DOCSLIB.ORG

Fluid Dynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fluid Dynamics Fluid Dynamics - Math 6750 Vorticity equation - Week 5 We consider the Navier-Stokes equations for an incompressible Newtonian fluid r · u = 0 (1) @u ρ + u · ru = −∇p + ρF + µ∆u (2) @t b 1 Bernoulli's Theorem Take the body force to be conservative, that is, Fb = −∇χ for some scalar potential χ. Using the vector identity 1 1 u · ru = (r × u) × u + rjuj2 = ! × u + rjuj2 2 2 we may rewrite the conservation of linear momentum equation (2) as @u 1 1 µ + ! × u = − rp − rχ − rjuj2 + ∆u @t ρ 2 ρ or @u + ! × u = −∇H + ν∆u; (3) @t p 1 2 where H = ρ + χ + 2 juj and ν = µ/ρ. If the flow is steady, then (3) reduces to ! × u = −∇H + ν∆u: (4) Taking the dot product of (4) against u gives u · rH = −u · (! × u) + ν∆u = νu · ∆u: (5) Consequently, H decreases in the flow direction when u·∆u < 0, i.e when the local net viscous force per unit volume tends to decelerate the fluid and work is done against viscous forces as an element of fluid moves along a streamtube; similarly H increases along the streamline when the net viscous force tends to accelerate the fluid. In the case of an ideal fluid (no viscosity), (5) reduces to u · rH = 0 which means the directional derivative of H along the flow field u is zero. This is known as Bernoulli streamline theorem. Theorem 1 (Bernoulli streamline thm). For a steady flow of an ideal fluid, subject to a p 1 conservative force F = −∇χ, the function H = + χ + juj2 is constant along a streamline. b ρ 2 In particular, an increase of the fluid velocity occurs simultaneously with a decrease in the (static) pressure or a decrease in the fluid's potential energy, along a given streamline. Here, p; χ, u depends on the particular point on the chosen streamline but the constant depends only on that streamline. We point out that the theorem says nothing more than H being constant along a given streamline, so H may have different constants on different streamlines. Recall that a flow is irrotational if ! ≡ 0. In this case, H is constant everywhere 1 Proposition 1. For a steady irrotational flow of an ideal fluid, subject to a conservative p 1 force F = −∇χ, the function H = + χ + juj2 is constant everywhere. b ρ 2 2 Vorticity Equation As it turns out, the net viscous force on an element of incompressible fluid is determined by the local gradients of vorticity since r × ! = r × (r × u) = r (r · u) − ∆u = −∆u: (6) This is rather surprising since we know from the Newtonian stress tensor that the viscous stress is generated solely by deformation of the fluid and is independent of the local vorticity, but the explanation is wholly a matter of kinematics. The rate of deformation tensor D and the vorticity ! play independent roles in the generation of stress, but certain spatial derivatives of D are identically related to certain derivatives of ! through the vector identity used in (6). It follows from (6) that the viscous distribution of vorticity is pivotal in understanding the evolution of large Reynolds number flow , i.e when the fluid viscosity is sufficiently small. The vorticity transport equation is obtained as follows. Taking the curl of Bernoulli's equation (3) and using the fact that r × (rf) = 0 for any C2 scalar function f, we find that @! + r × (! × u) = ν (r × ∆u) : @t We use the incompressibility condition and the fact that r · ! = 0 to cancel out terms. Applying the vector identity from (6) to ! reduces the viscous term into r × ∆u = −∇ × (r × !) = −∇ (r · !) + ∆! = ∆!: For the convection term, r × (! × u) = !r · u − ur · ! + u · r! − ! · ru = u · r! − ! · ru: Combining all the computations leads to the vorticity equation D! @! = + u · r! = ! · ru + ν∆! (7) Dt @t The pressure term has been eliminated in the vorticity equation, with the price that the vorticity equation contains both u and !. The terms in (7) can be interpreted: D! 1. The term is the familiar material derivative of !, describing the rate of change of Dt ! due to the convection of fluid. 2. The term ν∆! accounts for the diffusion of vorticity due to the viscous effects. 3. The term ! · ru represents the stretching or tilting of vortex tubes due to the folow velocity gradients. 2 4. The term ! (r · u) (which is absent due to the incompressibility condition) describes vortex stretching due to flow compressibility. In the case of a two-dimensional incompressible inviscid flow, i.e u = (u(x; y; t); v(x; y; t); 0), the vorticity only has one non-zero component ! = (0; 0;@xv − @yu) = !(x; y; t)e3 and @u ! · ru = ! = 0: @z D! Consequently, = 0 and so the vorticity ! of each individual fluid particle does not change Dt in time, i.e if ! = 0 at some time t0, then ! ≡ 0 for all time t ≥ t0. This also occurs for a unidirectional flow, say u = u(y; z; t)e1, since we then have @u @u ! = (0;@ u; −@ u) =) ! · ru = @ u − @ u = 0: z y z @y y @z If we further assume steady flow, then D! = u · r! = 0 Dt and we arrive at the following result Proposition 2. For a steady two-dimensional flow of an ideal fluid subject to a conservative body force, the vorticity ! is constant along a streamline. An immediate consequence of this is that steady flow past an aerofoil is generally irrota- tional. 3.
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]