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13 Vorticity Theorems
The incompressible flow momentum equations focus attention on and and explain the
flow pattern in terms of inertia, pressure, gravity, and viscous forces. Alternatively, one
can focus attention on and explain the flow pattern in terms of the rate of change,
deforming, and diffusion of by way of the vorticity equation. As will be shown, the
existence of generally indicates the viscous effects are important since fluid particles
can only be set into rotation by viscous forces. Thus, the importance of this topic is to
demonstrate that under most circumstances, an inviscid flow can also be considered
irrotational.
13.1
Vorticity Kinematics
Vorticity vector,
, or,
and
,
,
,
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A quantity intimately tied with vorticity is the circulation: Circulation
fluid rotation within a closed contour (or circuit) .
∮
∮
(Stokes Theorem: ∮
is the amount of
∫
∮
Fluid motion is called irrotational if
)
(
).
Vortex line = lines which are everywhere
tangent to the vorticity vector.
Next, we shall see that vorticity and vortex
lines must obey certain properties known as
the Helmholtz vorticity theorems, which
have great physical significance.
The first is the result of its very definition:
(
)
(vector identity: The divergence of the curl of any vector field is always zero)
i.e. the vorticity is divergence-free, which means that there can be no sources or sinks of
vorticity within the fluid itself.
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Helmholtz Theorem #1: a vortex line cannot end in the fluid. It must form a closed path
(smoke ring), end at a boundary, solid or free surface, or go to infinity.
The second follows from the first and using the
divergence theorem:
∫
∫
Application to a vortex tube results in the following
∫
∫
or –
∫
(minus sign due to outward normal),
∫
Helmholtz Theorem #2: The circulation around a given vortex line (i.e., the strength of
the vortex tube) is constant along its length.
This result can be put in the form of a simple one-dimensional incompressible continuity
equation. Define
and
as the average vorticity across and , respectively
which relates the vorticity strength to the cross sectional area changes of the tube.
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106
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13.2
Vortex dynamics
Consider the substantial derivative of the circulation assuming incompressible flow and
conservative body forces
∮
∮
∮
From the N-S equations we have
(
)
Also,
(
∮
∮ [
∮ [
(
)
(
∮
)
)
]
]
∮
∮
∮ [
]
Since the integration is around a closed contour and
∮
∮
(
(
and
)
are single-valued.
(
)
)
Implication: The circulation around a material loop of particles changes only if the net
viscous force on those particles gives a nonzero integral.
058:0160
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If
or
Chapters 3&4
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(i.e., inviscid or irrotational flow, respectively) then
The circulation of a material loop never changes.
Kelvins Circulation Theorem: for an ideal fluid (i.e. inviscid and incompressible) acted
upon by conservative forces (eg, gravity) the circulation is constant about any closed
material contour moving with the fluid.
Which leads to:
Helmholtz Theorem #3: No fluid particle can have rotation if it did not originally rotate.
Or, equivalently, in the absence of rotational forces, a fluid that is initially irrotational
remains irrotational. In general, we can conclude that vortices are preserved as time
passes. Only through the action of viscosity can they decay or disappear.
Kelvins Circulation Theorem and Helmholtz Theorem #3 are very important in the
study of inviscid flow. The important conclusion is reached that a fluid that is initially
irrotational remains irrotational, which is the justification for ideal-flow theory.
In a real viscous fluid, vorticity is generated by viscous forces. Viscous forces are
large near solid surfaces as a result of the no-slip condition. On the surface there is a
direct relationship between the viscous shear stress and the vorticity.
058:0160
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Chapters 3&4
108
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13.2.1 Vorticity Generation at a Wall
Consider a 1-D flow near a wall:
The viscous shear stress is given by:
(
)
i.e., the wall vorticity is directly proportional
to the wall shear stress.
NOTE: In this case, the only component of
is . Actually, this is a general result in that
it can be shown that surface is
perpendicular to the limiting streamline.
This analysis can be easily extended for general 3D flow:
at a fixed solid wall, where
is the rotation tensor.
It is true since at a wall with coordinate
,
and from continuity
Once vorticity is generated, its subsequent behavior is governed by the vorticity equation.
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13.2.2 Vorticity Transport Equation
Navier-Stokes equation for incompressible flow with constant density and viscosity:
( )
Body force (gravity) can be combined into .
The vorticity equation is obtained by taking the curl of this equation.
(
(
)
)
(
[
)
( )
[ (
( )
(
Vector calculus identity:
(
)
[
[
(
(
]
)
(
)
(
(
)]
)
)
[
[
] (
[
(
)
(
(
(
Vector calculus identity:
)
)
]
)
]
)
]
)
)]
(
)
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Fall 2012
(
)
Chapters 3&4
110
(
)
(
)
(
)
(
(
)
(
)
(
)
(
)
(
)
) ,
,
: rate of change of
(
) : rate of deforming vortex lines
: rate of viscous diffuision of
: stretching;
Note:
: turning
(1) Equation does not involve explicitly
) since is perpendicular to
(2) for 2-D flow (
deformation of , i.e.,
and there can be no
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13.2.3 Pressure Poisson Equation
To determine the pressure field in terms of the vorticity:
(
( )
)
(
)
[
( )
(
)
]
does not explicitly depend on viscosity
Pressure Poisson equation in vector form:
 p
 2       V V 

