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NASA NTRS1985

A study of the effects of numerical dissipation on the calculation of supersonic separated flows

An extensive investigation of the effect of numerical dissipation on the calculation of supersonic, separated flow over a rearward-facing step is carried out. The complete two-dimensional Navier-Stokes equations are solved by means of MacCormack's standard explicit, unsplit, time-dependent, finite difference method. A fourth-order numerical dissipation term is added explicitly. The magnitude of this term is progressively varied, and its consequences on the flowfield calculations are identified and studied. For a cold-wall, heat transfer case, numerical dissipation had a major effect on the results, particularly in the separated region. However, rather dramatically for an adiabatic wall case, numerical dissipation had virtually no effect on the results. The role of grid size on both the influence of numerical dissipation, and on the overall accuracy of the separated flow solutions is discussed.

Kuruvila, G.↗

NASA NTRS1987

Equivalent beam modeling using numerical reduction techniques

Numerical procedures that can accomplish model reductions for space trusses were developed. Three techniques are presented that can be implemented using current capabilities within NASTRAN. The proposed techniques accomplish their model reductions numerically through use of NASTRAN structural analyses and as such are termed numerical in contrast to the previously developed analytical techniques. Numerical procedures are developed that permit reductions of large truss models containing full modeling detail of the truss and its joints. Three techniques are presented that accomplish these model reductions with various levels of structural accuracy. These numerical techniques are designated as equivalent beam, truss element reduction, and post-assembly reduction methods. These techniques are discussed in detail.

Chapman, J. M.↗

NASA NTRS1987

On direct numerical simulations of turbulent reacting flows

A description of the emerging field of direct numerical simulations of turbulent, chemically reacting flows is presented. The types of direct numerical simulations, physical issues related to implementing the simulations, as well as the various numerical methods used are described. Examples are presented of recent applications of direct numerical simulations to a variety of problems, displaying both the potential of the method and also some of its limitations. Finally, our view of the potential role of direct numerical simulations in future research on turbulent, chemically reacting flows is presented.

Jou, W.-H.↗

NASA NTRS1986

Impact of turbulence modeling on numerical accuracy and efficiency of compressible flow simulations

Discussed is the numerical implementation of turbulence models used in viscous compressible flow simulations and their performance described with respect to numerical accuracy, efficiency and stability. The basic approach utilizes the Reynolds-averaged compressible Navier-Stokes equations in which the Reynolds stresses and heat fluxes are mathematically modeled by suitable turbulence models. The turbulence models investigated include zero-, one-, and two-equation eddy viscosity models. The flow fields investigated include theoretical and supersonic flows about two-dimensional and axisymmetric bodies. Discussions concerning the numerical implementation of models include differencing procedures and boundary conditions used to assume numerical stability and accuracy. Numerical performance is also evaluated by comparing computations with experimental results.

Coakley, Thomas J.↗

NASA NTRS1988

A comparison of numerical methods for the prediction of two-dimensional heat transfer in an electrothermal deicer pad

Transient, numerical simulations of the deicing of composite aircraft components by electrothermal heating have been performed in a 2-D rectangular geometry. Seven numerical schemes and four solution methods were used to find the most efficient numerical procedure for this problem. The phase change in the ice was simulated using the Enthalpy method along with the Method for Assumed States. Numerical solutions illustrating deicer performance for various conditions are presented. Comparisons are made with previous numerical models and with experimental data. The simulation can also be used to solve a variety of other heat conduction problems involving composite bodies.

Deicer↗

NASA NTRS1989

Assessment of numerical techniques for unsteady flow calculations

The characteristics of unsteady flow motions have long been a serious concern in the study of various fluid dynamic and combustion problems. With the advancement of computer resources, numerical approaches to these problems appear to be feasible. The objective of this paper is to assess the accuracy of several numerical schemes for unsteady flow calculations. In the present study, Fourier error analysis is performed for various numerical schemes based on a two-dimensional wave equation. Four methods sieved from the error analysis are then adopted for further assessment. Model problems include unsteady quasi-one-dimensional inviscid flows, two-dimensional wave propagations, and unsteady two-dimensional inviscid flows. According to the comparison between numerical and exact solutions, although second-order upwind scheme captures the unsteady flow and wave motions quite well, it is relatively more dissipative than sixth-order central difference scheme. Among various numerical approaches tested in this paper, the best performed one is Runge-Kutta method for time integration and six-order central difference for spatial discretization.

Hsieh, Kwang-Chung↗

NASA NTRS1990

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit.

