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NASA NTRS1990

Counterrotating prop-fan simulations which feature a relative-motion multiblock grid decomposition enabling arbitrary time-steps

Improvements are presented of a computer algorithm developed for the time-accurate flow analysis of rotating machines. The flow model is a finite volume method utilizing a high-resolution approximate Riemann solver for interface flux definitions. The numerical scheme is a block LU implicit iterative-refinement method which possesses apparent unconditional stability. Multiblock composite gridding is used to orderly partition the field into a specified arrangement of blocks exhibiting varying degrees of similarity. Block-block relative motion is achieved using local grid distortion to reduce grid skewness and accommodate arbitrary time step selection. A general high-order numerical scheme is applied to satisfy the geometric conservation law. An even-blade-count counterrotating unducted fan configuration is chosen for a computational study comparing solutions resulting from altering parameters such as time step size and iteration count. The solutions are compared with measured data.

Janus, J. Mark↗

NASA NTRS1990

Navier-Stokes analysis of transonic cascade flow

A new kind of C-type grid is proposed, this grid is non-periodic on the wake and allows minimum skewness for cascades with high turning and large camber. Reynolds-averaged Navier-Stokes equations are solved on this type of grid using a finite volume discretization and a full multigrid method which uses Runge-Kutta stepping as the driving scheme. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. A detailed numerical study is proposed for a highly loaded transonic blade. A grid independence analysis is presented in terms of pressure distribution, exit flow angles, and loss coefficient. Comparison with experiments clearly demonstrates the capability of the proposed procedure.

Arnone, A.↗

NASA NTRS1992

Numerical simulation of conservation laws

A new numerical framework for solving conservation laws is being developed. This new approach differs substantially from the well established methods, i.e., finite difference, finite volume, finite element and spectral methods, in both concept and methodology. The key features of the current scheme include: (1) direct discretization of the integral forms of conservation laws, (2) treating space and time on the same footing, (3) flux conservation in space and time, and (4) unified treatment of the convection and diffusion fluxes. The model equation considered in the initial study is the standard one dimensional unsteady constant-coefficient convection-diffusion equation. In a stability study, it is shown that the principal and spurious amplification factors of the current scheme, respectively, are structurally similar to those of the leapfrog/DuFort-Frankel scheme. As a result, the current scheme has no numerical diffusion in the special case of pure convection and is unconditionally stable in the special case of pure diffusion. Assuming smooth initial data, it will be shown theoretically and numerically that, by using an easily determined optimal time step, the accuracy of the current scheme may reach a level which is several orders of magnitude higher than that of the MacCormack scheme, with virtually identical operation count.

Chang, Sin-Chung↗

NASA NTRS1992

Navier-Stokes solution of transonic cascade flows using nonperiodic C-type grids

Arnone, Andrea↗

NASA NTRS1993

A Realizable Reynolds Stress Algebraic Equation Model

The invariance theory in continuum mechanics is applied to analyze Reynolds stresses in high Reynolds number turbulent flows. The analysis leads to a turbulent constitutive relation that relates the Reynolds stresses to the mean velocity gradients in a more general form in which the classical isotropic eddy viscosity model is just the linear approximation of the general form. On the basis of realizability analysis, a set of model coefficients are obtained which are functions of the time scale ratios of the turbulence to the mean strain rate and the mean rotation rate. The coefficients will ensure the positivity of each component of the mean rotation rate. These coefficients will ensure the positivity of each component of the turbulent kinetic energy - realizability that most existing turbulence models fail to satisfy. Separated flows over backward-facing step configurations are taken as applications. The calculations are performed with a conservative finite-volume method. Grid-independent and numerical diffusion-free solutions are obtained by using differencing schemes of second-order accuracy on sufficiently fine grids. The calculated results are compared in detail with the experimental data for both mean and turbulent quantities. The comparison shows that the present proposal significantly improves the predictive capability of K-epsilon based two equation models. In addition, the proposed model is able to simulate rotational homogeneous shear flows with large rotation rates which all conventional eddy viscosity models fail to simulate.

Shih, Tsan-Hsing↗

NASA NTRS1993

Computation of confined coflow jets with three turbulence models

A numerical study of confined jets in a cylindrical duct is carried out to examine the performance of two recently proposed turbulence models: an RNG-based K-epsilon model and a realizable Reynolds stress algebraic equation model. The former is of the same form as the standard K-epsilon model but has different model coefficients. The latter uses an explicit quadratic stress-strain relationship to model the turbulent stresses and is capable of ensuring the positivity of each turbulent normal stress. The flow considered involves recirculation with unfixed separation and reattachment points and severe adverse pressure gradients, thereby providing a valuable test of the predictive capability of the models for complex flows. Calculations are performed with a finite-volume procedure. Numerical credibility of the solutions is ensured by using second-order accurate differencing schemes and sufficiently fine grids. Calculations with the standard K-epsilon model are also made for comparison. Detailed comparisons with experiments show that the realizable Reynolds stress algebraic equation model consistently works better than does the standard K-epsilon model in capturing the essential flow features, while the RNG-based K-epsilon model does not seem to give improvements over the standard K-epsilon model under the flow conditions considered.

