Van Albada Limiter Research Articles

An Extended Eddy-Preserving Scheme for Strong Vortical Flows

Abstract The eddy-preserving limiter has been demonstrated to outperform the conventional van Albada limiter for monotone upstream-centered schemes for conservation laws (MUSCL) for vortical flows. It reduces the dissipation by inactivating the conventional van Albada limiter in the interpolation of velocity components on the swirl plane of the vortex. In this work, we extend the limiter for the interpolation of pressure since a minimum pressure often exists along the axis of a free vortex. Three-dimensional vortex advection cases are employed to demonstrate the effects of the novel scheme. An extended eddy-preserving limiter scheme has been demonstrated to further improve the preservation of the pressure extrema. Finally, the scheme is applied to flow through a draft tube and the numerical results are compared against experimental data.

An accuracy preserving limiter for the high-order discontinuous Galerkin method on unstructured grids

This paper develops a new limiter and applies it to the high-order discontinuous Galerkin (DG) method on unstructured grids. We extend the original second-order Van Albada limiter to the third-order accuracy based on the vector integral theorem. In order to achieve the high-order accuracy, we have derived the expressions for the second-order derivatives of the flow variables on the cell center. The developed limiter is differentiable, simple to be implemented and has a good property of convergence. Several numerical cases demonstrate that the new limiter can preserve the high-order numerical accuracy in smooth regions, and effectively control the nonphysical oscillations near the shock waves.

Numerical simulation of a bulb turbine with an eddy-preserving limiter scheme

The eddy-preserving limiter scheme has been demonstrated to outperform the conventional van Albada limiter for Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) for vortical flows. It reduces the dissipation by inactivating the conventional van Albada limiter in the interpolations of the velocity components on the swirl plane of the vortex. In this work, we extend the limiter for the interpolation of pressure, since a minimum pressure often exists along the axis of a free vortex. The conventional van Albada limiter is inactivated in the interpolation of pressure at smooth extrema. A 3D inviscid vortex advection case is employed to demonstrate the effects of the novel scheme. The extended eddy-preserving limiter scheme has been demonstrated to be able to further minimize the dissipation of the vortex than the original eddy-preserving limiter scheme. Finally, the scheme is applied to the BulbT test case and the numerical results are compared against experimental data.

A smoothed particle hydrodynamics numerical scheme with a consistent diffusion term for the continuity equation

A novel formulation for the diffusive term in the continuity equation is proposed to improve the stability of smoothed particle hydrodynamics weakly compressible scheme avoiding the introduction of empirical parameters. Densities at particle-particle interface have been computed by means of a first-order consistent total variational diminishing reconstruction and a one-dimensional Roe’s approximate Riemann solver is applied to add the correct amount of diffusion. Results of numerical tests also demonstrate that the proposed method is able to guarantee consistency both inside the fluid and close to the free surface. Furthermore, a numerical analysis of several flux limiter functions has been conducted, finding that the choice of this function is a critical point to guarantee the accuracy of the method. It has been assessed, through the monitoring of the internal energy, that the van Albada limiter is more effective in dissipating spurious density fluctuations.

A low-dissipation monotonicity-preserving scheme for turbulent flows in hydraulic turbines

The objective of this work is to improve the inherent dissipation of the numerical schemes under the framework of a Reynolds-averaged Navier-Stokes (RANS) simulation. The governing equations are solved by the finite volume method with the k-ω SST turbulence model. Instead of the van Albada limiter, a novel eddy-preserving limiter is employed in the MUSCL reconstructions to minimize the dissipation of the vortex. The eddy-preserving procedure inactivates the van Albada limiter in the swirl plane and reduces the artificial dissipation to better preserve vortical flow structures. Steady and unsteady simulations of turbulent flows in a straight channel and a straight asymmetric diffuser are demonstrated. Profiles of velocity, Reynolds shear stress and turbulent kinetic energy are presented and compared against large eddy simulation (LES) and/or experimental data. Finally, comparisons are made to demonstrate the capability of the eddy-preserving limiter scheme.

