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NASA NTRS1998

3D Navier-Stokes Time Accurate Solutions Using Multipartitioning Parallel Computation Methodology

A parallel CFD code solving 3D time accurate Navier-Stokes equations with multipartitioning parallel Methodology is being developed in collaboration with Ohio State University within the Air Vehicle Directorate, at Wright Patterson Air Force Base. The advantage of the multipartitioning parallel method is that the domain decomposition will not introduce domain boundaries for the implicit operators. A ring structure data communication is employed so that the implicit time accurate method can be implemented for multi-processors with the same accuracy as for the single processor. No sub-iteration is needed at the domain boundaries. The code has been validated for some typical unsteady flows, which include Coutte Flow, flow passing a cylinder. The code now is being employed for a large scale time accurate wall jet transient flow computation. 'ne preliminary results are promising. The mesh has been refined to capture more details of the flow field. The mesh refinement computation is in progress and would be difficult to successfully implement without the parallel computation techniques used. A modified version of the code with more efficient inversion of the diagonalized block matrix is currently being tested.

Zha, Ge-Cheng↗

NASA NTRS1999

Light-Emitting Diodes and Cool White Fluorescent Light Similarly Suppress Pineal Gland Melatonin and Maintain Retinal Function and Morphology in the Rat

Currently, the light sources most commonly used in animal habitat lighting are cool white fluorescent or incandescent lamps. We evaluated a novel light-emitting diode (LED) light source for use in animal habitat lighting by comparing its effectiveness to cool white fluorescent light (CWF) in suppressing pineal gland melatonin and maintaining normal retinal physiology and morphology in the rat. Results of pineal melatonin suppression experiments showed equal suppression of pineal melatonin concentrations for LED light and CWF light at five different light illuminances (100, 40, 10, 1 and 0.1 lux). There were no significant differences in melatonin suppression between LED and CWF light when compared to unexposed controls. Retinal physiology was evaluated using electroretinography. Results show no differences in a-wave implicit times and amplitudes or b-wave implicit times and amplitudes between 100-lux LED-exposed rats and 100-lux CWF-exposed rats. Results of retinal histology assessment show no differences in retinal thickness rod outer segment length and number of rod nuclei between rats exposed to 100-lux LED and 100-lux CWF for days. Furthermore, the retinal pigmented epithelium and rod outer segments of all eyes observed were in good condition and of normal thickness. This study indicates that LED light does not cause retinal damage and can suppress pineal melatonin at similar intensities as a conventional CWF light source. These data suggest that LED light sources may be suitable replacements for conventional light sources used in the lighting of rodent vivariums while providing many mechanical and economical advantages.

Holley, Daniel C.↗

NASA NTRS2001

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one-dimensional convection-diffusion-reaction (CDR) equations. First, accuracy, stability, conservation, and dense output are considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, N = 2, additive Runge-Kutta ARK2 methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms are of equal order to those of the elemental methods. Derived ARK2 methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z(exp [I]) goes to infinity, and retain high stability efficiency in the absence of stiffness, z(exp [I]) goes to zero. Extrapolation-type stage-value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK2 error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK2 methods while offering error control and stage-value predictors.

