Upper entropy bounds for transient forced convection

This article develops upper bounds for total entropy associated with convective heat transfer and transient fluid motion in an enclosure. Entropy production includes both friction and thermal irreversibilities due to fluid mixing in the enclosure. An integral formulation of entropy transport is developed in terms of the temperature excess (difference between the point-wise and spatially averaged temperature). The thermal irreversibility of entropy production is written in terms of the squared temperature excess. In this way, an upper entropy bound can be derived with respect to geometrical parameters and initial temperatures. Furthermore, this entropy bound is minimized by re-formulating the minimization problem in terms of a standard form of eigenvalue problem. Several example problems are considered and a spectral method is used to solve the governing energy equation. Theoretical predictions are compared successfully against numerical simulations for cases involving both Neumann and Dirichlet boundary conditions.     

An immersed boundary finite difference method for LES of flow around bluff shapes

A three‐dimensional numerical model using large eddy simulation (LES) technique and incorporating the immersed boundary (IMB) concept has been developed to compute flow around bluff shapes. A fractional step finite differences method with rectilinear non‐uniform collocated grid is employed to solve the governing equations. Bluff shapes are treated in the IMB method by introducing artificial force terms into the momentum equations. Second‐order accurate interpolation schemes for all sorts of grid points adjacent to the immersed boundary have been developed to determine the velocities and pressure at these points. To enforce continuity, the methods of imposition of pressure boundary condition and addition of mass source/sink terms are tested. It has been found that imposing suitable pressure boundary condition (zero normal gradient) can effectively reproduce the correct pressure distribution and enforce mass conservation around a bluff shape. The present model has been verified and applied to simulate flow around bluff shapes: (1) a square cylinder and (2) the Tsing Ma suspension bridge deck section model. Complex flow phenomena such as flow separation and vortex shedding are reproduced and the drag coefficient, lift coefficient, and pressure coefficient are calculated and analyzed. Good agreement between the numerical results and the experimental data are obtained. The model is proven to be an efficient tool for flow simulation around bluff bodies in time varying flows. Copyright © 2004 John Wiley & Sons, Ltd.     

Experimental investigation of a maximum entropy assumption for acceleration terms within a poly‐disperse moment framework

Moment transport methods are being developed to model poly‐dispersed multiphase flows by transporting statistical moments of the particle size–velocity joint probability density function (JPDF). A common feature of these methods is the requirement to reproduce or approximate the form of the JPDF from the transported moments for calculation of body force terms and other source terms. This paper examines the application of a maximum entropy technique against phase Doppler anemometry data sets from an electrostatically charged kerosene spray and also an automotive pressure swirl atomizer. An assessment of which moments are required to reproduce the JPDFs using a maximum entropy assumption to a sufficient level of accuracy is made. It is found that it is possible to reproduce the JPDFs to a high level of accuracy using a large number of moments; however, this incurs large computational overheads. If the moments to be transported are chosen on the basis of physical reasoning (such as the relationship between size and velocity due to drag) it is possible to reduce the number of moments to those which would be conserved via balance equations. This permits an approximation to the JPDF commensurate with the closure level of the moment transport method and thus the closure model method is naturally scalable with the degree of information from available conservation equations. Copyright © 2008 John Wiley & Sons, Ltd.     

Entropy dissipation scheme and minimums‐increase‐and‐maximums‐decrease slope limiter

In this paper, we continue to study the entropy dissipation scheme developed in former. We start with a numerical study of the scheme without the entropy dissipation term on the linear advection equation, which shows that the scheme is stable and numerical dissipation and numerical dispersion free for smooth solutions. However, the numerical results for discontinuous solutions show nonlinear instabilities near jump discontinuities. This is because the scheme enforces two related conservation properties in the computation. With this study, we design a so‐called ‘minimums‐increase‐and‐maximums‐decrease’ slope limiter in the reconstruction step of the scheme and delete the entropy dissipation in the linear fields and reduce the entropy dissipation terms in the nonlinear fields. Numerical experiments show improvements of the designed scheme compared with the results presented in former. However, the minimums‐increase‐and‐maximums‐decrease limiter is still not perfect yet, and better slope limiters are still sought. Copyright © 2011 John Wiley & Sons, Ltd.     

