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7/27/2019 Class 6 - Eqns of Motion
http://slidepdf.com/reader/full/class-6-eqns-of-motion 1/37
AOE 5104 Class 6
• Online presentations for next class:
– Equations of Motion 2
• Homework 2• Homework 3 (revised this morning) due
9/18
• d’Alembert
7/27/2019 Class 6 - Eqns of Motion
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Last class
Integral theorems…
R S
dS d n
R S
dS d nAA ..
R S
dS d nAA
C
S S d sAnA d..
…and their limitations
2D flow over airfoil with =0
C
.V = change in density in direction of V, multiplied by magnitude of V
Convective operator…
Irrotational and
Solenoidal Fields… 0.
0
A
7/27/2019 Class 6 - Eqns of Motion
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Class Exercise
1. Make up the most complex irrotational 3D velocityfield you can.
2223sin /3)2cos( z y x xy xe x k jiV ?
We can generate an irrotational field by taking the gradient of any
scalar field, since 0
I got this one by randomly choosing
z y xe x /132sin
And computingk jiV
z y x
Acceleration??
7/27/2019 Class 6 - Eqns of Motion
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2nd Order Integral Theorems
• Green’s theorem (1st form)
• Green’s theorem (2nd form)
Volume R
with Surface S
d
ndS
S R
S d dn
2
S
RS d d
n
-
n
22
These are both re-expressions of the divergence theorem.
7/27/2019 Class 6 - Eqns of Motion
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The Equations of Motion
7/27/2019 Class 6 - Eqns of Motion
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“Phrase of the Day”
Mutationem motus proportionalem esse vi
motrici impressae, & fieri secundum lineamrectam qua vis illa imprimitur.
Go Hokies?
7/27/2019 Class 6 - Eqns of Motion
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Supersonic Turbulent Jet Flow and
Near Acoustic Field
Freund at al. (1997)
Stanford Univ.
DNS
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Conservation Laws
• Conservation of mass
• Conservation of momentum
• Conservation of energy
0massof C.O.R.
ViscousPressureBodyof C.O.R. FFFmomentum
QWWWenergyof C.O.R. ViscousPressureBody
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Supersonic Turbulent Jet Flow and
Near Acoustic Field
Freund at al. (1997)
Stanford Univ.
DNS
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Conservation Laws
• Conservation of mass
• Conservation of momentum
• Conservation of energy
0massof C.O.R.
ViscousPressureBodyof C.O.R. FFFmomentum
QWWWenergyof C.O.R. ViscousPressureBody
Apply to the fluid material (not the space)
Experimental observations
Assumption: Fluid is a homogeneous continuum
7/27/2019 Class 6 - Eqns of Motion
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f low
x
y
z
x y z r i j k
o o o o x y z r i j k
1 o o, o
2 o o, o
3 o o, o
( , , )
( , , )
( , , )
x f x y z t
y f x y z t
z f x y z t
Position :
1) Lagrangian Method
Kinematics of Continua
1 o o, o
o o, o
2 o o, o 3 o o, o
( , , )
where partial derivative wrt time
holding ( , ) constant
( , , ) ( , , ),
x
y z
Df x y z t D
v Dt Dt
x y z
Df x y z t Df x y z t v v
Dt Dt
Velocity :
DTM
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2 2 2
1 o o, o 2 o o, o 3 o o, o
2 2 2
( , , ) ( , , ) ( , , ), ,
Concept is straightforward, but difficult to implement, often would produce more
information than we need or want, and
x y z
D f x y z t D f x y z t D f x y z t a a a
Dt Dt Dt
Acceleration :
doesn't fit the situation usually encountered
in fluid mechanics.
