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This chapter discusses numerical simulations of the incompressible Navier-Stokes equations. Exact Navier-Stokes solutions are presented that are used as benchmarks to validate new codes and evaluate the accuracy of a particular discretization. Some aspects of direct numerical simulation (DNS) and large-eddy simulation (LES) are discussed. The issue of stabilization at high Reynolds number is then presented using the concepts of dynamic subgrid modelling, over-integration, and spectral vanishing viscosity. A new parallel paradigm based on multi-level parallelism is introduced that can help realize adaptive refinement more easily. The final section includes a heuristic refinement method for Navier-Stokes equations.

This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.

Pressure-force-driven microfluidics is the foundation for volumetric scaling into the micro regime and, hence, for unique functionalities unachievable by macro-sized devices. The physics that govern the operation of pressure force driven microfluidics are just a subset of principles governing macro scale fluidics, and low Reynolds number fluid mechanics is essential for the design and operation of pressure force driven microfluidic devices. This is because microfluidics almost exclusively only deals with slow velocity and small sizes. This chapter first provides a brief introduction to the necessary microfluidics basics required for understanding pressure force driven microfluidics, primarily focusing on laminar, viscous flow and the importance of force scaling relative to device size. Device examples start with pressure-driven flows in straight microchannels, the simplest but most important geometry. Various other devices then follow, including passive and active valves, mixers, on-chip pressure sources, flow sensors and device packaging. Finally, the chapter ends with some current and future applications for pressure-force-driven microfluidics and the remaining challenges facing microfluidics designers.

This chapter presents three lectures intended as an introduction to locomotion at low Reynolds numbers. It covers swimming, crawling, and burrowing. For those who desire a more in-depth treatment, references throughout the text are included. These notes are written in an informal tone to match the lectures, which included a mix of chalk talks and PowerPoint presentations. Many of the PowerPoint slides have been reproduced here, although some copyrighted images have been omitted.

Flows at low Reynolds numbers (Re) are characterized by the dominance of viscosity and are encountered in small channels, at low flow velocities and for very viscous fluids. The linear Stokes equation, which governs such flows, is valid because the non-linear convective term of the Navier–Stokes equation is negligible: several general properties of these flows (reversibility, additivity and minimum dissipation) result from this linearity. Unlike in Chapter 8, the geometry of these flows is arbitrary. Flows around small objects like in the suspensions of particles and in porous media (and the model Hele–Shaw cell) are important applications. Non-linear terms need to be taken into account at large distances and this leads to Oseen’s equation.

By restricting our attention in this book to HIT, we have ruled out effects due to mean shear, system rotation, density stratification; and so on. This leaves us with a stark choice: deviation from Kolmogorov’s (1941) predictions for the energy spectrum (or second-order structure function) in stationary flows must be due to either the Reynolds number being finite (K41 is based on an assumption of very large Reynolds numbers) or internal intermittency, as was suggested later on by Kolmogorov, in 1962. Over the last few decades a veritable industry has grown up, based on the search for so-called intermittency corrections. Currently it is dominated by multiscale or multifractal models of turbulence. This activity tends to find a receptive audience, because many people seem to regard the K41 picture as being counter-intuitive, when one considers aspects of turbulence such as vortex-stretching, localness, and intermittency. Running counter to this belief in ‘intermittency corrections’ (or, increasingly, ‘anomalous exponents’) which has been dominant in recent times, there is a growing view that K41is an asymptotic theory, valid in the limit of infinite Reynolds number. In this school of thought, any deviations from K41 are due to finite viscosity. As a result, opinion in the turbulence community is deeply divided on this fundamental issue This chapter begins by considering the various criticisms of, or sources of unease about, K41. Then it reformulates the K41 spectral theory in terms of scale-invariance, the resolution of the scale-invariance paradox, and conservation of energy. This is followed by a discussion of the various theories which advocate finite-viscosity effects in explaining deviations from K41. We conclude with a brief discussion of the current situation.

