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3653 results in ebooks in fluid mechanics

Page 32 of 183

  • 8 - Introduction to Rotating Fluids

  • from Part II - Kinematics, Dynamics and Rheology
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 83-100

  • Summary

    Rotation of Earth plays a dominant role in the dynamics of its atmosphere and oceans on time scales of a day or more. In a rotating (non-inertial) frame, the dynamic effect of rotation is represented a virtual force, called the Coriolis force, in the Navier–Stokes equation. A dominant effect of rotation is to produce waves and flows that are themselves rotating relative to a frame of reference fixed to Earth. This tendency is most readily observed in the atmosphere, which contains large-scale rotating air masses called “highs” and “lows” by meteorologists, and an array of smaller and often stronger rotating systems, including hurricanes (or cyclones), tornadoes, water spouts, and dust devils. As we shall see, rotation of Earth plays a direct role in large-scale systems, but only an indirect role in the smaller systems.

    In this chapter we will modify the perturbation equations given in § 7.3.2 so that they apply more specifically to the atmosphere and oceans and in doing so will encounter a number of concepts and effects that are introduced by rotation. The Coriolis force complicates the analysis of rotating fluids in several ways; apart from introducing an extra term in the Navier–Stokes equation, it couples the horizontal components of this equation, so that motions are generally not unidirectional. It will take some care and effort to decipher the dynamic constraints that rotation imposes on the flow, so we shall proceed carefully, beginning in § 8.1 with the discussion of a number of rotational concepts. Then in § 8.2, we will present the equations governing flow in a rotating fluid. The rotation of the fluid itself, represented by the vorticity, is an important quantity of interest; the vorticity equation is presented and discussed in § 8.3. Often horizontal motions in a thin layer of fluid are independent of the vertical coordinate and can be quantified by vertically averaged equations; the vertically averaged continuity equation is developed in § 8.4. Large-scale motions in the atmosphere and oceans may be categorized as geostrophic or quasi-geostrophic. Geostrophic flows result from a balance between the Coriolis and pressure terms, as described in § 8.5, while quasi-geostrophic flows result from a balance between the inertial, Coriolis and pressure terms, as described in § 8.6.

  • Appendix B - Dimensions and Units

  • from Part VIII - Fundaments
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 405-411

  • Summary

    The fundaments of kinematics presented include:

  • • B.1: an introduction to dimensional analysis;

  • • B.2: an introduction to the international system of units; and

  • • B.3: a table of parameters relevant to waves and flows on Earth and their estimated magnitudes.

    • Dimensional Analysis

    This fundament provides an orientation to the process of dimensional analysis, whereby a problem is simplified by analyzing the dimensions of its variables and parameters. The theoretical basis for this process is provided by the Buckingham Pi theorem. An essential part of the process is the formation of a minimal number of dimensionless parameters, typically denoted by ∏, each of which is an algebraic combination of the dimensional parameters in the problem. One advantage of dimensional analysis is that we do not need a set of governing equations (though that is, of course, quite helpful). All that is needed is a “grocery list” of relevant parameters.

    A related procedure is non-dimensionalization, whereby a set of governing equations and the parameters within it are made dimensionless. Often the equations are significantly simplified by this procedure, making them more amenable to analysis and solution.

    Dimensional analysis employs the seven base dimensions (length, mass, time, temperature, electric current, amount of substance and luminous intensity) of the SI system. Mechanical systems involve only the first three (length, mass and time), denoted by L, M and T:

    L = length M= mass T = time.

    Commonly the dimensions of a variable are identified using square brackets; for example, [g] = L·T−2 means that the dimensions of acceleration are length divided by time squared.

