DOCSLIB.ORG

The Vorticity Equation on a Rotating Sphere and the Shallow Fluid Approximation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Vorticity Equation on a Rotating Sphere and the Shallow Fluid Approximation DISCRETE AND CONTINUOUS doi:10.3934/dcds.2019273 DYNAMICAL SYSTEMS Volume 39, Number 11, November 2019 pp. 6261{6276 THE VORTICITY EQUATION ON A ROTATING SPHERE AND THE SHALLOW FLUID APPROXIMATION Vikas S. Krishnamurthy∗ Faculty of Mathematics, University of Vienna Oskar-Morgenstern-Platz 1 1090 Vienna, Austria Abstract. The material conservation of vorticity in fluid flows confined to a thin layer on the surface of a large rotating sphere, is a central result of geophysical fluid dynamics. In this paper we revisit the conservation of vorticity in the context of global scale flows on a rotating sphere. Starting from the vorticity equation instead of the Euler equation, we examine the kinematical and dynamical assumptions that are necessary to arrive at this result. We argue that, in contrast to the planar case, a two-dimensional velocity field does not lead to a single component vorticity equation on the sphere. The shallow fluid approximation is then used to argue that only one component of the vorticity equation is significant for global scale flows. Spherical coordinates are employed throughout, and no planar approximation is used. 1. Introduction. The evolution of the vorticity field in a fluid flow is governed by the dynamical vorticity equation, and understanding the behavior of the vorticity often provides crucial information about the nature of the flow [1, 13]. The material conservation of vorticity [17] plays an important role in our understanding of many geophysical flows of interest [9, 21], see [10, 15,4,5] for some recent studies in this regard. The vorticity equation is obtained by eliminating the pressure (or geopotential) terms from the Euler equations, which are the governing equations of fluid flow. An alternative formulation is the streamfunction-vorticity formulation of the equations of motion [13], which also forms the basis for numerical approaches to geophysical problems [7]. The theoretical importance of vorticity may be contrasted with the fact that the vorticity is usually calculated from velocity measurements, rather than being measured directly [7]. In this paper, we focus on the vorticity equation itself, for a discussion of vortex solutions in geophysical flows, see [16], for a discussion of vortex motion on a sphere see [19, 11]. In the context of geophysical flows, the earth can be considered to be a sphere with a radius equal to the mean radius of the oblate spheroidal earth, R 6368 km [21,7, 20]. A spherical coordinate system is a natural choice to describe≈ flows on the surface of the earth. For an early study utilising spherical coordinates to study waves on a rotating sphere see [12], and for more recent studies with applications to geophysical flow features such as the Equatorial Undercurrent and the Antarctic 2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Vorticity equation, rotating sphere, two-dimensional flow, ideal fluid dynamics, curvature effects. The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin. ∗ Corresponding author. 6261 6262 VIKAS S. KRISHNAMURTHY Circumpolar Current, see [10,6]. We are mostly concerned with application of our discussion of the vorticity equation to large scale, global flows in the ocean, and for such flows the horizontal velocity scales are taken to be U = 0:1 m/s [21,7]. We can calculate the time scale T from the global length scale R and the velocity scale U to be T = R=U = 6:368 107 s, or equivalently T 2:02 years. This is in agreement with the time scale corresponding× to the large scale≈ motions (eg. gyres) in the ocean [20]. The earth rotates with an angular velocity Ω = 7:29 10−5 rad/s which corresponds to the duration of a sidereal day [7]. We evaluate× below a few numbers for later use in this paper. This first set of numbers is valid for large scale flows in the ocean: U 2 UΩ = 2:46 10−16 s−2 and = 1:14 10−12 s−2: (1) R2 × R × If we consider large scale flows in the atmosphere, then the velocity scale changes 5 to Ua = 10 m=s [21], and the time scale changes to Ta = 6:638 10 s 7:4 days. This time scale is in accordance with the time scale for large scale× motions≈ in the atmosphere (eg. cyclones) [7]. The numbers in (1) become U 2 U Ω a = 2:46 10−12 s−2 and a = 1:14 10−10 s−2: (2) R2 × R × The ratios of different length scales in geophysical flows are of central importance in theoretical analyses. Let H = 10 km be the typical vertical (i.e. radial) length scale for both oceanic and atmospheric flows. We can use the same length scale in both cases