The linear momentum of a moving fluid can be transported, like thermal energy or solute concentration, simultaneously by diffusion and convection: an important difference in these latter cases is that momentum is a vector quantity, whereas temperature and concentration are scalars. This chapter gives a simplified microscopic model of the viscosity which is the proportionality coefficient for momentum transfer and then compares the relative effectiveness of convection and diffusion mechanisms, which results in a definition of the Reynolds number and of similar dimensionless combinations for thermal and mass transport. Illustrations of the variation of flow properties with the Reynolds number for the flow in a tube and around obstacles are presented.

Effects of increasing fluid speed are analyzed. The Bernouilli and vorticity equations are derived, and the method of matching solutions is described for the Rankine vortex. Cases in which rotational flow is mandatory are explained, and bifurcations, hydraulic jumps, and transitions between stable and unstable behaviors are introduced. The ethical views of Hans Bethe and Edward Teller are contrasted. Other topics include potential flow around both cylinders and spheres and lessons that can be learnt about flow over a wavy streambed.

Remarkable changes in drag are described as fluid speed is increased, and methods to reduce drag at high speed are described. The Stokes paradox is considered, as are the effects of Reynolds number and roughness on drag. The use of conformal mappings to obtain flow streamlines is defined for problems including flow past an airfoil, flow past a step, and over a cavity.

This chapter presents the application of LBE to flows at moderate Reynolds numbers, typically hundreds to thousands. This is an important area of theoretical and applied fluid mechanics, one that relates, for instance, to the onset of nonlinear instabilities and their effects on the transport properties of the unsteady flow configuration. The regime of Reynolds numbers at which these instabilities take place is usually not very high, of the order of thousands, hence basically within reach of present day computer capabilities. Nonetheless, following the full evolution of these transitional flows requires very long-time integrations with short time-steps, which command substantial computational power. Therefore, efficient numerical methods are in great demand. Also of major interest are steady-state or pulsatile flows at moderate Reynolds numbers in complex geometries, such as they occur, for instance, in hemodynamic applications. The application of LBE to such flows will also briefly be mentioned

This chapter first discusses some of the physical properties of water, including viscosity, water pressure, and the surface films formed between water and air. With these principles in mind, it then explains how living in water affects the morphology and behaviour of aquatic insects that live in different habitats. In still water, buoyancy and the problem of maintaining position in the water column or on the substrate are perhaps the major challenges. Insects that live on the water surface must avoid sinking and they use various strategies that exploit the physical properties of surface films and hydrofuge or water-repellent body parts. The section on life in flowing waters begins with a discussion of some physical properties including Reynolds numbers, drag, shear stress, streamlining, laminar and turbulent flows, and boundary layers. These physical properties shape the various adaptations to flowing water, most of which centre on the need to reduce drag when moving and feeding, and to avoid accidental displacement.

This chapter examines how the physical properties of water influence and explain the great diversity of swimming performance and mechanisms - from the scale of spermatozoa on up to whales. The key parameters of inertia, viscosity and their manifestation in the critically important Reynolds number are explained and placed in the context of a range of swimming mechanisms, including undulatory movement and fin-based, jet-based, flagellar and ciliary propulsion. The air-water interface also presents an intriguing mechanical challenge for the many organisms that move on top of the water’s surface. The chapter concludes with a brief overview of the burgeoning field of biorobotic swimmers.

This concluding chapter highlights connections among various species of non-causal scientific explanation (including both explanations by constraint and dimensional explanations) and their connections to causal explanations and to explanations in mathematics. It examines some of the features that they all share by virtue of which they deserve to be grouped together as explanations. One such feature is the way that they can render what they explain not a coincidence. Another is the way that they make certain reducible properties (such as having the same Reynolds number or the same center of mass) into genuine respects of similarity. The distinction between natural and non-natural properties is drawn in both mathematics and science. Not every reducible property is a natural property; some reducible properties are mere conglomerates of more fundamental properties. In both math and science, there is a tight connection between naturalness and (both causal and non-causal) explanation.

