A locally DIV‐free projection scheme for incompressible flows based on non‐conforming finite elements

We present a projection scheme whose end‐of‐step velocity is locally pointwise divergence free, using a continuous ?₁ approximation for the velocity in the momentum equation, a first‐order Crouzeix–Raviart approximation at the projection step, and a ?₀ approximation for the pressure in both steps. The analysis of the scheme is done only for grids that guarantee the existence of a divergence free conforming ?₁ interpolant for the velocity. Optimal estimates for the velocity error in L²‐ and H¹‐norms are deduced. The numerical results demonstrate that these estimates should also hold on grids on which the continuous ?₁ approximation for the velocity locks. Since the end‐of‐step velocity is locally solenoidal, the scheme is recommendable for problems requiring good mass conservation. Copyright © 2005 John Wiley & Sons, Ltd.     

A spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for conjugate heat transfer applications

We present a spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for fluid–solid conjugate heat transfer applications. Using the discrete Boltzmann equation, we propose a numerical scheme for conjugate heat transfer applications on unstructured, non‐uniform grids. We employ a double‐distribution thermal lattice Boltzmann model to resolve flows with variable Prandtl (Pr) number. Based upon its finite element heritage, the spectral‐element discontinuous Galerkin discretization provides an effective means to model and investigate thermal transport in applications with complex geometries. Our solutions are represented by the tensor product basis of the one‐dimensional Legendre–Lagrange interpolation polynomials. A high‐order discretization is employed on body‐conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Thermal and hydrodynamic bounce‐back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. As a result, our scheme does not require tedious extrapolation at the boundaries, which may cause loss of mass conservation. We compare solutions of the proposed scheme with an analytical solution for a solid–solid conjugate heat transfer problem in a 2D annulus and illustrate the capture of temperature continuities across interfaces for conductivity ratio γ > 1. We also investigate the effect of Reynolds (Re) and Grashof (Gr) number on the conjugate heat transfer between a heat‐generating solid and a surrounding fluid. Steady‐state results are presented for Re = 5?40 and Gr = 10⁵?10⁶. In each case, we discuss the effect of Re and Gr on the heat flux (i.e. Nusselt number Nu) at the fluid–solid interface. Our results are validated against previous studies that employ finite‐difference and continuous spectral‐element methods to solve the Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive finite element method for an optimal control problem of Stokes flow with L2‐norm state constraint

An adaptive finite element approximation for an optimal control problem of the Stokes flow with an L²‐norm state constraint is proposed. To produce good adaptive meshes, the a posteriori error estimates are discussed. The equivalent residual‐type a posteriori error estimators of the H ¹‐error of state and L²‐error of control are given, which are suitable to carry out the adaptive multi‐mesh finite element approximation. Some numerical experiments are performed to illustrate the efficiency of the a posteriori estimators. Copyright © 2011 John Wiley & Sons, Ltd.     

An improved ghost‐cell immersed boundary method for compressible flow simulations

This study presents an improved ghost‐cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work, ghost cells are mirrored through the boundary described using a level‐set method to farther image points, incorporating a higher‐order extra/interpolation scheme for the ghost‐cell values. A sensor is introduced to deal with image points near the discontinuities in the flow field. Adaptive mesh refinement is used to improve the representation of the geometry efficiently in the Cartesian grid system. The improved ghost‐cell method is validated against four test cases: (a) double Mach reflections on a ramp, (b) smooth Prandtl–Meyer expansion flows, (c) supersonic flows in a wind tunnel with a forward‐facing step, and (d) supersonic flows over a circular cylinder. It is demonstrated that the improved ghost‐cell method can reach the accuracy of second order in L¹ norm and higher than first order in L^∞ norm. Direct comparisons against the cut‐cell method demonstrate that the improved ghost‐cell method is almost equally accurate with better efficiency for boundary representation in high‐fidelity compressible flow simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

Spatial accuracy and performance of a mixed‐order,explicit multi‐stage method for unsteady flows

