AOE 5104 Class 9

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AOE 5104 Class 9

• Online presentations for next class
• Kinematics 2 and 3
• Homework 4 (6 questions, 2 graded, 2 recitations,
  worth double, due 10/2)
• No office hours this week

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Kinematics

• Kinematics of Velocity

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The Equations of Motion
Differential Form (for a fixed volume element)
The Continuity equation
The Navier Stokes equations
The Viscous Flow Energy Equation
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Kinematic Concepts - Velocity

• Fluid Line. Any continuous string of fluid
  particles. Moves with flow. Cannot be broken.
  Fluid loop closed fluid line.
• Particle Path. Locus traced out by an individual
  fluid particle.

www.lavision.de
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Kinematic Concepts - Velocity

• Streamline. A line everywhere tangent to the
  velocity vector. Never cross, except at a
  stagnation point. No flow across a streamline.
• Streamsurface. Surface everywhere tangent to the
  velocity vector. Surface made by all the
  streamlines passing through a fixed curve in
  space. No flow through a stream surface. Infinite
  number of stream surfaces that contain a given
  streamline. A streamline must appear at the
  intersection of two stream surfaces.
• Streamtube. Streamsurface rolled so as to form a
  tube. No flow through tube wall.

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Francis turbine simulation ETH Zurich
http//www.cg.inf.ethz.ch/bauer/turbo/research_ga
llery.html
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Mathematical Description
V
1. Streamlines
Streamline
ds
2. Streamsurfaces Make up a function ?(x,y,z,t)
so that surfaces ? const. are streamsurfaces. ?
is called a streamfunction.

• 3. Relationship between 1 and 2
• Consider a streamline that sits at the
• intersection of two streamsurfaces.
• The two streamsurfaces must be
• described by two different streamfunctions,
  say ?1 and ?2
• At any point on the streamline the perpendicular
  to each streamsurface, and the velocity must all
  be normal to each other
• So, what about that mathematical relationship?

Flow
?2 const.
?1 const.
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Mathematical Description
where ? ?(x,y,z,t) and scalar
To find ? we take
So,
Steady flow ? ?,
Incompressible flow ? 1,
Unsteady flow streamlines largely meaningless
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Example 2D Flow Over An Airfoil
Find consistent relations for the steamfuncitons
(implicit or in terms of the velocity field).
Take
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Titan
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Example Spherical Flow
Flow takes place in spherical shells (no radial
velocity).
Find a set of streamfunctions.
Choose
er
e?
e?
r
?
r
?
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Kinematics of Vorticity
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Hermann Ludwig Ferdinand von Helmholtz (1821-1894)
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Vorticity ?

• ???V
• 2 ? circumferentially averaged angular velocity
  of the fluid particles
• Sum of rotation rates of 2 perpendicular fluid
  lines
• Non-zero vorticity doesnt imply spin
• ?.?0. Incompressible?
• Direction of ??

U
y
No spin, but a net rotation rate
Always true!
Can be anything compared to V that the curl
produces
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Circulation ?

• Macroscopic rotation of the fluid around loop C
• Non-zero circulation doesnt imply spin
• Connected to vorticity flux through Stokes
  theorem
• Stokes for a closed surface?

U
y
Open Surface S with Perimeter C
ndS
Net outflow of vorticity is zero
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Flow Past a Cookie-Tin
Top view
Side view
Re 4,000
Horseshoe vortex
Pictures are from An Album of Fluid Motion by
Van Dyke
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Large Eddy Simulation Re5000
George Constantinescu IIHR, U. Iowa
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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.
• 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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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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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

Earth Science and Engineering Imperial College UK
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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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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 William Thompson Kelvin (1824-1907)
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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....