A physically based flux limiter for QUICK calculations of advective scalar transport

Transient, advective transport of a contaminant into a clean domain will exhibit a moving sharp front that separates contaminated and clean regions. Due to ‘numerical diffusion’—the combined effects of ‘cross‐wind diffusion’ and ‘artificial dispersion’—a numerical solution based on a first‐order (upwind) treatment will smear out the sharp front. The use of higher‐order schemes, e.g. QUICK (quadratic upwinding) reduces the smearing but can introduce non‐physical oscillations in the solution. A common approach to reduce numerical diffusion without oscillations is to use a scheme that blends low‐order and high‐order approximations of the advective transport. Typically, the blending is based on a parameter that measures the local monotonicity in the predicted scalar field. In this paper, an alternative approach is proposed for use in scalar transport problems where physical bounds C_Low?C?C_High on the scalar are known a priori. For this class of problems, the proposed scheme switches from a QUICK approximation to an upwind approximation whenever the predicted upwind nodal value falls outside of the physical range [C_Low, C_High]. On two‐dimensional steady‐state and one‐dimensional transient test problems predictions obtained with the proposed scheme are essentially indistinguishable from those obtained with monotonic flux‐limiter schemes. An analysis of the modified equation explains the observed performance of first‐ and second‐order time‐stepping schemes in predicting the advective transport of a step. In application to the transient two‐dimensional problem of contaminate transport into a streambed, predictions obtained with the proposed flux‐limiter scheme agree with those obtained with a scheme from the literature. Copyright © 2007 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.     

The numerical study of roll‐waves in inclined open channels and solitary wave run‐up

In this paper, we introduce a finite‐volume kinetic BGK scheme and its applications to the study of roll and solitary waves. The current scheme is based on the numerical solution of the gas‐kinetic Bhatnagar–Gross–Krook model in the flux evaluation across each cell interface. An intrinsic connection between the BGK model and time‐dependent, non‐linear, non‐homogeneous shallow‐water equations enables us to solve shallow‐water equations automatically with our kinetic scheme. The analytical solution, experimental measurements, and numerical calculations for problems associated with roll‐waves down an inclined open channel and solitary waves incident on a sloped beach are also presented. Copyright © 2005 John Wiley & Sons, Ltd.     

LES of temporally evolving mixing layers by an eighth‐order filter scheme

An eighth‐order filter method for a wide range of compressible flow speeds (H. C. Yee and B. Sjogreen, Proceedings of ICOSAHOM09, June 22–26, 2009, Trondheim, Norway) is employed for large eddy simulations (LES) of temporally evolving mixing layers (TML) for different convective Mach numbers (M_c) and Reynolds numbers. The high‐order filter method is designed for accurate and efficient simulations of shock‐free compressible turbulence, turbulence with shocklets, and turbulence with strong shocks with minimum tuning of scheme parameters. The value of the M_c considered is for the TML range from the quasi‐incompressible regime to the highly compressible supersonic regime. The three main characteristics of compressible TML (the self‐similarity property, compressibility effects, and the presence of large‐scale structures with shocklets for high M_c) are considered for the LES study. The LES results that used the same scheme parameters for all studied cases agree well with experimental results and published direct numerical simulations (DNS). Published 2012. This article is a US Government work and is in the public domain in the USA.     

Dual-Family Viscous Shock Waves in n Conservation Laws with Application to Multi-Phase Flow in Porous Media

We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study S _{i, j} dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of S _{i, j} shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of S _{i, j} shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing S _{i, j} shocks and classical waves are described. As a physical application, all types of S _{i, j} shocks with i>j are detected and studied in a family of models for multi-phase flow in porous media.     

Computational analysis of dense gas shock tube flow

Nonclassical phenomena associated with the classical dynamics of real gases in a conventional shock tube are studied. A TVD predictor-corrector (TVD-MacCormack) scheme with reflective endwall boundary conditions is used for the one-dimensional Euler equations to simulate the evolution of the wave field of a van der Waals gas. Depending upon the initial conditions of the gas, wave fields are produced that contain nonclassical phenomena such as expansion shocks, composite waves, splitting shocks, etc. In addition, the interactions of waves reflected from the endwalls produce both classical and nonclassical phenomena. Wave field evolution is depicted using plots of the flow variables at specific times and withx-t diagrams.     

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.     

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.     

Fluid forces on a square cylinder in oscillating flows with non‐zero‐mean velocities

The unsteady forces on a square cylinder in sinusoidally oscillating flows with non‐zero‐mean velocities are investigated numerically by using a weakly compressible‐flow method with three‐dimensional large eddy simulations. The major parameters in the analysis are Keulegan–Carpenter number (KC) and the ratio between the amplitude and the mean velocities of the approaching flow (A_R). By varying the values of KC and A_R the resulting drag and lift of the cylinders are analyzed systematically at two selected approaching‐flow attack angles (0 and 22.5^°). In the case of the non‐zero attack angle, results show that both the drag and lift histories can be adequately described by Morison equations. However, Morison equations fail to correctly describing the lift history as the attack angle is zero. In addition, when the ratio of A_R/KC is near the Strouhal number of the bluff‐body flow, the resulting drag is promoted due to the occurrence of resonance. Based on the results of systematic analyses, finally, the mean and inertia force coefficients at the two selected attack angles are presented as functions of KC and A_R based on the Morison relationships. Copyright © 2008 John Wiley & Sons, Ltd.     

