AOE 5104 Class 10

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AOE 5104 Class 10

• Online presentations for next class
• Potential Flow 1
• Homework 4 due 10/2

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Kinematics of Vorticity
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Kinematic Concepts - Vorticity
Boundary layer growing on flat plate
Cylinder projecting from plate
Vortex line
n
dS
Vortex sheet
?
Vortex tube

• Vortex Line A line everywhere tangent to the
  vorticity vector. Vortex lines may not cross.
  Rarely are they streamlines. Thread together
  axes of spin of fluid particles. Given by ds??0.
  Could be a fluid line?
• Vortex sheet Surface formed by all the vortex
  lines passing through the same curve in space. No
  vorticity flux through a vortex sheet, i.e.
  ?.ndS0
• Vortex tube Vortex sheet rolled so as to form a
  tube.

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Spinning Cone
Spiral vortices formed on a 20cm-wide spinning
cone at 2.9m/s. Boundary layer is revealed.
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Implications (Helmholtz Vortex Theorems, Part 1)

• The strength of a vortex tube (defined as the
  circulation around it) is constant along the
  tube.
• The tube, and the vortex lines from which it is
  composed, can therefore never end. They must
  extend to infinity or form loops.
• The average vorticity magnitude inside a vortex
  tube is inversely proportional to the
  cross-sectional area of the tube

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Vortex Tube
Section 2
Since
n
dS
Section 1
So, we call ? The Vortex Tube Strength
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Implications (Helmholtz Vortex Theorems, Part 1)

• The strength of a vortex tube (defined as the
  circulation around it) is constant along the
  tube.
• The tube, and the vortex lines from which it is
  composed, can therefore never end. They must
  extend to infinity or form loops.
• The average vorticity magnitude inside a vortex
  tube is inversely proportional to the
  cross-sectional area of the tube

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Kinematic Concepts - Vorticity
Boundary layer growing on flat plate
Cylinder projecting from plate
Vortex line
n
dS
Vortex sheet
?
Vortex tube

• Vortex Line A line everywhere tangent to the
  vorticity vector. Vortex lines may not cross.
  Rarely are they streamlines. Thread together
  axes of spin of fluid particles. Given by ds??0.
  Could be a fluid line?
• Vortex sheet Surface formed by all the vortex
  lines passing through the same curve in space. No
  vorticity flux through a vortex sheet, i.e.
  ?.ndS0
• Vortex tube Vortex sheet rolled so as to form a
  tube.

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But, does the vortex tube travel along with the
fluid, or does it have a life of its own?
If it moves with the fluid, then the circulation
around the fluid loop shown should stay the same.
0
Same fluid loop at time tdt
(VdV)dt
Fluid loop C at time t
ds
Vdt
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So the rate of change of ? around the fluid loop
is Now, the momentum eq. tell us that
Viscous force per unit mass, say fv
Pressure force per unit mass
Body force per unit mass
So, in general
Body force torque
Viscous force torque
Pressure force torque
Same fluid loop at time tdt
(VdV)dt
Fluid loop at time t
Vdt
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Body Force Torque
Stokes Theorem
For gravity
So, body force torque is zero for gravity and for
any irrotational body force field
Therefore, body force torque is zero for most
practical situations
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Pressure Force Torque
If density is constant
So, pressure force torque is zero. Also true as
long as ? ?(p).

• Pressure torques generated by
• Curved shocks
• Free surface / stratification

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Shock in a CD Nozzle
Schlieren visualization Sensitive to in-plane
index of ref. gradient
Bourgoing Benay (2005), ONERA, France
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Viscous Force Torque

• Viscous force torques are non-zero where viscous
  forces are present ( e.g. Boundary layer, wakes)
• Can be really small, even in viscous regions at
  high Reynolds numbers since viscous force is
  small in that case
• The viscous force torques can then often be
  ignored over short time periods or distances

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Shock in a CD Nozzle
Schlieren visualization Sensitive to in-plane
index of ref. gradient
Bourgoing Benay (2005), ONERA, France
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Implications

• In the absence of body-force torques, pressure
  torques and viscous torques
• the circulation around a fluid loop stays
  constant Kelvins Circulation Theorem
• a vortex tube travels with the fluid material (as
  though it were part of it), or
• a vortex line will remain coincident with the
  same fluid line
• the vorticity convects with the fluid material,
  and doesnt diffuse
• fluid with vorticity will always have it
• fluid that has no vorticity will never get it