The Lagrangian Method is always used in solid mechanics :
DTM
P
o x
3 2
o o3
6
P y x lx
EI
3 2
o 1 o o o o o 2 o o o 3 o o o( , , , ), 3 ( , , , ), 0 ( , , , )6
P t x x f x y z t y x lx f x y z t z f x y z t
EI
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rad D
Dt t
t
Acceleration :v v
a v vG
1) Lagrangian Method
1 o o, o
2 o o, o
3 o o, o
( , , )
( , , )
( , , )
x f x y z t
y f x y z t
z f x y z t
Position :2) Eulerian Method
DTM
Position :
1 o o, o
2 o o, o
3 o o, o
( , , )
( , , )
( , , )
x
y
z
Df x y z t v
Dt
Df x y z t v
Dt
Df x y z t v
Dt
Velocity : ( , , , )
( , , , )
( , , , )
x
y
z
v x y z t
v x y z t
v x y z t
Velocity :
solve for position
as a function of time and “name”
express the velocity
as a function of time
and spatial position
denotes the derivative wrt time
holding the spatial position fixed,
often called the “local” derivative
WOW! big, big difference: velocity as
a function of time and spatial position,
not velocity as a function of time and
particle name
complication: laws governing motion
apply to particles (Lagrange), not to
positions in space
2
1 o o, o
2
2
2 o o, o
2
2
3 o o, o2
( , , )
( , , )
( , , )
x
y
z
D f x y z t a
Dt
D f x y z t a
Dt
D f x y z t a Dt
Acceleration :
skip this step and do not try to find
the positions of fluid particles
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Giuseppe Lodovico Lagrangia
(Joseph-Louis Lagrange)
born 25 January 1736 in Turin, Italy
died 10 April 1813 in Paris, France
Leonhard Paul Euler
born 15 April 1707 in Basel, Switzerlanddied 18 September 1783 in St. Petersburg, Russia
DTM
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Acceleration in the Eulerian Method:
x
y
z
r
d r
A fluid particle, represented as a blue dot in the figure,
moves from position to during the time interval .
Its velocity changes from ( , ) to ( , )
where be chosen
d dt
t d t dt
d dt
a
r r r
v r v r r
r MUST v
( , ) ( , ) D d t dt t
Dt dt
v v r r v r
( , ) ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , )= the derivative wrt time at a fixed location
( , ) ( , ) rad change in between two po
D d t dt t d t dt d t d t t
Dt dt dt
d t dt d t
dt t
d t t d
dt dt
v v r r v r v r r v r r v r r v ra
v r r v r r v
v r r v r v r vG ints in space at a fixed time
and are independent variables; so we are free to chose them anyway we want. In order
to follow a particle, we must chose : rad
dt
t d
d dt t
r
vr v a v vG
7/27/2019 Class 6 - Eqns of Motion
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.
If at a given instant we draw a line with the property that every point on
the line passed through the same reference point at some earlier time, the result
is known as a streakl ine .
It is called a streakline because, if the particles are dyed as they pass through
the common reference point, the result will be a line of dyed particles ( i.e., a
streak) through the flowfield.
If at a given instant the velocity is calculated at all points in the flowfield
and then a line is drawn with the property that the velocities of all of the particleslying on that line are tangent to it, the result is known as a streamline .
Streamlines are the velocity field lines. They provide a snapshot of the flowfield,
a picture at an instant. The surface formed by all the streamlines that pass
through a closed curve in space forms a stream tube.
1 o o, o
2 o o, o
3 o o, o
( , , )
( , , )
( , , )
x f x y z t
y f x y z t
z f x y z t
These equations define a line in terms of the
parameter, , when are constant.
Such a line is called a path l ine . t o o o, , x y z
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PerspectivesEulerian Perspective – the flow as as seen at fixed locations in space, or
over fixed volumes of space. (The perspective of most analysis.)
Lagrangian Perspective – the flow as seen by the fluid material. (Theperspective of the laws of motion.)
Control volume: finite fixed
region of space (Eulerian)
Coordinate: fixed point in space
(Eulerian)
Fluid system: finite piece of the fluid material
(Lagrangian)
Fluid particle: differentially small finite piece
of the fluid material (Lagrangian)
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III
II
I
flow
A system moving along in the flow occupies volumes
I and II at time t. During the next interval dt some of the system moves out of II into three and some
moves out of I into II. The rate of change of an
arbitrary property of the system, N , is given by the
following:
The Transport Theorem:
in II & III at in I & II at in II at in II at in III at in I at
II II
N t dt N t N t dt N t N t dt N t DN
Dt dt dt dt
dV dS t
v n
the unit vector
normal
to ABC , n
two triangular elements from the
family approximating the surface of volume II at time = t
the same two material elements,
but now approximating thesurface of volume III at time =
material that flowed through the
surface of volume II during the
interval and now fills volume III:
S dt v dN dt S v n
DTM
A B
C’
C
A’ B’
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Strategy
• Write down equations of motion for
Lagrangian rates of change seen by fluid
particle or system
• Derive relationship between Lagrangian
and Eulerian rates of change
• Substitute to get Eulerian equations of
motion
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Conservation of MassFrom a Lagrangian Perspective
Law: Rate of Change of Mass of Fluid Material = 0
For a Fluid Particle:
Volume d
Density
0.
0.
01
0
0
Dt
D
t
t t
d
d
t d
t
d
t
d
part
part part
part part
part
V
V
For a Fluid System:
d
Volume R
Density
= (x,y,z,t)
where part t Dt
D
is referred to as
the SUBSTANTIAL DERIVATIVE
(or total, or material, or Lagrangian…)
‘Seen by the
particle’
0
0
R
R sys
d Dt
D
d t
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AXIOMATA SIVE LEGES MOTUS
• Lex I. – Corpus omne perseverare in statuo suo quiescendi vel movendi
uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.
