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NASA NTRS1998

Nonlinear Interaction of Detuned Instability Waves in Boundary-Layer Transition: Amplitude Equations

The non-equilibrium critical-layer analysis of a system of frequency-detuned resonant-triads is presented. In this part of the analysis, the system of partial differential critical-layer equations derived in Part I is solved analytically to yield the amplitude equations which are analyzed using a combination of asymptotic and numerical methods. Numerical solutions of the inviscid non-equilibrium oblique-mode amplitude equations show that the frequency-detuned self-interaction enhances the growth of the lower-frequency oblique modes more than the higher-frequency ones. All amplitudes become singular at the same finite downstream position. The frequency detuning delays the occurrence of the singularity. The spanwise-periodic mean-flow distortion and low-frequency nonlinear modes are generated by the critical-layer interaction between frequency-detuned oblique modes. The nonlinear mean flow and higher harmonics as well as the primary instabilities become as large as the base mean flow in the inviscid wall layer in the downstream region where the distance from the singularity is of the order of the wavelength scale.

Lee, Sang Soo↗

NASA NTRS1984

Optical systolic solutions of linear algebraic equations

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Neuman, C. P.↗

NASA NTRS1980

Thermal Pollution Mathematical Model. Volume 3: User's Manual for One-Dimensional Numerical Model for the Seasonal Thermocline

A user's manual for a one dimensional thermal model to predict the temperature profiles of a deep body of water for any number of annual cycles is presented. The model is essentially a set of partial differential equations which are solved by finite difference methods using a high speed digital computer. The model features the effects of area change with depth, nonlinear interaction of wind generated turbulence and buoyancy, adsorption of radiative heat flux below the surface, thermal discharges, and the effects of vertical convection caused by discharge. The main assumption in the formulation is horizontal homogeneity. The environmental impact of thermal discharges from power plants is emphasized. Although the model is applicable to most lakes, a specific site (Lake Keowee, S.C.) application is described in detail. The programs are written in FORTRAN 5.

Lee, S. S.↗

NASA NTRS2018

Instability Wave–Streak Interactions in a High Mach Number Boundary Layer at Flight Conditions

The interaction of stationary streaks undergoing nonmodal growth with modally unstable instability waves in a hypersonic boundary-layer flow is studied using numerical computations. The geometry and flow conditions are selected to match a relevant trajectory location from the ascent phase of the HIFiRE-1 flight experiment; namely, a 7 degree half-angle, circular cone with 2.5 mm nose radius, freestream Mach number equal to 5.30, unit Reynolds number equal to 13.42 m -1 , and wall-to-adiabatic temperature ratio of approximately 0.35 over most of the vehicle. This paper investigates the nonlinear evolution of initially linear optimal disturbances that evolve into finite-amplitude streaks, followed by an analysis of the modal instability characteristics of the perturbed, streaky boundary-layer flow. The investigation is performed with stationary direct numerical simulations (DNS) and plane-marching parabolized stability equations (PSE), in conjunction with partial-differential-equation-based planar eigenvalue analysis. The overall effect of streaks is to reduce the peak amplification factors of instability waves, indicating a possible downstream shift in the onset of laminar-turbulent transition. The present study confirms previous findings that the mean flow distortion of the nonlinear streak perturbation reduces the amplification rates of the Mack-mode instability. More importantly, however, the present results demonstrate that the spanwise varying component of the streak can produce a larger effect on the Mack-mode amplification. The study with selected azimuthal wavenumbers for the stationary streaks reveals that a wavenumber of approximately 1.4 times larger than the optimal wavenumber is more effective in stabilizing the planar Mack-mode instabilities. In the absence of unstable first-mode waves for the present cold-wall condition, transition onset is expected to be delayed until the peak streak amplitude increases to nearly 35 percent of the freestream velocity, when intrinsic instabilities of the boundary-layer streaks begin to dominate the transition process. For streak amplitudes below that limit a significant net stabilization is achieved, yielding a potential transition delay that can exceed 100 percent of the length of the laminar region in the uncontrolled case.

Boundary layer stability↗

NASA NTRS1983

Aeroelastic loads prediction for an arrow wing. Task 3: Evaluation of the Boeing three-dimensional leading-edge vortex code

Two separated flow computer programs and a semiempirical method for incorporating the experimentally measured separated flow effects into a linear aeroelastic analysis were evaluated. The three dimensional leading edge vortex (LEV) code is evaluated. This code is an improved panel method for three dimensional inviscid flow over a wing with leading edge vortex separation. The governing equations are the linear flow differential equation with nonlinear boundary conditions. The solution is iterative; the position as well as the strength of the vortex is determined. Cases for both full and partial span vortices were executed. The predicted pressures are good and adequately reflect changes in configuration.

