Non-conservative Equations Research Articles

Numerical effects of the semi-conservative form of momentum equations for multi-dimensional two-phase flows

Some of the thermal hydraulics codes for multi-dimensional two-phase flow analysis use the non-conservative form of momentum equations for numerical convenience. From a mathematical point of view, these equations are equal to those in the conservative form. But, numerical integration of the non-conservative momentum equations over a control volume results in different solution characteristics, which may cause inaccurate solutions under some two-phase flow conditions. In this paper, a semi-conservative form of the momentum equations is suggested which is close to the conservative form but still maintains the feature of the non-conservative form. The numerical results of the semi-conservative and the non-conservative forms are compared against analytical solutions and the solutions of the FLUENT code that uses the conservative form. The results clearly showed that the semi-conservative form of the momentum equations provides better solutions than the non-conservative form, especially for heterogeneous two-phase flows.

Solid–fluid diffuse interface model in cases of extreme deformations

Diffuse interface methods have been recently proposed and successfully used for accurate compressible multi-fluid computations Abgrall [1]; Kapila et al. [20]; Saurel et al. [30]. These methods deal with extended systems of hyperbolic equations involving a non-conservative volume fraction equation and relaxation terms. Following the same theoretical frame, we derive here an Eulerian diffuse interface model for elastic solid-compressible fluid interactions in situations involving extreme deformations. Elastic effects are included following the Eulerian conservative formulation proposed in Godunov [16], Miller and Colella [23], Godunov and Romenskii [17], Plohr and Plohr [27] and Gavrilyuk et al. [14]. We apply first the Hamilton principle of stationary action to derive the conservative part of the model. The relaxation terms are then added which are compatible with the entropy inequality. In the limit of vanishing volume fractions the Euler equations of compressible fluids and a conservative hyperelastic model are recovered. It is solved by a unique hyperbolic solver valid at each mesh point (pure fluid, pure solid and mixture cell). Capabilities of the model and methods are illustrated on various tests of impacts of solids moving in an ambient compressible fluid.

Unique continuation from an arbitrary interior subdomain of the variable-coefficient Oseen equation

This paper presents a proof–based on a pointwise Carleman estimate for the Laplacian Δ from [I. Lasiecka, R. Triggiani, Z. Zhang, Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability, AMS Contemp. Math., 268 (2000) 227–326] or [I. Lasiecka, R. Triggiani, Z. Zhang, Global uniqueness, observability and stabilization of non-conservative Schrödinger equations via pointwise Carleman estimates. Part I: H 1 ( Ω ) -estimates, J. Inverse Ill-Posed Problems, 12 (1) (2004) 1–81]–of the following result: The only solution { u , p } of the Oseen problem { ( − ν 0 Δ ) u + L e ( u ) + ∇ p = λ u in Ω ; (a) div u ≡ 0 in Ω ; (b) u ≡ 0 in ω . (c) defined on a bounded Ω ⊂ R d , d = 2 , 3 with interior condition u ≡ 0 in an arbitrary sufficiently smooth subdomain ω ⊂ Ω , is the trivial solution u ≡ 0 , p ≡ const in Ω . The method of proof applies equally well to the corresponding Riemannian setting [where, in particular, Δ g is now the Laplace–Bertrami operator], by relying on the pointwise Carleman estimate for Δ g in this case from [R. Triggiani, P.F. Yao, Carleman estimates with no lower order terms for general Riemannian wave equations. Global uniqueness and observability in one shot, Appl. Math. Optim. 46 (2–3) (2002), 331–375] or [R. Triggiani, X. Xu, Pointwise Carleman estimates, global uniqueness, observability and stabilization for non-conservative Schrödinger equations on Riemannian manifolds at the H 1 ( Ω ) -level (with X. Xu), AMS Contemp. Math. 426 (2007), 339–404].

