LOCAL CONSERVATION EQUATIONS

From global conservation of mass:

A

0

dAnudVt

Apply this to a small fixed volume

x

z

y

dy

dz

dxFlux 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

, vu

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

w

zv

yu

xt

0

zw

yv

xu

zw

yv

xu

t

01 u

DtD

local version of continuity equation

01 u

DtD

If the density of a fluid parcel is constant

01

DtD

0

zw

yv

xuu Local conservation of

mass

fluid reacts instantaneously to changes in pressure - incompressible flow

CONSERVATION OF MOMENTUM

Momentum Theorem

A dAnuudVut

surface

A jij

body

dAndVg

Normal (pressure) and tangential (shear) forces

A jjii dAnuudVut

dAndVg A jiji

in tensor notation:

Use Divergence Theorem for tensors:

dAndVx jij

j

ij

to convert: dAndVgdAnuudV

tu

A jijiA jjii

0dV

xg

xuu

tu

j

iji

j

jii

Expanding the second term and combining:

j

iji

j

jii

j

ji x

gxuu

tu

xu

tu

j

iji

j

jii

j

ji x

gxuu

tu

xu

tu

0

j

iji

j

jiix

gxuu

tu

j

iji

i

xg

DtDu

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

For example, for x momentum:

zyxzuw

yuv

xuu

tu xzxyxx

j

iji

j

jiix

gxuu

tu

0

i

ixu

zw

yv

xuu

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

jiix

gxuu

tu

0

i

i

xu

j

iji

j

jiix

gxuu

tu

Conservation of momentumalso known as: Cauchy’s momentum 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

σij parameterizes the diffusive flux of momentum

i

j

j

iij x

uxu

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

312

0@0

DtD

xu

i

iii

1

2

2

1121212 x

uxup

And for incompressible flow:

1

111 2

xup

i

j

j

iij x

uxu

21

Strain rate tensor

For instance:

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

Rate of change in length, per unit length (strain rate) is:

AB

ABBAdt

xdtd

x

''11

u u+ (∂u/ ∂x)δx

xuxxdt

xux

xdt

11

LINEAR or NORMAL STRAIN

A B

A’ B’@ t + dt@ t

δx

x(u+ (∂u/ ∂x)δx) dt

change in length

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

xdtxv

dxydt

yu

dytdtdd 111Shear

strainrate is:

dα = CA / CB

xdtxv

dxydt

yu

dytdtdd 111

xv

yu

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

i

j

j

iij x

uxu

21

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

xdtxv

dxydt

yu

dytdtdd 111Shear

strain is:

dα = CA / CB

ydtyu

dyxdt

xv

dxtdtdd 111

21

21Rotation

rate is:

ydtyu

dyxdt

xv

dxtdtdd 111

21

21

zyu

xv

xzv

yw

yxw

zu

wvuzyx

kjiu

ˆˆˆ

Vorticity is twice rotation rate

j

iji

j

jiix

gxuu

tu

back to the momentum eq.:

ijijijjij

ij

xxp

x

322

ijijij p

j

ijij

jj

ij

xxp

x

ijijijij

312

iij

j xp

xp

ijijijjij

ij

xxp

x

322

i

ii

j

ij

ij

ij

xxxp

x

322

i

i

ii

j

j

i

jij

ij

xu

xxu

xu

xxp

x

32

212

0

j

j

ij

i

ij

ij

xu

xxu

xp

x

2

2

i

j

j

iij x

uxu

21

0

strain rate tensor

2

2

j

i

ij

ij

xu

xp

x

j

iji

j

jiix

gxuu

tu

back to Cauchy’s momentum eq.:

2

2

j

i

ii

i

xu

xpg

DtDu

upgDt

uD

2

Navier-StokesEquation(s)