# Euler Equations (euler + equation)

Distribution by Scientific Domains

Kinds of Euler Equations

 compressible euler equation

## Selected Abstracts

### Liquidity Constraints and Firms' Investment Return Behaviour

ECONOMICA, Issue 276 2002
Parantap Basu
We construct a production-based model, which compares the investment return behaviour of liquidity-constrained firms with that of unconstrained firms. The key testable implication that emerges from the model is that the investment returns of the constrained firms are predictable, while those of the unconstrained firms are not. We test this implication indirectly, verifying whether the capital stock and investment returns of the latter firms lead those of the former, and directly, via the estimation of an Euler equation. Our results are consistent with the model's prediction. [source]

### Euler deconvolution of the analytic signal and its application to magnetic interpretation

GEOPHYSICAL PROSPECTING, Issue 3 2004
P. Keating
ABSTRACT Euler deconvolution and the analytic signal are both used for semi-automatic interpretation of magnetic data. They are used mostly to delineate contacts and obtain rapid source depth estimates. For Euler deconvolution, the quality of the depth estimation depends mainly on the choice of the proper structural index, which is a function of the geometry of the causative bodies. Euler deconvolution applies only to functions that are homogeneous. This is the case for the magnetic field due to contacts, thin dikes and poles. Fortunately, many complex geological structures can be approximated by these simple geometries. In practice, the Euler equation is also solved for a background regional field. For the analytic signal, the model used is generally a contact, although other models, such as a thin dike, can be considered. It can be shown that if a function is homogeneous, its analytic signal is also homogeneous. Deconvolution of the analytic signal is then equivalent to Euler deconvolution of the magnetic field with a background field. However, computation of the analytic signal effectively removes the background field from the data. Consequently, it is possible to solve for both the source location and structural index. Once these parameters are determined, the local dip and the susceptibility contrast can be determined from relationships between the analytic signal and the orthogonal gradients of the magnetic field. The major advantage of this technique is that it allows the automatic identification of the type of source. Implementation of this approach is demonstrated for recent high-resolution survey data from an Archean granite-greenstone terrane in northern Ontario, Canada. [source]

### Do Actions Speak Louder Than Words?

JOURNAL OF MONEY, CREDIT AND BANKING, Issue 7 2009
Household Expectations of Inflation Based on Micro Consumption Data
inflation expectations; Consumer Expenditure Survey; Michigan Survey of Consumers; Survey of Professional Forecasters; Euler equation Survey data on household expectations of inflation are routinely used in economic analysis, yet it is not clear how accurately households are able to articulate their expectations in survey interviews. We propose an alternative approach to recovering households' expectations of inflation from their consumption expenditures. We show that these expectations measures have predictive power for consumer price index (CPI) inflation. They are better predictors of CPI inflation than household survey responses and more highly correlated with professional inflation forecasts, except for highly educated consumers, consistent with the view that more educated consumers are better able to articulate their expectations. We also document that households' inflation expectations respond to inflation news, as measured by the unpredictable component of inflation predictions in the Survey of Professional Forecasters. The response to inflation news tends to increase with households' level of education, consistent with the existence of constraints on household's ability to process this information. [source]

### CREDIT CRUNCH AND HOUSEHOLD WELFARE, THE CASE OF THE KOREAN FINANCIAL CRISIS,

THE JAPANESE ECONOMIC REVIEW, Issue 4 2008
SUNG JIN KANG
We examine how the credit crunch in Korea in the late 1990s affected household behaviour and welfare. Using 1996,1998 household panel data, we estimate a consumption Euler equation, augmented by endogenous credit constraints. Korean households coped with the negative shocks of the 1997 credit crunch by reducing consumption of luxury items while maintaining food, education and health related expenditures. Our results show that, in 1997,1998, during the crisis, the probability of facing credit constraints and the resulting expected welfare loss from the binding constraints increased significantly, suggesting the gravity of the credit crunch at the household level. [source]

