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NASA NTRS1979

An artificial viscosity method for the design of supercritical airfoils

A numerical technique is presented for the design of two-dimensional supercritical wing sections with low wave drag. The method is a design mode of the analysis code H which gives excellent agreement with experimental results and is widely used in the aircraft industry. Topics covered include the partial differential equations of transonic flow, the computational procedure and results; the design procedure; a convergence theorem; and description of the code.

Mcfadden, G. B.↗

NASA NTRS1992

Central difference TVD and TVB schemes for time dependent and steady state problems

We use central differences to solve the time dependent Euler equations. The schemes are all advanced using a Runge-Kutta formula in time. Near shocks, a second difference is added as an artificial viscosity. This reduces the scheme to a first order upwind scheme at shocks. The switch that is used guarantees that the scheme is locally total variation diminishing (TVD). For steady state problems it is usually advantageous to relax this condition. Then small oscillations do not activate the switches and the convergence to a steady state is improved. To sharpen the shocks, different coefficients are needed for different equations and so a matrix valued dissipation is introduced and compared with the scalar viscosity. The connection between this artificial viscosity and flux limiters is shown. Any flux limiter can be used as the basis of a shock detector for an artificial viscosity. We compare the use of the van Leer, van Albada, mimmod, superbee, and the 'average' flux limiters for this central difference scheme. For time dependent problems, we need to use a small enough time step so that the CFL was less than one even though the scheme was linearly stable for larger time steps. Using a total variation bounded (TVB) Runge-Kutta scheme yields minor improvements in the accuracy.

Jorgenson, P.↗

NASA NTRS1992

Central difference TVD and TVB schemes for time dependent and steady state problems

Jorgenson, P.↗

NASA NTRS1981

A multigrid algorithm for steady transonic potential flows around aerofoils using Newton iteration

The application of multigrid relaxation to transonic potential flow calculation was investigated. Conservative potential flows around aerofoils were taken as test problems. The solution algorithm was based on Newton iteration. It was found that the iteration to the circulation has to be kept outside the multigrid algorithm. To obtain meaningful norms of residuals, difference formulas with asymptotic scaling were introduced. Nonlinear instability problems were solved by upwind differencing using mass flux vector splitting instead of artificial viscosity or artificial density. It is suggested that the algorithms efficiency be increased by improving the iteration on the shock positions even though this is a highly nonlinear process.

Boestoel, J. W.↗

DOE OSTI2024

A modern concept of Lagrangian hydrodynamics

Here, we offer a modern interpretation of Lagrangian hydrodynamics as employed in Lagrangian simulations of compressible fluid flow. Our main result is to show that artificial viscosity, traditionally viewed as a numerical artifice to control unphysical oscillations in flows with shocks, actually represents a physical process and is necessary to derive accurate simulations in any compressible flow. We begin by reviewing the origins of two numerical devices, artificial viscosity and finite-volume methods. We proceed to construct a mathematical (PDE) model that incorporates those numerics and in which a new length scale, the observer, arises representing the discretization. Associated with that length scale, there are new inviscid fluxes that are the artificial viscosity as first formulated by Richtmyer and an artificial heat flux postulated by Noh but typically not included in Lagrangian codes. We discuss the connection of our results to bivelocity hydrodynamics. We conclude with some speculation as to the direction of future developments in multidimensional Lagrangian codes as computers get faster and have larger memories.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1989

Calculations of rotational flows using stream function

The stream function equation is solved for steady two-dimensional (and axisymmetric) rotational flows. Both finite differences and finite volumes discretization techniques are studied, using generalized body fitted coordinates and unstructured staggered grids, respectively. For inviscid transonic flows, a new artificial viscosity scheme which does not produce any artificial vorticity is introduced, for the stability of the mixed flow calculations and for capturing shocks. The solution of Euler equations, in primitive variables, are also considered. The effects of the artificial viscosity and numerical boundary conditions on the total enthalpy and the vorticity distributions are demonstrated.

Hafez, M.↗

NASA NTRS1993

Central difference TVD schemes for time dependent and steady state problems

A Runge-Kutta formula in time is presently used to advance schemes in which central differences are used to solve the time-dependent Euler equations; a second difference is added near shocks as an artificial viscosity to reduce the given scheme to a first-order upwind one at shocks. A matrix-valued dissipation is introduced and compared with the scalar viscosity; a connection is shown between this artificial viscosity and flux limiters. The use of various flux limiters for this central difference scheme is compared.

