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NASA NTRS1987

An adaptive grid, unsteady model for two-dimensional magnetohydrodynamic (MHD) flow

An adaptive grid finite difference method for solving multi-dimensional, time-dependent, magnetohydrodynamic (MHD) equations is developed. The method is capable of solving problems that include high gradients due to geometry, propagation of shock waves and unsteady boundary conditions. The grid generation technique is based on variational principles with direct control over grid concentration, smoothness and skewness. An example for a two-dimensional MHD simulation of the propagation of a solar-flare-generated shock wave in solar wind flow in the heliographic equatorial plane is selected for illustration of this method.

Panitchob, Supat↗

NASA NTRS1987

Study of instability and active control of boundary layers using Navier-Stokes equations

Spatial stability of a three-dimensional growing boundary layer is studied using Navier-Stokes equations. The finite difference method used is an explicit, predictor-corrector type which is fourth-order accurate in space and second-order in time. Active control of two-dimensional disturbances is simulated by localized surface heating. The computations show features of the flow relevant to early stages of transition such as lifting of vorticity away from the wall and localized regions of intense shear. Active control with single heating strip is only marginally effective, but may be improved by multiple strips.

Parikh, Paresh↗

NASA NTRS1987

Linear stability analysis of three-dimensional compressible boundary layers

A compressible stability analysis computer code is developed. The code uses a matrix finite-difference method for local eigenvale solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used inital-value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed that may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black-box stability code. Sample stability calculations are presented for the boundary layer profiles of an LFC swept wing.

Malik, Mujeeb R.↗

NASA NTRS1987

Concepts for radically increasing the numerical convergence rate of the Euler equations

Integral equation and finite difference methods have been developed for solving transonic flow problems using linearized forms of the transonic small disturbance and Euler equations. A key element is the use of a strained coordinate system in which the shock remains fixed. Additional criteria are developed to determine the free parameters in the coordinate straining; these free parameters are functions of the shock location. An integral equation analysis showed that the shock is located by ensuring that no expansion shocks exist in the solution. The expansion shock appears as oscillations in the solution near the sonic line, and the correct shock location is determined by removing these oscillations. A second objective was to study the ability of the Euler equation to model separated flow.

Nixon, David↗

NASA NTRS1987

Navier-Stokes computations for circulation control airfoils

Navier-Stokes computations of subsonic to transonic flow past airfoils with augmented lift due to rearward jet blowing over a curved trailing edge are presented. The approach uses a spiral grid topology. Solutions are obtained using a Navier-Stokes code which employs an implicit finite difference method, an algebraic turbulence model, and developments which improve stability, convergence, and accuracy. Results are compared against experiments for no jet blowing and moderate jet pressures and demonstrate the capability to compute these complicated flows.

Pulliam, Thomas H.↗

NASA NTRS1988

Liquid sprays and flow studies in the direct-injection diesel engine under motored conditions

A two dimensional, implicit finite difference method of the control volume variety, a two equation model of turbulence, and a discrete droplet model were used to study the flow field, turbulence levels, fuel penetration, vaporization, and mixing in diesel engine environments. The model was also used to study the effects of engine speed, injection angle, spray cone angle, droplet distribution, and intake swirl angle on the flow field, spray penetration and vaporization, and turbulence in motored two-stroke diesel engines. It is shown that there are optimum conditions for injection, which depend on droplet distribution, swirl, spray cone angle, and injection angle. The optimum conditions result in good spray penetration and vaporization and in good fuel mixing. The calculation presented clearly indicates that internal combustion engine models can be used to assess, at least qualitatively, the effects of injection characteristics and engine operating conditions on the flow field and on the spray penetration and vaporization in diesel engines.

Nguyen, Hung Lee↗

NASA NTRS1988

Solving time-dependent two-dimensional eddy current problems

Results of transient eddy current calculations are reported. For simplicity, a two-dimensional transverse magnetic field which is incident on an infinitely long conductor is considered. The conductor is assumed to be a good but not perfect conductor. The resulting problem is an interface initial boundary value problem with the boundary of the conductor being the interface. A finite difference method is used to march the solution explicitly in time. The method is shown. Treatment of appropriate radiation conditions is given special consideration. Results are validated with approximate analytic solutions. Two stringent test cases of high and low frequency incident waves are considered to validate the results.

