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3653 results in ebooks in fluid mechanics

Page 34 of 183

  • 22 - Gravitationally Forced Flows

  • from Part V - Non-Rotating Flows
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 218-226

  • Summary

    In the previous two chapters, we investigated the nature of flow in a horizontal channel assuming (except for the study of flash floods in § 21.4) resistance to flow is negligibly small, and we were able to classify flows as either sub- or super-critical and to investigate the smooth transition from sub- to super-critical and the abrupt transition from super- to sub-critical. We focused on the flow of water (as opposed to other fluids) because it has a very low value of viscosity and in many cases may be treated as inviscid. However, all real flows involve resistance which tends to degrade the mechanical energy. Typically channeled flows are maintained by the release of gravitational potential energy as the material moves downslope.

    In this chapter we consider the one-dimensional flow of a continuous material down a slope. The material might be water in a river, ice in a glacier, snow in an avalanche, debris in a rock slide, molten rock in a lava stream, mud in a landslide, muddy water in a turbidity current, etc. We will formulate the problem fairly generally in § 22.1, then specialize it to specific types of material in § 22.2.

    Formulation

    Most natural downslope flows, exemplified by rivers and glaciers, are much wider than they are deep and move along beds that are relatively flat. Typically a river is wider than deep by a factor of 100 (e.g., see Yalin, 1992). Also, rivers, glaciers and like flows typically are contained by banks that are raised and relatively steep. It follows that a first approximation of the flow bed is a shallow rectangle and that the velocity is independent of the cross-stream direction to dominant order.

    Let's consider flow of material in a straight, flat-bottomed channel (at z=0) of constant width; as previously, the downstream direction is x, the (unimportant) cross-stream direction is y and the (nearly) vertical direction is z. The height of the surface is represented by z = h(x, t). In this chapter we will focus on flows that are steady and invariant in the x direction, having h constant and v = u(z)1x. Our goal is to determine the vertical structure of the flow, u(z), driven by a down-gradient component of gravity.

  • 1 - Introduction

  • from Part I - Introductory Material
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 3-8

  • Summary

    This monograph is a survey of – and introduction to – the ways in which the atmosphere, oceans and various parts of the so-called solid Earth can deform and move. The most common type of deformation is a wave and the most common type of motion is flow. Our goals are to quantify and understand these deformations and motions. In order to achieve these goals, we will need to develop an appropriate set of procedures and tools.

    An appropriate procedure is mathematical modeling, which consists of three steps:

  • • Formulation of a mathematical model of a physical system. This is the crucial element in the procedure. If the model does not mimic the relevant physical processes, it will fail to yield useful results.

  • • Solution of the mathematical problem. This is the part stressed in most mathematics courses, but in fact is the most routine aspect of the procedure.

  • • Interpretation of the solution. This is the payoff; solution of a well-conceived model should yield new physical insight and make testable predictions.

    • We won't be following these steps formally, instead they will come naturally as we investigate each type of wave and flow.

    Our mathematical models will treat the physical system (that is, the portion of Earth under consideration) as a continuous body, using the mathematical apparatus of continuum mechanics. The concept of a continuous body is discussed in § 1.2. Continuum mechanics consists of three fundamental elements:

  • • Kinematics describes how bodies can move and deform. For example, it limits the types of deformation which leave a body competent (i.e., unfractured).

  • • Dynamics explains why bodies move and deform. They do so in response to external and internal forces, within the context of Newton's second law.

  • • Rheology quantifies the kinematic response to dynamic forces, taking into account the material properties of the body under investigation. Two identically shaped bodies, composed of water and of steel, respond quite differently to the same applied forces. Rheology encompasses the change of volume of a body induced by a change in pressure (compressive rheology) and the change of shape caused by a change in deviatoric stresses (shear rheology).

  • 11 - Deep-Water Waves

  • from Part III - Waves in Non-Rotating Fluids
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 119-133

  • Summary

    Water waves are fluid edge waves; they also are called surface waves or gravity waves. These waves arise from a much different mechanism than do edge waves in elastic bodies. This is evident from the fact that water waves travel much more slowly than do sound waves. However, they do have the same basic structure as elastic edge waves: harmonic variation in the horizontal direction and exponential decay with depth. Water waves occur on and near the free surface; they cannot occur in a closed container. These waves occur because gravity acts to pull down the water that stands higher than average and push aside water lying beneath, as illustrated in Figure 11.1. This figure is a “snapshot”; the wave might be progressive (moving either to the left or right) or standing (with fixed nodes) or some combination of these two limiting cases.

