Discretized Euler Equations Research Articles

A numerical method based on the Nambu bracket for the 3D vorticity equation

Abstract In this study, numerical methods for the 3D vorticity equation and Euler equation, which preserve the structure of Nambu mechanics for fluid dynamics, are investigated. Discrete vector fields of the stream function (or vector potential), velocity, and vorticity, as well as discrete counterparts of the gradient, curl, and divergence operators acting on them, are defined such that the structure of the de Rham complex in 3D Euclidean space is preserved. The inner products of the discrete vector fields are defined such that discrete counterparts of integration-by-parts formulae for the gradient, curl, and divergence operators hold. In addition, cross products of the discrete vector fields are introduced to define a skew-symmetric trilinear form. A discrete Nambu bracket, as well as the (kinetic) energy, helicity, and enstrophy of the discrete flow field, are defined straightforwardly. They are employed to derive a discrete vorticity equation in the same way as in the continuum setting. A discrete Euler equation is derived from the discrete vorticity equation based on the discrete counterpart of the Poincaré lemma, which holds under some typical conditions. It is proved that any solution to these discretized equations satisfies discrete analogues of the balances of energy, helicity, and enstrophy. Numerical experiments on a periodic array of rolls are conducted to examine the effectiveness of the method.

Asymptotics of perturbed discrete Euler equations in the critical case

We consider the difference equation Δ2y(k)+p(k)y(k+1)=0 under the condition limk→∞⁡k2p(k)=1/4; such a setting can cover various perturbations of the critical (double root) case in Euler type difference equations. We establish precise asymptotic formulae for all nonoscillatory solutions. An important role is played by the concept of the refined discrete regular variation, transformations of dependent and independent variable, and asymptotic theory of dynamic equations on time scales.

The endogenous grid method for discrete-continuous dynamic choice models with (or without) taste shocks

We present a fast and accurate computational method for solving and estimating a class of dynamic programming models with discrete and continuous choice variables. The solution method we develop for structural estimation extends the endogenous grid‐point method (EGM) to discrete‐continuous (DC) problems. Discrete choices can lead to kinks in the value functions and discontinuities in the optimal policy rules, greatly complicating the solution of the model. We show how these problems are ameliorated in the presence of additive choice‐specific independent and identically distributed extreme value taste shocks that are typically interpreted as “unobserved state variables” in structural econometric applications, or serve as “random noise” to smooth out kinks in the value functions in numerical applications. We present Monte Carlo experiments that demonstrate the reliability and efficiency of the DC‐EGM algorithm and the associated maximum likelihood estimator for structural estimation of a life‐cycle model of consumption with discrete retirement decisions. Life‐cycle model discrete and continuous choice Bellman equation Euler equation retirement choice endogenous grid‐point method nested fixed point algorithm extreme value taste shocks smoothed max function structural estimation C13 C63 D91

A Comparison between High-order Temporal Integration Methods Applied to the Discontinuous Galerkin Discretized Euler Equations

In this work we present a high-order Discontinuous Galerkin (DG) space approximation coupled with two high-order temporal integration methods for the numerical solution of time-dependent compressible flows. The time integration methods analyzed are the explicit Strong-Stability-Preserving Runge-Kutta (SSPRK) and the Two Implicit Advanced Step-point (TIAS) schemes. Their accuracy and efficiency are evaluated by means of an inviscid test case for which an exact solution is available. The study is carried out for several time-steps using different polynomial order approximations and several levels of grid refinement. The effect of mesh irregularities on the accuracy is also investigated by considering randomly perturbed meshes. The analysis of the results has the twofold objective of (i) assessing the performances of the temporal schemes in the context of the high-order DG discretization and(ii) determining if high-order implicit schemes can displace widely used high-order explicit schemes.