since, the average depth of the oceans is O(10 km) (and not o(10 km)), and the height of the atmospheric layer relevant to weather phenomena is also O(10 km) [7, 20]. The length scale H is much smaller than the radius of the earth R. We define the shallow fluid approximation as the case when in an expansion of the small parameter H " 1:57 10−3; (3) ≡ R ≈ × we only retain terms of the leading order. The large scale flow of a fluid on the sur- face of a stationary or rotating sphere can effectively be regarded as two-dimensional flow due to the smallness of ". The flow of water and air in the earth's oceans and atmosphere is a prototypical example of such flows. For our purposes, we consider the fluid flow to be incompressible since, the vari- ation of density in the ocean is very small, as is the variation in density in the first 10 km of the atmosphere above the surface of the earth [7, 20]. The effects of viscosity are neglected throughout this paper since the relevant Reynolds number is very large, O(1010), and the Ekman number is very small, O(10−15). This paper is organised as follows. We consider first fluid flow on the surface of a stationary sphere in 2. We discuss the general vorticity equation in an inertial ref- erence frame in 2.1 xand consider the two-dimensional planar vorticity equation in 2.2. In 2.3, wex consider two-dimensional flows on a stationary sphere and obtain thex requiredx form of the velocity field and the corresponding vorticity equation. We introduce scaling arguments to neglect the horizontal (i.e. zonal and meridional) components of the vorticity equation. In 3, we discuss fluid flow on the surface of a rotating sphere. The vorticity equationx for flow in a non-inertial reference frame is discussed in 3.1 and two-dimensional flow on a rotating sphere is considered in 3.2, once againx making use of scaling arguments. The streamfunction-vorticity formulationx of the governing equations is derived in 4. An appendix with vector x THE VORTICITY EQUATION ON A ROTATING SPHERE 6263 calculus identities and their application to vector calculus algebra in spherical co- ordinates has been provided at the end. These identities are used throughout the rest of this paper. 2. Flow on the surface of a stationary sphere. 2.1. Vorticity equation for flow in an inertial reference frame. Consider the three-dimensional flow of an inviscid fluid in an inertial frame of reference. We use vector notation and take the position vector in the inertial frame to be denoted by r, time by t, the density of the fluid by ρ = ρ(r; t), the velocity field by u = u(r; t), the pressure field by p = p(r; t) and the body forces acting on the fluid (per unit mass of the fluid) by F = F (r; t). The slight abuse of notation committed here by providing the same names to variables and their corresponding functions should not cause any confusion. Conservation of momentum for this fluid flow takes the form of the Euler equation @u rp + (u · r)u = + F ; (4) @t − ρ and conservation of mass takes the form of the `continuity equation' @ρ + r · (ρu) = 0: (5) @t Here r is the gradient operator in three dimensions. The reader may consult the standard textbooks of fluid dynamics for derivation and discussion of these formulas [1, 13]. We denote the material derivative operator i.e. the derivative following a fluid element, by D @ = + (u · r): (6) Dt @t The material derivative may act on any scalar or vector quantity of interest. We can rewrite (4) in terms of the material derivative as Du rp = + F ; (7) Dt − ρ with the material derivative acting on the velocity field in this case. We can also rewrite (5) using (6) and (A.1c) as Dρ + ρr · u = 0; (8) Dt with the material derivative acting on the density field in this case. The vorticity field ! = !(r; t) for a fluid flow is defined as ! = r u; (9) × and has the interpretation of being twice the local angular velocity in the fluid [13]. An equation of motion for the vorticity field may be derived by taking the curl of (4). Assuming that the derivatives are continuous allows us to apply the curl and the time derivative interchangeably, and we get @! rp + r (u · r)u = r + r F ; (10) @t × − × ρ × 6264 VIKAS S. KRISHNAMURTHY Using the standard vector calculus identities (A.1) we can rewrite some of the terms in this equation as 1 r (u · r)u = r r(u · u) + ! u (11a) × × 2 × = r (! u) (11b) × × = (u · r)! (! · r)u + !(r · u); (11c) − rp rρ rp r = × : (11d) − × ρ ρ2 Substituting into (10), we obtain the vorticity equation in an inertial frame of ref- erence: @! rρ rp + (u · r)! (! · r)u + !(r · u) = × + r F : (12) @t − ρ2 × The basic texts [1, 13, 21] may be consulted for discussion and interpretation of the various terms in the vorticity equation.
