Conductive heating of the solar wind. III - Breakdown of the Navier-Stokes equations, 4 June - 4 December 1965
Navier-Stokes equation in predicting solar wind flow pattern
SOLAR WIND↗
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An efficient explicit-implicit-characteristic method for solving the compressible Navier-Stokes equations
Explicit, implicit, and characteristic finite-difference methods are applied to solve model equations representative of the compressible Navier-Stokes equations. An approach is then formulated for solving the Navier-Stokes equation at high Reynolds numbers. The approach has drastically reduced the computation time required to obtain viscous flow solutions. Computational results for shock wave separated flows are presented.
Maccormack, R. W.↗
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.↗
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.↗
The relative merits of several numerical techniques for solving the compressible Navier-Stokes equations
Four explicit finite difference techniques designed to solve the time-dependent, compressible Navier Stokes equations are compared. These techniques are: (1) MacCormack, (2) modified Du Fort-Frankel, (3) modified hopscotch, and (4) Brailovskaya. The comparison was made numerically by solving the quasi-one dimensional Navier Stokes equations for the flow in a converging-diverging nozzle. Solutions with and without standing normal shock waves were computed for unit Reynolds numbers (based on total conditions) ranging from 45374 to 2269. The results indicate that all four techniques are comparable in accuracy; however, the modified hopscotch scheme is two to three times faster than the Brailovskaya and MacCormack schemes and three to six times faster than the modified Du Fort-Frankel scheme.
Holst, T. L.↗
Solutions of the Navier-Stokes equations for vortex breakdown
Steady solutions of the Navier-Stokes equations, in terms of velocity and pressure, for breakdown in an unconfined viscous vortex are obtained numerically using the artificial compressibility technique of Chorin combined with an ADI finite-difference scheme. Axisymmetry is assumed and boundary conditions are carefully applied at the boundaries of a large finite region in an axial plane while resolution near the axis is maintained by a coordinate transformation. The solutions, which are obtained for Reynolds numbers up to 200 based on the free-stream axial velocity and a characteristic core radius, show that breakdown results from the diffusion and convection of vorticity away from the vortex core which, because of the strong coupling between the circumferential and axial velocity fields in strongly swirling flows, can lead to stagnation and reversal of the axial flow near the axis.
Grabowski, W. J.↗
A rigorous solution of the Navier-Stokes equations for unsteady viscous flow at high Reynolds numbers around oscillating airfoils
A method based on the Navier-Stokes equations was developed for analyzing the unsteady incompressible viscous flow around oscillating airfoils at high Reynolds numbers. The Navier-Stokes equations have been integrated in their classical Helmholtz vorticity transport equation form, and the instantaneous velocity field at each time step was determined by the solution of Poisson's equation. A refined finite element was utilized to allow for a conformable solution of the stream function and its first space derivatives at the element interfaces. A corresponding set of accurate boundary conditions was applied; thus obtaining a rigorous solution for the velocity field. The details of the computational procedure and examples of computed results describing the unsteady flow characteristics around the airfoil are presented.
Bratanow, T.↗
Derivation of the Navier-Stokes equation
The proposed approach to the derivation of the Navier-Stokes equation is thought to be more plausible and easier to understand than other derivations that can be found in works on fluid mechanics. The tensor character of the stress is central to the derivation. In particular, a linear relation between stress and strain rate is assumed only for the shear, rather than for the full stress tensor as is done in most other derivations. An assumption for the shear is naturally simpler and easier to verify experimentally. The use of tensor analysis is shown to greatly simplify the derivation.
Deissler, R. G.↗
A finite element solution algorithm for the Navier-Stokes equations
A finite element solution algorithm is established for the two-dimensional Navier-Stokes equations governing the steady-state kinematics and thermodynamics of a variable viscosity, compressible multiple-species fluid. For an incompressible fluid, the motion may be transient as well. The primitive dependent variables are replaced by a vorticity-streamfunction description valid in domains spanned by rectangular, cylindrical and spherical coordinate systems. Use of derived variables provides a uniformly elliptic partial differential equation description for the Navier-Stokes system, and for which the finite element algorithm is established. Explicit non-linearity is accepted by the theory, since no psuedo-variational principles are employed, and there is no requirement for either computational mesh or solution domain closure regularity. Boundary condition constraints on the normal flux and tangential distribution of all computational variables, as well as velocity, are routinely piecewise enforceable on domain closure segments arbitrarily oriented with respect to a global reference frame.
Baker, A. J.↗
A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations
An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C deg-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1,000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and asymptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.