    The adiabatic incompressibility and kinematic viscosity of water are K ≈2×109 Pa and ν ≈ 10-6 m2 ·s-1. The pressure at the bottom of a typical ocean, which is 4200 m deep, is about 4 ×107 Pa. This compresses water by about 2%. Since we are interested in surface waves, it is safe to treat water as completely incompressible. Viscous effects are important only near solid boundaries (and then only within thin boundary layers) or if the water is turbulent (as it is in breakers and whitecaps). So it is safe to treat water as inviscid in the study of non-breaking waves.

    Waves on open water are almost always progressive (i.e., moving in a certain direction). Standing waves occur when two progressive waves travel past each other in opposite directions, such as commonly occurs near a vertical breakwater. We will consider progressive waves in this chapter, with standing waves considered in § 12.3.

    We will find in § 11.1 that the equation governing motions within water is not the wave equation, but Laplace's equation.1 The wave character of the solution comes from the boundary conditions at the surface. These conditions are developed in § 11.2.

    Water Wave Equation

    As with sound and seismic waves, let's focus our attention on plane waves traveling in the x direction, with motion and variation in both the horizontal (x) and vertical (z) directions.

  • 24 - Some Non-Rotating Turbulent Flows

  • from Part V - Non-Rotating Flows
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 242-250

  • Summary

    In this chapter we consider some non-rotating turbulent flows that are a bit more complicated than the flow of water down a slope considered in § 23.6. These include turbulent katabatic winds driven by thermal buoyancy (§ 24.1), avalanches driven by snow suspended in air (§ 24.2) and cumulonimbus clouds driven by the release of latent heat as water vapor condenses (§ 24.3).

    Turbulent Katabatic Winds

    In § 22.2.5, we investigate the katabatic wind down a slope in the case that the flow is laminar and, using typical parameter values, found that the flow is very likely turbulent rather than laminar. When flow is turbulent, the diffusivity coefficients for momentum and heat are not constant, but instead vary linearly with elevation. In this section, we will revisit the problem formulated in § 22.2.5, but with variable diffusivities, using Reynolds analogy to set the turbulent thermal diffusivity equal to the turbulent diffusivity of momentum.1

    The governing equations now are

    where z is elevation above the ground, u is the downslope speed, is the dimensionless perturbation temperature, is the reduced gravity, s the down-slope thermal gradient, is the temperature contrast, is the turbulent diffusivity, is a small dimensionless parameter is the velocity scale and is the roughness scale. As before, these equations are to be solved on the domain subject to the conditions u And as before, we can combine the two equations into a single complex equation, although the scalings are somewhat different. In the present case

    while the complex equation for W= T ∗ −iu/U is

    where is the dimensionless vertical distance and is the scaled boundary roughness, subject to conditions W(0) = 1 and W(∞) = 0.

    The problem for the turbulent katabatic winds is a bit more challenging than for the laminar winds because our complex ordinary differential equation now has a variable coefficient. We can get this equation in “standard” form by introducing a new independent variable; let

  • 35 - Lava Flows

  • from Part VII - Silicate Flows
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 367-378

  • Summary

    Commonly andesitic and rhyolitic magmas rise up discrete conduits, whereas basaltic magma often – at least during the early stages of an eruption phase – rises to the surface through a magmatic dike (also called a rift or fissure): a linear crack extending from a magma chamber at depth to the surface. Typically as magma erupts from a dike, outgassed volatiles (mainly water) produce a linear fire fountain composed of incandescent blobs of lava.

    After magma reaches the surface, its subsequent behavior depends largely whether the eruption is effusive or explosive. A discussion of the possible styles of lava flow is found in Griffiths (2000). Effusive eruptions lead to the production of lava which flows relatively gently downslope, initially in the form of a planar sheet. We will consider a simple model of lava sheet flow in the following section.

    On the other hand, explosive eruptions display a variety of behaviors depending on the speed, mass flow and composition of the erupting magma. Some of these behaviors are explored in § 35.2.