Load more

Recommended publications
  • The Vorticity Equation in a Rotating Stratified Fluid
    Chapter 7 The Vorticity Equation in a Rotating Stratified Fluid The vorticity equation for a rotating, stratified, viscous fluid » The vorticity equation in one form or another and its interpretation provide a key to understanding a wide range of atmospheric and oceanic flows. » The full Navier-Stokes' equation in a rotating frame is Du 1 +∧fu =−∇pg − k +ν∇2 u Dt ρ T where p is the total pressure and f = fk. » We allow for a spatial variation of f for applications to flow on a beta plane. Du 1 +∧fu =−∇pg − k +ν∇2 u Dt ρ T 1 2 Now uu=(u⋅∇ ∇2 ) +ω ∧ u ∂u 1 2 1 2 +∇2 ufuku +bω +g ∧=- ∇pgT − +ν∇ ∂ρt di take the curl D 1 2 afafafωω+=ffufu +⋅∇−+∇⋅+∇ρ∧∇+ ωpT ν∇ ω Dt ρ2 Dω or =−uf ⋅∇ +.... Dt Note that ∧ [ ω + f] ∧ u] = u ⋅ (ω + f) + (ω + f) ⋅ u - (ω + f) ⋅ u, and ⋅ [ω + f] ≡ 0. Terminology ωa = ω + f is called the absolute vorticity - it is the vorticity derived in an a inertial frame ω is called the relative vorticity, and f is called the planetary-, or background vorticity Recall that solid body rotation corresponds with a vorticity 2Ω. Interpretation D 1 2 afafafωω+=ffufu +⋅∇−+∇⋅+∇ρ∧∇+ ωpT ν∇ ω Dt ρ2 Dω is the rate-of-change of the relative vorticity Dt −⋅∇uf: If f varies spatially (i.e., with latitude) there will be a change in ω as fluid parcels are advected to regions of different f. Note that it is really ω + f whose total rate-of-change is determined.