Cooke, C. H.↗
A block iterative finite element algorithm for numerical solution of the steady-state, compressible Navier-Stokes equations
An iterative method for numerically solving the time independent Navier-Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss-Seidel principle in block form to the systems of the nonlinear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The continuous cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.
Cooke, C. H.↗
Computer solution of unsteady Navier-Stokes equations for an infinite hydrodynamic step bearing
Computer solution of unsteady Navier-Stokes equations for infinite hydrodynamic step bearing
Putre, H. A.↗
Numerical solution of supersonic three-dimensional free-mixing flows using the parabolic-elliptic Navier-Stokes equations
A numerical method is presented for solving the parabolic-elliptic Navier-Stokes equations. The solution procedure is applied to three-dimensional supersonic laminar jet flow issuing parallel with a supersonic free stream. A coordinate transformation is introduced which maps the boundaries at infinity into a finite computational domain in order to eliminate difficulties associated with the imposition of free-stream boundary conditions. Results are presented for an approximate circular jet, a square jet, varying aspect ratio rectangular jets, and interacting square jets. The solution behavior varies from axisymmetric to nearly two-dimensional in character. For cases where comparisons of the present results with those obtained from shear layer calculations could be made, agreement was good.
Hirsh, R. S.↗
Numerical solution of the Navier-Stokes equations for laminar, transonic flows
An implicit finite difference solution of the Navier-Stokes equations yielded time histories of the transonic laminar flow development about a circular cylinder and NACA-0018 airfoil. Reynolds numbers ranged from those corresponding to purely laminar flow to those corresponding to significant turbulence in the boundary layer. Body thermal conditions of an adiabatic wall and a specified body temperature were considered. Versatility in treating arbitrary bodies was incorporated by using numerically generated, body-fitted coordinate transformations. Solution of the simultaneous difference equations for the dependent variables was obtained using an accelerated Gauss-Seidel iterative scheme. Computational results are presented in the form of velocity vector fields, Mach number contours, aerodynamic coefficients, heat transfer rates at the body surface, and body temperature distributions. Truncation analyses of first and second derivative difference approximations resulted in general criteria for numerically generated coordinates so that flow near a body is more accurately represented.
Turner, L., III↗
Boundary conditions for implicit solutions to the compressible Navier-Stokes equations in finite computational domains
The admissible boundary conditions for the unsteady Navier-Stokes equations are insufficient to determine completely the flow variables at grid points on the boundaries of a finite computational domain. A general method is presented in which the boundary conditions are supplemented by conservatively differenced forms of the flow equations. For implicit algorithms, the method provides a fully implicit coupling between boundary and interior points. Implicit boundary-point computation schemes are given for most types of flow boundary encountered in internal and external flow problems. Convergence to steady state is generally faster than with locally explicit boundary-point schemes.
Thomas, P. D.↗
A numerical solution of the Navier-Stokes equations for supercritical fluid thermodynamic analysis
An explicit numerical solution of the compressible Navier-Stokes equations is applied to the thermodynamic analysis of supercritical oxygen in the Apollo cryogenic storage system. The wave character is retained in the conservation equations which are written in the basic fluid variables for a two-dimensional Cartesian coordinate system. Control-volume cells are employed to simplify imposition of boundary conditions and to ensure strict observance of local and global conservation principles. Non-linear real-gas thermodynamic properties responsible for the pressure collapse phenomonon in supercritical fluids are represented by tabular and empirical functions relating pressure and temperature to density and internal energy. Wall boundary conditions are adjusted at one cell face to emit a prescribed mass flowrate. Scaling principles are invoked to achieve acceptable computer execution times for very low Mach number convection problems. Detailed simulations of thermal stratification and fluid mixing occurring under low acceleration in the Apollo 12 supercritical oxygen tank are presented which model the pressure decay associated with de-stratification induced by an ordinary vehicle maneuver and heater cycle operation.
Heinmiller, P. J.↗
COMOC: a finite-element algorithm for the Navier-Stokes equations
Finite-element algorithm devised to facilitate solution of two-dimensional Navier-Stokes equations governing kinematics and thermodynamics of variable-viscosity, compressible, multiple-species fluid. Algorithm has been implemented into existing computer program system.
Baker, A. J.↗
An effective solution to the nonlinear, nonstationary Navier-Stokes equations for two dimensions
A sequence of approximate solutions for the nonlinear, nonstationary Navier-Stokes equations for a two-dimensional domain, from which explicit error estimates and rates of convergence are obtained, is described. This sequence of approximate solutions is based primarily on the Newton-Kantorovich method.
Gabrielsen, R. E.↗