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NASA NTRS1993

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 2: User's guide

A computer code called Proteus 3D was developed to solve the three-dimensional, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. This User's Guide describes the program's features, the input and output, the procedure for setting up initial conditions, the computer resource requirements, the diagnostic messages that may be generated, the job control language used to run the program, and several test cases.

Towne, Charles E.↗
NASA NTRS1993

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 3: Programmer's reference

A computer code called Proteus 3D was developed to solve the three-dimensional, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. The Programmer's Reference contains detailed information useful when modifying the program. The program structure, the Fortran variables stored in common blocks, and the details of each subprogram are described.

Towne, Charles E.↗
NASA NTRS1993

Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 3: Programmer's reference

A computer code called Proteus 2D was developed to solve the two-dimensional planar or axisymmetric, Reynolds-averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort was to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The governing equations are solved in generalized nonorthogonal body-fitted coordinates, by marching in time using a fully-coupled ADI solution procedure. The boundary conditions are treated implicitly. All terms, including the diffusion terms, are linearized using second-order Taylor series expansions. Turbulence is modeled using either an algebraic or two-equation eddy viscosity model. The thin-layer or Euler equations may also be solved. The energy equation may be eliminated by the assumption of constant total enthalpy. Explicit and implicit artificial viscosity may be used. Several time step options are available for convergence acceleration. The documentation is divided into three volumes. The Programmer's Reference contains detailed information useful when modifying the program. The program structure, the Fortran variables stored in common blocks, and the details of each subprogram are described.

Towne, Charles E.↗
NASA NTRS1981

Calculations of transonic potential flow over cascades

Transonic flow through a cascade was studied by using the full potential equation and the finite volume method of Jameson and Caughey. The C-type computational grid is generated by an electrostatic analogy and simple shearing transformation. The solution algorithm includes an option of using either an artificial density or an artificial viscosity formulation of the dissipative term. Using the developed code, flows through a cascade of NACA 0012 airfoils and flows through a cascade of shockless blades were computed. It is found that the designed flow through the shockless blade is accurately predicted, the artificial density formulation shows more tolerance to the mesh irregularity, and the C-type mesh does not extend very far upstream for a small pitch-cord ratio.

Jou, W. H.↗
NASA NTRS1983

Design of shock-free compressor cascades including viscous boundary layer effects

A computer code that generates shock-free transonic compressor cascade shapes while taking into account viscosity effects is developed. The mathematical model for the inviscid flow field is the full potential equation. The Kutta-Joukowski condition is satisfied by varying the free stream angle at downstream infinity. A boundary fitted computational grid of C-type is generated using a sequence of conformal mapping and nonorthogonal coordinate stretching and shearing transformations. The inviscid calculation is performed sequentially on up to four consecutively refined grids thereby accelerating the convergence of the solution process. The full potential equation is solved using a finite area technique and rotated, type-dependent finite differencing. Artificial viscosity of the first order is added in a fully conservative form. Shock-free cascade airfoil shapes are obtained using the fictitious gas concept of Sobieczky and the method of characteristics in the rheograph plane. Viscosity effects are incorporated via a boundary layer displacement thickness. The integral boundary layer code is based on Rotta's turbulence model and assumes transition region of zero length.

Dulikravich, G. S.↗
NASA NTRS1990

PROTEUS two-dimensional Navier-Stokes computer code, version 1.0. Volume 1: Analysis description

A new computer code was developed to solve the two-dimensional or axisymmetric, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The thin-layer or Euler equations may also be solved. Turbulence is modeled using an algebraic eddy viscosity model. The objective was to develop a code for aerospace applications that is easy to use and easy to modify. Code readability, modularity, and documentation were emphasized. The equations are written in nonorthogonal body-fitted coordinates, and solved by marching in time using a fully-coupled alternating direction-implicit procedure with generalized first- or second-order time differencing. All terms are linearized using second-order Taylor series. The boundary conditions are treated implicitly, and may be steady, unsteady, or spatially periodic. Simple Cartesian or polar grids may be generated internally by the program. More complex geometries require an externally generated computational coordinate system. The documentation is divided into three volumes. Volume 1 is the Analysis Description, and describes in detail the governing equations, the turbulence model, the linearization of the equations and boundary conditions, the time and space differencing formulas, the ADI solution procedure, and the artificial viscosity models.