We assess the spatial accuracy and performance of a mixed‐order, explicit multi‐stage method in which an inexpensive low‐order scheme is used for the initial stages, and a more expensive high‐order scheme is used for the final stage only. Compared with the use of a high‐order scheme for all stages, we observe that the mixed‐order scheme achieves comparable accuracy and convergence while providing a speed‐up of a factor of two on mesh sizes of O(10⁶ ? 10⁷) tetrahedron. For calculations with significant adaptive mesh refinement, a more modest speed‐up of 30% is obtained. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Acoustic upwinding for sub‐ and super‐sonic turbulent channel flow at low Reynolds number

A recently developed asymmetric implicit fifth‐order scheme with acoustic upwinding for the spatial discretization for the characteristic waves is applied to the fully compressible, viscous and non‐stationary Navier–Stokes equations for sub‐ and super‐sonic, mildly turbulent, channel flow (Re_τ=360). For a Mach number of 0.1, results are presented for uniform (32³, 64³ and 128³) and non‐uniform (expanding wall‐normal, 32³ and 64³) grids and compared to the (incompressible) reference solution found in (J. Fluid. Mech. 1987; 177 :133–166). The results for uniform grids on 128³ and 64³ nodes show high resemblance with the reference solution. Expanding grids are applied on 64³‐ and 32³‐node grids. The capability of the proposed technique to solve compressible flow is first demonstrated by increasing the Mach number to 0.3, 0.6 and 0.9 for isentropic flow on the uniform 64³‐grid. Next, the flow speed is increased to Ma=2. The results for the isothermal‐wall supersonic flows give very good agreement with known literature results. The velocity field, the temperature and their fluctuations are well resolved. This means that in all presented (sub‐ and super‐sonic) cases, the combination of acoustic upwinding and the asymmetric high‐order scheme provides sufficient high wave‐number damping and low wave‐number accuracy to give numerically stable and accurate results. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical simulation of two‐dimensional Kelvin–Helmholtz instability using weakly compressible smoothed particle hydrodynamics

The growth of the Kelvin–Helmholtz instability generated at the interface between two ideal gases is studied by means of a Smoothed Particle Hydrodynamics (SPH) scheme suitable for multi‐fluids. The SPH scheme is based on the continuity equation approach where the densities of SPH particles are evolved during the simulation, in combination with a momentum equation previously proposed in the literature. A series of simulations were carried out to investigate the influence of viscosity, smoothing, the thickness of density and velocity transition layers. It was found that the effective viscosity of the presented results are strongly dependent on the artificial viscosity parameter α_AV, with a linear dependence of 0.15. The utilisation of a viscosity switch is found to significantly reduce the spurious viscosity dependence to 1.68 × 10^?4 and generated qualitatively improved behaviour for inviscid fluids. The linear growth rate in the numerical solutions is found to be in satisfactory agreement with analytical expectations, with an average relative error 〈η_smooth〉=13%. In addition, the role played by velocity and density transition layers is also in general agreement with the analytical theory, except for the sharp‐velocity, finite‐density gradient cases where the larger growth rate than the classical growth rate is expected. We argue the inherited smoothing properties of the velocity field during the simulations are responsible for causing this discrepancy. Finally, the SPH results are in good agreement for finite velocity and density gradient scenarios, where an average relative error of 〈η_smooth〉=11.5% is found in our work. Copyright © 2015 John Wiley & Sons, Ltd.     

A discontinuous Galerkin‐based sharp‐interface method to simulate three‐dimensional compressible two‐phase flow

A numerical method for the simulation of compressible two‐phase flows is presented in this paper. The sharp‐interface approach consists of several components: a discontinuous Galerkin solver for compressible fluid flow, a level‐set tracking algorithm to follow the movement of the interface and a coupling of both by a ghost‐fluid approach with use of a local Riemann solver at the interface. There are several novel techniques used: the discontinuous Galerkin scheme allows locally a subcell resolution to enhance the interface resolution and an interior finite volume Total Variation Diminishing (TVD) approximation at the interface. The level‐set equation is solved by the same discontinuous Galerkin scheme. To obtain a very good approximation of the interface curvature, the accuracy of the level‐set field is improved and smoothed by an additional P_NP_M‐reconstruction. The capabilities of the method for the simulation of compressible two‐phase flow are demonstrated for a droplet at equilibrium, an oscillating ellipsoidal droplet, and a shock‐droplet interaction problem at Mach 3. Copyright © 2015 John Wiley & Sons, Ltd.     

