''13 ji
j
i
jiij
j
i uuxu
xg
xp
DtuD
Reynolds-Averaged Navier-Stokes Equations -- RANS
0
i
i
xu
4 equations; 7 unknowns
Similar situation as when we went from Cauchy’s eq to N-S eq
j
iji
i
xg
DtDu
1
2
2
j
i
ij
ij
xu
xp
x
2
21j
i
ii
i
xu
xpg
DtDu
j
ijji x
uAuu
'' Aj = eddy viscosity [m2/s]
xuAuu x
yuAuv y
zuAuw z
xvAvu x
yvAvv y
zvAvw z
xwAwu x
ywAwv y
zwAww z
j
ij
jiij
j
i
xuA
xg
xp
DtuD
31
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 uuDtD
and subtract momentum equation for mean flow:
DtuD i
yields the momentum equation for turbulent flow: Dt
Dui '
Turbulent Kinetic Energy (TKE) Equation
ijijoj
ijiijijij
oji eewg
xuuueuuuup
xu
dtd
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
uxue
21
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
zuwu
yuvu
xuuu
xuuu
j
iji
wgo
22
2422
i
j
j
i
i
j
j
iijij x
uxu
xu
xuee
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
zuuwwg
o0
In many ocean applications, the TKE balance is approximated as:
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
sm
At 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)
Spec
tral
Am
p (e
.g. m
2 /Hz)
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)
Spec
tral
Am
p (e
.g. m
2 /Hz)
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)
Spec
tral
Am
p (e
.g. m
2 /Hz)
023.0431 f
KSS ,
Wave number K (m-1)
S (m
3 s-2
)
3
2
sm
2
3
smS
m
K 1
3532 KS
(Monismith’s Lectures)
3532 KS
P
equilibrium range
inertialdissipating range
Kolmogorov’s K-5/3 law
P & small in inertial range -- vortex stretching
Hour from 00:00 on
Data from Ichetucknee River
-5/3
3532 2
UfS
325102 sm
(Monismith’s Lectures)
Kolmogorov’s K-5/3 law -- one of the most important results of turbulence theory