Nonlinear model reduction for unsteady discontinuous flows

We develop a new nonlinear reduced-order model (ROM) based on proper orthogonal decomposition (POD), which can be used for quantitative simulation of not only smooth flows, but also flows with strong discontinuities. The new model is derived using a Galerkin projection of the fully conservative, nonlinear discretized 2-D Euler equations onto the POD basis constructed for each conservative variable. This approach can be interpreted as a variant of the spectral method with a truncated set of basis functions. A system of ordinary differential equations (ODEs) derived using this model reduction technique resembles the major nonlinear and conservation properties of the original discretized Euler equations. The new reduced-order model also preserves the stability properties of the discrete full-order model equations, so that no additional stabilization is required unlike conventional POD-based models that are susceptible to numerical instabilities. The performance of the new POD ROM is evaluated for 2-D compressible unsteady inviscid flows over a wide range of Mach numbers including trans- and supersonic flows with strong shock waves.

Node-pair finite volume/finite element schemes for the Euler equation in cylindrical and spherical coordinates

A numerical scheme is presented for the solution of the compressible Euler equations in both cylindrical and spherical coordinates. The unstructured grid solver is based on a mixed finite volume/finite element approach. Equivalence conditions linking the node-centered finite volume and the linear Lagrangian finite element scheme over unstructured grids are reported and used to devise a common framework for solving the discrete Euler equations in both the cylindrical and the spherical reference systems. Numerical simulations are presented for the explosion and implosion problems with spherical symmetry, which are solved in both the axial–radial cylindrical coordinates and the radial–azimuthal spherical coordinates. Numerical results are found to be in good agreement with one-dimensional simulations over a fine mesh.

Critical Oscillation Constant for Difference Equations with Almost Periodic Coefficients

We investigate a type of the Sturm‐Liouville difference equations with almost periodic coefficients. We prove that there exists a constant, which is the borderline between the oscillation and the nonoscillation of these equations. We compute this oscillation constant explicitly. If the almost periodic coefficients are replaced by constants, our result reduces to the well‐known result about the discrete Euler equation.

First variation formula and conservation laws in several independent discrete variables

This paper sets the scene for discrete variational problems on an abstract cellular complex that serves as discrete model of R p and for the discrete theory of partial differential operators that are common in the Calculus of Variations. A central result is the construction of a unique decomposition of certain partial difference operators into two components, one that is a vector bundle morphism and other one that leads to boundary terms. Application of this result to the differential of the discrete Lagrangian leads to unique discrete Euler and momentum forms not depending either on the choice of reference on the base lattice or on the choice of coordinates on the configuration manifold, and satisfying the corresponding discrete first variation formula. This formula leads to discrete Euler equations for critical points and to exact discrete conservation laws for infinitesimal symmetries of the Lagrangian density, with a clear physical interpretation.

Structure-preserving discretization of incompressible fluids

The geometric nature of Euler fluids has been clearly identified and extensively studied over the years, culminating with Lagrangian and Hamiltonian descriptions of fluid dynamics where the configuration space is defined as the volume-preserving diffeomorphisms, and Kelvin’s circulation theorem is viewed as a consequence of Noether’s theorem associated with the particle relabeling symmetry of fluid mechanics. However computational approaches to fluid mechanics have been largely derived from a numerical–analytic point of view, and are rarely designed with structure preservation in mind, and often suffer from spurious numerical artifacts such as energy and circulation drift. In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form. Our approach approximates the group of volume-preserving diffeomorphisms using a finite-dimensional Lie group, and associated discrete Euler equations are derived from a variational principle with non-holonomic constraints. The resulting discrete equations of motion yield a structure-preserving time integrator with good long-term energy behavior and for which an exact discrete Kelvin’s circulation theorem holds.

Aeroelastic Damping Model Derived from Discrete Euler Equations

The prediction of flutter onset based on aerodynamic modeling using computational fluid dynamics can be made using an augmented system of equations. Computational times similar to those required for computational fluid dynamics steady-state calculations have been reported for wing test cases. However, for such methods to be fully useful,information aboutdamping must beobtainable without revertingto full-order time-domain simulation.This paper presents a method for computing damping based on a reduced-order modeling approach that systematically derives a two-degrees-of-freedom model from the full discrete system of equations. The method is based on a change of variables that employs the critical eigenvector of the aeroelastic system. The ability of this model to predict the damping for a model problem and for two wing test cases is shown.