    Lava Sheet Flow

    In this section we will develop a simple model of subaerial (that is, on land, as opposed to under the sea) lava sheet flow: a sheet of hot basaltic lava flowing steadily down a slope (in the x direction) from a magmatic dike.

    Our simple model will ignore

  • • variations in topography;

  • • horizontal variations in the lava surface;

  • • fluid inertia;

  • • loss of heat to the ground;

  • • radiative cooling;

  • • variations in the ambient temperature;

  • • effects of rain;

  • • the dynamic effect of crystals suspended in the liquid lava;

  • • buoyancy effects; and

  • • variations of thermal conductivity with temperature.

    • In order to simplify the analysis further, we will assume that the cool upper layer of lava is rafted along with the flow, rather than being anchored to the ground at the sides. The weight of this rafted layer is borne by the liquid beneath, so that there is a vertical hydrostatic balance. This rafted layer may deform as needed in order to adjust to the flow of the liquid beneath.

  • Part III - Waves in Non-Rotating Fluids

  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 101-102

  • Summary

    Our investigation of waves is divided into two parts depending whether rotation of the Earth has a part to play. In the present part we limit our attention to those waves unaffected by rotation of Earth, with those waves that are influenced by this rotation investigated in Part IV. We begin our investigation in the following chapter with an introduction to – and brief survey of – the types of waves that occur on and within Earth. The simplest type of wave, the sound wave, is quantified and discussed in § 9.3. The most common type of elastic wave is the seismic wave; these waves occur continuously as the brittle portions of Earth's crust rub against one another. Seismic waves that are unaffected by Earth's surface, called body waves, are considered in § 10.1 and § 10.2, while those that affected the surface are considered in § 10.3 and § 10.4. While body waves in fluids (we are thinking of air and water) are essentially sound waves, edge waves in fluids (particularly water) are distinctly different from elastic edge waves. We investigate deep-water waves from a mathematical perspective in Chapter 11 and in § 11.5 discuss how these waves occur and interact on the oceans. The rich variety of waves occurring in shallow water is investigated in Chapters 12 and 13. Our survey of non-rotating waves concludes in Chapter 14 with an investigation of capillary, interfacial and internal waves.

  • Preface

  • H. Aref, Virginia Polytechnic Institute and State University, S. Balachandar, University of Florida
  • Book:
    A First Course in Computational Fluid Dynamics
    Published online:
    03 October 2017
    Print publication:
    12 October 2017, pp vii-x

  • Summary

    This book has been in preparation for over a decade. Hassan Aref and I had been making substantial additions and revisions each year, in our desire to reach the perfect book for a first course in Computational Fluid Dynamics (CFD). I sincerely wish that we had completed the book a few years ago, so that Hassan was there when the book was published. Unfortunately this was not the case. September 9th 2011 was a tragic day for fluid mechanics. We lost an intellectual leader, a fearless pioneer and for me an inspirational mentor. It is quite amazing how my academic career intersected with Hassan's over the years. He was my Masters Thesis advisor at Brown University. But when Hassan left for the University of California at San Diego, I decided to stay and finish my PhD at Brown. A few years later, when I was an assistant professor at the Theoretical and Applied Mechanics Department at the University of Illinois at Urbana– Champaign, he was appointed as the head of the department. This is when he asked me to join him in this project of writing a non-traditional introductory book on CFD. The project was delayed when Hassan moved to Virginia Tech as the Dean and I moved to the University of Florida as the Chair of the Mechanical and Aerospace Engineering Department. His passing away a few years ago made me all the more determined to finish the book as a way to honor his memory. I am very glad that the book is now finished and can be a monument to his far-sighted vision. Decades ago when CFD was in its infancy he foresaw how powerful computers and numerical methods would dominate the field of fluid mechanics.

    Hassan and I shared a vision for this book. Our objective was to write something that would introduce CFD from the perspective of exploring and understanding the fascinating aspects of fluid flows. We wanted to target senior undergraduate students and beginning graduate students. We envisioned the student to have already taken a first level course in fluid mechanics and to be familiar with the mathematical foundations of differential equations.

