A numerical method for one‐dimensional compressible multiphase flows on moving meshes

This paper is devoted to the numerical approximation of a hyperbolic non‐equilibrium multiphase flow model with different velocities on moving meshes. Such goal poses several difficulties. The presence of different flow velocities in conjunction with cell velocities poses difficulties for upwinding fluxes. Another issue is related to the presence of non‐conservative terms. To solve these difficulties, the discrete equations method (J. Comput. Phys. 2003; 186 (2):361–396; J. Fluid. Mech. 2003; 495 :283–321; J. Comput. Phys. 2004; 196 :490–538; J. Comput. Phys. 2005; 205 :567–610) is employed and generalized to the context of moving cells. The complementary conservation laws, available for the mixture, are used to determine the velocities of the cells boundaries. With these extensions, an accurate and robust multiphase flow method on moving meshes is obtained and validated over several test problems with exact or experimental solutions. Copyright © 2007 John Wiley & Sons, Ltd.     

Remarks on the numerical solution of the adjoint quasi‐one‐dimensional Euler equations

We examine the numerical solution of the adjoint quasi‐one‐dimensional Euler equations with a central‐difference finite volume scheme with Jameson‐Schmidt‐Turkel (JST) dissipation, for both the continuous and discrete approaches. First, the complete formulations and discretization of the quasi‐one‐dimensional Euler equations and the continuous adjoint equation and its counterpart, the discrete adjoint equation, are reviewed. The differences between the continuous and discrete boundary conditions are also explored. Second, numerical testing is carried out on a symmetric converging–diverging duct under subsonic flow conditions. This analysis reveals that the discrete adjoint scheme, while being manifestly less accurate than the continuous approach, gives nevertheless more accurate flow sensitivities. Copyright © 2011 John Wiley & Sons, Ltd.     

A fully conservative high‐order upwind multi‐moment method using moments in both upwind and downwind cells

We propose a fully conservative high‐order upwind multi‐moment method for the conservation equation. The proposed method is based on a third‐order polynomial interpolation function and semi‐Lagrangian formulation and is a variant of the constrained interpolation profile conservative semi‐Lagrangian scheme with third‐order polynomial function method. The third‐order interpolation function is constructed based on three constraints in the upwind cell (two boundary values and a cell average) and a constraint in the downwind cell (a cell center value). The proposed method shows fourth‐order accuracy in a benchmark problem (sine wave propagation). We also propose a less oscillatory formulation of the proposed method. The less oscillatory formulation can minimize numerical oscillations. These methods were validated through scalar transport problems, and compressible flow problems (shock tube and 2D explosion problems). Copyright © 2016 John Wiley & Sons, Ltd.     

An eigenvector‐based linear reconstruction scheme for the shallow‐water equations on two‐dimensional unstructured meshes

This paper presents a new approach to MUSCL reconstruction for solving the shallow‐water equations on two‐dimensional unstructured meshes. The approach takes advantage of the particular structure of the shallow‐water equations. Indeed, their hyperbolic nature allows the flow variables to be expressed as a linear combination of the eigenvectors of the system. The particularity of the shallow‐water equations is that the coefficients of this combination only depend upon the water depth. Reconstructing only the water depth with second‐order accuracy and using only a first‐order reconstruction for the flow velocity proves to be as accurate as the classical MUSCL approach. The method also appears to be more robust in cases with very strong depth gradients such as the propagation of a wave on a dry bed. Since only one reconstruction is needed (against three reconstructions in the MUSCL approach) the EVR method is shown to be 1.4–5 times as fast as the classical MUSCL scheme, depending on the computational application. Copyright © 2006 John Wiley & Sons, Ltd.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element method for the one‐dimensional extended Boussinesq equations

A new finite element method for Nwogu's (O. Nwogu, ASCE J. Waterw., Port, Coast., Ocean Eng., 119 , 618–638 (1993)) one‐dimensional extended Boussinesq equations is presented using a linear element spatial discretisation method coupled with a sophisticated adaptive time integration package. The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J. Waterw., Port, Coast., Ocean Eng., 121 , 251–261 (1995)) by considering the truncation error at a node. Numerical tests with solitary and regular waves propagating in variable depth environments are compared with theoretical and experimental data. The accuracy of the results confirms the analytical prediction and shows that the new approach competes well with existing finite difference methods. The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured triangular meshes in two dimensions. This latter case is discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

