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NASA NTRS1991

Discrete shape sensitivity equations for aerodynamic problems

Intensive research and development in Computational Fluid Dynamics (CFD) has recently produced many powerful CFD codes to simulate complex aerodynamic phenomena. However, in order to enhance the usefulness of these CFD codes for design practice, development of design sensitivity equations compatible to these codes becomes very important. This paper represents a part of such an effort to develop a sensitivity analysis methodology that enables the sensitivity equations to be implemented into existing CFD codes with minimal code modification. The methodology is based upon a preelimination procedure which accounts for consistently linearized boundary conditions. Formulations of both the direct differentiation and the adjoint variable methods will be presented in the paper.

Hou, Gene J.-W.↗

NASA NTRS2014

Turbulence Modeling Verification and Validation

Computational fluid dynamics (CFD) software that solves the Reynolds-averaged Navier-Stokes (RANS) equations has been in routine use for more than a quarter of a century. It is currently employed not only for basic research in fluid dynamics, but also for the analysis and design processes in many industries worldwide, including aerospace, automotive, power generation, chemical manufacturing, polymer processing, and petroleum exploration. A key feature of RANS CFD is the turbulence model. Because the RANS equations are unclosed, a model is necessary to describe the effects of the turbulence on the mean flow, through the Reynolds stress terms. The turbulence model is one of the largest sources of uncertainty in RANS CFD, and most models are known to be flawed in one way or another. Alternative methods such as direct numerical simulations (DNS) and large eddy simulations (LES) rely less on modeling and hence include more physics than RANS. In DNS all turbulent scales are resolved, and in LES the large scales are resolved and the effects of the smallest turbulence scales are modeled. However, both DNS and LES are too expensive for most routine industrial usage on today's computers. Hybrid RANS-LES, which blends RANS near walls with LES away from walls, helps to moderate the cost while still retaining some of the scale-resolving capability of LES, but for some applications it can still be too expensive. Even considering its associated uncertainties, RANS turbulence modeling has proved to be very useful for a wide variety of applications. For example, in the aerospace field, many RANS models are considered to be reliable for computing attached flows. However, existing turbulence models are known to be inaccurate for many flows involving separation. Research has been ongoing for decades in an attempt to improve turbulence models for separated and other nonequilibrium flows. When developing or improving turbulence models, both verification and validation are important steps in the process. Verification insures that the CFD code is solving the equations as intended (no errors in the implementation). This is typically done either through the method of manufactured solutions (MMS) or through careful step-by-step comparisons with other verified codes. After the verification step is concluded, validation is performed to document the ability of the turbulence model to represent different types of flow physics. Validation can involve a large number of test case comparisons with experiments, theory, or DNS. Organized workshops have proved to be valuable resources for the turbulence modeling community in its pursuit of turbulence modeling verification and validation. Workshop contributors using different CFD codes run the same cases, often according to strict guidelines, and compare results. Through these comparisons, it is often possible to (1) identify codes that have likely implementation errors, and (2) gain insight into the capabilities and shortcomings of different turbulence models to predict the flow physics associated with particular types of flows. These are valuable lessons because they help bring consistency to CFD codes by encouraging the correction of faulty programming and facilitating the adoption of better models. They also sometimes point to specific areas needed for improvement in the models. In this paper, several recent workshops are summarized primarily from the point of view of turbulence modeling verification and validation. Furthermore, the NASA Langley Turbulence Modeling Resource website is described. The purpose of this site is to provide a central location where RANS turbulence models are documented, and test cases, grids, and data are provided. The goal of this paper is to provide an abbreviated survey of turbulence modeling verification and validation efforts, summarize some of the outcomes, and give some ideas for future endeavors in this area.

Rumsey, Christopher L.↗

NASA NTRS1995

Three-dimensional unstructured method for flows past bodies in 6-DOF relative motion

A three dimensional, unstructured-mesh methodology was developed to simulate unsteady flows past bodies in relative motion, where the trajectory was determined from the instantaneous aerodynamics. The method coupled the equations of fluid flow and those of rigid-body dynamics, and captured the time-dependent interference between stationary and moving boundaries. The unsteady, compressible Euler equations were solved on dynamic, unstructured meshes by an explicit, finite-volume, upwind method. The grid adaptation was performed within a window placed around the moving body. The Euler equations of dynamics were solved by a Runge-Kutta integration scheme. The flow solver and the adaptation scheme were validated by simulating the transonic, unsteady flow around a wing undergoing a forced, periodic pitching motion, then comparing the results with the experimental data. To validate the trajectory code, the six-degrees-of-freedom (DOF) motion of a store separating from a wing was computed using the experimentally determined force and moment fields, then comparing with an independently generated trajectory. Finally, the overall methodology was demonstrated by simulating the unsteady flowfield and the trajectory of a store dropped from a wing. The methodology, its computational cost notwithstanding, has proven to be accurate, automated, easy for dynamic gridding, and relatively efficient for the required man-hours.