Towne, Charles E.↗
NASA NTRS1972

On the numerical solution of two-dimensional, laminar compressible flows with imbedded shock waves.

The complete, time-dependent Navier-Stokes equations are expressed in conservation form and solved by employing an explicit finite difference numerical technique which incorporates artificial viscosity terms of the form first suggested by Rusanov for numerical stability in the vicinity of shock waves. Surface boundary conditions are developed in a consistent and unique manner through the use of a physically oriented extrapolation procedure. From numerical experimentation an extended range for the explicit stability parameter is established. Also employed is an additional convergence parameter which relates incremental spatial steps. Convergence of the transient solution to a steady state flow was obtained after 400 to 500 time steps.

Goodrich, W. D.↗
NASA NTRS1974

The computation of steady nozzle flow by a time-dependent method

The steady flow in two-dimensional and axisymmetric nozzles was computed using a time-dependent method. In this method the interior mesh points were computed using the MacCormack finite-difference scheme, while a characteristic scheme was used to calculate the boundary mesh points. No explicit artificial viscosity term was included. The fluid was assumed to be a perfect gas. This method was used to compute the flow in a 45 deg - 15 deg conical, converging-diverging nozzle, a 15 deg conical, converging nozzle, and a 10 deg conical, plug nozzle. Good agreement between the numerical solution and experimental data was found. In contrast to previous time-dependent methods, the computational times were less than one minute on a CDC 6600 computer.

Cline, M. C.↗
NASA NTRS1976

Time-asymptotic solutions of the Navier-Stokes equation for free shear flows using an alternating-direction implicit method

An uncoupled time asymptotic alternating direction implicit method for solving the Navier-Stokes equations was tested on two laminar parallel mixing flows. A constant total temperature was assumed in order to eliminate the need to solve the full energy equation; consequently, static temperature was evaluated by using algebraic relationship. For the mixing of two supersonic streams at a Reynolds number of 1,000, convergent solutions were obtained for a time step 5 times the maximum allowable size for an explicit method. The solution diverged for a time step 10 times the explicit limit. Improved convergence was obtained when upwind differencing was used for convective terms. Larger time steps were not possible with either upwind differencing or the diagonally dominant scheme. Artificial viscosity was added to the continuity equation in order to eliminate divergence for the mixing of a subsonic stream with a supersonic stream at a Reynolds number of 1,000.

Rudy, D. H.↗
NASA NTRS1976

Flux-corrected transport techniques for transient calculations of strongly shocked flows

New flux-corrected transport algorithms are described for solving generalized continuity equations. These techniques were developed by requiring that the finite difference formulas used ensure positivity for an initially positive convected quantity. Thus FCT is particularly valuable for fluid-like problems with strong gradients or shocks. Repeated application of the same subroutine to mass, momentum, and energy conservation equations gives a simple solution of the coupled time-dependent equations of ideal compressible fluid dynamics without introducing an artificial viscosity. FCT algorithms span Eulerian, sliding-rezone, and Lagrangian finite difference grids in several coordinate systems. The latest FCT techniques are fully vectorized for parallel/pipeline processing.

Boris, J. P.↗
NASA NTRS1976

Condensation of water vapor in rarefaction waves. I - Homogeneous nucleation

A detailed theoretical investigation has been made of the condensation of water vapor/carrier gas mixtures in the nonstationary rarefaction wave generated in a shock tube. It is assumed that condensation takes place by homogeneous nucleation. The equations of motion together with the nucleation rate and the droplet growth equations were solved numerically by the method of characteristics and Lax's method of implicit artificial viscosity. It is found that, for the case considered, the condensation wave formed by the collapse of the metastable nonequilibrium state is followed by a shock wave generated by the intersection of characteristics of the same family. The expansion is practically isentropic up to the onset of condensation. The condensation front accelerates in the x,t plane. The results of the computations for a chosen case of water vapor/nitrogen mixture are presented by plotting variations of pressure, nucleation rate, number density of critical clusters, and condensate mass-fraction along three particle paths. Some consideration is given to homogeneous condensation experiments conducted in a shock tube. Although a direct comparison of the present theoretical work and these experiments is not possible, several worthwhile interpretative features have resulted nevertheless.

