7. Atmospheric flows - Budapest University of Technology ... · PDF file- reversible compression; ... The dry adiabatic temperature ... ANSYS FLUENT + transformation system + customized
Post on 18-Feb-2018
223 Views
Preview:
Transcript
2014.02.11.
1
7. Atmospheric flows
Dr. Gergely KristófDept. of Fluid Mechanics, BME
April, 2012.
Energy balance of a moving volume of air in the atmosphere
Te atmosphere feels colder at the top of high mountain.Cold air does not flows down to sea level. Why?
The cold air flow would warm up due to the increasing pressure.(We cannot observe this phenomenon in any laboratory experiments.)
:wδ work due to:- reversible compression;- viscous dissipation.
:qδ heating due to:- turbulent transport,- radiation,- latent heat release
(phase change).
wqdu δδ += change of the internal energyof a unit mass of air.
The change of the thermodynamic state in a rapid vertical flow of dry air is dominated by the reversible compression work, therefore it is isentropic.
The potential temperature: Θ1
0
1
00
−−
=
=
κκ
κ
ρ
ρ
p
p
T
TFor isentropic flows:
pp
vp
c
R
c
cc=
−=
−
κ
κ 1
in which κ is the ratio of specific heats, thus
pc/R
pT
=
Pa105
Θ
The potential temperature of the atmospheric air in a point characterized by
temperature T and pressure p is defined as:
Θ would be the temperature if the air parcel would taken down to sea level.
2014.02.11.
2
Stable and unstable stratifications
Θ
zStableUnstable
On the basis of the potential temperature profile we can analyze the atmospheric stability:
Neutral
Vertical flows
z
Θ
Θ0Τ0
z
T
p0
z
Hydrostatics
gz
pρ−=
∂
∂
In order to solve this we need to find relation between p and ρ.
( )pf=ρ
Barotrophic relation:
const.=ρ homogenous atm.
const.00
== T,TR
pρ isothermal atm.
const.0
0 =
= m,
p
pm
ρρ polytrophic atm.
e.g. Θ=const.
( )zfT,TR
p==ρ
2014.02.11.
3
Problem #7.1
Please, calculate the pressure profile for a given (linear) temperature profile:
const.=∂
∂−=
z
Tγ
zTT γ−= 0
in which
,
.
To the solution
Assume, that in z=0: T=T0 and p=p0 !
The dry adiabatic temperature gradient
pc
R
p
p
T
T
=
00
The adiabatic relation:
p
.adiab
c
R
g
R=
γBy comparing the exponents:
km
K89
1000
89.
.
c
g
p.adiab ==== Γγwe obtain the adiabatic
temperature gradient:
We compare the T(p) relation of a linear temperature model with those of an adiabatic model.
g
R
p
p
T
T
γ
=
00
From the linear T profil of γ gradient we obtained:
γγ R
g
T
zTpp
−=
0
00
Thus the dry adiabatic T profile is a linear profile of Γ gradient.
Aha! T(z) must be also linear for an adiabatic
profile!
The standard atmosphere
Troposphere
Stratosphere
ΘT
T,Θ
z
11 km
km
K56.=γ
Standard (average)tropospherictemperature gradient:
K15.288T0 =
Pa1001325.1p 5
0 ⋅=
Referencevalues:
Γ
γ
Γ<
2014.02.11.
4
Unstable stratification
z
Θ
z
Θ
During the summer, when the surface is heavily heated:
Thermal convection:
Constant potential temperature.
The pressure gradient g
z
pρ−=
∂
∂
m
Pa12−≅The vertical component:
The vertical component:
m
Pa00260
km 1000
hPa 26.
x
p==
∂
∂
http://www.weatheronline.co.uk/map/vor/euro/d.htm
This high anisotropy causes problem in numerical solutions.
Buoyancy and the Boussinesq model
gz
p
dt
dwρρ −
∂
∂−=
The vertical component of theequation of motion:
gz
pρ−
∂
∂−=0We define the hydrostatic
state:
'ppp += 'ρρρ += 'TTT +=We decompose the profiles by using the hydrostatic state:
'T' βρρ −=The density perturbation can beexpressed in terms of the cubic heat expansion coefficient β:
g'z
'p
dt
dwρρ −
∂
∂−=
After subtracting the hydrostaticprofile we obtain:
T
'T'−=
ρ
ρ
1−= Tβ
:p constIf =
2014.02.11.