The improved space–time conservation element and solution element scheme for two‐dimensional dam‐break flow simulation

A new numerical scheme, namely space–time conservation element and solution element (CE/SE) method, has been used for the solution of the two‐dimensional (2D) dam‐break problem. Distinguishing from the well‐established traditional numerical methods (such as characteristics, finite difference, finite element, and finite‐volume methods), the CE/SE scheme has many non‐traditional features in both concept and methodology: space and time are treated in a unified way, which is the most important characteristic for the CE/SE method; the CEs and SEs are introduced, both local and global flux conservations in space and time rather than space only are enforced; an explicit scheme with a stagger grid is adopted. Furthermore, this scheme is robust and easy to implement. In this paper, an improved CE/SE scheme is extended to solve the 2D shallow water equations with the source terms, which usually plays a critical role in dam‐break flows. To demonstrate the accuracy, robustness and efficiency of the improved CE/SE method, both 1D and 2D dam‐break problems are simulated numerically, and the results are consistent with either the analytical solutions or experimental results. Copyright © 2011 John Wiley & Sons, Ltd.     

Preserving bounded and conservative solutions of transport in one‐dimensional shallow‐water flow with upwind numerical schemes: Application to fertigation and solute transport in rivers

This work intends to show that conservative upwind schemes based on a separate discretization of the scalar solute transport from the shallow‐water equations are unable to preserve uniform solute profiles in situations of one‐dimensional unsteady subcritical flow. However, the coupled discretization of the system is proved to lead to the correct solution in first‐order approximations. This work is also devoted to show that, when using a coupled discretization, a careful definition of the flux limiter function in second‐order TVD schemes is required in order to preserve uniform solute profiles. The work shows that, in cases of subcritical irregular flow, the coupled discretization is necessary but nevertheless not sufficient to ensure concentration distributions free from oscillations and a method to avoid these oscillations is proposed. Examples of steady and unsteady flows in test cases, river and irrigation are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

A discrete forcing method dedicated to moving bodies in two‐phase flow

The numerical simulation of interaction between structures and two‐phase flows is a major concern for many industrial applications. To address this challenge, the motion of structures has to be tracked accurately. In this work, a discrete forcing method based on a porous medium approach is proposed to follow a nondeformable rigid body with an imposed velocity by using a finite‐volume Navier‐Stokes solver code dedicated to multiphase flows and based on a two‐fluid approach. To deal with the action reaction principle at the solid wall interfaces in a conservative way, a porosity is introduced allowing to locate the solid and insuring no diffusion of the fluid‐structure interface. The volumetric fraction equilibrium is adapted to this novelty. Mass and momentum balance equations are formulated on a fixed Cartesian grid. Interface tracking is addressed in detail going from the definition of the porosity to the changes in the discretization of the momentum balance equation. This so‐called time‐ and space‐dependent porosity method is then validated by using analytical and elementary test cases.     

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.     

Evaluation of some high‐order shock capturing schemes for direct numerical simulation of unsteady two‐dimensional free flows

The present study addresses the capability of a large set of shock‐capturing schemes to recover the basic interactions between acoustic, vorticity and entropy in a direct numerical simulation (DNS) framework. The basic dispersive and dissipative errors are first evaluated by considering the advection of a Taylor vortex in a uniform flow. Two transonic cases are also considered. The first one consists of the interaction between a temperature spot and a weak shock. This test emphasizes the capability of the schemes to recover the production of vorticity through the baroclinic process. The second one consists of the interaction of a Taylor vortex with a weak shock, corresponding to the framework of the linear theory of Ribner. The main process in play here is the production of an acoustic wave. The results obtained by using essentially non‐oscillatory (ENO), total variation diminishing (TVD), compact‐TVD and MUSCL schemes are compared with those obtained by means of a sixth‐order accurate Hermitian scheme, considered as reference. The results are as follows; the ENO schemes agree pretty well with the reference scheme. The second‐order accurate Upwind‐TVD scheme exhibits a strong numerical diffusion, while the MUSCL scheme behavior is very sensitive to the value on the parameter β in the limiter function minmod. The compact‐TVD schemes do not yield improvement over the standard TVD schemes. Copyright © 2000 John Wiley & Sons, Ltd.     

