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NASA NTRS1994

An Adaptively-Refined, Cartesian, Cell-Based Scheme for the Euler and Navier-Stokes Equations

A Cartesian, cell-based scheme for solving the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal 'cut' cells are created. The geometry of the cut cells is computed using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded, with a limited linear reconstruction of the primitive variables used to provide input states to an approximate Riemann solver for computing the fluxes between neighboring cells. A multi-stage time-stepping scheme is used to reach a steady-state solution. Validation of the Euler solver with benchmark numerical and exact solutions is presented. An assessment of the accuracy of the approach is made by uniform and adaptive grid refinements for a steady, transonic, exact solution to the Euler equations. The error of the approach is directly compared to a structured solver formulation. A non smooth flow is also assessed for grid convergence, comparing uniform and adaptively refined results. Several formulations of the viscous terms are assessed analytically, both for accuracy and positivity. The two best formulations are used to compute adaptively refined solutions of the Navier-Stokes equations. These solutions are compared to each other, to experimental results and/or theory for a series of low and moderate Reynolds numbers flow fields. The most suitable viscous discretization is demonstrated for geometrically-complicated internal flows. For flows at high Reynolds numbers, both an altered grid-generation procedure and a different formulation of the viscous terms are shown to be necessary. A hybrid Cartesian/body-fitted grid generation approach is demonstrated. In addition, a grid-generation procedure based on body-aligned cell cutting coupled with a viscous stensil-construction procedure based on quadratic programming is presented.

Coirier, William John↗

NASA NTRS2002

Properties of the Residual Stress of the Temporally Filtered Navier-Stokes Equations

The development of a unifying framework among direct numerical simulations, large-eddy simulations, and statistically averaged formulations of the Navier-Stokes equations, is of current interest. Toward that goal, the properties of the residual (subgrid-scale) stress of the temporally filtered Navier-Stokes equations are carefully examined. Causal time-domain filters, parameterized by a temporal filter width 0 less than Delta less than infinity, are considered. For several reasons, the differential forms of such filters are preferred to their corresponding integral forms; among these, storage requirements for differential forms are typically much less than for integral forms and, for some filters, are independent of Delta. The behavior of the residual stress in the limits of both vanishing and in infinite filter widths is examined. It is shown analytically that, in the limit Delta to 0, the residual stress vanishes, in which case the Navier-Stokes equations are recovered from the temporally filtered equations. Alternately, in the limit Delta to infinity, the residual stress is equivalent to the long-time averaged stress, and the Reynolds-averaged Navier-Stokes equations are recovered from the temporally filtered equations. The predicted behavior at the asymptotic limits of filter width is further validated by numerical simulations of the temporally filtered forced, viscous Burger's equation. Finally, finite filter widths are also considered, and a priori analyses of temporal similarity and temporal approximate deconvolution models of the residual stress are conducted.

Pruett, C. D.↗

NASA NTRS1996

First-Order System Least-Squares for the Navier-Stokes Equations

This paper develops a least-squares approach to the solution of the incompressible Navier-Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier-Stokes equations as a first-order system by introducing a velocity flux variable and associated curl and trace equations. We show that the resulting system is well-posed, and that an associated least-squares principle yields optimal discretization error estimates in the H(sup 1) norm in each variable (including the velocity flux) and optimal multigrid convergence estimates for the resulting algebraic system.

Bochev, P.↗

NASA NTRS1986

The calculation of supersonic viscous flows using the parabolized Navier-Stokes equations

Solution of the parabolic Navier-Stokes (PNS) equations for supersonic flows is discussed, and compatibility of the PNS method with the triple-deck theory of Stewartson (1974) is demonstrated. Characteristic and stability analyses show that use of an appropriate filter on the pressure term in the x-momentum equation can suppress the departure solutions, giving the usual desired weak interaction solution. An Alternating Direction Explicit procedure, with minimal computer storage requirements compared to the full Navier-Stokes solvers, is proposed to calculate strongly interacting flows using a global iteration procedure for the PNS equations. The PNS equations are used to solve the hypersonic viscous interaction problem, and good agreement is found with experimental results.

Davis, R. T.↗

NASA NTRS1989

Algorithm For Solution Of Navier-Stokes Equations

Advantages of two previous algorithms combined. Steady-state Navier-Stokes equations put in implicit finite-difference form solved by approximate Newton iteration. LU-SSOR scheme is new relaxation method combining advantages of LU factorization with Gauss-Seidel relaxation. Vectorizable LU-SSOR scheme, based on central differences, requires scalar diagonal inversions. Application of scheme to approximate Newton iteration of Navier-Stokes equations yields set of equations requiring no implicit smoothing on left side. Only adaptive, total-variation-diminishing, flux-limited dissipation terms added to right side. Algorithm used to predict laminar, turbulent, and hypersonic flows.

