simulations on diffusion and transport...
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Simulations on diffusion and transport phenomena
・phenomena‐ Examples for transport and diffusion phenomena‐ Discuss about the common characteristics and nature among these examples
・physical models‐ Explain to the mathematical model about Brownian motion, turbulent diffusion, and turbulence transport
・numerical simulations‐ Examples for simulating one‐particle diffusion, thermal convection, and passive scalar turbulence
Field of computational applied science T. Watanabe
Homework (report) 1.Mention about the example of transport and diffusion phenomena
that you can observe at your ordinal life, engineering system, etc. Then explain about their characteristics by using the mathematics, pictures, painting, and so on.
2.Discuss about the importance and significance when the numerical simulations for examples mentioned in 1. are performed.
・using A4 size paper and summarizing within 1~2 sheets・making cover sheet and state your student number , nameand the date of submission・site for submission: 2号館4階422B室 (insert in the box)・deadline for submission July, 3(Fri.) 17:00
3.Please give your thoughts on this lecture.
Examples on transport and diffusion phenomena
• pollutant diffusion• come flying volcanic ash, yellow dust • water vapor, thermal (cloud physics)• thermal transport • combustion• chemical reaction flow• ・・・
http://www.nsc.go.jp/mext_speedi/index.html
緊急時迅速放射能影響予測ネットワークシステム(System for Prediction of Environmental Emergency Dose Information, SPEEDI)
Outflow and diffusion of a radioactive substance from a nuclear power plant at Fukushima
原子力施設から大量の放射性物質が放出されたり、あるいはそのおそれがあるという緊急時に、周辺環境における放射性物質の大気中濃度や被ばく線量などを、放出源情報、気象条件および地形データをもとに迅速に予測するシステム
Computational details on SPEEDI
Yellow dust
直径4ミクロン程度
環境省パンフレットより抜粋
Volcanic ash (Sakurajima, kagoshima)
桜島の噴火の衛星写真(Wikipediaより引用)
火山灰: 火山からの噴出物で直径2mm以下の大きさのものアイスランド島の火山噴火(2010年4月) ⇒ 経済活動への影響大
Birth of cloud
地球上の雲を概観した衛星画像(Wikipediaより引用)
雲粒: 直径1‐10μm程度⇒ 凝結、粒同士の衝突合併を繰り返し 0.2 mm程度まで成長
さらに成長すると雨粒(1mm)になり,重力により落下する.
Formation of cumulonimbus
・浮力により上昇した空気塊が凝結し、雲を作って
消滅するプロセスを繰り返しながら雲頂が高くなっていく。
・凝結の潜熱により雲内は暖められ,不安定な雲の成長が続く。鉛直方向に空気の性質は輸送される。
Complicated physical system interacting between fluid and particles with thermal transport
・ motions of air and vapor (fluid equations)・ variation of temperature (thermodynamics eq.)・ growth of droplets, deformation, merging, splitting(motions for ensemble of particles)
読売新聞 平成21年3月15日
気象庁HP:http://www.jma.go.jp/jma/kishou/know/whitep/1‐3‐1.html
Progress numerical weather prediction :Importance for understanding the meso‐scale atmospheric phenomena
・⊿T > ⊿Tc
・ simple example for thermal convectionupper plate : lower thermal sourceLower plate: higher thermal source : temperature difference
・⊿T < ⊿Tc
Thermal conduction (only) conduction+convection⇒Benard convection
TΔ
L
Thermal convection
Thermal convection
Formation of Benard cellin experiments
Simulation for turbulent thermal convectionJ. Schumacher, Phys. Rev. Lett. 100, 134502 (2008)
Common physics and nature・ Heat / mass transfer by fluid motions(transport・diffusion)・ complicated interaction between many particles and
surrounded fluid with heat transfer・ existence of motions with the wide range of temporal
and spatial scales
1.Numerical simulations for transport and diffusion of many particles in fluids
2.Numerical simulations for heat transfer model in turbulence
Let’s consider the role of fluid motions on transport/diffusion phenomena !
