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NASA NTRS1996

An Algebraic Multigrid Solver for Navier-Stokes Problems in the Discrete Second-Order Approximation

An algebraic multigrid scheme is presented for solving the discrete Navier-Stokes equations to second-order accuracy using the defect-correction method. Solutions have been obtained for problems involving both structured and unstructured meshes, with the resolution and resolution grading controlled by global and local mesh refinements. The solver is efficient and robust to the extent that no underrelaxation of variables has been required to ensure convergence, but rates of convergence can be improved with small amounts of underrelaxation of the velocity-pressure coupling. Provided that the computational mesh can resolve the flow field, convergence characteristics are almost mesh independent. Rates of convergence actually improve with refinement, asymptotically approaching mesh independent values. For extremely coarse meshes where dispersive truncation errors would be expected to prevent convergence (or even induce divergence), solutions can still be obtained by using explicit underrelaxation in the iterative cycle.

Webster, R.↗

NASA NTRS1998

Microwave Imaging on Metal Objects

This final report for the project discusses the attempts to model, using different methods, microwave image reconstruction. Maximum Entropy Method was not successful. Attempts to use Singular Value Decomposition (SVD) got some good results after initial failure. SVD is based upon a theory of linear algebra, to the effect that any M X N Matrix A whose number of rows M is greater than or equal to its number of columns, N can be written as the product of an M X N column-orthogonal matrix U, an N X N diagonal Matrix, W, with m positive or zero elements (the singular values) and the transposition of an N X N orthogonal matrix V. In microwave imaging, the scattered fields can be expressed by the induced current distribution. The SVD method required more contiguous computer memory than was available. Work was also done on the Conjugate Gradient Method (CGM), which didn't work well when tried earlier. It was found that separation of the imaginary part and the real part during calculation may work. This work was considered incomplete as of the end of the grant period.

Tolliver, C. L.↗

NASA NTRS2012

Algebraic Turbulence-Chemistry Interaction Model

The results of a series of Perfectly Stirred Reactor (PSR) and Partially Stirred Reactor (PaSR) simulations are compared to each other over a wide range of operating conditions. It is found that the PaSR results can be simulated by a PSR solution with just an adjusted chemical reaction rate. A simple expression has been developed that gives the required change in reaction rate for a PSR solution to simulate the PaSR results. This expression is the basis of a simple turbulence-chemistry interaction model. The interaction model that has been developed is intended for use with simple one-step global reaction mechanisms and for steady-state flow simulations. Due to the simplicity of the model there is very little additional computational cost in adding it to existing CFD codes.

Norris, Andrew T.↗

NASA NTRS1981

Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the bifurcation point. The plate is discretized by using displacement finite element (or finite difference) models. The special symmetries exhibited by the response of the anisotropic plate are used to reduce the size of the analysis region. The vector of unknown nodal parameters is expressed as a linear combination of a small number of basis vectors, and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the bifurcation point and the associated eigenmode of the panel. In the second stage of the bifurcation buckling mode is used to obtain a nonlinear solution in the vicinity of the bifurcation point and new (updated) sets of basis vectors and reduced equations are generated. In the third stage the reduced equations are used to trace the post-buckling paths. The effectiveness of the proposed technique for predicting the bifurcation and post-buckling behavior of plates is demonstrated by means of numerical examples for plates loaded by means of prescribed edge displacements.

Noor, A. K.↗

NASA NTRS1987

Advances in the computation of transonic separated flows over finite wings

The transonic flow around a low-aspect-ratio infinite wing in a wind tunnel is investigated by means of numerical simulations, with a focus on shock-induced separated flows. A coarse global grid with far-field boundaries matching those of the test section is subdivided into zones: the flow in the clustered zones near the wing is analyzed by solving the Reynolds-averaged Navier-Stokes equations, while that farther from the wing is modeled with the Euler equations. The results are presented graphically, and it is shown that a mushroomlike separated flow with two counterrotating vortices can be simulated when the correct shock strength is imposed (by careful selection of the artificial dissipation, the boundary conditions, the grid refinement, the algebraic turbulence model, and the geometry representation).