Nonlinear stability of discrete shocks for systems of conservation laws

In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m×m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the L ^p-norm for all p 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in L ^p (P 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.     

Improved third‐order weighted essentially nonoscillatory schemes with new smoothness indicators

In this article, we present two improved third‐order weighted essentially nonoscillatory (WENO) schemes for recovering their design‐order near first‐order critical points. The schemes are constructed in the framework of third‐order WENO‐Z scheme. Two new global smoothness indicators, τ_L3 and τ_L4, are devised by a nonlinear combination of local smoothness indicators (IS_k) and reference values (IS_G) based on Lagrangian interpolation polynomial. The performances of the proposed schemes are evaluated on several numerical tests governed by one‐dimensional linear advection equation or one‐ and two‐dimensional Euler equations. Numerical results indicate that the presented schemes provide less dissipation and higher resolution than the original WENO3‐JS and subsequent WENO3‐N scheme.     

A γ‐model BGK scheme for compressible multifluids

We present a γ‐model BGK scheme for the numerical simulation of compressible multifluids. The scheme is based on the incorporation of a conservative γ‐model scheme given in (J. Comput. Phys. 1996; 125 :150–160) into the gas kinetic BGK scheme (J. Comput. Phys. 1993; 109 :53–66, J. Comput. Phys. 1994; 114 :9–17), and is simple to implement. Several numerical examples presented in this paper validate the scheme in the application of compressible multimaterial flows. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimization of Conical Wings in Hypersonic Flow

A method of calculation is presented to determine conical wing shapes that minimize the coefficient of (wave) drag, C _D, for a fixed coefficient of lift, C _L, in steady, hypersonic flow. An optimization problem is considered for the compressive flow underneath wings at a small angle of attack δ and at a high free-stream Mach number M _∞ so that hypersonic small-disturbance (HSD) theory applies. A figure of merit, F=C _D/C _L ^3/2, is computed for each wing using a finite volume discretization of the HSD equations. A set of design variables that determine the shape of the wing is defined and adjusted iteratively to find a shape that minimizes F for a given value of the hypersonic similarity parameter, H= (M _∞δ)⁻², and planform area. Wings with both attached and detached bow shocks are considered. Optimal wings are found for flat delta wings and for a family of caret wings. In the flat-wing case, the optima have detached bow shocks while in the caret-wing case, the optimum has an attached bow shock. An improved drag-to-lift performance is found using the optimization procedure for curved wing shapes. Several optimal designs are found, all with attached bow shocks. Numerical experiments are performed and suggest that these optima are unique. Received 1 May 1998 and accepted 14 October 1998     

Validity ranges of interfacial wave theories in a two-layer fluid system

In the present paper, we endeavor to accomplish a diagram, which demarcates the validity ranges for interfacial wave theories in a two-layer system, to meet the needs of design in ocean engineering. On the basis of the available solutions of periodic and solitary waves, we propose a guideline as principle to identify the validity regions of the interfacial wave theories in terms of wave period T, wave height H, upper layer thickness d ₁, and lower layer thickness d ₂, instead of only one parameter–water depth d as in the water surface wave circumstance. The diagram proposed here happens to be Le Méhauté’s plot for free surface waves if water depth ratio r = d ₁/d ₂ approaches to infinity and the upper layer water density ρ ₁ to zero. On the contrary, the diagram for water surface waves can be used for two-layer interfacial waves if gravity acceleration g in it is replaced by the reduced gravity defined in this study under the condition of σ = (ρ ₂ − ρ ₁)/ρ ₂ → 1.0 and r > 1.0. In the end, several figures of the validity ranges for various interfacial wave theories in the two-layer fluid are given and compared with the results for surface waves. The project supported by the Knowledge Innovation Project of CAS (KJCX-YW-L02), the National 863 Project of China (2006AA09A103-4), China National Oil Corporation in Beijing (CNOOC), and the National Natural Science Foundation of China (10672056).     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

