Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical assessment of a shock‐detecting sensor for low dissipative high‐order simulation of shock–vortex interactions

Considering the importance of high‐order schemes implementation for the simulation of shock‐containing turbulent flows, the present work involves the assessment of a shock‐detecting sensor for filtering of high‐order compact finite‐difference schemes for simulation of this type of flows. To accomplish this, a sensor that controls the amount of numerical dissipation is applied to a sixth‐order compact scheme as well as a fourth‐order two‐register Runge–Kutta method for numerical simulation of various cases including inviscid and viscous shock–vortex and shock–mixing‐layer interactions. Detailed study is performed to investigate the performance of the sensor, that is, the effect of control parameters employed in the sensor are investigated in the long‐time integration. In addition, the effects of nonlinear weighting factors controlling the value of the second‐order and high‐order filters in fine and coarse non‐uniform grids are investigated. The results indicate the accuracy of the nonlinear filter along with the promising performance of the shock‐detecting sensor, which would pave the way for future simulations of turbulent flows containing shocks. Copyright © 2014 John Wiley & Sons, Ltd.     

A third‐order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time‐accurate solution of the compressible Navier–Stokes equations

A space and time third‐order discontinuous Galerkin method based on a Hermite weighted essentially non‐oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower‐upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third‐order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third‐order discontinous Galerkin methods, and less computing time than the lower‐order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Computational study of curved shock–vortex interaction

The interaction between a curved shock wave and a compressible vortex is numerically investigated. The investigation concentrates on the local deformation of the shock structure due to the shock–vortex interaction. The essentially non‐oscillatory (ENO) scheme is used to solve the unsteady two‐dimensional Euler equations. A curved shock wave is obtained by the diffraction of an initially planar shock wave around a right‐angled corner and then allowed to interact with a strong compressible vortex superimposed on the flow. The same vortex affects the shock wave differently depending on the placement of the vortex because of the varying strength of the shock wave. This effect could range from a non‐symmetric deformation of the shock wave to a local disruption in the shock structure depending on the strength of the shock wave in the interaction region. This process leading to a local disruption in the shock structure is analyzed in detail. It is shown that such a disruption in the shock structure can be predicted by simple one‐dimensional considerations. Copyright © 1999 John Wiley & Sons, Ltd.     

Low‐cost implicit schemes for all‐speed flows on unstructured meshes

Matrix‐free implicit treatments are now commonly used for computing compressible flow problems: a reduced cost per iteration and low‐memory requirements are their most attractive features. This paper explains how it is possible to preserve these features for all‐speed flows, in spite of the use of a low‐Mach preconditioning matrix. The proposed approach exploits a particular property of a widely used low‐Mach preconditioner proposed by Turkel. Its efficiency is demonstrated on some steady and unsteady applications. Copyright © 2008 John Wiley & Sons, Ltd.     

An anti‐diffusive volume of fluid method for interfacial fluid flows

A volume of fluid (VOF) method is developed combining a first‐order limited downwind scheme with higher order accurate schemes. The method is characterized by retaining a sharp fluid interface and a reduction in numerical diffusion near the interface, but avoids complicated geometrical reconstruction as occurs in most volume tracing algorithms. To demonstrate the accuracy and robustness of the method, a selection of numerical experiments are presented involving a pure advection problem, a water wave impact caused by a dam breaking and liquid sloshing in a partially filled tank. Copyright © 2011 John Wiley & Sons, Ltd.     

Time‐linearized time‐harmonic 3‐D Navier–Stokes shock‐capturing schemes

In the present paper, a numerical method for the computation of time‐harmonic flows, using the time‐linearized compressible Reynolds‐averaged Navier–Stokes equations is developed and validated. The method is based on the linearization of the discretized nonlinear equations. The convective fluxes are discretized using an O(Δx) MUSCL scheme with van Leer flux‐vector‐splitting. Unsteady perturbations of the turbulent stresses are linearized using a frozen‐turbulence‐Reynolds‐number hypothesis, to approximate eddy‐viscosity perturbations. The resulting linear system is solved using a pseudo‐time‐marching implicit ADI‐AF (alternating‐directions‐implicit approximate‐factorization) procedure with local pseudo‐time‐steps, corresponding to a matrix‐successive‐underrelaxation procedure. The stability issues associated with the pseudo‐time‐marching solution of the time‐linearized Navier–Stokes equations are discussed. Comparison of computations with measurements and with time‐nonlinear computations for 3‐D shock‐wave oscillation in a square duct, for various back‐pressure fluctuation frequencies (180, 80, 20 and 10 Hz), assesses the shock‐capturing capability of the time‐linearized scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