Helmholtz Vortex Theorems, Part 2
Body force torque
Viscous force torque
Pressure force torque
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Lord Kelvin (1824-1907) (William Thompson )
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Vorticity Transport Equation

• The kinematic condition for convection of vortex
  lines with fluid lines is found as follows

After a lot of math we get....
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Example Irrotational Flow?
Consider a vehicle moving at constant speed in
homogeneous medium (i.e. no free surfaces) under
the action of gravity, moving into a stationary
fluid.
Far ahead of the sub we have that (since V
const.)
Apparent uniform flow (V const.)
and the flow here is irrotational. Now, the flow
generates no body force or pressure torques and,
except in the vehicle boundary layer and wake, no
viscous torques.
http//www.lcp.nrl.navy.mil/ravi/par3d.html
So, the flow will remain irrotational everywhere
outside the boundary layer and wake, regardless
of how complex we make the sub shape.
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Example The Starting Vortex
Consider a stationary arifoil in a stationary
medium.
V 0
Now suppose the airfoil starts moving to left.
(Using the fact that that a lifting airfoil in
motion has a circulation about it).
?C remains 0 by Kelvins Theorem, if the loop
remains outside the airfoil wake, Thus a vortex
of circulation equal and opposite to that on the
airfoil must be shed.
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Starting Vortex
Pictures are from An Album of Fluid Motion by
Van Dyke
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Example Flow over a depression in a river bed
A river flows over a depression locally doubling
the depth. The river contains turbulence that is
too weak to change the overall flow pattern. An
turbulent eddy convects from upstream over the
depression. Estimate its strength in the
depression if the eddy is initially (a) vertical,
and (b) horizontal.
h
2h

• Solution Need to assume that the viscous torques
  are not significant for the eddy so that the
  fluid tube it occupies remains coincident with
  the same fluid tube.
• The vertical fluid tube occupied by the eddy will
  double in length in the depression and so (by
  continuity) it will halve its cross sectional
  area. By Helmholtz theorems, halving the cross
  sectional area of a vortex tube will double the
  vorticity. Hence ?22 ?1 in this case.
• The horizontal fluid tube containing the eddy
  will double its height and halve its length as it
  goes over the depression. Its cross-sectional
  area will thus double and by Helmhotz theorems
  the vorticity will halve. Hence ?2 ½?1 in this
  case.

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The River Avon nearing flood stage, Salisbury,
Wiltshire, UK
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Example Evolution of turbulence in a shear flow
Uky, kconst
Turbulence is convected and distorted in a shear
flow, as shown. Consider an eddy that at time t0
is vertically aligned and has a vorticity ?o.
Estimate the vorticity magnitude and angle of an
eddy, as functions of time.
y
Solution Need to assume that the viscous torques
are not significant for the eddy so that the
fluid tube it occupies remains coincident with
the same fluid tube. Also need too assume that
the eddy is to weak to influence the flow that is
convecting it.
Consider a segment of the eddy of length l. In
time t the top of the eddy will convect further
than the bottom by an amount equal to the
difference in the velocity (kl) times the time.
The angle of the fluid tube containing the eddy
will thus be The fluid tube also grows longer by
the factor The cross sectional area of the tube
thus reduces by this factor, and therefore the
vorticity increases by this factor, i.e.
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Eddies in a turbulent channel flow
DNS Simulation Thomas Bewley, Edward Hammond
Parviz Moin Stanford University
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Example Flow around a corner in a channel
Air flows through a duct with a 6o corner. An
initially vertical eddy is introduced into the
otherwise uniform flow upstream and convects
around the corner. Estimate its orientation
downstream.
6o
Solution Need to assume that the viscous torques
are not significant for the eddy so that the
fluid tube it occupies remains coincident with
the same fluid tube, and so that no vorticity is
generated in the turn. Also need to assume that
the eddy is too weak to influence the flow that
is convecting it. Consider two initially
perpendicular fluid lines, one in the streamwise
direction and one vertical, coincident with the
eddy. Since there is no vorticity component
perpendicular to the page, and no torques to
generate any, the sum of the rotation rates of
these two fluid lines must remain zero. The
streamwise fluid line will follow the
streamline and thus rotate counterclockwise
by 6o. The vertical fluid line and thus the
eddy must therefore rotate clockwise by 6o, as
shown.
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Turbulent flow in a pipe through a 180o bend.
Hitoshi Sugiyama, Utsunomiya University
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Ox Bows