### Geometry and a priori estimates for free boundary problems of the Euler's equation

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2008
Jalal Shatah
In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh-Taylor sign condition holds. We prove that as the surface tension tends to 0, when the Rayleigh-Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension. © 2007 Wiley Periodicals, Inc. [source]

### On the evolution of sharp fronts for the quasi-geostrophic equation

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 6 2005
José Luis Rodrigo
We consider the problem of the evolution of sharp fronts for the surface quasi-geostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two-dimensional Euler equation. The special interest of the quasi-geostrophic equation lies in its strong similarities with the three-dimensional Euler equation, while being a two-dimen-sional model. In particular, an analogue of the problem considered here, the evolution of sharp fronts for QG, is the evolution of a vortex line for the three-dimensional Euler equation. The rigorous derivation of an equation for the evolution of a vortex line is still an open problem. The influence of the singularity appearing in the velocity when using the Biot-Savart law still needs to be understood. We present two derivations for the evolution of a periodic sharp front. The first one, heuristic, shows the presence of a logarithmic singularity in the velocity, while the second, making use of weak solutions, obtains a rigorous equation for the evolution explaining the influence of that term in the evolution of the curve. Finally, using a Nash-Moser argument as the main tool, we obtain local existence and uniqueness of a solution for the derived equation in the C, case. © 2004 Wiley Periodicals, Inc. [source]

### Boundary Perturbation Methods for Water Waves

GAMM - MITTEILUNGEN, Issue 1 2007
David P. Nicholls
Abstract The most successful equations for the modeling of ocean wave phenomena are the free,surface Euler equations. Their solutions accurately approximate a wide range of physical problems from open,ocean transport of pollutants, to the forces exerted upon oil platforms by rogue waves, to shoaling and breaking of waves in nearshore regions. These equations provide numerous challenges for theoreticians and practitioners alike as they couple the difficulties of a free boundary problem with the subtle balancing of nonlinearity and dispersion in the absence of dissipation. In this paper we give an overview of, what we term, "Boundary Perturbation" methods for the analysis and numerical simulation of this "water wave problem". Due to our own research interests this review is focused upon the numerical simulation of traveling water waves, however, the extensive literature on the initial value problem and additional theoretical developments are also briefly discussed. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
C. Miehe
Abstract The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase-field. In this paper, we outline a thermodynamically consistent framework for phase-field models of crack propagation in elastic solids, develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. We start our investigation with an intuitive and descriptive derivation of a regularized crack surface functional that ,-converges for vanishing length-scale parameter to a sharp crack topology functional. This functional provides the basis for the definition of suitable convex dissipation functions that govern the evolution of the crack phase-field. Here, we propose alternative rate-independent and viscous over-force models that ensure the local growth of the phase-field. Next, we define an energy storage function whose positive tensile part degrades with increasing phase-field. With these constitutive functionals at hand, we derive the coupled balances of quasi-static stress equilibrium and gradient-type phase-field evolution in the solid from the argument of virtual power. Here, we consider a canonical two-field setting for rate-independent response and a time-regularized three-field formulation with viscous over-force response. It is then shown that these balances follow as the Euler equations of incremental variational principles that govern the multi-field problems. These principles make the proposed formulation extremely compact and provide a perfect base for the finite element implementation, including features such as the symmetry of the monolithic tangent matrices. We demonstrate the performance of the proposed phase-field formulations of fracture by means of representative numerical examples. Copyright © 2010 John Wiley & Sons, Ltd. [source]

### On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment,

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010
I. Kalashnikova
Abstract A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L2 inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well-posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well-posed and stable far-field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty-like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd. [source]