Numerical simulation of flows from free molecular regime to continuum regime by a DVM with streaming and collision processes

A discrete velocity method (DVM) with streaming and collision processes is presented in this work for simulation of flows from free molecular regime to continuum regime. The present scheme can be considered as a semi-Lagrangian like scheme. At first, we follow the conventional DVM to discretize the phase velocity space by a number of discrete velocities. Then, for each discrete velocity, the kinetic equation with BGK-Shakhov model is integrated in space and time within one time step. As a result, a simple algebraic formulation can be obtained, and its solution can be marched in time by the streaming and collision processes. However, differently from the conventional semi-Lagrangian scheme, the present scheme uses the MUSCL approach with van Albada limiter in the process of reconstructing the distribution function at the surrounding points of the cell center, and the transport distance is controlled in order to avoid extrapolation. This makes the present scheme be capable of simulating the hypersonic rarefied flows. In addition, as compared to the unified gas kinetic scheme (UGKS), the present scheme is simpler and easier for implementation. Thus, the computational efficiency can be improved accordingly. To validate the proposed numerical scheme, test examples from free molecular regime to continuum regime are simulated. Numerical results showed that the present scheme can predict the flow properties accurately even for hypersonic rarefied flows.

Predictor-Corrector LU-SGS Discontinuous Galerkin Finite Element Method for Conservation Laws

Efficient implicit predictor-corrector LU-SGS discontinuous Galerkin (DG) approach for compressible Euler equations on unstructured grids is investigated by adding the error compensation of high-order term. The original LU-SGS and GMRES schemes for DG method are discussed. Van Albada limiter is employed to make the scheme monotone. The numerical experiments performed for the transonic inviscid flows around NACA0012 airfoil, RAE2822 airfoil, and ONERA M6 wing indicate that the present algorithm has the advantages of low storage requirements and high convergence acceleration. The computational efficiency is close to that of GMRES scheme, nearly 2.1 times greater than that of LU-SGS scheme on unstructured grids for 2D cases, and almost 5.5 times greater than that of RK4 on unstructured grids for 3D cases.

A lattice Boltzmann model for simulation of compressible flows

SummaryThis paper presents a new model of lattice Boltzmann method for full compressible flows. On the basis of multi‐speed model, an extra potential energy distribution function is introduced to recover the full compressible Navier–Stokes equations with a flexible specific‐heat ratio and Prandtl number. The Chapman–Enskog expansion of the kinetic equations is performed, and the two‐dimension‐seventeen‐velocity density equilibrium distribution functions are obtained. The governing equations are discretized using the third order monotone upwind scheme for scalar conservation laws finite volume scheme. The van Albada limiter is used to avoid spurious oscillations. In order to verify the accuracy of this double‐distribution‐function model, the Riemann problems, Couette flows, and flows around a NACA0012 airfoil are simulated. It is found that the proposed lattice Boltzmann model is suitable for compressible flows, even for strong shock wave problem, which has an extremely large pressure ratio, 100,000. Copyright © 2014 John Wiley & Sons, Ltd.

Performance Evaluation of Two-Equation Turbulence Models for 3D Wing-Body Configuration

Numerical simulations of 3D aircraft configurations are performed in order to understand the effects of turbulence models on the prediction of aircraft's aerodynamic characteristics. An in-house CFD code that solves 3D RANS equations and two-equation turbulence model equations are used. The code applies Roe's approximated Riemann solver and an AF-ADI scheme. Van Leer's MUSCL extrapolation with van Albada's limiter is also adopted. Various versions of Menter's <TEX>$k-{\omega}$</TEX> SST turbulence models as well as Coakley's <TEX>$q-{\omega}$</TEX> model are incorporated into the CFD code. Menter's <TEX>$k-{\omega}$</TEX> SST models include the standard model, the 2003 model, the model incorporating the vorticity source term, and the model containing controlled decay. Turbulent flows over a wing are simulated in order to validate the turbulence models contained in the CFD code. The results from these simulations are then compared with computational results from the <TEX>$3^{rd}$</TEX> AIAA CFD Drag Prediction Workshop. Numerical simulations of the DLR-F6 wing-body and wing-body-nacelle-pylon configurations are conducted and compared with computational results of the <TEX>$2^{nd}$</TEX> AIAA CFD Drag Prediction Workshop. Aerodynamic characteristics as well as flow features are scrutinized with respect to the turbulence models. The results obtained from each simulation incorporating Menter's <TEX>$k-{\omega}$</TEX> SST turbulence model variations are compared with one another.

