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Results for “PARABOLIC EQUATION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
NASA NTRS1975

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

A numerical method is presented which is valid for integration of the parabolic-elliptic Navier-Stokes equations. The solution procedure is applied to the three-dimensional supersonic flow of a jet issuing into a supersonic free stream. Difficulties associated with the imposition of free-stream boundary conditions are noted, and a coordinate transformation, which maps the point at infinity onto a finite value, is introduced to alleviate these difficulties. Results are presented for calculations of a square jet and varying-aspect-ratio rectangular jets. The solution behavior varies from axisymmetry for the square jet to nearly two-dimensional for the high-aspect-ratio rectangle, although the computation always calculates the flow as though it were truly three-dimensional.

Hirsh, R. S.↗
NASA NTRS1985

Calculation of three-dimensional, viscous flow through turbomachinery blade passages by parabolic marching

The three-dimensional compressible Navier-Stokes equations are formulated in a rotating coordinate system, so as to include centrifugal and Coriolis forces. The equations are parabolized by using a previously calculated inviscid static pressure field. The thin layer Navier-Stokes approximation, which neglects streamwise diffusion, is used. A body-fitted coordinate system is used. The streamwise momentum equation is uncoupled from the cross-stream momentum equation by using contravariant momentum components, and then using the contravariant velocity components as primary unknowns. To reduce problems with small separated regions, the Reyhner and Flugge-Lotz approximation is used. The energy equation is included to allow for calculation of heat transfer. The flow may be laminar, or a simple eddy-viscosity turbulence may be used. A number of curved ducts and an axial stator were analyzed, including cases for which experimental data are available.

Katsanis, T.↗
NASA NTRS1985

Calculation of three-dimensional, viscous flow through turbomachinery blade passages by parabolic marching

The three-dimensional compressible Navier-Stokes equations are formulated in a rotating coordinate system, so as to include centrifugal and Coriolis forces. The equations are parabolized by using a previously calculated inviscid static pressure field. The thin layer Navier-Stokes approximation, which neglects streamwise diffusion, is used. A body-fitted coordinate system is used. The streamwise momentum equation is uncoupled from the cross-stream momentum equation by using contravariant momentum components, and then using the contravariant velocity components as primary unknowns. To reduce problems with small separating regions, the Reyhner and Flugge-Lotz approximation is used. The energy equation is included to allow for calculation of heat transfer. The flow may be laminar, or a simple eddy-viscosity turbulence may be used. A number of curved ducts and an axial stator were analyzed, including cases for which experimental data are available.

Katsanis, T.↗
NASA NTRS2016

Optimal Growth in Hypersonic Boundary Layers

The linear form of the parabolized linear stability equations is used in a variational approach to extend the previous body of results for the optimal, nonmodal disturbance growth in boundary-layer flows. This paper investigates the optimal growth characteristics in the hypersonic Mach number regime without any high-enthalpy effects. The influence of wall cooling is studied, with particular emphasis on the role of the initial disturbance location and the value of the spanwise wave number that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary-layer equations, mean flow solutions based on the full Navier-Stokes equations are used in select cases to help account for the viscous- inviscid interaction near the leading edge of the plate and for the weak shock wave emanating from that region. Using the full Navier-Stokes mean flow is shown to result in further reduction with Mach number in the magnitude of optimal growth relative to the predictions based on the self-similar approximation to the base flow.

Navier Stokes equations↗
NASA NTRS2016

Direct Numerical Simulation of Transition Due to Traveling Crossflow Vortices

Previous simulations of laminar breakdown mechanisms associated with stationary crossflow instability over a realistic swept-wing configuration are extended to investigate the alternate scenario of transition due to secondary instability of traveling crossflow modes. Earlier analyses based on secondary instability theory and parabolized stability equations have shown that this alternate scenario is viable when the initial amplitude of the most amplified mode of the traveling crossflow instability is greater than approximately 0.03 times the initial amplitude of the most amplified stationary mode. The linear growth predictions based on the secondary instability theory and parabolized stability equations agree well with the direct numerical simulation. Nonlinear effects are initially stabilizing but subsequently lead to a rapid growth followed by the onset of transition when the amplitude of the secondary disturbance exceeds a threshold value. Similar to the breakdown of stationary vortices, the transition zone is rather short and the boundary layer becomes completely turbulent across a distance of less than 15 times the boundary layer thickness at the completion of transition.