Kennedy, Christopher A.↗

NASA NTRS2001

The Use of Non-Standard Devices in Finite Element Analysis

A general mathematical description of the response behavior of thin-skin pneumatic envelopes and many other membrane and cable structures produces under-constrained systems that pose severe difficulties to analysis. These systems are mobile, and the general mathematical description exposes the mobility. Yet the response behavior of special under-constrained structures under special loadings can be accurately predicted using a constrained mathematical description. The static response behavior of systems that are infinitesimally mobile, such as a non-slack membrane subtended from a rigid or elastic boundary frame, can be easily analyzed using such general mathematical description as afforded by the non-linear, finite element method using an implicit solution scheme if the incremental uploading is guided through a suitable path. Similarly, if such structures are assembled with structural lack of fit that provides suitable self-stress, then dynamic response behavior can be predicted by the non-linear, finite element method and an implicit solution scheme. An explicit solution scheme is available for evolution problems. Such scheme can be used via the method of dynamic relaxation to obtain the solution to a static problem. In some sense, pneumatic envelopes and many other compliant structures can be said to have destiny under a specified loading system. What that means to the analyst is that what happens on the evolution path of the solution is irrelevant as long as equilibrium is achieved at destiny under full load and that the equilibrium is stable in the vicinity of that load. The purpose of this paper is to alert practitioners to the fact that non-standard procedures in finite element analysis are useful and can be legitimate although they burden their users with the requirement to use special caution. Some interesting findings that are useful to the US Scientific Balloon Program and that could not be obtained without non-standard techniques are presented.

Schur, Willi W.↗

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.↗

NASA NTRS1994

A Navier-Strokes Chimera Code on the Connection Machine CM-5: Design and Performance

We have implemented a three-dimensional compressible Navier-Stokes code on the Connection Machine CM-5. The code is set up for implicit time-stepping on single or multiple structured grids. For multiple grids and geometrically complex problems, we follow the 'chimera' approach, where flow data on one zone is interpolated onto another in the region of overlap. We will describe our design philosophy and give some timing results for the current code. A parallel machine like the CM-5 is well-suited for finite-difference methods on structured grids. The regular pattern of connections of a structured mesh maps well onto the architecture of the machine. So the first design choice, finite differences on a structured mesh, is natural. We use centered differences in space, with added artificial dissipation terms. When numerically solving the Navier-Stokes equations, there are liable to be some mesh cells near a solid body that are small in at least one direction. This mesh cell geometry can impose a very severe CFL (Courant-Friedrichs-Lewy) condition on the time step for explicit time-stepping methods. Thus, though explicit time-stepping is well-suited to the architecture of the machine, we have adopted implicit time-stepping. We have further taken the approximate factorization approach. This creates the need to solve large banded linear systems and creates the first possible barrier to an efficient algorithm. To overcome this first possible barrier we have considered two options. The first is just to solve the banded linear systems with data spread over the whole machine, using whatever fast method is available. This option is adequate for solving scalar tridiagonal systems, but for scalar pentadiagonal or block tridiagonal systems it is somewhat slower than desired. The second option is to 'transpose' the flow and geometry variables as part of the time-stepping process: Start with x-lines of data in-processor. Form explicit terms in x, then transpose so y-lines of data are in-processor. Form explicit terms in y, then transpose so z-lines are in processor. Form explicit terms in z, then solve linear systems in the z-direction. Transpose to the y-direction, then solve linear systems in the y-direction. Finally transpose to the x direction and solve linear systems in the x-direction. This strategy avoids inter-processor communication when differencing and solving linear systems, but requires a large amount of communication when doing the transposes. The transpose method is more efficient than the non-transpose strategy when dealing with scalar pentadiagonal or block tridiagonal systems. For handling geometrically complex problems the chimera strategy was adopted. For multiple zone cases we compute on each zone sequentially (using the whole parallel machine), then send the chimera interpolation data to a distributed data structure (array) laid out over the whole machine. This information transfer implies an irregular communication pattern, and is the second possible barrier to an efficient algorithm. We have implemented these ideas on the CM-5 using CMF (Connection Machine Fortran), a data parallel language which combines elements of Fortran 90 and certain extensions, and which bears a strong similarity to High Performance Fortran. We make use of the Connection Machine Scientific Software Library (CMSSL) for the linear solver and array transpose operations.

