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NASA NTRS1990

Applications of an adaptive unstructured solution algorithm to the analysis of high speed flows

An upwind cell-centered scheme for the solution of steady laminar viscous high-speed flows is implemented on unstructured two-dimensional meshes. The first-order implementation employs Roe's (1981) approximate Riemann solver, and a higher-order extension is produced by using linear reconstruction with limiting. The procedure is applied to the solution of inviscid subsonic flow over an airfoil, inviscid supersonic flow past a cylinder, and viscous hypersonic flow past a double ellipse. A detailed study is then made of a hypersonic laminar viscous flow on a 24-deg compression corner. It is shown that good agreement is achieved with previous predictions using finite-difference and finite-volume schemes. However, these predictions do not agree with experimental observations. With refinement of the structured grid at the leading edge, good agreement with experimental observations for the distributions of wall pressure, heating rate and skin friction is obtained.

Thareja, R. R.↗

NASA NTRS1989

Application of a lower-upper implicit scheme and an interactive grid generation for turbomachinery flow field simulations

A finite-volume lower-upper (LU) implicit scheme is used to simulate an inviscid flow in a turbine cascade. This approximate factorization scheme requires only the inversion of sparse lower and upper triangular matrices, which can be done efficiently without extensive storage. As an implicit scheme it allows a large time step to reach the steady state. An interactive grid generation program (TURBO), which is being developed, is used to generate grids. This program uses the control point form of algebraic grid generation which uses a sparse collection of control points from which the shape and position of coordinate curves can be adjusted. A distinct advantage of TURBO compared with other grid generation programs is that it allows the easy change of local mesh structure without affecting the grid outside the domain of dependence. Sample grids are generated by TURBO for a compressor rotor blade and a turbine cascade. The turbine cascade flow is simulated by using the LU implicit scheme on the grid generated by TURBO.

Choo, Yung K.↗

NASA NTRS1990

Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction

High order accurate finite-volume schemes for solving the Euler equations of gasdynamics are developed. Central to the development of these methods are the construction of a k-exact reconstruction operator given cell-averaged quantities and the use of high order flux quadrature formulas. General polygonal control volumes (with curved boundary edges) are considered. The formulations presented make no explicit assumption as to complexity or convexity of control volumes. Numerical examples are presented for Ringleb flow to validate the methodology.

Barth, Timothy J.↗

NASA NTRS1990

Three-dimensional shock-shock interactions on the scramjet inlet

The effects of shock impingement on the inlet of a scramjet engine are investigated numerically. The impinging shock is caused by the vehicle forebody. The interaction of this forebody shock with the inlet leading edge shock results in a very complex fully three-dimensional flowfield containing local regions of high pressure and intense heating. In the present investigation, this complex flowfield is calculated by solving the thin-layer Navier-Stokes equations using a finite-volume flux splitting technique due to van Leer. For zero or small sweep angles a Type IV interaction occurs while for moderate sweep of about 25 deg, a Type V interaction occurs. Both Type IV and Type V interactions are investigated.

Singh, D. J.↗

NASA NTRS1990

Multigrid solution of unsteady Navier-Stokes equations using a pressure method

A multigrid relaxation method is applied to a pressure-based implicit procedure to solve unseady, incompressible Navier-Stokes equations. The present multigrid method is a Correction Scheme according to Brandt. This method is used to solve the scalar matrices resulting from the finite-volume formulation and uses flux averaging as the restriction operator. The accuracy and computational efficiency are demonstrated with a steady state driven cavity flow and an unsteady flow over a circular cylinder case. The results are compared with single grid results using the OrthoMin conjugate gradient method and experimental data.

Jiang, Y.↗

NASA NTRS1990

Computational study for passive control of supersonic asymmetric vortical flows around cones

The unsteady, compressible, double and single thin-layer Navier-Stokes equations are used to numerically study for the passive control of steady and unsteady, supersonic asymmetric flows around circular and noncircular cones. The main computational scheme of the present study is the implicit upwind, flux-difference splitting, finite-volume scheme. Passive control of flow asymmetry is studied by using a vertical fin in the leeward plane of geometric symmetry and side strakes with and without thickness at different orientations. The study is focused on circular-section cones since they are the most potential section-shapes for strong flow asymmetry. Side-strakes passive control has been shown to be more efficient and practical than the vertical-fin passive control.

