Stable Interface Conditions for Discontinuous Galerkin Approximations of Navier-Stokes Equations

A study of boundary and interface conditions for Discontinuous Galerkin approximations of fluid flow equations is undertaken in this paper. While the interface flux for the inviscid case is usually computed by approximate Riemann solvers, most discretizations of the Navier-Stokes equations use an average of the viscous fluxes from neighboring elements. The paper presents a methodology for constructing a set of stable boundary/interface conditions that can be thought of as “viscous” Riemann solvers and are compatible with the inviscid limit.     

Boundary control of linearized Saint-Venant equations oscillating modes

The paper investigates the control of oscillating modes occurring in open-channels, due to the reflection of propagating waves on the boundaries. These modes are well represented by linearized Saint-Venant equations, a set of hyperbolic partial differential equations which describe the dynamics of one-dimensional open-channel flow around a given stationary regime. We use a distributed transfer function approach to compute a dynamic boundary controller that cancels the oscillating modes over all the canal pool. This result is recovered with a Riemann invariants approach in the case of a frictionless horizontal canal pool. The effect of a proportional boundary control on the poles of the transfer matrix is then characterized by a root locus, and we derive an asymptotic result for high frequencies closed-loop poles.     

A pressure-enthalpy finite element model for simulating hydrothermal reservoirs

A numerical model is presented for simulating single or two-phase flow and energy transport in hydrothermal reservoirs. The model is formulated via two non-linear equations for fluid pressure and enthalpy. Both equations are solved simultaneously using a new finite element technique which employs asymmetric weighting functions to overcome numerical oscillation. Non-linearity is treated by a modified Newton-Raphson scheme which takes into account derivative discontinuities in the non-linear coefficients. This scheme also treats unknown flux boundary conditions inplicitly, thus allowing larger time steps to be taken without inducing instability. The proposed model is applied to two test examples involving one-dimensional flow in both hot water and steam dominated reservoirs. Results indicate that the numerical technique presented is efficient and the model can be used to simulate both types of reservoirs.     

Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography

Many natural terrains have complicated surface topography. The simulation of steep-fronted flows that occur after heavy rainfall flash floods or as inundation from dyke breaches is usually based on the non-linear shallow water equations in hyperbolic conservation form. Particular challenges to numerical modellers are posed by the need to balance correctly the flux gradient and source terms in Godunov-type finite volume shock-capturing schemes and by the moving wet-dry boundary as the flood rises or falls. This paper presents a Godunov-type shallow flow solver on adaptive quadtree grids aimed at simulating flood flows as they travel over natural terrain. By choosing the stage and discharge as dependent variables in the hyperbolic non-linear shallow water equations, a new deviatoric formulation is derived that mathematically balances the flux gradient and source terms in cases where there are wet-dry fronts. The new formulation is more general in application than previous a priori approaches. Three benchmark tests are used to validate the solver, and include steady flow over a submerged hump, flow disturbances propagating over an elliptical-shaped hump, and free surface sloshing motions in a vessel with a parabolic bed. The model is also used to simulate the propagation of a flood due to a dam break over an initially dry floodplain containing three humps.     

Development of PML absorbing boundary conditions for computational aeroacoustics: A progress review

Recent advances in the development of perfectly matched layer (PML) as absorbing boundary conditions for computational aeroacoustics are reviewed. The PML methodology is presented as a complex change of variables. In this context, the importance of a proper space-time transformation in the PML technique for Euler equations is emphasized. A unified approach for the derivation of PML equations is offered that involves three essential steps. The three-step approach is illustrated in details for the PML of linear and non-linear Euler equations. Numerical examples are also given that include non-reflecting boundary conditions for a ducted channel flow and mixing layer roll-up vortices.     

A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients

In this work, a model for shallow water flows that accounts for the effects of horizontal density fluctuations is presented and derived. While the density is advected by the flow, a two-way feedback between the density gradients and the time evolution of the fluid is ensured through the pressure and source terms in the momentum equations. The model can be derived by vertically averaging the Euler equations while still allowing for density fluctuations in horizontal directions. The approach differs from multi-layer shallow water flows where two or more layers are considered, each of them having their own depth, velocity and constant density. A Roe-type upwind scheme is developed and the Roe matrices are computed systematically by going from the conservative to the quasi-linear form at a discrete level. Properties of the model are analyzed. The system is hyperbolic with two shock-wave families and a contact discontinuity associated to interfaces of regions with density jumps. This new field is degenerate with pressure and velocity as the corresponding Riemann invariants. We show that in some parameter regimes numerically recognizing such invariants across contact discontinuities is important to correctly compute the flow near those interfaces. We present a numerical algorithm that correctly captures all waves with a hybrid strategy. The method integrates the Riemann invariants near contact discontinuities and switches back to the conserved variables away from it to properly resolve shock waves. This strategy can be applied to any numerical scheme. Numerical solutions for a variety of tests in one and two dimensions are shown to illustrate the advantages of the strategy and the merits of the scheme.     

