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NASA NTRSDate not supplied

High-Fidelity CFD Verification Workshop 2024 Summary: Spalart-Allmaras QCR2000-R Turbulence Model

This paper summarizes solutions submitted for the Reynolds-averaged Navier-Stokes (RANS) test suite of the High-Fidelity CFD Verification Workshop. The goal of the workshop is to establish standards for verification of computational fluid dynamics (CFD) approaches to simulations of steady and unsteady turbulent flows. The RANS verification studies focus on a one-equation Spalart-Allmaras model with quadratic constitutive relation and rotation correction, SA-neg-QCR2000-R. The verification test cases are a two-dimensional subsonic flow around a Joukowski airfoil, a three-dimensional subsonic flow around an extruded NACA 0012 wing in a tunnel, and a subsonic flow around a wing-body configuration developed for verification of solvers participating in the 5 𝑡 ℎ High-Lift Prediction Workshop. The turbulencemodel formulation, geometry, flow conditions, grids, and reference solutions are described in detail. Solutions for the test cases are computed by seven established CFD solvers on adaptedand fixed-grid families using different discretization approaches. While some noticeable differences between solutions remain, the results achieved by contributing solvers show that different solutions computed for the same RANS model on different grid families can converge to a common limit with grid refinement. The apparent requirements for grid convergence are a well designed family of grids that provide sufficient resolution in important areas and a strong solver capable of deep iterative convergence on each grid. For each test case in the study, the variation between aerodynamic forces computed by different solvers on the finest grids of different families is less than 2%.

Boris Diskin↗
NASA NTRS1999

Solving Upwind-Biased Discretizations: Defect-Correction Iterations

This paper considers defect-correction solvers for a second order upwind-biased discretization of the 2D convection equation. The following important features are reported: (1) The asymptotic convergence rate is about 0.5 per defect-correction iteration. (2) If the operators involved in defect-correction iterations have different approximation order, then the initial convergence rates may be very slow. The number of iterations required to get into the asymptotic convergence regime might grow on fine grids as a negative power of h. In the case of a second order target operator and a first order driver operator, this number of iterations is roughly proportional to h-1/3. (3) If both the operators have the second approximation order, the defect-correction solver demonstrates the asymptotic convergence rate after three iterations at most. The same three iterations are required to converge algebraic error below the truncation error level. A novel comprehensive half-space Fourier mode analysis (which, by the way, can take into account the influence of discretized outflow boundary conditions as well) for the defect-correction method is developed. This analysis explains many phenomena observed in solving non-elliptic equations and provides a close prediction of the actual solution behavior. It predicts the convergence rate for each iteration and the asymptotic convergence rate. As a result of this analysis, a new very efficient adaptive multigrid algorithm solving the discrete problem to within a given accuracy is proposed. Numerical simulations confirm the accuracy of the analysis and the efficiency of the proposed algorithm. The results of the numerical tests are reported.

Diskin, Boris↗
NASA NTRS2012

Equivalent Source Method Applied to Launch Acoustic Simulations

Aeroacoustic simulations of the launch environment are described. A hybrid computational fluid dynamics (CFD)/computational aeroacoustic (CAA) approach is developed in order to accurately and efficiently predict the sound pressure level spectrum on the launch vehicle and surrounding structures. The high-fidelity CFD code LAVA (Launch Ascent and Vehicle Analysis), is used to generate pressure time history at select locations in the flow field. A 3D exterior Helmholtz solver is then used to iteratively determine a set of monopole sources which mimic the noise generating mechanisms identified by the CFD solver. The acoustic pressure field generated from the Helmholtz solver is then used to evaluate the sound pressure levels.

Housman, Jeffrey A.↗
NASA NTRS1982

Profiles of electrified drops and bubbles

Axisymmetric equilibrium shapes of conducting drops and bubbles, (1) pendant or sessile on one face of a circular parallel-plate capacitor or (2) free and surface-charged, are found by solving simultaneously the free boundary problem consisting of the augmented Young-Laplace equation for surface shape and the Laplace equation for electrostatic field, given the surface potential. The problem is nonlinear and the method is a finite element algorithm employing Newton iteration, a modified frontal solver, and triangular as well as quadrilateral tessellations of the domain exterior to the drop in order to facilitate refined analysis of sharply curved drop tips seen in experiments. The stability limit predicted by this computer-aided theoretical analysis agrees well with experiments.

Basaran, O. A.↗
NASA NTRS1991

An efficient advancing front algorithm for Delaunay triangulation

There has been some recent interest in fluid dynamics calculations on unstructured meshes. One method of unstructured mesh generation involves Delaunay triangulation. This method has certain advantages but it can be expensive to implement. Furthermore, there can be problems with crossing grid lines near boundaries. A method shown here avoids many of the robustness and efficiency problems previously associated with Delaunay triangulation. As an added feature, a simple algorithm is shown which allows removal of diagonal edges from cells that are nearly rectangular. This can result in significant savings in the cost per iteration of a flow solver using this grid.

