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NASA NTRS1993

Grid and aerodynamic sensitivity analyses of airplane components

An algorithm is developed to obtain the grid sensitivity with respect to design parameters for aerodynamic optimization. The procedure is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational B-Splines) for defining the wing-section geometry. An interactive algebraic grid generation technique, known as Two-Boundary Grid Generation (TBGG) is employed to generate C-type grids around wing-sections. The grid sensitivity of the domain with respect to geometric design parameters has been obtained by direct differentiation of the grid equations. A hybrid approach is proposed for more geometrically complex configurations such as a wing or fuselage. The aerodynamic sensitivity coefficients are obtained by direct differentiation of the compressible two-dimensional thin-layer Navier-Stokes equations. An optimization package has been introduced into the algorithm in order to optimize the wing-section surface. Results demonstrate a substantially improved design due to maximized lift/drag ratio of the wing-section.

Sadrehaghighi, Ideen↗

NASA NTRS1995

Grid Sensitivity and Aerodynamic Optimization of Generic Airfoils

An algorithm is developed to obtain the grid sensitivity with respect to design parameters for aerodynamic optimization. The procedure is advocating a novel (geometrical) parameterization using spline functions such as NURBS (Non-Uniform Rational B- Splines) for defining the airfoil geometry. An interactive algebraic grid generation technique is employed to generate C-type grids around airfoils. The grid sensitivity of the domain with respect to geometric design parameters has been obtained by direct differentiation of the grid equations. A hybrid approach is proposed for more geometrically complex configurations such as a wing or fuselage. The aerodynamic sensitivity coefficients are obtained by direct differentiation of the compressible two-dimensional thin-layer Navier-Stokes equations. An optimization package has been introduced into the algorithm in order to optimize the airfoil surface. Results demonstrate a substantially improved design due to maximized lift/drag ratio of the airfoil.

Sadrehaghighi, Ideen↗

NASA NTRS1972

Compressible boundary layer with normal pressure gradients: Quasi-similarity equations - Their properties at the wall and at sharp and blunt leading edges.

The failure of most viscous-inviscid interaction methods at strong interactions is attributed to the presence of a normal pressure gradient. A new theory is proposed for supersonic laminar boundary layers that can generate normal pressure gradients. The Navier-Stokes equations are reexamined by an order of magnitude analysis and all first and second order terms are retained. The approximation is found to be dependent not only on the boundary layer thickness but also on the ratio of the dimensionless viscosity and density. The equations are transformed into two quasi-similar, nonlinear, third order, ordinary integro-differential equations for the velocity and pressure as functions of a single transverse variable. The properties of the equations at the boundaries are discussed.

Seginer, A.↗

NASA NTRS1981

PAN AIR: A computer program for predicting subsonic or supersonic linear potential flows about arbitrary configurations using a higher order panel method. Volume 1: Theory document (version 1.1)

Panel aerodynamics (PAN AIR) is a system of computer programs designed to analyze subsonic and supersonic inviscid flows about arbitrary configurations. A panel method is a program which solves a linear partial differential equation by approximating the configuration surface by a set of panels. An overview of the theory of potential flow in general and PAN AIR in particular is given along with detailed mathematical formulations. Fluid dynamics, the Navier-Stokes equation, and the theory of panel methods were also discussed.

Magnus, A. E.↗

NASA NTRS1992

Three-dimensional boundary layer stability and transition

Nonparallel and nonlinear stability of a three-dimensional boundary layer, subject to crossflow instability, is investigated using parabolized stability equations (PSEs). Both traveling and stationary disturbances are considered and nonparallel effect on crossflow instability is found to be destabilizing. Our linear PSE results for stationary disturbances agree well with the results from direct solution of Navier-Stokes equations obtained by Spalart (1989). Nonlinear calculations have been carried out for stationary vortices and the computed wall vorticity pattern results in streamwise streaks which resemble remarkably well with the surface oil-flow visualizations in swept-wing experiments. Other features of the stationary vortex development (half-mushroom structure, inflected velocity profiles, vortex doubling, etc.) are also captured in our nonlinear calculations. Nonlinear interaction of the stationary amplitude of the stationary vortex is large as compared to the traveling mode, and the stationary vortex dominates most of the downstream development. When the two modes have the same initial amplitude, the traveling mode dominates the downstream development owing to its higher growth rate, and there is a tendency for the stationary mode to be suppressed. The effect of nonlinear wave development on the skin-friction coefficient is also computed.

Malik, M. R.↗

NASA NTRS1993

Numerical simulation of crossing/turbulent boundary layer interaction at Mach 8.3 comparison of zero and two-equation turbulence models

A 3D hypersonic crossing shock wave/turbulent boundary layer interaction is examined numerically. The test geometry consists of a pair of opposing sharp fins of angle alpha = 15 deg mounted on a flat plate. The freestream Mach number is 8.28. Two theoretical models are evaluated. The full 3D Reynolds-averaged Navier-Stokes equations are solved using the Baldwin-Lomax algebraic turbulent eddy viscosity model and the Rodi turbulence model. Computed results for both cases show good agreement with experiment for flat plate surface pressure and for pitot pressure and yaw angle profiles in the flowfield. General agreement is obtained for surface flow direction. Fair to poor agreement is obtained for surface heat transfer, indicating a need for more accurate turbulence models. The overall flowfield structure is similar to that observed in previous crossing shock interaction studies.

