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 1.

There
are
two
important
reference
frames,
Eulerian
and
Lagrangian.

Flows
can
 sometimes
be
more
easily
understood
through
judicious
use
of
one
or
the
other.

The
 material
derivative
is
a
very
important
operation
by
which
the
time
rate
of
change
of
a
 quantity
is
computed
in
the
Lagrangian
frame.
 
 2.

Conservation
Laws
–
two
laws
govern
fluid
flow:
conservation
of
mass
and
Newton’s
 Second
Law
of
Motion
(the
momentum
balance).


 
 3.
An
incompressible
flow
is
one
in
which
the
density
of
a
fluid
element
is
approximately
 constant.

Conservation
of
mass
then
implies
that

 
 
 
 
 
div
(v)
=
0,

 eq.(6.30)
 
 4.

A
Newtonian
fluid
is
one
in
which
the
shearing
stresses
are
linearly
related
to
the
 velocity
gradients.

The
constant
of
proportionality
is
the
viscosity,
µ.

It
is
a
measure
of
 molecular
friction
within
a
fluid.

 
 5.

In
this
course
we
consider,
pressure,
viscous
and
gravity
forces.

For
a
constant
 property
Newtonian
fluid,
the
momentum
balance
leads
to
the
Navier
Stokes
Equations,
 eqs.
(6.127).
 
 6.
There
are
seven
fundamental
units.

All
others
are
derived
units
that
are
determined
 through
dimensional
homogeneity
of
material
relations
(such
as
the
unit
of
Poise
for
 viscosity
through
Newton’s
Law
of
Viscosity),
definitions,
(such
as
the
Pascal
as
a
 force/area),
or
through
fundamental
laws
of
physics
(such
as
the
Newton
as
the
unit
of
 force
from
Newton’s
Second
Law).

 
 7.

Dimensional
analysis
is
the
study
of
the
relationship
between
quantities
based
solely
 on
a
consideration
of
the
dimensions
involved.

The
key
result
is
the
Buckingham
Pi
 Theorem,
which
gives
the
maximum
number
of
independent
dimensionless
parameters
 required
to
describe
this
relationship.

We
have
studied
three
methods
of
finding
Pi
 groups:
repeating
variables,
inspection,
and
scaling
equations.


 
 8.
Scaling
the
Navier
Stokes
equations
(without
the
gravity
force)
indicates
that
there
is
 one
dynamic
parameter,
the
Reynolds
number,
and
a
number
of
geometric
aspect
ratios
 (necessary
to
define
the
geometry).

Flows
with
the
same
Reynolds
number
are
said
to
 be
dynamically
similar;
those
with
the
same
aspect
ratio,
geometrically
similar.
 
 9.

Flow
in
pipes
is
characterized
by
the
flow
regime
(laminar,
transitional,
or
turbulent)
 and
whether
or
not
the
flow
is
developing
or
fully‐
developed.

The
relationship
between
 pressure
gradient,
pipe
diameter,
fluid
properties
and
flow
rate
is
given
by
the
 relationship
between
the
friction
factor,
f
,
the
Reynolds
number,
Re,

and
the
roughness
 parameter,
ε/
D,
Figure
8.20.
 
 10.

Flows
that
neglect
the
viscous
forces
are
known
as
inviscid
and
they
are
governed
by
 the
Euler
Equations
of
Motion,
eqs.
(6.5).

These
equations
lead
to
non‐physical
 predictions,
such
as
zero
drag
(D’Alembert’s
paradox)
and
a
violation
of
the
no‐slip
 condition.
Flows
at
high
Reynolds
number
exhibit
boundary
layer
behavior,
i.e.
the
 presence
of
thin
layers
near
boundaries
where
viscous
forces
balance
inertial
effects,
no
 matter
how
large
the
Reynolds
number.

 ME
152B
 Summary
Sheet
 11.
The
boundary
layer
approximations
are
an
important
simplification
of
the
Navier‐Stokes
 equations.
They
neglect
vertical
momentum,
use
the
pressure
gradient
in
the
streamwise
 direction
from
inviscid
theory,
and
approximate
the
viscous
force
by
the
shearing
in
the
 cross‐stream
direction.
These
equations
(without
the
pressure
gradient,
which
is
absent
for
 the
flat
plate)
are
given
in
Eqs.
(9.8‐9.9).

You
should
know
the
physical
ideas
that
go
into
 this
important
simplification.
 
 12.
Boundary
layer
flow
on
a
flat
plate
is
an
important
prototype
problem
that
is
analyzed
 analytically
for
laminar
flow
and
by
dimensional
analysis
for
turbulent
flow.

Transition
to
 turbulent
takes
place
at
a
Reynolds
number
based
on
distance
from
the
leading
edge
of

 Rex
=
2
–
5
x
105.

Important
results
include:
 
 Laminar
flow
 
 
 
 
 Turbulent
flow
 
 δ(x)
=
5
(ν
x/U)1/2
 
 
 
 δ(x)
=
0.37
(ν
/U)0.2
x
0.8
 
 ‐1/2
 
 
 
 
 Cf
=
1.328
Re
 Cf
=
0.072

Re
‐0.2
 
 13.