Sislian, J. P.↗
NASA NTRS1976

Computation of wave drag for transonic flow

The paper develops a method for calculating wave drag for two-dimensional transonic flow, with particular application to the prediction of the drag rise Mach number of a supercritical wing section. The method is based on a transonic similarity model which is defined by a normalized small perturbation equation and represents shock waves by the addition of an artificial viscosity term in the region of supersonic flow to the partial differential equation. The drag formula obtained allows the computer simulation of transonic wind tunnel data taking account of boundary layer and wall effects.

Garabedian, P. R.↗
NASA NTRS1977

A finite volume method for transonic potential flow calculations

It is proposed to solve the exact transonic potential flow equation on a mesh constructed from small volume elements, which can be conveniently packed around any reasonably smooth configuration. The calculation is performed on two sets of interlocking cells. The velocity and density are calculated in the primary cells, and a flux balance is then established in the secondary cells. The scheme is desymmetrized by the addition of artificial viscosity in the supersonic zone. Some results are included for a swept wing and a wing-cylinder combination.

Jameson, A.↗
NASA NTRS1978

Conservative implicit schemes for the full potential equation applied to transonic flows

Implicit approximate factorization techniques (AF) were investigated for the solution of matrix equations resulting from finite difference approximations to the full potential equation in conservation form. For transonic flows, an artificial viscosity, required to maintain stability in supersonic regions, was introduced by an upwind bias of the density. Two implicit AF procedures are presented and their convergence performance is compared with that of the standard transonic solution procedure, successive line overrelaxation (SLOR). Subcritical and supercritical test cases are considered. The results indicate that the AF schemes are substantially faster than SLOR.

Holst, T. L.↗
NASA NTRS1978

A finite element solution of transonic flow

The use of finite elements is explored in a field in which its use has previously not been deemed very feasible, that of transonic flow. The specific problem chosen is that of steady small-disturbance transonic flow. The nonlinear equations are formulated with an artificial viscosity term added to yield the proper domain of dependence and directional bias in supersonic regions and across imbedded shock waves. Justification is given for the problem and means of solution chosen, and the potential advantages of the finite element procedure over standard finite difference procedures are discussed. Several possible improvements on the method as presently derived are stated. Computational mesh requirements and certain mesh variations are described. Some results equivalent to finite difference calculations are given as a sample solution.

Tatum, K. E.↗
NASA NTRS1979

Transonic airfoil codes

Computer codes for the design and analysis of transonic airfoils are considered. The design code relies on the method of complex characteristics in the hodograph plane to construct shockless airfoil. The analysis code uses artificial viscosity to calculate flows with weak shock waves at off-design conditions. Comparisons with experiments show that an excellent simulation of two dimensional wind tunnel tests is obtained. The codes have been widely adopted by the aircraft industry as a tool for the development of supercritical wing technology.

Garabedian, P. R.↗
NASA NTRS1979

Fast, conservative schemes for the full potential equation applied to transonic flows

Implicit approximate factorization techniques (AF) are investigated for the solution of matrix equations resulting from finite-difference approximations to the full potential equation in conservation form. For transonic flows, an artificial viscosity, required to maintain stability in supersonic regions, is introduced by an upwind bias of the density. Two implicit AF procedures are presented, and their convergence performance is compared with that of the standard transonic solution procedure: successive line overrelaxation (SLOR). Subcritical and supercritical test cases are considered. Results indicate a substantial improvement in convergence rate for AF schemes relative to SLOR.

Holst, T. L.↗
NASA NTRS1979

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.↗