Zhu, J.↗

NASA NTRSDate not supplied

Comparison of Two Approaches to Constructing Second- and Third-Order Nodal-Gradient Cell-Centered Finite-Volume Methods for Mixed-Element Grids

In this paper, we introduce a third-order nodal-gradient cell-centered ﬁnite volume scheme applicable to mixed-element grids based on a single numerical ﬂux per face. Previously, we considered a split-quadrilateral-face approach, where we split each quadrilateral face of a non-tetrahedral cell into two triangles and then apply second- and third-order schemes designed for tetrahedral grids. This approach is relatively simple to implement but expensive because it requires two numerical ﬂux evaluations per quadrilateral face. In this study, we consider another approach, where the surface ﬂux quadrature is performed with a single numerical ﬂux per face with ﬂux corrections. It requires a 3×3 correction matrix to be stored at each face but is more economical than the split-quadrilateral-face approach. The two approaches are compared for three-dimensional ﬂows on irregular hexahedral grids, and their relative merits are discussed.

Third-order scheme↗

NASA NTRS2003

Nonlinear Aeroacoustics Computations by the Space-Time CE/SE Method

The Space-Time Conservation Element and Solution Element Method, or CE/SE Method for short, is a recently developed numerical method for conservation laws. Despite its second order accuracy in space and time, it possesses low dispersion errors and low dissipation. The method is robust enough to cover a wide range of compressible flows: from weak linear acoustic waves to strong discontinuous waves (shocks). An outstanding feature of the CE/SE scheme is its truly multi-dimensional, simple but effective non-reflecting boundary condition (NRBC), which is particularly valuable for computational aeroacoustics (CAA). In nature, the method may be categorized as a finite volume method, where the conservation element (CE) is equivalent to a finite control volume (or cell) and the solution element (SE) can be understood as the cell interface. However, due to its careful treatment of the surface fluxes and geometry, it is different from the existing schemes. Currently, the CE/SE scheme has been developed to a matured stage that a 3-D unstructured CE/SE Navier-Stokes solver is already available. However, in the present review paper, as a general introduction to the CE/SE method, only the 2-D unstructured Euler CE/SE solver is chosen and sketched in section 2. Then applications of the 2-D and 3-D CE/SE schemes to linear, and in particular, nonlinear aeroacoustics are depicted in sections 3, 4, and 5 to demonstrate its robustness and capability.

Loh, Ching Y.↗

NASA NTRS1993

Modeling void growth and movement with phase change in thermal energy storage canisters

A scheme was developed to model the thermal hydrodynamic behavior of thermal energy storage salts. The model included buoyancy, surface tension, viscosity, phases change with density difference, and void growth and movement. The energy, momentum, and continuity equations were solved using a finite volume formulation. The momentum equation was divided into two pieces. The void growth and void movement are modeled between the two pieces of the momentum equations. Results showed this scheme was able to predict the behavior of thermal energy storage salts.

Darling, Douglas↗

NASA NTRS1993

Modeling void growth and movement with phase change in thermal energy storage canisters

Darling, Douglas↗

NASA NTRS1981

Accelerated finite-volume calculation of transonic potential flows

A fully conservative finite-volume algorithm is used to calculate transonic potential flows past isolated airfoils and through two-dimensional channels. The difference equations are solved by a multigrid technique which uses an alternating direction implicit method as a smoothing algorithm. The finite-volume formulation provides a framework within which to treat flows past complicated geometries, while the multigrid/alternating direction scheme provides rapid convergence of the solution to very small residuals.

Jameson, A.↗

NASA NTRS1989

Numerical analysis of flow about a total temperature sensor

The unsteady flowfield about an airfoil-shaped inlet temperature sensor has been investigated using the thin-layer and full Navier-Stokes equations. A finite-volume formulation of the governing equations was used in conjunction with a Runge-Kutta time stepping scheme to analyze the flow about the sensor. Flow characteristics for this configuration were established at Mach numbers of 0.5 and 0.8 for different Reynolds numbers. The results were obtained for configurations of increasing complexity; important physical phenomena such as shock formation, boundary-layer separation, and unsteady wake formation were noted. Based on the computational results, recommendations for further study and refinement of the inlet temperature sensor were made.

