Author

# R. W. MacCormack

Bio: R. W. MacCormack is an academic researcher from Ames Research Center. The author has contributed to research in topics: Reynolds number & Turbulence. The author has an hindex of 6, co-authored 6 publications receiving 987 citations.

##### Papers

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01 Jan 1981TL;DR: In this paper, a second-order accurate method for solving viscous flow equations has been proposed that preserves conservation form, requires no block or scalar tridiagonal inversions, is simple and straightforward to program (estimated 10% modification for the update of many existing programs), and should easily adapt to current and future computer architectures.

Abstract: Although much progress has already been made In solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A new method for solving these equations has been devised that 1) is second-order accurate in space and time, 2) is unconditionally stable, 3) preserves conservation form, 4) requires no block or scalar tridiagonal inversions, 5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs), 6) is more efficient than present methods, and 7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 3 x 10(exp 5) to 3 x 10(exp 7) and at CFL numbers as high as 10(exp 3) are compared with theory and experiment.

427 citations

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TL;DR: In this paper, a second-order accurate method for solving viscous flow equations has been proposed that preserves conservation form, requires no block or scalar tridiagonal inversions, is simple and straightforward to program (estimated 10% modification for the update of many existing programs), and should easily adapt to current and future computer architectures.

Abstract: Although much progress has already been made In solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A new method for solving these equations has been devised that 1) is second-order accurate in space and time, 2) is unconditionally stable, 3) preserves conservation form, 4) requires no block or scalar tridiagonal inversions, 5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs), 6) is more efficient than present methods, and 7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 3 x 10(exp 5) to 3 x 10(exp 7) and at CFL numbers as high as 10(exp 3) are compared with theory and experiment.

326 citations

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TL;DR: In this paper, an efficient time-splitting, second-order accurate numerical scheme was used to solve the complete Navier-Stokes equations for supersonic and hypersonic laminar flow over a two-dimensional compression corner.

Abstract: An efficient time-splitting, second-order accurate, numerical scheme is used to solve the complete Navier-Stokes equations for supersonic and hypersonic laminar flow over a two-dimensional compression corner. A fine, exponentially stretched mesh spacing is used in the region near the wall for resolving the viscous layer. Good agreement is obtained between the present computed results and experimental measurement for a Mach number of 14.1 and a Reynolds number of 1.04 x 10(exp 5) with wedge angles of 15 deg, 18 deg, and 24 deg. The details of the pressure variation across the boundary layer are given, and a correlation between the leading edge shock and the peaks in surface pressure and heat transfer is observed.

135 citations

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TL;DR: In this paper, a simple eddy viscosity model is developed, and the interaction of a swept shock wave and a three-dimensional turbulent boundary layer is studied, and good agreement is obtained between the present results and experimental measurements for the case of a wedge with an angle of 6 deg on a flat-plate sidewall.

Abstract: A rapid numerical scheme is used to solve the complete mass-averaged Navier-Stokes equations for supersonic turbulent flow over a three-dimensional compression corner. A simple eddy viscosity model is developed, and the interaction of a swept shock wave and a three-dimensional turbulent boundary layer is studied. Good agreement is obtained between the present results and experimental measurements for the case of a wedge with an angle of 6 deg on a flat-plate sidewall. For the case of a 12-deg wedge angle, the computed results do not show the existence of a peak pressure found experimentally. However, the range of interaction, the plateau pressure, and the peak heat transfer are closely predicted for all cases. The high heat transfer near the axial corner is due to the thinning of the boundary layer and inflow of fresh high-momentum fluid. The heat transfer is relieved through pressure reduction and boundary-layer thickening.

74 citations

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TL;DR: In this article, a relaxation turbulence eddy viscosity model is incorporated to solve the complete Navier-Stokes equations for supersonic and hypersonic flows over a two-dimensionala l compression corner.

Abstract: A relaxation turbulence eddy viscosity model is incorporated to solve the complete Navier-Stokes equations for supersonic and hypersonic flows over a two-dimensiona l compression corner. The system of equations is solved by a time-split, second-order accurate numerical scheme. Details of the relaxation process are studied. In general, using the relaxation model, the eddy viscosity value in the outer layer of the separation region is reduced substantially, and good improvement in the prediction of upstream pressure propagation is obtained for a Mach number of 2.96 and Reynolds number of 10 7, with wedge angle of 25°. However, the application of the relaxation model to hypersonic flow at Mach number 8.66 and Reynolds number of 22 x 106, with highly cooled wall, shows unfavorable effects on heat transfer and skin friction in the reattachment and recompression regions.

32 citations

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TL;DR: In this paper, a two-equation turbulence model is proposed that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows.

Abstract: A comprehensive and critical review of closure approximations for two-equation turbulence models has been made. Particular attention has focused on the scale-determining equation in an attempt to find the optimum choice of dependent variable and closure approximations. Using a combination of singular perturbation methods and numerical computations, this paper demonstrates that: 1) conventional A:-e and A>w formulations generally are inaccurate for boundary layers in adverse pressure gradient; 2) using "wall functions'' tends to mask the shortcomings of such models; and 3) a more suitable choice of dependent variables exists that is much more accurate for adverse pressure gradient. Based on the analysis, a two-equation turbulence model is postulated that is shown to be quite accurate for attached boundary layers in adverse pressure gradient, compressible boundary layers, and free shear flows. With no viscous damping of the model's closure coefficients and without the aid of wall functions, the model equations can be integrated through the viscous sublayer. Surface boundary conditions are presented that permit accurate predictions for flow over rough surfaces and for flows with surface mass addition.

2,783 citations

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01 Jan 1996

TL;DR: An automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy, based on a new stabilised and bounded second-order differencing scheme proposed.