    In addition to the seven base dimensions, the SI system contains two supplemental units: plane and solid angle. These do not play a direct role in dimensional analysis and non-dimensionalization, but they do affect the numerical relationships among the dimensionless variables. The number of dimensionless parameters involved in a physical problem can easily be determined by use of the Buckingham Pi theorem which states that

    N∏ = Np −Nd,

    where N∏ is the number of dimensionless parameters, Np is the number of dimensional

    parameters and Nd is the number of base dimensions contained in all parameters.

  • 27 - Oceanic Currents

  • from Part VI - Flows in Rotating Fluids
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 277-291

  • Summary

    As we have previously noted, a major dynamic feature of the oceans is the thermocline, which separates a thin well-mixed top layer from a deep stably stratified layer beneath. The oceans have three types of large-scale motion that involve one or both of these layers, as summarized in Table 27.1.We have previously investigated the tides; in this chapter we will look into barotropic currents in the upper ocean (above the thermocline) driven by wind stress at the ocean surface and deep currents driven by subtle differences in temperature and salt content, beginning with the former.

    Figure 27.1 shows the major oceanic currents in the upper ocean, consisting primarily of gyres having ﹛clockwise, counterclockwise﹜ flow in the ﹛northern, southern﹜ hemisphere. There are five main gyres, occurring in the five major ocean basins: the northern Pacific, southern Pacific, northern Atlantic, southern Atlantic and (southern) Indian Oceans. In addition there are several minor gyres and currents near the equator and at high latitudes.

    The near-surface flows within an ocean basin are driven by the prevailing winds illustrated in Figure 26.1, but in a somewhat indirect manner. The prevailing winds are predominantly along lines of constant latitude, being easterly (from west to east) in the tropics and westerly (from east to west) in mid-latitudes. As we saw in § 25.5.4, these winds drive near-surface flows (Ekman transport) in the normal direction: north-south, along lines of constant longitude. These potential north-south flows have no dynamic outlet and instead the wind stresses result in deformation of the ocean surface which sets up a counterbalancing north-south pressure gradient. Near-surface flows arise due to a latitudinal variation of this dynamic balance, quantified by the curl of the wind stress, forming the great oceanic gyres illustrated in Figure 27.1. These flows are predominantly in the east-west direction in the open oceans. With the exception of the southern ocean surrounding Antarctica, these east-west flows are blocked by continents and are diverted in the north-south direction in thin boundary layers, predominantly on the western boundaries of the oceans, forming the western boundary currents.

  • 28 - Vortices

  • from Part VI - Flows in Rotating Fluids
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 292-314

  • Summary

    Vortices are important because they are an intrinsic feature of the most damaging storms that occur on earth: hurricanes and tornadoes. Also, vortices are an interesting example of nonlinear interaction between a rotating flow and the structure that spontaneously develops due to the dynamic constraint imposed by the Proudman–Taylor theorem.

    This chapter begins in the following section with a survey of various types of vortices and a brief discussion of their dynamics. The equations governing axisymmetric vortices are introduced in § 28.2, then two simple vortices are investigated in § 28.3. These simple vortices are the basis for the more realistic models that are developed in § 28.4. Finally in § 28.5 we briefly discuss hurricanes and consider their efficiency in converting heat to kinetic energy.

    Survey of Vortices

    A vortex is a swirling mass of fluid; it is a three-dimensional structure in which fluid flows roughly symmetrically about an axis. There are many kinds of atmospheric vortices, ranging in size and strength from the flow at a street corner that rustles leaves around in a circle to hurricanes that devastate coastlines. On the other hand, only one type of vortex occurs in water: the whirlpool.

    Vortices may be categorized as either barotropic or baroclinic. These two types are discussed in the following subsections.

    Barotropic Vortices

    Barotropic vortices arise as a nonlinear consequence of the instability of a basic-state shear. They often are abetted by topography and are distinguished from baroclinic vortices by the lack of a source of energy other than the kinetic energy of the flow. A familiar example of a barotropic vortex is that generated as wind blows past a building. A down-wind corner of the building is a singular point in the flow-field, where flow separation occurs, with a weak vortex forming in the lee of the building.