An application of the finite difference‐based lattice Boltzmann model to simulating flow‐induced noise

This paper presents a novel approach to simulate aerodynamically generated sounds by modifying the finite difference‐based lattice BGK compressible fluid model for the purpose of speeding up the calculation and also stabilizing the numerical scheme. With the model, aerodynamic sounds generated by a uniform flow around a two‐dimensional circular cylinder at Re = 150 are simulated. The third‐order‐accurate up‐wind scheme is used for the spatial derivatives, and the second‐order‐accurate Runge–Kutta method is applied for the time marching. The results show that we successively capture very small acoustic pressure fluctuations, with the same frequency of the Karman vortex street, much smaller than the whole pressure fluctuation around a circular cylinder. The propagation velocity of the acoustic waves shows that the points of peak pressure are biased upstream owing to the Doppler effect in the uniform flow. For the downstream, on the other hand, it is faster. It is also apparent that the amplitude of sound pressure is proportional to r^?1/2, r being the distance from the centre of the circular cylinder. Moreover, the edgetone generated by a two‐dimensional jet impinging on a wedge to predict the frequency characteristics of the discrete oscillations of a jet‐edge feedback cycle is investigated. The jet is chosen long enough to guarantee the parabolic velocity profile of the jet at the outlet, and the edge is of an angle of α = 23°. At a stand‐off distance w, the edge is inserted along the centreline of the jet, and a sinuous instability wave with real frequency is assumed to be created in the vicinity of the nozzle exit and to propagate towards the downstream. We have succeeded in capturing small pressure fluctuations resulting from periodic oscillation of jet around the edge. Copyright © 2006 John Wiley & Sons, Ltd.     

A further work on multi‐phase two‐fluid approach for compressible multi‐phase flows

This paper is to continue our previous work Niu (Int. J. Numer. Meth. Fluids 2001; 36 :351–371) on solving a two‐fluid model for compressible liquid–gas flows using the AUSMDV scheme. We first propose a pressure–velocity‐based diffusion term originally derived from AUSMDV scheme Wada and Liou (SIAM J. Sci. Comput. 1997; 18 (3):633—657) to enhance its robustness. The scheme can be applied to gas and liquid fluids universally. We then employ the stratified flow model Chang and Liou (J. Comput. Physics 2007; 225 :240–873) for spatial discretization. By defining the fluids in different regions and introducing inter‐phasic force on cell boundary, the stratified flow model allows the conservation laws to be applied on each phase, and therefore, it is able to capture fluid discontinuities, such as the fluid interfaces and shock waves, accurately. Several benchmark tests are studied, including the Ransom's Faucet problem, 1D air–water shock tube problems, 2D shock‐water column and 2D shock‐bubble interaction problems. The results indicate that the incorporation of the new dissipation into AUSM⁺‐up scheme and the stratified flow model is simple, accurate and robust enough for the compressible multi‐phase flows. Copyright © 2008 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.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Highly accurate solutions of the bifurcation structure of mixed‐convection heat transfer using spectral method

This paper is concerned with producing highly accurate solution and bifurcation structure using the pseudo‐spectral method for the two‐dimensional pressure‐driven flow through a horizontal duct of a square cross‐section that is heated by a uniform flux in the axial direction with a uniform temperature on the periphery. Two approaches are presented. In one approach, the streamwise vorticity, streamwise momentum and energy equations are solved for the stream function, axial velocity, and temperature. In the second approach, the streamwise vorticity and a combination of the energy and momentum equations are solved for stream function and temperature only. While the second approach solves less number of equations than the first approach, a grid sensitivity analysis has shown no distinct advantage of one method over the other. The overall solution structure composed of two symmetric and four asymmetric branches in the range of Grashof number (Gr) of 0–2 × 10⁶ for a Prandtl number (Pr) of 0.73 has been computed using the first approach. The computed structure is comparable to that found by Nandakumar and Weinitschke (1991) using a finite difference scheme for Grashof numbers in the range of 0–1×10⁶. The stability properties of some solution branches; however, are different. In particular, the two‐cell structure of the isolated symmetric branch that has been found to be unstable by the study of Nandakumar and Weinitschke is found to be stable by the current study. Copyright © 2002 John Wiley & Sons, Ltd.     

