The Urban Atmosphere: turbulence &
parameteriza7ons
Elie Bou-‐ZeidPrinceton University
Civil & Environmental EngineeringLecture 5
Turbulence & Leonardo Da Vinci
Leonardo Da Vinci already observed that sometimes fluid flow seems to “degenerate” into what looks like random motion
"ʺObserve the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the
water has eddying motions, one part of which is due to the principal current, the other to random and reverse motion."ʺ
Slide 2
Turbulence & Osborne Reynolds
Osborne Reynolds found that: If the viscous forces dominate: flow stays laminar
When inertial forces (the non-‐‑linear term) begin to dominate, flow transits to turbulence
He then characterized that Balance using the Reynolds number
Re UL UDorυ υ
=
e.g. 10 m building in a 1m/s wind
Where does Re come from?
Slide 3
What is turbulence? Inertial forces >> viscous forces
w = < w >L C = < C >L
probe
w = < w >T + w′ C = < C >T + C′
probe
0 10 20 30 40 50-0.5
0
0.5
time (s)ve
locity
(m/s)
0 10 20 30 40 50-0.5
0
0.5
time (s)
veloc
ity (m
/s)
Slide 4
Turbulent vortices Slide 5
Turbulent structures and mean flow
Stagna/on points Recircula/on region Horseshoe vortex
Slide 6
Turbulent plume dispersion Slide 7
Turbulent plume dispersion Slide 8
Reynolds decomposition for turbulence Reynolds proposed a very ingenious
idea: write the velocity u as a sum of the mean flow and the turbulent departure from the mean
He then split the Navier-Stokes equations along these lines to get one equation for the mean, and another for the turbulence
u u u′= +
ui∂u j∂ xi
u u u′= +
v = v + ′v p = p + ′pw = w + ′w ρ = ρ + ′ρ
what is ′u ?
Slide 9
Some Rules of Reynolds averaging
It really represents ensemble averaging: ergodicity allows us to equate that to /me and/or space averaging
Rules of the Reynolds decomposi/on are the same as any other sta/s/cal opera/on that involves averaging
if u = u + ′u
Where u is the mean of the series u, for example u is a series of velocity measurements
u = 1N
uii=1
N
∑ and similarly we have v = v + ′v
Then the following apply
u + v = u + v cu = cu if c is a constant
u =u uv = uv
while uv = uv + ′u ′v
Slide 10
Some Rules of Reynolds averaging dudt
= dudt
in practice not in theory, only with spectral gap, i.e. when mean changes much more slowly than turbulent fluctuations
′u = 0
′u ′v , ′u 2, ′v ′u 2, ′v 2 ′u 2 not necessarily equal to 0, most often ≠ 0 in turblence
urms = ′u2( )1/2
u ′v = u ′v = 0
Turbulence intensity = I = σ uu
= std(u)u
′u ′v is the covariance of u and v
and ′u ′v
σ uσ vis their correlation coefficient
Slide 11
Turbulence in Rivers
Flow around a boulder
Falls Missisippi & Illinois
Slide 12
Turbulence in Oceans
waves
wake of an island
wake of a hovercraft Tidal instability (U WA)
Slide 13
The scales of turbulence η L
Dissipation)range inertial)range energy)production)range
production = P dissipation = ε cascade
Slide 14
© 2011 Elie Bou-‐Zeid
The scales of turbulence Energy Production Range: Large scales – Affected by boundary conditions flow specific – Interacts with the mean flow : transforms the mean kinetic energy
(MKE) into Turbulent kinetic energy (TKE) – Produces almost all convective/advective fluxes
Inertial subrange: Intermediate scales – Does not interact with boundaries or mean flow flow independent – Receives the cascading TKE & passes it on to smaller scales – Turbulence is homogeneous and isotropic Kolmogorov theory
Dissipation rage: Smallest scales – Does not interact with boundaries or mean flow flow independent – Dissipates all the TKE into heat (low source of heat) – The smallest turbulent scale is called the Kolmogorov scale
Slide 15
Index (Einstein) notation to simplify things
In index notation an equation can have many “repeated” indices & many “free” indices
Each index, repeated or free, takes the values denoting all spatial (or other) dimensions e.g. in ui , i can be 1, 2, or 3 where 1 indicates the x-‐‑direction (u1 = u, x1 = x) 2 indicates the y-‐‑direction (u2 = v, x2 = y) 3 indicates the z-‐‑direction (u3 = w, x3 = z)
Slide 16
Index (Einstein) notation to simplify things
A free index, if used, should appear in every term of the equation, but only once.
