class 6 - eqns of motion

7/27/2019 Class 6 - Eqns of Motion

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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



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



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??



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.



The Equations of Motion



“Phrase of the Day”

Mutationem motus proportionalem esse vi

motrici impressae, & fieri secundum lineamrectam qua vis illa imprimitur.

Go Hokies?



Supersonic Turbulent Jet Flow and

Near Acoustic Field

Freund at al. (1997)

Stanford Univ.

DNS



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



Supersonic Turbulent Jet Flow and

Near Acoustic Field

Freund at al. (1997)

Stanford Univ.

DNS



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



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



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



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



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

http://en.wikipedia.org/wiki/File:Leonhard_Euler_2.jpg

http://en.wikipedia.org/wiki/File:Langrange_portrait.jpg



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



.

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



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)



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’



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



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



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.



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



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





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





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





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



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.



Isaac Newton

1642-1727

C f



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



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



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,



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



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



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




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




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



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)

class 6 - eqns of motion

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