1

      V  V   V    V  
2

1
V V    V  V   V    V 
2
1
  2  V  V      V  ω 
2
1
  2  V  V   ω    V   V    ω 
2
1
  2  V  V   ω  ω  V      V  
2
1
  2  V  V   ω  ω  V   2 V     V 


2
1
  2  V  V   V  2 V  ω  ω
2
  a  b   b    a   a   b 


  a  b   b    a   a   b 
   a   2a     a 
058:0160
Jianming Yang
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Pressure Poisson equation in tensor form:
 p
1
 2      2  V  V   V  2 V  ω  ω
2

 2  uk ek 
1 2
 u j e j    uk ek     ui ei  

   V     V 

2 xi xi 
x j x j
 2 uk 
uk  
u 
1 2

u
u


u




ei     lmn n el 
 ijk

j k jk 
i ik
2 xi xi
x j x j 
x j  
xm 
2

 2ui
u u 
1  u ju j 

 ui
   ijk  lmn k n   ei  el 
2 xi xi
x j x j 
x j xm 


 2ui
uk un 
1   

u
u

u



 j j   i x x  ijk lmn x x  il

2 xi  xi
j
j
j
m 


u j 
 2ui
uk un
1  

2
u

u








 j
 i
jm kn
jn km
2 xi 
xi 
x j x j
x j xm
 2ui
u u
u u
  u j 

  jm kn k n   jn km k n
uj
  ui
xi  xi 
x j x j
x j xm
x j xm
 u j u j
 2u j 
 2ui
u u u u j
 
 uj
 ui
 k k k

 x x
xi xi 
x j x j x j x j x j xk
i
 i

uk u j
x j xk
Chapters 3&4
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13.3
Kinematic Decomposition of flow fields
Previously, we discussed the decomposition of fluid motion into translation, rotation, and
deformation. This was done locally for a fluid element. Now we shall see that a global
decomposition is possible.
Helmholtz’s Decomposition: any continuous and finite vector field can be expressed as
the sum of the gradient of a scalar function plus the curl of a zero-divergence vector .
The vector vanishes identically if the original vector field is irrotational.
Then
And
If
The GDE for
Since
is the Laplace Equation
(
)
(
i.e.
The solution of this equation is
∫|
|
Thus,
∫|
|
Which is known as the Biot-Savart law.
)
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The Biot-Savart law can be used to compute the velocity field induced by a known
vorticity field. It has many useful applications, including in ideal flow theory (e.g., when
applied to line vortices and vortex sheets it forms the basis of computing the velocity
field in vortex-lattice and vortex-sheet lifting-surface methods).
The important conclusion from the Helmholtz decomposition is that any incompressible
flow can be thought of as the vector sum of rotational and irrotational components. Thus,
a solution for irrotational part
represents at least part of an exact solution. Under
certain conditions, high Re flow about slender bodies with attached thin boundary layer
and wake,
is small over much of the flow field such that
is a good approximation
to . This is probably the strongest justification for ideal-flow theory. (incompressible,
inviscid, and irrotational flow).
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The vorticity components in cylindrical coordinates are
So the flow is irrotational.
The flow is incompressible, steady, and irrotational, (also, the free surface is a
streamline), so the Bernoulli equation holds at the surface, where
Introduce
and