Lee, Min Eig↗

NASA NTRS1987

Nonplanar interface morphologies during unidirectional solidification of a binary alloy. II - Three-dimensional computations

A finite difference method is used to obtain three-dimensional steady-state solutions for nonplanar interface morphologies in order to study the situation of equal thermal properties in the crystal and melt with negligible latent heat release. Stable steady-state solutions corresponding to two-dimensional bands and three-dimensional hexagonal nodes, as well as to rectangular interface planiforms, are found using a model of an aluminum-chromium alloy with a distribution coefficient of greater than one. Hexagonal nodes are predicted near the onset of instability, in agreement with weakly nonlinear theory.

Mcfadden, G. B.↗

NASA NTRS1988

A Three-Dimensional, Time-Dependent Numerical Modeling of Super-Sonic, Super-Alfvenic MHD Flow

A three-dimensional, time-dependent, MHD disturbance propagating through a steady inhomogeneous MHD flow is analyzed by a numerical model based on the Lax-Wendroff finite difference method. The initial steady MHD flow is assumed to be supersonic and super-Alfvenic. The background steady MHD flow is obtained by dropping the time-dependent terms in the unsteady MHD equations. A transient MHD disturbance corresponding to a MHD shock is generated at the lower radial boundary surface. The resulting development and propagation of 3-D MHD disturbances interacting with the background steady MHD flow are numerically simulated in this first application to the field of interplanetary physics. Detailed descriptions of the mathematical formulation and numerical methodology are presented. Example computations are performed to demonstrate the validity of the formulation and the capabilities of the model.

Han, S. M.↗

NASA NTRS1988

Numerical investigation of three-dimensional flow separation using the boundary layer equations

The steady, incompressible, three-dimensional laminar and turbulent boundary-layer equations are solved in a streamline coordinate system and in a self-adaptive grid system using Matsuno's finite difference method. Techniques are described for calculating laminar and turbulent separation using the boundary-layer equations. Any type (bubble type or free vortex-layer type) of major separation line can be calculated at an angle of attack on ellipsoids of revolution by this boundary layer code. Results are presented for ellipsoids of revolution at angles of attack up to 45 degrees. Agreements with other numerical and experimental results are very good for laminar flows. Turbulent flows are also investigated with algebraic turbulence models proposed by Rotta and Cebeci and Smith. Good agreement with experimental results was obtained at a small angle of attack (10 degrees) but only qualitative agreement was obtained at a high angle of attack (30 degrees) for turbulent flow on a 6:1 ellipsoid of revolution.

Wie, Yong-Sun↗

NASA NTRS1986

Interface procedures for overlapping grids

The solution of partial differential equations by means of either implicit or conservative explicit finite-difference methods, on the one hand, or multicomponent overlapping grid schemes on the other, is presently subjected to interpolation at grid boundaries in an attempt to eliminate the time lag. The accuracy of the transient solution of a hyperbolic or parabolic partial differential equation is dependent on the procedures used to transfer information between grids in a composite grid system. The present techniques reduce numerical solution errors.

Mastin, C. Wayne↗

NASA NTRS1988

Developing fluid flow in a curved duct of square cross-section and its fully developed dual solutions

Developing fluid flow in a curved duct of square cross-section is studied numerically by a factored ADI finite-difference method on a staggered grid. A central-difference scheme with primitive variables is used inside the computational domain to reduce numerical diffusion. Two Reynolds numbers, 574 and 790, based upon a bulk velocity and hydraulic diameter are chosen for curvature ratios of 1/6.45 and 1/2.3, respectively. It is found that the secondary flow is far more complicated than expected, with the appearance of at least two pairs of vortices. Main-flow separation is also observed for the higher curvature ratio. Furthermore, it is observed that the flow develops into two quite different states downstream, depending upon the inlet conditions. Solutions of the fully developed Navier-Stokes equations is shown to be not unique beyond a certain critical Reynolds number. Developing flow seems to evolve into the fully developed state along a particular branch into which the fully developed solution bifurcates.

Soh, W. Y.↗

NASA NTRS1989

Calculation of steady and unsteady pressures at supersonic speeds with CAP-TSD

A finite difference technique is used to solve the transonic small disturbance flow equation making use of shock capturing to treat wave discontinuities. Thus the nonlinear effects of thickness and angle of attack are considered. Such an approach is made feasible by the development of a new code called CAP-TSD (Computational Aeroelasticity Program - Transonic Small Disturbance), and is based on a fully implicit approximate factorization (AF) finite difference method to solve the time dependent transonic small disturbance equation. The application of the CAP-TSD code to the calculation of low to moderate supersonic steady and unsteady flows is presented. In particular, comparisons with exact linear theory solutions are made for steady and unsteady cases to evaluate shock capturing and other features of the current method. In addition, steady solutions obtained from an Euler code are used to evaluate the small disturbance aspects of the code. Steady and unsteady pressure comparisons are made with measurements for an F-15 wing model and for the RAE tailplane model.

