1、1,Lecture 4: Equations governing Geophysical Flows with turbulence effects 第4章 考慮湍流效應(yīng)的地流控制方程 及邊界定義和渦度方程,2,4.1 時(shí)間平均的湍流模型,3,Laminar vs turbulent flow :Turbulence is flow dominated by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar.,4,5,Osborn

2、e Reynolds (1842-1912),Osborne Reynolds was born in Belfast, United Kingdom and died in Watchet in Somerset, England. He graduated from Cambridge University in 1867 after studying mathematics. In 1868 he became a professor of engineering at Owens College in Manchester,His early work was on magnetism

3、 and electricity but he soon concentrated on hydraulics and hydrodynamics. He also worked on electromagnetic properties of the sun and of comets, and considered tidal motions in rivers. After 1873 Reynolds concentrated mainly on fluid dynamics and it was in this area that his contributions were of w

4、orld leading importance.,Reynolds famously studied the conditions in which the flow of fluid in pipes transitioned from laminar to turbulent. From these experiments came the dimensionless Reynolds number for dynamic similarity . Reynolds also proposed what is now known as Reynolds-averaging of turbu

5、lent flows, where quantities such as velocity are expressed as the sum of mean and fluctuating components.,6,中文名稱： 雷諾數(shù) 英文名稱： Reynolds number 定義1： 在流體運(yùn)動(dòng)中慣性力對(duì)黏滯力比值的無量綱數(shù)Re=UL/ 。其中U為速度特征尺度，L為長(zhǎng)度特征尺度，為運(yùn)動(dòng)學(xué)黏性系數(shù)。,雷諾數(shù)小，意味著流體流動(dòng)時(shí)各質(zhì)點(diǎn)間的粘性力占主要地位，流體各質(zhì)點(diǎn)平行于管路內(nèi)壁有規(guī)則地流動(dòng)，呈層流流動(dòng)狀態(tài)。雷諾數(shù)大，意味著慣性力占主要地位，流體呈紊流流動(dòng)狀態(tài)，一般管道雷諾數(shù)Re2000為層

6、流狀態(tài)，Re4000為紊流狀態(tài)，Re20004000為過渡狀態(tài)。,7,Laminar Flow is also referred to as streamline or viscous flow. These terms are descriptive of the flow because, in laminar flow, layers of water flowing over one another at different speeds with virtually no mixing between layers, fluid particles move in definite

7、and observable paths or streamlines, (3) the flow is characteristic of viscous (thick) fluid or is one in which viscosity of the fluid plays a significant part. Turbulent Flow is characterized by the irregular movement of particles of the fluid. The particles travel in irregular paths with no observ

8、able pattern and no definite layers. Turbulence is flow dominated by recirculation, eddies, and apparent randomness.,8,Exercise :,The steady Incompressible viscous laminar flow in a channel .,Assuming, v=0, w=0, impressible, steady,Write the continuity equation, momentum equation, Write the no-slip

9、boundary conditions for r=r0. Try to find the expression for the velocity u(r) . 4) What is the maximum of velocity u ? (at r=r0),x,r0,9,10,Re=0.1,Re=50,Re=105,Dark area is the area where viscous effects are important,Turbulent flows are 3D, unsteady, rotational, viscous and chaotic fluid motions wi

10、th instability and nonlinearity. Turbulence consists of eddies in a wide range of time and length scales. Larger eddies carry most of the energy and is mainly responsible for the enhanced diffusivity and stresses. Larger eddies carry small eddies, and transfer kinetic energy to smaller ones. Ultimat

11、ely, the smallest eddies dissipate into heat by the action of molecular viscosity.,11,Momentum equation for turbulent flow 湍流流動(dòng),For N-S equation in previous lectures, we didnt consider turbulence and didnt separate the variables into time-average part and fluctuation part.,12,Color shades of the ins

12、tantaneous streamwise velocity in LES of turbulent flow in plane diffuser,Spatial and temporal fluctuations exist in turbulent flow. 時(shí)間和空間的脈動(dòng),13,14,Eddies and DNS for turbulence computations,The largest eddy size (l0) is comparable to the boundary-layer thickness. The smallest eddy size (h) is Kolmo

