High Reynolds Number Viscous Flows Research Articles

On the starting vortex generated by a translating and rotating flat plate

Abstract

Investigations of a compressible second order finite volume code towards the incompressible limit

The accuracy of many existing codes, originally implemented for transonic or supersonic problems, often degrades as the inflow Mach number tends to zero. At low Mach number, the Roe scheme or a similar formulated artificial viscosity scheme which is close to a Roe scheme presents an excess of artificial viscosity. As a result, several years ago a correction of these kinds of schemes was introduced by Turkel. Such correction yields a significant improvement of the accuracy of the solution for the low Mach regime. In particular, for inviscid low Mach number flows it was shown in the literature that the accuracy of the results obtained with the modification of Turkel is comparable with results obtained with CFD codes designed for incompressible flows. Unfortunately, a loss of robustness for the correction of Turkel is often observed when considering compressible high Reynolds number viscous flows. Typical examples investigated in aerodynamics are configurations at high angle of attack, for example an aircraft in landing configuration. Besides other complex flow phenomena, for these kinds of flows both strong locally compressible flows and large incompressible regions are observed. In particular for these kinds of flows the modification of the code needs to work reliably. Based on the work of Rossow and Swanson we are going to present a further modification for the extension of a compressible code towards the incompressible limit. Numerical examples demonstrate the superiority of the method when compared with the modification suggested by Turkel. Moreover, 2D and 3D examples are given demonstrating the applicability of the method to flows of locally compressible and large incompressible effects. Furthermore, the suggested method does not show severe robustness problems for these kinds of flows when compared with the non-modified scheme.

A preconditioned multigrid for simulation of atmospheric flow and wind turbine wakes

Absract This paper demonstrates a low Mach number preconditioning formulation for high Reynolds number, low Mach number atmospheric flows. The preconditioning is implemented together with multigrid approach into a multistage solver in order to provide an efficient and robust all purpose solver to be used for the simulation of high Reynolds number viscous flows. Several modification and scaling of the preconditioned parameter are introduced to ensure robust use of the scheme for this particular flow regime in presence of stagnation points and seperation regions. The simulations results of standard cases of atmospheric flow over hilly terrain with attached and separated flow are presented to demonstrate the superior convergence, accuracy and robustness of the developed preconditioned scheme. An immersed wind turbine model is also developed and embedded into the preconditioned solver which allows for the simulations of far wake behind wind turbines in wind farms. The use of the immersed model reduces the stringent grid requirements of full RANS simulations around wind turbines and allows one to efficiently predict the wake evolution within wind farms.

Kutta-Joukowski force expression for viscous flow

The Kutta Joukowski (KJ) theorem, relating the lift of an airfoil to circulation, was widely accepted for predicting the lift of viscous high Reynolds number flow without separation. However, this theorem was only proved for inviscid flow and it is thus of academic importance to see whether there is a viscous equivalent of this theorem. For lower Reynolds number flow around objects of small size, it is difficult to measure the lift force directly and it is thus convenient to measure the velocity flow field solely and then, if possible, relate the lift to the circulation in a similar way as for the inviscid KJ theorem. The purpose of this paper is to discuss the relevant conditions under which a viscous equivalent of the KJ theorem exists that reduces to the inviscid KJ theorem for high Reynolds number viscous flow and remains correct for low Reynolds number steady flow. It has been shown that if the lift is expressed as a linear function of the circulation as in the classical KJ theorem, then the freestream velocity must be corrected by a component called mean deficit velocity resulting from the wake. This correction is small only when the Reynolds number is relatively large. Moreover, the circulation, defined along a loop containing the boundary layer and a part of the wake, is generally smaller than that based on inviscid flow assumption. For unsteady viscous flow, there is an inevitable additional correction due to unsteadiness.