  • 4 - Spatial Discretization

  • H. Aref, Virginia Polytechnic Institute and State University, S. Balachandar, University of Florida
  • Book:
    A First Course in Computational Fluid Dynamics
    Published online:
    03 October 2017
    Print publication:
    12 October 2017, pp 118-165

  • Summary

    In the previous chapter we considered initial value ODEs, where the interest was in the computation of time evolution of one or more variables given their starting value at some initial time. There is no inherent upper time limit in integrating these initial value ODEs. Therefore numerical methods for their solution must be capable of accurate and stable long time integration. By contrast, in the case of two-point boundary value and eigenvalue problems for ODEs arising in fluid mechanics, the independent space variable has two well-defined end points with boundary conditions specified at both ends. The spatial domain between the two boundary points can be infinite, as in the case of Blasius boundary layer: see (1.6.5), where the spatial domain extends from the wall (η = 0) out to infinity (η → ∞). For such a problem it is possible to treat the space variable much like time in an initial value problem, and proceed with integration from one boundary to the other and then subsequently verify the boundary conditions at the other end. We shall consider numerical methods of this sort in the next chapter.

    An alternative approach is to discretize the entire domain between the two boundaries into a finite number of grid points and to approximate the dependent variables by their grid point values. This leads to a system of equations that can be solved simultaneously. Much like the time integration error considered in the previous chapter, here one encounters a discretization error. The discretization error arises from several sources: interpolation errors arise from approximating the function between grid points; differentiation errors arise in the approximation of first-, second- and higher-order derivatives; and integration errors arise from the numerical integration of a function based on its discretized values at the grid points. These errors are indeed interrelated and depend on the discretization scheme. This chapter will consider various discretization schemes. In particular, discrete approximations to the first- and second-derivative operators will be obtained. Errors arising from the different discretization schemes will be considered. The concept of discrete approximation to the first and second derivatives as matrix operators will be introduced. Finally, we will consider spatial discretization as a means to numerically integrate functions. The actual solution methodologies for two-point boundary value and eigenvalue problems for ODEs, using the tools developed in this chapter, are treated in Chapter 5.

  • 5 - Boundary Value and Eigenvalue ODEs

  • H. Aref, Virginia Polytechnic Institute and State University, S. Balachandar, University of Florida
  • Book:
    A First Course in Computational Fluid Dynamics
    Published online:
    03 October 2017
    Print publication:
    12 October 2017, pp 166-209

  • Summary

    We will begin this chapter with a discussion of two-point boundary value ODEs and then move on to two-point eigenvalue problems. The Blasius equation in Example 1.6.4 is one of the most celebrated two-point boundary value problems in fluid dynamics. Two-point boundary value problems form a challenging class of numerical computations with a wide literature. For an exposition of the theoretical ideas on two-point boundary value problems see Press et al. (1986), Chapter 16, or Keller (1968). Broadly speaking there are two classes of methods for such problems, shooting methods and relaxation methods. Shooting methods are related to integration schemes for initial value problems, while relaxation methods are based on discrete approximations to derivative operators. Thus, we are ready to tackle both these methods. In a shooting method you start from one end as in an initial value calculation using the actual boundary conditions specified at that boundary supplemented with a few other assumed (or guessed) ones which replace the actual boundary conditions of the ODE specified at the other end, and integrate (or “shoot”) towards the other boundary. In general when you eventually reach the other boundary, the boundary conditions there will not be satisfied (in other words you “miss” the target). Hence, you modify your guessed boundary conditions at the starting boundary (“aim” again) and integrate forward (“shoot” again). In this way you have in effect generated a mapping of initially guessed boundary conditions onto errors in matching the actual boundary conditions to be enforced at the other end point. Iterative procedures can now be invoked to converge to the desired value of the boundary conditions at the other end.

    In a relaxation method you write the ODE(s) as a system of finite difference equations, guess a solution, and then iteratively improve it by using a root finding procedure for a system of coupled nonlinear equations. This can obviously be a rather formidable task. Often a good initial guess is needed to converge towards the correct answer. Press et al. (1986) give the whimsical recommendation for dealing with two-point boundary value problems to “shoot first and relax later.” As we shall see below, in most problems of interest in fluid dynamics the relaxation method is very powerful, exhibiting rapid convergence towards the correct answer. Furthermore, unlike the shooting method, the relaxation method is easily extendible to boundary value problems in multi-dimensions.

