Anthony T. Patera

A spectral element method for fluid dynamics: Laminar flow in a channel expansion

A spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible Navier-Stokes equations. In the spectral element discretization, the computational domain is broken into a series of elements, and the velocity in each element is represented as a high-order Lagrangian interpolant through Chebyshev collocation points. The hyperbolic piece of the governing equations is then treated with an explicit collocation scheme, while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle. The implementation of the technique is demonstrated on a one-dimensional inflow-outflow advection-diffusion equation, and the method is then applied to laminar two-dimensional (separated) flow in a channel expansion. Comparisons are made with experiment and previous numerical work.

Secondary instability of wall-bounded shear flows

An analysis is given of a secondary instability that obtains in a wide class of wall-bounded parallel shear flows, including plane Poiseuille flow, plane Couette flow, flat-plate boundary layers, and pipe Poiseuille flow. In these flows it is shown that two-dimensional finite-amplitude waves are (exponentially) unstable to infinitesimal three-dimensional disturbances. This secondary instability seems to be the prototype of transitional instability in these flows in that it has the characteristic (convective) timescales observed in the typical transitions. In the case of plane Poiseuille flow, two-dimensional nonlinear equilibria and quasi-equilibria exist, and the stability of the secondary flow is determined by a three-dimensional linear eigenvalue calculation. In flows without equilibria (e.g. pipe flow), a time-dependent stability analysis is performed by direct spectral numerical calculation of the incompressible three-dimensional Navier–Stokes equations. The energetics and vorticity dynamics of the instability are discussed. It is shown that the two-dimensional wave mediates the transfer of energy from the mean flow to the three-dimensional perturbation but does not directly provide energy to the disturbance. The instability is of an inviscid character as it persists to high Reynolds numbers and grows on convective timescales. Maximum vorticity (inflexion-point) arguments predict some features of the instability like phase-locking of the two-dimensional and three-dimensional waves, but they do not explain its essential three-dimensionality. The inviscid vorticity dynamics of the instability shows that vortext-stretching and tilting effects are both required to explain the persistent exponential growth. The instability is not centrifugal in nature. The three-dimensional instability requires that a threshold two-dimensional amplitude be achieved (about 1% of the centreline velocity in plane Poiseuille flow): the growth rates are relatively insensitive to amplitude for moderate two-dimensional amplitudes. With moderate two-dimensional amplitudes, the critical Reynolds numbers for substantial three-dimensional growth are about 1000 in plane Poiseuille and Couette flows and several thousand in pipe Poiseuille flow. The asymptotic (as R approaches infinity) growth rate in plane Poiseuille flow is approximately 0·15 h / U 0 , where h is the half-channel width and U 0 is the centreline velocity. It is possible to make some progress identifying experimental features of transitional spot structure with aspects of the nonlinear two-dimensional/linear three-dimensional instability. The principal excitation of the eigenfunction of the three-dimensional (growing) disturbance is localized within a given periodicity length (in both the stream and cross-stream directions) near the vorticity maxima of the two-dimensional flow; its planwise structure corresponds to that of observed streaks in early transitional spots; its vortical structure resembles that of a streamwise vortex lifting off the wall. As the three-dimensional disturbance grows to finite amplitude, the flows become chaotic with statistical structure similar to that observed experimentally in moderate-Reynolds-number turbulent shear flows.

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow

In this paper we present a simple, general methodology for the generation of high-order operator decomposition (“splitting”) techniques for the solution of time-dependent problems arising in ordinary and partial differential equations. The new approach exploits operator integration factors to reduce multiple-operator equations to an associated series of single-operator initial-value subproblems. Two illustrations of the procedure are presented: the first, a second-order method in time applied to velocity-pressure decoupling in the incompressible Stokes problem; the second, a third-order method in time applied to convection-Stokes decoupling in the incompressible Navier-Stokes equations. Critical open questions are briefly described.

Domain Decomposition by the Mortar Element Method

The paper reviews recent results concerning the mortar element method, which allows for coupling variational discretizations of different types on nonoverlapping subdomains. The basic ideas and proofs are recalled on a model problem, and new extensions are presented.

Numerical investigation of incompressible flow in grooved channels. Part 1. Stability and self-sustained oscillations

Incompressible moderate-Reynolds-number flow in periodically grooved channels is investigated by direct numerical simulation using the spectral element method. For Reynolds numbers less than a critical value R c the flow is found to approach a stable steady state, comprising an ‘outer’ channel flow, a shear layer at the groove lip, and a weak re-circulating vortex in the groove proper. The linear stability of this flow is then analysed, and it is found that the least stable modes closely resemble Tollmien–Schlichting channel waves, forced by Kelvin–Helmholtz shear-layer instability at the cavity edge. A theory for frequency prediction based on the Orr–Sommerfeld dispersion relation is presented, and verified by variation of the geometric parameters of the problem. The accuracy of the theory, and the fact that it predicts many qualitative features of low-speed groove experiments, suggests that the frequency-selection process in these flows is largely governed by the outer, more stable flow (here a channel), in contrast to most current theories based solely on shear-layer considerations. The instability of the linear mode for R > R c is shown to result in self-sustained flow oscillations (at frequencies only slightly shifted from the originating linear modes), which again resemble (finite-amplitude) Tollmien-Schlichting modes driven by an unstable groove vortex sheet. Analysis of the amplitude dependence of the oscillations on degree of criticality reveals the transition to oscillatory flow to be a regular Hopf bifurcation.

A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations

We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second-order coercive linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic ‘energy’ reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine ‘truth-mesh’ discretization are then derived by appealing to a dual max min relaxation evaluated for optimally chosen adjoint and hybrid-flux candidate Lagrange multipliers generated by a K-element coarser ‘working-mesh’ approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomain-local, symmetric Neumann problems. The technique is illustrated for the convection-diffusion and linear elasticity equations.

A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly convergent global reduced-basis approximations - (Galerkin) projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate the output of interest and associated error bound - depends only on N (typically very small) and the parametric complexity of the problem. In this paper we develop new a posteriori error estimation procedures for noncoercive linear, and certain nonlinear, problems that yield rigorous and sharp error statements for all N. We consider three particular examples: the Helmholtz (reduced-wave) equation; a cubically nonlinear Poisson equation; and Burgers equation - a model for incompressible Navier-Stokes. The Helmholtz (and Burgers) example introduce our new lower bound constructions for the requisite inf-sup (singular value) stability factor; the cubic nonlin-earity exercises symmetry factorization procedures necessary for treatment of high-order Galerkin summations in the (say) residual dual-norm calculation; and the Burgers equation illustrates our accommodation of potentially multiple solution branches in our a posteriori error statement. Numerical results are presented that demonstrate the rigor, sharpness, and efficiency of our proposed error bounds, and the application of these bounds to adaptive (optimal) approximation.

A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations

We consider “Lagrangian” reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.

An isoparametric spectral element method for solution of the Navier-Stokes equations in complex geometry

Abstract An isoparametric spectral element technique is presented for solution of the two-dimensional Navier-Stokes equations in arbitrary (curvy) geometries. The spatial discretization is described, and the method is illustrated by solution of Poissons equation in a cylindrical annulus. A fractional-step Navier-Stokes solver based on a discretely consistent Poisson equation for the pressure is then presented, and exponential convergence in space is demonstrated for the case of (separated) flow between eccentric rotating cylinders. The technique is made efficient by the use of explicit collocation (rather than Galerkin) operators for the convective terms, and use of static condensation to solve the elliptic equations resulting from the Stokes problem, The algorithm is amenable to a high degree of parallelism, in both multiprocessor and vector-pipeline environments.