Singh, K. P.↗

DOE OSTI2024

SUNDIALS time integrators for exascale applications with many independent systems of ordinary differential equations

Many complex systems can be accurately modeled as a set of coupled time-dependent partial differential equations (PDEs). However, solving such equations can be prohibitively expensive, easily taxing the world’s largest supercomputers. One pragmatic strategy for attacking such problems is to split the PDEs into components that can more easily be solved in isolation. This operator splitting approach is used ubiquitously across scientific domains, and in many cases leads to a set of ordinary differential equations (ODEs) that need to be solved as part of a larger “outer-loop” time-stepping approach. The SUNDIALS library provides a plethora of robust time integration algorithms for solving ODEs, and the U.S. Department of Energy Exascale Computing Project (ECP) has supported its extension to applications on exascale-capable computing hardware. In this paper, we highlight some SUNDIALS capabilities and its deployment in combustion and cosmology application codes (Pele and Nyx, respectively) where operator splitting gives rise to numerous, small ODE systems that must be solved concurrently.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Fluid Dynamics in Curvilinear Coordinates without Fictitious Forces

The use of curvilinear coordinates is sometimes indicated by the inherent geometry of a fluid dynamics problem, but this introduces fictitious forces into the momentum equations that spoil the strict conservative form. If one is willing to work in three dimensions, these fictitious forces can be eliminated by solving for rectangular (Cartesian) momentum components on a curvilinear mesh. A thoroughly geometric approach to fluid dynamics on spacetime demonstrates this transparently, while also giving insight into a greater unity of the relativistic and nonrelativistic cases than is usually appreciated.

pseudo-forces↗

NASA NTRS1994

Fourth order difference methods for hyperbolic IBVP's

Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

Gustafsson, Bertil↗

NASA NTRS1999

High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining challenges facing the field of computational fluid dynamics. In structural mechanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the computation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order accuracy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence suggests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Center. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18, 1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25, 1998 at the NASA Ames Research Center in the United States. During this special course, lecturers from Europe and the United States gave a series of comprehensive lectures on advanced topics related to the high-order numerical discretization of partial differential equations with primary emphasis given to computational fluid dynamics (CFD). Additional consideration was given to topics in computational physics such as the high-order discretization of the Hamilton-Jacobi, Helmholtz, and elasticity equations. This volume consists of five articles prepared by the special course lecturers. These articles should be of particular relevance to those readers with an interest in numerical discretization techniques which generalize to very high-order accuracy. The articles of Professors Abgrall and Shu consider the mathematical formulation of high-order accurate finite volume schemes utilizing essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) reconstruction together with upwind flux evaluation. These formulations are particularly effective in computing numerical solutions of conservation laws containing solution discontinuities. Careful attention is given by the authors to implementational issues and techniques for improving the overall efficiency of these methods. The article of Professor Cockburn discusses the discontinuous Galerkin finite element method. This method naturally extends to high-order accuracy and has an interpretation as a finite volume method. Cockburn addresses two important issues associated with the discontinuous Galerkin method: controlling spurious extrema near solution discontinuities via "limiting" and the extension to second order advective-diffusive equations (joint work with Shu). The articles of Dr. Henderson and Professor Schwab consider the mathematical formulation and implementation of the h-p finite element methods using hierarchical basis functions and adaptive mesh refinement. These methods are particularly useful in computing high-order accurate solutions containing perturbative layers and corner singularities. Additional flexibility is obtained using a mortar FEM technique whereby nonconforming elements are interfaced together. Numerous examples are given by Henderson applying the h-p FEM method to the simulation of turbulence and turbulence transition.