Unsteady viscous jet flow into stationary surroundings

The unsteady constant-property uniform pressure flow of a viscous laminar axisymmetric jet into stationary surroundings is analyzed with the aid of the boundary layer approximations and an integral method. Numerical solutions of the resulting non-linear system of equations are presented for the response of an initially steady jet to monotonic mass flow variations which are imposed at the initial axial station of the jet. Significant departures from quasi-steady behaviour arise for sufficiently large streamwise distances and/or rates of change of the boundary conditions. Among these effects are overshoots in mass and momentum flux, a dynamic effect on flow entrainment, and a tendency toward discontinous behavior. The influence of the kinematic viscosity on the transient flow is examined, and an estimate is made of the applicability of quasi-steady approximations in this problem.     

Lie group analysis of stagnation-point flow of a nanofluid

This article concerns with a steady two-dimensional boundary layer flow of an electrically conducting incompressible nanofluid over a stretching sheet in a porous medium with internal heat generation/absorption. The transport model includes the effect of Brownian motion with thermophoresis in the presence of chemical reaction and magnetic field. Lie group analysis is applied to the governing equations. The transformed self similar non-linear ordinary differential equations along with the boundary conditions are solved numerically. The influences of various relevant parameters on the flow field, temperature and nanoparticle volume fraction as well as wall heat flux and wall mass flux are elucidated through graphs and tables.     

An isentropic flow algorithm with body fitted meshes

An efficient algorithm based on flux difference splitting is presented for the solution of the three-dimensional equations of isentropic flow in a generalised coordinate system, and with a general convex gas law. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The algorithm requires only one function evaluation of the gas law in each computational cell. The scheme has good shock capturing properties and the advantage of using body-fitted meshes. Numerical results are shown for Mach 3 flow of air past a circular cylinder. Furthermore, the algorithm also applies to shallow water flows by employing the familiar gas dynamics analogy.     

An implicit low-diffusive HLL scheme with complete time linearization: Application to cavitating barotropic flows

A numerical method for generic barotropic flows is presented, together with its application to the simulation of cavitating flows. A homogeneous-flow cavitation model is indeed considered, which leads to a barotropic state equation. The continuity and momentum equations for compressible flows are discretized through a mixed finite-element/finite-volume approach, applicable to unstructured grids. P1 finite elements are used for the viscous terms, while finite volumes for the convective ones. The numerical fluxes are computed by shock-capturing schemes and ad-hoc preconditioning is used to avoid accuracy problems in the low-Mach regime. A HLL flux function for barotropic flows is proposed, in which an anti-diffusive term is introduced to counteract accuracy problems for contact discontinuities and viscous flows typical of this class of schemes, while maintaining its simplicity. Second-order accuracy in space is obtained through MUSCL reconstruction. Time advancing is carried out by an implicit linearized scheme. For this HLL-like flux function two different time linearizations are considered; in the first one the upwind part of the flux function is frozen in time, while in the second one its time variation is taken into account. The proposed numerical ingredients are validated through the simulations of different flow configurations, viz. the Blasius boundary layer, a Riemann problem, the quasi-1D cavitating flow in a nozzle and the flow around a hydrofoil mounted in a tunnel, both in cavitating and non-cavitating conditions. The Roe flux function is also considered for comparison. It is shown that the anti-diffusive term introduced in the HLL scheme is actually effective to obtain good accuracy (similar to the one of the Roe scheme) for viscous flows and contact discontinuities. Moreover, the more complete time linearization is a key ingredient to largely improve numerical stability and efficiency in cavitating conditions.     

Coupled model of surface water flow, sediment transport and morphological evolution

This paper presents a mathematical model coupling water flow and sediment transport dynamics that enables calculating the changing surface morphology through time and space. The model is based on the shallow water equations for flow, conservation of sediment concentration, and empirical functions for bed friction, substrate erosion and deposition. The sediment transport model is a non-capacity formulation whereby erosion and deposition are treated independently and influence the sediment flux by exchanging mass across the bottom boundary of the flow. The resulting hyperbolic system is solved using a finite volume, Godunov-type method with a first-order approximate Riemann solver. The model can be applied both to short time scales, where the flow, sediment transport and morphological evolution are strongly coupled and the rate of bed evolution is comparable to the rate of flow evolution, or to relatively long time scales, where the time scale of bed evolution associated with erosion and/or deposition is slow relative to the response of the flow to the changing surface and, therefore, the classical quasi-steady approximation can be invoked. The model is verified by comparing computed results with documented solutions. The developed model can be used to investigate a variety of problems involving coupled flow and sediment transport including channel initiation and drainage basin evolution associated with overland flow and morphological changes induced by extreme events such as tsunami.     

Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition

The hybrid spectral difference methods (HSD) for the Laplace and Helmholtz equations in exterior domains are proposed. We consider the fictitious domain method with the absorbing boundary conditions (ABCs). The HSD method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The fictitious domain is composed of two subregions; the Cartesian grid region and the boundary layer region in which the radial grid is imposed. The boundary layer region with the radial grid makes it easy to implement the discrete radial ABC. The discrete radial ABC is a discrete version of the Bayliss–Gunzburger–Turkel ABC without pertaining any radial derivatives. Numerical experiments confirming efficiency of our numerical scheme are provided.     