Numerical simulation of a single bubble by compressible two‐phase fluids

AbstractThe present work deals with the numerical investigation of a collapsing bubble in a liquid–gas fluid, which is modeled as a single compressible medium. The medium is characterized by the stiffened gas law using different material parameters for the two phases. For the discretization of the stiffened gas model, the approach of Saurel and Abgrall is employed where the flow equations, here the Euler equations, for the conserved quantities are approximated by a finite volume scheme, and an upwind discretization is used for the non‐conservative transport equations of the pressure law coefficients. The original first‐order discretization is extended to higher order applying second‐order ENO reconstruction to the primitive variables. The derivation of the non‐conservative upwind discretization for the phase indicator, here the gas fraction, is presented for arbitrary unstructured grids. The efficiency of the numerical scheme is significantly improved by employing local grid adaptation. For this purpose, multiscale‐based grid adaptation is used in combination with a multilevel time stepping strategy to avoid small time steps for coarse cells. The resulting numerical scheme is then applied to the numerical investigation of the 2‐D axisymmetric collapse of a gas bubble in a free flow field and near to a rigid wall. The numerical investigation predicts physical features such as bubble collapse, bubble splitting and the formation of a liquid jet that can be observed in experiments with laser‐induced cavitation bubbles. Opposite to the experiments, the computations reveal insight to the state inside the bubble clearly indicating that these features are caused by the acceleration of the gas due to shock wave focusing and reflection as well as wave interaction processes. While incompressible models have been used to provide useful predictions on the change of the bubble shape of a collapsing bubble near a solid boundary, we wish to study the effects of shock wave emissions into the ambient liquid on the bubble collapse, a phenomenon that may not be captured using an incompressible fluid model. Copyright © 2009 John Wiley & Sons, Ltd.

Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients

In this paper, two main results are proved. We consider a nonconservative linear Cauchy problem with discontinuous coefficients accross a noncharacteristic hypersurface. We introduce then a viscous perturbation of the problem; the viscous solution u^eps depends of the small positive parameter eps. This problem, obtained by small viscous perturbation, is parabolic for fixed positive eps. We prove then, under stability assumptions, the convergence, when eps vanishes, of u^eps towards the solution of a well-posed limit hyperbolic problem. Our first result is obtained, in the multi-D framework, for piecewise smooth coefficients and states the convergence of u^\eps towards an unique solution. Our second result shows that, in the expansive nonconservative scalar case, that is to say for sign(x a(x))>0, our viscous approach successfully singles out a solution. Even for scalar, piecewise constant 1-D nonconservative hyperbolic equations, this result is new and not treated during our analysis performed on systems. For both results, an asymptotic analysis of the convergenceis performed at any order, containing a boundary layer analysis.

Simulated morphological landscape of polymer single crystals by phase field model

The novel phase field model with the "polymer characteristic" was established based on a nonconserved spatiotemporal Ginzburg-Landau equation (TDGL model A). Especially, we relate the diffusion equation with the crystal growth faces of polymer single crystals. Namely, the diffusion equations are discretized according to the diffusion coefficient of every lattice site in various crystal growth faces and the shape of lattice is selected based on the real proportion of the unit cell dimensions. Spatiotemporal growth of syndiotactic polypropylene single crystals during isothermal crystallization has been investigated theoretically based on this phase field model. Two dimensional numerical calculations are performed to elucidate the faceted single crystal growth including square, rectangular, lozenge-shaped, and hexagonal single crystals. Our simulated patterns are in good agreement with the experimental morphologies, and the physical origin of polymer single crystal growth is discussed.

Discontinuous Galerkin finite element method for shallow two-phase flows

We present a discontinuous Galerkin finite element method for a depth-averaged two-phase flow model. This model contains nonconservative products for which we developed a discontinuous Galerkin finite element formulation in Rhebergen et al. [S. Rhebergen, O. Bokhove, J.J.W. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys. 227 (2008) 1887]. The goal is to qualitatively validate the model against a laboratory experiment and to show the abilities of the model to capture physical phenomena. To be able to perform these test cases, a WENO slope limiter is investigated in conjunction with a discontinuity detector to detect regions where spurious oscillations appear.

CONTINUUM LIMIT OF SELF-DRIVEN PARTICLES WITH ORIENTATION INTERACTION

The discrete Couzin–Vicsek algorithm (CVA), which describes the interactions of individuals among animal societies such as fish schools is considered. In this paper, a kinetic (mean-field) version of the CVA model is proposed and its formal macroscopic limit is provided. The final macroscopic model involves a conservation equation for the density of the individuals and a non-conservative equation for the director of the mean velocity and is proved to be hyperbolic. The derivation is based on the introduction of a non-conventional concept of a collisional invariant of a collision operator.

