Reynolds Method to Diagnosize Symptoms of Infected Flows....
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
Reynolds Averaged NS Equations
Averaging of x-momentum Equation
uμ x
pvu
τ
uρ
uμ x
pvu
τ
uρ
uUuU
τ
uUρρ
'
uU
uμ x
pvu
τ
uρ
vVuUvu
vuVuvUVU
vuVuvUVU
vuVU
kwujvuiuuvu ˆˆˆ
z
wu
y
vu
x
uu
z
wu
y
vu
x
uu
z
wu
y
vu
x
uuVU
uμ x
pvu
τ
uρ
uUu
uU
U
Reynolds Averaged Steady Turbulent Momentum Equations
uμ x
pvu
τ
uρ
Ux
P
z
wu
y
vu
x
uuVU
z
wu
y
vu
x
uuU
x
PVU
Reynolds averaged x-momentum equation for steady incompressible turbulent flow
The Reynolds View of Cross Correlation
Reynolds averaged y-momentum equation for steady incompressible turbulent flow
Reynolds averaged z-momentum equation for steady incompressible turbulent flow
z
ww
y
vw
x
uwW
z
PVW
z
wv
y
vv
x
uvV
y
PVV
Reynolds Averaged Navier Stokes equations
0
z
W
y
V
x
U
z
wu
y
vu
x
uuU
x
PVU
z
ww
y
vw
x
uwW
z
PVW
z
wv
y
vv
x
uvV
y
PVV
Reynolds Stress Tensor
2
2
2
wvwuw
wvvuv
wuvuu
R
This is usually called the Reynolds stress tensor
2
2
2
wwvwu
wvvvu
wuvuu
R
Reynolds stresses : total 9 - 6 are unknown
Total 4 equations and 4 + 6 = 10 unknowns
Time averaged Infected Navier Stokes Equation
z
wu
y
vu
x
u
z
U
y
U
x
Uv
dx
dp
z
UW
y
UV
x
UU
''''1 2'
2
2
2
2
2
2
For all the Three Momentum Equations, turbulent stress tensor:
)'()''()''(
)''()'()''(
)''()''()'(
2
2
2
infected,
wvwwu
wvvuv
wuvuu
zzzyzx
yzyyyx
xzxyxx
ij
Reynolds stresses
• Performing the Reynolds Averaging Process, new terms has arisen, namely the Reynolds-stress tensor:
''infection, jiijij uu
• This brings us at the turbulent closure problem, the fact that we have more unknowns than equations.
– Three velocities + pressure + six Reynolds-stresses
– Three momentum equations + the continuity equation
• To close the problem, we need additional equations to solve infected flow.
• Derivations of Reynolds-stress conservation Equations
Derivation of Conservation Equations for Reynolds Stresses
• Introduces new unknowns (22 new unknowns)
Simplified Reynolds Averaged Navier Stokes equations
0
z
W
y
V
x
U
Ux
PVU t
Wz
PVW t
Vy
PVV t
4 equations 5 unknowns → We need one more ???
Modeling of Turbulent Viscosity
μ Fluid property – often called laminar viscosity
tμ Flow property – turbulent viscosity
......
-k
-k
-k
Re
3
2
1Re
-k
Eq.
Two
Eq.-One
TKEM
constantMVM
μon based Models
t
t
fk
kl
l
Curvature
Buoyancy
Low
Layer
Layer
Layer
bounded
wall
Free
High
lengthmixing
MVM: Mean velocity modelsTKEM: Turbulent kinetic energy equation models
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