Mathematical modeling for atmospheric turbulence Haroldo Fraga de Campos Velho Laboratório de Computação e Matemática Aplicada – LAC Instituto Nacional de Pesquisas Espaciais – INPE E-mail:
[email protected] Web-page: http://www.lac.inpe.br/~haroldo
Haroldo F. de Campos Velho (LAC-INPE) Senior Researcher / Scientific computing RESEARCH FIELDS: • Inverse problems (many applications, improving methodologies) • Data assimilation: New method: based on neural network • Atmospheric turbulence parameterization: Last results: Taylor’s approach for turbulence in clouds New model for convective boundary layer growth Cosmological evolution as turbulent-like dynamics
Presetation outline • Turbulence and Geoffrey Ingram Taylor • Turbulence representation - PBL: convective, stable, neurtral - Transition boundary layers - Meso-scale BRAMS model with new parameterization
• Turbulence in the cloud (strato-cumulus) • Counter-gradient term • Parameteriization considering intermittency • Final remarks
Rerences: • G.I. Taylor (1921): Diffusion by Continous Movements, Proceedings London Mathematical Society, 20, 196-211.
• G.A. Degrazia, O.L.L. Moraes (1992): Boundary-Layer Meteorology, 58, 205-214 (SBL). G.A. Degrazia, H.F. de Campos Velho, J.C. Carvalho (1997): Beiträge zur Physic der Atmosphäre, 70 (1), 57-64 (CBL). G.A. Degrazia, A. Goulart, D. Anfossi, H.F. de Campos Velho, P. Lukaszcyk (2003) Il Nuovo Cimento, 26 C(1), 39-51 – residual boundary layer. • H.F. de Campos Velho et al. (2001): Physica A, 295, 219-223 – intermittency. • G.P. Almeida, A.A. Costa, H.F. de Campos Velho, J.C.P. de Oliveira (2006): Atmospheric Resarch, 80(1), 105-132 (cloud) • A. B. Nunes, H.F. de Campos Velho, P. Satyamurty, G.A. Degrazia, • A. G. Goulart, U. Rizza (2010): Boundary-Layer Meteorology, 58, 205-214 – PBL growth.
Old ideas on turbulence • Leonardo da Vinci (~ 1510): ‘‘... thus the water has eddying motions, one part of which is due the principal current, the other to the random and reverse motion.’’
• Osborne Reynolds (1883): flow decomposition
! ! ! ⎧ v( r , t ) = v (r , t ) + v' (r , t ) ⎨ ! ⎩v (r , t ) : average flow (similar to laminar flow)
• Lewis Fry Richardson (1922): - Cascade: ‘‘Big whirls have little whirls ...’’ - Universality: statistical behaviour is independent of external geometry, fluid nature, mecanism of injection of energy, ...
• G.I. Taylor (1921): statistical theory
Old ideas on turbulence • Leonardo da Vinci (~ 1510): ‘‘... Então a água tem movimentos vorticais, uma parte das quais é devido a corrente principal e outra ao movimento reverso aleatório.”
Old ideas on turbulence • Osborne Reynolds (1883): flow decomposition
Old ideas on turbulence • Osborne Reynolds (1883): flow decomposition
Old ideas on turbulence
Idéias antigas da turbulência • Lewis Fry Richardson (1922): o Cascata de energia: “Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity. o Universalidade: - o comportamento estatístico é independente da geometria, - natureza do fluido - mecanismo de injeção de energia ... L. F. Richardson: pesquisador inglês, matemático, físico, meteorologista, psicólogo e pacifista. Pioneiro da moderna previsão numérica do tempo e aplicação de técnicas similares ao estudo das causas da guerra e sua prevenção. Pioneiro na técnica dos fractais, solução de equações lineares, método de extrapolação de Richardson.