    Whirlpools – vortices in water – commonly form in rivers and near narrow channels that have strong tidal flows by the same mechanism: flow separation at an irregular boundary, with the whirlpool being a region of strong re-circulation in the lee of the obstacle; they rarely occur in open waters.

  • Appendix A - Mathematics

  • from Part VIII - Fundaments
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 381-404

  • Summary

    The fundaments of mathematics include:

  • • A.1: a brief introduction to the algebra of vectors and tensors in three dimensions;

  • • A.2: a summary of the calculus of three-dimensional vectors;

  • • A.3: an introduction to curvilinear coordinate systems;

  • • A.4: an introduction to Taylor series;

  • • A.5: an introduction to Fourier series and integrals;

  • • A.6: classification of several simple types of linear, second-order partial differential equations;

  • • A.7: a listing of the Greek symbols that are commonly used in applied mathematics; and

  • • A.8: an introduction to scalar and vector potentials.

    • These mathematical fundaments will employ the orthogonal reference coordinates system introduced in § 2.1 (and visualized in Figure 2.1) and will use extensively the associated unit vectors 11, 12 and 13. These vectors are arranged in a right-hand configuration such that 11 × 12 = 13. Any or all of these will be designated by 1 i, with i = 1, 2 and/or 3, as the situation requires. These vectors satisfy the orthogonality relation 1 i •1 j = δij, where δij is the Kronecker delta.

    Vectors and Tensors

    This fundament is not a general summary, but instead focuses on vectors and tensors in three-dimensional space, specifically vectors having three elements (or components) and tensors of rank two having nine elements.

    Definitions, Notation and Representation

    A tensor is an ordered set of nk numbers, where k and n are non-negative integers, that obeys certain tensor transformation rules. The integer k is the rank of the tensor. The A tensor of rank

    A tensor of rank 0 is a scalar: a number.

    A tensor of rank 1 is a vector: an ordered set of n numbers. Each number is a component of the vector. (Typically n = 3)

    Tensors may be thought of as square arrays in k dimensions. A matrix is a related two-dimensional array of size m by n, where m and n are positive integers. A matrix is square if m = n. A tensor of rank two behaves algebraically the same as a square matrix, and the tensor elements may be written out and manipulated in matrix-display form.

    In this document, a scalar is denoted by a lower case letter in italics (e.g., x).

  • Part II - Kinematics, Dynamics and Rheology

  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 21-22

  • Summary

    In this part we develop the mathematical apparatus necessary for the quantification of waves and flows. As we noted in the introduction, this apparatus consists of three components. The first, kinematics, places a constraint on the possible motions of the particles comprising a continuous body in order that the body indeed remains continuous; this constraint is developed in Chapter 3. The second component, dynamics, essentially applies Newton's second law to each particle in the body. As we see in Chapter 4, this gives us a vector equation, called the momentum equation, relating the acceleration of a fluid particle to the forces acting on it. The third component of our apparatus, rheology, is a set of equations describing how the material of a body deforms or flows in response to forces. Our rheological equations include the change of volume induced by a change in pressure, as explained in Chapter 5, and the change of shape induced by changes in deviatoric stresses, as explained in Chapter 6.

    The primary mathematical quantities of interest are the deformation (or its time derivative the velocity), the strain tensor (or its time derivative the rate-of-strain tensor) and the stress tensor. While stress and strain are used fairly interchangeably in colloquial usage, they have precise and distinct meanings in the present context. Stress quantifies the applied force and strain (or rate of strain) quantifies the deformation (or flow) that occurs in response to that force. More specifically, the stress tensor describes the structure of the forces acting within a body, while the strain (and rate-of-strain) tensor describes the structure of the internal deformation.