A high‐resolution method for compressible two‐fluid flows and simulation of three‐dimensional shock–bubble interactions

A high‐resolution method is developed to capture the material interfaces of compressible two‐fluid flows in multiple dimensions. A fluid mixture model system with single velocity and pressure is used, and viscous effect can also be taken into account. A consistent thermodynamic law based on the assumption of pressure equilibrium is employed to describe the thermodynamic behaviors of the pure fluids and mixture of two components. The splitting and unsplit Eulerian formulations of piecewise parabolic method are extended to numerically integrate the hyperbolic part of the model system, whereas the system of diffusion equations is solved using an explicit, central difference scheme. The block‐structured adaptive mesh refinement (AMR) capability is built in the hydrodynamic code to locally improve grid resolution. The resulting method is verified to be at least second‐order accurate in space. Numerical results show that the discontinuities, particularly contact discontinuities, can be resolved sharply. The use of AMR allows flow features at disparate scales to be resolved sufficiently. In addition, three‐dimensional shock–bubble interactions are simulated to investigate effects of Mach number on bubble evolution. The flow structures including those peculiar to three‐dimensional bubble are resolved correctly, and some physical phenomena with increasing Mach number are reported. Copyright © 2012 John Wiley & Sons, Ltd.     

Third‐order‐accurate semi‐implicit Runge–Kutta scheme for incompressible Navier–Stokes equations

A semi‐implicit three‐step Runge–Kutta scheme for the unsteady incompressible Navier–Stokes equations with third‐order accuracy in time is presented. The higher order of accuracy as compared to the existing semi‐implicit Runge–Kutta schemes is achieved due to one additional inversion of the implicit operator I‐τγL, which requires inversion of tridiagonal matrices when using approximate factorization method. No additional solution of the pressure‐Poisson equation or evaluation of Navier–Stokes operator is needed. The scheme is supplied with a local error estimation and time‐step control algorithm. The temporal third‐order accuracy of the scheme is proved analytically and ascertained by analysing both local and global errors in a numerical example. Copyright © 2005 John Wiley & Sons, Ltd.     

Robustness of the hybrid DRP‐WENO scheme for shock flow computations

This paper describes a new variant of hybrid scheme that is constructed by a wave‐capturing scheme and a nonoscillatory scheme for flow computations in the presence of shocks. The improved fifth‐order upwind weighted essentially nonoscillatory scheme is chosen to be conjugated with the seven‐point dispersion‐relation‐preserving scheme by means of an adaptive switch function of grid‐point type. The new hybrid scheme can achieve a better resolution than the hybrid scheme which is based on the classical weighted essentially scheme. Ami Harten's multiresolution analysis algorithm is applied to density field for detecting discontinuities and setting point values of the switch function adaptively. Moreover, the tenth‐order central filter is applied in smooth part of the flow field for damping dispersion errors. This scheme can promote overall computational efficiency and yield oscillation‐free results in shock flows. The resolution properties and robustness of the new hybrid scheme are tested in both 1D and 2D linear and nonlinear cases. It performs well for computing flow problems with rich structures of weak/strong shocks and large/small vortices, such as the shock‐boundary layer interaction problem in a shock tube, which illustrates that it is very robust and accurate for direct numerical simulation of gas‐dynamics flows. Copyright © 2011 John Wiley & Sons, Ltd.     