For an equation with 1 free index, the index will denote a vector equation where the value of the free index tells us what direction we are considering. For the momentum equation for example with a free index i, i=1 denotes the x-‐‑momentum equation while i=2 denotes the y-‐‑momentum equation
An equation with no free indices will be a scalar equation An equation with 2 free indices will be a tensor equation
Slide 17
Index (Einstein) notation to simplify things Repeated indices – If the term has a repeated index, the repetition means one should
replace the repeated index with all the coordinates (x and y and z) and sum the terms, for example:
– A repeated (dummy) index, should appear twice, or not at all, in
each term
– A dummy index cannot appear more than twice in a single term,
but we can have more than one dummy index in a term (uiui)(vivi)
∂ui∂xi
=∂u1∂x1
+∂u2∂x2
+∂u3∂x3
= ∇.u
is a repeated index, no free indices is a free index, no repeated indices
i i
i
u x iT x i∂ ∂∂ ∂
Slide 18
Equations in index notation ∂ρ∂t
+∇. ρu( ) = 0 ⎯ →⎯ ∂ρ∂t +
∂ ρui( )∂xi
= 0
∂u
∂t+ u.∇u= − 1
ρ∇p +ν∇2u
− fc k
×U+ 1−
′θvθv
⎛
⎝⎜⎞
⎠⎟g
∂ui∂t
+ u j∂ui∂x j
= − 1ρ∂p∂xi
+ν∂2ui∂x j
2+ fcεij3u j −δ i3 1−
′θvθv
⎛
⎝⎜⎞
⎠⎟g
εijk is the alternating unit tensor =1 for even permutatins of (i,j,k):123,231,312−1 for odd permutatins of (i,j,k):132,213,321
0 if any index is repeated, e.g. 112,233,333
⎧
⎨⎪⎪
⎩⎪⎪
δ ij is the Kronecker delta = 1 if i = j0 if i ≠ j
⎧⎨⎪
⎩⎪
Slide 19
Decomposing the con/nuity equa/on ∂ui∂xi
= 0
∂ ui + ui′( )∂xi
=∂ ui( )∂xi
+∂ ui′( )∂xi
= 0 (1)
Apply the Reynolds Averaging
∂ ui( )∂xi
+∂ ui′⎛⎝⎜
⎞⎠⎟
∂xi= 0
∂ui∂xi
= 0 (2)
Subtract the mean equation from the full equation (1)-(2)
∂ui′
∂xi= 0 (3)
Slide 20
Decomposing the momentum equa/on ∂ui∂t
+ u j∂ui∂x j
= − 1ρ∂p∂xi
+ν∂2ui∂x j
2+ fcεij3u j −δ i3 1−
θv′
θv
⎛
⎝⎜⎜
⎞
⎠⎟⎟g
Apply the Reynolds decomposition to every term
∂ ui + ui′( )∂t
+ u j + u j′( ) ∂ ui + ui′( )
∂x j
= − 1ρ
∂ p + p′( )∂xi
+ν∂2 ui + ui′( )
∂x j2
+ fcεij3 u j + u j′( )−δ i3 1− θv′θv⎛
⎝⎜⎜
⎞
⎠⎟⎟g
Apply Reynolds averaging on the whole equation , i.e. on every term
Noting that terms with the mean of a perturbation are zero∂ ui′
∂t, u j′
∂ui∂x j
, fcεij3 u j′ = 0
∂ui∂t
+ u j∂ui∂x j
= − 1ρ∂p∂xi
+ν∂2ui∂x j
2+ fcεij3u j −δ i3g −
∂ ui′u j′⎛⎝⎜
⎞⎠⎟
∂x jThis is the equation for the mean flow, the equation of conservation of mean momentum
Slide 21
Decomposition of the energy conservation equation
Mean Conservation of Energy
∂θ∂t
+ u j∂θ∂x j
=κ ∂2θ∂x j2 −
1ρcp
∂Ri∂xi
−Leρcp
E −∂ θ ′u j′⎛⎝⎜
⎞⎠⎟
∂x j
∂θ∂t
+ u j∂θ∂x j
=κ ∂2θ
∂x j2−
1ρcp
∂Ri∂xi
−Leρcp
E
Slide 22
Turbulent flux terms?
