FLOWING FLUIDS AND PRESSURE VARIATION

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Chapter 4 FLOWING FLUIDS AND PRESSURE VARIATION
Fluid Mechanics, Spring Term 2009
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Pressure differences are (often) the forces that
move fluids
e.g., pressure is low at the center of a
hurricane. For your culture The force balance
for hurricanes is the pressure force vs. the
Coriolis force. Thats why hurricanes on the
northern hemisphere always spin
counter-clockwise. In a tornado, the balance is
between pressure force and centrifugal
acceleration a tornado can spin either way
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Lagrangian and Eulerian Descriptions of Fluid
Motion
Eulerian Observer stays at a fixed point in
space. Lagrangian Observer moves along with a
given fluid particle.
We consider first the Lagrangian case. The
position of a fluid particle at a given time t
can be written as a Cartesian vector
For a complete description of a flow, we need to
know at every point. Usually one
references each particle to its initial position
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For fluid mechanics, the more convenient
description is usually the Eulerian one
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For your Culture In solid mechanics, we often
use Lagrangian methods. We dont need to follow
every particle, but just some small volumes.
When displacements get large (e.g. in fluid
flow), the deforming grid gets problematic. But
we sometimes use mixed approaches, e.g.
Lagrangian tracers in a Eulerian frame.
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Streamlines and Flow Patterns
Figure 4.3 (p. 84)
Streamlines are used for visualizing the flow.
Several streamlines make up a flow pattern. A
streamline is a line drawn through the flow field
such that the flow vector is tangent to it at
every point at a given instant in time.
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Uniform vs. Non-Uniform Flow
Using s as the spatial variable along the path
(i.e., along a streamline) Flow is uniform if
Examples of uniform flow
Note that the velocity along different
streamlines need not be the same! (in these
cases it probably isnt).
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Examples of non-uniform flow

• Converging flow speed increases along each
  streamline.
• Vortex flow Speed is constant along each
  streamline, but the direction of the velocity
  vector changes.

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Steady vs. Unsteady Flow
For steady flow, the velocity at a point or along
a streamline does not change with time
Any of the previous examples can be steady or
unsteady, depending on whether or not the flow is
accelerating
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Turbulent flow in a jet
Turbulence is associated with intense mixing and
unsteady flow.
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Flow around an airfoil Partly laminar, i.e.,
flowing past the object in layers
(laminae). Turbulence forms mostly downstream
from the airfoil. (Flow becomes more turbulent
with increased angle of attack.)
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Flow inside a pipe
Laminar Turbulent
Turbulent flow is nearly constant across a
pipe. Flow in a pipe becomes turbulent either
because of high velocity, because of large pipe
diameter, or because of low viscosity.
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Methods for Developing Flow Patterns (i.e.,
finding the velocity field)
Analytical Methods The governing equations
(mostly the Navier-Stokes equation) are
non-linear. Closed-form solutions to these
equations only exist for special, strongly
simplified cases. Computational Methods The
authors of the book seem to feel that this is not
overly useful. As a numerical modeler I
disagree. For experimental methods, it is often
difficult to find materials that scale properly
to large scales (e.g., entire oceans or the
Earths mantle). Experimental Methods Very
useful for complicated flows, especially flows
that involve turbulence. While the corrects
physical laws are known, time and space
resolution make turbulence tricky in numerical
models.
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Pathline, Streakline, and Streamline
Important concepts in flow visualization In
steady flows, all 3 are the same. Pathline The
line (or path) that a given fluid particle
takes. Streakline The line formed by all fluid
particles that have passed through a given fixed
point. Streamline A continuous line that is
tangent to the velocity vectors everywhere along
its path (at a given moment in time).
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xxxxx
xxxxx
Streakline at t t0
Streakline at t gt t0
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Acceleration Normal and Tangential Components
Velocity can be written as
where V(s,t) is the speed and is a unit
vector tangential to the velocity.
The derivative of the speed is (since ds/dt V)
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The time derivative of the unit vector is
non-zero because the direction of the unit vector
changes. The centripetal acceleration is
so that the total acceleration becomes
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Acceleration in Cartesian Coordinates
This is probably one of the most fundamental
concepts of the course, but it is not very
intuitive!
or
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From last page, we had
This derivative is called the full derivative or
material derivative. It is often written D/Dt
instead of d/dt. It can apply to other quantities
as well.
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For a simpler example, lets look at the material
derivative of temperature T in one dimension
T(x,t)
At a given point x0, a change in temperature can
be caused by two different mechanisms

