Geoscience Reference
In-Depth Information
velocity,withzeromean.Theaverageofproducts,saytwodifferentcomponentsof
theinstantaneouslocalflow,is
u
i
u
j
=
u
i
u
j
+
u
i
u
j
(2.8)
and, for example, the average of the
j
th component of the advection term in the
Eulerequation(2.7)is
1
u
i
∂
u
j
∂
x
i
+
∂
u
i
u
j
u
i
∂
u
j
=
(2.9)
∂
x
i
∂
x
i
Thelasttermisthecontributionofturbulencetothemomentumbalance,andmaybe
interpretedas a symmetrickinematicstresstensor(withunitsofvelocitysquared):
τ
ij
=
−
u
i
u
j
(2.10)
called the
Reynolds stress tensor
. Rigorously, the Reynolds stress decomposition
impliessteady flow, butpractically it providesausefulapproximationprovidedthe
time scale over which the mean flow varies is long compared with the scales of
typical turbulent eddies effecting transfer in the flow, i.e., that a
spectral gap
be-
tweenthemeanflowandturbulenceexists. Similarly,measurementsaremostoften
made as time series at particularlocationsin a flow as the fluid streams past, rather
thanasa “snapshot”overtheentiredomainataparticularmoment.Inthiscase,the
frozen-field (Taylor's) hypothesis holds that the turbulence structure moving past
the measurement site is representative of the spatial structure, and that the covari-
ance(in time) matrixof deviatoryvelocitycomponentsis representativeof the true
Reynoldsstress tensor. Notethat Taylor'shypothesisrequiresthatturbulentveloci-
tiesbeconsiderablysmallerthanthemeanvelocity(TennekesandLumley1972),a
conditionsometimesviolatedin oceanicflows.
Droppingtheanglebrackets,letquantitieswithoutprimesdenotethe“mean”val-
ues, allowed to vary with time under the stipulations noted above. The Boussinesq
formoftheEulerequationthenbecome
p
ρ
−
g
ρ
ρ
∂
u
t
+
u
·
∇
u
=
−
∇
k
+
∇
·
∼
(2.11)
∂
where
is the Reynoldsstress tensor. Note that despite ignoringmolecularviscos-
ity in deriving (2.11), we have recovered friction in the form of a deviatory stress
associatedwithnonlinearturbulentfluctuationsfrominstabilitiesin theflow.
The terminology of Reynolds stress is often ambiguous—in most cases, we are
concerned with the vertical variation of the horizontal deviatoric stress compo-
nents,i.e.
∼
1
WhereappropriatetheEinsteinsummationconventionisadopted,whereinrepeatedindicesimply
summation.