A numerical study on a Cartesian-based body-fitted adaptive grid method

ABSTRACT

A hybrid Cartesian-based body-fitted adaptive grid method for compressible Navier–Stokes equations is implemented and investigated. In this method, the body-fitted structured grids are generated around the geometries, and the left regions are filled with Cartesian grids. To transfer the data between the different grids, the donor cell searching technique is adopted. An unstructured data-based finite volume update procedure is used, and least squares method is suggested to retain the second order in the overlap region. The moving shock waves with different speeds and vortex passing through the interfaces of the hybrid Cartesian grid are used to explore the accuracy and conservation. A new technique is presented to deal with the non-physical stagnation of slowly moving shock wave around the interface of grid. Numerical examples are presented to demonstrate the results. The three-dimensional extension has also been shown by a benchmark problem.     

A numerical study of turbulent flow over a two‐dimensional hill

Calculations of mean velocities and Reynolds stresses are reported for the recirculating flow established in the wake of two‐dimensional polynomial‐shaped obstacles that are symmetrical about a vertical axis and mounted in the water channel downstream of a fully developed channel flow for Re=6×10⁴. The study involves calculations of mean and fluctuating flow properties in the streamwise and spanwise directions and include comparisons with experimental data [Almeida GP, Durão DFG, Heitor MV. Wake flows behind two‐dimensional model hills. Experimental Thermal and Fluid Science 1993; 7: 87–101] for flow around a single obstacle with data resulting from the interaction of consecutive obstacles, using two versions of the low‐Reynolds number differential second‐moment (DSM) closure model. The results include analysis of the turbulent stresses in local flow co‐ordinates and reveal flow structure qualitatively similar to that found in other turbulent flows with a reattachment zone. It is found that the standard isotropization of production model (IPM), based on that proposed by Gibson and Launder [Ground effects on pressure fluctuations in the atmospheric boundary layer. Journal of Fluid Mechanics 1978; 86(3): 191–511], with the incorporation of the wall reflection model of Craft and Launder [New wall‐reflection model applied to the turbulent impinging jet. AIAA Journal 1992; 32(12): 2970–2972] predicts the mean velocities quite well, but underestimates the size of the recirculation region and turbulent quantities in the shear layer. These inadequacies are circumvented by adopting a new cubic Reynolds stress closure scheme based on that more recently developed by Craft and Launder [A Reynolds stress closure designed for complex geometries. International Journal of Heat and Fluid Flow 1996; 17: 245–254] which satisfies the two component limit (TCL) of turbulence. In this model the geometry‐specific quantities, such as the wall‐normal vector or wall distance, are replaced by invariant dimensionless gradient indicators. Also, the model captures the diverse behaviour of the different components of the stress dissipation, ε_ij, near the wall and uses a novel decomposition for the fluctuating pressure terms. Copyright © 2001 John Wiley & Sons, Ltd.     

Application of differential quadrature method to solve entry flow of viscoelastic second‐order fluid

The entry flow of viscoelastic second‐order fluid between two parallel plates is discussed. The governing equations of vorticity and the streamfunction are expanded with respect to a small parameter that characterizes the elasticity of the fluid by means of the standard perturbation method. By using the differential quadrature method with only a few grid points, high‐accurate numerical solutions are obtained. The numerical results show a lot of the features of a viscoelastic second‐order fluid. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical method for the evaluation of hydrodynamic forces of translating bodies under a free surface

This paper presents a numerical method to evaluate the hydrodynamic forces of translating bodies under a free surface. Both steady and unsteady problems are considered. Analytical and numerical studies are carried out based on the Havelock wave‐source function and the integral equation method. Two main problems arising inherently in the proposed solution method are overcome in order to facilitate the numerical implementation. The first lies in evaluating the Havelock function, which involves integrals with highly oscillatory kernels. Particular integration contours leading to non‐oscillatory integrands are derived a priori so that the integrals can be evaluated efficiently. The second problem lies in evaluating singular kernels in the boundary integral equation. The corresponding non‐singular formulation is derived using some theorems of potential theory, including the Gauss flux theorem and the property related to the equipotential body. The subsequent formulation is amenable to the solution by directly using the standard quadrature formulas without taking another special treatment. This paper also attempts to enhance the computational efficiency by presenting an interpolation method used to evaluate matrix elements, which are ascribed to a discretization procedure. In addition to the steady case, numerical examples consist of cases involving a submerged prolate spheroid, which is originally idle and then suddenly moves with a constant speed and a constant acceleration. Also systematically studied is the variation of hydrodynamic forces acting on the spheroid for various Froude numbers and submergence depths. Copyright © 2000 John Wiley & Sons, Ltd.     

