Optimal control of unsteady compressible viscous flows

The control of complex, unsteady flows is a pacing technology for advances in fluid mechanics. Recently, optimal control theory has become popular as a means of predicting best case controls that can guide the design of practical flow control systems. However, most of the prior work in this area has focused on incompressible flow which precludes many of the important physical flow phenomena that must be controlled in practice including the coupling of fluid dynamics, acoustics, and heat transfer. This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady two‐dimensional compressible Navier–Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counter‐rotating viscous vortices above an infinite wall which, due to the self‐induced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wall‐normal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case. Copyright © 2002 John Wiley & Sons, Ltd.     

Global linear stability analysis of time‐averaged flows

The global linear stability analysis (LSA) of stationary/steady flows has been applied to various flows in the past and is fairly well understood. The LSA of time‐averaged flows is explored in this paper. It is shown that the LSA of time‐averaged flows can result in useful information regarding its stability. The method is applied to study flow past a cylinder at Reynolds number (Re) beyond the onset of vortex shedding. Compared with the direct numerical simulation, LSA of the Re=100 steady flow severely underpredicts the vortex shedding frequency. However, the LSA of the time‐averaged flow results in the correct value of the non‐dimensional frequency, St, of the associated instability. Copyright © 2007 John Wiley & Sons, Ltd.     

Parallel finite element computation of unsteady incompressible flows

A parallel semi-explicit iterative finite element computational procedure for modelling unsteady incompressible fluid flows is presented. During the procedure, element flux vectors are calculated in parallel and then assembled into global flux vectors. Equilibrium iterations which introduce some ‘local implicitness’ are performed at each time step. The number of equilibrium iterations is governed by an implicitness parameter. The present technique retains the advantages of purely explicit schemes, namely (i) the parallel speed-up is equal to the number of parallel processors if the small communication overhead associated with purely explicit schemes is ignored and (ii) the computation time as well as the core memory required is linearly proportional to the number of elements. The incompressibility condition is imposed by using the artificial compressibility technique. A pressure-averaging technique which allows the use of equal-order interpolations for both velocity and pressure, this simplifying the formulation, is employed. Using a standard Galerkin approximation, three benchmark steady and unsteady problems are solved to demonstrate the accuracy of the procedure. In all calculations the Reynolds number is less than 500. At these Reynolds numbers it was found that the physical dissipation is sufficient to stabilize the convective term with no need for additional upwind-type dissipation. © 1998 John Wiley & Sons, Ltd.     

On the use of sensitivity analysis in model reduction to predict flows for varying inflow conditions

The proper orthogonal decomposition (POD)‐based model reduction method is more and more successfully used in fluid flows. However, the main drawback of this methodology rests in the robustness of these reduced order models (ROMs) beyond the reference at which POD modes have been derived. Any variation in the flow or shape parameters within the ROM fails to predict the correct dynamics of the flow field. To broaden the spectrum of these models, the POD modes should have the global characteristics of the flow field over which the predictions are required. Mixing of snapshots with varying parameters is one way to improve the global nature of the modes but is computationally demanding because it requires full‐order solutions for a number of parameter values in order to assemble atextitrich enough database on which to perform POD. Instead, we have used sensitivity analysis (SA) to include the flow and shape parameters influence during the basis selection process to develop more robust ROMs for varying viscosity (Reynolds number), changing orientation and shape definition of bodies. This study aims at extending these ideas to inflow conditions to demonstrate the effectiveness of the proposed approach in capturing the effect of varying inflow on the dynamics of the flow over an elliptic cylinder. Numerical experiments show that the newly derived models allow for a more accurate representation of the flows when exploring the parameter space. Copyright © 2011 John Wiley & Sons, Ltd.     

On an attempt to simplify the Quartapelle–Napolitano approach to computation of hydrodynamic forces in open flows

In this paper we are interested in the Quartapelle–Napolitano approach to calculation of forces in viscous incompressible flows in exterior domains. We study the possibility of deriving a simpler formulation of this approach which might lead to a more convenient expression for the hydrodynamic force, but conclude that such a simplification is, within the family of approaches considered, impossible. This shows that the original Quartapelle–Napolitano formula is in fact “optimal” within this class of approaches.     