    [Show full text]
  • Scaling the Vorticity Equation
    Vorticity Equation in Isobaric Coordinates To obtain a version of the vorticity equation in pressure coordinates, we follow the same procedure as we used to obtain the z-coordinate version: ∂ ∂ [y-component momentum equation] − [x-component momentum equation] ∂x ∂y Using the p-coordinate form of the momentum equations, this is: ∂ ⎡∂v ∂v ∂v ∂v ∂Φ ⎤ ∂ ⎡∂u ∂u ∂u ∂u ∂Φ ⎤ ⎢ + u + v +ω + fu = − ⎥ − ⎢ + u + v +ω − fv = − ⎥ ∂x ⎣ ∂t ∂x ∂y ∂p ∂y ⎦ ∂y ⎣ ∂t ∂x ∂y ∂p ∂x ⎦ which yields: d ⎛ ∂u ∂v ⎞ ⎛ ∂ω ∂u ∂ω ∂v ⎞ vorticity equation in ()()ζ p + f = − ζ p + f ⎜ + ⎟ − ⎜ − ⎟ dt ⎝ ∂x ∂y ⎠ p ⎝ ∂y ∂p ∂x ∂p ⎠ isobaric coordinates Scaling The Vorticity Equation Starting with the z-coordinate form of the vorticity equation, we begin by expanding the total derivative and retaining only nonzero terms: ∂ζ ∂ζ ∂ζ ∂ζ ∂f ⎛ ∂u ∂v ⎞ ⎛ ∂w ∂v ∂w ∂u ⎞ 1 ⎛ ∂p ∂ρ ∂p ∂ρ ⎞ + u + v + w + v = − ζ + f ⎜ + ⎟ − ⎜ − ⎟ + ⎜ − ⎟ ()⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ∂t ∂x ∂y ∂z ∂y ⎝ ∂x ∂y ⎠ ⎝ ∂x ∂z ∂y ∂z ⎠ ρ ⎝ ∂y ∂x ∂x ∂y ⎠ Next, we will evaluate the order of magnitude of each of these terms. Our goal is to simplify the equation by retaining only those terms that are important for large-scale midlatitude weather systems. 1 Scaling Quantities U = horizontal velocity scale W = vertical velocity scale L = length scale H = depth scale δP = horizontal pressure fluctuation ρ = mean density δρ/ρ = fractional density fluctuation T = time scale (advective) = L/U f0 = Coriolis parameter β = “beta” parameter Values of Scaling Quantities (midlatitude large-scale motions) U 10 m s-1 W 10-2 m s-1 L 106 m H 104 m δP (horizontal) 103 Pa ρ 1 kg m-3 δρ/ρ 10-2 T 105 s -4 -1 f0 10 s β 10-11 m-1 s-1 2 ∂ζ ∂ζ ∂ζ ∂ζ ∂f ⎛ ∂u ∂v ⎞ ⎛ ∂w ∂v ∂w ∂u ⎞ 1 ⎛ ∂p ∂ρ ∂p ∂ρ ⎞ + u + v + w + v = − ζ + f ⎜ + ⎟ − ⎜ − ⎟ + ⎜ − ⎟ ()⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ∂t ∂x ∂y ∂z ∂y ⎝ ∂x ∂y ⎠ ⎝ ∂x ∂z ∂y ∂z ⎠ ρ ⎝ ∂y ∂x ∂x ∂y ⎠ ∂ζ ∂ζ ∂ζ U 2 , u , v ~ ~ 10−10 s−2 ∂t ∂x ∂y L2 ∂ζ WU These inequalities appear because w ~ ~ 10−11 s−2 the two terms may partially offset ∂z HL one another.
    [Show full text]
  • 1 RELATIVE VORTICITY EQUATION Newton's Law in a Rotating Frame in Z-Coordinate
    RELATIVE VORTICITY EQUATION Newton’s law in a rotating frame in z-coordinate (frictionless): ∂U + U ⋅∇U = −2Ω × U − ∇Φ − α∇p ∂t ∂U ⎛ U ⋅ U⎞ + ∇⎜ ⎟ + (∇ × U) × U = −2Ω × U − ∇Φ − α∇p ∂t ⎝ 2 ⎠ Applying ∇ × to both sides, and noting ω ≡ ∇ × U and using identities (the underlying tilde indicates vector): 1 A ⋅∇A = ∇(A ⋅ A) + (∇ × A) × A 2 ∇ ⋅(∇ × A) = 0 ∇ × ∇γ = 0 ∇ × (γ A) = ∇γ × A + γ ∇ × A ∇ × (F × G) = F(∇ ⋅G) − G(∇ ⋅ F) + (G ⋅∇)F − (F ⋅∇)G So, ∂ω ⎡ ⎛ U ⋅ U⎞ ⎤ + ∇ × ⎢∇⎜ ⎟ ⎥ + ∇ × (ω × U) = −∇ × (2Ω × U) − ∇ × ∇Φ − ∇ × (α∇p) ∂t ⎣ ⎝ 2 ⎠ ⎦ ⇓ ∂ω + ∇ × (ω × U) = −∇ × (2Ω × U) − ∇α × ∇p − α∇ × ∇p ∂t Using S to denote baroclinicity vector, S = −∇α × ∇p , then, ∂ω + ω(∇ ⋅ U) − U(∇ ⋅ω) + (U ⋅∇)ω − (ω ⋅∇)U = −2Ω(∇ ⋅ U) + U∇ ⋅(2Ω) − (U ⋅∇)(2Ω) + (2Ω ⋅∇)U + S ∂t ∂ω + ω(∇ ⋅ U) + (U ⋅∇)ω − (ω ⋅∇)U = −2Ω(∇ ⋅ U) − (U ⋅∇)(2Ω) + (2Ω ⋅∇)U + S ∂t A little rearrangement: ∂ω = − (U ⋅∇)(ω + 2Ω) − (ω + 2Ω)(∇ ⋅ U) + [(ω + 2Ω)⋅∇]U + S ∂t tendency advection convergence twisting baroclinicity RELATIVE VORTICITY EQUATION (Vertical Component) Take k ⋅ on the 3-D vorticity equation: ∂ζ = −k ⋅(U ⋅∇)(ω + 2Ω) − k ⋅(ω + 2Ω)(∇ ⋅ U) + k ⋅[(ω + 2Ω)⋅∇]U + k ⋅S ∂t 1 ∇p In isobaric coordinates, k = , so k ⋅S = 0 . Working through all the dot product, you ∇p should get vorticity equation for the vertical component as shown in (A7.1). The same vorticity equations can be easily derived by applying the curl to the horizontal momentum equations (in p-coordinates, for example). ∂ ⎡∂u ∂u ∂u ∂u uv tanφ uω f 'ω ∂Φ ⎤ − ⎢ + u + v + ω = + + + fv − ⎥ ∂y
    [Show full text]
  • Euler's Equation
    Chapter 5 Euler’s equation Contents 5.1 Fluid momentum equation ........................ 39 5.2 Hydrostatics ................................ 40 5.3 Archimedes’ theorem ........................... 41 5.4 The vorticity equation .......................... 42 5.5 Kelvin’s circulation theorem ....................... 43 5.6 Shape of the free surface of a rotating fluid .............. 44 5.1 Fluid momentum equation So far, we have discussed some kinematic properties of the velocity fields for incompressible and irrotational fluid flows. We shall now study the dynamics of fluid flows and consider changes S in motion due to forces acting on a fluid. We derive an evolution equation for the fluid momentum by consider- V ing forces acting on a small blob of fluid, of volume V and surface S, containing many fluid particles. 5.1.1 Forces acting on a fluid The forces acting on the fluid can be divided into two types. Body forces, such as gravity, act on all the particles throughout V , Fv = ρ g dV. ZV Surface forces are caused by interactions at the surface S. For the rest of this course we shall only consider the effect of fluid pressure. 39 40 5.2 Hydrostatics Collisions between fluid molecules on either sides of the surface S pro- duce a flux of momentum across the boundary, in the direction of the S normal n. The force exerted on the fluid into V by the fluid on the other side of S is, by convention, written as n Fs = −p n dS, ZS V where p(x) > 0 is the fluid pressure. 5.1.2 Newton’s law of motion Newton’s second law of motion tells that the sum of the forces acting on the volume of fluid V is equal to the rate of change of its momentum.