Yee, H. C.↗

NASA NTRS1990

Numerical and experimental simulations of the postbuckling response of laminated anisotropic panels

Numerical simulations are used in conjuction with experiments to study the buckling and postbuckling responses and failure initiation of flat unstiffened composite panels. The numerical simulations are conducted using two-dimensional shear-flexible finite elements. The effect of the laminate stacking sequence on the buckling and postbuckling responses is studied. Correlation between numerical and experimental results is good through buckling, but the numerical models overestimate the postbuckling stiffness of the panels. To explain the discrepancies in the postbuckling stiffnesses, analytic sensitivity derivatives are calculated and are used to study the sensitivity of the buckling and postbuckling responses to variations in different material and lamination parameters of the panel. Experimental results indicate that failure occurs along a nodal line. Numerical results show that the location of failure initiation corresponds to that of the maximum transverse shear strain energy density in the panel, which occurs at the edge of the panel at a nodal line.

Noor, Ahmed K.↗

NASA NTRS1991

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I - The dynamics of time discretization and its implications for algorithm development in computational fluid dynamics

Yee, H. C.↗

NASA NTRS1993

High order hybrid numerical simulations of two dimensional detonation waves

In order to study multi-dimensional unstable detonation waves, a high order numerical scheme suitable for calculating the detailed transverse wave structures of multidimensional detonation waves was developed. The numerical algorithm uses a multi-domain approach so different numerical techniques can be applied for different components of detonation waves. The detonation waves are assumed to undergo an irreversible, unimolecular reaction A yields B. Several cases of unstable two dimensional detonation waves are simulated and detailed transverse wave interactions are documented. The numerical results show the importance of resolving the detonation front without excessive numerical viscosity in order to obtain the correct cellular patterns.

Cai, Wei↗

NASA NTRS1996

A Numerical Simulation of Scattering from One-Dimensional Inhomogeneous Dielectric Random Surfaces

In this paper, an efficient numerical solution for the scattering problem of inhomogeneous dielectric rough surfaces is presented. The inhomogeneous dielectric random surface represents a bare soil surface and is considered to be comprised of a large number of randomly positioned dielectric humps of different sizes, shapes, and dielectric constants above an impedance surface. Clods with nonuniform moisture content and rocks are modeled by inhomogeneous dielectric humps and the underlying smooth wet soil surface is modeled by an impedance surface. In this technique, an efficient numerical solution for the constituent dielectric humps over an impedance surface is obtained using Green's function derived by the exact image theory in conjunction with the method of moments. The scattered field from a sample of the rough surface is obtained by summing the scattered fields from all the individual humps of the surface coherently ignoring the effect of multiple scattering between the humps. The statistical behavior of the scattering coefficient sigma(sup 0) is obtained from the calculation of scattered fields of many different realizations of the surface. Numerical results are presented for several different roughnesses and dielectric constants of the random surfaces. The numerical technique is verified by comparing the numerical solution with the solution based on the small perturbation method and the physical optics model for homogeneous rough surfaces. This technique can be used to study the behavior of scattering coefficient and phase difference statistics of rough soil surfaces for which no analytical solution exists.

Sarabandi, Kamal↗

NASA NTRS1997

Advances in Numerical Boundary Conditions for Computational Aeroacoustics

Advances in Computational Aeroacoustics (CAA) depend critically on the availability of accurate, nondispersive, least dissipative computation algorithm as well as high quality numerical boundary treatments. This paper focuses on the recent developments of numerical boundary conditions. In a typical CAA problem, one often encounters two types of boundaries. Because a finite computation domain is used, there are external boundaries. On the external boundaries, boundary conditions simulating the solution outside the computation domain are to be imposed. Inside the computation domain, there may be internal boundaries. On these internal boundaries, boundary conditions simulating the presence of an object or surface with specific acoustic characteristics are to be applied. Numerical boundary conditions, both external or internal, developed for simple model problems are reviewed and examined. Numerical boundary conditions for real aeroacoustic problems are also discussed through specific examples. The paper concludes with a description of some much needed research in numerical boundary conditions for CAA.

Tam, Christopher K. W.↗

NASA NTRS2000

Object Based Numerical Zooming Between the NPSS Version 1 and a 1-Dimensional Meanline High Pressure Compressor Design Analysis Code

