Vorticity And Circulation 574o2t

Introduction Class 15

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Circulation and Vorticity: Two primary measures of rotation in a fluid The presentation illustrates the concepts of vorticity and circulation, and shows how these concepts can be useful in understanding fluid flows.

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Vorticity: The tendency to spin about an axis; Microscopic measure of rotation at any point in the fluid The vorticity is defined as the curl of the velocity vector:

w=ΔxV 

Thus each point in the fluid has an associated vector vorticity, and the whole fluid space may be thought of as being threaded by vortex lines which are everywhere tangent to the local vorticity vector.

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These vortex lines represent the local axis of spin of the fluid particle at each point. In two dimensions, the vorticity is the sum of the angular velocities of any pair of mutuallyperpendicular, infinitesimal fluid lines ing through the point in question

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For rigid (unbending) body rotation, every line perpendicular to the axis of rotation has the same angular velocity: therefore the vorticity is the same at every point, and is twice the angular velocity. Vorticity is related to the moment of momentum of a small spherical fluid particle about its own center of mass.

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In physics, angular momentum, moment of momentum, or rotational momentum is a vector quantity that represents the product of a body's rotational inertia  and rotational velocity about a particular axis. The angular momentum of a system of particles (e.g. a rigid body) is the sum of angular moments of the individual particles.

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For a rigid body rotating around an axis of symmetry (e.g. the blades of a ceiling fan), the angular momentum can be expressed as the product of the body's moment of inertia, (I), (i.e. a measure of an object's resistance to changes in its rotation rate) and its angular velocity ω:

L=Iω In this way, angular momentum is sometimes described as the rotational analog of linear momentum.

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For the case of an object that is small compared with the radial distance to its axis of rotation (planet orbiting in a circle around the Sun), the angular momentum can be expressed as its  linear momentum, (mv), crossed by its  position from the origin, (r). Thus, the angular momentum (L) of a particle with respect to some point of origin is:

L

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r x mv

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Angular momentum is  conserved in a system where there is no net external  torque, and its conservation helps explain many diverse (various) phenomenas.

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The conservation of angular momentum is used extensively in analyzing what is called central force motion. If the net force on some body is directed always toward some fixed point, the center, then there is no torque on the body with respect to the center, and so the angular momentum of the body about the center is constant.

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The conservation of angular momentum explains the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. By bringing part of mass of her body closer to the axis she decreases her body's moment of inertia. Because angular momentum is constant in the absence of external torques, the angular velocity (rotational speed) of the skater has to increase.

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An example of angular momentum conservation: A spinning figure skater reduces her  moment of inerti a  by pulling in her arms, causing her rotation rate to increase.

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The angular momentum of a particle of mass (m) with respect to a chosen origin is given by L = mvr sin θ or more formally by the vector product L=rxp The direction is given by the right hand rule  which would give (L) the direction out of the diagram. For an orbit, angular momentum is  conserved, and this leads to one of  Kepler's laws. For a circular orbit, (L ) becomes L = mvr

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In astronomy, Kepler's laws of planetary motion are three scientific laws describing  orbital motion, each giving a description of the  motion of planets around the Sun. Kepler's laws are: The orbit of every planet is an ellipse with the Sun at one of the two foci(crucial, important) points. A line ing a planet and the Sun sweeps out equal areas during equal intervals of time. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis  of its orbit.

(1) The orbits are ellipses, with focal points ƒ1 and ƒ2 for the first planet and ƒ1 and ƒ3 for the second planet. The Sun is placed in focal point ƒ1. (2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2. (3) The total orbit times for planet 1 and planet 2 have a ratio  a13/2 : a23/2.

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The angular momentum of a rigid object is defined as the product of the  moment of inertia and the angular velocity. It is analogous to linear momentum and is subject to the fundamental constraints of the conservation of angular momentum  principle if there is no external torque on the object. Angular momentum is a vector quantity. It is derivable from the expression for the  angular momentum of a particle

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Angular momentum and linear momentum are examples of the  parallels between linear and rotational motion. They have the same form and are subject to the fundamental constraints of conservation laws, the  conservation of momentum and the conservation of angular momentum .