### On the computation of steady-state compressible flows using a discontinuous Galerkin method

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2008
Hong Luo
Abstract Computation of compressible steady-state flows using a high-order discontinuous Galerkin finite element method is presented in this paper. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries. Particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities. A physics-based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave. A recently developed, fast, low-storage p -multigrid method is used for solving the governing compressible Euler equations to obtain steady-state solutions. The method is applied to compute a variety of compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy of the developed discontinuous Galerkin method for computing compressible steady-state flows. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Multigrid convergence acceleration for implicit and explicit solution of Euler equations on unstructured grids

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9 2010
Ali Ramezani
Abstract The multigrid method is one of the most efficient techniques for convergence acceleration of iterative methods. In this method, a grid coarsening algorithm is required. Here, an agglomeration scheme is introduced, which is applicable in both cell-center and cell-vertex 2 and 3D discretizations. A new implicit formulation is presented, which results in better computation efficiency, when added to the multigrid scheme. A few simple procedures are also proposed and applied to provide even higher convergence acceleration. The Euler equations are solved on an unstructured grid around standard transonic configurations to validate the algorithm and to assess its superiority to conventional explicit agglomeration schemes. The scheme is applied to 2 and 3D test cases using both cell-center and cell-vertex discretizations. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### Numerical simulation of a single bubble by compressible two-phase fluids

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010
Siegfried Müller
Abstract The 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. [source]

### Toward accurate hybrid prediction techniques for cavity flow noise applications

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2009
W. De Roeck
Abstract A large variety of hybrid computational aeroacoustics (CAA) approaches exist differing from each other in the way the source region is modeled, in the way the equations are used to compute the propagation of acoustic waves in a non-quiescent medium, and in the way the coupling between source and acoustic propagation regions is made. This paper makes a comparison between some commonly used numerical methods for aeroacoustic applications. The aerodynamically generated tonal noise by a flow over a 2D rectangular cavity is investigated. Two different cavities are studied. In the first cavity (L/D=4, M=0.5), the sound field is dominated by the cavity wake mode and its higher harmonics, originating from a periodical vortex shedding at the cavity leading edge. In the second cavity (L/D=2, M=0.6), shear-layer modes, due to flow-acoustic interaction phenomena, generate the major components in the noise spectrum. Source domain modeling is carried out using a second-order finite-volume large eddy simulation. Propagation equations, taking into account convection and refraction effects, are solved using high-order finite-difference schemes for the linearized Euler equations and the acoustic perturbation equations. Both schemes are compared with each other for various coupling methods between source region and acoustic region. Conventional acoustic analogies and Kirchhoff methods are rewritten for the various propagation equations and used to obtain near-field acoustic results. The accuracy of the various coupling methods in identifying the noise-generating mechanisms is evaluated. In this way, this paper provides more insight into the practical use of various hybrid CAA techniques to predict the aerodynamically generated sound field by a flow over rectangular cavities. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### On CFL evolution strategies for implicit upwind methods in linearized Euler equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2009
H. M. Bücker
Abstract In implicit upwind methods for the solution of linearized Euler equations, one of the key issues is to balance large time steps, leading to a fast convergence behavior, and small time steps, needed to sufficiently resolve relevant flow features. A time step is determined by choosing a Courant,Friedrichs,Levy (CFL) number in every iteration. A novel CFL evolution strategy is introduced and compared with two existing strategies. Numerical experiments using the adaptive multiscale finite volume solver QUADFLOW demonstrate that all three CFL evolution strategies have their advantages and disadvantages. A fourth strategy aiming at reducing the residual as much as possible in every time step is also examined. Using automatic differentiation, a sensitivity analysis investigating the influence of the CFL number on the residual is carried out confirming that, today, CFL control is still a difficult and open problem. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### A general Riemann solver for Euler equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2008
Hao Wu
Abstract In this paper, we present a general Riemann solver which is applied successfully to compute the Euler equations in fluid dynamics with many complex equations of state (EOS). The solver is based on a splitting method introduced by the authors. We add a linear advection term to the Euler equations in the first step, to make the numerical flux between cells easy to compute. The added linear advection term is thrown off in the second step. It does not need an iterative technique and characteristic wave decomposition for computation. This new solver is designed to permit the construction of high-order approximations to obtain high-order Godunov-type schemes. A number of numerical results show its robustness. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### High-order ENO and WENO schemes for unstructured grids