Yoon, Seokkwan↗

NASA NTRS1995

The space-time solution element method: A new numerical approach for the Navier-Stokes equations

This paper is one of a series of papers describing the development of a new numerical method for the Navier-Stokes equations. Unlike conventional numerical methods, the current method concentrates on the discrete simulation of both the integral and differential forms of the Navier-Stokes equations. Conservation of mass, momentum, and energy in space-time is explicitly provided for through a rigorous enforcement of both the integral and differential forms of the governing conservation laws. Using local polynomial expansions to represent the discrete primitive variables on each cell, fluxes at cell interfaces are evaluated and balanced using exact functional expressions. No interpolation or flux limiters are required. Because of the generality of the current method, it applies equally to the steady and unsteady Navier-Stokes equations. In this paper, we generalize and extend the authors' 2-D, steady state implicit scheme. A general closure methodology is presented so that all terms up through a given order in the local expansions may be retained. The scheme is also extended to nonorthogonal Cartesian grids. Numerous flow fields are computed and results are compared with known solutions. The high accuracy of the scheme is demonstrated through its ability to accurately resolve developing boundary layers on coarse grids. Finally, we discuss applications of the current method to the unsteady Navier-Stokes equations.

Scott, James R.↗

NASA NTRS1986

Pseudo-time algorithms for the Navier-Stokes equations

A pseudo-time method is introduced to integrate the compressible Navier-Stokes equations to a steady state. This method is a generalization of a method used by Crocco and also by Allen and Cheng. We show that for a simple heat equation that this is just a renormalization of the time. For a convection-diffusion equation the renormalization is dependent only on the viscous terms. We implement the method for the Navier-Stokes equations using a Runge-Kutta type algorithm. This permits the time step to be chosen based on the inviscid model only. We also discuss the use of residual smoothing when viscous terms are present.

Swanson, R. C.↗

NASA NTRS1974

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.↗

NASA NTRS1976

Numerical solution of the steady-state Navier-Stokes equations for hypersonic flow about blunt axisymmetric bodies

Anderson, E. C.↗

NASA NTRS1983

Heating analysis of bent-nose biconics at high angles of attack using the parabolized Navier-Stokes equations

A method based on the Parabolized Navier-Stokes equations is used to calculate the flow field and heat transfer of lifting entry vehicles. The method is based on the Bean and Warming implicit algorithm and uses a new procedure for preventing departure solutions. Calculations are carried out for blunt on-axis and bent biconics, assuming a perfect gas and laminar flow, and compared with available heat transfer, surface pressure and shock shape measurements for a range of Mach numbers and angles of attack. In all calculations presented here, the starting solution is obtained from available inviscid and boundary layer codes. Good agreement with experiment is indicated. Thus, the method provides an accurate and rather inexpensive procedure for calculating three-dimensional flows at supersonic Mach numbers.

Stephenson, B. L.↗

NASA NTRS1983

Turbulence modeling methods for the compressible Navier-Stokes equations

Turbulence modeling methods for the compressible Navier-Stokes equations, including several zero- and two-equation eddy-viscosity models, are described and applied. Advantages and disadvantages of the models are discussed with respect to mathematical simplicity, conformity with physical theory, and numerical compatibility with methods. A new two-equation model is introduced which shows advantages over other two-equation models with regard to numerical compatibility and the ability to predict low-Reynolds-number transitional phenomena. Calculations of various transonic airfoil flows are compared with experimental results. A new implicit upwind-differencing method is used which enhances numerical stability and accuracy, and leads to rapidly convergent steady-state solutions.

Coakley, T. J.↗

NASA NTRS1985

Numerical solutions of 3-dimensional Navier-Stokes equations for closed bluff-bodies

The Navier-Stokes equations are solved numerically. These equations are unsteady, compressible, viscous, and three-dimensional without neglecting any terms. The time dependency of the governing equations allows the solution to progress naturally for an arbitrary initial guess to an asymptotic steady state, if one exists. The equations are transformed from physical coordinates to the computational coordinates, allowing the solution of the governing equations in a rectangular parallelepiped domain. The equations are solved by the MacCormack time-split technique which is vectorized and programmed to run on the CDc VPS 32 computer. The codes are written in 32-bit (half word) FORTRAN, which provides an approximate factor of two decreasing in computational time and doubles the memory size compared to the 54-bit word size.