Idéias antigas da tubulência • Lewis Fry Richardson (1922): o A moderna previsão numérica do tempo, segue o esquema idealizado por Richardson. o “Computadores” para Richardson significava homens que calculavam ... (profissional de contabilidade é contador, aquele que trabalha é o trabalhador, aquele que faz computações é o computador)
Idéias antigas da turbulência • Lewis Fry Richardson: o Número de Richardson (razão entre energia potencial e cinética):
gh Ri = 2 u o Extrapolação de Richardson: - Análise numérica: é um método de aceleração sequencial - Aplicações: (i) integração de Romberg (ii) algortimo de Bulirsch–Stoer para solução de EDO
From the wikipedia
A Teoria de Geoffrey Ingram Taylor • Fenômenos associados com o nome de G. I. Taylor * Taylor cone * Taylor dispersion * Taylor number * Taylor vortex * Taylor–Couette flow * Rayleigh–Taylor instability * Taylor-Proudman theorem * Taylor-Green vortex * Taylor microscale
Introduction to the Taylor’s theory Turbulence and Reynolds Hypothesis: The dynamical (turbulent) variable can be understood a sum of a mean stream and a deviation: ϕ = ϕ + ϕ ' ; being ϕ a mean stream (similar to laminar flow), and ϕ ' a fluctuation (deviation) with zero mean: ϕ ' = 0 . Dealing with average process the Navie-Stokes and the diffusion equations become
dvi ∂p ∂ ρ =− − 2ε ikj Ωk v j + ρgδ 3i + ∑ σ ij − ρ vi ' v j ' dt ∂xi ∂ x j j
[
( )
∂ v jα dα ∂α ∂ = +∑ =∑ dt ∂t j ∂x j j ∂x j
]
⎡ ⎤ ∂α − v j 'α '⎥ ⎢K m ∂x j ⎢⎣ ⎥⎦
⎡ ⎤ ∂α ∂α First order closure: ⎢ K m − v j 'α '⎥ ≈ − v j 'α ' = K ij ∂x j ∂x j ⎢⎣ ⎥⎦
Reynolds’ fluxes
eddy diffusivity
Fick’s Law: From a mass differential balance:
ε
c( x + ε ) − c( x) ≈ ε
ΔA u1
u2
considering ε as a small length. Assuming ε ≈ lm, where lm is the characterisc length, the mean free path in Brownian movement and gas theory. For net flux F:
F = F2 − F1 = −u m [c( x + l m ) − c( x)] ∂c = −(u m l m ) ∂x
mass concentration : c ≡ g cm -3 volume mass time m Flux : F = = = u.c g s -1 cm -2 area At net flux : F = F1 − F2 = [u1c1 − u 2 c2 ] ≈ u m [c1 − c2 ]
[
]
[
(mass or particles)
∂c ∂x
]
above it was assumed : u1 ≈ u 2
diffusivity
Generalizing:
