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NASA NTRS1975

Modal equations for cellular convection

We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possible useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of 'modal equations' is analyzed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.

Gough, D. O.↗

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 NTRS1977

Solar sources of the interplanetary magnetic field and solar wind

Open magnetic field lines, those which extend from the solar photosphere to interplanetary space, are traced in the current-free (potential field) approximation using measured photospheric fields as a boundary condition. It is found that (1) only a relatively small fraction of the photospheric area connects via open field lines to the interplanetary magnetic field; (2) those photospheric areas which do contribute open field lines lie beneath coronal holes and within the boundaries of the holes as projected onto the photosphere or else between loop systems of an active region; (3) the interplanetary magnetic field in the plane of the sun's equator, essentially the field in the ecliptic plane, may connect to photospheric regions of high latitude; and (4) the fastest solar wind streams are correlated with those magnetic flux tubes which expand least in cross-sectional area over the distance between the photosphere and the coronal height where the solar wind begins.

Levine, R. H.↗

NASA NTRS1977

Nonlinear singularly perturbed optimal control problems with singular arcs

A third order, nonlinear, singularly perturbed optimal control problem is considered under assumptions which assure that the full problem is singular and the reduced problem is nonsingular. The separation between the singular arc of the full problem and the optimal control law of the reduced one, both of which are hypersurfaces in state space, is of the same order as the small parameter of the problem. Boundary layer solutions are constructed which are stable and reach the outer solution in a finite time. A uniformly valid composite solution is then formed from the reduced and boundary layer solutions. The value of the approximate solution is that it is relatively easy to obtain and does not involve singular arcs. To illustrate the utility of the results, the technique is used to obtain an approximate solution of a simplified version of the aircraft minimum time-to-climb problem. A numerical example is included.

Ardema, M. D.↗

NASA NTRS1978

Practical calculation of laminar and turbulent bled-off boundary layers

Bleed-off of boundary layer material is shown to be an effective means for reducing drag by conserving the laminar boundary layer and preventing separation of the turbulent boundary layer. The case in which the two effects of bleed-off overlap is examined. Empirical methods are extended to the case of bleed-off. Laminar and turbulent boundary layers are treated simultaneously and the approximation differential equations are solved without an uncertain error. The case without bleed-off is also treated.

Eppler, R.↗

NASA NTRS1979

Nonlinear singularly perturbed optimal control problems with singular arcs

Singular perturbation techniques are studied for dealing with singular arc problems by analyzing a relatively low-order but otherwise general system. This system encompasses many flight mechanic problems including Goddard's problem and a version of the minimum time-to-climb problem. Boundary layer solutions are constructed which are stable and reach the outer solution in a finite time. A uniformly valid composite solution is then formed from the reduced and boundary layer solutions. The value of the approximate solution is that it is relatively easy to obtain and does not involve singular arcs. To illustrate the utility of the results, the technique is used to obtain an approximate solution of a simplified version of the aircraft minimum time-to-climb problem.

Ardema, M. D.↗

NASA NTRS1981

The distant bow shock and magnetotail of Venus - Magnetic field and plasma wave observations

An examination of the magnetic field and plasma wave data obtained by the Pioneer Venus orbiter in the wake region behind Venus discloses a well developed bow shock whose location is similar to that observed on previous missions in contrast to the dayside bow shock. Venus also has a well developed magnetotail in which the field strenght is enhanced over magnetosheath values and in which the magnetic field is aligned approximately with the solar wind direction. The boundary between magnetosheath and magnetotail is also marked by a change in the plasma wave spectrum.

Russell, C. T.↗

NASA NTRS1982

Stability of Goertler vortices in boundary layers

A formal analysis of Goertler-type instability is presented. The boundary-layer and disturbance equations are formulated in a general, orthogonal, curvilinear system of coordinates constructed from the inviscid flow over a curved surface. Effects of curvature on the boundary-layer flow are analyzed. The basic approximation for the disturbance equations is presented and solved numerically. Previous analyses are discussed and compared with our analysis. It is shown that the general system of coordinates developed in this analysis and the correct order-of-magnitude analysis of the disturbance velocities with two velocity scales leads to a rational foundation for future work in Goertler vortices.

Floryan, J. M.↗

NASA NTRS1984

Application of a fractional-step method to incompressible Navier-Stokes equation

A numerical method for computing three dimensional, time dependent incompressible flows is presented. The method is based on a fractional step, or time-splitting, scheme in conjunction with the approximate-factorization technique. The use of velocity boundary conditions for the intermediate velocity field leads to inconsistent numerical solutions. Appropriate boundary conditions for the intermediate velocity field are derived and tested. Numerical solutions for flow inside a driven cavity and over a backward-facing step are presented and compared with experimenal data and other numerical results.