Narayanswami, N.↗

NASA NTRS1977

Over-relaxation applied to the MacCormack finite-difference scheme

An over-relaxation procedure is applied to the MacCormack finite-difference scheme in order to reduce the computation time required to obtain a steady-state solution. The implementation of this acceleration procedure to an existing computer program using the regular MacCormack method is extremely simple and does not require additional storage. The over-relaxation procedure does not alter the steady-state solution, which is second-order accurate. The method is first applied to Burgers' equation. A stability condition and an expression for the increase in the rate of convergence are derived. The method is then applied to the calculation of the hypersonic viscous flow over a flat plate, using the complete Navier-Stokes equations, and the inviscid flow over a wedge. Reductions in computing time by factors of 3 and 1.5, respectively, are obtained by over-relaxation.

Desideri, J.-A.↗

NASA NTRS1985

Application of a viscous-inviscid interaction method to predict transonic separated flows

A viscous-inviscid interaction numerical procedure is used to predict turbulent transonic separated flows over axisymmetric bodies. The viscous flow solutions are obtained by solving the boundary-layer equations inversely in a coupled manner by a finite-difference scheme; the inviscid flow is determined by a finite-difference solution to the full-potential equation in conservative form. Parameters affecting the convergence rate of the interaction procedure are discussed. The predictions of a new turbulence model proposed by Johnson and King are compared with experimental data and numerical solutions to the Navier-Stokes equations for flows with large regions of separation.

Lee, D. S.↗

NASA NTRS1985

The collisional dynamics of particulate disks

It is shown that the use of the Krook equation greatly simplifies the discussion of the collisional dynamics of particulate disks. Collisions are assumed to be inelastic from the outset. A simple heuristic argument is used to compute the effective collision rate in a disk of spherical particles with a power-law distribution of sizes. For Saturn's rings, the effective collision rate for momentum transport is substantially lower than that conventionally estimated on the basis of an observed optical depth at visual wavelengths. The closure of the vertically integrated set of moment equations without discarding the third-order moments at the outset is discussed, allowing for the possibility of a bent disk. In the limit that the collision frequency is much larger than the orbit frequency, the usual Navier-Stokes equations of viscous hydrodynamics for a thin disk are recovered with an explicit expression for the shear viscosity. The method is generalized to include the effects of gravitational scattering.

Shu, F. H.↗

NASA NTRS1988

An explicit Runge-Kutta method for unsteady rotor/stator interaction

A quasi-three-dimensional rotor/stator analysis has been developed for blade-to-blade flows in turbomachinery. The analysis solves the unsteady Euler or thin-layer Navier-Stokes equations in a body-fitted coordinate system. It accounts for the effects of rotation, radius change, and stream-surface thickness. The Baldwin-Lomax eddy-viscosity model is used for turbulent flows. The equations are integrated in time using a four-stage Runge-Kutta scheme with a constant timestep. Results are shown for the first stage of the Space Shuttle Main Engine high pressure fuel turbopump. Euler and Navier-Stokes results are compared on the scaled single- and multi-passage machine. The method is relatively fast and the quasi-three-dimensional formulation is applicable to a wide range of turbomachinery geometries.

Jorgenson, Philip C. E.↗

NASA NTRS1988

An explicit Runge-Kutta method for unsteady rotor/stator interaction

Jorgenson, Philip C. E.↗

NASA NTRS1992

An incremental strategy for calculating consistent discrete CFD sensitivity derivatives

In this preliminary study involving advanced computational fluid dynamic (CFD) codes, an incremental formulation, also known as the 'delta' or 'correction' form, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For typical problems in 2D, a direct solution method can be applied to these linear equations which are associated with aerodynamic sensitivity analysis. For typical problems in 2D, a direct solution method can be applied to these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods appear to be needed for future 3D applications; however, because direct solver methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form result in certain difficulties, such as ill-conditioning of the coefficient matrix, which can be overcome when these equations are cast in the incremental form; these and other benefits are discussed. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two laminar sample problems: (1) transonic flow through a double-throat nozzle; and (2) flow over an isolated airfoil.

Korivi, Vamshi Mohan↗

NASA NTRS1993

Methodology for sensitivity analysis, approximate analysis, and design optimization in CFD for multidisciplinary applications

In this study involving advanced fluid flow codes, an incremental iterative formulation (also known as the delta or correction form) together with the well-known spatially-split approximate factorization algorithm, is presented for solving the very large sparse systems of linear equations which are associated with aerodynamic sensitivity analysis. For smaller 2D problems, a direct method can be applied to solve these linear equations in either the standard or the incremental form, in which case the two are equivalent. Iterative methods are needed for larger 2D and future 3D applications, however, because direct methods require much more computer memory than is currently available. Iterative methods for solving these equations in the standard form are generally unsatisfactory due to an ill-conditioning of the coefficient matrix; this problem can be overcome when these equations are cast in the incremental form. These and other benefits are discussed. The methodology is successfully implemented and tested in 2D using an upwind, cell-centered, finite volume formulation applied to the thin-layer Navier-Stokes equations. Results are presented for two sample airfoil problems: (1) subsonic low Reynolds number laminar flow; and (2) transonic high Reynolds number turbulent flow.