Viscous regularization of the full set of nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations

A viscous regularization technique, based on the local entropy residual, was proposed by Delchini et al. (2015) to stabilize the nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations using an artificial viscosity technique. This viscous regularization is modulated by the local entropy production and is consistent with the entropy minimum principle. However, Delchini et al. (2015) only based their work on the hyperbolic parts of the Grey Radiation‐Hydrodynamic equations and thus omitted the relaxation and diffusion terms present in the material energy and radiation energy equations. Here, we extend the theoretical grounds for the method and derive an entropy minimum principle for the full set of nonequilibrium‐diffusion Grey Radiation‐Hydrodynamic equations. This further strengthens the applicability of the entropy viscosity method as a stabilization technique for radiation‐hydrodynamic shock simulations. Radiative shock calculations using constant and temperature‐dependent opacities are compared against semi‐analytical reference solutions, and we present a procedure to perform spatial convergence studies of such simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

Energy relaxation method for chemical non‐equilibrium flow computations

In 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 & Sons, Ltd.     

Calculation of curved open channel flow using physical curvilinear non‐orthogonal co‐ordinates

There are two main difficulties in numerical simulation calculations using FD/FV method for the flows in real rivers. Firstly, the boundaries are very complex and secondly, the generated grid is usually very non‐uniform locally. Some numerical models in this field solve the first difficulty by the use of physical curvilinear orthogonal co‐ordinates. However, it is very difficult to generate an orthogonal grid for real rivers and the orthogonal restriction often forces the grid to be over concentrated where high resolution is not required. Recently, more and more models solve the first difficulty by the use of generalized curvilinear co‐ordinates (ξ,η). The governing equations are expressed in a covariant or contra‐variant form in terms of generalized curvilinearco‐ordinates (ξ,η). However, some studies in real rivers indicate that this kind of method has some undesirable mesh sensitivities. Sharp differences in adjacent mesh size may easily lead to a calculation stability problem oreven a false simulation result. Both approaches used presently have their own disadvantages in solving the two difficulties that exist in real rivers. In this paper, the authors present a method for two‐dimensional shallow water flow calculations to solve both of the main difficulties, by formulating the governing equations in a physical form in terms of physical curvilinear non‐orthogonal co‐ordinates (s,n). Derivation of the governing equations is explained, and two numerical examples are employed to demonstrate that the presented method is applicable to non‐orthogonal and significantly non‐uniform grids. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical solution for the second‐order wave interaction with porous structures

This study mathematically formulates the fluid field of a water‐wave interaction with a porous structure as a two‐dimensional, non‐linear boundary value problem (bvp) in terms of a generalized velocity potential. The non‐linear bvp is reformulated into an infinite set of linear bvps of ascending order by Stokes perturbation technique, with wave steepness as the perturbation parameter. Only the first‐ and second‐order linear bvps are retained in this study. Each linear bvp is transformed into a boundary integral equation. In addition, the boundary element method (BEM) with linear elements is developed and applied to solve the first‐ and second‐order integral equations. The first‐ and second‐order wave profiles, reflection and transmission coefficients, and the amplitude ratio of the second‐order components are computed as well. The numerical results correlate well with previous analytical and experimental results. Numerical results demonstrate that the second‐order component can be neglected for a deep water‐wave and may become significant for an intermediate depth wave. Copyright © 1999 John Wiley & Sons, Ltd.     

Mathematical Modeling of Slurry Infiltration and Particle Dispersion in Saturated Sand

This paper proposes a new computational method for the analysis of slurry infiltration in saturated sand considering the relationship among soil deformation, slurry seepage and particle dispersion. The nonlinear governing equations for slurry infiltration are derived, and the corresponding variational principles based on time increment are established. The finite element method is employed to solve the problem. The computational results are validated with the reported test data, which shows that this method is much better than the traditional Herzig method in predicting the particle deposition. The proposed method is demonstrated through an example of slurry infiltration in slurry trench.     

Shock‐capturing with natural high‐frequency oscillations

This paper explores the potential of a newly developed conjugate filter oscillation reduction (CFOR) scheme for shock‐capturing under the influence of natural high‐frequency oscillations. The conjugate low‐ and high‐pass filters are constructed based on the principle of the discrete singular convolution (DSC), a local spectral method. The accuracy and resolution of the DSC basic algorithm are accessed with a one‐dimensional advection equation. Two Euler systems, the advection of an isotropic vortex flow and the interaction of shock–entropy wave are utilized to demonstrate the utility of the CFOR scheme. Computational accuracy and order of approximation are examined and compared with the literature. Some of the best numerical results are obtained for the shock–entropy wave interaction. Numerical experiments indicate that the CFOR scheme is stable, conservative and reliable for the numerical simulation of hyperbolic conservation laws. Copyright © 2003 John Wiley & Sons, Ltd.     