    [Show full text]
  • A Vectors, Tensors, and Their Operations
    A Vectors, Tensors, and Their Operations A.1 Vectors and Tensors A spatial description of the fluid motion is a geometrical description, of which the essence is to ensure that relevant physical quantities are invariant under artificially introduced coordinate systems. This is realized by tensor analysis (cf. Aris 1962). Here we introduce the concept of tensors in an informal way, through some important examples in fluid mechanics. A.1.1 Scalars and Vectors Scalars and vectors are geometric entities independent of the choice of coor- dinate systems. This independence is a necessary condition for an entity to represent some physical quantity. A scalar, say the fluid pressure p or den- sity ρ, obviously has such independence. For a vector, say the fluid velocity u, although its three components (u1,u2,u3) depend on the chosen coordi- nates, say Cartesian coordinates with unit basis vectors (e1, e2, e3), as a single geometric entity the one-form of ei, u = u1e1 + u2e2 + u3e3 = uiei,i=1, 2, 3, (A.1) has to be independent of the basis vectors. Note that Einstein’s convention has been used in the last expression of (A.1): unless stated otherwise, a repeated index always implies summation over the dimension of the space. The inner (scalar) and cross (vector) products of two vectors are familiar. If θ is the angle between the directions of a and b, these operations give a · b = |a||b| cos θ, |a × b| = |a||b| sin θ. While the inner product is a projection operation, the cross-product produces a vector perpendicular to both a and b with magnitude equal to the area of the parallelogram spanned by a and b.Thusa×b determines a vectorial area 694 A Vectors, Tensors, and Their Operations with unit vector n normal to the (a, b) plane, whose direction follows from a to b by the right-hand rule.
    [Show full text]
  • Viscous Vorticity Equation (VISVE) for Turbulent 2-D Flows with Variable Density and Viscosity
    Journal of Marine Science and Engineering Article VIScous Vorticity Equation (VISVE) for Turbulent 2-D Flows with Variable Density and Viscosity Spyros A. Kinnas Ocean Engineering Group, Department of Civil, Architectural, and Environmental Engineering, The University of Texas at Austin, Austin, TX 78712, USA; [email protected]; Tel.: +1-512-475-7969 Received: 23 January 2020; Accepted: 4 March 2020; Published: 11 March 2020 Abstract: The general vorticity equation for turbulent compressible 2-D flows with variable viscosity is derived, based on the Reynolds-Averaged Navier-Stokes (RANS) equations, and simplified versions of it are presented in the case of turbulent or cavitating flows around 2-D hydrofoils. Keywords: vorticity; viscous flow; turbulent flow; mixture model; cavitating flow 1. Introduction The vorticity equation has been utilized by several authors in the past to analyze the viscous flow around bodies. Vortex element (or particle, or blub) and vortex-in-cell methods have been used for several decades for the analysis of 2-D or 3-D flows, as described in Chorin [1], Christiansen [2], Leonard [3], Koumoutsakos et al. [4], Ould-Salihi et al. [5], Ploumhans et al. [6], Cottet and Poncet [7], and Cottet and Poncet [8]. Those methods essentially decouple the vortex dynamics (convection and stretching) from the effects of viscosity. In recent years, the VIScous Vorticity Equation (VISVE), has been solved by using a finite volume method, without decoupling the vorticity dynamics from the effects of viscosity. This method has been applied to 2-D and 3-D hydrofoils, cylinders, as well as propeller blades as described in Tian [9], Tian and Kinnas [10], Wu et al.
    [Show full text]
  • Streamfunction-Vorticity Formulation
    Streamfunction-Vorticity Formulation A. Salih Department of Aerospace Engineering Indian Institute of Space Science and Technology, Thiruvananthapuram { March 2013 { The streamfunction-vorticity formulation was among the first unsteady, incompressible Navier{ Stokes algorithms. The original finite difference algorithm was developed by Fromm [1] at Los Alamos laboratory. For incompressible two-dimensional flows with constant fluid properties, the Navier{Stokes equations can be simplified by introducing the streamfunction y and vorticity w as dependent variables. The vorticity vector at a point is defined as twice the angular velocity and is w = ∇ ×V (1) which, for two-dimensional flow in x-y plane, is reduced to ¶v ¶u wz = w · kˆ = − (2) ¶x ¶y For two-dimensional, incompressible flows, a scalar function may be defined in such a way that the continuity equation is identically satisfied if the velocity components, expressed in terms of such a function, are substituted in the continuity equation ¶u ¶v + = 0 (3) ¶x ¶y Such a function is known as the streamfunction, and is given by V = ∇ × ykˆ (4) In Cartesian coordinate system, the above relation becomes ¶y ¶y u = v = − (5) ¶y ¶x Lines of constant y are streamlines (lines which are everywhere parallel to the flow), giving this variable its name. Now, a Poisson equation for y can be obtained by substituting the velocity components, in terms of streamfunction, in the equation (2). Thus, we have ∇2y = −w (6) where the subscript z is dropped from wz. This is a kinematic equation connecting the streamfunction and the vorticity. So if we can find an equation for w we will have obtained a formulation that automatically produces divergence-free velocity fields.