• Lex II.
– Mutationem motus proportionalem esse vi motrici impressae, &fieri secundum lineam rectam qua vis illa imprimitur.
• Lex III. – Actioni contrariam semper & æqualem esse reactionem: sive
corporum duorum actiones in se mutuo semper esse æquales &in partes contrarias dirigi.
• Corol. I. – Corpus viribus conjunctis diagonalem parallelogrammi eodem
tempore describere, quo latera separatis.
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Conservation of MomentumFrom a Lagrangian Perspective (Fluid Particle)
Law: Rate of Change of Momentum = Fbody+Fpressure+Fviscous
Dt
Dd
t d
t
d
part part
VVV
ROC of Momentum
Fbody: d f
dy dx
dz j
i
k P
Net …
density
volume d
velocity V
body force per unit mass f
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2
dy
y
yy
yy
2
dz
z
zy
zy
2
dy
y
p p
Elemental Volume, Surface Forces
x, i
y, j
z, k
2
dy
y
p p
Sides of volume have lengths d x, d y, d z
2dy
y
yy
yy
2
dz
z
zy
zy
• Volume d = d xd yd z
• Density
• Velocity V
2
2
dx
y
dx
y
xy
xy
xy
xy
On front
and rear
faces
y -component
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Conservation of MomentumFrom a Lagrangian Perspective (Fluid Particle)
Law: Rate of Change of Momentum = Fbody+Fpressure+Fviscous
Dt
Dd
t d
t
d
part part
VVV
ROC of Momentum
Fbody:
Fpressure
:
d f
d p so
d y
pdxdz dy
y
p pdxdz dy
y
p pcomponent y
pressure
F
j j j2
1
2
1dy
dx
dz j
i
k P
2
dy
y
p p
P
x, i
y, j
z, k
2
dy
y
p p
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Conservation of MomentumFrom a Lagrangian Perspective (Fluid Particle)
Law: Rate of Change of Momentum = Fbody+Fpressure+Fviscous
Dt
Dd
t d
t
d
part part
VVV
ROC of Momentum
Fbody:
Fpressure
:
Fviscous:
d f
d p so
d y
pdxdz dy
y
p pdxdz dy
y
p pcomponent y
pressure
F
j j j2
1
2
1
d z y x
dxdydz z
dxdydz z
dydz dx xdydz dx x
dxdz dy y
dxdz dy y
component y
zy yy xy zy
zy
zy
zy
xy
xy
xy
xy
yy
yy
yy
yy
j j j
j j
j j
21
21
2
1
2
1
21
21.. .
Likewise for
x and z
dy dx
dz j
i
k P
2
dz
z
zy
zy
P
x, i
y, j
z, k
2dy
y yy
yy
2
dz
z
zy
zy
2
dy
y
yy
yy
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Conservation of MomentumFrom a Lagrangian Perspective (Fluid Particle)
Law: Rate of Change of Momentum = Fbody+Fpressure+Fviscous
Dt
Dd
t d
t
d
part part
VVV
ROC of Momentum
Fbody:
Fpressure:
Fviscous:
d f
d p so
d y
pdxdz dy
y
p pdxdz dy
y
p pcomponent y
pressure
F
j j j2
1
2
1
d z y x
dxdydz z
dxdydz z
dydz dx xdydz dx x
dxdz dy y
dxdz dy y
component y
zy yy xy zy
zy
zy
zy
xy
xy
xy
xy
yy
yy
yy
yy
j j j
j j
j j
21
21
2
1
2
1
21
21.. .
k τ jτiτf V
).().().( z y x p
Dt
D So,
k jiτ
k jiτ
k jiτ
zz yz xz z
zy yy xy y
zx yx xx x
where
Likewise for
x and z
dy dx
dz j
i
k P
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AXIOMS CONCERNING MOTION
• Law 1. – Every body continues in its state of rest or of uniform motion in a
straight line, unless it is compelled to change that state by forcesimpressed upon it.
• Law 2.
– Change of motion is proportional to the motive force impressed;and is in the same direction as the line of the impressed force.
• Law 3. – For every action there is always an opposed equal reaction; or,
the mutual actions of two bodies on each other are always equal and directed to opposite parts.
• Corollary 1. – A body, acted on by two forces simultaneously, will describe the
diagonal of a parallelogram in the same time as it would describethe sides by those forces separately.
