LOCAL CONSERVATION EQUATIONS

From global conservation of mass:

A

0

dAnudVt

Apply this to a small fixed volume

x

z

y

dy

dz

dx

Flux of mass in (kg/s) = dzdyu Flux of mass out (kg/s) = dzdyu

dzdydxux

Net Flux of mass in ‘x’ = dzdydxux

Net Flux of mass in ‘y’ = dzdydxvy

Net Flux of mass in ‘z’ = dzdydxwz

dxux

u

, u

, w

, v

u

Mass per area per time(kg/(m2 s)

dAnu

Net Flux of mass in x, y and z = dzdydxwz

dzdydxvy

dzdydxux

dzdydxu

dVu

dAnudVu

DIVERGENCE Theorem – relates integral over a volume to the integral over a closed area surrounding the volume

Other forms of the DIVERGENCE Theorem

dAndV

θ is any scalar

dAndVx jij

j

ij

for any tensor

dAnu

0

dAnudVt

0

dVudVt

0

dVut

From global mass conservation:

dAnudVu

Using the DIVERGENCE Theorem

0

dVut

0

ut

0

wz

vy

uxt

0

zw

yv

xu

zw

yv

xu

t

01

uDtD

local version of continuity equation

01

uDtD

If the density of a fluid parcel is constant

01

DtD

0

zw

yv

xu

u Local conservation of

mass

fluid reacts instantaneously to changes in pressure - incompressible flow

CONSERVATION OF MOMENTUM

Momentum Theorem

AdAnuudVu

t

surface

A jij

body

dAndVg

Normal (pressure) and tangential (shear) forces

A jjii dAnuudVut

dAndVgA jiji

in tensor notation:

Use Divergence Theorem for tensors:

dAndVx jij

j

ij

to convert:

dAndVgdAnuudV

tu

A jijiA jjii

0dVx

gx

uu

tu

j

iji

j

jii

Expanding the second term:

j

iji

j

jii

j

ji x

gx

uu

tu

x

u

tu

j

iji

j

jii

j

ji x

gx

uu

tu

x

u

tu

0

j

iji

j

jii

xg

x

uu

tu

j

iji

i

xg

tDu

Local Momentum EquationValid for a continuous medium (solid or liquid)

For example, for x momentum:

zyxzu

wyu

vxu

utu xzxyxx

j

iji

j

jii

xg

x

uu

tu

0

i

i

xu

zw

yv

xu

u

4 equations, 12 unknowns; need to relate variables to each other

Simulation of wind blowing past a building (black square) reveals the vortices that are shed downwind of the building; dark orange represents the highest air speeds, dark blue the lowest. As a result of such vortex formation and shedding, tall buildings can experience large, potentially catastrophic forces.

j

iji

j

jii

xg

x

uu

tu

0

i

i

x

u

j

iji

j

jii

xg

x

uu

tu

Conservation of momentumalso known as: Cauchy’s equation

Relation between stress and strain rate

4 equations, 12 unknowns; need to relate flow field and stress tensor

For a fluid at rest, there’s only pressure acting on the fluid, and we can write:

ijijij p

p is pressure and δij is Kronecker’s delta, which is 1 @ i = j, and 0 @ i = j ;The minus sign in front of p is needed for consistency with tensor sign convention

σij is the “deviatoric” part of the stress tensor

parameterizes the diffusive flux of momentum

i

j

j

iij x

u

x

u

For an incompressible Newtonian fluid, the deviatoric tensor can be written as:

Another way of representing the deviatoric tensor, a more general way, is:

ijijijij

3

12

0@0

Dt

D

x

u

i

iii

1

2

2

1121212 x

u

x

up

And for incompressible flow:

1

111 2

x

up

i

j

j

iij x

u

x

u

2

1

Strain rate tensor

For instance:

j

iji

j

jii

xg

x

uu

tu

back to the momentum eq.:

ijijijjij

ij

xx

p

x

3

22

ijijij p

j

ijij

jj

ij

xx

p

x

ijijijij

3

12

iij

j x

p

x

p

ijijijjij

ij

xx

p

x

3

22

i

ii

j

ij

ij

ij

xxx

p

x

3

22

i

i

ii

j

j

i

jij

ij

x

u

xx

u

x

u

xx

p

x

3

2

2

12

0

j

j

ij

i

ij

ij

x

u

xx

u

x

p

x

2

2

i

j

j

iij x

u

x

u

2

1

0

2

2

j

i

ij

ij

x

u

x

p

x

j

iji

j

jii

xg

x

uu

tu

back to the momentum eq.:

2

2

j

i

ii

i

x

u

x

pg

Dt

Du

upgDt

uD

2

Navier-StokesEquation(s)

Strain rates – strain, or deformation, consists of LINEAR and SHEAR strain

Rate of change in length, per unit length is:

AB

ABBA

dtx

Dt

D

x

''11

u u+ (∂u/ ∂x)δx

u dt

LINEAR or NORMAL STRAIN

A B

A’ B’

@ t + dt@ t

δx

(u+ (∂u/ ∂x)δx) dt

x

uxxdt

x

ux

xdt

11

'''' AABBABBA

SHEAR STRAIN

u

v+ (∂v/ ∂x)δx

u dt

Bδx

(u+ (∂u/ ∂y)δy) dt

δy

u+ (∂u/ ∂y)δy

v

v dt

(v+ (∂v/ ∂x)δx)dt

dα

dβ

C A

xdt

x

v

dxydt

y

u

dytdt

dd 111Shear strain is:

dα = CA / CB

xdt

x

v

dxydt

y

u

dytdt

dd 111

x

v

y

u

LINEAR and SHEAR strains can be used to describe fluid deformationIn terms of the STRAIN RATE TENSOR:

i

j

j

iij x

u

x

u

2

1

the diagonal terms are the normal strain rates

the off-diagonal terms are half the shear strain rates

This tensor is symmetric

VORTICITY (Rotation Rate) vs SHEAR STRAIN

u

v+ (∂v/ ∂x)δx

u dt

Bδx

(u+ (∂u/ ∂y)δy) dt

δy

u+ (∂u/ ∂y)δy

v

v dt

(v+ (∂v/ ∂x)δx)dt

dα

dβ

C A

xdt

x

v

dxydt

y

u

dytdt

dd 111Shear strain is:

dα = CA / CB

ydt

y

u

dyxdt

x

v

dxtdt

dd 111

2

1Vorticity is:

ydt

y

u

dyxdt

x

v

dxtdt

dd 111

2

1

zy

u

x

v

xz

v

y

w

yx

w

z

u

wvu

zyx

kji

u

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