Li, Fei↗
NASA NTRS1982

Propagation of wide bandwidth signals through strongly turbulent ionized media

Analytic and numerical techniques are presented which directly address the problem of propagation of wide bandwidth signals through random ionized media. This work is applicable to the problems of satellite communication and space based radar observation through a disturbed ionospheric propagation channel that would result from a high altitude chemical release or nuclear detonation. An analytic solution is presented for the two-position, two-frequency mutual coherence function for spherical wave propagation in the strong scattering limit. This solution is used to derive simple expressions for the power impulse response function and to determine its relationship to the important parameters of decorrelation distance and coherence bandwidth which describe the disturbed propagation channel. Results for mean time delay and time delay jitter are presented and compared to direct simulation results and to other theoretical calculations. A numerical/analytical solution to the parabolic wave equation is presented in the form of a multiple phase-screen (MPS) propagation simulation. In this very general solution technique, the random medium is divided into a finite number of layers. The field fluctuation through each layer is obtained by replacing the layer by a centrally located thin phase-screen, whose statistical properties are determined from the statistics of the electron-density irregularities. The waveform then propagates from phase-screen to phase-screen via an exact solution to the Fresnel-Kirchnoff equation. For wide bandwidth waveforms, numerical solutions are obtained at a number of discrete frequencies centered about the carrier and then time-domain solutions are obtained by Fourier transform techniques. Detailed results are presented for a number of interesting cases including propagation of a 20 MHz bandwidth signal through a finite barium cloud at a carrier frequency of 100 MHz. One of the major uses of the MPS propagation simulation has been to provide realizations of the received signal after propagation through a disturbed channel. The MPS simulation obtains a general solution of the parabolic wave equation under both weak and strong scattering conditions. A second technique to directly obtain realizations of wide bandwidth waveforms is presented here. This technique is limited to the case of strong scattering but requires only a fraction of the computer resources needed for MPS signal generation. Detailed comparisons of the two signal generation techniques are presented.

Random Media↗
NASA NTRS1979

Numerical prediction of three-dimensional juncture region flow using the parabolic Navier-Stokes equations

A numerical solution algorithm is established for prediction of subsonic turbulent three-dimensional flows in aerodynamic configuration juncture regions. A turbulence closure model is established using the complete Reynolds stress. Pressure coupling is accomplished using the concepts of complementary and particular solutions to a Poisson equation. Specifications for data input juncture geometry modification are presented.

Baker, A. J.↗
NASA NTRS1983

A modified Dodge algorithm for the parabolized Navier-Stokes equations and compressible duct flows

A revised version of Dodge's split-velocity method for numerical calculation of compressible duct flow has been developed. The revision incorporates balancing of massflow rates on each marching step in order to maintain front-to-back continuity during the calculation. Qualitative agreement with analytical predictions and experimental results has been obtained for some flows with well-known solutions.

Cooke, C. H.↗
NASA NTRS1986

A coupled marching procedure for the partially parabolized Navier-Stokes equations

A coupled finite-difference formulation is described for solving a reduced form of the compressible Navier-Stokes equations by a multiple space marching procedure. The properties of the equations are discussed and the solution algorithm presented. The scheme is used to compute incompressible flows by taking the incompressible limit (M about 0.1) of the compressible formulation. A separate Poisson equation is not required for the pressure. Results are compared with experimental data and other numerical predictions for low Reynolds number channel inlet flow, flow over a rearward-facing step in a channel, and flow near the trailing edge of a flat plate.

Liu, Xuezong↗
NASA NTRS1992

Implementation of Vigneron's streamwise pressure gradient approximation in the PNS equations

For single sweep parabolized Navier-Stokes solvers, the streamwise pressure gradient must be modified in the subsonic region to eliminate numerical instabilities. The accuracy of this modification on the solution of the parabolized Navier-Stokes equations with Vigneron's technique is shown to depend on how the numerical approximation of the pressure gradient is formed. A simple test case of supersonic laminar flow over a flat plate is computed with two different numerical methods for solving the PNS equations. Significant errors in the temperature profile and skin friction coefficient are demonstrated using a fully conservative differencing treatment of Vigneron's splitting for the pressure gradient typically used in parabolized Navier-Stokes solvers. The physical reason for this error is discussed. An alternate formulation is demonstrated which minimizes these errors.

Morrison, Joseph H.↗
NASA NTRS1996

Finite Difference Time Marching in the Frequency Domain: A Parabolic Formulation for the Convective Wave Equation

An explicit finite difference iteration scheme is developed to study harmonic sound propagation in ducts. To reduce storage requirements for large 3D problems, the time dependent potential form of the acoustic wave equation is used. To insure that the finite difference scheme is both explicit and stable, time is introduced into the Fourier transformed (steady-state) acoustic potential field as a parameter. Under a suitable transformation, the time dependent governing equation in frequency space is simplified to yield a parabolic partial differential equation, which is then marched through time to attain the steady-state solution. The input to the system is the amplitude of an incident harmonic sound source entering a quiescent duct at the input boundary, with standard impedance boundary conditions on the duct walls and duct exit. The introduction of the time parameter eliminates the large matrix storage requirements normally associated with frequency domain solutions, and time marching attains the steady-state quickly enough to make the method favorable when compared to frequency domain methods. For validation, this transient-frequency domain method is applied to sound propagation in a 2D hard wall duct with plug flow.

Baumeister, K. J.↗
NASA NTRS2002

Stochastic Modeling of Laminar-Turbulent Transition

Stochastic versions of stability equations are developed in order to develop integrated models of transition and turbulence and to understand the effects of uncertain initial conditions on disturbance growth. Stochastic forms of the resonant triad equations, a high Reynolds number asymptotic theory, and the parabolized stability equations are developed.

Rubinstein, Robert↗