Manro, M. E.↗

NASA NTRS1989

Exponential Finite-Difference Technique

Report discusses use of explicit exponential finite-difference technique to solve various diffusion-type partial differential equations. Study extends technique to transient-heat-transfer problems in one dimensional cylindrical coordinates and two and three dimensional Cartesian coordinates and to some nonlinear problems in one or two Cartesian coordinates.

Handschuh, Robert F.↗

NASA NTRS1992

Comparison of level-1 and level-2 buckling and postbuckling solutions

The accuracy of the solutions obtained with the Level-1 computational modules of DISDECO (Delft Interactive Shell Design Code) are investigated by comparing them with solutions obtained with ANILISA, a Level-2 computational module. Whereas the Level-1 solutions are based on the use of trigonometric functions to represent the out-of-plane displacement W and involve Galerkin type approximations to reduce the governing partial differential equations to simple algebraic expressions, the Level-2 solutions employ a truncated Fourier expansion in the circumferential direction and solve the resulting ordinary differential equations forming a nonlinear eigenvalue problem by a Stodola like technique. This involves the numerical integration of the set of ordinary differential equations, which in turn makes it possible to satisfy the specified boundary conditions exactly. It is shown that in many applications the Level-1 predictions concerning the critical buckling load of the perfect structure and its imperfection sensitivity are quite accurate and reliable. For those cases where this is not true, it is shown how the pre and postprocessing facilities of DISDECO can be used efficiently to arrive at a reliable prediction. This involves always the use of the Level-2 computational module ANILISA with its rigorous prebuckling solution and its feature which makes it possible to satisfy the specified boundary conditions exactly.

Arbocz, J.↗

NASA NTRS1990

On a class of unsteady three-dimensional Navier Stokes solutions relevant to rotating disc flows: Threshold amplitudes and finite time singularities

A class of exact steady and unsteady solutions of the Navier Stokes equations in cylindrical polar coordinates is given. The flows correspond to the motion induced by an infinite disc rotating with constant angular velocity about the z-axis in a fluid occupying a semi-infinite region which, at large distances from the disc, has velocity field proportional to (x,-y,O) with respect to a Cartesian coordinate system. It is shown that when the rate of rotation is large, Karman's exact solution for a disc rotating in an otherwise motionless fluid is recovered. In the limit of zero rotation rate a particular form of Howarth's exact solution for three-dimensional stagnation point flow is obtained. The unsteady form of the partial differential system describing this class of flow may be generalized to time-periodic equilibrium flows. In addition the unsteady equations are shown to describe a strongly nonlinear instability of Karman's rotating disc flow. It is shown that sufficiently large perturbations lead to a finite time breakdown of that flow whilst smaller disturbances decay to zero. If the stagnation point flow at infinity is sufficiently strong, the steady basic states become linearly unstable. In fact there is then a continuous spectrum of unstable eigenvalues of the stability equations but, if the initial value problem is considered, it is found that, at large values of time, the continuous spectrum leads to a velocity field growing exponentially in time with an amplitude decaying algebraically in time.

Hall, Philip↗

NASA NTRS1996

The Evolution of Finite Amplitude Wavetrains in Plane Channel Flow

We consider a viscous incompressible fluid flow driven between two parallel plates by a constant pressure gradient. The flow is at a finite Reynolds number, with an 0(l) disturbance in the form of a traveling wave. A phase equation approach is used to discuss the evolution of slowly varying fully nonlinear two dimensional wavetrains. We consider uniform wavetrains in detail, showing that the development of a wavenumber perturbation is governed by Burgers equation in most cases. The wavenumber perturbation theory, constructed using the phase equation approach for a uniform wavetrain, is shown to be distinct from an amplitude perturbation expansion about the periodic flow. In fact we show that the amplitude equation contains only linear terms and is simply the heat equation. We review, briefly, the well known dynamics of Burgers equation, which imply that both shock structures and finite time singularities of the wavenumber perturbation can occur with respect to the slow scales. Numerical computations have been performed to identify areas of the (wavenumber, Reynolds number, energy) neutral surface for which each of these possibilities can occur. We note that the evolution equations will breakdown under certain circumstances, in particular for a weakly nonlinear secondary flow. Finally we extend the theory to three dimensions and discuss the limit of a weak spanwise dependence for uniform wavetrains, showing that two functions are required to describe the evolution. These unknowns are a phase and a pressure function which satisfy a pair of linearly coupled partial differential equations. The results obtained from applying the same analysis to the fully three dimensional problem are included as an appendix.