    Waves and flows occur as changes to the static state of a body, so before tackling waves and flows, we need to determine the static state and develop the equations governing small deviations from this state; this is accomplished in Chapter 7. The statics and dynamics of rotating fluids are of particular importance in geophysical fluid dynamics; the basic concepts of – and equations governing – rotating fluids are considered in Chapter 8.

  • 33 - Overview of Volcanic 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 354-358

  • Summary

    As we noted in § 30.3, there are three types of volcanic flows:

  • • at mid-ocean spreading centers;

  • • beneath subduction zones; and

  • • at hotspot volcanoes.

    • Volcanic flows at mid-ocean spreading centers are passive upwellings that fill the void created as the lithospheric slabs on either side are pulled away. Due to the parabolic shape of the cooling slabs, the upwelling speed is greatest, the motion is closest to adiabatic and the temperature is hottest beneath the spreading center. As can be seen from Figure 7.3, the mantle adiabat, when extrapolated to the top of the mantle, is greater than the melting temperature. This means that the upwelling material becomes partially molten. The viscosity of molten silicates is far smaller than the viscosity characterizing sub-solidus creep and the density of melt is less than that of the solidified remainder. It follows that the molten silicate (now called magma), percolates upward, collecting in magma chambers. The molten material in these chambers wells upward to the surface, where it forms basaltic pillow lava. Since this flow occurs in an expanding environment, the magma is not confined and out-gassing of volatiles is a quiet affair.

    The situation regarding out-gassing is quite different for the andesitic volcanism that occurs above subduction zones. This volcanism originates at the top of the subducting slabs due to a combination of viscous-dissipative heating and the presence of water. Most andesitic volcanoes occur on the rim of the Pacific Ocean, forming the so-called ring of fire. As the lithospheric slabs trundle their way from spreading center to subduction zone, minerals in the top of the slab become hydrated; that is, they incorporate water into their crystalline structure. After subduction, these hydrated minerals break down, releasing the water from its crystalline cage. This liberated water does two things. First, it lubricates the top of the slab, permitting it to slip beneath the adjoining crustal material.

  • 4 - Dynamics and the Stress Tensor

  • from Part II - Kinematics, Dynamics and Rheology
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 35-47

  • Summary

    In the previous chapter we investigated the ways in which a continuous body can deform and move. These deformations and motions are induced by three types of forces acting on the body, as explained in § 4.1. This chapter focuses on the quantification of the internal contact force, F C, and its relation to the stress tensor, denoted by S. The stress tensor describes the state of stress within a continuous body as explained in § 4.2; it is the dynamic counterpart to the strain and rate-of-strain tensors of kinematics.

    In the study of kinematics and dynamics, we have many choices. For example, in kinematics we are free to choose the location and orientation of our reference coordinate system. This system may be inertial or non-inertial; it may be arbitrarily oriented or aligned with the principal axes of the strain or rate-of-strain tensor. Similarly, in dynamics we are free to define what is meant by a body. Most of the time, we think of a body as a complete physical entity, with a physically defined boundary. However, we are free to define as a “body” any arbitrary portion of a physical body, such as the parcel introduced in § 1.2. We will use this freedom to our advantage in what follows. In the process of defining an arbitrary portion of a physical body as a “body,” we must of necessity define its surfaces. The stresses on these defined surfaces are manifestations of the internal state of stress of the body at that location.

    Forces

    The deformations and motions of a continuous body are induced by three types of forces acting on the body:

  • • Body forces, denoted by FB , are applied directly to each internal particle of the body. Examples of body forces are the gravitational and electromagnetic (Lorentz) forces. The body force due to gravity is a known constant when studying the atmosphere or oceans and is a prescribed function of radius when studying mantle convection.

  • • Virtual forces, denoted by FV , arise from the use of a non-inertial frame of reference that is either accelerating, rotating or both. Examples of virtual forces are the centrifugal force due to the use of a rotating coordinate system and the Coriolis force, due to motion observed in a rotating frame of reference.