Kaynak, Unver↗

NASA NTRS1980

A mixed-mode crack analysis of rectilinear anisotropic solids using conservation laws of elasticity

A very simple and convenient method of analysis for studying two-dimensional mixed-mode crack problems in rectilinear anisotropic solids is presented. The analysis is formulated on the basis of conservation laws of anisotropic elasticity and of fundamental relationships in anisotropic fracture mechanics. The problem is reduced to a system of linear algebraic equations in mixed-mode stress intensity factors. One of the salient features of the present approach is that it can determine directly the mixed-mode stress intensity solutions from the conservation integrals evaluated along a path removed from the crack-tip region without the need of solving the corresponding complex near-field boundary value problem. Several examples with solutions available in the literature are solved to ensure the accuracy of the current analysis. This method is further demonstrated to be superior to other approaches in its numerical simplicity and computational efficiency. Solutions of more complicated and practical engineering problems dealing with the crack emanating from a circular hole in composites are presented also to illustrate the capacity of this method.

Wang, S. S.↗

NASA NTRS1988

On the validation of a code and a turbulence model appropriate to circulation control airfoils

A computer code for calculating flow about a circulation control airfoil within a wind tunnel test section has been developed. This code is being validated for eventual use as an aid to design such airfoils. The concept of code validation being used is explained. The initial stages of the process have been accomplished. The present code has been applied to a low-subsonic, 2-D flow about a circulation control airfoil for which extensive data exist. Two basic turbulence models and variants thereof have been successfully introduced into the algorithm, the Baldwin-Lomax algebraic and the Jones-Launder two-equation models of turbulence. The variants include adding a history of the jet development for the algebraic model and adding streamwise curvature effects for both models. Numerical difficulties and difficulties in the validation process are discussed. Turbulence model and code improvements to proceed with the validation process are also discussed.

Viegas, J. R.↗

NASA NTRS1999

Advanced Multigrid Solvers for Fluid Dynamics

The main objective of this project has been to support the development of multigrid techniques in computational fluid dynamics that can achieve "textbook multigrid efficiency" (TME), which is several orders of magnitude faster than current industrial CFD solvers. Toward that goal we have assembled a detailed table which lists every foreseen kind of computational difficulty for achieving it, together with the possible ways for resolving the difficulty, their current state of development, and references. We have developed several codes to test and demonstrate, in the framework of simple model problems, several approaches for overcoming the most important of the listed difficulties that had not been resolved before. In particular, TME has been demonstrated for incompressible flows on one hand, and for near-sonic flows on the other hand. General approaches were advanced for the relaxation of stagnation points and boundary conditions under various situations. Also, new algebraic multigrid techniques were formed for treating unstructured grid formulations. More details on all these are given below.

Brandt, Achi↗

NASA NTRS2004

Textbook Multigrid Efficiency for Leading Edge Stagnation

A multigrid solver is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work which is a small (less than 10) multiple of the operation count in evaluating the discrete residuals. TME in solving the incompressible inviscid fluid equations is demonstrated for leading- edge stagnation flows. The contributions of this paper include (1) a special formulation of the boundary conditions near stagnation allowing convergence of the Newton iterations on coarse grids, (2) the boundary relaxation technique to facilitate relaxation and residual restriction near the boundaries, (3) a modified relaxation scheme to prevent initial error amplification, and (4) new general analysis techniques for multigrid solvers. Convergence of algebraic errors below the level of discretization errors is attained by a full multigrid (FMG) solver with one full approximation scheme (F.4S) cycle per grid. Asymptotic convergence rates of the F.4S cycles for the full system of flow equations are very fast, approaching those for scalar elliptic equations.

Diskin, Boris↗

NASA NTRS1975

Numerical methods for evaluating the derivatives of eigenvalues and eigenvectors

Two numerical methods are presented for computing the derivatives of eigenvalues and eigenvectors which do not require complete solution of the eigenvalue problem if only a few derivatives are sought. The 'iterative' method may be used to find the first derivative of one or all of the eigenvectors together with the second derivative of their eigenvalues in a self-adjoint system. If the left- and right-hand eigenvectors are known, the first derivative of the eigenvector corresponding to the largest eigenvalue and the second derivative of the largest eigenvalue may be obtained for a nonself-adjoint system. The 'algebraic' method may be used to find all orders of the derivatives, provided they exist, without requiring the left-hand eigenvectors.