Hewitt, R. E.↗

NASA NTRS1983

Spectral methods for partial differential equations

Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized.

Hussaini, M. Y.↗

NASA NTRS1984

Spectral methods for partial differential equations

Hussaini, M. Y.↗

NASA NTRS1984

Analysis of lateral and torsional vibration characteristics of beams and shafts with end located rotational masses

Partial differential equations are derived for free lateral and torsional vibration of a uniform free-free beam with a rotational mass attached to each extremity. For appropriate boundary conditions, nonlinear algebraic equations are obtained using a symbolic manipulation computer program, the solutions of which enable the computation of the neutral frequencies and mode-shapes. The mode-shapes are linear combinations of trigonometric and hyperbolic sine and cosine functions. A computer program is written for the numerical solution of the algebraic equations mentioned above, which can compute the natural frequencies, mode-shapes, and node points for any given set of parameters, for any given number of modes.

Robertson, D. K.↗

NASA NTRS1975

Second-order non-iterative ADI solution of non-linear partial differential equations

A new method for the solution of non-linear partial differential equations by an ADI procedure is described. Although the method is second order accurate in time, it does not require either iterations or predictor corrector methods to overcome the nonlinearity of the equations. Thus the computational effort required for the solution of the non-linear problem becomes similar to that required for the linear case. The method is applied to a two-dimensional 'extended Burgers equation'. Linear stability is studied, and some numerical solutions obtained. The improved accuracy obtained by the 2nd order truncation error is clearly manifested.

Wolfshtein, M.↗

NASA NTRS1980

Numerical solutions of transonic flows by parametric differentiation and integral equation techniques

The paper discusses results of an exploratory study of the advantages obtained by combining integral equation methods with the method of parametric differentiation in the treatment of transonic flow problems. In the proposed method, the nonlinear unsteady transonic flow equation for small perturbations is transformed into a linear equation by the use of the method of parametric differentiation. The linear equation is split into a pair of weakly coupled partial differential equations by writing the transformed perturbation potential as the sum of a steady component and an unsteady component. The solution of the steady equation as an integral equation is based on Ogana's treatment (1978). As a test case, the formulation is applied to predict the steady transonic flow over a nonlifting parabolic-arc airfoil.

Sivaneri, N. T.↗

NASA NTRS1990

Simulation of the collapse and dissipation of Langmuir wave packets

Particle-in-cell (PIC) simulations and Zakharov's partial differential equations (PDEs) are used to investigate the collapse of isolated Langmuir wave packets in two dimensions. Collapse thresholds are determined numerically, and the roles of enhanced Langmuir wave damping and nonlinearities not included in the standard Zakharov equations are discussed. The Langmuir wave and ion dynamics in PIC simulations are compared with the predictions of PDE simulations incorporating enhanced Langmuir damping. Electron heating and coherent acceleration in the PIC simulations are discussed and compared with predictions of the transit-time theory.

Newman, D. L.↗

NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given; the equations are used to study the spacial distributions of the density and velocity.

Deissler, R. G.↗

NASA NTRS1975

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitation and rotation terms in the equations are of first order in the space variables, the pressure gradient terms are of second order, and the turbulent viscosity term is of third order. The presence of a turbulent viscosity insures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the intial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial inflow plays an important role and allows collapse to occur even when the rotation is large. An approximate solution of the governing partial differential equations is also given, in order to study the spacial distributions of the density and velocity.

Deissler, R. G.↗

NASA NTRS1976

Gravitational collapse of a turbulent vortex with application to star formation

The gravitational collapse of a rotating cloud or vortex is analyzed by expanding the dependent variables in the equations of motion in two-dimensional Taylor series in the space variables. It is shown that the gravitational and rotational terms in the equations are of first order in the space variables, the pressure-gradient terms are of second order, and the turbulent-viscosity term is of third order. The presence of turbulent viscosity ensures that the initial rotation is solid-body-like near the origin. The effect of pressure on the collapse process is found to depend on the shape of the initial density disturbance at the origin. Dimensionless collapse times, as well as the evolution of density and velocity, are calculated by solving numerically the system of nonlinear ordinary differential equations resulting from the series expansions. The axial flow is always inward and allows collapse to occur (axially) even when the rotation is large. An approximate solution of the governing partial differential equations is also given in order to study the spatial distributions of the density and velocity.

Deissler, R. G.↗