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About this book

Instabilities of fluid flows and the associated transitions between different possible flow states provide a fascinating set of problems that have attracted researchers for over a hundred years. This book addresses state-of-the-art developments in numerical techniques for computational modelling of fluid instabilities and related bifurcation structures, as well as providing comprehensive reviews of recently solved challenging problems in the field.

Table of Contents


Novel Methods and Approaches


Order-of-Magnitude Speedup for Steady States and Traveling Waves via Stokes Preconditioning in Channelflow and Openpipeflow

Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton’s method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier–Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10–50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.
Laurette S. Tuckerman, Jacob Langham, Ashley Willis

Time-Stepping and Krylov Methods for Large-Scale Instability Problems

With the ever increasing computational power available and the development of high-performances computing, investigating the properties of realistic very large-scale nonlinear dynamical systems has become reachable. It must be noted however that the memory capabilities of computers increase at a slower rate than their computational capabilities. Consequently, the traditional matrix-forming approaches wherein the Jacobian matrix of the system considered is explicitly assembled become rapidly intractable. Over the past two decades, so-called matrix-free approaches have emerged as an efficient alternative. The aim of this chapter is thus to provide an overview of well-grounded matrix-free methods for fixed points computations and linear stability analyses of very large-scale nonlinear dynamical systems.
J.-Ch. Loiseau, M. A. Bucci, S. Cherubini, J.-Ch. Robinet

Spatial and Temporal Adaptivity in Numerical Studies of Instabilities, with Applications to Fluid Flows

In this article we discuss how to formulate numerical methods for calculating branches of steady solutions, periodic orbits and bifurcations of partial differential equations that are adaptive in both space and time. The methods are implemented within the open-source software framework oomph-lib and examples of their use in current research problems of fluid flow past cylinders and free surface flows on rotating cylinders are presented.
Andrew L. Hazel

Unstable Periodically Forced Navier–Stokes Solutions–Towards Nonlinear First-Principle Reduced-Order Modeling of Actuator Performance

We advance the computation of physical modal expansions for unsteady incompressible flows. Point of departure is a linearization of the Navier–Stokes equations around its fixed point in a frequency domain formulation. While the most amplified stability eigenmode is readily identified by a power method, the technical challenge is the computation of more damped higher-order eigenmodes. This challenge is addressed by a novel method to compute unstable periodically forced solutions of the linearized Navier–Stokes solution. This method utilizes two key enablers. First, the linear dynamics is transformed by a complex shift of the eigenvalues amplifying the flow response at the given frequency of interest. Second, the growth rate is obtained from an iteration procedure. The method is demonstrated for several wake flows around a circular cylinder, a fluidic pinball, i.e. the wake behind a cluster of cylinders, a wall-mounted cylinder, a sphere and a delta wing. The example of flow control with periodic wake actuation and forced physical modes paves the way for applications of physical modal expansions. These results encourage Galerkin models of three-dimensional flows utilizing Navier–Stokes based modes.
Marek Morzyński, Wojciech Szeliga, Bernd R. Noack

On Acceleration of Krylov-Subspace-Based Newton and Arnoldi Iterations for Incompressible CFD: Replacing Time Steppers and Generation of Initial Guess

We propose two techniques aimed at improving the convergence rate of steady state and eigenvalue solvers preconditioned by the inverse Stokes operator and realized via time-stepping. First, we suggest a generalization of the Stokes operator so that the resulting preconditioner operator depends on several parameters and whose action preserves zero divergence and boundary conditions. The parameters can be tuned for each problem to speed up the convergence of a Krylov-subspace-based linear algebra solver. This operator can be inverted by the Uzawa-like algorithm, and does not need a time-stepping. Second, we propose to generate an initial guess of steady flow, leading eigenvalue and eigenvector using orthogonal projection on a divergence-free basis satisfying all boundary conditions. The approach, including the two proposed techniques, is illustrated on the solution of the linear stability problem for laterally heated square and cubic cavities.
Alexander Gelfgat

Reviews of Methods, Approaches, and Problems


Stationary Flows and Periodic Dynamics of Binary Mixtures in Tall Laterally Heated Slots