5
Acoustic filteringThe density need to depend on the pressure, but
we need to eliminate the acoustic waves. (Acoustic effects require very small time stepping.)
)z(ρρ ≅
0=⋅∇ )v( ρ
0=∂
∂+
∂
∂+
∂
∂+
∂
∂
z
w
y
v
x
u
t
ρρρρThe continuity equation for compressible fluid:
Since the average density is a function of the altitude, this is a more complex continuity equation we normally use in incompressible flow models.
Coriolis force
x
y
N
S
Ω
wvu
kji
vC zyx ΩΩΩΩ 22 −=×−=
xyz
zxy
yzx
vuC
uwC
wvC
ΩΩ
ΩΩ
ΩΩ
22
22
22
−=
−=
−=
0≅
−=
≅
z
y
x
C
fuC
fvC φΩ sinf 2=in which
)sin(
)cos(
z
y
x
φΩΩ
φΩΩ
Ω
=
=
= 0
φ the geographic latitude
Inertial forces in a rotating frame:- Centrifugal force- Coriolis force
― Taken into account in g.
― Effects only moving bodies.
the average of w is 0.
the buoyancy forceprevails in vertical
flows
Geostrophic wind
0=∂
∂+
∂
∂
y
pv
x
pu gg 0=∇⋅ pvg
rthus
Steady flow with the assumptions below:
gz
p
fuy
p
fvx
p
g
g
−∂
∂−=
−∂
∂−=
+∂
∂−=
ρ
ρ
ρ
10
10
10 No convective acceleration:
streamlines are approximated with parallel lines. Strain stresses are neglected.
Hydrostatic equilibrium in z direction.
On the North hemisphere:
The pressure gradient is perpendicular to the direction of the equilibrium flow.
... self-consistent specification of the boundary conditions.
2014.02.11.
6
Gradient windSame as the geostrophic wind excepting, that circular streamlines are assumed.
+__ +
cyclone anticyclone
On the North hemisphere:g
z
p
fuy
p
y
vv
x
vu
fvx
p
y
uv
x
uu
gg
gg
g
gg
gg
g
−∂
∂−=
−∂
∂−=
∂
∂+
∂
∂
+∂
∂−=
∂
∂+
∂
∂
ρ
ρ
ρ
10
1
1The centrifugal must be taken into account:
Problem #7.2
a) Calculate the below non-dimensional pressure gradient for a gradient windin cylindrical system of coordinates:
gvf
r
p
ρ∂
∂
r is the distance from the center of the cyclone and vg=f(r).
b) What is the magnitude of the non-dimensional pressure gradient for a geostrophic wind?
To the solution
Ekman spiral
The phenomenon is described by a model similar to the geostrophic wind model, butforces rising from the turbulent stresses must be taken into account:
∂
∂
∂
∂+
∂
∂
∂
∂=
3m
Nj
z
vv
zi
z
uv
zF ttturb
rrr
45°
vgpressure gradient
~30 m
~100 m
Wind direction changes rapidly at the top of the boundary layer (in the outer layer).When approaching the ground the wind direction turns towards the decreasing pressure.
The solution for a medium latitude caseon the North hemisphere:
2014.02.11.
7
Velocity magnitude in the boundary layer
The Monyin-Obuhovprofile:
+=
L
z
z
zln
u
uβ
κ 0*
1z0: roughness height;κ: Von Kármán const;L: M-O scale.
pc
H
uL
ρκ 0
2*
T
g−=
H0: heat flux(H0 ~ -dΘ/dz)
+ _
Two important effects must be taken into account: surface roughness and thermal stratification.
=
νκ*
*
91 uzln
u
u
The profile of a constant density flow past a flat plate for comparison:
We can note that, the Reynolds number does not count in atmospheric flows.
Summary
• Thermal stratification;• Adiabatic compression and expansion
due to vertical flows;• Variation of density in vertical flows
due to the hydrostatic pressure;• Coriolis force;• Turbulence in stratified medium;• Moisture transport and phase
changes;• Surface energy balance involving the
radiation heat transport, the heat storage and a number of other complex phenomena.