Transonic Shock Problem for the Euler System in a Nozzle

In this paper, we study the well-posedness problem on transonic shocks for steady ideal compressible flows through a two-dimensional slowly varying nozzle with an appropriately given pressure at the exit of the nozzle. This is motivated by the following transonic phenomena in a de Laval nozzle. Given an appropriately large receiver pressure P _r , if the upstream flow remains supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the flow is compressed and slowed down to subsonic speed, and the position and the strength of the shock front are automatically adjusted so that the end pressure at exit becomes P _r , as clearly stated by Courant and Friedrichs [Supersonic flow and shock waves, Interscience Publishers, New York, 1948 (see section 143 and 147)]. The transonic shock front is a free boundary dividing two regions of C ^2,α flow in the nozzle. The full Euler system is hyperbolic upstream where the flow is supersonic, and coupled hyperbolic-elliptic in the downstream region Ω₊ of the nozzle where the flow is subsonic. Based on Bernoulli’s law, we can reformulate the problem by decomposing the 3 × 3 Euler system into a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system on u ₂ with a value given at a point, and an algebraic equation on (ρ, u ₁, u ₂) along a streamline. In terms of this reformulation, we can show the uniqueness of such a transonic shock solution if it exists and the shock front goes through a fixed point. Furthermore, we prove that there is no such transonic shock solution for a class of nozzles with some large pressure given at the exit. This research was supported in part by the Zheng Ge Ru Foundation when Yin Huicheng was visiting The Institute of Mathematical Sciences, The Chinese University of Hong Kong. Xin is supported in part by Hong Kong RGC Earmarked Research Grants CUHK-4028/04P, CUHK-4040/06P, and Central Allocation Grant CA05-06.SC01. Yin is supported in part by NNSF of China and Doctoral Program of NEM of China.     

On Discontinuity Waves in Linear Piezoelectricity

A body composed of a linear piezoelectric medium is considered. It is shown that the condition of local propagation for a singular hypersurface S of any given order r, with r≥1, can be expressed in terms of a suitable acoustic tensor. This tensor does not depend on the order r and coincides with the one used for plane progressive waves in the homogeneous case. Thus, just as in Linear Elasticity, the laws of propagation of such discontinuity waves are the same as those for plane progressive waves. For any r≥1 singular hypersurfaces are characteristic for the linear piezoelectric partial differential equations, whereas for r=0 singular hypersurfaces may be non-characteristic for such equations. A condition is written which characterizes the strong waves of order 0 that are characteristic. For the latter waves the aforementioned acoustic tensor can be used to express the condition of local propagation. This revised version was published online in July 2006 with corrections to the Cover Date.     

A numerical analysis of a class of generalized Boussinesq‐type equations using continuous/discontinuous FEM

An improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time‐dependent varying seabed are included. Thus, high‐order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher‐order models, an extra O(μ²ⁿ⁺²) term (n ∈ ?) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth‐order models. The discretization of the spatial variables is made using continuous P₂ Lagrange elements. A predictor‐corrector scheme with an initialization given by an explicit Runge–Kutta method is also used for the time‐variable integration. Moreover, a CFL‐type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Generalized lattice‐BGK concept for thermal and chemically reacting flows at low Mach numbers

The lattice‐BGK method has been extended by introducing additional, free parameters in the original formulation of the lattice‐BGK methods. The relationship between these parameters and the macroscopic moment equations is analysed by Taylor series and Chapman–Enskog expansion. The parameters are determined from the macroscopic moment equations by comparisons with the governing equations to be modelled. Extensions are presented for the Navier–Stokes equations at low Mach numbers in Cartesian or axisymmetric coordinates with constant or variable density, for scalar convection–diffusion equations and for equations of Poisson type. The generalized lattice‐BGK concept is demonstrated by two applications of chemical engineering. These are the computation of chemically reacting flow through an axisymmetric reactor and of the transport and deposition of particles to filters under the action of different forces. Copyright © 2006 John Wiley & Sons, Ltd.     

A new r‐ratio formulation for TVD schemes for vertex‐centered FVM on an unstructured mesh

The r‐ratio is a parameter that measures the local monotonicity, by which a number of high‐resolution and TVD schemes can be formed. A number of r‐ratio formulations for TVD schemes have been presented over the last few decades to solve the transport equation in shallow waters based on the finite volume method (FVM). However, unlike structured meshes, the coordinate directions are not clearly defined on an unstructured mesh; therefore, some r‐ratio formulations have been established by approximating the solute concentration at virtual nodes, which may be estimated from different assumptions. However, some formulations may introduce either oscillation or diffusion behavior within the vertex‐centered (VC) framework. In this paper, a new r‐ratio formulation, applied to an unstructured grid in the VC framework, is proposed and compared with the traditional r‐ratio formulations. Through seven commonly used benchmark tests, it is shown that the newly proposed r‐ratio formulation obtains better results than the traditional ones with less numerical diffusion and spurious oscillation. Moreover, three commonly used TVD schemes—SUPERBEE, MINMOD, and MUSCL—and two high‐order schemes—SOU and QUICK—are implemented and compared using the new r‐ratio formulation. The new r‐ratio formulation is shown to be sufficiently comprehensive to permit the general implementation of a high‐resolution scheme within the VC framework. Finally, the sensitivity test for different grid types demonstrates the good adaptability of this new r‐ratio formulation. Copyright © 2015 John Wiley & Sons, Ltd.