in which either the displacement thickness or the skin friction is specified; the pressure is deduced from the solution. A vorticity-stream-function formulation of the boundary layer equations is used to eliminate the unknown pressure. Solutions of the resulting finite difference equations, in which the flow direction is taken into account, are obtained by several global iteration schemes which are stable and have unconditional diagonal dominance. Results are compared with Klineberg and Steger's separated boundary layer calculations, and with Briley's solution of Navier-Stokes equations for a separated region. In addition, an approximate technique is presented in which the streamwise convection of vorticity is set equal to zero in the reversed flow region; such a technique results in a quick forward marching procedure for separated flows.

Carter, J. E.↗
NASA NTRS1976

A numerical solution of the flow field over a transonic airfoil including strong-shock-induced flow separation

An approximate numerical method has been developed for computing the entire transonic flow field over an airfoil, including the boundary layer, the shock-wave/boundary-layer interaction, and the shock-induced-separation. The inviscid flow, the turbulent boundary layer in the weak interaction zone, and the strong interaction elliptic region are solved separately. The inviscid flow is represented by the solution of the transonic small-disturbance equation, while the boundary layer is solved by the generalized Galerkin method, and the strong interaction region (including the wake) is computed from the time-averaged Navier-Stokes equations. A second-order interaction is computed iteratively using the inviscid pressure distribution as a boundary condition for the boundary layer and Navier-Stokes codes, and the viscous displacement thickness to modify the airfoil geometry for the inviscid computation. The computed pressure distributions and shock-wave profiles are compared with experimental data. Agreement is very good except that the pressure recovery on the trailing edge is greater than the experimental data.

Seginer, A.↗
NASA NTRS1977

Introductory Remarks

Advanced transonic configurations, such as the supercritical wing, are inherently more sensitive to Reynolds number than earlier configurations because the pressure recovery gradients imposed on the boundary layer are generally steeper. The results of two-dimensional supercritical airfoil investigations and theoretical calculations show this effect. A technique for approximately simulating full-scale Reynolds number characteristics at present wind-tunnel Reynolds numbers for near-cruise conditions is described. The transition strip, which in the past has been located near the leading edge of the wing, is rearward so that the relative displacement thickness of the boundary layer at the trailing edge of the wing is the same as might be expected on a full-scale configuration with the transition near the leading edge. Two-dimensional wind-tunnel results indicate that the technique provides a very good simulation of airfoil characteristics at full-scale Reynolds number. The need for sorting the effects of Reynolds number and aeroelasticity, which can be done in the NTF, is also addressed.

Whitcomb, R.↗
NASA NTRS1977

Representation of turbulent shear stress by a product of mean velocity differences

A quadratic form in the mean velocity for the turbulent shear stress is presented. It is expressed as the product of two velocity differences whose roots are the maximum velocity in the flow and a cutoff velocity below which the turbulent shear stress vanishes. Application to pipe and channel flows yields the centerline velocity as a function of pressure gradient, as well as the velocity profile. The flat plate, boundary-layer problem is solved by a system of integral equations to obtain friction coefficient, displacement thickness, and momentum-loss thickness. Comparisons are made with experiment.

Braun, W. H.↗
NASA NTRS1977

An approximate calculation of the strong interaction on a transonic airfoil

This paper describes an interactive computational scheme that calculates the transonic flow field over an airfoil. The inviscid flow, the turbulent boundary layer in the weak interaction region, and the elliptic flow in the strong interaction region are computed separately, and the solutions are matched iteratively in a direct, second-order interaction scheme. The pressure distribution, obtained from a transonic, small-perturbation solution, is prescribed on the boundary layer and the Navier-Stokes equations and the ensuing displacement thicknesses are used to modify the inviscid flow. The best overall results are obtained by using a turbulence model that eliminates the eddy viscosity in the strong interaction region. Computed pressure distributions are compared with experimental data and with Navier-Stokes solutions. In weak interactions, the agreement with experimental data is very good, as it is also in strong interactions except for the pressures in the separated-flow region. There the agreement is qualitative only, and the pressure recovery is too high. The results in both cases are superior or equal to those of the Navier-Stokes solutions, and the present method is significantly faster.

Seginer, A.↗
NASA NTRS1977

Analytical study of viscous effects on transonic flow over boattail nozzles

An analytical study has been made of the effects of viscous/inviscid interactions on the transonic flow over boattail nozzles. The theoretical method couples a relaxation solution of the full potential transonic flow equations with a conventional boundary-layer solution to account for displacement-thickness effects. Surface pressures calculated on circular-arc boattails with solid jet-plume simulators show good agreement with experiment for free-stream Mach numbers less than 0.90. For separated flows, an empirical discriminating streamline model of the separation bubble gives good results until the onset of shock-induced separation, which occurs typically at a Mach number of about 0.90.