Let us now detail the stability properties of an Ekman layer introduced in Part I, page 11. First we will recall how to compute the critical Reynolds number. Then we will describe briefly what happens at larger Reynolds numbers. The first step in the study of the stability of the Ekman layer is to consider the linear stability of a pure Ekman spiral of the form where U∞ is the velocity away from the layer and ζ is the rescaled vertical component ζ = x3/√εν. The corresponding Reynolds number is Let us consider the Navier–Stokes–Coriolis equations, linearized around uE The problem is now to study the (linear) stability of the 0 solution of the system (LNSCε). If u=0 is stable we say that uE is linearly stable, if not we say that it is linearly unstable. Numerical results show that u=0 is stable if and only if Re<Rec where Rec can be evaluated numerically. Up to now there is no mathematical proof of this fact, and it is only possible to prove that 0 is linearly stable for Re<Re1 and unstable for Re>Re2 with Re1<Rec<Re2, Re1 being obtained by energy estimates and Re2 by a perturbative analysis of the case Re=∞. We would like to emphasize that the numerical results are very reliable and can be considered as definitive results, since as we will see below, the stability analysis can be reduced to the study of a system of ordinary differential equations posed on the half-space, with boundary conditions on both ends, a system which can be studied arbitrarily precisely, even on desktop computers (first computations were done in the 1960s by Lilly).

This book investigates high-Reynolds number flows, and analyses flows that can be described in the framework of Prandtl’s 1904 classical boundary-layer theory, including Blasius’s boundary layer on a flat plate, Falkner–Skan solutions for the boundary layer on a wedge surface, and other applications of Prandtl’s theory. It then discusses separated flows, and considers the so-called ‘self-induced separation’ in supersonic flow, and which led to the ‘triple-deck model’. It also presents Sychev’s 1972 theory of the boundary-layer separation in an incompressible fluid flow past a circular cylinder. It discusses the triple-deck flow near the trailing edge of a flat plate, and then considers the incipience of the separation at corner points of the body surface in subsonic and supersonic flows. It covers the Marginal Separation theory—a special version of the triple-deck theory—and describes the formation and bursting of short separation bubbles at the leading edge of a thin aerofoil.

Here some solutions of Navier–Stokes equations are found.The flow of a fluid along a pipe (Poisseuille flow) and that between two rotating cylinders (Couette flow) are the simplest. In the limit of large viscosity (small Reynolds number) the equations become linear: Stokes equations. Flow past a sphere is solved in detail. It is used to calculate the drag on a sphere, a classic formula of Stokes. An exact solution of the Navier–Stokes equation describing a dissipating vortex is also found. It is seen that viscosity cannot be ignored at the boundary or at the core of vortices.

This chapter presents the main ideas behind the application of LB methods to the simulation of turbulent flows. The attention is restricted to the case of direct numerical simulation, in which all scales of motion within the grid resolution are retained in the simulation. Turbulence modeling, in which the effect of unresolved scales on the resolved ones is taken into account by various forms of modeling, will be treated in a subsequent chapter.

Chapter 3 focuses on the high-Reynolds number flow of an incompressible fluid near the trailing edge of a flat plate. It begins with Goldstein’s (1930) solution for a viscous wake behind the plate, and shows that the displacement effect of the wake produces a singular pressure gradient near the trailing edge. It further shows that this singularity leads to a formation triple-deck viscous-inviscid interaction region that occupies a small vicinity of the trailing edge. A detailed analysis of the flow in each tier of the triple-deck structure is conducted based on the asymptotic analysis of the Navier–Stokes equations. As a result, the so-called ‘interaction problem’ is formulated. It concludes with the numerical solution of so-called ‘interaction problem’.