Von Lavante, Ernst↗

NASA NTRS1990

Flowfield computation of entry vehicles

The equations governing the multidimensional flow of a reacting mixture of thermally perfect gasses were derived. The modeling procedures for the various terms of the conservation laws are discussed. A numerical algorithm, based on the finite-volume approach, to solve these conservation equations was developed. The advantages and disadvantages of the present numerical scheme are discussed from the point of view of accuracy, computer time, and memory requirements. A simple one-dimensional model problem was solved to prove the feasibility and accuracy of the algorithm. A computer code implementing the above algorithm was developed and is presently being applied to simple geometries and conditions. Once the code is completely debugged and validated, it will be used to compute the complete unsteady flow field around the Aeroassist Flight Experiment (AFE) body.

Prabhu, Dinesh K.↗

NASA NTRS1990

Multigrid solution of unsteady Navier-Stokes equations using a pressure method

A multigrid relaxation method is applied to a pressure-based implicit procedure to solve unseady, incompressible Navier-Stokes equations. The present multigrid method is a Correction Scheme according to Brandt. This method is used to solve the scalar matrices resulting from the finite-volume formulation and uses flux averaging as the restriction operator. The accuracy and computational efficiency are demonstrated with a steady state driven cavity flow and an unsteady flow over a circular cylinder case. The results are compared with single grid results using the OrthoMin conjugate gradient method and experimental data.

Jiang, Y.↗

NASA NTRS1993

Computational analysis of methods for reduction of induced drag

The purpose of this effort was to perform a computational flow analysis of a design concept centered around induced drag reduction and tip-vortex energy recovery. The flow model solves the unsteady three-dimensional Euler equations, discretized as a finite-volume method, utilizing a high-resolution approximate Riemann solver for cell interface flux definitions. The numerical scheme is an approximately-factored block LU implicit Newton iterative-refinement method. Multiblock domain decomposition is used to partition the field into an ordered arrangement of blocks. Three configurations are analyzed: a baseline fuselage-wing, a fuselage-wing-nacelle, and a fuselage-wing-nacelle-propfan. Aerodynamic force coefficients, propfan performance coefficients, and flowfield maps are used to qualitatively access design efficacy. Where appropriate, comparisons are made with available experimental data.

Janus, J. M.↗

NASA NTRS1992

Progress in multi-dimensional upwind differencing

Multi-dimensional upwind-differencing schemes for the Euler equations are reviewed. On the basis of the first-order upwind scheme for a one-dimensional convection equation, the two approaches to upwind differencing are discussed: the fluctuation approach and the finite-volume approach. The usual extension of the finite-volume method to the multi-dimensional Euler equations is not entirely satisfactory, because the direction of wave propagation is always assumed to be normal to the cell faces. This leads to smearing of shock and shear waves when these are not grid-aligned. Multi-directional methods, in which upwind-biased fluxes are computed in a frame aligned with a dominant wave, overcome this problem, but at the expense of robustness. The same is true for the schemes incorporating a multi-dimensional wave model not based on multi-dimensional data but on an 'educated guess' of what they could be. The fluctuation approach offers the best possibilities for the development of genuinely multi-dimensional upwind schemes. Three building blocks are needed for such schemes: a wave model, a way to achieve conservation, and a compact convection scheme. Recent advances in each of these components are discussed; putting them all together is the present focus of a worldwide research effort. Some numerical results are presented, illustrating the potential of the new multi-dimensional schemes.

Vanleer, Bram↗

NASA NTRS1988

Improved 3-D turbomachinery CFD algorithm

The building blocks of a computer algorithm developed for the time-accurate flow analysis of rotating machines are described. The flow model is a finite volume method utilizing a high resolution approximate Riemann solver for interface flux definitions. This block LU implicit numerical scheme possesses apparent unconditional stability. Multi-block composite gridding is used to orderly partition the field into a specified arrangement. Block interfaces, including dynamic interfaces, are treated such as to mimic interior block communication. Special attention is given to the reduction of in-core memory requirements by placing the burden on secondary storage media. Broad applicability is implied, although the results presented are restricted to that of an even blade count configuration. Several other configurations are presently under investigation, the results of which will appear in subsequent publications.

Janus, J. Mark↗

NASA NTRS1989

A set of strongly coupled, upwind algorithms for computing flows in chemical nonequilibrium

Two new algorithms have been developed to predict the flow of viscous, hypersonic, chemically reacting gases over three-dimensional bodies. Both take advantage of the benefits of upwind differencing, Total Variation Diminishing (TVD) techniques and of a finite-volume framework, but obtain their solution in two separate manners. The first algorithym is a time-marching scheme, and is generally used to obtain solutions in the subsonic portions of the flow field. The second algorithm is a much less expensive, space-marching scheme and can be used for the computation of the larger, supersonic portion of the flow field. Both codes compute their interface fluxes with a new temporal Riemann solver and the resulting schemes are made fully implicit including the chemical source terms.

Molvik, Gregory A.↗