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 classIntegral theorems…

R S

dSd n

R S

dSd nAA ..

R S

dSd nAA

C

SSd 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 Exercise1. Make up the most complex irrotational 3D velocity

field you can.

2223sin /3)2cos( zyxxyxe x kjiV ?

We can generate an irrotational field by taking the gradient of any scalar field, since 0

I got this one by randomly choosing

zyxe x /132sin And computing

kjiVzyx

Acceleration??

2nd Order Integral Theorems• Green’s theorem (1st form)

• Green’s theorem (2nd form)

Volume Rwith Surface S

d

ndS

SR

Sd dn

2

S

RSd d

n-

n22

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 lineam

rectam 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

0 mass of C. O. R.

ViscousPressureBody of C. O. R. FFFmomentum

QWW Wenergy of 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

0 mass of C. O. R.

ViscousPressureBody of C. O. R. FFFmomentum

QWW Wenergy of C. O. R. ViscousPressureBody

Apply to the fluid material (not the space)

Experimental observations

Assumption: Fluid is a homogeneous continuum

flow

x

y

z

x y z r i j k

o o o ox 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 Dv

Dt Dtx y z

Df x y z t Df x y z tv v

Dt Dt

Velocity :

DTM

2 2 21 o o, o 2 o o, o 3 o o, o

2 2 2

( , , ) ( , , ) ( , , ), ,

Concept is straightforward, but difficult to implement, often would produce moreinformation 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 ta a a

Dt Dt Dt

Acceleration :

doesn't fit the situation usually encounteredin fluid mechanics.

The Lagrangian Method is always used in solid mechanics :

DTM

P

ox

3 2o o3

6Py x lxEI

3 2o 1 o o o o o 2 o o o 3 o o o( , , , ), 3 ( , , , ), 0 ( , , , )

6P t

x x f x y z t y x lx f x y z t z f x y z tEI

radDDt t

t

Acceleration :v va v vG

1) Lagrangian Method

1 o o, o

2 o o, o

3 o o, o

( , , ) ( , , )

( , , )

x f x y z ty f x y z tz 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 tv

DtDf x y z t

vDt

Df x y z tv

Dt

Velocity : ( , , , ) ( , , , )

( , , , )

x

y

z

v x y z tv x y z tv 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

21 o o, o

2

22 o o, o

2

23 o o, o

2

( , , )

( , , )

( , , )

x

y

z

D f x y z ta

DtD f x y z t

aDt

D f x y z ta

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, Italydied 10 April 1813 in Paris, France

Leonhard Paul Euler

born 15 April 1707 in Basel, Switzerlanddied 18 September 1783 in St. Petersburg, Russia

DTM

Acceleration in the Eulerian Method:

x

y

z

rdr

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 dtd dt

a

r r r

v r v r rr MUST v

( , ) ( , )D d t dt tDt 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 tDt dt dt

d t dt d tdt t

d t t ddt 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 dtt

rvr 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 streakline.

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 particles lying 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 ty f x y z tz f x y z t

These equations define a line in terms of the parameter, , when are constant.Such a line is called a pathline.

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. (The perspective 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 tDNDt dt dt dt

dV dSt

v nthe 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 the surface of volume III at time =

material that flowed through the surface of volume II during the interval and now fills volume III:

S dtv 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 Mass From 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

DtD

t

ttd

d

td

td

td

part

partpart

partpart

part

V

V

For a Fluid System:

d

Volume R

Density =(x,y,z,t)

whereparttDt

D

is referred to as

the SUBSTANTIAL DERIVATIVE(or total, or material, or Lagrangian…)

‘Seen by the particle’

0

0

R

Rsys

dDtD

dt

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 Momentum From a Lagrangian Perspective (Fluid Particle)

Law: Rate of Change of Momentum = Fbody+Fpressure+Fviscous

DtDd

td

td

partpart

VVV

ROC of Momentum

Fbody: df

dydx

dzji

kP

Net …density volume dvelocity Vbody force per unit mass f

2dy

yyy

yy

2dz

zzy

zy

2dy

ypp

Elemental Volume, Surface Forces

x, i

y, j

z, k

2dy

ypp

Sides of volume have lengths dx, dy, dz

2dy

yyy

yy

2dz

zzy

zy

• Volume d = dxdydz• Density • Velocity V

2

2dx

y

dxy

xyxy

xyxy

On front and rear faces

y-component

Conservation of Momentum From a Lagrangian Perspective (Fluid Particle)

Law: Rate of Change of Momentum = Fbody+Fpressure+Fviscous

DtDd

td

td

partpart

VVV

ROC of Momentum

Fbody:

Fpressure:

df

dpso

dypdxdzdy

yppdxdzdy

yppcomponenty

pressure

F

jjj 21

21

dydx

dzji

kP

2dy

ypp

P

x, i

y, j

z, k

2dy

ypp

Conservation of Momentum From a Lagrangian Perspective (Fluid Particle)

Law: Rate of Change of Momentum = Fbody+Fpressure+Fviscous

DtDd

td

td

partpart

VVV

ROC of Momentum

Fbody:

Fpressure:

Fviscous:

df

dpso

dypdxdzdy

yppdxdzdy

yppcomponenty

pressure

F

jjj 21

21

dzyx

dxdydzz

dxdydzz

dydzdxx

dydzdxx

dxdzdyy

dxdzdyy

componenty

zyyyxyzyzy

zyzy

xyxy

xyxy

yyyy

yyyy

jjj

jj

jj

21

21

21

21

21

21...