We
have
touc
hed
upon
thes
e to
pics
Most important atmospheric flow related physical phenomena beyond the scope of basic level fluid mechanics:
CFD based atmospheric simulations
Gergely Kristóf Ph.D., Miklós Balogh, Norbert Rácz29-th March 2009.
2014.02.11.
8
model conversion interface
Advantages of a CFD based model
grid refinement
• Either the surface geometry isdescribed in high details or (alternatively) meso-scaleeffects are taken into account.
• Bidirectional interface is a source of numerical errors eg.it can cause partial reflection.
Meso scale model
CFD
CFD(with some changes)
• Higher accuracy• No limits on geometrical precision• Flexible meshing • Advanced turbulent models• Easy customization• Advanced pre- and post
processing tools
Methodology
ANSYS FLUENT + transformation system
+ customized source terms
Mathematical description
z~,~,p~,T~
,~ vρTransformed variables
0=⋅∇ v~
( ) ( ) ( ) Fgτvvv +−+⋅∇+−∇=⊗⋅∇+∂
∂000 ρρρρ ~p~~~~
t
( ) ( ) ( ) Ttpp ST~
KT~
c~T~
ct
+∇⋅∇=⋅∇+∂
∂00 ρρ v
( ) ( ) kbkk
t SGGkk~kt
+−++
∇⋅∇=⋅∇+
∂
∂ερ
σ
µρρ 000 v
( ) ( ) ++
−+
∇⋅∇=⋅∇+
∂
∂
εν
ερερε
σ
µερερ
ε kCSC~
t
t2
201000 v
( )000 TT~~ −−= βρρρ
εεεε
SGCk
C b ++ 31
Customized volume sources
2014.02.11.
9
zTT 0 γ−=γ
γ−=
R
g
0
00
T
zTpp
z
0 e ζ−ρ=ρ
m100/C65.0 °=γ
K15.288T0 = Pa1001325.1p 5
0 ⋅=3
0 m/kg225.1=ρ
2553.5)R/(g =γ14 m10 −−=ζ
Standard ISA profile Approximate profile
Error bound is within
0.4% below 4000 m.
Equilibrium profiles up to the height of 11 km
TTT~
T 0 +−=
ρ+ρ−ρ=ρ 0~
( )z~1Ln1
z ζ−ζ
−=
z0 ew~w~wζ=
ρ
ρ=
Transformation expressions
pp~epp~p z
0
+⋅=+⋅ρ
ρ= ζ−
Summary of source terms
In the energy equation:
In the transport equation of turbulent kinetic energy
In turbulent dissipation equation:
In momentum equation: Jw~fvSu l00 ρρ −=
fuSv 0ρ−=
φΩ sinf 2=
φΩ cos2=l
( ) ( )( ) ( )20
100
120 1 w~p~JJugTT
~JuJSw ρζρβρ −++−+−= −−
ll
( )Jw~cSJS pT γΓρΘ −−= 0
( )γΓµ
β −−=t
tk
PrgS
( )γΓµ
βε
εεε −−=t
t
Prg
kCCS 31
( ) 11
−−= z~J ζ
2014.02.11.
10
Related publications[1] Kristóf G, Rácz N, Balogh M: Adaptation of Pressure Based CFD Solvers for Mesoscale Atmospheric Problems,
Boundary-Layer Meteorol, 2008.[2] N.Rácz, G.Kristóf, T.Weidinger, M.Balogh: Simulation of gravity waves and model validation to laboratory
experiments, CD, Urban Air Quality Conf. Cyprus, 2007.[3] G.Kristóf, N.Rácz, M.Balogh: Adaptation of pressure based CFD solvers to urban heat island convection
problems, CD, Urban Air Quality Conf. Cyprus, 2007.
[4] G.Kristóf, N.Rácz, Tamás Bányai, Norbert Rácz: Development of computational model for urban heat island convection using general purpose CFD solver, ICUC6, 6-th Int.Conf.on Urban Climate, Göteborg, pp. 822-825., 2006.