Nature of “flows” -turbulence-
Turbulence
• Complicated and irregular flow pattern (⇔laminar )
• Instability of flow (non‐linear)• Energy dissipation
(non‐equilibrium)• Strong ability of mixing
and transfer
Visualization of grid generated turbulent flow
Reynolds number Re : Non‐dimensional parameter chara
νUL
ReU : characteristic vel. L : characteristic lengthν: kinematic viscosity
U~2m/s, L~1m/s, ν~1.5×10‐5 m2/s
Re >> 1 : turbulent (airplane, car, etc…)Re < 1 : laminar (microbe)
例: flow around the walking man
Re ~105
・flow around the circular cylider (種子田定俊著: 画像から学ぶ流体力学(朝倉書店))
Re=1404, 大小様々な渦を含む乱雑な流れ。後流ではカルマン渦列が形成されている。
D. J. Trriton, Physical Fluid Dynamics, Oxford Sci. Pub. , pp 29
(日本流体力学会編、「流れの可視化」 朝倉書店, p155)
St Christopher and the vortex A Karman vortex in the wake of St Christopher’s heels
T. Mizota et al. Nature 404, p.226
Direct numerical simulation of turbulence
fuuuu
21 νp
ρt
0 uContinuty equation
Navier-Stokes eq.
Equations of motion for fluid
:, txu:ρ :ν
:, tp xFluid velocity Pressure Density (=const.) viscosity
zyx
kji
・Initial and boundary conditions → solving the partial differential equation
・discretization of variables hlhl tt uxuxu ,,
,2,1,0 lxΔlx l ,2,1,0 htΔht h
・differential method, spectral method, etc…
1. Vortical structures in turbulence
sωωIsosurface of
vorticity:
uω
3D turbulence
2.Vortical structures in turbulence
vorticitysmall 0 large
2D turbulence
1.Particle diffusion in fluids
Particle diffusion : Brownian motion
ttζdttdm Rvv
Drag force Random force due to the water molecules
tdttd vx
Equation of motion for Brownian particle in fluid
fluid
particle
tR
:tv :tx Position vector Velocity vector
:m Mass of particle
Temporal evolutions of Brownian particles
Trajectories of four particles2,000 particles (animation)
Particle diffusion: Brownian motionStatistical law: time dependence of mean squared displacement
Dtxtx 20 2
:representing ensemble average
ζTkD B
D: diffusion coefficient 12~ TlD
DlT 2~ー> time scale that cluster of particles extends up to the scale l
lT
Particle diffusion by turbulence
ttttζdttdm Rxuvv
,
Stokes drag force Random forcedue to water molecules
tdttd vx
Equation of motion for single particle in turbulent flow
:, txu Velocity vector at x,time t
Fluctuating randomly in space and time
t,xu
粒子 tR
Simulation on turbulent diffusion of particles
D=400 D=1000 D=4000
09.0tS 22.0tS 9.0tS
D=density of particle / density of fluid
dssvxtxt
0
0
Evaluation of diffusion coefficient
2021lim xtxt
Dt
t
t t
τdτsvsvτt
sdsdsvsvxtx
0
0 0
2
2
0
tvdttdx
τdτsvsvD
0
Diffusion coefficient
τdτsτsxussxuζTkD B
0,,
0, tRttutvζdttdvm x
ζtRttxutv ,
Molecular diffusion Turbulent diffusion
Light particle :
Diffusion by thermal motions << Diffusion by turbulent flows
smO 21310 smO 2110
(water(20℃)、particle diameter = 1μm、Re=104)
2.Heat and mass transfer in fluids
Basic equation for thermal convection
0 u
・equation of motion for fluid
Tz
Raptu
PrReRe1
22uuu
TTtT 2
PrRe1
u
advection diffusion
Buoyancy term
κ
νPr
3TgLΔRa
Prandlenumber
Rayleigh
number:体積膨張率 g:重力加速度 ν:動粘度
Rayleigh’s linear stability theoryRayleigh number:
流体中での対流に関する無次元数レイリー数が大きくなると対流が発生する。対流の擾乱が成長も減衰もしないような臨界状態を臨界レイリー数と呼ぶ。
Critical Rayleigh number:
⇒no convection
⇒onset of convection
2
322 )(k
kRaC
3TgLΔRa
cRa
5.657cRacRaRa
cRaRa
●領域のサイズ Lx=8 , Ly=1
●格子点の数 Na=160 ,Nb=20
●上下の壁の温度 T1=1 , T2=0
●レイノルズ数 Re=1
●プラントル数 Pr=1
●レイリー数 Ra=600,700,2000,10000
●すべり境界条件(上下) 周期境界条件(左右)
Numerical simulations for thermal convection
Results for above critical Ra
T V
Ra=700
t=1.0 t=1.0
t=25.0
t=30.0
t=25.0
t=30.0
t=40.0 t=40.035.2267.2
38
k
Simulation results for higher Ra numbers
Ra=2000
T V
Ra=10000
Passive scalar transport under mean gradient
・ Existence of mean scalar gradients
distribution of Temperature (T), salinity (S) inside the sea (normal direction)
104.0 CmT 101.0 mkggS
Schmit number ,Plandle number
CS
310OSC
10~1 OOPr : heat
: salinity
Passive scalar: Any motions of scalar quantitydoes not affect to fluid motions
Passive scalar under uniform mean gradient
TκTtT 2 uAdvection – diffusion eq.
Fluctuation of does not affect to fluid motion tθ ,x
Scalar field
mean gradient
xGxx tθtT ,,
G,0,0G
32 Guθκθ
tθ
u
T
z
fluctuation
計算機A 計算機B
計算機D 計算機C
Image of parallel computation
Large-scale simulationParallel computation by using 「supercomputer」
A B
D C
「Kei」 ( CCP2012,Kobe )
Large-scale simulation for turbulent transport
Mixing by turbulent vortices
103 102048 NGrid points:
Process numbers: pe=128
Total memory:2TB(=2000GB)
Computation time:5000(h)=208days
Buoyancy term=0
Vortical structures (green) and temperature sheets(blue)
Visualization of numerical results of turbulent transport
vorticity(z)(color) and scalar gradient (x‐y plane)(black)
Visualization of numerical results of turbulent transport
Structures in 2D slice
x
z
zx ,,
・z-direction
・x-direction
-1
263Rcase G
海上保安庁海洋情報部 http://www1.kaiho.mlit.go.jp/
distribution of temperature at sea surface
44
2D slice by 20483 DNS
2
0,, yx
L
45
L
5
2D slice by 20483 DNS 0,, yx
46
905
10
2D slice by 20483 DNS 0,, yx
Summary of lecture• Physics of transport and diffusion phenomena
‐Mathematical models of particle motion in fluid and its numerical simulations‐Model for heat and mass transfer in turbulent flow(Simulations for scalar field advected by fluid motion)
• Diffusion of the ensemble of particles by turbulence(molecular diffusion vs. Turbulent diffusion)
• Heat transfer by turbulence(cliff structures in temperature field, correlation with vortices )
Effective use of the transfer ability of turbulence Diffusion properties by turbulece
Importance for better understanding of the flow properties
Application for constructing more reliable simulation models (weather prediction, pollutant diffusion)
Homework (report) 1.Mention about the example of transport and diffusion phenomena
that you can observe at your ordinal life, engineering system, etc. Then explain about their characteristics by using the mathematics, pictures, painting, and so on.
2.Discuss about the importance and significance when the numerical simulations for examples mentioned in 1. are performed.
・using A4 size paper and summarizing within 1~2 sheets・making cover sheet and state your student number , nameand the date of submission・site for submission: 2号館4階422B室 (insert in the box)・deadline for submission July, 3(Fri.) 17:00
3.Please give your thoughts on this lecture.