An extension of the SIMPLE based discontinuous Galerkin solver to unsteady incompressible flows

In this paper, we present a SIMPLE based algorithm in the context of the discontinuous Galerkin method for unsteady incompressible flows. Time discretization is done fully implicit using backward differentiation formulae (BDF) of varying order from 1 to 4. We show that the original equation for the pressure correction can be modified by using an equivalent operator stemming from the symmetric interior penalty (SIP) method leading to a reduced stencil size. To assess the accuracy as well as the stability and the performance of the scheme, three different test cases are carried out: the Taylor vortex flow, the Orr‐Sommerfeld stability problem for plane Poiseuille flow and the flow past a square cylinder. (1) Simulating the Taylor vortex flow, we verify the temporal accuracy for the different BDF schemes. Using the mixed‐order formulation, a spatial convergence study yields convergence rates of k + 1 and k in the L²‐norm for velocity and pressure, respectively. For the equal‐order formulation, we obtain approximately the same convergence rates, while the absolute error is smaller. (2) The stability of our method is examined by simulating the Orr–Sommerfeld stability problem. Using the mixed‐order formulation and adjusting the penalty parameter of the symmetric interior penalty method for the discretization of the viscous part, we can demonstrate the long‐term stability of the algorithm. Using pressure stabilization the equal‐order formulation is stable without changing the penalty parameter. (3) Finally, the results for the flow past a square cylinder show excellent agreement with numerical reference solutions as well as experiments. Copyright © 2015 John Wiley & Sons, Ltd.     

GENSMAC3D: a numerical method for solving unsteady three‐dimensional free surface flows

A numerical method for solving three‐dimensional free surface flows is presented. The technique is an extension of the GENSMAC code for calculating free surface flows in two dimensions. As in GENSMAC, the full Navier–Stokes equations are solved by a finite difference method; the fluid surface is represented by a piecewise linear surface composed of quadrilaterals and triangles containing marker particles on their vertices; the stress conditions on the free surface are accurately imposed; the conjugate gradient method is employed for solving the discrete Poisson equation arising from a velocity update; and an automatic time step routine is used for calculating the time step at every cycle. A program implementing these features has been interfaced with a solid modelling routine defining the flow domain. A user‐friendly input data file is employed to allow almost any arbitrary three‐dimensional shape to be described. The visualization of the results is performed using computer graphic structures such as phong shade, flat and parallel surfaces. Results demonstrating the applicability of this new technique for solving complex free surface flows, such as cavity filling and jet buckling, are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

Pressure segregation methods based on a discrete pressure Poisson equation. An algebraic approach

In this paper, we introduce some pressure segregation methods obtained from a non‐standard version of the discrete monolithic system, where the continuity equation has been replaced by a pressure Poisson equation obtained at the discrete level. In these methods it is the velocity instead of the pressure the extrapolated unknown. Moreover, predictor–corrector schemes are suggested, again motivated by the new monolithic system. Key implementation aspects are discussed, and a complete stability analysis is performed. We end with a set of numerical examples in order to compare these methods with classical pressure‐correction schemes. Copyright © 2007 John Wiley & Sons, Ltd.     

An unsteady incompressible Navier–Stokes solver for large eddy simulation of turbulent flows

An unsteady incompressible Navier–Stokes solver that uses a dual time stepping method combined with spatially high‐order‐accurate finite differences, is developed for large eddy simulation (LES) of turbulent flows. The present solver uses a primitive variable formulation that is based on the artificial compressibility method and various convergence–acceleration techniques are incorporated to efficiently simulate unsteady flows. A localized dynamic subgrid model, which is formulated using the subgrid kinetic energy, is employed for subgrid turbulence modeling. To evaluate the accuracy and the efficiency of the new solver, a posteriori tests for various turbulent flows are carried out and the resulting turbulence statistics are compared with existing experimental and direct numerical simulation (DNS) data. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of incompressible massively separated viscous flows using unsteady Navier–Stokes equations

The unsteady incompressible Navier–Stokes equations are formulated in terms of vorticity and stream-function in generalized curvilinear orthogonal co-ordinates to facilitate analysis of flow configurations with general geometries. The numerical method developed solves the conservative form of the vorticity transport equation using the alternating direction implicit method, whereas the streamfunction equation is solved by direct block Gaussian elimination. The method is applied to a model problem of flow over a backstep in a doubly infinite channel, using clustered conformal co-ordinates. One-dimensional stretching functions, dependent on the Reynolds number and the asymptotic behaviour of the flow, are used to provide suitable grid distribution in the separation and reattachment regions, as well as in the inflow and outflow regions. The optimum grid distribution selected attempts to honour the multiple length scales of the separated flow model problem. The asymptotic behaviour of the finite differenced transport equation near infinity is examined and the numerical method is carefully developed so as to lead to spatially second-order-accurate wiggle-free solutions, i.e. with minimum dispersive error. Results have been obtained in the entire laminar range for the backstep channel and are in good agreement with the available experimental data for this flow problem, prior to the onset of three-dimensionality in the experiment.     