Merriam, Marshal L.↗
NASA NTRS1992

Optimum design of ninety degree bends

An algorithm for the optimum design of an internal flow component to obtain the maximum pressure rise is presented. Maximum pressure rise in a duct with simultaneous turning and diffusion is shown to be related to the control of flow separation on the passage walls. Such a flow is usually associated with downstream conditions that are desirable in turbomachinery and propulsion applications to ensure low loss and stable performance. The algorithm requires the solution of an 'adjoint' problem in addition to the 'direct' equations governing the flow in a body, which in the present analysis are assumed to be the laminar Navier-Stokes equations. The theoretical framework and computational algorithms presented in this study are for the steady Navier-Stokes equations. A procedure is developed for the numerical solution of the adjoint equations. This procedure is coupled with a direct solver in a design iteration loop, that provides a new shape with a higher pressure rise. This procedure is first validated for the design of optimum plane diffusers in two-dimensional flow. The direct Navier-Stokes and the 'adjoint' equations are solved using a finite volume formulation for spatial discretization in an artificial compressibility framework. A simplified version of the above approach is then utilized to design ninety degree diffusing bends. Calculations were carried out for a mean radius ratio at inlet of 2.5 and Reynolds numbers varying from 100 to 500. While at this stage laminar flows is assumed, it is shown that a similar approach can be conceived for turbulent flows.

Modi, Vijay↗
NASA NTRS1994

An implicit numerical scheme for the simulation of internal viscous flows on unstructured grids

The Navier-Stokes equations are solved numerically for two-dimensional steady viscous laminar flows. The grids are generated based on the method of Delaunay triangulation. A finite-volume approach is used to discretize the conservation law form of the compressible flow equations written in terms of primitive variables. A preconditioning matrix is added to the equations so that low Mach number flows can be solved economically. The equations are time marched using either an implicit Gauss-Seidel iterative procedure or a solver based on a conjugate gradient like method. A four color scheme is employed to vectorize the block Gauss-Seidel relaxation procedure. This increases the memory requirements minimally and decreases the computer time spent solving the resulting system of equations substantially. A factor of 7.6 speed up in the matrix solver is typical for the viscous equations. Numerical results are obtained for inviscid flow over a bump in a channel at subsonic and transonic conditions for validation with structured solvers. Viscous results are computed for developing flow in a channel, a symmetric sudden expansion, periodic tandem cylinders in a cross-flow, and a four-port valve. Comparisons are made with available results obtained by other investigators.

Jorgenson, Philip C. E.↗
NASA NTRS2009

Three-dimensional Finite Element Formulation and Scalable Domain Decomposition for High Fidelity Rotor Dynamic Analysis

This paper has two objectives. The first objective is to formulate a 3-dimensional Finite Element Model for the dynamic analysis of helicopter rotor blades. The second objective is to implement and analyze a dual-primal iterative substructuring based Krylov solver, that is parallel and scalable, for the solution of the 3-D FEM analysis. The numerical and parallel scalability of the solver is studied using two prototype problems - one for ideal hover (symmetric) and one for a transient forward flight (non-symmetric) - both carried out on up to 48 processors. In both hover and forward flight conditions, a perfect linear speed-up is observed, for a given problem size, up to the point of substructure optimality. Substructure optimality and the linear parallel speed-up range are both shown to depend on the problem size as well as on the selection of the coarse problem. With a larger problem size, linear speed-up is restored up to the new substructure optimality. The solver also scales with problem size - even though this conclusion is premature given the small prototype grids considered in this study.

Datta, Anubhav↗
NASA NTRS1984

A semi-direct procedure using a local relaxation factor and its application to an internal flow problem

Generally, fast direct solvers are not directly applicable to a nonseparable elliptic partial differential equation. This limitation, however, is circumvented by a semi-direct procedure, i.e., an iterative procedure using fast direct solvers. An efficient semi-direct procedure which is easy to implement and applicable to a variety of boundary conditions is presented. The current procedure also possesses other highly desirable properties, i.e.: (1) the convergence rate does not decrease with an increase of grid cell aspect ratio, and (2) the convergence rate is estimated using the coefficients of the partial differential equation being solved.

Chang, S. C.↗
NASA NTRS1999

TLNS3D/CDISC Multipoint Design of the TCA Concept

This paper presents the work done to date by the authors on developing an efficient approach to multipoint design and applying it to the design of the HSR TCA (High Speed Research Technology Concept Aircraft) configuration. While the title indicates that this exploratory study has been performed using the TLNS3DMB flow solver and the CDISC (Constrained Direct Iterative Surface Curvature) design method, the CDISC method could have been used with any flow solver, and the multipoint design approach does not require the use of CDISC. The goal of the study was to develop a multipoint design method that could achieve a design in about the same time as 10 analysis runs.