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2007
W. R. Wolf
Abstract This work describes the implementation and analysis of high-order accurate schemes applied to high-speed flows on unstructured grids. The class of essentially non-oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third- and fourth-order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2-D Euler equations in a cell centred finite volume context. High-order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge,Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high-order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high-speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd. [source]

### Hierarchic multigrid iteration strategy for the discontinuous Galerkin solution of the steady Euler equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9-10 2006
Koen Hillewaert
Abstract We study the efficient use of the discontinuous Galerkin finite element method for the computation of steady solutions of the Euler equations. In particular, we look into a few methods to enhance computational efficiency. In this context we discuss the applicability of two algorithmical simplifications that decrease the computation time associated to quadrature. A simplified version of the quadrature free implementation applicable to general equations of state, and a simplified curved boundary treatment are investigated. We as well investigate two efficient iteration techniques, namely the classical Newton,Krylov method used in computational fluid dynamics codes, and a variant of the multigrid method which uses interpolation orders rather than coarser tesselations to define the auxiliary coarser levels. Copyright © 2005 John Wiley & Sons, Ltd. [source]

### MUSTA schemes for multi-dimensional hyperbolic systems: analysis and improvements

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005
V. A. Titarev
Abstract We develop and analyse an improved version of the multi-stage (MUSTA) approach to the construction of upwind Godunov-type fluxes whereby the solution of the Riemann problem, approximate or exact, is not required. The new MUSTA schemes improve upon the original schemes in terms of monotonicity properties, accuracy and stability in multiple space dimensions. We incorporate the MUSTA technology into the framework of finite-volume weighted essentially nonoscillatory schemes as applied to the Euler equations of compressible gas dynamics. The results demonstrate that our new schemes are good alternatives to current centred methods and to conventional upwind methods as applied to complicated hyperbolic systems for which the solution of the Riemann problem is costly or unknown. Copyright © 2005 John Wiley & Sons, Ltd. [source]

### On the practical importance of the SSP property for Runge,Kutta time integrators for some common Godunov-type schemes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2005
David I. Ketcheson
Abstract We investigate through analysis and computational experiment explicit second and third-order strong-stability preserving (SSP) Runge,Kutta time discretization methods in order to gain perspective on the practical necessity of the SSP property. We consider general theoretical SSP limits for these schemes and present a new optimal third-order low-storage SSP method that is SSP at a CFL number of 0.838. We compare results of practical preservation of the TVD property using SSP and non-SSP time integrators to integrate a class of semi-discrete Godunov-type spatial discretizations. Our examples involve numerical solutions to Burgers' equation and the Euler equations. We observe that ,well-designed' non-SSP and non-optimal SSP schemes with SSP coefficients less than one provide comparable stability when used with time steps below the standard CFL limit. Results using a third-order non-TVD CWENO scheme are also presented. We verify that the documented SSP methods with the number of stages greater than the order provide a useful enhanced stability region. We show by analysis and by numerical experiment that the non-oscillatory third-order reconstructions used in (Liu and Tadmor Numer. Math. 1998; 79:397,425, Kurganov and Petrova Numer. Math. 2001; 88:683,729) are in general only second- and first-order accurate, respectively. Copyright © 2005 John Wiley & Sons, Ltd. [source]