A distributed Lagrange multiplier/fictitious domain method for flows around moving rigid bodies: Application to particulate flow

This article discusses the application of a Lagrange multiplier‐based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the Navier–Stokes equations around moving rigid bodies; the rigid body motions are due to hydrodynamical forces and gravity. The solution method combines finite element approximations, time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. The study concludes with the presentation of numerical results concerning four test problems, namely the simulation of an incompressible viscous flow around a NACA0012 airfoil with a fixed center but free to rotate, then the sedimentation of 200 and 1008 cylinders in a two‐dimensional channel, and finally the sedimentation of two spherical balls in a rectangular cylinder. Copyright © 1999 John Wiley & Sons, Ltd.     

Three‐dimensional finite element modelling of coupled free/porous flows: applications to industrial and environmental flows

Conjunctive modelling of free/porous flows provides a powerful and cost‐effective tool for designing industrial filters used in the process industry and also for quantifying surface–subsurface flow interactions, which play a significant role in urban flooding mechanisms resulting from sea‐level rise and climate changes. A number of well‐established schemes are available in the literature for simulation of such regimes; however, three‐dimensional (3D) modelling of such flow systems still presents numerical and practical challenges. This paper presents the development of a fully 3D, transient finite element model for the prediction and quantitative analyses of the hydrodynamic behaviour encountered in industrial filtrations and environmental flows represented by coupled flows. The weak‐variational formulation in this model is based on the use of C⁰ continuous equal‐order Lagrange polynomial functions for velocity and pressure fields represented by 3D hexahedral finite elements. A mixed UVWP finite element scheme based on the standard Galerkin technique satisfying the Ladyzhenskaya–Babuska–Brezzi stability criterion through incorporation of an artificial compressibility term in the continuity equation has been employed for the solution of coupled partial differential equations. We prove that the discretization generates unified stabilization for both the Navier–Stokes and Darcy equations and preserves the geometrical flexibility of the computational grids. A direct node‐linking procedure involving the rearrangement of the global stiffness matrix for the interface elements has been developed by the authors, which is utilized to couple the governing equations in a single model. A variety of numerical tests are conducted, indicating that the model is capable of yielding theoretically expected and accurate results for free, porous and coupled free/porous problems encountered in industrial and environmental engineering problems representing complex filtration (dead‐end and cross‐flow) and interacting surface–subsurface flows. The model is computationally cost‐effective, robust, reliable and easily implementable for practical design of filtration equipments, investigation of land use for water resource availability and assessment of the impacts of climatic variations on environmental catastrophes (i.e. coastal and urban floods). The model developed in this work results from the extension of a multi‐disciplinary project (AEROFIL) primarily sponsored by the European aerospace industries for development of a computer simulation package (Aircraft Cartridge Filter Analysis Modelling Program), which was successfully utilized and deployed for designing hydraulic dead‐end filters used in Airbus A380.Copyright © 2012 John Wiley & Sons, Ltd.     

Fast solvers for models of ICEO microfluidic flows

We demonstrate the performance of a fast computational algorithm for modeling the design of a microfluidic mixing device. The device uses an electrokinetic process, induced charge electroosmosis (J. Fluid Mech. 2004; 509 ), by which a flow through the device is driven by a set of polarizable obstacles in it. Its design is realized by manipulating the shape and orientation of the obstacles in order to maximize the amount of fluid mixing within the device. The computation entails the solution of a constrained optimization problem in which function evaluations require the numerical solution of a set of partial differential equations: a potential equation, the incompressible Navier–Stokes equations, and a mass‐transport equation. The most expensive component of the function evaluation (which must be performed at every step of an iteration for the optimization) is the solution of the Navier–Stokes equations. We show that by using some new robust algorithms for this task (SIAM J. Sci. Comput. 2002; 24 :237–256; J. Comput. Appl. Math. 2001; 128 :261–279), based on certain preconditioners that take advantage of the structure of the linearized problem, this computation can be done efficiently. Using this computational strategy, in conjunction with a derivative‐free pattern search algorithm for the optimization, applied to a finite element discretization of the problem, we are able to determine optimal configurations of microfluidic devices. Copyright © 2009 John Wiley & Sons, Ltd.     

Modeling of Steady Flows in a Channel by Navier–Stokes Variational Inequalities

A mathematical model of a steady viscous incompressible fluid flow in a channel with exit conditions different from the Dirichlet conditions is considered. A variational inequality is derived for the formulated subdifferential boundary-value problem, and the structure of the set of its solutions is studied. For two-ption on the low Reynolds number is proved. In the three-dimensional case, a class of constraints on the tangential component of velocity at the exit, which guarantees solvability of the variational inequality, is found.     