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Isaac Newton
1642-1727
C f
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Conservation of EnergyFrom a Lagrangian Perspective (Fluid Particle)
Law: Rate of Change of Energy = Wbody+Wpressure+Wviscous+Q
• Total energy is internal energy + kinetic energy= e + V 2 /2 per unit mass
• Rate of work (power) = force x velocity indirection of force
• Fourier’s law to gives rate of heat added byconduction
Dt
V e Dd
t
V ed
part
)()( 2
212
21
ROC of Energy
Wbody d Vf . dy dx
dz j
i
k P
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2
dy
y
vv
2
dy
y
p p
Elemental Volume, Surface Force
Work and Heat Transfer
x, i
y, j
z, k
2
dy
y
p p
Sides of volume have lengths d x, d y, d z
• Volume d = d xd yd z
• Density
• Velocity V
y -contributions
2
dy
y
vv
Viscous work
requires expansion
of v velocity on all
six sides
2
dy
y
y
T k
y
T k
Velocity
components
u, v, w
2
dy
y
y
T k
y
T k
C ti f E
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Conservation of EnergyFrom a Lagrangian Perspective (Fluid Particle)
Law: Rate of Change of Energy = Wbody+Wpressure+Wviscous+Q
Dt
V e Dd
t
V ed
part
)()( 2
212
21
ROC of Energy
Wbody
Wpressure
Wviscous
d Vf .
d p ).( V
d wvu z y x )).().().(( τττ
Q:
d T k Q so
d y
T
k ydxdz dy y
T
k y y
T
k dxdz dy y
T
k y y
T
k oncontributi y
).(
2
1
2
1
).().().().().(.)( 2
21
T k wvu p Dt
V e D z y x
τττVVf So,
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Equations for Changes Seen From
a Lagrangian Perspective
0 = d
Dt
D
R
S
z y x
S R R
dS ).( + ).( + ).( +dS p-d = d Dt
Dk nτ jnτinτnf V
dS T).k( +dS .++ p-+d .=d )2
V +(e
Dt
D
S S
z y x
R
2
R
nVk nτ jnτinτnf V ).().().(
Differential Form (for a particle)
Integral Form (for a system)
V.
Dt
D
k τ jτiτf V
).().().( z y x p Dt
D
).().().().().(.)( 2
21
T k wvu p Dt
V e D z y x
τττVVf
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Conversion from Lagrangian to
Eulerian rate of change - Derivative
x
y
z ( x(t),y(t),z(t),t )
.Vt
z w
yv
xu
t t
z
z t
y
yt
x
xt
Dt
D
t part The Substantial Derivative
Time Derivative Convective Derivative
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Conversion from Lagrangian to
Eulerian rate of change - Integral
x
y
z
The Reynolds
Transport
Theorem
S R
R
R
R
R
R R R sys
dS d t α
d t
d t
d d Dt
D
Dt
Dd d
Dt
D
Dt
d Dd
Dt
D= d
t
nV
V
VV
V
.
).(
..
.
.Vt Dt
D
Volume R
Surface S
Apply
Divergence
Theorem
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Equations for Changes Seen From
a Lagrangian Perspective
0 = d
Dt
D
R
S
z y x
S R R
dS ).( + ).( + ).( +dS p-d = d Dt
Dk nτ jnτinτnf V
dS T).k( +dS .++ p-+d .=d )2
V +(e
Dt
D
S S
z y x
R
2
R
nVk nτ jnτinτnf V ).().().(
Differential Form (for a particle)
Integral Form (for a system)
V.
Dt
D
k τ jτiτf V
).().().( z y x p Dt
D
).().().().().(.)( 2
21
T k wvu p Dt
V e D z y x
τττVVf
part t Dt
D
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Equations for Changes Seen From
an Eulerian Perspective
Differential Form (for a fixed volume element)
Integral Form (for a system)0 =dS d
t S R
nV.
S
z y x
S R R
dS ).( + ).( + ).( +dS p-d = dS d t
k nτ jnτinτnf nVVV
).(
dS T).k( +dS .++ p-+d .=dS V +ed )t
V +e
S S
z y x
RS
22
R
nVk nτ jnτinτnf VnV ).().().(.)(
)(212
1
V.
Dt
D
k τ jτiτf V
).().().( z y x p
Dt
D
).().().().().(.)( 2
21
T k wvu p Dt
V e D z y x
τττVVf
.V
t Dt
D
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Equivalence of Integral and
Differential Forms
0 = dS d t
S R
nV.
d =dS RS
VnV ..
0.
d
t RV
0. V
t 0..
VV
t
V.
Dt
D
Cons. of mass
(Integral form)
Divergence
Theorem
Conservation of
mass for any
volume R
Then we get or
Cons. of mass
(Differential form)