Barth, Timothy↗

NASA NTRS2005

CFD Multiphysics Tool

The recent bold initiatives to expand the human presence in space require innovative approaches to the design of propulsion systems whose underlying technology is not yet mature. The space propulsion community has identified a number of candidate concepts. A short list includes solar sails, high-energy-density chemical propellants, electric and electromagnetic accelerators, solar-thermal and nuclear-thermal expanders. For each of these, the underlying physics are relatively well understood. One could easily cite authoritative texts, addressing both the governing equations, and practical solution methods for, e.g. electromagnetic fields, heat transfer, radiation, thermophysics, structural dynamics, particulate kinematics, nuclear energy, power conversion, and fluid dynamics. One could also easily cite scholarly works in which complete equation sets for any one of these physical processes have been accurately solved relative to complex engineered systems. The Advanced Concepts and Analysis Office (ACAO), Space Transportation Directorate, NASA Marshall Space Flight Center, has recently released the first alpha version of a set of computer utilities for performing the applicable physical analyses relative to candidate deep-space propulsion systems such as those listed above. PARSEC, Preliminary Analysis of Revolutionary in-Space Engineering Concepts, enables rapid iterative calculations using several physics tools developed in-house. A complete cycle of the entire tool set takes about twenty minutes. PARSEC is a level-zero/level-one design tool. For PARSEC s proof-of-concept, and preliminary design decision-making, assumptions that significantly simplify the governing equation sets are necessary. To proceed to level-two, one wishes to retain modeling of the underlying physics as close as practical to known applicable first principles. This report describes results of collaboration between ACAO, and Embry-Riddle Aeronautical University (ERAU), to begin building a set of level-two design tools for PARSEC. The "CFD Multiphysics Tool" will be the propulsive element of the tool set. The name acknowledges that space propulsion performance assessment is primarily a fluid mechanics problem. At the core of the CFD Multiphysics Tool is an open-source CFD code, HYP, under development at ERAU. ERAU is renowned for its undergraduate degree program in Aerospace Engineering the largest in the nation. The strength of the program is its applications-oriented curriculum, which culminates in one of three two-course Engineering Design sequences: Aerospace Propulsion, Spacecraft, or Aircraft. This same philosophy applies to the HYP Project, albeit with fluid physics modeling commensurate with graduate research. HYP s purpose, like the Multiphysics Tool s, is to enable calculations of real (three-dimensional; geometrically complex; intended for hardware development) applications of high speed and propulsive fluid flows.

Perrell, Eric R.↗

DOE OSTI2023

MARBLES (Multi-scale Adaptively Refined Boltzmann LatticE Solver) [SWR-23-37]

MARBLES (Multi-scale Adaptively Refined Boltzmann LatticE Solver) is an open-source computational fluid dynamics package powered by the lattice Boltzmann equations and built on AMReX. In the lattice Boltzmann method, local collisions between meso-scale fictitious particles drive the governing equations which enables MARBLES to easily simulate flow around complex and/or moving geometry without the generation of a body-conforming mesh. Using AMReX data structures and operations ensures a high level of computational performance and parallel scaling on heterogenous architectures while also naturally supporting locally enhanced grid resolution and fidelity through automatic mesh refinement. New domains and problem definitions are easily specified through an input file with examples and guidance on all options and variables provided in the MARBLES documentation.

Henry de Frahan, Marc↗

DOE OSTI2023

MFiX: Fractional-Step Method Implementation

A comprehensive, multiphase computational fluid dynamics (CFD) simulation solves several coupled transport equations including continuity, momentum, species, and energy. Chemical reactions further couple these equations through heats of reaction and rates of formation of products and rates of destruction of reactants. A fractional-step method separates changes attributed to chemical reactions from transport phenomena like convection and diffusion. When the governing equations are split into the transport and reacting components, efficient and independent methodologies can be exploited to solve the different systems. Specifically, discretization of field variable transport equations results in large, sparse matrices which are loosely coupled. These systems are solved in succession using iterative techniques that take advantage of the matrix structure. In contrast, chemical reactions tightly couple field variables locally within the domain (e.g., within a single computational cell) resulting in low dimensional but dense, nonlinear systems that are better solved using direct integration techniques.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1981

Numerical solution of compressible viscous flows at high Reynolds numbers

A new numerical method which was used to reduce the computation time required in fluid dynamics to solve the Navier-Stokes equations at flight Reynolds numbers is described. The method is the implicit analogue of the explicit finite different method. It uses this as its first stage, while the second stage removes the restrictive stability condition by recasting the difference equations in an implicit form. The resulting matrix equations to be solved are either upper or lower block bidiagonal equations. The new method makes it possible and practical to calculate many important three dimensional, high Reynolds number flow fields on computers.

Maccormack, R. W.↗

NASA NTRS1988

Self-excitation of auroral arcs in a three-dimensionally coupled magnetosphere-ionosphere system

This paper presents the first full three-dimensional dynamic simulation of auroral arc formation. The magnetospheric and ionospheric dynamics are represented by one-fluid magnetohydrodynamic equations and two-fluid weakly ionized plasma equations, respectively. The feedback coupling between magnetospheric Alfven waves and ionospheric density waves results in a spontaneous generation of longitudinally elongated striations of field-aligned currents and ionospheric electron densities, which compare very well with many features of quiet auroral arcs.