! ! F = − D ∇c
Taylor’s equation Particle moviments:
Remembering:
v(t ) =
dx(t ) ⇒ x(t ) = ∫0t v(t ' ) dt ' dt
diffusivitty = v(t ).x(t )
v(t ) x(t ) = x(t ) taking the average:
- then:
dx(t ) d ⎡ 1 2 ⎤ t = ⎢ x (t )⎥ = ∫0 v(t )v(t ' ) dt ' dt dt ⎣ 2 ⎦
(self-)correlation function
d ⎡1 2 ⎤ t x (t )⎥ = ∫0 v(t )v(t ' ) dt ' ⎢ dt ⎣ 2 ⎦
The correlation function R(τ) in a homogeneous statistical process is independent of the parameter ‘‘t’’: t = t’ + τ. In this case (homogeneous) R(τ) is given by
R(τ ) = v(t ' )v(t '+τ ) = v 2 ρ (τ ) Substituting and integrating by parts:
(Obs. : R(0) = v 2 ⇒ ρ (0) = 1) x 2 (t ) = 2v 2 ∫0t (t − τ ) ρ (τ ) dτ
Deriving the Taylor’s equation From the average displacement:
2
x (t ) = 2∫
t τ
v(t )v(t ' ) dt 'dτ ∫ 0 0
2 t τ = 2v ρ (t ' )dt ' dτ 0 0
∫∫
Integrating by parts (being: dw = dτ and u (τ ) =
x
2
2 t τ (t ) = 2v ρ (t ' )dt ' dτ 0 0
∫∫
∫0 ρ (t ' )dt ' ⇒ du dτ = ρ (τ )):
t ⎫⎪ t ⎤ = v ⎨ τ ∫ ρ (t ' )dt ' − ∫ τρ (τ )dτ ⎬ ⎥⎦ 0 0 ⎪⎩⎢⎣ 0 ⎪⎭
⎪⎡ 2⎧
τ
t ⎫ ⎤ = 2v ⎨ t ∫ ρ (τ )dτ − 0 − ∫ τρ (τ )dτ ⎬ ⎥⎦ 0 ⎩⎢⎣ 0 ⎭ 2 ⎧⎡
t
τ
2 t = 2v (t − τ ) ρ (τ )dτ 0
∫
Taylor’s equation (spectral version) Fourier transform:
Φ(ω ) = 1 2
R (τ ) =
1 π
∫
∫ +∞
−∞
+∞
R (τ ) e iωτ d τ −∞
Φ(ω ) e −iωτ d ω ⇒ R (0) =
1 2
∫
+∞ −∞
Φ(ω ) d ω
If the correlation is independent of parameter τ (homogeneous turbulence):
v(t )v(t + τ ) = v(0)v(τ ) = v(−τ )v(0) ⇒ R(τ ) = R(−τ ) (even function) the spectral equation becomes:
Φ(ω ) =
1
π
odd function
[cos ωτ + sin ωτ ]dτ = 2 ∫0∞ R(τ ) cos ωτ dτ
+∞ ∫−∞ R(τ )
π
In meteorology, the frequency is given in Hz (s-1) instead rad. s-1: n = ω/2π. Hence, the spectra will be: S(n) = 2π Φ(2πn). Then, the velocities variances are:
vi2 = σ i2 = R(0) = ∫0∞ S Li (n) dn = ∫0∞ nS Li (n) d (ln n)
Considering the 1st form of Taylor’s equation
xi2
⎡∞ S Li (n) ⎤ (t − τ )⎢ ∫ 2 cos(2π nτ )dn⎥ dτ = ⎢⎣ 0 σ i ⎥⎦ sin 2 (nπ t ) 2 ∞ = σ i ∫0 FLi (n) dn 2 (nπ ) xi2 = 2v 2 t 2 2σ i ∫ 0
From the Batchelor’s relation:
Kαα =
d ⎡1 2 ⎤ xi ⎥ ⎢ dt ⎣ 2 ⎦
⎛ α = x, y , z ⎞ ⎜⎜ ⎟⎟ ⎝ i = u , v, w ⎠
the fundamental equation is obtained:
Kαα
σ i2 ∞ sin (2π nt ) = F ( n ) dn ∫0 Li 2 2π n
The Lagrangian form for the eddy diffusivity.