Uncertainty quantification of limit-cycle oscillations

Different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit-cycle oscillation, subject to parametric variability. The aeroelastic system is that of a rigid airfoil, supported by pitch and plunge structural coupling, with nonlinearities in the component in pitch. The nonlinearities are adjusted to permit the formation of a either a subcritical or supercritical branch of limit-cycle oscillations. Uncertainties are specified in the cubic coefficient of the torsional spring and in the initial pitch angle of the airfoil. Stochastic projections of the time-domain and cyclic equations governing system response are carried out, leading to both intrusive and non-intrusive computational formulations. Non-intrusive formulations are examined using stochastic projections derived from Wiener expansions involving Haar wavelet and B-spline bases, while Wiener–Hermite expansions of the cyclic equations are employed intrusively and non-intrusively. Application of the B-spline stochastic projection is extended to the treatment of aerodynamic nonlinearities, as modeled through the discrete Euler equations. The methodologies are compared in terms of computational cost, convergence properties, ease of implementation, and potential for application to complex aeroelastic systems.

Application of proper orthogonal decomposition to the discrete Euler equations

AbstractThe response of a fluid moving above a panel to localized oscillation of the panel is predicted using reduced‐order modelling (ROM) with the proper orthogonal decomposition technique. The flow is assumed to be inviscid and is modelled with the Euler equations. These non‐linear equations are discretized with a total‐variation diminishing algorithm and are projected onto an energy‐optimal subspace defined by an energy‐threshold criterion applied to a modal representation of time series data. Results are obtained for a bump oscillating in a Mach 1.2 flow. ROM is found to reduce the degrees of freedom necessary to simulate the flowfield by three orders of magnitude while preserving solution accuracy. Other observed benefits of ROM include increased allowable time step and robustness to variation of oscillation amplitude. Published in 2002 by John Wiley & Sons, Ltd.

A two-dimensional linearized unsteady Euler scheme for pulse response calculations

An unsteady linearized Euler scheme for use on moving meshes is presented. This is derived from a scheme for the full non-linear unsteady Euler equations. This scheme is based on the Jameson cell-centred scheme, but is time-accurate and includes the necessary terms to account for grid motion. It is assumed that the unsteadiness in the flow and mesh is small. Using this assumption the discrete unsteady Euler equations are linearized about the full, non-linear, steady mean flow. The resulting equations are solved in the work presented here using a dual-time scheme. In the basic scheme no assumptions are made about the form of the perturbations other than that they are small. This permits the direct calculation of non-periodic flows, e.g. pulse responses. Linear pulse responses are a useful tool as they can be used to calculate the flow due to general inputs. The equations that would result from the assumption of harmonic flow are also derived. Results are presented for heave, pitch and ramp test cases and compared to full non-linear Euler results calculated using a dual-time scheme.

The finite volume flic method and its stability analysis

In this study, a general finite volume Fluid-in-cell method (FVFLIC) for solving the Navier—Stokes equations is introduced. The stability of the numerical method is then analysed by directly using two-dimensional Euler equations instead of a linear model equation. This direct approach to the analysis of non-linear stability is based on the Taylor expansion of the discretized Euler equations and some basic principals that have been used for analysing linear model equations. The exact forms of numerical viscosity or truncation errors are derived and discussed. The influences of the numerical viscosity as well as the artificial viscosity on numerical solutions are investigated. Results from this analysis can be used to construct appropriate artificial viscosity terms. Based on the above methodology, a stability criterion is proposed for the calculation of time steps for general three-dimensional computation using non-orthogonal grids.

Condition improvement for point relaxation in multigrid, subsonic Euler-flow computations

Insight is given into the conditions of derivative matrices to be inverted in point-relaxation methods for 1-D and 2-D, first-order upwind-discretized Euler equations. Speed regimes are found where ill-conditioning of these matrices occurs; 1-D flow equations appear to be less well conditioned than 2-D flow equations. The ill-conditioning is easily improved by adding regularizing matrices to the derivative matrices. A smoothing analysis is made of point Gauss-Seidel relaxation applied to discrete Euler equations conditioned by such an additive matrix. The method is successfully applied to a very low-subsonic, steady, 2-D stagnation flow.