Bennett, Robert M.↗

NASA NTRS1989

Development and application of a program to calculate transonic flow around an oscillating three-dimensional wing using finite difference procedures

A finite difference method for solving the unsteady transonic flow about harmonically oscillating wings is investigated. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equation for small disturbances. The differential equation for the unsteady potential is linear with spatially varying coefficients and with the time variable eliminated by assuming harmonic motion. Difference equations are derived for harmonic transonic flow to include a coordinate transformation for swept and tapered planforms. A pilot program is developed for three-dimensional planar lifting surface configurations (including thickness) for the CRAY-XMP at Boeing Commercial Airplanes and for the CYBER VPS-32 at the NASA Langley Research Center. An investigation is made of the effect of the location of the outer boundaries on accuracy for very small reduced frequencies. Finally, the pilot program is applied to the flutter analysis of a rectangular wing.

Weatherill, Warren H.↗

NASA NTRS1989

Control-volume based Navier-Stokes equation solver valid at all flow velocities

A control-volume based finite difference method to solve the Reynolds averaged Navier-Stokes equations is presented. A pressure correction equation valid at all flow velocities and a pressure staggered grid layout are used in the method. Example problems presented herein include: a developing laminar channel flow, developing laminar pipe flow, a lid-driven square cavity flow, a laminar flow through a 90-degree bent channel, a laminar polar cavity flow, and a turbulent supersonic flow over a compression ramp. A k-epsilon turbulence model supplemented with a near-wall turbulence model was used to solve the turbulent flow. It is shown that the method yields accurate computational results even when highly skewed, unequally spaced, curved grids are used. It is also shown that the method is strongly convergent for high Reynolds number flows.

Kim, S.-W.↗

NASA NTRS1989

Rotordynamic coefficients for labyrinth seals calculated by means of a finite difference technique

The compressible, turbulent, time dependent and three dimensional flow in a labyrinth seal can be described by the Navier-Stokes equations in conjunction with a turbulence model. Additionally, equations for mass and energy conservation and an equation of state are required. To solve these equations, a perturbation analysis is performed yielding zeroth order equations for centric shaft position and first order equations describing the flow field for small motions around the seal center. For numerical solution a finite difference method is applied to the zeroth and first order equations resulting in leakage and dynamic seal coefficients respectively.

Nordmann, R.↗

NASA NTRS1989

Numerical studies of the effects of jet-induced mixing on liquid-vapor interface condensation

Numerical solutions of jet-induced mixing in a partially full cryogenic tank are presented. An axisymmetric laminar jet is discharged from the central part of the tank bottom toward the liquid-vapor interface. Liquid is withdrawn at the same volume flow rate from the outer part of the tank. The jet is at a temperature lower than the interface, which is maintained at a certain saturation temperature. The interface is assumed to be flat and shear-free and the condensation-induced velocity is assumed to be negligibly small compared with radial interface velocity. Finite-difference method is used to solve the nondimensional form of steady state continuity, momentum, and energy equations. Calculations are conducted for jet Reynolds numbers ranging from 150 to 600 and Prandtl numbers ranging from 0.85 to 2.65. The effects of above stated parameters on the condensation Nusselt and Stanton numbers which characterize the steady-state interface condensation process are investigated. Detailed analysis to gain a better understanding of the fundamentals of fluid mixing and interface condensation is performed.

Lin, Chin-Shun↗

NASA NTRS1989

Grid sensitivity capability for large scale structures

The considerations and the resultant approach used to implement design sensitivity capability for grids into a large scale, general purpose finite element system (MSC/NASTRAN) are presented. The design variables are grid perturbations with a rather general linking capability. Moreover, shape and sizing variables may be linked together. The design is general enough to facilitate geometric modeling techniques for generating design variable linking schemes in an easy and straightforward manner. Test cases have been run and validated by comparison with the overall finite difference method. The linking of a design sensitivity capability for shape variables in MSC/NASTRAN with an optimizer would give designers a powerful, automated tool to carry out practical optimization design of real life, complicated structures.

Nagendra, Gopal K.↗