[5] G. Kristóf, T. Weidinger, T. Bányai, N. Rácz, T.Gál, J.Unger: A városi hősziget által generált konvekció modellezése általános célú áramlástani szoftverrel - példaként egy szegedi alkalmazással, III. Magyar Földrajzi Konferencia, Budapest, 2006., Bp, CD
[6] Kristóf G., Rácz N., Bányai T., Gál T., Unger J., Weidinger T.: A városi hősziget által generált konvekció modellezése általános célú áramlástani szoftverrel− összehasonlítás kisminta kísérletekkel A 32. Meteorológiai Tudományos Napok előadásai. Országos Meteorológiai Szolgálat, Bp., 2006
[7] Dr. Lajos T., Dr. Kristóf G., Dr. Goricsán I., Rácz N.: Városklíma vizsgálatok a BME Áramlástan Tanszékén, hősziget numerikus szimulációja VAHAVA projekt (A globális klímaváltozás: hazai hatások és válaszok) zárókonferenciája Bp. CD, 2006
[8] Rácz N. és Kristóf G.: Hősziget cirkuláció kisminta méréseinek összehasonlítása saját fejlesztésű LES modellel Egyetemi Meteorológiai Füzetek No. 20 ELTE Meteorológiai Tanszék, Bp. 173-176, 2006.
[9] M. Balogh, G. Kristóf:Automated Grid Generation for Atmospheric Dispersion Simulations, pp.1-6., MICROCAD konferencia, Miskolc, 2007.
Two validation examples
Comparison with the results of water tank experiments
Experimetal setup
Uniformly stratified salt water:ρ= 1.03- 1.00 g/cm3
Typical towing speeds:U = 1-15 cm/s
Brunt-Vaisala frequency range:N= 1.09-1.55 1/s
Reexp ≈ 102-103
Studied obstacle heights:h= 20mm, 40mm
Gyüre, B. and Jánosi, I.M., 2003. Stratified flow over asymmetric and double bell-shaped obstacles. Dynamics of Atmospheres and Oceans 37, 155-170.
z
gN
∂
ρ∂
ρ−=
2014.02.11.
11
z(x) = a exp(-b|x|2γ )different parameterset for positive andnegative x
Numerical mesh
U/Nh = 1.4
U/Nh = 0.3
Gravity wavesGyüre, B. and Jánosi, I.M., 2003. Stratified flow over asymmetric and double bell-shaped obstacles. Dynamics of Atmospheres and Oceans 37, 155-170.
Simulation vs. experimental data
Wave length Amplitude
2014.02.11.
12
A.Cenedese, P.Monti: Interaction between an Inland Urban Heat Island and a Sea-Breeze Flow: A Laboratory Study, 2003.
Experimental setup
920 k prismatic cells
Streamlines colored byvelocity magnitude
Fine structure of the thermal boundary layer
Large eddy simulation in FLUENT
PIV results(Cenedese &Monti 2003)
CFD results
Thermal convectionA.Cenedese, P.Monti: Interaction between an Inland Urban
Heat Island and a Sea-Breeze Flow: A Laboratory Study, 2003.
2014.02.11.
13
Comparison of velocity and temperature profiles
PIV
CFD
PIV
CFD
Some more application examples
Full scale simulations
Meso scale atmospheric dispersion
Orography of Pilis mountain
Evolution of surface concentration
2014.02.11.
14
Wind speed: 3m/sInjection velocity: 5 m/s
Chimney height 180 mStandard (stable) temperature profile
Micro-scale atmospheric dispersionStreamlines colored by temperature
Kelvin-Helmholtz instability
Com domain: 25 km x 5.5 kmTemperature difference 20 °C
Satellite image about Guadalupe island
First CFD results
Von Kármán vortices behind a volcanic island
2014.02.11.
15
Targeted application areas• Local circulation modeling:
– Urban heat island convection, ventilation of cities;
– See breeze;– Valley breeze.
• Power generation and pollution control:– Assessment of wind power potential,
optimization of wind farms;– Plumes emitted by cooling towers
and chimneys;– Dispersion of pollutant in the urban
atmosphere.• Research of meteorological phenomena:
– Gravity waves;– Cloud formation;– Flow around high mountain.
• Simulation of disasters:– Large scale fires (e.g. in forest
fires or town fires);– Volcanic plumes.
top related