∫
t 0
(t − τ ) ρ (τ ) d τ
Using the Gifford-Hay & Pasquill’s assumption:
a relation can be obtained for the spectra:
ρ Li ( β iτ ) = ρi (τ ) ;
nFLi (n) = β i nFi ( β i n)
β i = TLi Ti
and a Eulerian form for eddy diffusivity follows:
Kαα
σ i2 β i2 ∞ sin (2π nt β i ) = dn ∫0 Fi (n) 2 2π n
An asymptotic form can also be derived (long travel times, t → ∞):
Kαα =
σ i2 β i Fi (0) 4
Deriving the Taylor’s equation
xi2 = σ i2 ∫
∞
0
Análise assintótica:
sin 2 (nπ t ) FLi (n) dn 2 (nπ )
t →0
t →∞
Brownian motion: Einstein’s relation distance between two particles ~
t
Turbulence parameters: From the Taylor’s theory is easy to show that:
Now, using well establishied expressions:
xi2 = vi2 t 2
for t → 0
xi2 = vi2 TLi t
for t → ∞
Kαα =
σ i2
d ⎡1 2 ⎤ 2 x ( t ) = σ i i TLi = σ i li ⎢ ⎥ dt ⎣ 2 ⎦ ∞
= ∫ S i (n) dn 0
⎡ σ β F ( 0) ⎤ Eddy diffusivity : K zz = σ i ⎢ i i i ⎥ 4 ⎣ ⎦ β F ( 0) Lagrangian time - scale : TLi = i i 4 σ β F ( 0) Mixing length : li = i i i 4
Degrazia et al. (1998): BLM, 86, 525-534
Teoria da camada limite • Paradoxo de D’Alembert (1752): A força de arraste sobre um corpo submerso é nula!
Teoria da camada limite • Solução do paradoxo: a teoria da camada limite Ludwig Prandtl (físico alemão) sugeriu em 1904 que os efeitos de uma fina camada limite viscosa poderia ser a razão do arraste.
Teoria da camada limite • Solução do paradoxo: a teoria da camada limite Theodore von Kármán (húngaro) : esteira de vórtices.
Teoria da camada limite • Solução do paradoxo: a teoria da camada limite Theodore von Kármán (húngaro): esteira de vórtices. Exemplos no oceano (costa do Chile: próximo as ilhas Juan Fernandez)
Hipóteses de Kolmogorov (K-41): • Hipótese 1: Para Re muito altos, os movimentos de pequena escala da turbulência são estatísticamente isotrópicos. • Hipótese 2: Para Re muito altos, a estatistica das pequenas escalas são determinadas unicamente pela viscosidade (ν) e pela taxa de dissipação de energia (ε). Com estes parâmetros uma escala de comprimento é definida 14
η = (ν ε ) 3
É a escala de comprimento de Kolmogorov (η). • Hipotese 3: Para Re muito altos, as estatísticas das escalas no intervalo η INPE/ISIATA
Multifractal Approach for Dispersion Models: An evaluation with LES data
Large Eddy Simulation Dispersion Data Closure hypothesis (Smagorinsky model)
⎡ ∂ui
τ ij = − K M ⎢
⎢⎣ ∂x j
Eddy viscosity
K M = c1le1/ 2
Eddy diffusivity
τ ci = − K c
Schmidt number
K Sc = M KC
∂C ∂xi
+
∂u j ⎤ ⎥ ∂xi ⎥⎦
LES filtered conservation equation ∂u C ∂τ ∂C = − i − ci + S ∂t ∂xi ∂xi
Last equation is solved numerically with pseudo-spectral techniques (FFT for any horizzontal derivative and Crank-Nicholson for any vertical derivative)
LES filtered diffusion equation ∂u C 1 ∂ ⎡ ∂C ∂C ⎤ =− i + K ⎢ M ⎥+S ∂t ∂xi Sc ∂xi ⎣ ∂xi ⎦ :>INPE/ISIATA
Multifractal Approach for Dispersion Models: An evaluation with LES data
Results and Comparison
Ground Level Concentration 20:500 secs
60
0
0
1000
Ground Level Concentration 700:2400 secs
6
0 0
1000
5000
:>INPE/ISIATA
Multifractal Approach for Dispersion Models: An evaluation with LES data
Results and Comparison
Mean Concentration 20:200 secs
:>INPE/ISIATA
Multifractal Approach for Dispersion Models: An evaluation with LES data
Results and Comparison
Mean Concentration 700:2400 secs
:>INPE/ISIATA
Multifractal Approach for Dispersion Models: An evaluation with LES data
Results and Comparison
Mean Concentration 20:3000 secs
:>INPE/ISIATA
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