AOE 5104 Class 7

• Online presentations for next class:– Kinematics 1

• Homework 3

• Class next Tuesday will be given by Dr. Aurelien Borgoltz

• daVinci (Aaron Marcus, Justin Ratcliff)

Claude-Louis Navier (February 10, 1785 in Dijon - August 21, 1836 in Paris)

Pump flow

VisEng Ltd http://www.viseng.com/consult/flowvis.html

Equations for Changes Seen From a Lagrangian Perspective

0 = d Dt

D

R

S

zyx

SRR

dS ).( + ).( + ).( +dS p- d = d Dt

DknτjnτinτnfV

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

V + (eDt

D

SS

zyx

R

2

R nVknτjnτinτnfV ).().().(

Differential Form (for a particle)

Integral Form (for a system)

V. Dt

D

kτjτiτfV

).().().( zyxpDt

D

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

21

TkwvupDt

VeDzyx

τττVVf

O

Pump flow

VisEng Ltd http://www.viseng.com/consult/flowvis.html

Conversion from Lagrangian to Eulerian rate of change - Derivative

x

y

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

.Vt

zw

yv

xu

tt

z

zt

y

yt

x

xt

t part

Time Derivative Convective Derivative

.VtDt

D

The Substantial Derivative

Conversion from Lagrangian to Eulerian rate of change - Integral

x

y

z

The Reynolds Transport Theorem

SR

R

R

R

R

RRRsys

dSdt

α

dt

dt

ddDt

DDt

Ddd

Dt

D

Dt

d Dd

Dt

D= d

t

nV

V

VV

V

.

).(

..

.

.VtDt

D

Volume RSurface S

Apply Divergence Theorem

SRR

dSdt

α = d

Dt

DnV.

Equations for Changes Seen From a Lagrangian Perspective

0 = d Dt

D

R

S

zyx

SRR

dS ).( + ).( + ).( +dS p- d = d Dt

DknτjnτinτnfV

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

V + (eDt

D

SS

zyx

R

2

R nVknτjnτinτnfV ).().().(

Differential Form (for a particle)

Integral Form (for a system)

V. Dt

D

kτjτiτfV

).().().( zyxpDt

D

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

21

TkwvupDt

VeDzyx

τττVVf

parttDt

D

SRR

dSdt

α = d

Dt

DnV.

.VtDt

D

Equations for Changes Seen From an Eulerian Perspective

Differential Form (for a fixed point in space)

Integral Form (for a fixed control volume)0 = dSd t SR

nV.

S

zyx

SRR

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

knτjnτinτnfnVVV

).(

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

V+ e

SS

zyx

RS

22

R

nVknτjnτinτnfVnV ).().().(.)()(

212

1

V. Dt

D

kτjτiτfV

).().().( zyxpDt

D

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

21

TkwvupDt

VeDzyx

τττVVf

.VtDt

D

Equivalence of Integral and Differential Forms

0 = dSd t SR

nV.

d =dSR

S VnV ..

0.

d

tRV

0.

Vt

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)

Constitutive Relations - Closing the Equations of Motion

Could we solve, in principle, the equations we have derived for a particular flow?

Equations for Changes Seen From an Eulerian Perspective

Differential Form (for a fixed point in space)

Integral Form (for a fixed control volume)0 = dSd t SR

nV.

S

zyx

SRR

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

knτjnτinτnfnVVV

).(

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

V+ e

SS

zyx

RS

22

R

nVknτjnτinτnfVnV ).().().(.)()(

212

1

V. Dt

D

kτjτiτfV

).().().( zyxpDt

D

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

21

TkwvupDt

VeDzyx

τττVVf

.VtDt

D

Constitutive Relations

• Equations of motion– 5 eqns: Mass (1), Momentum (3), Energy (1)

– 13 unknowns: p, , u, v, w, T, 6 , e

• Need 8 more equations!

