LOCAL CONSERVATION EQUATIONS From global conservation of mass: A 0 dA n u dV t Apply this to a small fixed volume
Feb 23, 2016
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:
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
ix
gt
Du
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 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
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:
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
2
2
j
i
ij
ij
xu
xp
x
j
iji
j
jiix
gxuu
tu
back to the momentum eq.:
2
2
j
i
ii
i
xu
xpg
DtDu
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
ABBAdt
xDtD
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
xuxxdt
xux
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
xdtxv
dxydt
yu
dytdtdd 111Shear
strain 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
21Vorticity
is:
ydtyu
dyxdt
xv
dxtdtdd 111
21
zyu
xv
xzv
yw
yxw
zu
wvuzyx
kjiu
ˆˆˆ