Turbulence
is
described
by
the
Reynolds
Decomposition
of
quantities
into
mean

 and
fluctuating
components.
Time‐averaging
the
Navier
Stokes
equations
show
that
 momentum
transfer
by
viscous
stresses
is
augmented
by
Reynolds
stresses,
the
most
 important
of
which
in
boundary
layer
flows
is
ρ <u’v’>,
where
<>
denotes
time
averaging.

 
 14.
Turbulent
velocity
profiles
are
blunter
and
steeper
than
their
laminar
counterparts
due
 to
the
transverse
momentum
transfer
due
to
Reynolds
stresses.

Looking
close
to
the
wall,
 and
using
dimensional
analysis
involving
the
wall
shear
stress,
τW,
ν
and
ρ,
results
in
so‐ called
‘wall
variables’
 
 u+
=
u/u*;

where
u*
=
(τW/
ρ)1/2


the
so‐called
‘friction
velocity’
 
 y+
=
y/( ν/u*) Data
indicate
that
the
velocity
profile
in
the
laminar
sublayer
is
given
by
u+
=
y+,
and
in
the
 outer
region
by
u+
=
2.5
ln(y+)
+
5.0:
the
buffer
layer
is
a
transition
between
the
two.

 
 15.
At
high
Re,
we
divide
the
flow
into
a
boundary
layer
near
the
no
slip
surface,
and
outer
 regions
where
the
flow
is
steady,
incompressible,
inviscid,
and
irrotational
and
the
velocity
 is
given
by
the
gradient
of
a
velocity
potential.
Such
flows
are
called
potential
flows.

The
 velocity
potential
satisfies
Laplace’s
equation
and
the
pressure
is
given
by
Bernoulli’s
eq.,

 
 
 
 
 p
+
½
ρ
V2
=
constant
 You
should
know
the
assumptions
that
go
into
deriving
Bernoulli’s
equation
in
this
form.



 
 16.
Flow
around
a
cylinder
is
a
prototype
for
flow
around
bluff
bodies
at
high
Reynolds
 numbers.
Potential
flow
around
a
cylinder
predicts
a
high
pressure
at
the
forward
 stagnation
point
(from
Bernoulli’s
equation),
a
symmetric
pressure
profile,
complete
 pressure
recovery
at
the
rear,
and
no
drag.

Experimental
data
show
incomplete
pressure
 recovery
and
a
finite
drag.

The
discrepancy
is
explained
by
flow
separation,
which
is
due
to
 the
combination
of
adverse
pressure
gradient
and
frictional
losses
in
the
boundary
layer.
 
 17.

The
drag
force
on
an
object
is
due
to
a
combination
of
pressure
drag
and
viscous
(or
 frictional)
drag.

Using
dimensional
analysis,
the
drag
coefficient,
defined
below,
depends
on
 Re,
aspect
ratios,
and
geometry:
 
 
 CD
=
D/(1/2
ρ
V2
A)
=
CD(Re,
roughness,
geometry)
 
 Streamlined
bodies
are
characterized
by
attached
boundary
layers
and
most
of
the
drag
is
 due
to
frictional
drag.

Bluff
bodies
are
characterized
by
boundary
layer
separation
in
 regions
of
adverse
pressure
gradients
and
most
of
the
drag
is
due
to
pressure
drag.


Drag
 coefficients
for
a
large
number
of
objects
are
available.

Study
those
for
flow
around
smooth
 spheres
and
cylinders
(Fig.
9.21),
other
smooth
shapes
(Fig.
9.22),
important
complicated
 geometries
(Fig.
9.30),
rough
pipes
(Fig.
8.20),
rough
flat
plates
(Fig.
9.15)
and
rough
 spheres
(Fig.
9.25).

Understand
how
the
drag
force
is
related
to
the
flow
structure
and
 know
how
to
solve
problems
involving
drag
forces
using
these
figures.

 18.

Free
Surface
Flows
are
those
that
are
influenced
by
gravity
acting
at
a
free
surface.

 Scaling
the
Navier‐Stokes
equations
shows
that
the
Froude
number
appears
in
the
 dimensionless
equations
–
see
equation
(7.35).

In
our
book,

 
 
 Fr
=
V
/(gL)1/2
,
where
V,
L
are
a
characteristic
velocity
and
length.
 The
Froude
number
measures
the
ratio
of
inertial
effects
to
the
body
force
of
gravity.
 Wave
propagation:

A
free
surface
wave
propagates
over
a
pool
of
still
liquid
at
a
speed
that
 is
determined
by
conservation
of
mass
and
the
momentum
balance.

In
general,
c
=
c(g,
λ,
y)
 where
λ
is
the
wavelength
and
y
the
depth
of
the
fluid.