### Further experiences with computing non-hydrostatic free-surface flows involving water waves

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005
Marcel Zijlema
Abstract A semi-implicit, staggered finite volume technique for non-hydrostatic, free-surface flow governed by the incompressible Euler equations is presented that has a proper balance between accuracy, robustness and computing time. The procedure is intended to be used for predicting wave propagation in coastal areas. The splitting of the pressure into hydrostatic and non-hydrostatic components is utilized. To ease the task of discretization and to enhance the accuracy of the scheme, a vertical boundary-fitted co-ordinate system is employed, permitting more resolution near the bottom as well as near the free surface. The issue of the implementation of boundary conditions is addressed. As recently proposed by the present authors, the Keller-box scheme for accurate approximation of frequency wave dispersion requiring a limited vertical resolution is incorporated. The both locally and globally mass conserved solution is achieved with the aid of a projection method in the discrete sense. An efficient preconditioned Krylov subspace technique to solve the discretized Poisson equation for pressure correction with an unsymmetric matrix is treated. Some numerical experiments to show the accuracy, robustness and efficiency of the proposed method are presented. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### Generation of Arbitrary Lagrangian,Eulerian (ALE) velocities, based on monitor functions, for the solution of compressible fluid equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005
B. V. Wells
Abstract A moving mesh method is outlined based on the use of monitor functions. The method is developed from a weak conservation principle. From this principle a conservation law for the mesh position is derived. Using the Helmholtz decomposition theorem, this conservation law can be converted into an elliptic equation for a mesh velocity potential. The moving mesh method is discretized using standard finite elements. Once the mesh velocities are obtained an arbitrary Lagrangian,Eulerian (ALE) (Journal of Computational Physics 1974; 14:227) fluid solver is used to update the solution on the adaptive mesh. Results are shown for the compressible Euler equations of gas dynamics in one and two spatial dimensions. Two monitor functions are used, the fluid density (which corresponds to a Lagrangian description), and a function which includes the density gradient. A variety of test problems are considered. Copyright © 2005 John Wiley & Sons, Ltd. [source]

### On the adjoint solution of the quasi-1D Euler equations: the effect of boundary conditions and the numerical flux function

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8-9 2005
G. F. Duivesteijn
Abstract This work compares a numerical and analytical adjoint equation method with respect to boundary condition treatments applied to the quasi-1D Euler equations. The effect of strong and weak boundary conditions and the effect of flux evaluators on the numerical adjoint solution near the boundaries are discussed. Copyright © 2005 John Wiley & Sons, Ltd. [source]

### Two-dimensional anisotropic Cartesian mesh adaptation for the compressible Euler equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2004
W. A. Keats
Abstract Simulating transient compressible flows involving shock waves presents challenges to the CFD practitioner in terms of the mesh quality required to resolve discontinuities and prevent smearing. This paper discusses a novel two-dimensional Cartesian anisotropic mesh adaptation technique implemented for transient compressible flow. This technique, originally developed for laminar incompressible flow, is efficient because it refines and coarsens cells using criteria that consider the solution in each of the cardinal directions separately. In this paper, the method will be applied to compressible flow. The procedure shows promise in its ability to deliver good quality solutions while achieving computational savings. Transient shock wave diffraction over a backward step and shock reflection over a forward step are considered as test cases because they demonstrate that the quality of the solution can be maintained as the mesh is refined and coarsened in time. The data structure is explained in relation to the computational mesh, and the object-oriented design and implementation of the code is presented. Refinement and coarsening algorithms are outlined. Computational savings over uniform and isotropic mesh approaches are shown to be significant. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### Numerical simulation of dense gas flows on unstructured grids with an implicit high resolution upwind Euler solver