Numerical assessments of high-order accurate shock capturing schemes: Kelvin–Helmholtz type vortical structures in high-resolutions

This paper investigates performance of extensions of the state-of-the-art high-resolution shock capturing schemes by solving hyperbolic conservation laws in gas dynamics. Such numerical schemes used for the integration of compressible flow simulations should provide accurate solutions for the long time integrations these flows require. To this end, several joint solvers are developed within the framework of the reconstruction and flux-splitting approaches using the underlying MUSCL and WENO frameworks. The numerical assessments include testing and evaluation of various interpolation procedures, flux-limiters, Riemann solvers, flux-splitting schemes as well as their formal order of accuracy. A three-stage optimal TVD Runge–Kutta time stepping is employed for temporal integration. The modular development of these joint solvers provides an ease in characterizing the solution procedures. The performances of these high-resolution solvers are compared for several carefully selected two-dimensional Riemann problems including shock and rarefaction waves as well as joint discontinuities. Based on solutions obtained by all forms of five-point stencil schemes, we demonstrate that the reconstruction based WENO scheme with Roe solver is more accurate than all the versions of the flux-splitting WENO solvers tested in this study. We also show that results are highly dependent on the choice of the flux limiter. Performing benchmark quality high-resolution computations, it is shown that the Euler equations discretized by the fifth-order WENO scheme produce solutions which convect vorticity and create small-scale vortical flow structures which are usually associated with the high Reynolds number viscous flows. Surprisingly, it is found that these Kelvin–Helmholtz instability like vortical structures are not captured in any form of the third-order five-point stencil schemes.

振動する翼周りの流れに対するハイブリッド型非構造移動格子有限体積法(流体工学,流体機械)

Computing a flow around a pitching airfoil, the unstructured moving-grid finite-volume method is developed in this paper. The method has been proposed for various moving boundary problems. However, it is limited for compressible inviscid flows. In the case of calculating a viscous flow at high Reynolds number as a flow around a pitching airfoil, it is necessary to solve a boundary layer. So, a hybrid gird system of a prismatic mesh for boundary layer region and tetrahedral mesh for other regions is suitable. In this paper, the unstructured hybrid moving-grid finite-volume method is formulated in four dimensional space-time unified domain to satisfy a geometric conservation law on such a moving grid in addition to satisfy a physical conservation law. The method is apply to a flow around a pitching NACA0012 airfoil, then the results show that the method is effective for a high Reynolds number viscous flow around a moving body.

Compressible large eddy simulations of wall-bounded turbulent flows using a semi-implicit numerical scheme for low Mach number aeroacoustics

Large eddy simulations (LES) of low-speed, wall-bounded turbulent flows were conducted by numerically integrating the compressible Navier–Stokes equations in a generalized curvilinear coordinate system. An efficient numerical scheme based on a third-order additive semi-implicit Runge–Kutta method for time advancement and a sixth-order accurate, compact finite-difference scheme for spatial discretization were used. The convective terms in the wall-normal direction were treated implicitly to remove the time-step limitation associated with the use of fine meshes in the near-wall region for high Reynolds number viscous flows. The dynamic Smagorinsky subgrid-scale eddy viscosity model was used to close the filtered equations. Generalized characteristic-based non-reflecting boundary conditions were used together with grid stretching and enhanced damping in the exit zone. The accuracy and efficiency of the numerical scheme was assessed by simple acoustic model problems and by comparing LES predictions for fully developed turbulent channel flow and turbulent separated flow in an asymmetric diffuser to previous direct numerical simulation (DNS) and experimental data, respectively. LES predictions for both flows were in reasonable agreement with the DNS and experimental mean velocity and turbulence statistics. The findings suggest that the numerical approach employed here offers comparable accuracy to similar recent studies at approximately one-third of the computational cost and may provide both an accurate and efficient way to conduct computational aeroacoustics studies for low Mach number, confined turbulent flows.

