Numerical solution of a nonlinear integral equation by the use of step functions. III - Continuous solutions
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Yoshikawa, K. K.↗
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A flowfield segmentation method for the numerical solution of viscous flow problems
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Wu, J. C.↗
Analytic numerical solutions for shock waves
Study of weak solutions of simple wave equation, inviscid Burgers equation, and Euler equations has resulted in technique for accurate prediction of shock waves occurring in inviscid supersonic flows.
Maccormack, R. W.↗
Numerical solutions of the unsteady Navier-Stokes equations for arbitrary bodies using boundary-fitted curvilinear coordinates
A method of automatic body-fitted curvilinear coordinate generation is described and used to construct a finite-difference solution of the full incompressible time-dependent Navier-Stokes equations for the unsteady laminar viscous flow arbitrary two-dimensional airfoils or any other two-dimensional body. A method of controlling the spacing of the coordinate lines encircling the body is developed in order to treat high Reynolds number flows, since the coordinate lines must concentrate near the surface to a greater degree as the Reynolds number increases. Multiple airfoils and submerged hydrofoils are treated as illustrative examples. The solution shows good agreement with the Blasius boundary layer solution for the flow past a semi-infinite flat plate.
Thompson, J. F.↗
Laminar supersonic flow over a backstep - A numerical solution at higher Reynolds numbers
The Allen-Cheng solution of the flow over a backward facing step is extended to Reynolds numbers up to 16,000 and to inflow boundary-layer height ratios as low as 0.1 by moving the downstream boundary into the recompression region and by smoothing the resulting errors. The boundary conditions in the supersonic outer flow and the downstream boundary conditions in the wake are determined by an extrapolation procedure. Computational results are compared with relevant experimental data. Fair agreement is found between the calculated base pressures and the experimental values, whereas agreement between heat transfer rates appears to be qualitative only.
Kronzon, Y.↗
Numerical solution for subcritical flows by a transonic integral equation method
The method presented makes use of a division of the region of integration into closed rectangular elements. The velocity is taken to be constant in each element. The integral equation is reduced to a matrix equation which can be solved by an appropriate iteration approach. The derivation and solution of the matrix equation are discussed and the matrix elements are considered. The described concepts were implemented for a nonlifting parabolic-arc airfoil.
Ogana, W.↗
Numerical solutions of the triple-deck equations for laminar trailing-edge stall
The problem of determining the effect of laminar boundary layers on the lift of thin wings in subsonic flow at high Reynolds numbers is considered. The boundary value problem is formulated in the framework of the triple-deck theory of Brown and Stewartson. The resulting fourth-order boundary value was solved by an iterative finite-difference technique. An inverse iteration procedure provides proper treatment of the trailing-edge singularity, and asymptotic far-field expansions and coordinate stretchings are used to deal with the problem of the slow algebraic decay of the solution.
Chow, R.↗
Numerical solution of the steady-state Navier-Stokes equations for hypersonic flow about blunt axisymmetric bodies
The steady-state Navier-Stokes equations are solved for hypersonic flow about blunt axisymmetric bodies. The equations of motion are solved by successive approximations using an implicit finite-difference scheme. The results are compared with viscous shock-layer theory, experimental data, and time-dependent solutions of the Navier-Stokes equations. It is demonstrated that viscous shock-layer theory is sufficiently accurate for the range of flight conditions normally encountered by entry vehicles.
Anderson, E. C.↗
A numerical solution of the Navier-Stokes equations for chemically nonequilibrium, merged stagnation shock layers on spheres and two-dimensional cylinders in air
Results of solving the Navier-Stokes equations for chemically nonequilibrium, merged stagnation shock layers on spheres and two-dimensional cylinders are presented. The effects of wall catalysis and slip are also examined. The thin shock layer assumption is not made, and the thick viscous shock is allowed to develop within the computational domain. The results show good comparison with existing data. Due to the more pronounced merging of shock layer and boundary layer for the sphere, the heating rates for spheres become higher than those for cylinders as the altitude is increased.