Solution of shallow-water equations using least-squares finite-element method

A least-squares finite-element method (LSFEM) for the non-conservative shallow-water equations is pre- sented. The model is capable of handling complex topography, steady and unsteady flows, subcritical and supercritical flows, and flows with smooth and sharp gradient changes. Advantages of the model include: (1) sources terms, such as the bottom slope, surface stresses and bed frictions, can be treated easily without any special treatment; (2) upwind scheme is no needed; (3) a single approximating space can be used for all variables, and its choice of approximating space is not subject to the Ladyzhenskaya-Babuska-Brezzi (LBB) condition; and (4) the resulting system of equations is symmetric and positive-definite (SPD) which can be solved efficiently with the preconditioned conjugate gradient method. The model is verified with flow over a bump, tide induced flow, and dam-break. Computed results are compared with analytic solutions or other numerical results, and show the model is conservative and accurate. The model is then used to simulate flow past a circular cylinder. Important flow characteristics, such as variation of water surface around the cylinder and vortex shedding behind the cylinder are investigated. Computed results compare well with experiment data and other numerical results.

Integration of unsteady friction models in pipe flow simulations

Various approaches to integrate instantaneous acceleration models are examined. The basic physical model consists of the one-phase flow water hammer equations with the unsteady wall friction model of Brunone. The numerical scheme is based on characteristic upwind finite differences, representing an extension of the Godunov schemes to the non-conservative hyperbolic equations. The most accurate solutions result from the basic version of this method, which takes into account the effects of spatial and temporal derivatives of the unsteady friction term on the eigenvalues of the hyperbolic equations. A convergence analysis shows that second-order accuracy is obtained on smooth solutions with this approach, but only if the discontinuity in the unsteady friction correlation is removed. Two simpler numerical schemes treat the unsteady friction term as a source term and neglect its influence on the eigenvalues. The results obtained with the simplified schemes converge to the exact solutions, but exhibit larger numerical diffusivity than the basic version of the method.

Macroscopic limit of self-driven particles with orientation interaction

The discrete Couzin–Vicsek algorithm (CVA) has been proposed to model the interactions of individuals among animal societies such as schools of fish. In this Note, we propose a kinetic (mean-field) version of the CVA model and provide its formal macroscopic limit. The final macroscopic model involves a conservation equation for the density of the individuals and a non-conservative equation for the director of the mean velocity. The result is based on the introduction of a non-conventional concept of a collisional invariant of the collision operator. To cite this article: P. Degond, S. Motsch, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations

We present space- and space–time discontinuous Galerkin finite element (DGFEM) formulations for systems containing nonconservative products, such as occur in dispersed multiphase flow equations. The main criterium we pose on the weak formulation is that if the system of nonconservative partial differential equations can be transformed into conservative form, then the formulation must reduce to that for conservative systems. Standard DGFEM formulations cannot be applied to nonconservative systems of partial differential equations. We therefore introduce the theory of weak solutions for nonconservative products into the DGFEM formulation leading to the new question how to define the path connecting left and right states across a discontinuity. The effect of different paths on the numerical solution is investigated and found to be small. We also introduce a new numerical flux that is able to deal with nonconservative products. Our scheme is applied to two different systems of partial differential equations. First, we consider the shallow water equations, where topography leads to nonconservative products, in which the known, possibly discontinuous, topography is formally taken as an unknown in the system. Second, we consider a simplification of a depth-averaged two-phase flow model which contains more intrinsic nonconservative products.

Numerical simulation for two-phase flows using hybrid scheme

The present paper is devoted to the computation of two-phase flows using two-fluid approach. Two difficulties arise for the model: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. To deal with the aforementioned difficulties, the non-conservative volume fraction equation is discretized by WENO scheme, and the CWENO-type central-upwind scheme is applied to solve the mass, momentum and energy equations. Computational results are eventually provided and discussed.

Multifluid numerical approximations based on a multipressure formulation

In the present work, we consider the numerical approximations of multifluid compressible flows. We suppose that the flow is composed by nonmixing compressible fluids. We propose a dissipative model insuring, for the mixture, the conservation of the mass, the momentum and the total energy. Entropy balances between phases are described by additional nonconservative equations. At the limit of vanishing viscosity the model is still consistent with the entropy balances. The nonlinear projection scheme is used to preserve, at the discrete level, the main properties of the model. Numerical experiments are conducted to validate the proposed models for the multicomponent shock tube, the bubble–shock interaction and the tank filling problems.