A theoretical Taylor–Galerkin model for first‐order hyperbolic equation

This paper presents a class of Taylor–Galerkin (TG) finite‐element models for solving the first‐order hyperbolic equation which admits discontinuities. Five parameters are introduced for purposes of controlling stability, monotonicity and accuracy. In this paper, the total variation diminishing concept and the theory of M‐matrix are applied to construct a monotonic TG model for capturing discontinuities. To avoid making the scheme overly diffusive, we apply a flux‐corrected transport (FCT) technique of Boris and Book to overcome the difficulty with anti‐diffusive flux. In smooth flow regions, our strategyof developing the temporal and spatial high‐order TG finite‐element model is based on modified equation analysis. In regions where discontinuity is encountered, we resort to two dispersively more accurate models to make the prediction accuracy as high as that obtained in smooth cases. These models are developed using the entropy‐increasing principle and the theory of group velocity. Guided by this theory, a slower group velocity should be used ahead of the shock. To avoid a train of post‐shocks, free parameters should be chosen properly to obtain a group velocity which takes on a larger value than the exact phase velocity. In this paper, we also apply the entropy‐increasing principle to determine free parameters introduced in the finite‐element model. Under the entropy‐increasing requirement, it is mandatory that coefficients of the even and odd derivative terms shown in the modified equation should change signs alternatively in order to avoid non‐physical wiggles. Several benchmark problems have been investigated to confirm the integrity of these proposed characteristic models. Copyright © 2003 John Wiley & Sons, Ltd.     

An entropy fixed cell‐centered Lagrangian scheme

On the basis of the work [P.‐H. Maire, R. Abgrall, J. Breil, J. Ovadia, SIAM J. Sci. Comput. 29 (2007), 1781–1824], we present an entropy fixed cell‐centered Lagrangian scheme for solving the Euler equations of compressible gas dynamics. The scheme uses the fully Lagrangian form of the gas dynamics equations, in which the primary variables are cell‐centered. And using the nodal solver, we obtain the nodal viscous‐velocity, viscous‐pressures, antidissipation velocity, and antidissipation pressures of each node. The final nodal velocity is computed as a weighted sum of viscous‐velocity and antidissipation velocity, so do nodal pressures, whereas these weights are calculated through the total entropy conservation for isentropic flows. Consequently, the constructed scheme is conservative in mass, momentum, and energy; preserves entropy for isentropic flows, and satisfies a local entropy inequality for nonisentropic flows. One‐ and two‐dimensional numerical examples are presented to demonstrate theoretical analysis and performance of the scheme in terms of accuracy and robustness.Copyright © 2013 John Wiley & Sons, Ltd.     

A depth‐integrated non‐hydrostatic finite element model for wave propagation

This paper presents a two‐dimensional finite element model for simulating dynamic propagation of weakly dispersive waves. Shallow water equations including extra non‐hydrostatic pressure terms and a depth‐integrated vertical momentum equation are solved with linear distributions assumed in the vertical direction for the non‐hydrostatic pressure and the vertical velocity. The model is developed based on the platform of a finite element model, CCHE2D. A physically bounded upwind scheme for the advection term discretization is developed, and the quasi second‐order differential operators of this scheme result in no oscillation and little numerical diffusion. The depth‐integrated non‐hydrostatic wave model is solved semi‐implicitly: the provisional flow velocity is first implicitly solved using the shallow water equations; the non‐hydrostatic pressure, which is implicitly obtained by ensuring a divergence‐free velocity field, is used to correct the provisional velocity, and finally the depth‐integrated continuity equation is explicitly solved to satisfy global mass conservation. The developed wave model is verified by an analytical solution and validated by laboratory experiments, and the computed results show that the wave model can properly handle linear and nonlinear dispersive waves, wave shoaling, diffraction, refraction and focusing. Copyright © 2013 John Wiley & Sons, Ltd.     