Within NASA's High Performance Computing and Communication (HPCC) program, NASA Glenn Research Center is developing an environment for the analysis/design of propulsion systems for aircraft and space vehicles called the Numerical Propulsion System Simulation (NPSS). The NPSS focuses on the integration of multiple disciplines such as aerodynamics, structures, and heat transfer, along with the concept of numerical zooming between 0- Dimensional to 1-, 2-, and 3-dimensional component engine codes. The vision for NPSS is to create a "numerical test cell" enabling full engine simulations overnight on cost-effective computing platforms. Current "state-of-the-art" engine simulations are 0-dimensional in that there is there is no axial, radial or circumferential resolution within a given component (e.g. a compressor or turbine has no internal station designations). In these 0-dimensional cycle simulations the individual component performance characteristics typically come from a table look-up (map) with adjustments for off-design effects such as variable geometry, Reynolds effects, and clearances. Zooming one or more of the engine components to a higher order, physics-based analysis means a higher order code is executed and the results from this analysis are used to adjust the 0-dimensional component performance characteristics within the system simulation. By drawing on the results from more predictive, physics based higher order analysis codes, "cycle" simulations are refined to closely model and predict the complex physical processes inherent to engines. As part of the overall development of the NPSS, NASA and industry began the process of defining and implementing an object class structure that enables Numerical Zooming between the NPSS Version I (0-dimension) and higher order 1-, 2- and 3-dimensional analysis codes. The NPSS Version I preserves the historical cycle engineering practices but also extends these classical practices into the area of numerical zooming for use within a companies' design system. What follows here is a description of successfully zooming I-dimensional (row-by-row) high pressure compressor results back to a NPSS engine 0-dimension simulation and a discussion of the results illustrated using an advanced data visualization tool. This type of high fidelity system-level analysis, made possible by the zooming capability of the NPSS, will greatly improve the fidelity of the engine system simulation and enable the engine system to be "pre-validated" prior to commitment to engine hardware.

Follen, G.↗

NASA NTRS2000

Numerical Calculation of the Drag Force Acting on a Solid Particle Pushed by a Solid/Liquid Interface

The distribution of insoluble particles in a metal casting depends primarily on the interaction of the particles with the solid/liquid interface (SLI) during the solidification process. The balance of the forces acting on the particle essentially determines whether a particle will be engulfed or pushed by the SLI. An important component of this force balance is the drag force generated by the particle motion in front of the SLI. Previously developed mathematical models for particle/SLI interaction made use of steady-state solutions of this force provided by the lubrication theory. However, our numerical model based on the SLI tracking approach shows that the steady-state approach is inappropriate to model the interaction process and that at steady-state the theoretical solution underestimates the drag force. It was found that regression analysis of steady-state numerical solutions for cylindrical particles moving normal to a flat SLI gives a relationship of the form: Abstract The distribution of insoluble particles in a metal casting depends primarily on the interaction of the particles with the solid/liquid interface (SLI) during the solidification process. The balance of the forces acting on the particle essentially determines whether a particle will be engulfed or pushed by the SLI. An important component of this force balance is the drag force generated by the particle motion in front of the SLI. Previously developed mathematical models for particle/SLI interaction made use of steady-state solutions of this force provided by the lubrication theory. However, our numerical model based on the SLI tracking approach shows that the steady-state approach is inappropriate to model the interaction process and that at steady-state the theoretical solution underestimates the drag force. It was found that regression analysis of steady-state numerical solutions for cylindrical particles moving normal to a flat SLI gives a relationship of the form: F(sub D, sup num) =sqoare root of 3(pi)(eta)V(sub p)(R(sub p)/d)(sup 10(gamma)/3). This is to compared to the solution provided by the lubrication theory: F(sub D), sup theor) = 3 square root of 2(pi)(eta)V(sub p)(R(sub p)/d)(sub 3/2), where F(sub D) is the drag force, eta is the dynamic viscosity of the fluid, V(sub p) is the particle velocity, R(sub p) is the particle radius, gamma is Euler's constant (gamma approx. = 0.577), and d is the width of the gap between the particle and the SLI. The domain on which the equations proposed by the lubrication theory are relevant is clearly identified in the paper. The numerical model was then validated against the classical lubrication theory within the domain of its validity.

Catalina, Adrian V.↗

NASA NTRS2000

Recommendations for Achieving Accurate Numerical Simulation of Tip Clearance Flows in Transonic Compressor Rotors

The tip clearance flows of transonic compressor rotors are important because they have a significant impact on rotor and stage performance. While numerical simulations of these flows are quite sophisticated. they are seldom verified through rigorous comparisons of numerical and measured data because these kinds of measurements are rare in the detail necessary to be useful in high-speed machines. In this paper we compare measured tip clearance flow details (e.g. trajectory and radial extent) with corresponding data obtained from a numerical simulation. Recommendations for achieving accurate numerical simulation of tip clearance flows are presented based on this comparison. Laser Doppler Velocimeter (LDV) measurements acquired in a transonic compressor rotor, NASA Rotor 35, are used. The tip clearance flow field of this transonic rotor was simulated using a Navier-Stokes turbomachinery solver that incorporates an advanced k-epsilon turbulence model derived for flows that are not in local equilibrium. Comparison between measured and simulated results indicates that simulation accuracy is primarily dependent upon the ability of the numerical code to resolve important details of a wall-bounded shear layer formed by the relative motion between the over-tip leakage flow and the shroud wall. A simple method is presented for determining the strength of this shear layer.

