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NASA NTRS1971

Contribution to methods for calculating the flow about thin lifting wings at transonic speeds: Analytic expressions for the far field

The problem of determining the small-disturbance flow about two-dimensional airfoils at transonic speeds has been successfully treated by the process of matching a numerical solution of the near field to analytic expressions for the far field. The three-dimensional problem, it would appear, can be treated in a similar way with the aid of algorithms adapted to high-speed and high-capacity computers. The far-field potential for both lifting and nonlifting three-dimensional wings at transonic speeds is developed herein for a subsonic free stream. This potential could be used for a three-dimensional-wing computation similar to the computation made for the two-dimensional wing.

Klunker, E. B.↗

NASA NTRS1974

Three-dimensional crack problems

Methods for solving three dimensional crack problems are discussed. The stress intensity factor for the case of uniform pressure on the surface crack is analysis. An analysis of concentrated forces acting on the crack faces parallel to the crack front of a half-plane crack was made. Two dimensional calculations were made using the exact solution for the stress field around an elliptical hole in a uniformly stretched plate. The strain energy density was computed in the neighborhood of the ends of narrow, but sharp, ellipses.

Hartranft, R. J.↗

NASA NTRS1974

Comparison of approximate and numerical analyses of nonlinear combustion instability

At the present time, there are three general analytical techniques available to study problems of unsteady motions in rocket motors: linear stability analysis; approximate nonlinear analysis, founded on examining the behavior of coupled normal modes; and numerical calculations based on the conservation equations for one-dimensional flows. The last two yield the linear results as a limit. It is the main purpose of this paper to check the accuracy of the approximate analysis against the numerical analysis for some special cases. The results provide some justification for using the approximate analysis to study three-dimensional problems.

Culick, F. E. C.↗

NASA NTRS1975

Particle motion in atmospheric boundary layers of Mars and Earth

To study the eolian mechanics of saltating particles, both an experimental investigation of the flow field around a model crater in an atmospheric boundary layer wind tunnel and numerical solutions of the two- and three-dimensional equations of motion of a single particle under the influence of a turbulent boundary layer were conducted. Two-dimensional particle motion was calculated for flow near the surfaces of both Earth and Mars. For the case of Earth both a turbulent boundary layer with a viscous sublayer and one without were calculated. For the case of Mars it was only necessary to calculate turbulent boundary layer flow with a laminar sublayer because of the low values of friction Reynolds number; however, it was necessary to include the effects of slip flow on a particle caused by the rarefied Martian atmosphere. In the equations of motion the lift force functions were developed to act on a single particle only in the laminar sublayer or a corresponding small region of high shear near the surface for a fully turbulent boundary layer. The lift force functions were developed from the analytical work by Saffman concerning the lift force acting on a particle in simple shear flow.

White, B. R.↗

NASA NTRS1975

Further numerical calculations of the circulation of the atmosphere of Venus

Results of two-dimensional simulations of the deep circulation of Venus are presented. They prove that the high surface temperature can only be explained by the greenhouse effect, and that Goody and Robinson's (1966) dynamical model is not valid. Analytical radiative equilibrium solutions for a semi-grey atmosphere, both with and without an internal heat source, are presented. It is shown that the greenhouse effect is sufficient to produce the high surface temperature if the thermal optical depth is much greater than 100 and the ratio of solar to thermal optical depths does not exceed 0.005. A two-dimensional version of a three-dimensional model is used to test the validity of the new mechanism proposed by Gierasch (1975) to explain the 4-day circulation. Numerical experiments with large horizontal viscosities failed to show strong zonal velocities even for the case of large Prandtl numbers. It is observed that the dissipation of angular momentum introduced by the strong horizontal diffusion more than compensates for the upward transport of angular momentum due to the Hadley cell.

De Rivas, E. K.↗

NASA NTRS1975

A subvortex technique for the close approach to a discretized vortex sheet

The close-approach problem associated with flow calculation methods based on vortex-lattice theory was examined numerically using two-dimensional discretized vortex sheets. The analysis first yields a near-field radius of approximately the distance apart of the vortices in the lattice; only within this distance from the sheet are the errors arising from the discretization significant. Various modifications to the discrete vortices are then considered with the objective of reducing the errors. This leads to a near-field model in which a vortex splits into an increasing number of subvortices as it is approached. The subvortices, whose strengths vary linearly from the vortex position, are evenly distributed along an interpolated curve passing through the basic vortices. This subvortex technique can be extended to the three-dimensional case and is efficient because the number of vortices is effectively increased, but only where and when needed.