Liu, C. H.↗

NASA NTRS1990

Modeling supersonic combustion using a fully-implicit numerical method

A fully-implicit finite-volume algorithm for two-dimensional axisymmetric flows has been coupled to a detailed hydrogen-air reaction mechanism (13 species and 33 reactions) so that supersonic combustion phenomena may be investigated. Numerical computations are compared with ballistic-range shadowgraphs of Lehr (1972) that exhibit two discontinuities caused by a blunt body as it passes through a premixed stoichiometric hydrogen-air mixture. The suitability of the numerical procedure for simulating these double-front flows is shown. The requirements for the physical formulation and the numerical modeling of these flowfields are discussed. Finally, the sensitivity of these external flowfields to changes in certain key reaction rate constants is examined.

Maccormack, Robert W.↗

NASA NTRS1989

Advances in fluid dynamics

Various papers on advances in fluid dynamics are presented. Individual topics addressed include: nonunique solutions of the Euler equations; numerical solution of the biharmonic equation using a spectral multigrid method; finite-volume shape and placement for efficiency, accuracy, and physics; NND schemes and their application to numerical simulation of complex plume flow problems; symbolic analysis and computational algorithm construction; and velocity and mass flux distribution measurements of spherical glass beads in air flow in a 90-deg vertical-to-horizontal bend. Also discussed are: hyperbolic solution adaptive mesh generation; progress and future directions in CFD; numerical simulation of shock focusing; microbubble formation and splitting in a turbulent boundary layer for turbulence reduction; numerical study of spherical Taylor-Couette flow, trajectories of vortex lines beneath a free surface or above a plane; and a mathematical model for response of the coronary circulation to high sustained gravitational force fields.

Ballhaus, W. F., Jr.↗

NASA NTRS1990

CFD spinoff - Computational electromagnetics for radar cross section (RCS) studies

A finite-volume discretization procedure derived from proven CFD methods is used to solve the conservation form of the time-domain Maxwell's equations, in order to compute EM scattering from layered objects. This time-domain approach handles both single-frequency/continuous wave and broadband-frequency/pulse incident excitation. Arbitrarily shaped objects are modeled by means of a body-fitted coordinate transformation; complex internal/external structures with many material layers are treated through the implementation of a multizone framework capable of handling any type of zonal boundary condition. Results are presented for various two- and three-dimensional problems.

Shankar, Vijaya↗

NASA NTRS1990

Computation of vortex-dominated flow for a delta wing undergoing pitching oscillation

The conservative, unsteady Euler equations for the flow relative to a moving frame of reference are used to solve for the three-dimensional steady and unsteady flows around a sharp-edged delta wing. The resulting equations are solved by using an implicit, approximately factored, finite-volume scheme. Implicit second-order and explicit second- and fourth-order dissipations are added to the scheme. The boundary conditions are explicitly satisfied. The grid is generated by locally using a modified Joukowski transformation in crossflow planes at the grid-chord stations. The computational applications cover a steady flow around a delta wing, whose results serve as the initial conditions for the unsteady flow around a pitching delta wing at a large mean angle of attack. The steady results are compared with the experimental data, and the unsteady results are compared with results of a flux-difference splitting scheme.

Kandil, Osama A.↗

NASA NTRS1990

Numerical computation of viscous flow about unconventional airfoil shapes

A new two-dimensional computer code was developed to analyze the viscous flow around unconventional airfoils at various Mach numbers and angles of attack. The Navier-Stokes equations are solved using an implicit, upwind, finite-volume scheme. Both laminar and turbulent flows can be computed. A new nonequilibrium turbulence closure model was developed for computing turbulent flows. This two-layer eddy viscosity model was motivated by the success of the Johnson-King model in separated flow regions. The influence of history effects are described by an ordinary differential equation developed from the turbulent kinetic energy equation. The performance of the present code was evaluated by solving the flow around three airfoils using the Reynolds time-averaged Navier-Stokes equations. Excellent results were obtained for both attached and separated flows about the NACA 0012 airfoil, the RAE 2822 airfoil, and the Integrated Technology A 153W airfoil. Based on the comparison of the numerical solutions with the available experimental data, it is concluded that the present code in conjunction with the new nonequilibrium turbulence model gives excellent results.