Rudisill, C. S.↗

NASA NTRS2004

Textbook Multigrid Efficiency for Leading Edge Stagnation

A multigrid solver is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work which is a small (less than 10) multiple of the operation count in evaluating the discrete residuals. TME in solving the incompressible inviscid fluid equations is demonstrated for leading-edge stagnation flows. The contributions of this paper include (1) a special formulation of the boundary conditions near stagnation allowing convergence of the Newton iterations on coarse grids, (2) the boundary relaxation technique to facilitate relaxation and residual restriction near the boundaries, (3) a modified relaxation scheme to prevent initial error amplification, and (4) new general analysis techniques for multigrid solvers. Convergence of algebraic errors below the level of discretization errors is attained by a full multigrid (FMG) solver with one full approximation scheme (FAS) cycle per grid. Asymptotic convergence rates of the FAS cycles for the full system of flow equations are very fast, approaching those for scalar elliptic equations.

Diskin, Boris↗

NASA NTRS1985

Computational aspects of unsteady flows

The calculation of unsteady flows and the development of numerical methods for solving unsteady boundary layer equations and their application to the flows around important configurations such as oscillating airfoils are presented. A brief review of recent work is provided with emphasis on the need for numerical methods which can overcome possible problems associated with flow reversal and separation. The zig-zag and characteristic box schemes are described in this context, and when embodied in a method which permits interaction between solutions of inviscid and viscous equations, the characteristic box scheme is shown to avoid the singularity associated with boundary layer equations and prescribed pressure gradient. Calculations were performed for a cylinder started impulsively from rest and oscillating airfoils. The results are presented and discussed. It is conlcuded that turbulence models based on an algebraic specification of eddy viscosity can be adequate, that location of translation is important to the calculation of the location of flow separation and, therefore, to the overall lift of an oscillating airfoil.

Cebeci, T.↗

NASA NTRS1989

A skin friction law for compressible turbulent flow

An algebraic skin friction law is derived for adiabatic, compressible, equilibrium, turbulent boundary layer flow. An outer solution in terms of the Clauser defect stream function is matched to an inner empirical expression composed of compressible laws of the wall and wake. The modified Crocco temperature-velocity relationship and the Clauser eddy viscousity model are used in the outer solution. The skin friction law pertains for all pressure gradients in the incompressible through supersonic range and for small pressure gradients in the hypersonic range. Excellent comparisons with experiment are obtained in the appropriate parameter ranges. The application to numerical computation is discussed.

Barnwell, Richard W.↗

NASA NTRS1989

Some Navier-Stokes calculations for the CAST-10 airfoil

Results of the simulation of the viscous flow past the CAST-10 airfoil were shown for different flow conditions. Since the experiments provide only surface pressures and force coefficients, the comparison to the numerical results relies on these. Good agreement of these results is found for the lower Mach number cases except for the shock position. As numerical experiments indicate, this seems to be due to the turbulent shock boundary layer interaction which is not correctly modelled by the algebraic turbulence model employed. For the lower Mach number case the influence of the transition location was investigated, too. Changing the transition location at the lower surface has much more influence on the pressure distribution than changing it on the upper side. For the higher Mach number case the double shock structure found in the experiment for the lower Reynolds numbers was not reproduced by the numerical solutions. The reason for this is unknown though it may be due to the turbulence modelling. For the higher Reynolds number a better resolution of the boundary layer is needed in the computation in order to recover the experimental pressure plateau; but then the shock position is still found downstream of the experimental one.

Schwamborn, D.↗

NASA NTRS1992

Overview of the relevant CFD work at Thiokol Corporation

An in-house developed proprietary advanced computational fluid dynamics code called SHARP (Trademark) is a primary tool for many flow simulations and design analyses. The SHARP code is a time dependent, two dimensional (2-D) axisymmetric numerical solution technique for the compressible Navier-Stokes equations. The solution technique in SHARP uses a vectorizable implicit, second order accurate in time and space, finite volume scheme based on an upwind flux-difference splitting of a Roe-type approximated Riemann solver, Van Leer's flux vector splitting, and a fourth order artificial dissipation scheme with a preconditioning to accelerate the flow solution. Turbulence is simulated by an algebraic model, and ultimately the kappa-epsilon model. Some other capabilities of the code are 2-D two-phase Lagrangian particle tracking and cell blockages. Extensive development and testing has been conducted on the 3-D version of the code with flow, combustion, and turbulence interactions. The emphasis here is on the specific applications of SHARP in Solid Rocket Motor design. Information is given in viewgraph form.