• Information about the fluid is needed

• Constitutive relations– Thermodynamics: p, , T, e

– Viscous stress relations

p = p(,T) and e = e(,T)

Newtonian fluid

Newtonian (Isotropic) Fluid• Viscous Stress is Linearly Proportional to Strain Rate• Relationship is isotropic (the same in all directions)

ij

zzzyzx

yxyyyx

xzxyxx

Stress, is a tensor…

…and so has some basic properties when we rotate the coordinate system used to represent it like…• Principal axes… axis directions for

which all off shear stresses are zero

• Tensor invariants… combinations of elements that don’t change with the axis directions

zz

yy

xx

00

00

00

)(

)(

)(3

1

3

1

3

1

CubictDeterminanIII

QuadraticII

LinearI

i j jiijjjii

i ii

But what is strain rate (or rate of deformation?)

(Symmetric so yx= xy, yz= zx, xz= zx)

Distortion of a Particle in a Flow

M>1, accelerating, expanding flow

Total change

= rotation

+ dilation

+ shear deformation

Physically

Rate of deformationor strain rate

“Cauchy Stokes Decomposition”

Distortion of a Particle in a Flow

rrrV

rr

VV

d

z

v

y

w

x

w

z

u

z

v

y

w

y

u

x

v

x

w

z

u

y

u

x

v

d

z

wy

vx

u

d

z

v

y

w

x

w

z

u

z

v

y

w

y

u

x

v

x

w

z

u

y

u

x

v

d

dd

d

dz

dy

dx

z

w

y

w

x

wz

v

y

v

x

vz

u

y

u

x

u

dw

dv

du

d

0

0

0

00

00

00

0

0

0

21

21

21

21

21

21

21

21

21

21

21

21

V+dV

V

Deformation is represented by dV×time so rate of deformation is given by dV

Total change

= rotation + dilation + shear deformation

Mathematically

Rate of deformation or strain rate

Newtonian (Isotropic) Fluid

z

w

z

v

y

w

x

w

z

u

z

v

y

w

y

v

y

u

x

v

x

w

z

u

y

u

x

v

x

u

toalproportionLLYISOTROPICA

zzzyzx

yxyyyx

xzxyxx

21

21

21

21

21

21

z

wy

vx

u

toalproportionLLYISOTROPICA

zz

yy

xx

00

00

00

00

00

00

V.22

x

u

z

w

y

v

x

u

x

uxx

So

So Each stress = Const.× Corresponding strain + Const. × First invariant of comp. rate component strain rate tensor

Or And likewise fory and z.

Stokes’ Hypothesis

• Stokes hypothesized that the total normal viscous stress xx+yy+zz should be zero, so that they can’t behave like an extra pressure (i.e. he wanted to simplify things so that the total pressure felt anywhere in the flow would be the same as the pressure used in the thermodynamic relations).

• This implies =-⅔µ and remains controversial• With this, and in general (non-principal) axes,

we finally have

y

u

x

vx

u

xy

xx

V.2 32 and likewise for yy and zz

and likewise for yz and xz

V.22

x

u

z

w

y

v

x

u

x

uxx

The Equations of MotionDifferential Form (for a fixed volume element)

V. Dt

D

).(2)()(f

)().(2)(f

)()().(2f

31

31

31

V

V

V

z

w

zy

w

z

v

yx

w

z

u

xz

p

Dt

Dw

y

w

z

v

zy

v

yy

u

x

v

xy

p

Dt

Dv

x

w

z

u

zy

u

x

v

yx

u

xx

p

Dt

Du

z

y

x

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

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

31

31

31

221

VV

VVVf

z

ww

y

w

z

vv

x

w

z

uu

zy

w

z

vw

y

vv

y

u

x

vu

y

x

w

z

uw

y

u

x

vv

x

uu

xTkp

Dt

VeD

The Continuity equation

The Navier Stokes’ equations

The Viscous Flow Energy Equation

These form a closed set when two thermodynamic relations are

specified

Leonhard Euler1707-1783

Assumptions made / Info encodedAssumption/Law Mass NS VFEEConservation of massConservation of momentumConservation of energyContinuumNewtonian fluidIsotropic viscosityStokes´ HypothesisFourier´s Law of Heat conductionNo heat addition except by conduction

Summary

• Conservations laws

• Lagrangian and Eulerian perspectives

• Equations of motion dervied from a Lagrangian perspective

• The Substantial Derivative and the Reynolds Transport Theorem connect Lagrangian with Eulerian

• Constitutive Relations provide information about the fluid material

• Assumptions