Likewise for x and z

dydx

dzji

kP

2dz

zzy

zy

P

x, i

y, j

z, k

2dy

yyy

yy

2dz

zzy

zy

2dy

yyy

yy

Conservation of Momentum From a Lagrangian Perspective (Fluid Particle)

Law: Rate of Change of Momentum = Fbody+Fpressure+Fviscous

DtDd

td

td

partpart

VVV

ROC of Momentum

Fbody:

Fpressure:

Fviscous:

df

dpso

dypdxdzdy

yppdxdzdy

yppcomponenty

pressure

F

jjj 21

21

dzyx

dxdydzz

dxdydzz

dydzdxx

dydzdxx

dxdzdyy

dxdzdyy

componenty

zyyyxyzyzy

zyzy

xyxy

xyxy

yyyy

yyyy

jjj

jj

jj

21

21

21

21

21

21...

kτjτiτfV ).().().( zyxpDtD

So,

kjiτkjiτkjiτ

zzyzxzz

zyyyxyy

zxyxxxx

where

Likewise for x and z

dydx

dzji

kP

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 forces impressed 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 describe the sides by those forces separately.

Isaac Newton1642-1727

Conservation of Energy From a Lagrangian Perspective (Fluid Particle)

Law: Rate of Change of Energy = Wbody+Wpressure+Wviscous+Q

• Total energy is internal energy + kinetic energy = e + V2/2 per unit mass

• Rate of work (power) = force x velocity in direction of force

• Fourier’s law to gives rate of heat added by conduction

DtVeDd

tVed

part

)()( 2212

21

ROC of Energy

Wbody dVf . dydx

dzji

kP

2dy

yvv

2dy

ypp

Elemental Volume, Surface Force Work and Heat Transfer

x, i

y, j

z, k

2dy

ypp

Sides of volume have lengths dx, dy, dz

• Volume d = dxdydz• Density • Velocity V

y-contributions

2dy

yvv

Viscous work requires expansion of v velocity on all six sides

2dy

yyTk

yTk

Velocity components u, v, w

2dy

yyTk

yTk

Conservation of Energy From a Lagrangian Perspective (Fluid Particle)

Law: Rate of Change of Energy = Wbody+Wpressure+Wviscous+Q

DtVeDd

tVed

part

)()( 2212

21

ROC of Energy

Wbody

Wpressure

Wviscous

dVf .

dp ).( V

dwvu zyx )).().().(( τττ

Q:

dTkQso

dyTk

ydxdzdy

yTk

yyTkdxdzdy

yTk

yyTkoncontributiy

).(

21

21

).().().().().(.)( 2

21

TkwvupDt

VeDzyx

τττVVfSo,

Equations for Changes Seen From a Lagrangian Perspective

0 = d DtD

R

S

zyxSRR

dS ).( + ).( + ).( +dS p- d = d DtD knτjnτinτnfV

dS T).k(+dS . + + p- + d . = d )2

V + (eDtD

SSzyx

R

2

R nVknτjnτinτnfV ).().().(

Differential Form (for a particle)

Integral Form (for a system)

V. DtD

kτjτiτfV ).().().( zyxpDtD

).().().().().(.)( 2

21

TkwvupDt

VeDzyx

τττVVf

Conversion from Lagrangian to Eulerian rate of change - Derivative

x

y

z(x(t),y(t),z(t),t)

.Vt

zw

yv

xu

ttz

zty

ytx

xt

DtD

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

SR

R

R

R

R

RRRsys

dSdtα

dt

dt

ddDt

DDt

DddDt

D

Dt

d Dd DtD= d

t

nV

V

VV

V

.

).(

..

.

.VtDt

D

Volume RSurface S

Apply Divergence Theorem

Equations for Changes Seen From a Lagrangian Perspective

0 = d DtD

R

S

zyxSRR

dS ).( + ).( + ).( +dS p- d = d DtD knτjnτinτnfV

dS T).k(+dS . + + p- + d . = d )2

V + (eDtD

SSzyx

R

2

R nVknτjnτinτnfV ).().().(

Differential Form (for a particle)

Integral Form (for a system)

V. DtD

kτjτiτfV ).().().( zyxpDtD

).().().().().(.)( 2

21

TkwvupDt

VeDzyx

τττVVf

parttDtD

Equations for Changes Seen From an Eulerian Perspective

Differential Form (for a fixed volume element)

Integral Form (for a system)0 = dSd t SR

nV.

Szyx

SRR

dS ).( + ).( + ).( + dS p- d = dSd t

knτjnτinτnfnVVV ).(

dS T).k(+dS . + + p- + d . =dSV+ ed )t

V+ e

SSzyx

RS

22

R

nVknτjnτinτnfVnV ).().().(.)(

)(212

1

V. DtD

kτjτiτfV ).().().( zyxpDtD

).().().().().(.)( 2

21

TkwvupDt

VeDzyx

τττVVf

.VtDt

D

Equivalence of Integral and Differential Forms

0 = dSd t SR

nV.

d =dSR

S VnV ..

0.

dtR

V

0. V

t0..

VV

t

V. DtD

Cons. of mass (Integral form)

Divergence Theorem

Conservation of mass for any volume R

Then we get or

Cons. of mass (Differential form)