A conservative semi‐implicit method for coupled surface–subsurface flows in regional scale

A semi‐implicit method for coupled surface–subsurface flows in regional scale is proposed and analyzed. The flow domain is assumed to have a small vertical scale as compared with the horizontal extents. Thus, after hydrostatic approximation, the simplified governing equations are derived from the Reynolds averaged Navier–Stokes equations for the surface flow and from the Darcy's law for the subsurface flow. A conservative free‐surface equation is derived from a vertical integral of the incompressibility condition and extends to the whole water column including both, the surface and the subsurface, wet domains. Numerically, the horizontal domain is covered by an unstructured orthogonal grid that may include subgrid specifications. Along the vertical direction a simple z‐layer discretization is adopted. Semi‐implicit finite difference equations for velocities and a finite volume approximation for the free‐surface equation are derived in such a fashion that, after simple manipulation, the resulting discrete free‐surface equation yields a single, well‐posed, mildly nonlinear system. This system is efficiently solved by a nested Newton‐type iterative method that yields simultaneously the pressure and a non‐negative fluid volume throughout the computational grid. The time‐step size is not restricted by stability conditions dictated by friction or surface wave speed. The resulting algorithm is simple, extremely efficient, and very accurate. Exact mass conservation is assured also in presence of wetting and drying dynamics, in pressurized flow conditions, and during free‐surface transition through the interface. A few examples illustrate the model applicability and demonstrate the effectiveness of the proposed algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

This paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

In this paper we present a class of semi‐discretization finite difference schemes for solving the transient convection–diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection–diffusion (CD) equation to the inhomogeneous steady convection–diffusion‐reaction (CDR) equation after using different time‐stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one‐dimensional framework. For the sake of increasing accuracy, the exact solution for the one‐dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one‐dimensional problem. Development of the proposed time‐stepping schemes is rooted in the Taylor series expansion. All higher‐order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection–diffusion‐reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

High‐order shock capturing schemes for turbulence calculations

This work investigates high‐order central compact methods for simulating turbulent supersonic flows that include shock waves. Several different types of previously proposed characteristic filters, including total variation diminishing, monotone upstream‐centered scheme for conservation laws, and weighted essentially non‐oscillatory filters, are investigated in this study. Similar to the traditional shock capturing schemes, these filters can eliminate the numerical instability caused by large gradients in flow fields, but they also improve efficiency compared with classical shock‐capturing schemes. Adding the nonlinear dissipation part of a classical shock‐capturing scheme to a central scheme makes the method suitable for incorporation into any existing central‐based high‐order subsonic code. The amount of numerical dissipation to add is sensed by means of the artificial compression method switch. In order to improve the performance of the characteristic filters, we propose a hybrid approach to minimize the dissipation added by the characteristic filter. Through several numerical experiments (including a shock/density wave interaction, a shock/vortex interaction, and a shock/mixing layer interaction) we show that our hybrid approach works better than the original method, and can be used for future turbulent flow simulations that include shocks. Copyright © 2009 John Wiley & Sons, Ltd.     

Computation of internal high-speed separated flow with modified B–L and J–K models

The existence of shock–turbulent boundary layer interactions lead to very complicated flow phenomena and pose a challenge for numerical simulation. In this paper, two turbulence models, the Baldwin–Lomax (B–L) model and the Johnson–King (J–K) model, which were originally developed for simple external flow simulation, are modified to model complex high-speed internal separated flows. The full Navier–Stokes solver used in this paper is based on a cell-centered finite volume method and multistepping time marching scheme. Both implicit residual smoothing and local time stepping techniques are incorporated to accelerate the convergence rate. To ensure the numerical stability with the present explicit scheme, a point-implicit treatment to the source term in the ordinary differential equation (ODE) of the J–K model has been developed and has proved to be very effective in modeling such a complex flow. An arc-bump channel flow case has been studied. Comparisons of computed results with experimental data show that the present solver, with the modified turbulence models, predicts the shock and the flow separation very well. The J–K model is found to predict the size of the separation bubble with a higher accuracy. © 1998 John Wiley & Sons, Ltd.     