Momentum‐based approximation of incompressible multiphase fluid flows

We introduce a time stepping technique using the momentum as dependent variable to solve incompressible multiphase problems. The main advantage of this approach is that the mass matrix is time‐independent making this technique suitable for spectral methods. A level set method is applied to reconstruct the fluid properties such as density. We also introduce a stabilization method using an entropy‐viscosity technique and a compression technique to limit the flattening of the level set function. We extend our algorithm to immiscible conducting fluids by coupling the incompressible Navier‐Stokes and the Maxwell equations. We validate the proposed algorithm against analytical and manufactured solutions. Results on test cases such as Newton's bucket problem and a variation thereof are provided. Surface tension effects are tested on benchmark problems involving bubbles. A numerical simulation of a phenomenon related to the industrial production of aluminium is presented at the end of the paper.     

A novel efficient hybrid DSMC–dynamic collision limiter algorithm for multiscale transitional flows

A new 2D parallel multispecies polyatomic particle–based hybrid flow solver is developed by coupling the Direct Simulation Monte Carlo (DSMC) method with a novel Dynamic Collision Limiter (DCL) approach to solve multiscale transitional flows. The hybrid DSMC‐DCL solver can solve nonequilibrium multiscale flows with length scales ranging from continuum to rarefied. The DCL method, developed in this work, dynamically assigns different number of collisions in cells, which is based on the local value of K‐S parameter such that the number of collisions per time step is limited in near‐equilibrium flow regions. Present hybrid solver uses the Kolmogorov‐Smirnov statistical test as the continuum breakdown parameter, based on which, the solution domain is decomposed into near‐equilibrium and nonequilibrium flow regions. Direct Simulation Monte Carlo is used where nonequilibrium flow regions are encountered, while the DCL method is used where flow regions are found to be in near‐equilibrium state. In this work, we have studied hypersonic flow of nitrogen over a blunt body with an aerospike and supersonic flow of argon through a micronozzle. The results obtained by the hybrid DSMC‐DCL solver are compared and shown to agree well with the experimental data and with those obtained from DSMC, with significant savings in the computational cost.     

Analysis of a second‐order upwind method for the simulation of solute transport in 2D shallow water flow

A two‐dimensional model for the simulation of solute transport by convection and diffusion into shallow water flow over variable bottom is presented. It is based on a finite volume method over triangular unstructured grids. A first‐order upwind technique, a second order in space and time and an extended first‐order method are applied to solve the non‐diffusive terms in both the flow and solute equations and a centred implicit discretization is applied to the diffusion terms. The stability constraints are studied and the form to avoid oscillatory results in the solute concentration in the presence of complex flow situations is detailed. Some comparisons are carried out in order to show the performance in terms of accuracy of the different options. Copyright © 2007 John Wiley & Sons, Ltd.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

Towards entropy detection of anomalous mass and momentum exchange in incompressible fluid flow

An entropy‐based approach is presented for assessment of computational accuracy in incompressible flow problems. It is shown that computational entropy can serve as an effective parameter in detecting erroneous or anomalous predictions of mass and momentum transport in the flow field. In the present paper, the fluid flow equations and second law of thermodynamics are discretized by a Galerkin finite‐element method with linear, isoparametric triangular elements. It is shown that a weighted entropy residual is closely related to truncation error; this relationship is examined in an application problem involving incompressible flow through a converging channel. In particular, regions exhibiting anomalous flow behaviour, such as under‐predicted velocities, appear together with analogous trends in the weighted entropy residual. It is anticipated that entropy‐based error detection can provide important steps towards improved accuracy in computational fluid flow. Copyright © 2002 John Wiley & Sons, Ltd.     

Solution of magnetohydrodynamic flow in a rectangular duct by Chebyshev collocation method

In this study, matrix representation of the Chebyshev collocation method for partial differential equation has been represented and applied to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Numerical solution of velocity and induced magnetic field is obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. The Chebyshev collocation method is used with a reasonable number of collocations points, which gives accurate numerical solutions of the MHD flow problem. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H≤1000. Copyright © 2010 John Wiley & Sons, Ltd.     

A theory of mixtures

This paper is concerned with a dynamical theory of mixtures, composed of n reactive constituents in relative motion to each other. The theory is developed in terms of the constituent ingredients using a balance of energy and an entropy production inequality for each constituent of the mixture, together with invariance requirements under superposed rigid body motions of the whole mixture. The balance of energy and the entropy production inequality for each of the constituents, which include contributions arising from interactions, combine to yield a single energy equation and a single entropy production inequality in terms of the ingredients of the mixture as a whole; the relations between the thermodynamical variables of the mixture and those of its constituents depend, in general, on the past history of the temperature and the kinematic variables. Full thermodynamical restrictions are deduced, and the theory is applied to the special case of a mixture of two ideal fluids.