Jespersen, Dennis C.↗

NASA NTRS1995

A Data Parallel Multizone Navier-Stokes Code

We have developed a data parallel multizone compressible Navier-Stokes code on the Connection Machine CM-5. The code is set up for implicit time-stepping on single or multiple structured grids. For multiple grids and geometrically complex problems, we follow the "chimera" approach, where flow data on one zone is interpolated onto another in the region of overlap. We will describe our design philosophy and give some timing results for the current code. The design choices can be summarized as: 1. finite differences on structured grids; 2. implicit time-stepping with either distributed solves or data motion and local solves; 3. sequential stepping through multiple zones with interzone data transfer via a distributed data structure. We have implemented these ideas on the CM-5 using CMF (Connection Machine Fortran), a data parallel language which combines elements of Fortran 90 and certain extensions, and which bears a strong similarity to High Performance Fortran (HPF). One interesting feature is the issue of turbulence modeling, where the architecture of a parallel machine makes the use of an algebraic turbulence model awkward, whereas models based on transport equations are more natural. We will present some performance figures for the code on the CM-5, and consider the issues involved in transitioning the code to HPF for portability to other parallel platforms.

Jespersen, Dennis C.↗

NASA NTRS1995

Fractional Step and Pseudocompressibility Methods for the Solution of Incompressible Navier-Stokes Equations

The fractional step and the pseudocompressibility methods for the solution of the incompressible Navier-Stokes equations are outlined. The fractional step method is based on finite-volume formulation and uses the pressure and the volume fluxes across the faces of each cell as dependent variables. The momentum equations are solved implicitly and the Poisson equation for the pressure is solved by using the multigrid method. The pseudocompressibility approach uses an implicit-higher-order-upwind differencing scheme for the convective terms together with the Gauss-Seidel line relaxation method. The dependent variables in the pseudocompressibility approach are the pressure and the cartesian velocity components in unstaggered mesh orientation. The 90-degree square duct flow, the wing-tip vortex wake flow and unsteady turbulent flows over an oscillating NACA 0015 airfoil are computed using both the fractional step and the pseudocompressibility methods. The results obtained from two different schemes are compared against experimental measurements.

Kiris, Cetin↗

NASA NTRS1995

Numerical Simulations of High Enthalpy Pulse Facilities

Axisymmetric flows within shock tubes and expansion tubes are simulated including the effects of finite rate chemistry and both laminar and turbulent boundary layers. The simulations demonstrate the usefulness of computational fluid dynamics for characterizing the flows in high enthalpy pulse facilities. The modeling and numerical requirements necessary to simulate these flows accurately are also discussed. Although there is a large body of analysis which explains and quantifies the boundary layer growth between the shock and the interface in a shock tube, there is a need for more detailed solutions. Phenomena such as thermochemical nonequilibrium. or turbulent transition behind the shock are excluded in the assumptions of Mirels' analysis. Additionally there is inadequate capability to predict the influence of the boundary layer on the expanded gas behind the interface. Quantifying the gas in this region is particularly important in expansion tubes because it is the location of the test gas. Unsteady simulations of the viscous flow in shock tubes are computationally expensive because they must follow features such as a shock wave over the length of the facility and simultaneously resolve the small length scales within the boundary layer. As a result, efficient numerical algorithms are required. The numerical approach of the present work is to solve the axisymmetric gas dynamic equations using an finite-volume formulation where the inviscid fluxes are computed with a upwind TVD scheme. Multiple species equations are included in the formulation so that finite-rate chemistry can be modeled. The simulations cluster grid points at the shock and interface and translate this clustered grid with these features to minimize numerical errors. The solutions are advanced at a CFL number of less than one based on the inviscid gas dynamics. To avoid limitations on the time step due to the viscous terms, these terms are treated implicitly. This requires a block tri-diagonal matrix inversion along each line of cells normal to the wall. The cost of this inversion is more than offset by the larger allowable time step. The source terms representing the finite-rate chemical kinetics are also treated implicitly. An algebraic turbulence model for compressible flow is used. The flow in a low pressure shock tube is computed and the results are compared with Mirels'analysis. The driven gas is nitrogen at 70 Pa, and the incident shock speed is approximately 2.9 km/sec so that there is little dissociation. The simulations include a laminar boundary layer and are run until the limiting flow regime is achieved. At this limit, the shock and interface travel at the same velocity because the amount of driven gas between these two features remains the same: the mass flow across the shock is equal to the mass of gas being entrained at the interface by the boundary layer. Simulations with several grids are presented to establish the grid independence of the solution, Good agreement is achieved between Mirels' correlations and the computations. This is expected since the flow conditions are chosen to be consistent with the assumptions used in Mirels' analysis. This comparison adds credibility to the numerical approach and highlights some of the differences between the theory and the detailed simulations. In addition, simulations of the HYPULSE expansion tube are presented for two operating conditions and the computations are compared to experimental data. The operating gas for both cases is nitrogen. One test condition is at a total enthalpy of 15.2 MJ/Kg and a relatively low pressure of 2 kPa. This case is characterized by a laminar boundary layer and significant chemical nonequilibrium. in the acceleration gas. The second test condition is at a total enthalpy of 10.2 MJ/Kg and a pressure of 38 kPa and is characterized by a turbulent boundary layer. The simulations compare well with experiment and reveal that the nonuniformity in pressure observed during the test time is related to variations in the boundary layer displacement thickness.