A Petrov–Galerkin finite element model for one‐dimensional fully non‐linear and weakly dispersive wave propagation

A new finite element method is presented to solve one‐dimensional depth‐integrated equations for fully non‐linear and weakly dispersive waves. For spatial integration, the Petrov–Galerkin weighted residual method is used. The weak forms of the governing equations are arranged in such a way that the shape functions can be piecewise linear, while the weighting functions are piecewise cubic with C²‐continuity. For the time integration an implicit predictor–corrector iterative scheme is employed. Within the framework of linear theory, the accuracy of the scheme is discussed by considering the truncation error at a node. The leading truncation error is fourth‐order in terms of element size. Numerical stability of the scheme is also investigated. If the Courant number is less than 0.5, the scheme is unconditionally stable. By increasing the number of iterations and/or decreasing the element size, the stability characteristics are improved significantly. Both Dirichlet boundary condition (for incident waves) and Neumann boundary condition (for a reflecting wall) are implemented. Several examples are presented to demonstrate the range of applicabilities and the accuracy of the model. Copyright © 2001 John Wiley & Sons, Ltd.     

Fourth‐order exponential finite difference methods for boundary value problems of convective diffusion type

Methods based on exponential finite difference approximations of h⁴ accuracy are developed to solve one and two‐dimensional convection–diffusion type differential equations with constant and variable convection coefficients. In the one‐dimensional case, the numerical scheme developed uses three points. For the two‐dimensional case, even though nine points are used, the successive line overrelaxation approach with alternating direction implicit procedure enables us to deal with tri‐diagonal systems. The methods are applied on a number of linear and non‐linear problems, mostly with large first derivative terms, in particular, fluid flow problems with boundary layers. Better accuracy is obtained in all the problems, compared with the available results in the literature. Application of an exponential scheme with a non‐uniform mesh is also illustrated. The h⁴ accuracy of the schemes is also computationally demonstrated. Copyright © 2001 John Wiley & Sons, Ltd.     

High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

In this paper, we present a class of high‐order accurate cell‐centered arbitrary Lagrangian–Eulerian (ALE) one‐step ADER weighted essentially non‐oscillatory (WENO) finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two‐dimensional unstructured triangular meshes. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space–time Galerkin predictor allows the schemes to be high order accurate also in time by using an element‐local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained either (i) as an arithmetic average among the states surrounding the node, as suggested by Cheng and Shu, or (ii) as a solution of multiple one‐dimensional half‐Riemann problems around a vertex, as suggested by Maire, or (iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional Harten–Lax–van Leer Riemann solver recently proposed by Balsara et al. Once the vertex velocity and thus the new node location have been determined by the node solver, the local mesh motion is then constructed by straight edges connecting the vertex positions at the old time level tⁿ with the new ones at the next time level t^{n + 1}. If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unnecessary, as the ALE fluxes already properly take into account the rezoned geometry. In this sense, our scheme falls into the category of direct ALE methods. Furthermore, the geometric conservation law is satisfied by the scheme by construction. We apply the high‐order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic magnetohydrodynamic equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration. Copyright © 2014 John Wiley & Sons, Ltd.     