Convergence Analysis of the Defect-Correction Iteration for Hyperbolic Problems

This paper explains some of the convergence behaviour of iterative implicit and defect-correction schemes for the solution of the discrete steady Euler equations. Such equations are also commonly solved by (pseudo) time integration, the steady solution being achieved as the limit (for $t \to \infty $) of the solution of a time-dependent problem. Implicit schemes are then often chosen for their favourable stability properties, permitting large timesteps for efficiency. An important class of implicit schemes involving first- and second-order accurate upwind discretisations is considered. In the limit of an infinite timestep, these implicit schemes approach defect-correction algorithms. Thus our analysis is informative for both types of construction.Simple scalar linear model problems are introduced for the one-dimensional and two-dimensional cases. These model problems are analyzed in detail by both Fourier and matrix analyses. The convergence behaviour appears to be strongly dependent on a parameter $\beta...

Modification of the Euler equations for ‘‘vorticity confinement’’: Application to the computation of interacting vortex rings

A new ‘‘vorticity confinement’’ method is described which involves adding a term to the momentum conservation equations of fluid dynamics. This term depends only on local variables and is zero outside vortical regions. The partial differential equations with this extra term admit solutions that consist of Lagrangian-like confined vortical regions, or covons, in the shape of two-dimensional (2-D) vortex ‘‘blobs’’ and three-dimensional (3-D) vortex filaments, which convect in a constant external velocity field with a fixed internal structure, without spreading, even if the equations contain diffusive terms. Solutions of the discretized equations on a fixed Eulerian grid show the same behavior, in spite of numerical diffusion. Effectively, the new term, together with diffusive terms, constitute a new type of regularization of the inviscid equations which appears to be very useful in the numerical solution of flow problems involving thin vortical regions. The discretized Euler equations with the extra term can be solved on fairly coarse, Eulerian computational grids with simple low-order (first- or second-) accurate numerical methods, but will still yield concentrated vortices which convect without spreading due to numerical diffusion. Since only a fixed grid is used with local variables, the vorticity confinement method is quite general and can automatically accommodate changes in vortex topology, such as merging. Applications are presented for incompressible flow in 3D, where pairs of thin vortex rings interact and, in some cases, merge.

Equivalence of Discrete Euler Equations and Discrete Hamiltonian Systems

Erbe and Yan recently presented a discrete linear Hamiltonian system. Their system is a special case of the discrete Hamiltonian system Δy(n − l) = Hz(n, y(n), z(n − l))Δz(n − l) = −Hy(n, y(n), z(n − l)), where Δy(n − 1) = y(n) − y(n − 1). Under certain implicit solvability hypotheses, these systems are equivalent to the discrete Euler equation ƒy(n, yn, Δyn − l) = Δƒr(n, yn, Δyn − l). A Reid Roundabout Theorem for linear recurrence relations −Knyn+1 + Bnyn − KTn−1yn−1 = 0 is shown to imply the corresponding result obtained by Erbe and Yan for discrete linear Hamiltonian systems. Furthermore, discrete linear Hamiltonian systems are shown to have a symplectic transition matrix.

Design of optimally smoothing multistage schemes for the euler equations

AbstractA recently derived local preconditioning of the Euler equations is shown to be useful in developing multistage schemes suited for multigrid use. The effect of the preconditioning matrix on the spatial Euler operator is to equalize the characteristic speeds. When applied to the discretized Euler equations, the preconditioning has the effect of strongly clustering the operator's eigenvalues in the complex plane. This makes possible the development of explicit marching schemes that effectively damp most high‐frequency Fourier modes, as desired in multigrid applications. The technique is the same as developed earlier for scalar convection schemes: placement of the zeros of the amplification factor of the multistage scheme in locations where eigenvalues corresponding to high‐frequency modes abound.

Finite difference solutions of the Euler equations in the vicinity of sharp edges

Attempts have been made to explain why finite difference solutions of the Euler equations can describe flows with large vortical structures around sharp-edged bodies. The present paper is concerned with the influence of a singular sharp edge on the truncation error for a set of discretized Euler equations. An analysis is conducted of the distribution of the truncation error of one finite difference approximation of the Euler equations near a sharp edge of a thin plate. The analysis leads to a determination of the size of the region of the neighborhood of such a singularity. Attention is given to the consistency of a discretization of the Euler equations, and numerical experiments.