  • 6 - Shear Rheology

  • from Part II - Kinematics, Dynamics and Rheology
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 54-66

  • Summary

    Shear rheology, commonly referred to simply as rheology, is the scientific study of the deformational response (strain and rate of strain) of a body to an applied force (stress), often called the stress–strain relation, expressed symbolically as f (S,E, E) = ∅. The response depends principally on the composition of the body, although the temperature and pressure can be important. (For example, the rheology of butter depends rather strongly on the temperature.) If the stress is independent of the rate of strain, the body is elastic and the symbolic relation simplifies to f (S,E) = ∅. Most elastic materials deform only slightly in response to an applied stress and the strain response to stress is adequately described by a linear relationship, obeying Hooke's law; this relationship is developed in § 6.1.

    If the stress is independent of the strain, the body is fluid and the symbolic relation simplifies to f (S,E)=∅. If this relationship is linear, the fluid is said to be Newtonian. The two most commonly encountered fluids, liquid water and air, are Newtonian. However, for many geological materials the relation between stress and flow is nonlinear and the material is non-Newtonian. In general, there are two types of nonlinear rheology: shear thinning and shear thickening. A shear-thinning fluid deforms more readily as the shearing force is increased. Familiar examples of such material are latex paint and quicksand. On the other hand, a shear thickening fluid becomes more resistant to flow as the shearing stress is increased. Many geological materials, including ice (in glaciers), lavas and Earth's silicate mantle behave as shear-thinning fluids. This behavior is investigated in § 6.2.

    The classification of materials as either elastic or fluid is simplistic, and somewhat misleading. For viscoelastic materials the stress depends on both strain and rate of strain. In reality a material may exhibit elastic, fluid or even more complex behavior depending on its thermodynamic state, the magnitude of the stresses applied and the time scale of interest. Most solid materials, as they are heated toward their melting point, develop more fluid-like behavior.1 Many materials (such as emulsions likemayonnaise and whipped cream) behave elastically when the applied stress is small, but change their rheological character to fluid-like or viscoelastic at higher stresses.

  • 5 - Some Thermodynamics

  • from Part II - Kinematics, Dynamics and Rheology
  • David E. Loper, Florida State University
  • Book:
    Geophysical Waves and Flows
    Published online:
    26 October 2017
    Print publication:
    26 October 2017, pp 48-53

  • Summary

    We have investigated how a body may move and deform; this is kinematics, codified in the equation of conservation of mass. We also have investigated the distribution of stresses within a body which may cause it to deform; this is dynamics, codified in the equation of conservation of momentum. Versions of these conservation equations for elastic and fluid bodies are presented in § 4.7. We now turn our attention to the manner in which a body responds to the stresses; this is rheology. Commonly, rheology is taken to mean a change of body shape (called shear rheology), but it also encompasses the change of volume or density (without change of shape) induced by a change of pressure (called compression rheology). This latter response is determined by the equation of state, giving the density ρ in terms of appropriate independent thermodynamic variables. Generally, the equation of state is presented in differential form (as in § 5.2), but for an ideal gas, it is a simple algebraic equation.

    As explained in Appendix E.5, in a system composed of one constituent the density is a function of two independent thermodynamic variables, with some freedom in the choice of those variables. Two useful choices of independent variables are temperature T and pressure p, or pressure and specific entropy s. It should be noted that the density and pressure are unique variables, in that they play leading roles in both thermodynamics and dynamics. The differential form of the equation of state for density is presented in § 5.2, following a discussion in the next section of some new adjectives that we encounter while delving into the dynamics and thermodynamics of geophysical fluids. The thermodynamic presentation in this chapter is rather abbreviated; for more detail, refer to Appendix E.

    New Adjectives

    In our study of geophysical fluids, we will encounter a number of new adjectives. It may be helpful to define these clearly at this point, before plunging into the dynamic and thermodynamic details. These adjectives include:

Page 32 of 183