Chwalowski, Pawel↗

NASA NTRS2001

Turbulence Modeling Workshop

Advances in turbulence modeling are needed in order to calculate high Reynolds number flows near the onset of separation and beyond. To this end, the participants in this workshop made the following recommendations. (1) A national/international database and standards for turbulence modeling assessment should be established. Existing experimental data sets should be reviewed and categorized. Advantage should be taken of other efforts already under-way, such as that of the European Research Community on Flow, Turbulence, and Combustion (ERCOFTAC) consortium. Carefully selected "unit" experiments will be needed, as well as advances in instrumentation, to fill the gaps in existing datasets. A high priority should be given to document existing turbulence model capabilities in a standard form, including numerical implementation issues such as grid quality and resolution. (2) NASA should support long-term research on Algebraic Stress Models and Reynolds Stress Models. The emphasis should be placed on improving the length-scale equation, since it is the least understood and is a key component of two-equation and higher models. Second priority should be given to the development of improved near-wall models. Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) would provide valuable guidance in developing and validating new Reynolds-averaged Navier-Stokes (RANS) models. Although not the focus of this workshop, DNS, LES, and hybrid methods currently represent viable approaches for analysis on a limited basis. Therefore, although computer limitations require the use of RANS methods for realistic configurations at high Reynolds number in the foreseeable future, a balanced effort in turbulence modeling development, validation, and implementation should include these approaches as well.

Rubinstein, R.↗

NASA NTRS1999

Turbulence Model Comparisons and Reynolds Number Effects Over a High-Speed Aircraft at Transonic Speeds

This paper gives the results of a grid study, a turbulence model study, and a Reynolds number effect study for transonic flows over a high-speed aircraft using the thin-layer, upwind, Navier-Stokes CFL3D code. The four turbulence models evaluated are the algebraic Baldwin-Lomax model with the Degani-Schiff modifications, the one-equation Baldwin-Barth model, the one-equation Spalart-Allmaras model, and Menter's two-equation Shear-Stress-Transport (SST) model. The flow conditions, which correspond to tests performed in the NASA Langley National Transonic Facility (NTF), are a Mach number of 0.90 and a Reynolds number of 30 million based on chord for a range of angle-of-attacks (1 degree to 10 degrees). For the Reynolds number effect study, Reynolds numbers of 10 and 80 million based on chord were also evaluated. Computed forces and surface pressures compare reasonably well with the experimental data for all four of the turbulence models. The Baldwin-Lomax model with the Degani-Schiff modifications and the one-equation Baldwin-Barth model show the best agreement with experiment overall. The Reynolds number effects are evaluated using the Baldwin-Lomax with the Degani-Schiff modifications and the Baldwin-Barth turbulence models. Five angles-of-attack were evaluated for the Reynolds number effect study at three different Reynolds numbers. More work is needed to determine the ability of CFL3D to accurately predict Reynolds number effects.

Rivers, Melissa B.↗

NASA NTRSDate not supplied

Transition Prediction for the 30P30N Airfoil and the CRM-HL using FUN3D

The ability to accurately model transition over high-lift configurations can lead to improved agreement between stationary Reynolds-Averaged Navier-Stokes (RANS) computations and experimental measurements. In this work, selected transport-equation-based RANS models for transition prediction in the NASA FUN3D flow solver are used to simulate flow over the 30P30N three-element airfoil and the High-Lift Common Research Model (CRM-HL). The primary RANS-based transition model is the Langtry-Menter (LM) 𝜸-Re𝜽𝒕 model coupled with the Menter shear-stress transport model. Additionally, the existing capability for transition prediction based on linear stability correlations is used to allow the modeling of transition over multiple elements of the 30P30N configuration. This approach pairs an algebraic transition model with the Spalart-Allmaras (SA) turbulence model that relies on an imposed transition location in FUN3D and the parabolized stability equations as implemented in the LASTRAC code. Predictions based on these different transition prediction approaches and models are compared with each other, experimental data, and previous simulations by evaluating the lift and drag coefficients, the surface-pressure/skin-friction distributions, and the locations of transition onset. Results are presented for multiple angles of attack for the 30P30N airfoil and CRM-HL. For the CRM-HL, a grid resolution study is performed for the LM transition model and the SA turbulence model.

High Lift↗