∂ ui′u j′( )∂x j
: Effect of turbulence on mean flow
ui′u j′ : turbulent stress applied on the mean flow
acts almost like a very strong viscosity
∂ θ′u j′( )∂x j
: Effect of turbulence on mean temperature field
θ′u j′ : turbulent heat flux, acts almost like a very strong diffusivity
Similarly : C′u j′ : turbulent flux of pollutants, water vapor,etc.
Slide 23
Why are the turbulent flux terms important – 1 ?
Climate, weather and other models cannot solve the full equa/ons: too demanding computa/onally
They solve the mean equa/ons only
But in these equa/ons we have these addi/onal unknowns, the turbulent fluxes, that we need to model: turbulence closure problem
Slide 24
Simplest turbulence closure: K-‐theory
The K theory follows conven/onal turbulence modeling approaches and postulates that turbulence acts as a strong (eddy) viscosity for momentum and strong (eddy) diffusivity for heat and scalars, for example
u′w′ = −Kmdudz
v′w′ = −Kmdvdz
Km = eddy viscosity
θ ′w′ = −Khdθdz
q′w′ = −Kqdqdz
Kh,q = eddy diffusivities
Slide 25
Why are the turbulent flux terms important – 2 ?
At the surface, these turbulent fluxes are the fluxes of heat, water vapor, momentum from the surface to the atmosphere
We need to model (special surface closure) or measure them near the ground to describe urban-‐atmosphere interac/ons
Slide 26
Vertical surface fluxes of sensible heat or any scalar
27 Slide 27
How are they related to H and LE? H = ρcp ′w ′θ = vertical (dynamic) heat flux
′w ′θ = H ρcp= vertical kinematic heat flux
LE =Lv × E = ρLv ′w ′q = vertical (dynamic) latent heat fluxLv ′w ′q = vertical kinematic latent heat fluxτ s = surface stress ≡ vertical dynamic momentum fluxes
=ρ ′u ′w2+ ′v ′w
2( )1/2 = ρu*2 > 0u*
2 ≡ friction velocity > 0
′u ′w , ′v ′w = vertical kinematic flux of u,v-momentum
Kinematic fluxes are equivalent to Reynolds fluxes in the ABL and represent the transport of a variable per unit area per unit time, normalized by average mass per unit volume (ρ) or average heat capacity per unit volume (ρcp)
28 Slide 28
Similarity for the ASL flow over rough wall
PHW W LτΔ =z
U = ū ρ, µ of the fluid
Flow over rough wall under steady state conditions: what is U?
ΔP
Surface stress τs
We can reformulate the question as: what is / ?Without changing anythingabout the problem fundamentals
dU dz
Slide 29
ΔP : kg m−1s−2 U : ms−1
ρ : kg m−3 µ : kg m−1 s−1
z : m τ : kg m−1 s−2
Similarity for the ASL flow over rough wall
z U = ū
ρ, µ of the fluid
4 parameters – 3 dimensions: we need 1 non-‐‑dimensional variable
ΔP
Surface stress τs
Slide 30
Outside of the viscous sublayer the viscous force is small
ΔP : kg m−1s−2 U : ms−1
ρ : kg m−3 µ : kg m−1 s−1
z : m τ : kg m−1 s−2
τ / ρz dU / dz
= ??