  • 1 - CFD in Perspective

  • H. Aref, Virginia Polytechnic Institute and State University, S. Balachandar, University of Florida
  • Book:
    A First Course in Computational Fluid Dynamics
    Published online:
    03 October 2017
    Print publication:
    12 October 2017, pp 1-34

  • Summary

    Computational fluid dynamics, usually abbreviated CFD, is the subject in which the calculation of solutions to the basic equations of fluid mechanics is achieved using a computer. In this sense, CFD is a subfield of fluid mechanics, but one that relies heavily on developments in fields outside of fluid mechanics proper, in particular numerical analysis and computer science. CFD is sometimes called numerical fluid mechanics.

    It is a prevailing view that the computer revolution has produced a “third mode” of scientific investigation, often called scientific computing or computational science, that today enters on a par with the more established modes of (theoretical) analysis and (laboratory) experimentation. CFD is this new “third mode” of doing fluid mechanics.

    The Nature of CFD

    Our objective is to explore CFD from a basic science perspective. We take as a given the power and elegance of analytical reasoning in fluid mechanics, and the precision and persuasiveness of well-executed experiments and observations. We wish to demonstrate that CFD has ingredients from both these traditional ways of doing fluid mechanics, and, furthermore, injects a set of unique viewpoints, challenges and perspectives. The subject has many very new aspects, due in large part to the rapid and recent development of computer technology. Yet it also has considerable historical and intellectual depth with many contributions of great beauty and profundity that derive directly from cornerstone developments in mathematical analysis and physical theory.

    There is often a very special flavor to a CFD investigation. Sometimes this arises from the unique combination of trying to describe a continuum using a finite set of states (and, thus, ideas from discrete mathematics). On other occasions one has the challenge to address a question that is still too difficult for conventional analysis and is largely inaccessible to laboratory experiments. An example of this kind is whether the three-dimensional Euler equations for incompressible flow generate singularities in a finite time starting from smooth initial data. In a computation we can often impose a constraint or symmetry that is very difficult to achieve or maintain experimentally, such as exactly two-dimensional flow. In a computation we can increase the nonlinearity in the problem gradually, from a regime conforming to a linearized description, that is usually amenable to analysis, to a weakly nonlinear regime (where analysis typically is much harder), and finally, to a fully nonlinear range (where analytical understanding may be altogether lacking).

  • 7 - Partial Differential Equations

  • H. Aref, Virginia Polytechnic Institute and State University, S. Balachandar, University of Florida
  • Book:
    A First Course in Computational Fluid Dynamics
    Published online:
    03 October 2017
    Print publication:
    12 October 2017, pp 244-292

  • Summary

    In this chapter we turn, finally, to subjects that are considered more mainstream CFD, viz., the variety of methods in use for discretizing partial differential equations (PDEs) on uniform grids in space and time, thus making them amenable to numerical computations. This is a very wide field on which many volumes have been (and will be) written. There are many different methods that have been advanced and promoted over the years. Many of the more well-established methods and schemes are named for their originators, although substantial leeway exists for improvements in detail and implementation. It is also not difficult to come away with feelings of helplessness and confusion concerning which method to choose and why. Our goal in this chapter will be to shed light on some of the guiding principles in constructing these methods.

    A digression on semantics is in order. The words method and scheme are both in use for numerical procedures. It may be of interest to recall their precise meaning as given, for example, by The Merriam–Webster Dictionary:

    Method: from the Greek meta (with) ' hodos (way).

    (1) a procedure or process for achieving an end;

    (2) orderly arrangement; plan.

    Scheme

    (1) a plan for doing something; esp. a crafty plot.

    (2) a systematic design.

    Although both these terms seem to apply for numerical procedures, it would seem from these definitions that the term method is somewhat more desirable than scheme! But we have used these terms interchangeably, perhaps with a slight emphasis on methods, and we will continue to do so.

    Definitions and Preliminaries

    Before we begin to consider numerical methods for PDEs, here we start by defining and classifying the PDEs that we wish to solve. In a PDE the quantity being solved, such as fluid velocity or temperature, is a function of multiple independent variables, such as time and space (t and x) or multiple space directions (x and y). As mathematical entities, PDEs come in many different forms with varying properties, and they are a very rich topic in their own right. Here we will stay focused on only those classes of PDEs that we generally encounter in fluid mechanical applications.

Page 34 of 183