Download flowing fluids and pressure variation!

Pressure differences are (often) the forces that move fluids!
Chapter 4!
!
FLOWING FLUIDS AND
PRESSURE VARIATION!
Fluid Mechanics, Spring Term 2014!
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:!
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. That s why hurricanes
in 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…!
For fluid mechanics, the more convenient description is
usually the Eulerian one:!
For your Culture : In solid mechanics, we often use
Lagrangian methods. We don t need to follow every particle ,
but just some small volumes.!
Streamlines and Flow Patterns!
Figure 4.1 (p. 78)!
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.!
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.!
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 non-uniform flow:!
Examples of uniform flow:!
a)  Converging flow: speed increases along each
streamline.!
Note that the velocity along different streamlines need
not be the same! (in these cases it probably isn t).!
b)  Vortex flow: Speed is constant along each
streamline, but the direction of the velocity vector
changes.!
Steady vs. Unsteady Flow!
Turbulent flow in a jet!
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:!
Turbulence is associated with intense mixing and unsteady flow.!
Flow inside a pipe:!
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.)!
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.!
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 Earth s mantle).!
!
Experimental Methods: Very useful for complicated flows,
especially flows that involve turbulence. While the correct
physical laws are known, time and space resolution make
turbulence tricky in numerical models.!
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).!
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. !
xxxxx!
Streakline at t = t0!
xxxxx!
Streakline at t > t0!
The derivative of the speed is (since ds/dt = V):!
The time derivative of the unit vector is non-zero because the
direction of the unit vector changes. The centripetal
acceleration is!
Acceleration in Cartesian Coordinates!
This is probably one of the most fundamental concepts of
the course, but it is not very intuitive!!
so that the total acceleration becomes!
or!
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.!
!
!
!
!
!
The full time derivative describes the change in time of a
certain property as we move along with the fluid.!
For a simpler example, let s look at the material
derivative of temperature T in one dimension: T(x,t)!
At a fixed point x0, a change
in temperature can be
caused by two different
mechanisms:!
1)  The temperature of the local fluid particle changes
(e.g., due to heat conduction, radioactive heating,
etc…):!
2)  All fluid particles keep their temperature, but the
velocity u brings a new particle to x0 which has a
different temperature:!
The changes are called local change and convective
change (the convective change is also called advective
change )!
Example 4.1 (p. 85): Find the acceleration half-way
through the nozzle!
Velocity is given as!
= local temperature change!
= local acceleration in x!
= convective temperature change!
= convective acceleration in x!
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…!
Lagrangian reference frame:!
In the Lagrangian frame (moving along with a fluid
particle), there is no convective acceleration. Because
you re staying with a given particle, no other particle can
come in and bring with it a different velocity.!
Eulerian:!
(same in x and y)!
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.!
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.!
Euler’s Equation!
Euler s equation is Newton s 2nd Law applied to a
continuous fluid.!
Recall Newton s 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…)!
Now we shrink the fluid element to !
Uniform acceleration of a tank of liquid:!
(partial derivative since p may be a
function of other coord.s and time)!
and!
so that!
Horizontal balance: !
Euler s
equation!
(force balance in
a moving fluid)!
Vertical balance:
(hydrostatic!)!
Derivation of the Bernoulli Equation:!
!
Start with Euler s equation applied along a pathline:!
Assume steady flow (
now full derivatives.!
Recall what we just did to get the Bernoulli equation:!
!
1)  Assume steady flow (don t apply this to anything else!)!
2)  Integrate forces (per volume) along a pathline.!
); all s-derivatives are
Integrating with respect to s, we get!
Bernoulli s eqn.!
Application of the Bernoulli Equation: Stagnation Tube!
•  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.!
From the geometry of the flow we know (we observe or else
we just assume; we haven t really derived this) that at both
points there is no vertical acceleration. The vertical balance
is thus hydrostatic:!
Apply to points 1 and 2!
(same depth z):!
Point 2 is a stagnation point
(velocity is zero)!
The stagnation tube is a simple device for measuring velocity.!
Notice that we had to
have a free surface at the
top of the fluid.!
!
If the flow is in a
pressurized pipe, we
don t know the pressure
at point 1.!
!
Hence the Pitot tube…!
Rotation and Vorticity!
Rotation of a fluid element in a rotating tank of fluid!
(solid body rotation).!
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,
pp. 96-99.!
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.!
The net rate of
rotation of the
bisector is!
As!
And similarly !
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)!
The property more frequently used is the vorticity!
Vortices!
Vortex with irrotational flow (free vortex):!
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 the flow.!
In polar coordinates, the rotation rate is (see p. 104 for details)!
(V is function of r, only)!
Solid body rotation (forced vortex):!
!
!
or
at every point. !
!
!
(note: here ω is the angular velocity, not vorticity!)!
Forced vortex (interior) and!
free vortex (outside):!
!
Good approximation to
naturally occurring !
vortices such as!
tornadoes.!
A paddle wheel does not rotate
in a free vortex!!
We can find the pressure variation in different vortices!
!
!
In general: !
!
1)  Solid body rotation: !
Euler s equation for
any vortex:!
2)  Free vortex (irrotational):!
Application to forced vortex (solid body rotation):!
with !
Pressure as function of!
z and r!
p = 0 gives free surface!