Wilson, Gregory J.↗

NASA NTRS1997

Euler/Navier-Stokes Methods

Implicit methods have been the workhorse for the Euler and Navier-Stokes equations for the last 25 years. The ground breaking work of Dr. Joe Steger in implementing such techniques in practical Euler and Navier-Stokes codes provided the basis for all the success in this area. This presentation will highlight his contribution and technical excellence in the area of implicit methods for CFD.

Pulliam, Tom↗

NASA NTRS2002

Graphite Ablation and Thermal Response Simulation Under Arc-Jet Flow Conditions

The Two-dimensional Implicit Thermal Response and Ablation program, TITAN, was developed and integrated with a Navier-Stokes solver, GIANTS, for multidimensional ablation and shape change simulation of thermal protection systems in hypersonic flow environments. The governing equations in both codes are demoralized using the same finite-volume approximation with a general body-fitted coordinate system. Time-dependent solutions are achieved by an implicit time marching technique using Gauess-Siedel line relaxation with alternating sweeps. As the first part of a code validation study, this paper compares TITAN-GIANTS predictions with thermal response and recession data obtained from arc-jet tests recently conducted in the Interaction Heating Facility (IHF) at NASA Ames Research Center. The test models are graphite sphere-cones. Graphite was selected as a test material to minimize the uncertainties from material properties. Recession and thermal response data were obtained from two separate arc-jet test series. The first series was at a heat flux where graphite ablation is mainly due to sublimation, and the second series was at a relatively low heat flux where recession is the result of diffusion-controlled oxidation. Ablation and thermal response solutions for both sets of conditions, as calculated by TITAN-GIANTS, are presented and discussed in detail. Predicted shape change and temperature histories generally agree well with the data obtained from the arc-jet tests.

Chen, Y.-K.↗

NASA NTRS2002

Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations

Additive Runge-Kutta (ARK) methods are investigated for application to the spatially discretized one- dimensional convection-diffusion-reaction (CDR) equations. Accuracy, stability, conservation, and dense-output are first considered for the general case when N different Runge-Kutta methods are grouped into a single composite method. Then, implicit-explicit, (N = 2), additive Runge-Kutta (ARK(sub 2)) methods from third- to fifth-order are presented that allow for integration of stiff terms by an L-stable, stiffly-accurate explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method while the nonstiff terms are integrated with a traditional explicit Runge-Kutta method (ERK). Coupling error terms of the partitioned method are of equal order to those of the elemental methods. Derived ARK(sub 2) methods have vanishing stability functions for very large values of the stiff scaled eigenvalue, z['] yields -infinity, and retain high stability efficiency in the absence of stiffness, z['] yield 0. Extrapolation-type stage- value predictors are provided based on dense-output formulae. Optimized methods minimize both leading order ARK(sub 2) error terms and Butcher coefficient magnitudes as well as maximize conservation properties. Numerical tests of the new schemes on a CDR problem show negligible stiffness leakage and near classical order convergence rates. However, tests on three simple singular-perturbation problems reveal generally predictable order reduction. Error control is best managed with a PID-controller. While results for the fifth-order method are disappointing, both the new third- and fourth-order methods are at least as efficient as existing ARK(sub 2) methods.