The
result
for
c
is
given
by

 eq.
(10.4)
and
shown
in
Figure
10.5,
with
limiting
relationships
for
“shallow”
and
“deep
 water”.
Wave
propagation
in
a
liquid
moving
with
a
characteristic
speed
V
is
governed
by
 the
Froude
number,

Fr
=
V/c.

Three
cases
can
be
distinguished:


 Fr
<
1
(“subcritical
flow”),
where
waves
propagate
both
upstream
and
downstream),

 Fr
>
1,
(“supercritical
flow”)
where
waves
propagate
only
in
the
downstream
direction,
and
 Fr
=
1
(“critical
flow”),
which
separates
the
two.
 Flow
down
slopes:

We
studied
the
laminar
flow
(“Nusselt
film
flow”),
turbulent
flow,
and
 the
result
for
rough
bottomed
surfaces
and/or
very
long
fetches:
V
=
C
(g
h
sin(θ))1/2.
 Hydraulic
jumps:
Hydraulic
jumps
are
sudden
discontinuous
changes
in
the
height
and
 velocity
of
a
flowing
liquid
with
a
free
surface.
They
are
possible
because
a
given
 volumetric
flow
rate
can
be
accommodated
by
either
a
fast,
shallow
layer
or
a
slow(er),
 deep(er)
layer.

The
necessary
condition
for
a
hydraulic
jump,
established
by
applying
 conservation
of
mass
and
using
Bernoulli’s
equation,
is
that
the
upstream
flow
be
 supercritical
(Fr
>
1)
and
the
downstream
flow
be
subcritical
(Fr
<
1).


 
 19.

Similarity
and
Modeling
are
based
on
the
principle
that
if
the
geometry
is
the
same
and
 the
dimensionless
dynamic
parameters
are
the
same,
then
the
dimensionless
flow
and
any
 secondary
quantities,
(e.g.
the
drag
coefficient
or
the
Strouhal
number),
arealso
the
same.
 Geometric
similarity
means
the
shape
and
all
the
aspect
ratios
are
the
same.
 Dynamic
similarity
means
that
the
Reynolds
numbers
are
the
same
and,
if
a
free
surface
is
 involved,
the
Froude
numbers
are
the
same.

We
discussed
several
types
of
models:
scaled‐ down,
full
scale,
and
scaled‐up
models.

Dynamic
similarity
for
a
scaled
model
involves
 changing
the
viscosity
if
both
the
Reynolds
and
Froude
numbers
are
involved.
 
 20.
Bernoulli’s
Equation
may
be
derived
under
the
assumptions
of
inviscid,
steady
flow
by
 applying
force
balances
along
and
perpendicular
to
a
streamline,
resulting
in
eqs.
(3.7)
and
 (3.14).

Several
applications
of
Bernoulli’s
equation
include
Pitot
tubes,
free
jets,
flow
over
 weirs,
the
orifice
flow
meter,
and
draining
a
tank.

Predictions
using
Bernoulli’s
equation
 are
often
inaccurate
and
have
to
be
corrected
for
time
dependence
and
viscous
effects
by
 introducing
empirical
“discharge
coefficients”
 
 21.

Minor
Losses
in
pipe
flows
refers
to
any
pressure
(head)
loss
that
is
due
to
elbows,
 return
bends,
valves,
couplings,
etc.

The
pressure
loss
can
be
a
very
large
fraction
of
the
 total
head
loss,
and
so
the
name
‘minor
losses’
is
misleading.

Minor
losses
are
described
 through
the
use
of
empirical
“loss
coefficients”
as
follows:
 
 
 
 
 p1‐p2
=
KL
(1/2
ρ
V2)

 In
general
KL
=
KL(geometry,
Re),
but
for
high
Reynolds
numbers,
it
is
taken
to
be
only
a
 function
of
geometry.

Many
graphs
and
charts
of
KL
are
given
in
the
book.

 
 22.

Compressible
flows
are
ones
in
which
the
density
of
is
not
constant.
The
most
 important
features
of
compressible
flows
are
the
ability
to
propagate
sounds
waves
and
the
 occurrence
of
shock
waves.
The
speed
of
sound
is
given
by
c
=
(∂p/∂ρ)S


The
Mach
number
 of
a
flow
is
M
=
V/c.

All
the
results
of
incompressible
flows
are
modified
by
an
additional
 dependence
on
the
Mach
number.

The
Mach
number
plays
the
role
in
compressible
flow
 that
the
Froude
numbers
does
in
free
surface
flows:
M>1
is
supercritical
(waves
propagate
 in
one
direction),
M<1
is
subcritical
(waves
propagate
in
more
than
one
direction),
and

 M
=
1
separates
the
two.

Hydraulic
jumps
find
analogies
in
shock
waves,
where
the
flow
is
 supercritical
upstream
and
subcritical
downstream
of
shocks.

There
are
further
 mathematical
and
physical
analogies
between
compressible
flows
and
free
surface
ripples.
 ...
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This note was uploaded on 03/23/2009 for the course ME 152B taught by Professor Homsy during the Winter '09 term at UCSB.

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