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''1
3 jij
i
jiij
j
i uux
u
xg
x
p
Dt
uD
Reynolds-Averaged Navier-Stokes Equations -- RANS
0
i
i
x
u
4 equations; 7 unknowns
Similar situation as when we went from Cauchy’s eq to N-S eq
j
iji
i
xg
Dt
Du
1
2
2
j
i
ij
ij
x
u
x
p
x
2
21
j
i
ii
i
x
u
x
pg
Dt
Du
j
ijji x
uAuu
'' Aj = eddy viscosity [m2/s]
x
uAuu x
y
uAuv y
z
uAuw z
x
vAvu x
y
vAvv y
z
vAvw z
x
wAwu x
y
wAwv y
z
wAww z
j
ij
jiij
j
i
x
uA
xg
x
p
Dt
uD3
1
Turbulence Closure
jjj uAA
Turbulent Kinetic Energy (TKE)
An equation to describe TKE is obtained by:
multiplying the momentum equation for turbulent flow times the turbulent flow itself (scalar product)
and then do ensemble averages
Total flow = Mean plus turbulent parts = 'uU
Same for a scalar: '
Start with momentum equation (balance) for total flow: 'ii uuDt
D
and subtract momentum equation for mean flow:
Dt
uD i
yields the momentum equation for turbulent flow: Dt
Dui '
Turbulent Kinetic Energy (TKE) Equation
ijijoj
i
jiijijijoj
i eewg
x
uuueuuuup
xu
dt
d
221 2
212
21
Multiplying turbulent flow momentum equation times ui and dropping the primes (all lower case letters are turbulent or fluctuating variables)
2
21
221
221
221
wdtd
vdtd
udtd
udtd
i
Total changes of TKE Transport of TKE Shear Production
Buoyancy Production
ViscousDissipation
i
j
j
iij x
u
xu
e21
fluctuating strain rate
Transport of TKE. Has a flux divergence form and represents spatial transport of TKE. The first two terms are transport of turbulence by turbulence itself: pressure fluctuations (waves) and turbulent transport by eddies; the third term is viscous transport
z
uwu
y
uvu
x
uuu
x
uuu
j
iji
wg
o
22
24
22
i
j
j
i
i
j
j
iijij x
u
x
u
x
u
x
uee
interaction of Reynolds stresses with mean shear;
represents gain of TKE
represents gain or loss of TKE, depending on covarianceof density and w fluctuations
represents loss of TKE
http://apollo.lsc.vsc.edu/classes/met455/notes/section4/1.html
Injective range -- large scales where forcing injects the energy
Inertial range -- where the time required for energy transfer is shorter than the dissipative time and the energy is thus conserved and transported to smaller scales.
Dissipative range -- where the energy dissipation overcomes the transfer and the cascade is stopped.
Turbulence Production and Cascade
http://math.unice.fr/~musacchi/tesi/node9.html
KL
Inertial range
“Big whorls have little whorlsThat feed on their velocity;And little whorls have lesser whorls,And so on to viscosity.” (Lewis F Richardson, 1920)
The largest scales of turbulent motion (energy containing scales) are set by geometry:- depth of channel- distance from boundary
The rate of energy transfer to smaller scales can be estimated from scaling:
u velocity of the eddies containing energyl is the length scale of those eddies
u2 kinetic energy of eddies
l / u turnover time
u2 / (l / u ) rate of energy transfer = u3 / l ~
3
2
s
mAt any intermediate scale l, 31l~lu
But at the smallest scales LK,
413
L Kolmogorov length scale
Typically, 32616 1010 smkgW so that m~LK
310
Turbulence Cascade has a well defined structure – Kolmogorov’s K-5/3 law
Spectral power
Time (secs)
S
Frequency (Hertz)
Sp
ectr
al A
mp
(e.
g.
m2/H
z)
T = 30 s
N
n
Nknin
N
n
tnfin
eyt
eytY k
1
2
1
2
S = sin(2 π t /30)
033.0301 f
Time (secs)
S
Frequency (Hertz)
Sp
ectr
al A
mp
(e.
g.
m2/H
z)
S = sin(2 π t /30) + sin(2 π t /12)
0833.0121 f
S = sin(2 π t /30) + sin(2 π t /12) + sin(2 π t /43)
Time (secs)
S
Frequency (Hertz)
Sp
ectr
al A
mp
(e.
g.
m2/H
z)
023.0431 f
(Monismith’s Lectures)
3532 KS
P
equilibrium range
inertialdissipating range
Kolmogorov’s K-5/3 law
P & small in inertial range -- vortex stretching
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