Jorgenson, P.↗

NASA NTRS1988

Improving the accuracy of central difference schemes

General difference approximations to the fluid dynamic equations require an artificial viscosity in order to converge to a steady state. This artificial viscosity serves two purposes. One is to suppress high frequency noise which is not damped by the central differences. The second purpose is to introduce an entropy-like condition so that shocks can be captured. These viscosities need a coefficient to measure the amount of viscosity to be added. In the standard scheme, a scalar coefficient is used based on the spectral radius of the Jacobian of the convective flux. However, this can add too much viscosity to the slower waves. Hence, it is suggested that a matrix viscosity be used. This gives an appropriate viscosity for each wave component. With this matrix valued coefficient, the central difference scheme becomes closer to upwind biased methods.

Turkel, Eli↗

NASA NTRS1980

An artificial energy method for calculating flows with shocks

The artificial-viscosity method, first proposed by von Neumann and Richtmyer, introduces an artificial viscous pressure term in regions of compression such that an increase in entropy occurs in shock transition zones. The paper describes how dissipative flows can be induced by reducing the total energy available for adiabatic processes in shock zones. A class of inviscid fluid flows, called semiflows, is described in which the flows exhibit thermodynamic differences. Induced dissipative flows modify the pressure in regions of compression in a manner analogous to the artificial-viscosity method and for a gas, the effect is equivalent to suitably modifying the gas constant in the equation of state. By employing MacCormack's method and the usual non-adiabatic equations, numerical solutions of a Riemann problem are compared with the modified artificial energy method, showing that the dissipation effect predicted by the analytical formulation is reflected in the numerical method as well.

Rose, M. E.↗

NASA NTRS1990

Three-dimensional compressible jet-in-crossflow calculations using improved viscosity models and adapted grid

Previous calculations of jet-in-crossflow problems have been sensitive to the turbulence and artificial viscosity models and to the grid. Consequently, the eddy viscosity model in the PARC3D code was modified to consider the turbulent jet by switching from the Baldwin-Lomax (1978) model to an axisymmetric jet model. A modified artificial viscosity model has been utilized and evaluated in this study as well. The new model includes cell size scaling and a directional dependence in the coefficients. Computational results from PARC3D demonstrate the effects of the viscosity models on the pressure distribution fore and aft of the jet and the ability of the adaptive grid scheme to adjust to the three-dimensional gradients around the jet.

Lytle, John K.↗

NASA NTRS1994

Canonical-variables multigrid method for steady-state Euler equations

A novel approach for the solution of inviscid flow problems for subsonic compressible flows is described. The approach is based on canonical forms of the equations, in which subsystems governed by hyperbolic operators are separated from those governed by elliptic ones. The discretizations used as well as the iterative techniques for the different subsystems are inherently different. Hyperbolic parts, which describe, in general, propagation phenomena, are discretized using upwind schemes and are solved by marching techniques. Elliptic parts, which are directionally unbiased, are discretized using h-elliptic central discretizations, and are solved by pointwise relaxations together with coarse grid acceleration. The resulting discretization schemes introduce artificial viscosity only for the hyperbolic parts of the system; thus a smaller total artificial viscosity is used, while the multigrid solvers used are much more efficient. Solutions of the subsonic compressible Euler equations are achieved at the same efficiency as the full potential equation.

Taasan, Shlomo↗

DOE OSTI2020

A matrix-free hyperviscosity formulation for high-order ALE hydrodynamics

The numerical approximation of compressible hydrodynamics is at the core of high-energy density (HED) multiphysics simulations as shocks are the driving force in experiments like inertial confinement fusion (ICF). In this work, we describe our extension of the hyperviscosity technique, originally developed for shock treatment in finite difference simulations, for use in arbitrarily high-order finite element methods for Lagrangian hydrodynamics. Hyperviscosity enables shock capturing while preserving the high-order properties of the underlying discretization away from the shock region. Specifically, we compute a high-order term based on a product of the mesh length scale to a high power scaled by a hyper-Laplacian operator applied to a scalar field. We then form the total artificial viscosity by taking a non-linear blend of this term and a traditional artificial viscosity term. We also present a matrix-free formulation for computing the finite element based hyper-Laplacian operator. Such matrix-free methods have superior performance characteristics compared to traditional full matrix assembly approaches and offer advantages for GPU based HPC hardware. We demonstrate the numerical convergence of our method and its application to complex, multi-material ALE simulations on high-order (curved) meshes.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1988