Dynamic crack analysis in piezoelectric solids with non-linear electrical and mechanical boundary conditions by a time-domain BEM

This paper presents advanced transient dynamic crack analysis in two-dimensional (2D), homogeneous and linear piezoelectric solids using non-linear mechanical and electrical crack-face boundary conditions. Stationary cracks in infinite and finite piezoelectric solids subjected to impact loadings are considered. For this purpose a time-domain boundary element method (TDBEM) is developed. A Galerkin-method is implemented for the spatial discretization, while a collocation method is applied for the temporal discretization. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the generalized crack-opening-displacements (CODs) numerically. An iterative solution algorithm is developed to consider the non-linear semi-permeable electrical crack-face boundary conditions. Furthermore, an additional iteration scheme for crack-face contact analysis is implemented at time-steps when a physically meaningless crack-face intersection occurs. Several numerical examples are presented and discussed to show the effects of the electrical crack-face boundary conditions on the dynamic intensity factors.     

A comparison of conservative upwind difference schemes for the euler equations

Numerical results are presented and compared for three conservative upwind difference schemes for the Euler equations when applied to two standard test problems. This includes consideration of the effect of treating part of the flux balance as a source, and a comparison of different averaging of the flow variables. Two of the schemes are also shown to be equivalent in their implementation, while being different in construction and having different approximate Jacobians.     

A Discontinuous Spectral Element Model for Boussinesq-Type Equations

We present a discontinuous spectral element model for simulating 1D nonlinear dispersive water waves, described by a set of enhanced Boussinesq-type equations. The advective fluxes are calculated using an approximate Riemann solver while the dispersive fluxes are obtained by centred numerical fluxes. Numerical computation of solitary wave propagation is used to prove the exponential convergence.     

A discrete Boltzmann equation model for two-phase shallow granular flows

In this paper a Discrete Boltzmann Equation model (hereinafter DBE) is proposed as solution method of the two-phase shallow granular flow equations, a complex nonlinear partial differential system, resulting from the depth-averaging procedure of mass and momentum equations of granular flows. The latter, as e.g. a debris flow, are flows of mixtures of solid particles dispersed in an ambient fluid.The reason to use a DBE, instead of a more conventional numerical model (e.g. based on Riemann solvers), is that the DBE is a set of linear advection equations, which replaces the original complex nonlinear partial differential system, while preserving the features of its solutions. The interphase drag function, an essential characteristic of any two-phase model, is accounted for easily in the DBE by adding a physically based term. In order to show the validity of the proposed approach, the following relevant benchmark tests have been considered: the 1D simple Riemann problem, the dam break problem with the wet–dry transition of the liquid phase, the dry bed generation and the perturbation of a state at rest in 2D. Results are satisfactory and show how the DBE is able to reproduce the dynamics of the two-phase shallow granular flow.     

A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws

We present a strict Lyapunov function for hyperbolic systems of conservation laws that can be diagonalized with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel     

On different flux splittings and flux functions in WENO schemes for balance laws

In this paper we focus our attention on obtaining well-balanced schemes for balance laws by using Marquina’s flux in combination with the finite difference and finite volume WENO schemes. We consider also the Rusanov flux splitting and the HLL approximate Riemann solver. In particular, for the presented numerical schemes we develop corresponding discretizations of the source term, based on the idea of balancing with the flux gradient. When applied to the open-channel flow and to the shallow water equations, we obtain the finite difference WENO scheme with Marquina’s flux splitting, which satisfies the approximate conservation property, and also the balanced finite volume WENO scheme with Marquina’s solver satisfying the exact conservation property. Finally, we also present an improvement of the balanced finite difference WENO scheme with the Rusanov (locally Lax–Friedrichs) flux splitting, we previously developed in [Vuković S, Sopta L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J Comput Phys 2002;179:593–621].     

Rosenbrock methods with an explicit first stage

Traditional Rosenbrock methods suffer from order reduction when applied to partial differential equations with non-homogeneous boundary conditions and source terms. The paper studies a family of Rosenbrock schemes with an explicit first stage. This structure allows one to construct algorithms with high stage orders, and which do not suffer from order reduction. The paper discusses additional order conditions needed for linear stability, for using inexact Jacobians, and implementation aspects. Second- and third-order practical schemes are constructed, and their application to one- and two-dimensional partial differential equations test problems confirm the theoretical findings.     

A Runge-Kutta discontinuous Galerkin method for the Euler equations

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. Like the traditional gas-kinetic schemes, our proposed RKDG method does not need to use the characteristic decomposition or the Riemann solver in computing the numerical flux at the surface of the finite elements. The integral term containing the non-linear flux can be computed exactly at the microscopic level. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods give higher resolution in solving problems with smooth solutions. Moreover, shock and contact discontinuities can be well captured by using the proposed methods.