Surface flux models: The Monin-‐Obukhov similarity theory
τ s / ρ
z du / dz= cst Recall that u* = τ / ρ
This is the appropriate velocity scale in the ASLu*
z du / dz= cst = κ 0.4 is the von karman constant
⇒ du =u*κ
1z
dz
du0
u∫ =
u*κ
1z
dzz0
z
∫
why z0 (it is called the surface roughness length)? see next slide
uu*
=1κ
lnzz0
⎛⎝⎜
⎞⎠⎟
This is called log-law or law-of-the-wall.
It applies for any wall-bounded flow over smooth or rough surfaces.e.g.measure the wind profile at 6 points in the first 20 meters, you can extrapolate to 150 m
Slide 31
Log-profile of velocity & z0 : smooth wall
Viscous sublayer ~ few mm
Real profile
Fictitious extension of log profile
Roughness elements
z0 is an integration constants that indirectly represents the viscous friction forces at the surface
32
ln(z)"
Slide 32
Log-profile of velocity & z0 : rough wall
Real profile
Fictitious extension of log profile
Roughness elements
z0 is an integration constants that indirectly represents the pressure/form drag due to small roughness elements as well as viscous friction
Viscous sublayer ~ few mm
33
ln(z)"
Slide 33
Log-profile of velocity & d : very rough walls Over surfaces with large roughness elements, we introduce the displacement height d. This is needed when the height of the roughness element h z
uu*
= 1κ
ln z − dz0
⎛
⎝⎜⎞
⎠⎟
for vegetation height h, z0 ≈1
10h & d ≈ 2
3h
for buildings of height h, z0 ≈ ??h , d ≈ ??husually depends on building density
What is d for buildings that aresodense that they are almost touching each other?
34
Slide 34
Real profile
Fictitious extension of log profile
Viscous sublayer ~ few mm
Log-profile of velocity & d : very rough walls
Roughness elements
d represents the average height above the surface of the large roughness elements. We ignore it when d
Model for d and z0
Slide 36
AT=Ad,on next slide
λp = plan area covered by building/total lot area = Ap / AT zd = d = displacement height zH =H = average building height
λp
Model for d and z0 (Macdonal, Griffiths and Hall, Atmospheric Environment, 32(11), pp. 1857-‐1864, 1998)
Area density = λ = λp = plan area covered by building/total lot area = Ap / Ad Frontal density = λf
= total fontal (perpend. to wind) area of buildings/total lot area = Af / Ad A= empirical fieng parameter H = average height of buildings CD= drag coefficient of 1 building = 1.2 κ = 0.4 = von Karman constant Slide 37
Monin and Obukhov: stability effects Monin and Obukhov postulated that the effect of all the fluxes can be lumped into one variable the ratio of the mechanical/shear generation of TKE over thebouyant production or destruction
38
Slide 38
Turbulent Kinetic Energy equation
∂e∂t
+ u j∂e∂x j
= δ i3gρui′θ ′
⎛⎝⎜
⎞⎠⎟ − ui
′u j′∂ui∂x j
−∂u j′e∂x j
− 1ρ∂ui′ ′p∂xi
−υ∂ ′ui∂x j
⎛
⎝⎜
⎞
⎠⎟
2
