scales of motion, reynolds averaging september 22
DESCRIPTION
Figure 8.1 in StewartTRANSCRIPT
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Scales of Motion, Scales of Motion, Reynolds Reynolds averagingaveraging
September 22September 22
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xFvxP
zuw
yuv
xuu
tu
sin21
Recall the momentum Recall the momentum equation for a fluid:equation for a fluid:
Fx is a frictional forcer per unit mass
This includes all friction, including winds
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Figure 8.1 in Stewart
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Frictional StressesFrictional Stresses Internal stress (frictional stress)Internal stress (frictional stress) Molecular friction stress related to Molecular friction stress related to
velocity shearvelocity shear
zuTxz
Molecular Viscosity
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uF
zu
yu
xuF
x
x
2)(
2
2
2
2
2
2)( ;
del squareKinematic molecular
viscosity
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uy
x
02
2
yu
constant stress, no
acceleration in fluid
uy
x
02
2
yu
changing stress,
acceleration of fluid
“curvature of velocity field”
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zT
yT
xT
zu
zyu
yxu
xF xzxyxxx
1
For an incompressible fluid, the frictional force per unit mass takes the form:
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TurbulenceTurbulence Turbulence comes from the non-linear terms Turbulence comes from the non-linear terms
in the momentum equationsin the momentum equations
ufvxp
zuw
yuv
xuu
tu
dtdu 21
local advective, non-linear
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Reynolds NumberReynolds Number Reynolds Number Reynolds Number RRee, is the ratio of the , is the ratio of the
non-linear terms to the viscous termsnon-linear terms to the viscous terms
UL
LULUU
xuxuu
ViscouslinearNonRe
22
2
U: typical velocity scale u and ∂u ≈ U
L: typical distance scale ∂x ≈ L
LU
xuu
2
22
2
LU
xu
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UL
Pipe flow
U = average velocity in center
L = radius of pipe-Flow is not turbulent (non-linear terms are not important) if Re<1000 in pipes
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Turbulence Turbulence In more complicated flows, non-linear terms are In more complicated flows, non-linear terms are
important when Rimportant when Ree≈ 1 or larger≈ 1 or larger Flows become fully turbulent if RFlows become fully turbulent if Ree>10>1055
Gulf Stream: U~1 m/s L~100 kmGulf Stream: U~1 m/s L~100 km ν≈ν≈1010-6 -6 mm22ss-1 -1 RRee≈10≈101111 => turbulent => turbulent flowflow
Non-linear effects are strong compared to Non-linear effects are strong compared to molecular frictionmolecular friction
In open ocean, molecular friction is usually In open ocean, molecular friction is usually ignored! Turbulent (non-linear) terms are the ignored! Turbulent (non-linear) terms are the important ones!important ones!
Turbulent fluctuations act like molecular friction – Turbulent fluctuations act like molecular friction – dissipate (redistribute) energy and other propertiesdissipate (redistribute) energy and other properties
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Turbulent stressesTurbulent stresses
pppwwwvvvuuu
mean over some period
“eddy”
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0uby definition
T
T
TuTudt
tu
Ttu
dttuT
uu
0
0
)0()(1
)(1
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Non-linear terms in the momentum equation can be written as:
xuu
xuu
xuuuu
xuu
xuu
xuu
xuu
xuuuu
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Using the previous equation, the continuity equation splits into two equations
0
0
xw
xv
xu
dxw
dxv
dxu
The x-component of the of the momentum equation becomes
wuzu
zvu
yu
yuu
xu
x
vxP
DtuD
sin21
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At large Reynolds number, is very small compared to other terms. The addition force per unit mass due to turbulence is:
wuz
vuy
uux
Fx
Reynolds Stresses
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Reynolds StressesReynolds Stresses
we assume turbulence acts like molecular
viscocity:
zuATwu zxz
kinematic eddy viscosity
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2
2
zuA
zuA
zzT
zzxz
assume that Az is either constant or that it varies more slowly in the z direction than ∂ū/∂z. Similar for horizontal components but Az is much less than Ax,y=AH
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Figure 8.4 in Stewart
The buoyancy force acting on the displaced parcel is:
2gVF ‘
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The acceleration of the displaced parcel is:
)( 2g
mFa
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Stability EquationStability Equation
dzdE
1
Stability is defined such that:
E > 0 stable
E = 0 neutral stability
E < 0 unstable
Influence of stability is expressed by a stability frequency N
(also known as Brunt-Vaisala frequency):gEN 2
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Figure 8.6 in Stewart
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Figure 8.7 in Stewart
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Richardson NumberRichardson Number The relative importance of static stability The relative importance of static stability
and dynamic instability is expressed by and dynamic instability is expressed by the Richardson Number:the Richardson Number:
2zugERi
Ri > 0.25 Stable
Ri < 0.25 Velocity shear enhances turbulence