3. Vorticity

Question 1

Vorticity

Definition

Answer 1

-vorticity is defined as:
|ω = ∇ x |u
-it is a measure of the local rotation or spin in a flow

Question 2

Vorticity in 2D Flow

Answer 2

|u = (u,v,0)
|ω = ∇ x (u,v,0) = (0,0,ω)
-where;
ω = ∂v/∂x - ∂u/∂y

Question 3

Vorticity and Streamfunction

Answer 3

|ω = (0,0, ∂v/∂x - ∂u/∂y)
-but for a 2D flow:
u = ∂/∂y , v = -∂/∂x
-sub in for scalar omega;
ω = -∂²ψ/∂x² - ∂ψ²/∂y²
-so omega in terms of the stream function:
ω = -∇²ψ

Question 4

Vorticity in a Shear Flow

Answer 4

|u = (ky, 0, 0)
|ω = (0,0,-k)
-the vorticity is not a measure of global rotation since shear flow has no global rotation but does have non-zero vorticity
-instead locally, fluid particles rotate because he top is moving faster than the bottom, these local spinning elements are known as vorticity rollers

Question 5

Physical Meaning of Vorticity

Answer 5

-vorticity can be thought of by picturing a crossed pair of small vanes that float with the fluid and are free to rotate

Question 6

Solid Body Rotation

Answer 6

-for |u = |Ω x ||r with constant angular velocity Ω and |r = (x,y,z), vorticity is given by:
|ω = ∇ x (|Ωx|r)
= |Ω(∇. |r) - (|Ω.∇)|r
= 3Ω - Ω = 2Ω
-so the vorticity is twice the local rotation rate
-crossed vanes placed in the flow would move in a circle and rotate locally

Question 7

Vorticity in a Line Vortex Flow

Answer 7

-let ur=0, uθ=k/r, uz=0, where k is a positive constant
-streamfunction:
∂ψ/∂r = -k/r
=> ψ = -k lnr
-so streamlines are circles
-however |ω = |0, so the vorticity is 0 everywhere in the flow except at r=0 where the functions |u and |ω are not defined
-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

Question 8

Streamfunction and Vorticity

Answer 8

-in 2D flow, the vorticity 
|ω = (0,0,ω)
-where ω = ∂v/∂x - ∂u/∂y
-in addition if the flow is incompressible:
u = ∂ψ/∂y ,  v = - ∂ψ/∂x
-sub in:
ω = - ∇² ψ
|ω = - ∇² ψ ^ez

Question 9

Rankine Vortex

u

Answer 9

uθ = {Ω/2 r , ir r≤a

or Ωa²/2r, if r>a}

Question 10

Rankine Vortex

Vorticity

Answer 10

ω = {Ω, if r≤a, OR 0, if r>a}

Question 11

Circulation

Answer 11

-consider a closed curve C in the flow
-the circulation around C is the line integral of the tangential velocity around C:
Γ = ∮|u . d|l

Question 12

Circulation and Stokes Theorem

Answer 12

Γ = ∮|u . d|l
= ∫ (∇x|u) . ^n dS
= ∫ |ω . ^n dS
-so the circulation around C is equal to the vorticity flux through the surface S, it is the strength of the vortex tube

Question 13

Simple Vortex

Circulation

Answer 13

|u = k/r ^eθ
Γ = 2πk
-where circulation is calculated around a circle of radius r

Question 14

Rankine Vortex

Circulation

Answer 14

uθ = {Ω/2 r , ir r≤a
or Ωa²/2r, if r>a}
Γ = {πΩr² , for r≤a
πΩa² , for r<a></a>