Watanabe, Kunihiko↗

NASA NTRS1990

Experimental and analytical study of close-coupled ventral nozzles for ASTOVL aircraft

Flow in a generic ventral nozzle system was studied experimentally and analytically with a block version of the PARC3D computational fluid dynamics program (a full Navier-Stokes equation solver) in order to evaluate the program's ability to predict system performance and internal flow patterns. For the experimental work a one-third-size model tailpipe with a single large rectangular ventral nozzle mounted normal to the tailpipe axis was tested with unheated air at steady-state pressure ratios up to 4.0. The end of the tailpipe was closed to simulate a blocked exhaust nozzle. Measurements showed about 5 1/2 percent flow-turning loss, reasonable nozzle performance coefficients, and a significant aftward axial component of thrust due to flow turning loss, reasonable nozzle performance coefficients, and a significant aftward axial component of thrust due to flow turning more than 90 deg. Flow behavior into and through the ventral duct is discussed and illustrated with paint streak flow visualization photographs. For the analytical work the same ventral system configuration was modeled with two computational grids to evaluate the effect of grid density. Both grids gave good results. The finer-grid solution produced more detailed flow patterns and predicted performance parameters, such as thrust and discharge coefficient, within 1 percent of the measured values. PARC3D flow visualization images are shown for comparison with the paint streak photographs. Modeling and computational issues encountered in the analytical work are discussed.

Mcardle, Jack G.↗

NASA NTRS1990

Flow studies in close-coupled ventral nozzles for STOVL aircraft

Flow in a generic ventral nozzle system was studied experimentally and analytically with the PARC3D computational fluid dynamics program (a full Navier-Stokes equations solver) in order to evaluate the program's ability to predict system performance and internal flow patterns. A generic model of a tailpipe with a rectangular ventral nozzle, about one-third of full size, was tested with unheated air at steady-state pressure ratios up to 4.0. Measurements showed about 5.5 percent flow-turning loss and reasonable nozzle performance coefficients. The flow turned more than the designed 90 deg, causing an aftward axial component in the total thrust. Flow behavior into and through the ventral duct is discussed and illustrated with paint streak flow visualization photographs. PARC3D graphic images are shown for comparison with the experiment photographs. The program successfully predicted internal flow patterns; it also computed thrust and discharge coefficients within 1 percent of measured values.

Mcardle, Jack G.↗

NASA NTRS1991

Computerized reduction of elementary reaction sets for combustion modeling

If the entire set of elementary reactions is to be solved in the modeling of chemistry in computational fluid dynamics, a set of stiff ordinary differential equations must be integrated. Some of the reactions take place at very high rates, requiring short time steps, while others take place more slowly and make little progress in the short time step integration. The goal is to develop a procedure to automatically obtain sets of finite rate equations, consistent with a partial equilibrium assumptions, from an elementary set appropriate to local conditions. The possibility of computerized reaction reduction was demonstrated. However, the ability to use the reduced reaction set depends on the ability of the CFD approach in incorporate partial equilibrium calculations into the computer code. Therefore, the results should be tested on a code with partial equilibrium capability.

Wikstrom, Carl V.↗

NASA NTRS1990

Experimental and analytical study of close-coupled ventral nozzles for ASTOVL aircraft

Mcardle, Jack G.↗

NASA NTRS1936

Similitude in hydrodynamic tests involving planing

The problems of using models in planing tests are addressed. If one passes from the model to a hull of linear dimensions n times greater, the speeds are connected by the law of mechanical similitude. The normal forces given by the hydrodynamic equations (perfect fluid) also follow the law of dynamic similitude (Reech's method) and are multiplied by n(exp 3). A series of tests were performed and the actual results were compared to theoretical results.

Gruson, M F↗

NASA NTRS1995

Computational fluid dynamic analysis of hybrid rocket combustor flowfields

Computational fluid dynamic analyses of the Navier-Stokes equations coupled with solid-phase pyrolysis, gas-phase combustion, turbulence and radiation are performed to study hybrid rocket combustor flowfields. The computational study is closely co-ordinated with a companion experimental program using a planar slab burner configuration with HTPB as fuel and gaseous oxygen. Computational predictions agree reasonably well with measurement data of fuel regression rates and surface temperatures. Additionally, most of the parametric trends predicted by the model are in general agreement with experimental trends. The computational model is applied to extend the results from the lab-scale to a full-scale axisymmetric configuration. The numerical predictions indicate that the full-scale configuration burns at a slower rate than the lab-scale combustor under identical specific flow rate conditions. The results demonstrate that detailed CFD analyses can play a useful role in the design of hybrid combustors.

Venkateswaran, S.↗