Numerical Simulation of Rotor Flow Field Based on Overset Grids and Several Spatial and Temporal Discretization Schemes

A numerical method based on solutions of Euler/Navier-Stokes (N-S) equations is developed for calculating the flow field over a rotor in hover. Jameson central scheme, van Leer flux-vector splitting scheme, advection upwind splitting method (AUSM) scheme, upwind AUSM/van Leer scheme, AUSM+ scheme and AUSMDV scheme are implemented for spatial discretization, and van Albada limiter is also applied. For temporal discretization, both explicit Runge-Kutta method and implicit lower-upper symmetric Gauss-Seidel (LU-SGS) method are attempted. Simultaneously, overset grid technique is adopted. In detail, hole-map method is utilized to identify intergrid boundary points (IGBPs). Furthermore, aimed at identification issue of donor elements, inverse-map method is implemented. Eventually, blade surface pressure distributions derived from numerical simulation are validated compared with experimental data, showing that all the schemes mentioned above have the capability to predict the rotor flow field accurately. At the same time, vorticity contours are illustrated for analysis, and other characteristics are also analyzed.

A comparative study of TVD‐limiters—well‐known limiters and an introduction of new ones

SUMMARYThis paper gives a comparative study of TVD‐limiters for standard explicit Finite Volume schemes. In contrast to older studies, it includes also unsymmetrical limiter functions that depend on the local CFL‐number. We classify the limiters and show how to extend these families of limiters. We introduce a new member of the Superbee family, which is adapted to Roe's linear third‐order scheme. Based on an idea by Serna and Marquina, new smooth limiters are introduced, which turn the van Leer and van Albada limiters into complete classes of limiters. The comparison of the limiters is done with some standard test cases. The results clarify the influence of the chosen limiter on the quality of the numerical results. Compared with ENO or WENO schemes, they also show the high resolution, which can be obtained by a CFL‐number‐dependent limiter when the grid is not highly refined. Copyright © 2010 John Wiley &amp; Sons, Ltd.

A COMPARATIVE STUDY OF FINITE VOLUME AND FINITE ELEMENT ON SOME TRANSCRITICAL FREE SURFACE FLOW PROBLEMS

This paper presents details of finite volume and finite element numerical models based on unstructured triangular meshes that are used to solve the two-dimensional nonlinear shallow water equations (SWEs). The finite volume scheme uses Roe's approximate Riemann solver to evaluate the convection terms. Second order accuracy is achieved by means of the MUSCL approach with MinMod and VanAlbada limiters. The finite element model utilizes the Lax–Wendroff two-step scheme, which is second-order in space and time. The models are validated and their relative performance compared for several benchmark problems, including a hydraulic jump, and flows in converging and converging–diverging channels.

Energy relaxation method for chemical non‐equilibrium flow computations

AbstractIn this paper, an algorithm for chemical non‐equilibrium hypersonic flow is developed based on the concept of energy relaxation method (ERM). The new system of equations obtained are studied using finite volume method with Harten–Lax–van Leer scheme for contact (HLLC). The original HLLC method is modified here to account for additional species and split energy equations. Higher order spatial accuracy is achieved using MUSCL reconstruction of the flow variables with van Albada limiter. The thermal equilibrium is considered for the analysis and the species data are generated using polynomial correlations. The single temperature model of Dunn and Kang is used for chemical relaxation. The computed results for a flow field over a hemispherical cylinder at Mach number of 16.34 obtained using the present solver are found to be promising and computationally (25%) more efficient. The present solver captures physically correct solution as the entropy conditions are satisfied automatically during the computations. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Efficient and Robust Reynolds-Stress Model Computation of Three-Dimensional Compressible Flows