I II III IV V VI VIII: local storage of TKE also called tendencyII: advection of TKE by mean flowIII: bouyant producation of destruction of TKEIV: shear production of TKEV: mean turbulent transport termVI: mean pressure transport termVII: = ε = molecular dissipation
39
TKE = e = ′ui ′ui =′u 2 + ′v 2 + ′w 2
2
Monin and Obukhov: stability effects Monin and Obukhov postulated that the effect of all the fluxes can be lumped into one variable the ratio of the mechanical/shear generation of TKE over thebouyant production or destruction
−u*
3 /κ zgθv
w′θv′⎛⎝⎜
⎞⎠⎟= −
θvu*3
κ z g w′θv′⎛⎝⎜
⎞⎠⎟
where w′θv′ is called the bouyancy flux
Hence they formulated with this ratio a length scale representing effect of stability, this is now a new parameter to dimensional analysis formulations of a "similarity solution". The stability parameter they presented is :
ζ= zL
= −κ z g w′θv′
⎛⎝⎜
⎞⎠⎟
θvu*3
where L = −θvu*
3
κ g w′θv′⎛⎝⎜
⎞⎠⎟
is the Obukhov length scale
40
Slide 40
With surface heat and moisture fluxes 1 3 1 2/ : : ( ) : : :dU dz s kgm z d m kgm s L mρ τ− − − −−
⇒ 2 non-dimensional numbers τ / ρ
z − d( )dU / dz ANDz − d
L
κ z − d( )u*
dudz
= φmz − d
L⎛⎝⎜
⎞⎠⎟ = φm ς( ) ς =
z − dL
: stability parameter
How do you deterimne φm ?
upon integration from the point of zero velocity (at z0 )to z
u =u*κ
lnz − d
z0
⎛⎝⎜
⎞⎠⎟− Ψm
z − dL
⎛⎝⎜
⎞⎠⎟ +Ψm
z0L
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
whereΨm (ς ) =1−φm (x)
x0
ς
∫ dx
upon integration from z1 to z2
u 2 − u1 =u*κ
lnz2z1
⎛
⎝⎜⎞
⎠⎟− Ψm (ς2 ) +Ψm (ς1)
⎡
⎣⎢⎢
⎤
⎦⎥⎥
41
Slide 41
Similarly for the profiles of q and θ κ (z − d0 )u*
dudz
= φm(ς ) momentum
κu*(z − d0 )w 'θ '
dθdz
= φh (ς ) heat
κu*(z − d0 )w 'q '
dqdz
= φv (ς ) water vapor,CO2
Under neutral condition (no heat/bouyancy flux at the surface), all φ 's =1
42
z0
z0,h
z0,v
Slide 42
How to compute Ψ or Φ ? Experiments
, , , ,
, , , ,
0 1: ( ) 1 5 ( ) 5
1 : ( ) 6 ( ) 5 5ln( )m v h m v h
m v h m v h
ς φ ς ς ψ ς ςς φ ς ψ ς ς
< < = + → = −
< = → = − −
( )
( )
( )
1/ 22,
21/ 4
,
21/ 4
0 : ( ) ( ) 1 16
1( ) 2ln , 1 162
1 1( ) 2ln ln 2arctan( ) , 1 162 2 2
m v h
v h
m
x x
x x x x
ς φ ς φ ς ς
ψ ς ς
πψ ς ς
−< = = −
⎛ ⎞+→ = = −⎜ ⎟⎝ ⎠
⎛ ⎞+ +⎛ ⎞→ = + − + = −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠
Stable ABL, Brutsaert 2005:
Unsatble ABL, Brutsaert 1982:
43
Slide 43
Monin-Obukhov Similarity Theory (MOST) for the ASL
The log-‐‑law and MOST only apply in the Atmospheric Surface Layer
Technically, they only apply for steady state condition and for very large homogeneous surfaces
In practice, they could be applied when d/dt
Variation of velocity profiles with stability
45
Slide 45
The diurnal cycle, a sequence of ABLs
46
Slide 46
Convective and Stable ABLs 47
Slide 47
Recommended Readings
Tennekes and Lumley, A first course in Turbulence
Stull, Boundary Layer Meteorology
Wyngaard, Turbulence in the Atmosphere
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