VanZante, Dale E.↗

NASA NTRS1999

Entropy Splitting and Numerical Dissipation

A rigorous stability estimate for arbitrary order of accuracy of spatial central difference schemes for initial-boundary value problems of nonlinear symmetrizable systems of hyperbolic conservation laws was established recently by Olsson and Oliger (1994) and Olsson (1995) and was applied to the two-dimensional compressible Euler equations for a perfect gas by Gerritsen and Olsson (1996) and Gerritsen (1996). The basic building block in developing the stability estimate is a generalized energy approach based on a special splitting of the flux derivative via a convex entropy function and certain homogeneous properties. Due to some of the unique properties of the compressible Euler equations for a perfect gas, the splitting resulted in the sum of a conservative portion and a non-conservative portion of the flux derivative. hereafter referred to as the "Entropy Splitting." There are several potential desirable attributes and side benefits of the entropy splitting for the compressible Euler equations that were not fully explored in Gerritsen and Olsson. The paper has several objectives. The first is to investigate the choice of the arbitrary parameter that determines the amount of splitting and its dependence on the type of physics of current interest to computational fluid dynamics. The second is to investigate in what manner the splitting affects the nonlinear stability of the central schemes for long time integrations of unsteady flows such as in nonlinear aeroacoustics and turbulence dynamics. If numerical dissipation indeed is needed to stabilize the central scheme, can the splitting help minimize the numerical dissipation compared to its un-split cousin? Extensive numerical study on the vortex preservation capability of the splitting in conjunction with central schemes for long time integrations will be presented. The third is to study the effect of the non-conservative proportion of splitting in obtaining the correct shock location for high speed complex shock-turbulence interactions. The fourth is to determine if this method can be extended to other physical equations of state and other evolutionary equation sets. If numerical dissipation is needed, the Yee, Sandham, and Djomehri (1999) numerical dissipation is employed. The Yee et al. schemes fit in the Olsson and Oliger framework.

Yee, H. C.↗

NASA NTRS2001

The Numerical Analysis of a Turbulent Compressible Jet

A numerical method to simulate high Reynolds number jet flows was formulated and applied to gain a better understanding of the flow physics. Large-eddy simulation was chosen as the most promising approach to model the turbulent structures due to its compromise between accuracy and computational expense. The filtered Navier-Stokes equations were developed including a total energy form of the energy equation. Subgrid scale models for the momentum and energy equations were adapted from compressible forms of Smagorinsky's original model. The effect of using disparate temporal and spatial accuracy in a numerical scheme was discovered through one-dimensional model problems and a new uniformly fourth-order accurate numerical method was developed. Results from two- and three-dimensional validation exercises show that the code accurately reproduces both viscous and inviscid flows. Numerous axisymmetric jet simulations were performed to investigate the effect of grid resolution, numerical scheme, exit boundary conditions and subgrid scale modeling on the solution and the results were used to guide the three-dimensional calculations. Three-dimensional calculations of a Mach 1.4 jet showed that this LES simulation accurately captures the physics of the turbulent flow. The agreement with experimental data was relatively good and is much better than results in the current literature. Turbulent intensities indicate that the turbulent structures at this level of modeling are not isotropic and this information could lend itself to the development of improved subgrid scale models for LES and turbulence models for RANS simulations. A two point correlation technique was used to quantify the turbulent structures. Two point space correlations were used to obtain a measure of the integral length scale, which proved to be approximately 1/2 D(sub j). Two point space-time correlations were used to obtain the convection velocity for the turbulent structures. This velocity ranged from 0.57 to 0.71 U(sub j).

DeBonis, James R.↗

NASA NTRS1994

Numerical Generation of Asymmetric Flows about Slender Bodies of Revolution at Extreme Incidence

It is well known that slender bodies of revolution will develop an asymmetric, unsteady flow pattern in experimental tests, if the angle of incidence to the oncoming stream is above a critical value. It has been suggested that the origin of these asymmetric flows may stem from geometric imperfections of the model being tested, or from disturbances in the oncoming stream. In numerical simulations, it is possible to generate bodies of revolution which are perfectly symmetric about their longitudinal axis, and to impose uniform flow conditions which are free from disturbances. The current work presents numerical simulations of the flow about an ogive-cylinder configuration at 40 and 60 degree angle of incidence. These simulations. were performed using numerical algorithms which are also symmetric about the lateral plane of the cylinder body. The flowfields at 40 degree angle of attack were seen to remain symmetric to the round-off accuracy of the computer. At 60 degree angle of attack, a lateral force coefficient developed of O(1) which progressed to an alternate vortex shedding in time. The nature of this lateral force generation and vortex shedding was dependent on the choice of numerical algorithm. The origin of the asymmetries observed in the 60 degree angle of attack computations were traced to round-off errors in the implicit block-matrix inverter. A means of inverting the implicit operator matrices, which maintains the symmetry of the overall numerical algorithm was implemented.

Murman, Scott M.↗