Three-dimensional unsteady flow simulations: Alternative strategies for a volume-averaged calculation

This work builds on a SIMPLE-type code to produce two numerical codes of greatly improved speed and accuracy for solution of the Navier–Stokes equations. Both implicit and explicit codes employ an improved QUICK (quadratic upstream interpolation for convective kinematics) scheme to finite difference convective terms for non-uniform grids. The PRIME (update pressure implicit, momentum explicit) algorithm is used as the computational procedure for the implicit code. Use of both the ICCG (incomplete Cholesky decomposition, conjugate gradient) method and the MG (multigrid) technique to enhance solution execution speed is illustrated. While the implicit code is first-order in time, the explicit is second-order accurate. Two- and three-dimensional forced convection and sidewall-heated natural convection flows in a cavity are chosen as test cases. Predictions with the new schemes show substantial computational savings and very good agreement when compared to previous simulations and experimental data.     

Evaluation of an energy relaxation method for the simulation of unsteady,viscous, real gas flows

This study investigates a new energy relaxation method designed to capture the dynamics of unsteady, viscous, real gas flows governed by the compressible Navier–Stokes equations. We focus on real gas models accounting for inelastic molecular collisions and yielding temperature‐dependent heat capacities. The relaxed Navier–Stokes equations are discretized using a mixed finite volume/finite element method and a high‐order time integration scheme. The accuracy of the energy relaxation method is investigated on three test problems of increasing complexity: the advection of a periodic set of vortices, the interaction of a temperature spot with a weak shock, and finally, the interaction of a reflected shock with its trailing boundary layer in a shock tube. In all cases, the method is validated against benchmark solutions and the numerical errors resulting from both discretization and energy relaxation are assessed independently. Copyright © 2004 John Wiley & Sons, Ltd.     

The Lagrangian approach of advective term treatment and its application to the solution of Navier—Stokes equations

A one-dimensional transport test applied to some conventional advective Eulerian schemes shows that linear stability analyses do not guarantee the actual performances of these schemes. When adopting the Lagrangian approach, the main problem raised in the numerical treatment of advective terms is a problem of interpolation or restitution of the transported function shape from discrete data. Several interpolation methods are tested. Some of them give excellent results and these methods are then extended to multi-dimensional cases. The Lagrangian formulation of the advection term permits an easy solution to the Navier-Stokes equations in primitive variables V, p, by a finite difference scheme, explicit in advection and implicit in diffusion. As an illustration steady state laminar flow behind a sudden enlargement is analysed using an upwind differencing scheme and a Lagrangian scheme. The importance of the choice of the advective scheme in computer programs for industrial application is clearly apparent in this example.     

Finite‐element/level‐set/operator‐splitting (FELSOS) approach for computing two‐fluid unsteady flows with free moving interfaces

The present work is devoted to the study on unsteady flows of two immiscible viscous fluids separated by free moving interface. Our goal is to elaborate a unified strategy for numerical modelling of two‐fluid interfacial flows, having in mind possible interface topology changes (like merger or break‐up) and realistically wide ranges for physical parameters of the problem. The proposed computational approach essentially relies on three basic components: the finite element method for spatial approximation, the operator‐splitting for temporal discretization and the level‐set method for interface representation. We show that the finite element implementation of the level‐set approach brings some additional benefits as compared to the standard, finite difference level‐set realizations. In particular, the use of finite elements permits to localize the interface precisely, without introducing any artificial parameters like the interface thickness; it also allows to maintain the second‐order accuracy of the interface normal, curvature and mass conservation. The operator‐splitting makes it possible to separate all major difficulties of the problem and enables us to implement the equal‐order interpolation for the velocity and pressure. Diverse numerical examples including simulations of bubble dynamics, bifurcating jet flow and Rayleigh–Taylor instability are presented to validate the computational method. Copyright © 2004 John Wiley & Sons, Ltd.     

A simple and efficient error analysis for multi‐step solution of the Navier–Stokes equations

A simple error analysis is used within the context of segregated finite element solution scheme to solve incompressible fluid flow. An error indicator is defined based on the difference between a numerical solution on an original mesh and an approximated solution on a related mesh. This error indicator is based on satisfying the steady‐state momentum equations. The advantages of this error indicator are, simplicity of implementation (post‐processing step), ability to show regions of high and/or low error, and as the indicator approaches zero the solution approaches convergence. Two examples are chosen for solution; first, the lid‐driven cavity problem, followed by the solution of flow over a backward facing step. The solutions are compared to previously published data for validation purposes. It is shown that this rather simple error estimate, when used as a re‐meshing guide, can be very effective in obtaining accurate numerical solutions. Copyright © 2002 John Wiley & Sons, Ltd.