    [Show full text]
  • Circulation and Vorticity
    Lecture 4: Circulation and Vorticity • Circulation •Bjerknes Circulation Theorem • Vorticity • Potential Vorticity • Conservation of Potential Vorticity ESS227 Prof. Jin-Yi Yu Measurement of Rotation • Circulation and vorticity are the two primary measures of rotation in a fluid. • Circul ati on, whi ch i s a scal ar i nt egral quantit y, i s a macroscopic measure of rotation for a finite area of the fluid. • Vorticity, however, is a vector field that gives a microscopic measure of the rotation at any point in the fluid. ESS227 Prof. Jin-Yi Yu Circulation • The circulation, C, about a closed contour in a fluid is defined as thliihe line integral eval uated dl along th e contour of fh the component of the velocity vector that is locally tangent to the contour. C > 0 Î Counterclockwise C < 0 Î Clockwise ESS227 Prof. Jin-Yi Yu Example • That circulation is a measure of rotation is demonstrated readily by considering a circular ring of fluid of radius R in solid-body rotation at angular velocity Ω about the z axis . • In this case, U = Ω × R, where R is the distance from the axis of rotation to the ring of fluid. Thus the circulation about the ring is given by: • In this case the circulation is just 2π times the angular momentum of the fluid ring about the axis of rotation. Alternatively, note that C/(πR2) = 2Ω so that the circulation divided by the area enclosed by the loop is just twice thlhe angular spee dfifhid of rotation of the ring. • Unlike angular momentum or angular velocity, circulation can be computed without reference to an axis of rotation; it can thus be used to characterize fluid rotation in situations where “angular velocity” is not defined easily.
    [Show full text]
  • 6 Fundamental Theorems: Vorticity and Circulation
    EOSC 512 2019 6 Fundamental Theorems: Vorticity and Circulation In GFD, and especially the study of the large-scale motions of the atmosphere and ocean, we are particularly interested in the rotation of the fluid. As a consequence, again assuming the motions are of large enough scale to feel the e↵ects of (in particular, the di↵erential) rotation of the outer shell of rotating, spherical planets: The e↵ects of rotation play a central role in the general dynamics of the fluid flow. This means that vorticity (rotation or spin of fluid elements) and circulation (related to a conserved quantity...) play an important role in governing the behaviour of large-scale atmospheric and oceanic motions. This can give us important insight into fluid behaviour that is deeper than what is derived from solving the equations of motion (which is challenging enough in the first place). In this section, we will develop two such theorems and principles related to the conservation of vorticity and circulation that are particularly useful in gaining insight into ocean and atmospheric flows. 6.1 Review: What is vorticity? Vorticity was previously defined as: @uk !i = ✏ijk (6.1) @xj Or, in vector notation: ~! = ~u r⇥ ˆi ˆj kˆ @ @ @ = (6.2) @x @y @z uvw @w @v @u @w @v @u = ˆi + ˆj + ˆi @y − @z @z − @x @x − @y ✓ ◆ ✓ ◆ ✓ ◆ Physically, the vorticity is two times the local rate of rotation (or “spin”) of a fluid element. Here, it is important to distinguish between circular motion and the rotation of the element. Here, the fluid element moving from A to B on the circular path has no vorticity, while the fluid element moving from C to D has non-zero vorticity.