Thepurposeofthiswork isthedevelopmentofanefe cientand robustimplicitmethodology fortheintegration of the three-dimensional compressible Favre ‐Reynolds-averaged Navier ‐Stokes equations with near-wall Reynoldsstressclosure.Thee vemean-e owandseventurbulencetransportequationsarediscretizedinspace,ona structured multiblock grid,using an O(¢x 3)e nitevolumeupwind-biased MUSCL schemewithVanLeere ux vectorsplitting and Van Albada limiters. Time integration is based on an implicit dual-time-stepping procedure with alternating direction implicit subiterations. A particular treatment of the approximate Jacobians used in the subiterations substantiallydiminishesthecomputing timerequirementsoftheimplicitphase,sothatthecomputationaloverhead of the Reynolds-stress seven-equation closure, compared to a two-equation closure, is less than 30% per iteration. For steady e ow computations local time steps are used, for both the subiterations and the time marching, with Courant‐Friedrichs‐Lewynumbersof O(10‐20)fortheO(1‐20)subiterationsneededand O(100‐500)forthetime marching. The robustness of the method is ensured by appropriate positivity, boundedness, and Reynolds-stress realizability constraints. The numerical method is illustrated by computing (and conducting numerical studies ) for a high subsonic (M » 0:7) three-dimensional e ow with large separation and for a transonic three-dimensional shock-wave/boundary-layer interaction.

New two- and three-dimensional non-oscillatory central finite volume methods on staggered Cartesian grids

We present a 3D finite volume generalization of the 1-dimensional Lax–Friedrichs and Nessyahu–Tadmor schemes for hyperbolic equations on Cartesian grids. The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is by-passed thanks to the use of the staggered Lax–Friedrichs scheme, is extended here to a two-step, 3-dimensional non-oscillatory centered scheme in finite volume formulation. Piecewise linear cell interpolants using several van Leer-type limiting techniques to estimate the gradient (van Leer, van Albada, SuperBee, MinMod) lead to a non-oscillatory spatial resolution of order superior to 1. The fact that the expected second-order resolution is not fully attained in 3D is investigated first by considering an alternate dual grid (in 2D), and by using the original van Albada limiter in 3D. Numerical results for a linear advection problem with continuous and discontinuous initial conditions in 2D and 3D show the accuracy and stability of the method. A comparison is made between the 2D Arminjon–Stanescu–Viallon and Jiang–Tadmor formulations and the new one. A new simple projection method is used for the gradients in the new 2D scheme. We also include results for the 3D Euler system (channel with a forward facing step).

Near-Wall k-e Computation of Transonic Turbomachinery Flows with Tip Clearance

Acomputational methodforthenumericalintegration oftheFavre ‐Reynolds averaged, three-dimensionalcompressible Navier ‐Stokes equations in axial turbomachinery, using the Launder ‐Sharma near-wall k‐ turbulence closure, is developed. The mean e ow and turbulence transport equations are discretized using a e nite volume method based on MUSCL Van Leere ux-vector splitting with Van Albada limiters and are integrated in time using a fully coupled, approximately factored, implicit backward Euler method. The resulting scheme is robust and was found stable for local time steps of Courant ‐Friedrichs‐Lewy number (CFL)=20. The computational domain is discretized using a basic H ‐O‐H grid. The tip-clearance gap is discretized using a e ne O-type grid. The radial distribution ofnodeswithin thetip-clearancegap is independent of theblade-rowO grid, and a bufferoverlap grid is used to convey information. Boundary conditions at periodicity boundaries and at domain interfaces are treated using e ve phantom nodes. This procedure ensures stability at high CFL. Results are presented for the NASA 37 rotor, at an operating point near surge. Computations are compared with measurements both for blade-to-blade Mach number contours and pitchwise distributions and for radial distributions downstream of the blades. Results are obtained using three grids of 10 6 , 2 ££ 10 6 , and 3 ££ 10 6 points, with 21, 31, and 41 radial stations within the tip-clearancegap,respectively,demonstratingthatresultsaregridindependent.Comparisonwithmeasurementsis satisfactory, with theexception of pressureratio overestimation due to unsatisfactory prediction ofe ow separation by the turbulence model.