Kim, J.↗

NASA NTRS1986

Semiempirical methods for computing turbulent flows

Two semiempirical theories which provide a basis for determining the turbulent friction and heat exchange near a wall are presented: (1) the Prandtl-Karman theory, and (2) the theory utilizing an equation for the energy of turbulent pulsations. A comparison is made between exact numerical methods and approximate integral methods for computing the turbulent boundary layers in the presence of pressure, blowing, or suction gradients. Using the turbulent flow around a plate as an example, it is shown that, when computing turbulent flows with external turbulence, it is preferable to construct a turbulence model based on the equation for energy of turbulent pulsations.

Belov, I. A.↗

NASA NTRS1986

A comparison of computed and experimental surface pressure and heating on 70 deg sphere cones at angles of attack to 20 deg

A method for handling the flow fields around 70 deg spherical cones at angles of attack up to 20 deg is described. Surface pressure and heating results are compared with existing wind tunnel data. Computed surface pressures are found to be in excellent agreement with experimental data, while the surface heating from approximate and finite difference forms of the boundary layer equations are in good agreement with the data. The effect of bluntness ratio on surface pressures and heating is very small. Body corner radius, while not having a significant impact on body aerodynamics, produced large changes in surface pressure and heating distribution.

Weilmuenster, K. J.↗

NASA NTRS1985

Topics in spectral methods

After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.

Gottlieb, D.↗

NASA NTRS1985

Application of a fractional-step method to incompressible Navier-Stokes equations

Kim, J.↗

NASA NTRS1987

Model-size reduction for the analysis of symmetric structures with asymmetric boundary conditions

A simple computational procedure is presented for reducing the size of the analysis model for a symmetric structure with asymmetric boundary conditions to that of the corresponding structure with symmetric boundary conditions. The procedure is based on approximating the asymmetric response of the structure by a linear combination of symmetric and antisymmetric global approximation vectors (or modes). The key elements of the procedure are (1) restructuring the governing finite-element equations to delineate the contributions to the symmetric and antisymmetric components of the asymmetric response, (2) successive application of the finite element method and the classical Rayleigh-Ritz technique. The finite-element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh-Ritz technique. The effectiveness of the computational procedure is demonstrated by means of numerical examples of linear static problems of shells, and its potential for solving nonlinear problems is discussed.

Noor, Ahmed K.↗

NASA NTRS1987

Similarity rule for sidewall boundary-layer effects in airfoil testing

The simplified form of the similarity rule for sidewall boundary layer effects presented envisions the generation of changes in both airfoil thickness and freestream Mach number by the sidewall boundary layer. Within the small-disturbance approximation, this approach encompasses the methods of both Barnwell (1980) and Sewall (1982), and can accordingly be applied at speeds ranging from the lowest to transonic Mach numbers.

Murthy, A. V.↗

NASA NTRS1992

Calculation of unsteady transonic flows with mild separation by viscous-inviscid interaction

This paper presents a method for calculating viscous effects in two- and three-dimensional unsteady transonic flow fields. An integral boundary-layer method for turbulent viscous flow is coupled with the transonic small-disturbance potential equation in a quasi-steady manner. The viscous effects are modeled with Green's lag-entrainment equations for attached flow and an inverse boundary-layer method for flows that involve mild separation. The boundary-layer method is used stripwise to approximate three-dimensional effects. Applications are given for two-dimensional airfoils, aileron buzz, and a wing planform. Comparisons with inviscid calculations, other viscous calculation methods, and experimental data are presented. The results demonstrate that the present technique can economically and accurately calculate unsteady transonic flow fields that have viscous-inviscid interactions with mild flow separation.

Howlett, James T.↗

NASA NTRS1992

An implicit Navier-Stokes code for turbulent flow modeling

This paper presents a numerical approach to calculating turbulent flows employing advanced turbulence models. The main features include a line-by-line Gauss-Seidel algorithm using Roe's approximate Riemann solver, TVD numerical schemes, implicit boundary conditions and a decoupled turbulence-model solver. Based on the problems tested so far, the method has consistently demonstrated its ability in offering accuracy, boundedness and a fast rate of convergence to steady-state solution.

Huang, P. G.↗

NASA NTRS1992

The disconnection event of 16.0 March 1986 in comet Halley

From kinematic extrapolation of tail/nucleus distance measurements on photographic images in the International Halley Watch (IHW) archive, we calculated the disconnection time of the 16-19 March 1986 event to be 16.0 (plus or minus 0.1) March. The solar wind conditions around Comet Halley at the time of the DE, inferred by corotation of IMP-8 satellite data to the comet, were such that (1) Comet Halley had just crossed the interplanetary magnetic field (IMF) sector boundary; (2) the solar wind density was approximately 8 cm(exp -3); (3) the solar wind speed was approximately 600 km/sec; (4) the IMF magnitude was approximately 8 nT. Given these conditions, we conclude that the most likely cause of the 16.0 March DE was front-side magnetic reconnection, as described in the model of Niedner and Brandt (1978).

Randall, Cora E.↗