Kennedy, Christopher A.↗

NASA NTRS2003

Advanced MHD Algorithm for Solar and Space Science: lst Year Semi Annual Progress Report

We report progress for the development of MH4D for the first and second quarters of FY2004, December 29, 2002 - June 6, 2003. The present version of MH4D can now solve the full viscous and resistive MHD equations using either an explicit or a semi-implicit time advancement algorithm. In this report we describe progress in the following areas. During the two last quarters we have presented poster at the EGS-AGU-EUG Joint Assembly in Nice, France, April 6-11, 2003, and a poster at the 2003 International Sherwood Theory Conference in Corpus Christi, Texas, April 28-30 2003. In the area of code development, we have implemented the MHD equations and the semi-implicit algorithm. The new features have been tested.

Schnack, Dalton D.↗

NASA NTRS2003

MOBY, A Radiometric Buoy for Performance Monitoring and Vicarious Calibration of Satellite Ocean Color Sensors: Measurement and Data Analysis Protocols

The Marine Optical Buoy (MOBY) is the centerpiece of the primary ocean measurement site for calibration of satellite ocean color sensors based on independent in situ measurements. Since late 1996, the time series of normalized water-leaving radiances L(sub WN)(lambda) determined from the array of radiometric sensors attached to MOBY are the primary basis for the on-orbit calibrations of the USA Sea-viewing Wide Field-of-view Sensor (SeaWiFS), the Japanese Ocean Color and Temperature Sensor (OCTS), the French Polarization Detection Environmental Radiometer (POLDER), the German Modular Optoelectronic Scanner on the Indian Research Satellite (IRS1-MOS), and the USA Moderate Resolution Imaging Spectrometer (MODIS). The MOBY vicarious calibration L(sub WN)(lambda) reference is an essential element in the international effort to develop a global, multi-year time series of consistently calibrated ocean color products using data from a wide variety of independent satellite sensors. A longstanding goal of the SeaWiFS and MODIS (Ocean) Science Teams is to determine satellite-derived L(sub WN)(labda) with a relative combined standard uncertainty of 5 %. Other satellite ocean color projects and the Sensor Intercomparison for Marine Biology and Interdisciplinary Oceanic Studies (SIMBIOS) project have also adopted this goal, at least implicitly. Because water-leaving radiance contributes at most 10 % of the total radiance measured by a satellite sensor above the atmosphere, a 5 % uncertainty in L(sub WN)(lambda) implies a 0.5 % uncertainty in the above-atmosphere radiance measurements. This level of uncertainty can only be approached using vicarious-calibration approaches as described below. In practice, this means that the satellite radiance responsivity is adjusted to achieve the best agreement, in a least-squares sense, for the L(sub WN)(lambda) results determined using the satellite and the independent optical sensors (e.g. MOBY). The end result of this approach is to implicitly absorb unquantified, but systematic, errors in the atmospheric correction, incident solar flux, and satellite sensor calibration into a single correction factor to produce consistency with the in situ data.

Clark, Dennis K.↗

NASA NTRS2003

A Variational Analysis of Divergence Profiles Based upon Column-Integrated Mass, Moisture and Energetic Constraints with Satellite-Derived Boundary Fluxes