Ahmed, S.↗

NASA NTRS1989

Application of unstructured grid methods to steady and unsteady aerodynamic problems

The purpose is to describe the development of unstructured grid methods which have several advantages when compared to methods which make use of structured grids. Unstructured grids, for example, easily allow the treatment of complex geometries, allow for general mesh movement for realistic motions and structural deformations of complete aircraft configurations which is important for aeroelastic analysis, and enable adaptive mesh refinement to more accurately resolve the physics of the flow. Steady Euler calculations for a supersonic fighter configuration to demonstrate the complex geometry capability; unsteady Euler calculations for the supersonic fighter undergoing harmonic oscillations in a complete-vehicle bending mode to demonstrate the general mesh movement capability; and vortex-dominated conical-flow calculations for highly-swept delta wings to demonstrate the adaptive mesh refinement capability are discussed. The basic solution algorithm is a multi-stage Runge-Kutta time-stepping scheme with a finite-volume spatial discretization based on an unstructured grid of triangles in 2D or tetrahedra in 3D. The moving mesh capability is a general procedure which models each edge of each triangle (2D) or tetrahedra (3D) with a spring. The resulting static equilibrium equations which result from a summation of forces are then used to move the mesh to allow it to continuously conform to the instantaneous position or shape of the aircraft. The adaptive mesh refinement procedure enriches the unstructured mesh locally to more accurately resolve the vortical flow features. These capabilities are described in detail along with representative results which demonstrate several advantages of unstructured grid methods. The applicability of the unstructured grid methodology to steady and unsteady aerodynamic problems and directions for future work are discussed.

Batina, John T.↗

NASA NTRS1989

Navier-Stokes solutions for flows related to store separation

The objective is developing CFD capabilities to obtain solutions for viscous flows about generic configurations of internally and externally carried stores. The emphasis is placed on the supersonic flow regime with extensions being made to the transonic regime. The project is broken into four steps: (1) Cavity flows for internal carriage configurations; (2) High angle of attack flows, which may be experienced during the separation of the stores: (3) Flows about a body near a flat plate for external carriage configurations; and (4) Flows about a body inside or in the proximity of a cavity. Three-dimensional unsteady cavity flow solutions are obtained by an explicit, MacCormack algorithm, EMCAV3, for open, close, and transitional cavities. High angle of attack flows past cylinders are solved by an implicit, upwind algorithm. All the results compare favorably with the experimental data. For flows about multiple body configurations, the Chimera embedding scheme is modified for finite-volume and multigrid algorithms, MaGGiE. Then a finite volume, implicit, upwind, multigrid Navier-Stokes solver which uses on overlapped/embedded and zonal grids, VUMXZ3, is developed from the CFL3D code. Supersonic flows past a cylinder near a flat plate are computed using this code. The results are compared with the experimental data. Currently the VUMXZ3 code is being modified to accomplish step 4 of this project. Wind tunnel experiments are also being conducted for validation purposes.