The steady and oscillatory dynamics of binary fluids contained in slots heated by the side is studied by using continuation methods, and stability analysis. The bifurcation points on the branches of solutions are determined with precision by calculating their spectra for a large range of Rayleigh numbers. It will be seen that continuation and stability methods are a powerful tool to analyze the origin of the hydrodynamic instabilities leading to steady and time periodic flows, and their dynamics. The role played by the shear stresses of the steady field, and the solutal and thermal buoyancies, at the onset of the oscillations is studied by means of the energy equation of the perturbations. With the parameters used, it is found that the shear is always the main responsible for the instabilities, and that the work done by the two buoyancies can even help to stabilize the fluid. The results also show that binary mixtures of Prandtl number order one, like pure gases, present multiple stable periodic flows coexisting in the same range of parameters, since several unstable leading multipliers remain attached to the unit circle and go back into it. However, at lower Prandtl numbers only the first branch of periodic orbits bifurcating directly from the steady state is found to be stable, because some of the unstable multipliers of the other branches quickly increase their modulus and never re-enter the unit circle.
Juan Sánchez Umbría, Marta Net

A Brief History of Simple Invariant Solutions in Turbulence

When studying fluid mechanics in terms of instability, bifurcation and invariant solutions one quickly finds out how little can be done by pen and paper. For flows on sufficiently simple domains and under sufficiently simple boundary conditions, one may be able to predict the parameter values at which the base flow becomes unstable and the basic properties of the secondary flow. On more complicated domains and under more realistic boundary conditions, such questions can usually only be addressed by numerical means. Moreover, for a wide class of elementary parallel shear flows the base flow remains stable in the presence of sustained turbulent motion. In such flows, secondary solutions often appear with finite amplitude and completely unconnected to the base flow. Only using techniques from computational dynamical systems can such behaviour be explained. Many of these techniques, such as for the detection and classification of bifurcations and for the continuation in parameters of equilibria and time-periodic solutions, were developed in the late 1970s for dynamical systems with few degrees of freedom. The application to fluid dynamics or, to be more precise, to spatially discretized Navier–Stokes flow, is far from straightforward. In this historical review chapter, we follow the development of this field of research from the valiant naivety of the early 1980s to the open challenges of today.
Lennaert van Veen

The Lid-Driven Cavity

The lid-driven cavity is an important fluid mechanical system serving as a benchmark for testing numerical methods and for studying fundamental aspects of incompressible flows in confined volumes which are driven by the tangential motion of a bounding wall. A comprehensive review is provided of lid-driven cavity flows focusing on the evolution of the flow as the Reynolds number is increased. Understanding the flow physics requires to consider pure two-dimensional flows, flows which are periodic in one space direction as well as the full three-dimensional flow. The topics treated range from the characteristic singularities resulting from the discontinuous boundary conditions over flow instabilities and their numerical treatment to the transition to chaos in a fully confined cubical cavity. In addition, the streamline topology of two-dimensional time-dependent and of steady three-dimensional flows are covered, as well as turbulent flow in a square and in a fully confined lid-driven cube. Finally, an overview on various extensions of the lid-driven cavity is given.
Hendrik C. Kuhlmann, Francesco Romanò

Instabilities in the Wake of an Inclined Prolate Spheroid

We investigate the instabilities, bifurcations and transition in the wake behind a 45-degree inclined 6:1 prolate spheroid, through a series of direct numerical simulations (DNS) over a wide range of Reynolds numbers (Re) from 10 to 3000. We provide a detailed picture of how the originally symmetric and steady laminar wake at low Re gradually looses its symmetry and turns unsteady as Re is gradually increased. Several fascinating flow features have first been revealed and subsequently analysed, e.g. an asymmetric time-averaged flow field, a surprisingly strong side force etc. As the wake partially becomes turbulent, we investigate a dominating coherent wake structure, namely a helical vortex tube, inside of which a helical symmetry alteration scenario was recovered in the intermediate wake, together with self-similarity in the far wake.
Helge I. Andersson, Fengjian Jiang, Valery L. Okulov

