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