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2004
P. Colonna
Abstract The study of the dense gas flows which occur in many technological applications demands for fluid dynamic simulation tools incorporating complex thermodynamic models that are not usually available in commercial software. Moreover, the software mentioned can be used to study very interesting phenomena that usually go under the name of ,non-classical gasdynamics', which are theoretically predicted for high molecular weight fluids in the superheated region, close to saturation. This paper presents the numerical methods and models implemented in a computer code named zFlow which is capable of simulating inviscid dense gas flows in complex geometries. A detailed description of the space discretization method used to approximate the Euler equations on unstructured grids and for general equations of state, and a summary of the thermodynamic functions required by the mentioned formulation are also given. The performance of the code is demonstrated by presenting two applications, the calculation of the transonic flow around an airfoil computed with both the ideal gas and a complex equation of state and the simulation of the non-classical phenomena occurring in a supersonic flow between two staggered sinusoidal blades. Non-classical effects are simulated in a supersonic flow of a siloxane using a Peng,Robinson-type equation of state. Siloxanes are a class of substances used as working fluids in organic Rankine cycles turbines. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### Reduced order state-space models from the pulse responses of a linearized CFD scheme

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2003
Ann L. Gaitonde
This paper describes a method for obtaining a time continuous reduced order model (ROM) from a system of time continuous linear differential equations. These equations are first put into a time discrete form using a finite difference approximation. The unit sample responses of the discrete system are calculated for each system input and these provide the Markov parameters of the system. An eigenvalue realization algorithm (ERA) is used to construct a discrete ROM. This ROM is then used to obtain a continuous ROM of the original continuous system. The focus of this paper is on the application of this method to the calculation of unsteady flows using the linearized Euler equations on moving meshes for aerofoils undergoing heave or linearized pitch motions. Applying a standard cell-centre spatial discretization and taking account of mesh movement a continuous system of differential equations is obtained which are continuous in time. These are put into discrete time form using an implicit finite difference approximation. Results are presented demonstrating the efficiency of the system reduction method for this system. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Numerical simulation of vortical ideal fluid flow through curved channel

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2003
N. P. Moshkin
Abstract A numerical algorithm to study the boundary-value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co-ordinate system. The convergence of the finite-difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka,Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two-dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Equilibrium real gas computations using Marquina's scheme

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003
Youssef Stiriba
Abstract Marquina's approximate Riemann solver for the compressible Euler equations for gas dynamics is generalized to an arbitrary equilibrium equation of state. Applications of this solver to some test problems in one and two space dimensions show the desired accuracy and robustness. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Moving meshes, conservation laws and least squares equidistribution

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1-2 2002
M. J. Baines
Abstract In this paper a least squares measure of a residual is minimized to move an unstructured triangular mesh into an optimal position, both for the solution of steady systems of conservation laws and for functional approximation. The result minimizes a least squares measure of an equidistribution norm, which is a norm measuring the uniformity of a fluctuation monitor. The minimization is carried out using a steepest descent approach. Shocks are treated using a mesh with degenerate triangles. Results are shown for a steady-scalar advection problem and two flows governed by the Euler equations of gasdynamics. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### Some recent finite volume schemes to compute Euler equations using real gas EOS

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2002
T. Gallouët
Abstract This paper deals with the resolution by finite volume methods of Euler equations in one space dimension, with real gas state laws (namely, perfect gas EOS, Tammann EOS and Van Der Waals EOS). All tests are of unsteady shock tube type, in order to examine a wide class of solutions, involving Sod shock tube, stationary shock wave, simple contact discontinuity, occurrence of vacuum by double rarefaction wave, propagation of a one-rarefaction wave over ,vacuum', , Most of the methods computed herein are approximate Godunov solvers: VFRoe, VFFC, VFRoe ncv (,, u, p) and PVRS. The energy relaxation method with VFRoe ncv (,, u, p) and Rusanov scheme have been investigated too. Qualitative results are presented or commented for all test cases and numerical rates of convergence on some test cases have been measured for first- and second-order (Runge,Kutta 2 with MUSCL reconstruction) approximations. Note that rates are measured on solutions involving discontinuities, in order to estimate the loss of accuracy due to these discontinuities. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2001
Victoria Dolean
Abstract We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two-dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo-time step requires the solution of a sparse linear system for the flow variables. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson-type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes. Copyright © 2001 John Wiley & Sons, Ltd. [source]