Maskew, B.↗

NASA NTRS1974

Computation of steady axisymmetric flow using a one-dimensional time-dependent method

An iterative numerical method for computing steady, three dimensional, viscous, compressible flow fields, about aerodynamic bodies was studied. In order to develop the iterative method economically, the primary emphasis was directed towards supersonic, axisymmetric flow. However, the technique readily extends to three spatial dimensions. The viscous flow field about a cone-cylinder-flare body was calculated and compared to existing experimental data. Numerical predictions of the cone boundary layer and the flow field shock wave structure agreed with corresponding measurements. A separation was calculated at the cylinder-flare junction in six iterations; however, the size of the vortex did not correspond to the measured size. It was estimated that fifty iterations would be required to properly define the vortex. It was concluded that the iteration technique is of limited value for plane two dimensional and axisymmetrix flows, but of great value for three-dimensional flows.

Walitt, L.↗

NASA NTRS1976

Comparison of predictions and under-the-wing EBF noise data

Detailed three-dimensional free field noise data were obtained by NASA for an under-the-wing externally blown model three-flap wing, and for a similar slotless wing. Spatial (polar and azimuthal) and spectral characteristics of these data are presented. These data are compared with predictions from some published EBF noise calculation methods. Methods include the totally empirical ANOPP and GELAC procedures, and a semi-empirical noise component method. The latter method adds the separately computed dipole noise due to fluctuating lift and drag, trailing edge noise, and quadrupole noise due to the deflected jet. Each of these components is calculated for the local geometry and flow conditions.

Fink, M. R.↗

NASA NTRS1976

Numerical Studies of Three-dimensional Breakdown in Trailing Vortex Wakes

Finite element, three dimensional relaxation methods are used to calculate the development of vortex wakes behind aircraft for a considerable downstream distance. The inclusion of a self-induction term in the solution, dependent upon local curvature and vortex core radius, permits calculation of finite lifetimes for systems for which infinite life would be predicted two dimensionally. The associated computer program is described together with single-pair, twin-pair, and multiple-pair studies carried out using it. It is found, in single-pair studies, that there is a lower limit to the wavelengths at which the Crow-type of instability can occur. Below this limit, self-induction effects cause the plane of the disturbance waves to rotate counter to the vortex direction. Self induction in two dimensionally generated twin spiral waves causes an increase in axial length which becomes more marked with decreasing initial wavelength. The time taken for vortex convergence toward the center plane is correspondingly increased. The limited parametric twin-pair study performed suggests that time-to-converge increases with increasing flap span. Limited studies of Boeing 747 configurations show correct qualitative response to removal of the outer flap and to gear deployment, as compared with wind tunnel and flight test experience.

Evans, P. F.↗

NASA NTRS1976

Numerical solution of supersonic three-dimensional free-mixing flows using the parabolic-elliptic Navier-Stokes equations

A numerical method is presented for solving the parabolic-elliptic Navier-Stokes equations. The solution procedure is applied to three-dimensional supersonic laminar jet flow issuing parallel with a supersonic free stream. A coordinate transformation is introduced which maps the boundaries at infinity into a finite computational domain in order to eliminate difficulties associated with the imposition of free-stream boundary conditions. Results are presented for an approximate circular jet, a square jet, varying aspect ratio rectangular jets, and interacting square jets. The solution behavior varies from axisymmetric to nearly two-dimensional in character. For cases where comparisons of the present results with those obtained from shear layer calculations could be made, agreement was good.

Hirsh, R. S.↗

NASA NTRS1976

Three-dimensional potential flow over hills and oval mounds

An analysis was made of the potential flow behavior for an initially uniform flow passing over a single axisymmetric hill, an oval mound, and a combination of two hills. Small perturbation theory was used, and the resulting Laplace equation for the perturbation velocity potential was solved by using either a product solution or a Green's function. The three dimensional solution is of interest in calculating the pressure distribution around obstacles, the flow of pollutants carried by the wind, and the augmentation of wind velocity for windmill siting. The augmentation in velocity at the top of a hill was found to be proportional to the hill height relative to a characteristic width dimension of the hill. An axisymmetric hill produced about 20 percent less velocity increase than a two dimensional ridge having the same cross-sectional profile.

Siegel, R.↗

NASA NTRS1977

Calculation of laminar and turbulent boundary layers for two-dimensional time-dependent flows

A general method for computing laminar and turbulent boundary layers for two-dimensional time-dependent flows is presented. The method uses an eddy-viscosity formulation to model the Reynolds shear-stress term and a very efficient numerical method to solve the governing equations. The model was applied to steady two-dimensional and three-dimensional flows and was shown to give good results. A discussion of the numerical method and the results obtained by the present method for both laminar and turbulent flows are discussed. Based on these results, the method is efficient and suitable for solving time-dependent laminar and turbulent boundary layers.

