3.1 Definition Physical meaning Streamfunction and vorticity The Rankine vortex Circulation...

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1 Chapter 3 Vorticity Contents 3.1 Definition Physical meaning Streamfunction and vorticity The Rankine vortex Circulation Examples of vortex lines (vortices) Definition We have already considered u, which is a measure of the local expansion or contraction of the fluid. ( u = for incompressible fluids.) The vorticity, ω = u, (3.1) is a measure of the local rotation or spin in a flow. It is a concept of central importance in fluid dynamics. 3.2 Physical meaning In the simplest case of a 2-D flow, u(x,y,t) = v x u u(x,y,t) v(x,y,t), the vorticity ω = u = v is perpendicular to the plane of motion; its magnitude is ω = x u. Consider the two short fluid line-elements AB and AC. The vertical differential velocity is 19

2 2 3.2 Physical meaning B δu δy A δx δv C δv = v C v A = v(x+δx,y) v(x,y) δx v x So, v x (Taylor theorem). is the angular velocity of the fluid line-element AC. Similarly, the horizontal differential velocity So, u δu = u B u A = u(x,y +δy) u(x,y) δy u. is the angular velocity of the fluid line element AB. ω Thus, 2 = 1 ( v 2 x u ) represents the average angular velocity of the two fluid line-elements AB and AC. This could be measured using a crossed pair of small vanes that float with the fluid. Example 3.1 (Solid body rotation) For u = Ω r, with constant angular velocity Ω and r = x y z, ω = (Ω r) = Ω( r) (Ω )r, = 3Ω Ω = 2Ω. Ω So, the vorticity is twice the local rotation rate which is also global here. Example 3.2 (Shear flow) For u = ky, v =, the vorticity ω = k with ω = k. Vorticity rollers The vorticity is not a measure of global rotation: a shear flow has no global rotation but a non-zero vorticity. Example 3.3 (Line vortex flow) Let u r =, u θ = k/r and u z = where k is a positive constant. (This is a crude model for a bath-plug vortex see examples 2.2 and 2.3)

3 Chapter 3 Vorticity 21 ψ Streamfunction: r = k r the streamlines are circles. ψ = klnr (const. = ). So, However, from ω = [ 1 r r (ru θ) 1 r ] θ u r ê z = 1 k =, r rêz one finds that the vorticity is zero everywhere in the flow except at r = where the functions u and ω are not defined. (In fact ω can be defined as a Dirac delta distribution.) Although the flow is rotating globally, there is no local rotation. Crossed vanes placed in the flow would move in a circle, but without spinning. 3.3 Streamfunction and vorticity In a two-dimensional flow, the vorticity ω = coordinates. If, in addition, the fluid is incompressible, u = ψ More generally (Cartesian or plane polar coordinates), ω where ω = v x u in Cartesian and v = ψ, so that x ω = 2 ψ x 2 2 ψ 2 = 2 ψ. (3.2) u = (ψê z ) ω = ( (ψê z )), Since ψ/ z = for a 2-D flow, one finds again = ( (ψê z )) 2 (ψê z ), ( ) = z ψ 2 ψê z. ω = 2 ψê z (3.3) 3.4 The Rankine vortex Example 3.3, with u θ = k/r, is a crude model for the bath-plug vortex: infinite vorticity concentrated in r = (singularity). In real vortices the vorticity is spread over a small area. Consider an azimuthal flow, u r = u z = and u θ = f(r), in cylindrical coordinates, such that { Ω if r a (Ω constant), ω = if r > a

4 Circulation Hence, ψ ψ(r) and 2 ψ = 1 r r a: ( d r dψ ) { Ω if r a, = dr dr if r > a. ( d r dψ ) = rω dψ dr dr dr = Ω 2 r+ B r = u θ. We require u θ to be bounded at r =. So, B = and, for r a, u θ = Ω 2 r, ψ = Ω 4 r2. r > a: ( d r dψ ) = u θ = dψ dr dr dr = A r. The continuity of u θ at r = a implies that A = Ω 2 a2. So, dψ dr = Ωa2 gives ψ = Ωa2 4 2r ψ = Ωa2 lnr +D, and applying the continuity of ψ at r = a now ( 2 1+2ln r ). a So, and u θ r 1/r Ω 4 r2 if r a, ψ = Ωa2 ( 1+2ln r ) if r > a. 4 a Ω r if r a : solid body rotation, 2 u θ = Ωa 2 if r > a : bath-plug flow. 2r ω Ω a r There is still a discontinuity in vorticity but the flow is quite adequate for predicting the shape of the water surface. a r 3.5 Circulation Consider a closed curve C in the flow. The circulation around C is the line integral of the tangential velocity around C: Γ = u dl. (3.4) C

5 Chapter 3 Vorticity 23 n C S dl 11 By Stokes theorem, for any surface S spanning the curve C, Γ = u dl = ( u) nds = C S u S ω nds. (3.5) So, the circulation around C is equal to the vorticity flux through the surface S: it is the strength of the vortex tube. Example 3.4 (The Rankine vortex) The circulation around a circle of radius r is given by Γ = 2π u θ (r)rdθ = 2πru θ (r) = { πωr 2 if r a, πωa 2 if r > a. When the vorticity is concentrated in thin filaments (tubes) as it is in the Rankine vortex, it is useful to think in term of vortex. 3.6 Examples of vortex lines (vortices) i. Bath-plug vortex. The shape of the free surface of water can be modelled using the Rankin vortex. Note that the sense of rotation is not determined by the rotation of the Earth! ii. Vortices behind aeroplanes.

6 Examples of vortex lines (vortices) The characteristic vapour trails left by aircraft are vortex lines shed from the wing tips. (The vortices have low pressure, so vapour water condenses there.) These vortices decay very slowly and are a danger for small aircraft flying behind large ones. iii. Horseshoe vortex & downwash behind chimneys. Vortex lines in shear flows above ground level travel with the air and can be stretched and bent by tall buildings and chimneys. This results in a downwards flow behind chimneys, dragging pollutant down to ground level. iv. Vortex rings. Smoke rings and underwater bubble rings are examples of vortex rings.

7 Chapter 3 Vorticity 25 v. von Kármán vortex street. In certain conditions, a flow past an obstacle (e.g. a cylinder, an island) produces a series of line vortices.

8 Examples of vortex lines (vortices)