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Sometimes the word ''rotation" is used as a synonym for vorticity, but this does not mean that a flow has to be curved for vorticity to be present. For instance, picture below shows water flowing in a straight channel.

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The streamlines are essentially straight and parallel to the side wall. But the rotation of the arrow shows that vertical vorticity is present. Near the wall is a viscous boundary layer in which the velocity increases with distance from the wall

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Examine the fluid cross: One leg moves downstream parallel to the wall while the other leg rotates counterclockwise owing to the non-uniform velocity distribution. Thus there is a net vorticity, and the vorticity meter turns counterclockwise.

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On the other hand, the flow may be without rotation even though the streamlines are curved. Fig. below shows in plan view a tank for producing a sink vortex in which the streamlines are tight spirals and nearly circular.

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As shown in Fig. the vorticity meter moves in a circular path but does not rotate. It moves in pure translation-as would a com needle

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Consider a fluid cross at a point on a circular streamline (Fig. below). Leg A follows the streamline, hence it rotates counterclockwise. Since the angular momentum of the fluid is conserved as it flows toward the drain, the tangential velocity varies inversely with the radius.

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Thus the velocity of the inner part of leg B is greater than the velocity of the outer part, and leg B turns clockwise. The clockwise turning rate of B is just equal and opposite to the counterclockwise turning rate of A. Hence the vorticity is zero. The vorticity meter, in averaging the rotations of legs A and B, translates, without rotation, on a circular trajectory

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Crocco's theorem is a fluid dynamics theorem relating the velocity, vorticity, and stagnation pressure (or entropy) of a potential flow. Because stagnation pressure loss, there are three popular forms for writing Crocco's theorem: 1) Stagnation pressure:  

V x ω = 1/ρ Δp

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Because stagnation pressure loss and entropy generation can be viewed as essentially the same thing 2) Entropy:

2) quantity of movement:

   

(V) - is velocity, (ω)  is vorticity,  (ρ) - is density, (Δp) - is stagnation pressure, (T) -  is temperature,  (s) -  is entropy, and  (n)  is the direction normal to the streamlines.

For the special case of steady motion of an incompressible, inviscid fluid acted on by conservative body forces, Crocco's theorem has the form  V x ω =1/ρ x Δp  ρ = p + ½ρV² + ρ U where (V) is the vector velocity, (ω) the vector vorticity, (ρ) the density, and (Δp) – gradient of stagnation pressure. 

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The stagnation pressure (p), is the sum of the static pressure (p), the dynamic pressure (ρV²/2), and the potential energy per unit volume (ρ U) associated with the conservative body-force field. In fluid dynamics, stagnation pressure (or total pressure) is the static pressure at a stagnation point in a fluid flow (in stagnation point the fluid velocity is zero and all kinetic energy has been converted into pressure energy). Stagnation pressure is equal to the sum of the free-stream dynamic pressure and freestream static pressure

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When a flow is two-dimensional in the plane of the paper, the vorticity vector is normal to the paper while the velocity vector lies in the paper and along the streamline (Fig. below). By Crocco's theorem, the gradient of stagnation pressure is normal to both the velocity vector and the vorticity vector; thus it lies in the plane of the paper and normal to V. Consequently the stagnation pressure, (p ̻), is constant along each streamline and varies between streamlines only if vorticity is present.

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To illustrate, consider again the straight boundary layer of Fig. The static pressure is uniform across the boundary layer but the velocity is variable. Thus the stagnation pressure is variable, and, vorticity is present.

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The velocity gradient is strongest near the wall and so is the gradient of stagnation pressure. Wben the vorticity meter is near the wall, the rate of spin is relatively large. With the vorticity meter farther out in the boundary layer, the rate of spin is smaller.