Wilmoth, R. G.↗
NASA NTRS1977

Investigations related to the inviscid-viscous interaction in transonic flows about finite 3-D wings

Inviscid small-disturbance theory has been shown to predict three-dimensional transonic flows about finite wings reasonably well as long as viscous effects are negligible. In order to include these effects, the inviscid small-disturbance solution of Bailey and Ballhaus (1975) has been combined with a finite-difference solution for Prandtl's boundary-layer equations. This solution employs the conditionally stable Krause (1968) scheme, implicit in the direction normal to the wall, to cope with the domain-of-dependence problem that arises for reverse cross flow. To be consistent with the inviscid-flow solution, the boundary layer is computed in the representative wing planform plane which is transformed into rectangular shape in the computational domain. The flow has been assumed turbulent, and a scalar eddy-viscosity model is adopted. The interaction between inviscid and viscous flow is modeled with the help of the displacement surface which is added to the geometric wing shape. Sample distributions of displacement thickness for swept wings are presented for weak and strong interaction cases.

Kordulla, W.↗
NASA NTRS1978

The effect of the transport property models on the shuttle boundary layer

Theoretical solutions for the nonsimilar, laminar boundary-layer were computed for four points along the shuttle entry trajectory. Since the boundary layer is that region of the flow field where the effects of viscosity and of thermal conductivity are most important, numerical solutions for the boundary layer were generated using different models for the transport properties. These solutions indicate that the displacement thickness and the heat-transfer rates are very sensitive to changes in the models for thermal conductivity and for specific heat. Thus, the solutions are sensitive to the assumed transport-property model.

Bertin, J. J.↗
NASA NTRS1978

Aerodynamic heating in gaps of thermal protection system tile arrays in laminar and turbulent boundary layers

An experimental heat-transfer investigation was conducted on two staggered arrays of metallic tiles in laminar and turbulent boundary layers. This investigation was conducted for two purposes. The impingement heating distribution where flow in a longitudinal gap intersects a transverse gap and impinges on a downstream blocking tile was defined. The influence of tile and gap geometries was analyzed to develop empirical relationships for impingement heating in laminar and turbulent boundary layers. Tests were conducted in a high temperature structures tunnel at a nominal Mach number of 7, a nominal total temperature of 1800 K, and free-stream unit Reynolds numbers from 1.0 x 10 million to 4.8 x 10 million per meter. The test results were used to assess the impingement heating effects produced by parameters that include gap width, longitudinal gap length, slope of the tile forward-facing wall, boundary-layer displacement thickness, Reynolds number, and local surface pressure.

Avery, D. E.↗
NASA NTRS1978

Inverse boundary-layer theory and comparison with experiment

Inverse boundary layer computational procedures, which permit nonsingular solutions at separation and reattachment, are presented. In the first technique, which is for incompressible flow, the displacement thickness is prescribed; in the second technique, for compressible flow, a perturbation mass flow is the prescribed condition. The pressure is deduced implicitly along with the solution in each of these techniques. Laminar and turbulent computations, which are typical of separated flow, are presented and comparisons are made with experimental data. In both inverse procedures, finite difference techniques are used along with Newton iteration. The resulting procedure is no more complicated than conventional boundary layer computations. These separated boundary layer techniques appear to be well suited for complete viscous-inviscid interaction computations.

Carter, J. E.↗
NASA NTRS1978

A combined potential and viscous flow solution for V/STOL engine inlets

A potential flow routine and a viscous boundary layer routine have been combined into a single routine for estimating the flow in and around subsonic inlets. In this combined routine, the viscous flow solution about the inlet body is obtained by adding the viscous displacement thickness to the inlet geometry. Combination of the two flow solutions has resulted in cost savings, both in preparation time and in computer time. This routine is a useful tool in optimizing lip shapes for V/STOL inlets. The method of combining the routine, comparison with NASA test data, and utilization of the routine for V/STOL inlet design are presented.

Ybarra, A. H.↗
NASA NTRS1978

Computation of hypersonic laminar viscous flow past spinning sharp and blunt cones at high angle of attack

Computational results, obtained with a parabolic Navier-Stokes marching code, are presented for hypersonic viscous flow past spinning sharp and blunt cones at angle of attack. The code takes into account the asymmetries in the flow field resulting from spinning motion and computes the asymmetric shock shape, crossflow and streamwise shear, heat transfer, crossflow separation, and vortex structure. The Magnus force and moments are also computed. Comparisons are made with other theoretical analyses based on boundary-layer and boundary-region equations, and an anomaly is discovered in the displacement thickness contribution to the Magnus force when compared with boundary-layer results. In addition, a new criterion for defining crossflow separation behind spinning bodies is introduced which generalizes the Moore-Rott-Sears criterion for two-dimensional unsteady separation. A condition which characterizes the onset of separation in the flow field is defined.

Agarwal, R.↗
NASA NTRS1978

A simplified Mach number scaling law for helicopter rotor noise

Mach number scaling laws are derived for the rotational and the high-frequency broadband noise from helicopter rotors. The rotational scaling law is obtained directly from the theory of Lowson and Ollerhead (1969) by exploiting the properties of the dominant terms in the expression for the complex Fourier coefficients of sound radiation from a point source. The scaling law for the high-frequency broadband noise is obtained by assuming that the noise sources are acoustically compact and computing the instantaneous pressure due to an element on an airfoil where vortices are shed. Experimental results on the correlation lengths for stationary airfoils are extended to rotating airfoils. On the assumption that the correlation length varies as the boundary layer displacement thickness, it is found that the Mach number scaling law contains a factor of Mach number raised to the exponent 5.8. Both scaling laws were verified by model tests.

Aravamudan, K. S.↗