Campbell, Richard L.↗
NASA NTRSDate not supplied

Numerical Behaviour of a Smooth Local Correlation-based Transition Model in a Newton-Krylov Flow Solver

The numerical behaviour of transport-equation-based transition models, including both iterative and grid convergence, is influenced by the source terms. Transition models contain source terms that are large and highly nonlinear, and can be destabilizing in a strong implicit solver. Linearization strategies with varying levels of coupling are evaluated in conjunction with a source-term time step restriction to determine best-practices for solving the SA-sLM2015smooth local correlation-based transition model in an implicit Newton-Krylov flow solver. Achieving deep iterative convergence facilitates a detailed investigation of the grid convergence of these free-transition simulations, which are evaluated relative to fully-turbulent simulations performed using the Spalart-Allmaras turbulence model. Simulations of the NLF0416 general aviation airfoil, VA-2 supercritical airfoil, and NASA CRM-NLF wing-body geometry are performed over a range of grid levels. The results demonstrate that both a fully-coupled linearization strategy and a source-term time step restriction improve nonlinear convergence as the complexity of the free-transition simulations increases. In general, additional grid resolution is required for free-transition simulations relative to fully-turbulent simulations in order to achieve a similar level of accuracy, with the grid convergence of free-transition simulations sensitive to the streamwise grid spacings in the transition regions.

AATT↗
NASA NTRS1977

Fast methods incorporating direct elliptic solvers for nonlinear applications in fluid dynamics

Semidirect methods are discussed, their present role, as well as some developments for their application in computational fluid dynamics. A semidirect method is a computational scheme that uses a fast, direct, elliptic solver as the driving algorithm for the iterative solution of finite difference equations. Specific subtopics include: (1) direct Cauchy Riemann solvers for first order elliptic equations; (2) application of the semidirect method to the mixed elliptic hyperbolic problem of steady, inviscid transonic flow; and (3) the treatment of interior conditions, such as those on an airfoil or wing, in semidirect methods.

Martin, E. D.↗
NASA NTRS1983

A semi-direct solver for compressible 3-dimensional rotational flow

An iterative procedure is presented for solving steady inviscid 3-D subsonic rotational flow problems. The procedure combines concepts from classical secondary flow theory with an extension to 3-D of a novel semi-direct Cauchy-Riemann solver. It is developed for generalized coordinates and can be exercised using standard finite difference procedures. The stability criterion of the iterative procedure is discussed along with its ability to capture the evolution of inviscid secondary flow in a turning channel.

Chang, S. C.↗
NASA NTRS1995

Detailed modeling analysis for soot formation and radiation in microgravity gas jet diffusion flames

Radiation heat transfer in combustion systems has been receiving increasing interest. In the case of hydrocarbon fuels, a significant portion of the radiation comes from soot particles, justifying the need for detailed soot formation model and radiation transfer calculations. For laminar gas jet diffusion flames, results from this project (4/1/91 8/22/95) and another NASA study show that flame shape, soot concentration, and radiation heat fluxes are substantially different under microgravity conditions. Our emphasis is on including detailed soot transport models and a detailed solution for radiation heat transfer, and on coupling them with the flame structure calculations. In this paper, we will discuss the following three specific areas: (1) Comparing two existing soot formation models, and identifying possible improvements; (2) A simple yet reasonably accurate approach to calculating total radiative properties and/or fluxes over the spectral range; and (3) Investigating the convergence of iterations between the flame structure solver and the radiation heat transfer solver.

Ku, Jerry C.↗
NASA NTRS1996

Multigrid Acceleration of Time-Accurate Navier-Stokes Calculations

A numerical scheme to solve the unsteady Navier-Stokes equations is described. The scheme is fully implicit in time and is unconditionally stable (at least for first- and second-order discretizations of the physical time derivatives). With unconditional stability, the choice of the time step is based on the physical phenomena to be resolved rather than limited by numerical stability. This is especially important for high Reynolds number viscous flows, where the spatial variation of grid cell size can be as much as six orders of magnitude. A multigrid-multiblock, steady-state, three-dimensional Navier-Stokes solver, TLNS3D, was modified to iteratively invert the equations at each physical time step. The implementation of this procedure in TLNS3D is discussed. The implications of applying several popular turbulence models to unsteady flow are also considered. Numerical results are presented to show the application of the scheme to various two-dimensional turbulent flows. The results of a three-dimensional laminar flow calculation are also given.

Melson, N. Duane↗
NASA NTRS1983

A semi-direct solver for compressible three-dimensional rotational flow

An iterative procedure is presented for solving steady inviscid 3-D subsonic rotational flow problems. The procedure combines concepts from classical secondary flow theory with an extension to 3-D of a novel semi-direct Cauchy-Riemann solver. It is developed for generalized coordinates and can be exercised using standard finite difference procedures. The stability criterion of the iterative procedure is discussed along with its ability to capture the evolution of inviscid secondary flow in a turning channel.

Chang, S.-C.↗