A physically consistent model for artificial dissipation in transonic potential flow computations

The effect that artificial dissipation has on numerical solutions of the transonic Full Potential Equation (FPE) are investigated by comparing the artificially dissipative FPE to a Physically Dissipative Potential (PDP) equation. Analytic expressions were derived from the variables C and M sub c that are used in the artificial density formulation. It was shown that these new values generate artificial dissipation which is equivalent to the physical dissipation existing in the PDP equation. The new expression for the variables C and M sub c can easily be incorporated into the existing full potential codes which are based either on the artificial density or on the artificial viscosity formulation. A comparison of Physically Dissipative Potential (PDP), Artificial Density or Viscosity (ADV), Artificial Mass Flux (AMF), and ADV with variable C and M sub c formulation (MCC) is also presented.

Dulikravich, George S.↗

NASA NTRS1986

Influence of numerical dissipation in computing supersonic vortex-dominated flows

Steady supersonic vortex-dominated flows are solved using the unsteady Euler equations for conical and three-dimensional flows around sharp- and round-edged delta wings. The computational method is a finite-volume scheme which uses a four-stage Runge-Kutta time stepping with explicit second- and fourth-order dissipation terms. The grid is generated by a modified Joukowski transformation. The steady flow solution is obtained through time-stepping with initial conditions corresponding to the freestream conditions, and the bow shock is captured as a part of the solution. The scheme is applied to flat-plate and elliptic-section wings with a leading edge sweep of 70 deg at an angle of attack of 10 deg and a freestream Mach number of 2.0. Three grid sizes of 29 x 39, 65 x 65 and 100 x 100 have been used. The results for sharp-edged wings show that they are consistent with all grid sizes and variation of the artificial viscosity coefficients. The results for round-edged wings show that separated and attached flow solutions can be obtained by varying the artificial viscosity coefficients. They also show that the solutions are independent of the way time stepping is done. Local time-stepping and global minimum time-steeping produce same solutions.

Kandil, O. A.↗

NASA NTRS1993

Proteus two-dimensional Navier-Stokes computer code, version 2.0. Volume 1: Analysis description

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. This is the Analysis Description, and presents the equations and solution procedure. 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 are described in detail.

Towne, Charles E.↗

NASA NTRS1993

Proteus three-dimensional Navier-Stokes computer code, version 1.0. Volume 1: Analysis description

A computer code called Proteus 3D has been developed to solve the three dimensional, Reynolds averaged, unsteady compressible Navier-Stokes equations in strong conservation law form. The objective in this effort has been to develop a code for aerospace propulsion applications that is easy to use and easy to modify. Code readability, modularity, and documentation have been emphasized. The governing equations are solved in generalized non-orthogonal 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 is the Analysis Description, and presents the equations and solution procedure. It 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 NTRS1993

Hypersonic flow computations around re-entry vehicles

The development of an algorithm for the solution of the compressible Euler equations at high Mach numbers on unstructured tetrahedral meshes is described. The basic algorithm is constructed in the form of a central difference scheme plus an explicit added artificial viscosity based upon fourth order differences of the solution. The stability of the solution in the vicinity of strong gradients is preserved by the incorporation of an additional artificial viscosity based upon a second order difference. Higher order accuracy is regained by using the ideas of flux corrected transport to limit the amount of added viscosity. The solution is advanced to steady state by means of an explicit multi-stage time-stepping method. The computational efficiency of the complete process is improved by incorporating an unstructured multigrid acceleration procedure. A number of flows of practical interest are analyzed to demonstrate the numerical performance of the proposed approach.

Peraire, J.↗

NASA NTRS1988

Accuracy versus convergence rates for a three dimensional multistage Euler code

Using a central difference scheme, it is necessary to add an artificial viscosity in order to reach a steady state. This viscosity usually consists of a linear fourth difference to eliminate odd-even oscillations and a nonlinear second difference to suppress oscillations in the neighborhood of steep gradients. There are free constants in these differences. As one increases the artificial viscosity, the high modes are dissipated more and the scheme converges more rapidly. However, this higher level of viscosity smooths the shocks and eliminates other features of the flow. Thus, there is a conflict between the requirements of accuracy and efficiency. Examples are presented for a variety of three-dimensional inviscid solutions over isolated wings.

Turkel, Eli↗