    [Show full text]
  • Chapter 7 Several Forms of the Equations of Motion
    CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium, the equations of motion for compress- ible flow become the famous Navier-Stokes equations. In Cartesian coordinates, ,l , ------ +0------- ()lUi = ,t ,xi ,lUi , ------------- + -------- ()lUiU j + Pbij – lGi – ,t ,x j , .-------- ()2µSij – ()()23 µµ– v bijSkk = 0 . (7.1) ,x j ,l()ek+ , ------------------------ + ------- ()lUiht–g,()T ,xi – lGiUi – ,t ,xi , ------- ()2µU S – ()()23 µµ– b U S = 0 j ij v ij j kk ,xi The Navier-Stokes equations are the foundation of the science of fluid mechanics. With the inclusion of an equation of state, virtually all flow solving revolves around finding solutions of the Navier-Stokes equations. Most exceptions involve fluids where the relation between stress and rate-of-strain is nonlinear such as polymers, or where the equation of state is not very well understood (for example supersonic flow in water) or rarefied flows where the Boltzmann equation must be used to explicitly account for particle collisions. bjc 7.1 4/1/13 The momentum equation expressed in terms of vorticity The equations can take on many forms depending on what approximations or assumptions may be appropriate to a given flow. In addition, transforming the equations to different forms may enable one to gain insight into the nature of the solutions. It is essential to learn the many different forms of the equations and to become practiced in the manipulations used to transform them. 7.1.1 INCOMPRESSIBLE NAVIER-STOKES EQUATIONS If there are no body forces and the flow is incompressible, ¢ U = 0 the Navier-Stokes equations reduce to what is probably their most familiar form.
    [Show full text]
  • FLUID VORTICES FLUID MECHANICS and ITS Applicanons Volume 30
    FLUID VORTICES FLUID MECHANICS AND ITS APPLICAnONS Volume 30 Series Editor: R. MOREAU MADYLAM Ecole Nationale Superieure d'Hydraulique de Grenoble BOlte Postale 95 38402 Saint Martin d'Heres Cedex, France Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advance­ ment. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological ~nd ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list of related mechanics titles, see final pages. Fluid Vortices Edited by SHELDON 1. GREEN Department ofMechanical Engineering, University ofBritish Colwnbia, Vancouver, Canada. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
    [Show full text]
  • A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes
    University of Pennsylvania ScholarlyCommons Department of Earth and Environmental Departmental Papers (EES) Science 12-2010 A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes Federico Falcini Douglas J. Jerolmack University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/ees_papers Part of the Environmental Sciences Commons, Geomorphology Commons, and the Sedimentology Commons Recommended Citation Falcini, F., & Jerolmack, D. J. (2010). A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes. Journal of Geophysical Research: Earth Surface, 115 (F4), F04038-. http://dx.doi.org/10.1029/2010JF001802 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ees_papers/76 For more information, please contact [email protected]. A Potential Vorticity Theory for the Formation of Elongate Channels in River Deltas and Lakes Abstract Rivers empty into oceans and lakes as turbulent sediment-laden jets, which can be characterized by a Gaussian horizontal velocity profile that spreads and decays downstream because of shearing and lateral mixing at the jet margins. Recent experiments demonstrate that this velocity field controls river-mouth sedimentation patterns. In nature, diffuse jets are associated with mouth bar deposition forming bifurcating distributary networks, while focused jets are associated with levee deposition and the growth of elongate channels that do not bifurcate. River outflows from elongate channels are similar in structure to cold filaments observed in ocean currents, where high potential vorticity helps to preserve coherent structure over large distances. Motivated by these observations, we propose a hydrodynamic theory that seeks to predict the conditions under which elongate channels form.
    [Show full text]