• The temperature of the local fluid particle
  changes (e.g., due to heat conduction,
  radioactive heating, etc)
• All fluid particles keep their temperature, but
  the velocity u brings a new particle to x0 which
  has a different temperature

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The changes are called local change and
convective change (the convective change is
also called advective change)
local temperature change
local acceleration in x
convective temperature change
convective acceleration in x
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Example 4.1 (p. 94) Find the acceleration
half-way through the nozzle
Velocity is given as
Taking the x-derivative of u, and multiplying it
times u gives
Just plug in the values and x 0.5L to get the
answer
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Lagrangian reference frame
In the Lagrangian frame (moving along with a
fluid particle), there is no convective
acceleration. Because youre staying with a
given particle, no other particle can come in and
bring with it a different velocity.
Eulerian
The convective terms may be seen as a correction
due to the fact that new particles with different
properties are moving into our observation volume.
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Note that conservation laws naturally apply in
the Lagrangian frame A conserved quantity such
as total energy E remains constant in a given
material volume. An Eulerian observer sees
different material volumes flow past, each of
them possibly with different E.
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Eulers Equation
Eulers equation is Newtons 2nd Law applied to a
continuous fluid. Recall Newtons 2nd law for a
particle (balance in l-direction)
This law is fundamental and thus also applies to
fluid particles
(There may be additional forces)
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Now we shrink the fluid element to
(partial derivative since p may be a function of
other coord.s and time)
so that
and
Eulers equation (force balance in a moving fluid)
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Uniform acceleration of a tank of liquid (Fig.
4.13)
Horizontal balance
Vertical balance
(hydrostatic!)
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Derivation of the Bernoulli Equation Start with
Eulers equation applied along a pathline
Assume steady flow ( ) all
s-derivatives are now full derivatives.
Integrating with respect to s, we get
Bernoullis eqn.
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• Recall what we just did to get the Bernoulli
  equation
• Assume steady flow (dont apply this to anything
  else!)
• Integrate forces (per volume) along a pathline.

• The integral of force along a distance gives us
  energy
• Note that the velocity term is the kinetic
  energy per unit volume.
• Also note that energy / volume has same units as
  pressure.
• The kinetic energy / volume is also known as
  kinetic pressure.
• Along a streamline in steady, inviscid flow, the
  sum of piezometric pressure plus kinetic pressure
  is constant.

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Application of the Bernoulli Equation Stagnation
Tube
Apply to points 1 and 2 (same depth z)
Point 2 is a stagnation point (velocity is zero)
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From the geometry of the flow we know (we observe
or else we just assume we havent really derived
this) that at both points there is no vertical
acceleration. The vertical balance is thus
hydrostatic
The stagnation tube is a simple device for
measuring velocity.
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Notice that we had to have a free surface at the
top of the fluid. If the flow is in a
pressurized pipe, we dont know whether the
piezometric head measured by l is due to static
pressure in the pipe or due to flow. Hence the
Pitot tube
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Pitot tube
Look carefully the Pitot tube is really 2 tubes
in 1. If we know the static pressure at point 1
and the dynamic pressure at point 2, we have all
we need to find the velocity. For more details
see the book, p. 102.
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Rotation and Vorticity
Rotation of a fluid element in a rotating tank of
fluid (solid body rotation).
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Rotation of fluid element in flow between moving
and stationary parallel plates
You can think of the cruciforms as small paddle
wheels that are free to rotate about their
center. If the paddle wheel rotates, the flow is
rotational at that point.
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As
And similarly
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The net rate of rotation of the bisector is
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The rotation rate we just found was that about
the z-axis hence, we may call it
and similarly
The rate-of-rotation vector is
Irrotational flow requires (i.e.,
for all 3 components)
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The property more frequently used is the vorticity
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Vortices
A vortex is the motion of many fluid particles
around a common center. The streamlines are
concentric circles. Choose coordinates such that
z is perpendicular to flow. In polar coordinates,
the vorticity is (see p. 112 for details)
(V is function of r, only)
Solid body rotation (forced vortex)
or
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Vortex with irrotational flow (free vortex)
A paddle wheel does not rotate in a free vortex!
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Forced vortex (interior) and free vortex
(outside) Good approximation to naturally
occurring vortices such as tornadoes.
Eulers equation for any vortex
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• We can find the pressure variation in different
  vortices
• (lets assume constant height z)
• In general
• Solid body rotation
• Free vortex (irrotational)

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Application to forced vortex (solid body
rotation)
with
Pressure as function of z and r
p 0 gives free surface