A conservative extension of the method of characteristics for 1-D shallow flows

The method of characteristics (MOC) has been used for a long time in open channels and pipes flows. It is based on non-conservative equations, and hence it cannot be used directly for solving discontinuous shallow flows. In this paper we develop a conservative version of the MOC scheme for 1-D shallow flows by imposing the conservation law at the interpolation step. The conservation property of the scheme ensures the production of an accurate shock modeling and enables the MOC scheme to simulate dam-break type flows. By using a proper interpolation function, the proposed method can also produce quite accurate low-oscillatory results. A number of challenging test cases show considerable improvement compared to the traditional non-conservative MOC scheme in the case of dam-break type and trans-critical flow simulations.

Periodic Liénard-type delay equations with state-dependent impulses

Conditions are obtained for Liénard-type equations with delay and state-dependent impulses to admit an absolutely continuous periodic solution with first derivative of bounded variation (and consequently with Lebesgue integrable second derivative). The results are applied to Josephson's equation and the nonconservative forced pendulum equation.

New solutions for the one-dimensional nonconservative inviscid Burgers equation

The propagation of travelling waves is a relevant physical phenomenon. As usual the understanding of a real propagating wave depends upon a correct formulation of a idealized model. Discontinuous functions, Dirac- δ measures and their distributional derivatives are, respectively, idealizations of sharp jumps, localized high peaks and single sharp localised oscillations. In the present paper we study the propagation of distributional travelling waves for Burgers inviscid equation. This will be afforded by our theory of distributional products, and is based on a rigorous and consistent concept of solution we have introduced in [C.O.R. Sarrico, Distributional products and global solutions for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003) 641–656]. Our approach exhibit Dirac- δ travelling solitons (they are just the “infinitesimal narrow solitons” of Maslov, Omel'yanov and Tsupin [V.P. Maslov, O.A. Omel'yanov, Asymptotic soliton-form solutions of equations with small dispersion, Russian Math. Surveys 36 (1981) 73–149; V.P. Maslov, V.A. Tsupin, Necessary conditions for the existence of infinitely narrow solitons in gas dynamics, Soviet Phys. Dokl. 24 (1979) 354–356]) and also solutions which are not measures such as for instance u ( x , t ) = b + δ ′ ( x − b t ) , a wave of constant speed b. Moreover, for signals with two jump discontinuities we have, in our setting, the propagation of more solitons and more values for the signal speed are allowed than those afforded within classical framework.

Electron transport coefficients in O2 magnetron discharges

Electron transport coefficients required for the modelling of bulk electron transport in an O2 magnetron discharge are calculated from solution of the non-conservative Boltzmann equation. The influence on the transport coefficients (rate coefficients, drift velocity vector elements, diffusion tensor elements) of the electric and magnetic field strengths and their orientation with respect to each other have been systematically investigated over ranges consistent with practical operation. The results represent the first multi-term solution of the non-conservative Boltzmann's equation for electric and magnetic fields at arbitrary orientations.

Difference scheme for two-phase flow

A numerical method for two-phase flow with hydrodynamics behavior was considered. The nonconservative hyperbolic governing equations proposed by Saurel and Gallout were adopted. Dissipative effects were neglected but they could be included in the model without major difficulties. Based on the opinion proposed by Abgrall that “a two phase system, uniform in velocity and pressure at t=0 will be uniform on the same variable during its temporal evolution”, a simple accurate and fully Eulerian numerical method was presented for the simulation of multiphase compressible flows in hydrodynamic regime. The numerical method relies on Godunov-type scheme, with HLLC and Lax-Friedrichs type approximate Riemann solvers for the resolution of conservation equations, and nonconservative equation. Speed relaxation and pressure relaxation processes were introduced to account for the interaction between the phases. Test problem was presented in one space dimension which illustrated that our scheme is accurate, stable and oscillation free.

On the use of symmetrizing variables for vacuums

The paper is devoted to the computation of shallow-water equations (or Euler equations) in using an approximate Godunov scheme called VFRoe, when the flow may include dry areas (or very low density regions). This is achieved with the help of symmetrizing variables. Overall we are able to insure the discrete preservation of positive variables on interfaces, and at the same time to compute vacuum occurence or propagation of shock waves over a near-vacuum. A short section is also dedicated to the non-conservative hyperbolic equations arising within the setting of one-equation or two-equation turbulent compressible models. Many numerical tests confirm the capabilities of the scheme, and measuring the L 1 error norm in particular cases enables us to specify the actual rate of convergence.