A 5‐equation,transient, hyperbolic, 1‐dimensional model for slug capturing in pipes

A novel numerical scheme for slug capturing in pipes using a 1‐dimensional transient hyperbolic 5‐equation 2‐fluid model is presented. Previous work has shown that 1‐dimensional 2‐fluid models are able to capture slug flow automatically. In this work, a similar approach is further developed using a new numerical scheme, applied to a hyperbolic 5‐equation 2‐fluid model. Starting from a finite volume discretisation of a 5‐equation 2‐fluid hyperbolic model and adding appropriate closure relations, a second‐order code is implemented and applied to air‐water flows in horizontal pipes, simulating the 2‐phase to 1‐phase flow process. The code is evaluated in some common standard test cases. A slug capturing application is also discussed. We show, in an air/water horizontal pipe, slug initiation, growth, and development. Moreover, a grid refinement analysis is performed showing that the method is grid independent and we show the code capability to take into account eventual surface tension effects, through the instantaneous pressure relaxation process. Finally, a prediction of flow regime transitions is shown and compared with a well‐known theoretical flow pattern map in addition to a preliminary comparison of computed slug characteristics against well‐known empirical correlations.     

Optimization of the ADER‐DG method in GPU applied to linear hyperbolic PDEs

We optimized the Arbitrary accuracy DErivatives Riemann problem (ADER) ‐ Discontinuous Galerkin (DG) numerical method using the CUDA‐C language to run the code in a graphic processing unit (GPU). We focus on solving linear hyperbolic partial–differential equations where the method can be expressed as a combination of precomputed matrix multiplications becoming a good candidate to be used on the GPU hardware. Moreover, the method is arbitrarily high order involving intensive work on local data, a property that is also beneficial for the target hardware. We compare our GPU implementation against CPU versions of the same method observing similar convergence properties up to a threshold where the error remains fixed. This behavior is in agreement with the CPU version, but the threshold is slightly larger than in the CPU case. We also observe a big difference when considering single and double precisions where in the first case, the threshold error is significantly larger. Finally, we did observe a speed‐up factor in computational time that depends on the order of the method and the size of the problem. In the best case, our novel GPU implementation runs 23 times faster than the CPU version. We used three partial–differential equation to test the code considering the linear advection equation, the seismic wave equation, and the linear shallow water equation, all of them considering variable coefficients. Copyright © 2015 John Wiley & Sons, Ltd.     

A non‐iterative numerical approach for two‐dimensional viscous flow problems governed by the Falker–Skan equation

In this paper, a non‐iterative numerical approach for two‐dimensional laminar viscous flow over a semi‐infinite flat plane, governed by the Falkner–Skan equation is proposed. This approach can solve the non‐linear Falkner–Skan equation without any iteration and verifies that a direct numerical approach could be proposed even for non‐linear problems. Furthermore, this approach can also provide a family of iterative formulae, so that it logically contains traditional iterative techniques. Copyright © 2001 John Wiley & Sons, Ltd.     

GENSMAC3D: a numerical method for solving unsteady three‐dimensional free surface flows

A numerical method for solving three‐dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier–Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user‐friendly input data file is employed to allow almost any arbitrary three‐dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Modelling of two‐dimensional laminar flow using finite element method

In this paper, a Galerkin weighted residual finite element numerical solution method, with velocity material time derivative discretisation, is applied to solve for a classical fluid mechanics system of partial differential equations modelling two‐dimensional stationary incompressible Newtonian fluid flow. Classical examples of driven cavity laminar flow and laminar flow past a cylinder are presented. Numerical results are compared with data found in the literature. Copyright © 1999 John Wiley & Sons, Ltd.