Computation of unsteady nozzle flow resulting from fluctuating back-pressure using euler equations

In this paper is presented a numerical method for the integration of the unsteady two-dimensional Euler equations using a third-order upwind-biased scheme with Van Leer flux-vector-splitting and Van Albada limiters, with MUSCL space-discretization, and an explicit two-stage Runge-Kutta time-integration procedure. The method is applied to the numerical computation of flows in a transonic nozzle with fluctuating back-pressure, and compared with available experimental data. The effects of frequency on the shock-wave response are studied in detail. Despite the use of an inviscid flow model the unsteady pressures are quite satisfactorily predicted over the range of frequencies studied.

An implicit mixed finite-volume-finite-element method for solving 3D turbulent compressible flows

The development of new aeronautic projects require accurate and efficient simulations of compressible flows in complex geometries. It is well known that most flows of interest are at least locally turbulent and that the modelling of this turbulence is critical for the reliability of the computations. A turbulence closure model which is both cheap and reasonably accurate is an essential part of a compressible code. An implicit algorithm to solve the 2D and 3D compressible Navier-Stokes equations on unstructured triangular/tetrahedral grids has been extended to turbulent flows. This numerical scheme is based on second-order finite element-finite volume discretization: the diffusive and source terms of the Navier-Stokes equations are computed using a finite element method, while the other terms are computed with a finite volume method. Finite volume cells are built around each node by means of the medians. The convective fluxes are evaluated with the approximate Riemann solver of Roe coupled with the van Albada limiter. The standard κ-e model has been introduced to take into account turbulence. Implicit integration schemes with efficient numerical methods (CGS, GMRES and various preconditioning techniques) have also been implemented

Near-Wall Reynolds-Stress Three-Dimensional Transonic Flow Computation

A computational method for the Favre-Reynolds-averaged three-dimensional compressible Navier-Stokes equations using near-wall Reynolds-stress closure is described. The near-wall Reynolds-stress closure uses the Launder-Shima formulation for the Reynolds stresses and the Jones-Launder-Sharma modified dissipation (e*) equation. The mean-flow and turbulence-transport equations are discretized using a finite volume method based on MUSCL Van Leer flux-vector-splitting with Van Albada limiters. The mean-flow and turbulence equations are integrated in time using a fully coupled approximately factored implicit backward Euler method. The resulting scheme is robust and achieves optimal convergence with local-time-step Courant-Friedrichs-Lewy = 50. The turbulence closure is validated by comparison with classic flat-plate boundary-layer data. Results are presented for the three-dimensional Delery transonic channel test case and compared with k-e computations. An analysis of the limitations of the closure is attempted, and possible improvements are suggested.

Implicit computation of three-dimensional compressible Navier-Stokes equations using k-epsilon closure

A computational method for the numerical integration of the Favre-Reynolds-averaged three-dimensional compressible Navier-Stokes equations using the Launder-Sharma near-wall k-e turbulence closure is developed. The mean flow and turbulence transport equations are discretized using a finite volume method based on MUSCL Van Leer flux-vector splitting with Van Albada limiters. The mean flow and turbulence equations are integrated in time using a fully coupled approximately factored implicit backward Euler method. The resulting scheme is robust and was found stable for local-time steps with Courant-Friedrichs-Lewy number equal to 50. Higher time steps are possible but not optimal for convergence. Results are presented for the three Delery transonic channel test-cases. Although these test-cases are nominally two dimensional, three-dimensional computations presented quantify the important three-dimensional effects induced by the sidewall boundary layers. Finally computational results are compared with the experiment for a geometrically three-dimensional transonic nozzle.