A diagnostic study is made of the mean global divergent circulation based upon a constrained least action principle that minimizes column-integrated divergent kinetic energy subject to constraints on mass, moisture, available potential energy (ape) and total kinetic energy. The concept of gross moist stability was incorporated in the prescription of Lagrange weight function associated with the ape constraint in order to simulate the net effects of cumulus convective heating in the tropics. The variational analyses were validated satisfactorily with the original NCEP/NCAR-reanalysis divergence fields for the Septembers of 1987 and 1988. Further analyses show that in the tropical ascending regions, the analyzed divergences are dominated by the mass and ape constraints; the moisture constraint is implicitly satisfied while the kinetic energy constraint is highly dependent on the ape constraint. In the subtropical descending regions, the analyzed divergences are dominated by the mass, moisture and kinetic energy constraints; the ape constraint is implicitly satisfied. When the constraint integrals were blended with the satellite-derived boundary flux data from GPCP precipitation and ERBE/SRB radiation estimates, the newly analyzed divergences are significantly stronger than the reanalysis divergences in the areas where the estimates of precipitation rates are higher. With few exceptions, the increases in upper-layer divergences are coupled with nearly equal increases in lower-layer convergences.

Lu, Huei-Lin↗

NASA NTRS2002

Evaluation of Transport in the Lower Tropical Stratosphere in a Global Chemistry and Transport Model

A general circulation model (GCM) relies on various physical parameterizations and provides a solution to the atmospheric equations of motion. A data assimilation system (DAS) combines information from observations with a GCM forecast and produces analyzed meteorological fields that represent the observed atmospheric state. An off-line chemistry and transport model (CTM) can use winds and temperatures from a either a GCM or a DAS. The latter application is in common usage for interpretation of observations from various platforms under the assumption that the DAS transport represents the actual atmospheric transport. Here we compare the transport produced by a DAS with that produced by the particular GCM that is combined with observations to produce the analyzed fields. We focus on transport in the tropics and middle latitudes by comparing the age-of-air inferred from observations of SF6 and CO2 with the age-of-air calculated using GCM fields and DAS fields. We also compare observations of ozone, total reactive nitrogen, and methane with results from the two simulations. These comparisons show that DAS fields produce rapid upward tropical transport and excessive mixing between the tropics and middle latitudes. The unrealistic transport produced by the DAS fields may be due to implicit forcing that is required by the assimilation process when there is bias between the GCM forecast and observations that are combined to produce the analyzed fields. For example, the GCM does not produce a quasi-biennial oscillation (QBO). The QBO is present in the analyzed fields because it is present in the observations, and systematic implicit forcing is required by the DAS. Any systematic bias between observations and the GCM forecast used to produce the DAS analysis is likely to corrupt the transport produced by the analyzed fields. Evaluation of transport in the lower tropical stratosphere in a global chemistry and transport model.

Douglass, Anne R.↗

NASA NTRS2004

Computations of Wall Distances Based on Differential Equations

The use of differential equations such as Eikonal, Hamilton-Jacobi and Poisson for the economical calculation of the nearest wall distance d, which is needed by some turbulence models, is explored. Modifications that could palliate some turbulence-modeling anomalies are also discussed. Economy is of especial value for deforming/adaptive grid problems. For these, ideally, d is repeatedly computed. It is shown that the Eikonal and Hamilton-Jacobi equations can be easy to implement when written in implicit (or iterated) advection and advection-diffusion equation analogous forms, respectively. These, like the Poisson Laplacian term, are commonly occurring in CFD solvers, allowing the re-use of efficient algorithms and code components. The use of the NASA CFL3D CFD program to solve the implicit Eikonal and Hamilton-Jacobi equations is explored. The re-formulated d equations are easy to implement, and are found to have robust convergence. For accurate Eikonal solutions, upwind metric differences are required. The Poisson approach is also found effective, and easiest to implement. Modified distances are not found to affect global outputs such as lift and drag significantly, at least in common situations such as airfoil flows.

Tucker, Paul G.↗

NASA NTRS1999

Numerical Solution of Incompressible Navier-Stokes Equations Using a Fractional-Step Approach

A fractional step method for the solution of steady and unsteady incompressible Navier-Stokes equations is outlined. The method is based on a finite volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (3rd and 5th) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds Numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when 5th-order upwind differencing and a modified production term in the Baldwin-Barth one-equation turbulence model are used with adequate grid resolution.

Kiris, Cetin↗