Johnston, K. D.↗
The numerical solution of viscous flows at high Reynolds number
A review is presented of implicit and hybrid methods applicable to solving viscous flows at high Reynolds numbers. Flows within axisymmetric channels containing stationary shock waves, past blunt-nosed lifting airfoils, past sharp-nosed symmetric airfoils with buffet, past three-dimensional compression ramps with side walls, and past ogive- and hemisphere-cylinders at angle of attack have been examined. Reynolds numbers as high as 10 to the 9th power have been used.
Maccormack, R. W.↗
Numerical solution for perpendicular sonic hydrogen injection into a ducted supersonic airstream
This note discusses a computer program being developed to study the flow field near opposing perpendicular fuel injectors in scramjets. The MacCormack time-split, finite difference relaxation technique was used to solve the full two-dimensional compressible Navier-Stokes equations along with energy and species equations. By using this technique, a program was developed to consider the turbulent nonreacting flow of hydrogen and air in a rectangular duct. A damping term, proportional to the second derivative of pressure and temperature, was used to produce a stable solution behind the hydrogen jet in the neighborhood of the recompression shock. A case using actual conditions encountered in current scramjet design was analyzed, with results agreeing qualitatively with experimental observations.
Drummond, J. P.↗
Artificial compressibility methods for numerical solutions of transonic full potential equation
New methods for transonic flow computations based on the full potential equation in conservation form are presented. The idea is to modify slightly the density (due to the artificial viscosity in the supersonic region), and solve the resulting elliptic-like problem iteratively. It is shown that standard discretization techniques (central differencing) as well as some standard iterative procedures (SOR, ADI, and explicit methods) are applicable to the modified transonic mixed-type equation. Calculations of transonic flows around cylinders and airfoils are discussed with special emphasis on the explicit methods that are suitable for vector processing on the STAR 100 computer.
Hafez, M.↗
Numerical solution of supersonic laminar flow over an inclined body of revolution
A mixed explicit-implicit scheme is used to solve the time-dependent thin-layer approximation of the Navier-Stokes equations for a supersonic laminar flow over an inclined body of revolution. Test cases for Mach 2.8 flow over a cylinder with 15 deg flare angle at angles of attack of 0, 1, and 4 deg are calculated. Good agreement is obtained between the present computed results and experimental measurements of surface pressure. A pair of vortices on the leeward and a peak in the normal force distribution near the flared juncture are predicted; the role of circumferential communication is discussed.
Hung, C. M.↗
Application of numerical solutions to non-linear problems of fluid mechanics obtained by the method of least squares and the finite element method to unsteady Navier-Stokes equations
A method of calculating viscous fluid flows having an average Reynolds number is presented.
Mantel, B.↗
Boundary conditions for the numerical solution of elliptic equations in exterior regions
Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace's equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used.
Bayliss, A.↗
Stellar convection. II - A multimode numerical solution for convection in spheres
A stable, equilibrium solution for convection in a self-gravitating sphere of Boussinesq fluid was computed by using a modal analysis in which the theta, phi dependence of the fluid is expanded in a set of 168 spherical harmonics. The temperature, velocity, and convective flux of the fluid as well as the kinetic and thermal energy spectra as functions of wavelength are computed. The spectra are found to be in agreement with both experimental observations and analytic scaling laws. The dynamics of the energy cascade is examined by computing the ratio of the amount of energy dissipation at a particular wavelength to the amount of energy produced at that same wavelength. It is found that there is only a slight cascade of kinetic energy to smaller wavelengths but a large cascade of thermal energy.
Marcus, P. S.↗
Numerical Solution for Navier-Stokes Equations
Carefully selected blend of computational techniques solves complete set of equations for viscous, unsteady, hypersonic flow in general curvilinear coordinates. New algorithm has tested computation of axially directed flow about blunt body having shape similar to that of such practical bodies as wide-body aircraft or artillery shells. Method offers significant computational advantages because of conservation-law form of equations and because it reduces amount of metric data required.
Warsi, Z. U. A.↗
Nonlinear grid error effects on numerical solution of partial differential equations
Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.
Dey, S. K.↗