Cebeci, T.↗

NASA NTRS1978

A new method for designing shock-free transonic configurations

A new method for the design of shock-free supercritical airfoils, wings, and three-dimensional configurations is described. Results illustrating this procedure in two and three dimensions are given. They include modifications to part of the upper surface of an NACA 64A410 airfoil that will maintain shock-free flow over a range of Mach numbers for a fixed lift coefficient, and the modifications required on part of the upper surface of a swept wing with an NACA 64A410 root section to achieve shock-free flow. While the results are given for inviscid flow, the same procedures can be employed iteratively with a boundary layer calculation in order to achieve shock-free viscous designs. With a shock-free pressure field the boundary layer calculation will be reliable and not complicated by the difficulties of shock-wave boundary-layer interaction.

Sobieczky, H.↗

NASA NTRS1978

Rotor-airfoil flight investigation - Preliminary results

A flight investigation has been conducted to study the behavior of three advanced-technology airfoils in the three-dimensional, unsteady-flow environment of the helicopter main rotor. Three sets of instrumented main-rotor blades were flown on an AH-1G helicopter. Each set employed one of three airfoils developed with significantly different design technologies. Data are given on performance, rotor loads, and the measurements of chordwise pressure distribution at 90 percent blade radius. The pressure data are compared with theoretical distributions calculated for two-dimensional, steady flow.

Morris, C. E. K., Jr.↗

NASA NTRS1979

Techniques for interpretation of geoid anomalies

For purposes of geological interpretation, techniques are developed to compute directly the geoid anomaly over models of density within the earth. Ideal bodies such as line segments, vertical sheets, and rectangles are first used to calculate the geoid anomaly. Realistic bodies are modeled with formulas for two-dimensional polygons and three-dimensional polyhedra. By using Fourier transform methods the two-dimensional geoid is seen to be a filtered version of the gravity field, in which the long-wavelength components are magnified and the short-wavelength components diminished.

Chapman, M. E.↗

NASA NTRS1980

Three-dimensional interactions and vortical flows with emphasis on high speeds

Diverse kinds of three-dimensional regions of separation in laminar and turbulent boundary layers are discussed that exist on lifting aerodynamic configurations immersed in flows from subsonic to hypersonic speeds. In all cases of three dimensional flow separation, the assumption of continuous vector fields of skin-friction lines and external-flow streamlines, coupled with simple topology laws, provides a flow grammar whose elemental constituents are the singular points: nodes, foci, and saddles. Adopting these notions enables one to create sequences of plausible flow structures, to deduce mean flow characteristics, expose flow mechanisms, and to aid theory and experiment where lack of resolution in numerical calculations or wind tunnel observation causes imprecision in diagnosing the three dimensional flow features.

Peake, D. J.↗

NASA NTRS1980

An improved panel method for the solution of three-dimensional leading-edge vortex flows. Volume 1: Theory document

An improved panel method for the solution of three dimensional flow and wing and wing-body combinations with leading edge vortex separation is presented. The method employs a three dimensional inviscid flow model in which the configuration, the rolled-up vortex sheets, and the wake are represented by quadratic doublet distributions. The strength of the singularity distribution as well as shape and position of the vortex spirals are computed in an iterative fashion starting with an assumed initial sheet geometry. The method calculates forces and moments as well as detail surface pressure distributions. Improvements include the implementation of improved panel numerics for the purpose of elimination the highly nonlinear effects of ring vortices around double panel edges, and the development of a least squares procedure for damping vortex sheet geometry update instabilities. A complete description of the method is included. A variety of cases generated by the computer program implementing the method are presented which verify the mathematical assumptions of the method and which compare computed results with experimental data to verify the underlying physical assumptions made by the method.

Johnson, F. T.↗

NASA NTRS1980

On the stability of three-dimensional compressible nonparallel boundary layers

A compressible linear stability theory is presented for three-dimensional nonuniform boundary layers. The amplitude, phase, and wavenumber equations which govern the motion of the disturbance are obtained by using the method of multiple scales. Group velocity trajectories are used to identify the disturbance growth direction. The spatial stability theory is applied to the flow on a laminar flow control supercritical sweptback wing of infinite span. Three different methods are used to calculate the absolute maximum logarithmic amplitude ratio N. In the front crossflow instability region, the method of maximum spatial growth rate predicts large difference in N compared with the method of fixed wavelength and the method of fixed spanwise component of wavelength. This difference decreases in the middle streamwise instability region, and almost vanishes in the rear crossflow instability region. Compressibility of the medium reduces N by about 15% in both the front and rear regions, and by about 40% in the middle region of the wing. Nonuniformity of the medium has large effects specially in the rear region.

El-Hady, N. M.↗