Assessment of the finite volume method applied to the v2 − f model

The objective of this paper is to assess the accuracy of low‐order finite volume (FV) methods applied to the v² ? f turbulence model of Durbin (Theoret. Comput. Fluid Dyn. 1991; 3 :1–13) in the near vicinity of solid walls. We are not (like many others) concerned with the stability of solvers ‐ the topic at hand is simply whether the mathematical properties of the v² ? f model can be captured by the given, widespread, numerical method. The v² ? f model is integrated all the way up to solid walls, where steep gradients in turbulence parameters are observed. The full resolution of wall gradients imposes quite high demands on the numerical schemes and it is not evident that common (second order) FV codes can fully cope with such demands. The v² ? f model is studied in a statistically one‐dimensional, fully developed channel flow where we compare FV schemes with a highly accurate spectral element reference implementation. For the FV method a higher‐order face interpolation scheme, using Lagrange interpolation polynomials up to arbitrary order, is described. It is concluded that a regular second‐order FV scheme cannot give an accurate representation of all model parameters, independent of mesh density. To match the spectral element solution an extended source treatment (we use three‐point Gauss–Lobatto quadrature), as well as a higher‐order discretization of diffusion is required. Furthermore, it is found that the location of the first internal node need to be well within y⁺=1. Copyright © 2009 John Wiley & Sons, Ltd.     

Improved MPS method with variable‐size particles

Using variable‐size particles in the moving particle semi‐implicit method (MPS) could lead to inaccurate predictions and/or numerical instability. In this paper, a variable‐size particle moving particle semi‐implicit method (VSP‐MPS) scheme is proposed for the MPS method to achieve more reliable simulations with variable‐size particles. To improve stability and accuracy, a new gradient model is developed based on a previously developed MPS scheme that requires no surface detection MPS. The dynamic particle coalescing and splitting algorithm is revised to achieve dynamic multi‐resolution. A cubic spline function with additional function is employed as the kernel function. The effectiveness of the VSP‐MPS method is demonstrated by three verification examples, that is, a hydrostatic pressure problem, a complicated free surface flow problem with large deformation, and a dynamic impact problem. The new VSP‐MPS scheme with variable‐size particles is found to have balanced efficiency and accuracy that is suitable for simulating large systems with complex flow patterns. Copyright © 2015 John Wiley & Sons, Ltd.     

A parametric finite‐difference method for shallow sea waves

This paper presents a parametric finite‐difference scheme concerning the numerical solution of the one‐dimensional Boussinesq‐type set of equations, as they were introduced by Peregrine (J. Fluid Mech. 1967; 27 (4)) in the case of waves relatively long with small amplitudes in water of varying depth. The proposed method, which can be considered as a generalization of the Crank‐Nickolson method, aims to investigate alternative approaches in order to improve the accuracy of analogous methods known from bibliography. The resulting linear finite‐difference scheme, which is analysed for stability using the Fourier method, has been applied successfully to a problem used by Beji and Battjes (Coastal Eng. 1994; 23 : 1–16), giving numerical results which are in good agreement with the corresponding results given by MIKE 21 BW (User Guide. In: MIKE 21, Wave Modelling, User Guide. 2002; 271–392) developed by DHI Software. Copyright © 2006 John Wiley & Sons, Ltd.     

A high‐order Petrov–Galerkin finite element method for the classical Boussinesq wave model

A high‐order Petrov–Galerkin finite element scheme is presented to solve the one‐dimensional depth‐integrated classical Boussinesq equations for weakly non‐linear and weakly dispersive waves. Finite elements are used both in the space and the time domains. The shape functions are bilinear in space–time, whereas the weighting functions are linear in space and quadratic in time, with C⁰‐continuity. Dispersion correction and a highly selective dissipation mechanism are introduced through additional streamline upwind terms in the weighting functions. An implicit, conditionally stable, one‐step predictor–corrector time integration scheme results. The accuracy and stability of the non‐linear discrete equations are investigated by means of a local Taylor series expansion. A linear spectral analysis is used for the full characterization of the predictor–corrector inner iterations. Based on the order of the analytical terms of the Boussinesq model and on the order of the numerical discretization, it is concluded that the scheme is fourth‐order accurate in terms of phase velocity. The dissipation term is third order only affecting the shortest wavelengths. A numerical convergence analysis showed a second‐order convergence rate in terms of both element size and time step. Four numerical experiments are addressed and their results are compared with analytical solutions or experimental data available in the literature: the propagation of a solitary wave, the oscillation of a flat bottom closed basin, the oscillation of a non‐flat bottom closed basin, and the propagation of a periodic wave over a submerged bar. Copyright © 2008 John Wiley & Sons, Ltd.