Global Galerkin Method for Stability Studies in Incompressible CFD and Other Possible Applications

In this paper the author reviews methodology of a version of the global Galerkin that was developed and applied in a series of his earlier publications. The method is based on divergence-free basis functions satisfying all the linear and homogeneous boundary conditions. The functions are defined as linear superpositions of the Chebyshev polynomials of the first and second types that are combined in divergence free vectors. The description and explanations of treatment of boundary conditions inhomogeneities and singularities are given. Possible implementation for steady state solvers, path-continuation, stability solvers and straight-forward integration in time are discussed. The most important results obtained using the approach are briefly reviewed and possible future applications are deliberated.
Alexander Gelfgat

Some Recently Solved Problems


Instabilities in Extreme Magnetoconvection

Thermal convection in an electrically conducting fluid (for example, a liquid metal) in the presence of a static magnetic field is considered in this chapter. The focus is on the extreme states of the flow, in which both buoyancy and Lorentz forces are very strong. It is argued that the instabilities occurring in such flows are often of unique and counter-intuitive nature due to the action of the magnetic field, which suppresses conventional turbulence and gives preference to two-dimensional instability modes not appearing in more conventional convection systems. Tools of numerical analysis suitable for such flows are discussed.
Oleg Zikanov, Yaroslav Listratov, Xuan Zhang, Valentin Sviridov

A Mathematical and Numerical Framework for the Simulation of Oscillatory Buoyancy and Marangoni Convection in Rectangular Cavities with Variable Cross Section

It is often assumed that two-dimensional flow can be used to model with an acceptable degree of approximation the preferred mode of instability of thermogravitational flows and thermocapillary flows in laterally heated shallow cavities for a relatively wide range of substances and conditions (essentially pure or compound semiconductor materials in liquid state for the case of buoyancy convection and molten oxide materials or salts and a variety of organic liquids for the case of Marangoni convection). In line with the general spirit of this book, such assumption is challenged by comparing two-dimensional and three-dimensional results expressly produced for such a purpose. More precisely, we present a general mathematical and numerical framework specifically developed to (1) explore the sensitivity of such phenomena to geometrical “irregularities” affecting the liquid container and (2) take advantage of a reduced number of spatial degrees of freedom when this is possible. Sudden variations in the shape of the container are modelled as a single backward-facing or forward-facing step on the bottom wall or a combination of both features. The resulting framework is applied to a horizontally extended configuration with undeformable free top liquid-gas surface (representative of the Bridgman crystal growth technique) and for two specific fluids pertaining to the above-mentioned categories of materials, namely molten silicon (Pr  <  1) and silicone oil (Pr  >  1. The assumption of flat interface is justified on the basis of physical reasoning and a scaling analysis. The overall model proves successful in providing useful insights into the stability behaviour of these fluids and the departure from the approximation of two-dimensional flow. It is shown that the presence of a topography in the bottom wall can lead to a variety of situations with significant changes in the emerging waveforms.
Marcello Lappa

Continuation for Thin Film Hydrodynamics and Related Scalar Problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution of a single scalar field like a density or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn–Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift–Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto–Sivashinsky equation, convective Allen–Cahn and Cahn–Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues.
S. Engelnkemper, S. V. Gurevich, H. Uecker, D. Wetzel, U. Thiele

Numerical Bifurcation Analysis of Marine Ice Sheet Models

The climate variability associated with the Pleistocene Ice Ages is one of the most fascinating puzzles in the Earth Sciences still awaiting a satisfactory explanation. In particular, the explanation of the dominant 100 kyr period of the glacial cycles over the last million years is a long-standing problem. Based on bifurcation analyses of low-order models, many theories have been suggested to explain these cycles and their frequency. The new aspect in this contribution is that, for the first time, numerical bifurcation analysis is applied to a two-dimensional marine ice sheet model with a dynamic grounding line. In this model, we find Hopf bifurcations with an oscillation period of about 100 kyr which may be relevant to glacial cycles.
T. E. Mulder, H. A. Dijkstra, F. W. Wubs
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