Baysal, Oktay↗

NASA NTRS1989

A comparative study of Navier-Stokes codes for high-speed flows

A comparative study was made with four different codes for solving the compressible Navier-Stokes equations using three different test problems. The first of these cases was hypersonic flow through the P8 inlet, which represents inlet configurations typical of a hypersonic airbreathing vehicle. The free-stream Mach number in this case was 7.4. This 2-D inlet was designed to provide an internal compression ratio of 8. Initial calculations were made using two state-of-the-art finite-volume upwind codes, CFL3D and USA-PG2, as well as NASCRIN, a code which uses the unsplit finite-difference technique of MacCormack. All of these codes used the same algebraic eddy-viscosity turbulence model. In the experiment, the cowl lip was slightly blunted; however, for the computations, a sharp cowl leading edge was used to simplify the construction of the grid. The second test problem was the supersonic (Mach 3.0) flow in a three-dimensional corner formed by the intersection of two wedges with equal wedge angles of 9.48 degrees. The flow in such a corner is representative of the flow in the corners of a scramjet inlet. Calculations were made for both laminar and turbulent flow and compared with experimental data. The three-dimensional versions of the three codes used for the inlet study (CFL3D, USA-PG3, and SCRAMIN, respectively) were used for this case. For the laminar corner flow, a fourth code, LAURA, which also uses recently-developed upwind technology, was also utilized. The final test case is the two-dimensional hypersonic flow over a compression ramp. The flow is laminar with a free-stream Mach number of 14.1. In the experiment, the ramp angle was varied to change the strength of the ramp shock and the extent of the viscous-inviscid interaction. Calculations were made for the 24-degree ramp configuration which produces a large separated-flow region that extends upstream of the corner.

Rudy, David H.↗

NASA NTRS1990

Development of a time-dependent incompressible Navier-Stokes solver based on a fractional-step method

The development, validation and application of a fractional step solution method of the time-dependent incompressible Navier-Stokes equations in generalized coordinate systems are discussed. A solution method that combines a finite-volume discretization with a novel choice of the dependent variables and a fractional step splitting to obtain accurate solutions in arbitrary geometries was previously developed for fixed-grids. In the present research effort, this solution method is extended to include more general situations, including cases with moving grids. The numerical techniques are enhanced to gain efficiency and generality.

Rosenfeld, Moshe↗

NASA NTRS1990

Asymmetric supersonic flow around cones with noncircular sections

The unsteady, compressible, single and double thin-layer, Navier-Stokes equations are used to solve for steady and unsteady, asymmetric, supersonic flow around pointed bodies with noncircular sections at high incidence and zero side slip. The equations are solved by using an implicit, upwind, flux-difference splitting finite-volume scheme. Since the flow is locally conical, the solutions are presented on a cross-flow plane at the axial station of unit. The grid is generated by using a modified Joukowski transformation. The computational applications cover noncircular sections with elliptic and diamond shapes. Unsteady asymmetric vortex shedding has been captured at large angles of attack. It is shown that for the same flow conditions and same cross-section fineness ratio, the diamond-section cones develop less flow asymmetry than the elliptic-section cones. Passive control of flow asymmetry was also demonstrated computationally.

Kandil, Osama A.↗

NASA NTRS1990

A numerical study of hypersonic stagnation heat transfer predictions at a coordinate singularity

The problem of grid induced errors associated with a coordinate singularity on heating predictions in the stagnation region of a three-dimensional body in hypersonic flow is examined. The test problem is for Mach 10 flow over an Aeroassist Flight Experiment configuration. This configuration is composed of an elliptic nose, a raked elliptic cone, and a circular shoulder. Irregularities in the heating predictions in the vicinity of the coordinate singularity, located at the axis of the elliptic nose near the stagnation point, are examined with respect to grid refinement and grid restructuring. The algorithm is derived using a finite-volume formulation. An upwind-biased total-variation diminishing scheme is employed for the inviscid flux contribution, and central differences are used for the viscous terms.

Grasso, Francesco↗

NASA NTRS1990

Navier-Stokes computations for aerodynamic configurations at high angles of attack

Navier-Stokes computations at high angles of attack over aerodynamic configurations are presented using an implicit finite-volume algorithm. The spatial differencing is upwind-biased for the convective and pressure terms and central for the shear stress and heat transfer terms. The equations are relaxed to steady state with a spatially factored implicit algorithm. In order to treat general geometries, a multiblock patched-grid framework is implemented. Applications and detailed comparisons with experimental data are made for two simple but representative geometric shapes: (1) a highly swept delta wing and (2) a prolate spheroid of 6:1 length-to-diameter. Recent extensions of the algorithm to compute the flow over an F-18 forebody-strake configuration are shown, including comparisons with wind tunnel and flight test results. Comparisons across the range of Reynolds number for all cases indicate that either low Reynolds number or high Reynolds number flows are simulated well, but that flows at intermediate Reynolds number cannot be simulated accurately without a detailed knowledge of the transitional zone between laminar and turbulent flow.

Thomas, J. L.↗