Improvements in the reliability and quality of unstructured hybrid mesh generation

AbstractThis paper presents a reliable and automated approach to the generation of unstructured hybrid grids comprised of tetrahedra, prisms and pyramids for high Reynolds number viscous flow simulations. To enhance robustness, the hybrid mesh generation process starts with the formation of an isotropic tetrahedral grid. Prismatic layers are then added on no‐slip walls fully automatically by obeying user‐specified boundary conditions and three parameters: the number of the layers, an initial layer thickness normal to the walls, and a stretching factor. Topological modifications to the original isotropic tetrahedral elements are prohibited during the layer generation process. The tetrahedral elements near no‐slip walls are shifted inward and the resulting gap between the tetrahedra and the walls is filled up with prismatic elements. To enhance the quality of the prismatic layers around sharp corners, two normals are evaluated for the marching process in these regions. The addition of prismatic elements is locally stopped if negative‐volume elements are created or not enough space is left. An angle‐based smoothing method ensures that the quality of the tetrahedral elements is retained for a reasonable computational cost. The method is demonstrated for two scaled experimental supersonic airplane models designed at the National Aerospace Laboratory of Japan (NAL). Numerical results are compared with wind tunnel experimental data. Copyright © 2004 John Wiley & Sons, Ltd.

Improvement of Drag Prediction for Transonic Transport Aircraft Configuration Using Hybrid Unstructured Navier-Stokes Solver.

In general, an unstructured grid technique has a good flexibility for a complex geometry, and already its usefulness has been demonstrated for full aircraft computations. In the case of three-dimensional, high Reynolds number viscous flow calculations, however, the very fine and stretched grids are required to resolve accurately thin boundary layers developed along the body surface, and so we often have some difficulties in high quality grid generation. A hybrid unstructured grid technique is incorporated into CASPER, a CFD-based design system developed at TRDI-JDA. In this paper, transonic and high Reynolds number flows around an ONERA Model M5 configuration are computed using the CASPER. To validate the present code, with respect to pressure distributions, longitudinal forces and moment coefficients, and transition lines, the present computed results are quantitatively compared with the other computed results and wind-tunnel testing data. Furthermore, some approaches are discussed for reliable drag prediction using CFD method.

On numerical techniques in CFD

Numerical techniques play an important role in CFD. Some of them are reviewed in this paper. The necessity of using high order difference scheme is demonstrated for the study of high Reynolds number viscous flow. Physical guide lines are provided for the construction of these high order schemes. To avoid unduly ad hoc tremtment in the boundary region the use of compact scheme is recommended because it has a small stencil size compared with the traditional finite difference scheme. Besides preliminary Fourier analysis shows the compact scheme can also yield better space resolution which makes it more suitable to study flow with multiscales e.g. turbulence. Other approaches such as perturbation method and finite spectral method are also emphasized. Typical numerical simulations were carried out. The first deals with Euler equations to show its capabilities to capture flow discontinuity. The second deals with Navier-Stokes equations studying the evolution of a mixing layer, the pertinent structures at different times are shown. Asymmetric break down occurs and also the appearance of small vortices.

Applications of unstructured hybrid grid method to high-Reynolds number viscous flows

An unstructured hybrid grid method is discussed for its capability to compute three-dimensional compressible viscous flows of complex geometry. A hybrid of prismatic and tetrahedral grids is used to accurately resolve the wall boundary layers for high-Reynolds number viscous flows. The Navier-Stokes equations for compressible flows are solved by a finite volume, cell-vertex scheme. The LU-SGS implicit time integration method is used to reduce the computational time for very fine grids in boundary layer regions. Two kinds of one-equation turbulence models are evaluated here for their accuracy. The method is applied to computations of transonic flows around the ONERA M5 airplane and ONERA M6 wing, and supersonic shock/boundary layer interacting flows inside a scramjet inlet to validate the accuracy and efficiency of the method

Multigrid Strategies for Viscous Flow Solvers on Anisotropic Unstructured Meshes

Unstructured multigrid techniques for relieving the stiffness associated with high-Reynolds number viscous flow simulations on extremely stretched grids are investigated. One approach consists of employing a semi-coarsening or directional-coarsening technique, based on the directions of strong coupling within the mesh, in order to construct more optimal coarse grid levels. An alternate approach is developed which employs directional implicit smoothing with regular fully coarsened multigrid levels. The directional implicit smoothing is obtained by constructing implicit lines in the unstructured mesh based on the directions of strong coupling. Both approaches yield large increases in convergence rates over the traditional explicit full-coarsening multigrid algorithm. However, maximum benefits are achieved by combining the two approaches in a coupled manner into a single algorithm. An order of magnitude increase in convergence rate over the traditional explicit full-coarsening algorithm is demonstrated, and convergence rates for high-Reynolds number viscous flows which are independent of the grid aspect ratio are obtained. Further acceleration is provided by incorporating low-Mach-number preconditioning techniques, and a Newton-GMRES strategy which employs the multigrid scheme as a preconditioner. The compounding effects of these various techniques on speed of convergence is documented through several example test cases.