Abolhassani, J. S.↗

NASA NTRS1985

Analysis of two-dimensional incompressible flow past airfoils using unsteady Navier-Stokes equations

The conservative form of the unsteady Navier-Stokes equations in terms of vorticity and stream function in generalized curvilinear coordinates are used to analyze the flow structure of steady separation and unsteady flow with massive separation. The numerical method solves the discretized equations using an ADI-BGE method. The method is applied to a symmetric 12 percent thick Joukowski airfoil. A conformal clustered grid is generated; several 1-D stretching transformations are used to obtain a grid that attempts to resolve many of the multiple scales of the unsteady flow with massive separation, while maintaining the transformation metrics to be smooth and continuous in the entire flow field. Detailed numerical results are obtained for three flow configurations (1) Re = 1000, alpha = 5 deg, (2) Re = 1000, alpha = 15 deg, (3) Re = 10,000, alpha = 5 deg. No artificial dissipation was added; however, lack of a fine grid in the normal direction has presently led to results which are considered qualitative, especially for case (3).

Ghia, K. N.↗

NASA NTRS1986

Analysis of two-dimensional incompressible flow past airfoils using unsteady Navier-Stokes equations

The conservative form of the unsteady Navier-Stokes equations in terms of vorticity and stream function in generalized curvilinear coordinates are used to analyze the flow structure of steady separation and unsteady flow with massive separation. The numerical method solves the discretized equations using an ADI-BGE method. The method is applied to a symmetric 12 percent thick Joukowski airfoil. A conformal clustered grid is generated; several 1-D stretching transformations are used to obtain a grid that attempts to resolve many of the multiple scales of the unsteady flow with massive separation, while maintaining the transformation metrics to be smooth and continuous in the entire flow field. Detailed numerical results are obtained for three flow configurations (1) Re = 1000, alpha = 5 deg., (2) Re =1000, alpha = 15 deg., (3) Re = 10,000, alpha = 5 deg. No artificial dissipation was added; however, lack of a fine grid in the normal direction has presently led to results which are considered qualitative, especially for case (3).

Ghia, K. N.↗

NASA NTRS1992

Three-dimensional computational study of asymmetric flows using Navier-Stokes equations

The unsteady, compressible, thin-layer Navier-Stokes equations are used to obtain three-dimensional, asymmetric, vortex-flow solutions around cones and cone-cylinder configurations. The equations are solved using an implicit, upwind, flux-difference splitting, finite-volume scheme. The computational applications cover asymmetric flows around a 5 semi-apex angle cone of unit length at various Reynolds number. Next, a cylindrical afterbody of various length is added to the conical forebody to study the effect of the length of cylindrical afterbody on the flow asymmetry. All the asymmetric flow solutions are obtained by using a short-duration side-slip disturbance.

Cheung, Y. K.↗

NASA NTRS1989

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

A high-order Lagrangian-decoupling method is presented for the unsteady convection-diffusion and incompressible Navier-Stokes equations. The method is based upon: (1) Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem; (2) implicit high-order backward-differentiation finite-difference schemes for integration along characteristics; (3) finite element or spectral element spatial discretizations; and (4) mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high order accuracy, and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.

Ho, Lee-Wing↗

NASA NTRS2000

Numerical Simulations of Homogeneous Turbulence Using Lagrangian-Averaged Navier-Stokes Equations

The Lagrangian-averaged Navier-Stokes (LANS) equations are numerically evaluated as a turbulence closure. They are derived from a novel Lagrangian averaging procedure on the space of all volume-preserving maps and can be viewed as a numerical algorithm which removes the energy content from the small scales (smaller than some a priori fixed spatial scale alpha) using a dispersive rather than dissipative mechanism, thus maintaining the crucial features of the large scale flow. We examine the modeling capabilities of the LANS equations for decaying homogeneous turbulence, ascertain their ability to track the energy spectrum of fully resolved direct numerical simulations (DNS), compare the relative energy decay rates, and compare LANS with well-accepted large eddy simulation (LES) models.

Mohseni, Kamran↗

NASA NTRS1994

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations

A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, N-sided 'cut' cells are created using polygon-clipping algorithms. The grid is stored in a binary-tree structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded: a gradient-limited, linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The more robust of a series of viscous flux functions is used to provide the viscous fluxes at the cell interfaces. Adaptively-refined solutions of the Navier-Stokes equations using the Cartesian, cell-based approach are obtained and compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

Coirier, William J.↗