Analysis of super compact finite difference method and application to simulation of vortex–shock interaction

Turbulence and aeroacoustic noise high‐order accurate schemes are required, and preferred, for solving complex flow fields with multi‐scale structures. In this paper a super compact finite difference method (SCFDM) is presented, the accuracy is analysed and the method is compared with a sixth‐order traditional and compact finite difference approximation. The comparison shows that the sixth‐order accurate super compact method has higher resolving efficiency. The sixth‐order super compact method, with a three‐stage Runge–Kutta method for approximation of the compressible Navier–Stokes equations, is used to solve the complex flow structures induced by vortex–shock interactions. The basic nature of the near‐field sound generated by interaction is studied. Copyright © 2001 John Wiley & Sons, Ltd.     

Comparative study of implicit and subgrid‐scale model large‐eddy simulation techniques for low‐Reynolds number airfoil applications

A computational study of a high‐fidelity, implicit large‐eddy simulation (ILES) technique with and without the use of the dynamic Smagorinsky subgrid‐scale (SGS) model is conducted to examine the contributions of the SGS model on solutions of transitional flow over the SD7003 airfoil section. ILES without an SGS model has been shown in the past to produce comparable and sometimes favorable results to traditional SGS‐based large‐eddy simulation (LES) when applied to canonical turbulent flows. This paper evaluates the necessity of the SGS model for low‐Reynolds number airfoil applications to affirm the use of ILES without SGS‐modeling for a broader class of problems such as those pertaining to micro air vehicles and low‐pressure turbines. It is determined that the addition of the dynamic Smagorinsky model does not significantly affect the time‐mean flow or statistical quantities measured around the airfoil section for the spatial resolutions and Reynolds numbers examined in this study. Additionally, the robustness and reduced computational cost of ILES without the SGS model demonstrates the attractiveness of ILES as an alternative to traditional LES. Published 2012. This article is a US Government work and is in the public domain in the USA.     

The anti‐dissipative,non‐monotone behavior of Petrov–Galerkin upwinding

The Petrov–Galerkin method has been developed with the primary goal of damping spurious oscillations near discontinuities in advection dominated flows. For time‐dependent problems, the typical Petrov–Galerkin method is based on the minimization of the dispersion error and the simultaneous selective addition of dissipation. This optimal design helps to dampen the oscillations prevalent near discontinuities in standard Bubnov–Galerkin solutions. However, it is demonstrated that when the Courant number is less than 1, the Petrov–Galerkin method actually amplifies undershoots at the base of discontinuities. This is shown in an heuristic manner, and is demonstrated with numerical experiments with the scalar advection and Richards' equations. A discussion of monotonicity preservation as a design criterion, as opposed to phase or amplitude error minimization, is also presented. The Petrov–Galerkin method is further linked to the high‐resolution, total variation diminishing (TVD) finite volume method in order to obtain a monotonicity preserving Petrov–Galerkin method.     

Simulation of unsteady turbulent flows around moving aerofoils using the pseudo‐time method

The pseudo‐time formulation of Jameson has facilitated the use of numerical methods for unsteady flows, these methods have proved successful for steady flows. The formulation uses iterations through pseudo‐time to arrive at the next real time approximation. This iteration can be used in a straightforward manner to remove sequencing errors introduced when solving mean flow equations together with another set of differential equations (e.g. two‐equation turbulence models or structural equations). The current paper discusses the accuracy and efficiency advantages of removing the sequencing error and the effect that building extra equations into the pseudo‐time iteration has on its convergence characteristics. Test cases used are for the turbulent flow around pitching and ramping aerofoils. The performance of an implicit method for solving the pseudo‐steady state problem is also assessed. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of incompressible two‐phase flows with a Boussinesq–Scriven interface stress tensor

We consider the numerical simulation of a three‐dimensional two‐phase incompressible flow with a viscous interface. The simulation is based on a sharp interface Navier–Stokes model and the Boussinesq–Scriven constitutive law for the interface viscous stress tensor. In the recent paper [Soft Matter 7, 7797–7804, 2011], a model problem with a spherical droplet in a Stokes Poiseuille flow with a Boussinesq–Scriven law for the surface viscosity has been analyzed. In that paper, relations for the droplet migration velocity are derived. We relate the results obtained with our numerical solver for the two‐phase Navier–Stokes model to these theoretical relations. Copyright © 2013 John Wiley & Sons, Ltd.