A new adaptive GMRES algorithm for achieving high accuracy

GMRES(k) is widely used for solving non-symmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram–Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram–Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications. © 1998 John Wiley & Sons, Ltd.

A 3-D finite-volume method for the Navier-Stokes equations with adaptive hybrid grids

This paper describes the development and application of a numerical scheme for the three-dimensional Navier-Stokes equations of viscous flow using hybrid (prismatic/tetrahedral) grids. Employment of prisms is a relatively new approach towards complex geometry high Reynolds number viscous flow computations. The body surface is covered with triangles, which provides geometric flexibility, while the structure of the mesh in the direction normal to the surface provides better resolution of the viscous stresses. The irregular areas between different prismatic layers covering the surfaces of the domain are filled with tetrahedral elements. Their triangular faces match the corresponding triangular faces of the prisms. A dual adaptation scheme is developed which employs both directional and isotropic local refinement/coarsening of the prisms and tetrahedra. The structure of the prisms is preserved by avoiding interfaces with mid-edge nodes. Spatial discretization consists of a finite-volume, node-based scheme that is of the central-differencing type. The solution is marched in time using a Taylor series expansion following the Lax-Wendroff approach. The scheme employs a dual cells arrangement for evaluation of the viscous terms, which has the property of suppressing odd-even modes in the solution. A storage-efficient data structure is employed, which utilizes the structure of the prismatic grid in one of the directions.

Numerical simulation of viscous transonic flows over an airfoil

High Reynolds number viscous transonic flow is described based on an interaction of the potential outer flow with the boundary layer and wake. Following the procedure of Lighthill (1958), the solutions in these domains are matched to each other through boundary conditions. The solution to the complete problem is obtained iteratively through successive computations of the flows in the outer and inner domains. Both old and new algorithms are used for the iteration process and subsequent problem solution. Results are given for all the airfoils from the Experimental Data Base for Computer Program Assessment (AGARD-AR-138, 1979). A comparison of these results with experimental data shows the degree of agreement between these unbounded airfoil flow simulations and real transonic flow over the central part of a straight wing.

High resolution schemes for steady flow computation

With the introduction of the concept of total variation diminishing scheme (TVD), a variety of numerical schemes using this approach have emerged. For steady state calculations, two particular TVD schemes have proved popular, i.e., the Yee symmetric and the Osher-Chakravarthy upwind TVD schemes. When applied to Euler equations, these two schemes give almost identical results. However, when they are employed to solve Navier-Stokes equations, the authors found dramatic differences especially when high Reynolds number viscous flow is tackled. In one viscous flow calculation, the Yee scheme gave an “unrepresentative” result while Osher-Chakravarthy scheme gave the “physical” result. The paper demonstrates that the numerical dissipation embedded in the schemes may be the cause. Modifications, therefore, are suggested to make Yee's scheme less dissipative so that it is much more suitable for viscous flow calculations. The numerical experiments do favor the modified scheme. Osher-Chakravarthy TVD scheme and the modified Yee scheme are recommended for viscous flow calculation at high Reynolds number.

An implicit method for three dimensional viscous flow with application to cones at angle of attack

Abstract An iteration method for solving the implicit difference equations associated with three nonlinear parabolic differential equations is derived and analyzed. The method is applied to the high Reynolds number viscous flow around a cone at high angle of attack. The requirements which must be met to ensure convergence of the iterations are obtained. In addition, an analysis of the stability of the difference equations is presented and discussed. The numerical results are compared with experimental data for a 10° cone at 12° angle of attack, and a 5·6° cone at 8° angle of attack. The agreement is very good.