Taylor, Arthur C., III↗

NASA NTRS1993

The helical decomposition and the instability assumption

Direct numerical simulations show that the triadic transfer function T(k,p,q) peaks sharply when q (or p) is much smaller than k. The triadic transfer function T(k,p,q) gives the rate of energy input into wave number k from all interactions with modes of wave number p and q, where k, p, q form a triangle. This observation was thought to suggest that energy is cascaded downscale through non-local interactions with local transfer and that there was a strong connection between large and small scales. Both suggestions were in contradiction with the classical Kolmogorov picture of the energy cascade. The helical decomposition was found useful in distinguishing between kinematically independent interactions. That analysis has gone beyond the question of non-local interaction with local transfer. In particular, an assumption about the statistical direction of triadic energy transfer in any kinematically independent interaction was introduced (the instability assumption). That assumption is not necessary for the conclusions about non-local interactions with local transfer recalled above. In the case of turbulence under rapid rotation, the instability assumption leads to the prediction that energy is transferred in spectral space from the poles of the rotation axis toward the equator. The instability assumption is thought to be of general validity for any type of triad interactions (e.g. internal waves). The helical decomposition and the instability assumption offer detailed information about the homogeneous statistical dynamics of the Navier-Stokes equations. The objective was to explore the validity of the instability assumption and to study the contributions of the various types of helical interactions to the energy cascade and the subgrid-scale eddy-viscosity. This was done in the context of spectral closures of the Direct Interaction or Quasi-Normal type.

Waleffe, Fabian A.↗

NASA NTRS1978

Numerical solution of flow problems using body-fitted coordinate systems

The technique of boundary-fitted coordinate systems is based on a method of automatic numerical generation of a general curvilinear coordinate system having a coordinate line coincident with each boundary of a general multi-connected region containing any number of arbitrarily shaped bodies. Once the curvilinear coordinate system is generated, any partial differential system of interest can be solved on this coordinate system by transforming the equations and solving the resulting system in finite difference approximation on the rectangular transformed plane. This method of automatic body-fitted curvilinear coordinate generation is used to construct finite-difference solutions of the full, time dependent Navier-Stokes equations for the unsteady viscous flow about arbitrary two-dimensional airfoils, or any other two-dimensional bodies. Finally, initial results for three-dimensional applications are also presented.

Thompson, J. F.↗

NASA NTRS1984

Three-dimensional viscous flows with large secondary velocity

A new system of approximation equations is derived for three-dimensional steady viscous compressible flows in which a primary-flow direction is present, but in which both transverse velocity components can be large. Previous approaches which address simplification of the steady Navier-Stokes equations are discussed, and a new approach is proposed. The transverse velocity vector which corrects a given potential flow has been decomposed into potential and rotational components. It is found that the potential-velocity vector may be assumed small, whereas the rotational-velocity vector may be assumed small, whereas the rotational velocity vector and hence the composite secondary flow can be of order unity. This assumption leads to a system of governing equations whose characteristic polynomial has a non-elliptic form for arbitrary Mach numbers. The resulting non-elliptic approximation equations can be solved as an initial/boundary-value problem. Computed results confirm the small scalar-potential approximation.

Briley, W. R.↗

NASA NTRS1984

Flowfield scaling of a swept compression corner interaction A comparison of experiment and computation

Over the past decade, there has been some important progress in understanding the problems of three-dimensional (3D) shock wave/turbulent boundary layer interactions. However, the problems are by no means solved. The present investigation has the objective to determine the flowfield structure of a swept compression corner interaction and to perform a test of an equation considered by Settles and Bogdonoff (1982). The experimental data obtained in a 20 x 20 cm high Reynolds number supersonic wind tunnel are compared with the predictions of a state-of-the-art numerical solution of the Navier-Stokes equations. It is found that a 3D scaling law for Reynolds number effects, previously established for interaction 'footprints', is equally valid when applied to the present flowfield.

Settles, G. S.↗

NASA NTRS1987

Free-wake analysis of a rotor in hover

A numerical method based on the axisymmetric, incompressible Navier-Stokes equations is combined with a lifting surface code to predict the vortex wake of hovering rotors. The lifting surface code, AMI Hover, is used to obtain the circulation distribution on the blade. This circulation distribution is fed into the Navier-Stokes code to compute the vortex wake under this specified circulation distribution. An iteration approach is used between these two codes to converge the circulation distribution and the shape of the vortex wake. A relaxation scheme is developed to resolve the instability encountered among the tip vortices. A reconcentration scheme is used to solve the diffusion problem due to the strong artificial viscosity. The results from the present method are compared with experimental data obtained by smoke-flow visualization and hot-wire measurements for several rotor blade configurations. The comparisons show that the present method is able to predict the complex wake system shed by a hovering rotor.

Chen, C. S.↗