Abstract: The accuracy of numerical simulation algorithms is one of main concerns in modern Computational Fluid Dynamics. Development of new and more accurate mathematical models requires an insight into the problem of numerical errors. In order to construct an estimate of the solution error in Finite Volume calculations, it is first necessary to examine its sources. Discretisation errors can be divided into two groups: errors caused by the discretisation of the solution domain and equation discretisation errors. The first group includes insufficient mesh resolution, mesh skewness and non-orthogonality. In the case of the second order Finite Volume method, equation discretisation errors are represented through numerical diffusion. Numerical diffusion coefficients from the discretisation of the convection term and the temporal derivative are derived. In an attempt to reduce numerical diffusion from the convection term, a new stabilised and bounded second-order differencing scheme is proposed. Three new methods of error estimation are presented. The Direct Taylor Series Error estimate is based on the Taylor series truncation error analysis. It is set up to enable single-mesh single-run error estimation. The Moment Error estimate derives the solution error from the cell imbalance in higher moments of the solution. A suitable normalisation is used to estimate the error magnitude. The Residual Error estimate is based on the local inconsistency between face interpolation and volume integration. Extensions of the method to transient flows and the Local Residual Problem error estimate are also given. Finally, an automatic error-controlled adaptive mesh refinement algorithm is set up in order to automatically produce a solution of pre-determined accuracy. It uses mesh refinement and unrefinement to control the local error magnitude. The method is tested on several characteristic flow situations, ranging from incompressible to supersonic flows, for both steady-state and transient problems.

1,418 citations

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TL;DR: In this paper, an automatic grid generation program is employed, and because an implicit finite-difference algorithm for the flow equations is used, time steps are not severely limited when grid points are finely distributed.

Abstract: Finite-difference procedures are used to solve either the Euler equations or the "thin-layer" Navier-Stokes equations subject to arbitrary boundary conditions. An automatic grid generation program is employed, and because an implicit finite-difference algorithm for the flow equations is used, time steps are not severely limited when grid points are finely distributed. Computational efficiency and compatibility to vectorized computer processors is maintained by use of approximate factorization techniques. Computed results for both inviscid and viscous flow about airfoils are described and compared to viscous known solutions.

691 citations

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TL;DR: The field of computational fluid dynamics during recent years has developed sufficiently to initiate some changes in traditional methods of aerodynamic design, and numerical simulations offer the potential of mending many ills of wind-tunnel and turbomachinery experiments and of providing thereby important new technical capabilities for the aerospace industry.

Abstract: Introduction E is an honor and challenge to present the Dryden Lecture ..i Research for 1979. Since my topic concerns a new trend in fluid mechanics, it should not be surprising that some aspects of this paper involve basic mechanics of turbulence, a field enriched by numerous contributions of Dr. Hugh L. Dryden. Having worked in related fields of fluid mechanics during past years, and long respected both his professional contributions and personal integrity, it is a special pleasure to present this Dryden lecture. The field of computational fluid dynamics during recent years has developed sufficiently to initiate some changes in traditional methods of aerodynamic design. Both computer power and numerical algorithm efficiency are simultaneously improving with time, while the energy resource for driving large wind tunnels is becoming progressively more valuable. Partly for these reasons it has been advocated that the impact of computational aerodynamics on future methods of aircraft design will be profound. ' Qualitatively, the changes taking place are not foreign to past experience in other fields of engineering. For example, trajectory mechanics and neutron transport mechanics already have been largely revolutionized by the computer. Computations rather than experiments now provide the principal source of detailed information in these fields. The amount of reactor experimentation required has been much reduced over former years; experiments now are performed mainly on clear, physically describable arrays of elements aimed at further confirmation of computational techniques; and better designs are achieved than with former experimental methods alone. Similar changes in the relative roles of experimental and computational aerodynamics are anticipated in the future. There are three compelling motivations for vigorously developing computational aerodynamics. One is to provide important new technological capabilities that cannot be provided by experimental facilities. Because of their fundamental limitations, wind tunnels have rarely been able to simulate, for example, Reynolds numbers of aircraft flight, flowfield temperatures around atmosphere entry vehicles, aerodynamics of probes entering planetary atmospheres, aeroelastic distortions present in flight, or the propulsiveexternal flow interaction in flight. In addition, transonic wind tunnels are notoriously limited by wall and support interference; and stream nonuniformities of wind tunnels severely affect laminar-turbulent transition. Moreover, the dynamic-aerodynamic interaction between vehicle motion in flight and transition-dependent separated flow also is inaccessible to wind-tunnel simulation. In still different ways ground facilities for turbomachinery experiments are limited in their ability, for example, to simulate flight inlet-flow nonuniformities feeding into a compressor stage, or to determine detailed flowfields between rotating blades. Numerical flow simulations, on the other hand, have none of these fundamental limitations, but have their own: computer speed and memory. These latter limitations are fewer, but previously have been much more restrictive overall because the full Navier-Stokes equations are of such great complexity that only highly truncated and approximate forms could be handled in the past. In recent years the Navier-Stokes equations have begun to yield under computational attack with the largest current computers. Since the fundamental limitations of computational speed and memory are rapidly decreasing with time, whereas the fundamental limitations of experimental facilities are not, numerical simulations offer the potential of mending many ills of wind-tunnel and turbomachinery experiments, and of providing thereby important new technical capabilities for the aerospace industry. A second compelling motivation concerns energy conservation. The large developmental wind tunnels require large amounts of energy, whereas computers require comparatively

689 citations

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TL;DR: In this paper, the Euler equations for steady transonic inviscidions are used to test the stability of a transonic system with respect to a large number of potential orders of magnitude.

510 citations