13、gorovs scales (much larger than molecular length scales) The larger Re, the greater l0/h,Eddy velocity,Dissipation of turbulent kinetic energy,Eddy size,Direct numerical simulations (DNS) resolve the smallest eddies Wind u = 10 ms-1, n = 1.5 x 10-5 m2s-1, h = 1.5 x 10-6 m 1 km3 volume: we need (1000

14、/1.5 x 10-6)3 = 2.963 x 1026 grid points! A 2 GHz single-processor computer: 4.695 x 109 years! (exactly the age of the Earth!),15,Geophysical flow is large-Reynolds-number turbulent flow.Reynolds-averaged Navier-Stokes equations for impressible flow雷諾平均方程,Large-scale time-averaged flow,Turbulent fl

15、uctuations,Total instantaneous,Substituting into the governing equations,16,Reynolds stress represents an average flux of x-momentum due to small-scale motions across the y-surface,Reynolds stress tensor (雷諾應(yīng)力) How to model them to close the equations?,17,18,New governing equation for geophysical fl

16、ows with turbulent stress,今后簡(jiǎn)化方程, 默認(rèn)為是雷諾時(shí)間平均項(xiàng),How to model Eddy (turbulent viscosity) coefficient AM,To model the Reynolds stresses, we imagine that they can be related to large-scale motions through eddy-viscosity.,AM105 m2s-1 for free atmosphere compared to 1.5 x 10-5 m2s-1 for kinematic viscosity

17、,19,Boussinesqs eddy viscosity model,Effect of turbulence for momentum/scalar/energy transport.,湍流中湍流輸運(yùn)粘性輸運(yùn) (AM is turbulent kinematic viscosity),20,Turbulent momentum transport,Turbulent energy transport,Water vapor or pollutant transport,湍流交換系數(shù),湍流動(dòng)量交換系數(shù),湍流熱量輸運(yùn)通量密度,湍流水汽通量密度,湍流動(dòng)量輸運(yùn)通量密度,湍流動(dòng)量交換系數(shù)AM的各向

18、異性(變化劇烈) For atmosphere near ground, vertical AV10 m2s-1 and Horizontal AH104KV They decrease as z increase, For seawater, AV10-3-10-1 m2 s-1 ; AH105KV,21,u,u,Horizontal Reynolds stress,Vertical Reynolds stress,垂直方向受下面剛體邊界和 重力場(chǎng)的作用, 都抑制垂直方向的湍流發(fā)展,22,Six components of Reynolds stresses,23,governing equ

19、ation for geophysical flows with turbulent stress and Boussinesq approximation (2),今后簡(jiǎn)化方程, 默認(rèn)為是雷諾時(shí)間平均項(xiàng),24,4.2 量綱分析,Scale analysis in geophysical flows (z),HL; WU, V,25,10,For large scale,Hydrostatic balance,Scale analysis in geophysical flows (x ,y),HL; WU, V; For large scale WL/(UH)1 or 1,26,1,for

20、large scale motion,Vertical/horizontal Ekman number,27,羅斯貝數(shù) (Ro, Rossby number):慣性力/科氏力 The ratio of inertial force to Coriolis force Ro=U/(L*f)=(U2/L)/(f *U); f = 2 sin Ro1, Coriolis force can be neglected; Ro1，both are important; Ro1，inertial force can be neglected,Scales when rotation effect is i

21、mportant,大尺度 地轉(zhuǎn)平衡關(guān)系式 (二力平衡),Geostropic flow (地轉(zhuǎn)流2) between Coriolis term and pressure gradient term,31,10,Ekman number 埃克曼數(shù),Ek=n/(2f H2 ) = (n U/H2 )/(2f U) 粘性力/科氏力 The ratio of viscous forces in a fluid to Coriolis force arising from planetary rotation,H is vertical length scale of a phenomenon, n

22、is the kinematic viscosity運(yùn)動(dòng)粘性系數(shù) f = 2 sin is the Coriolis frequency, the latitude,Ek1 viscosity can be neglected Ek1 viscosity is important For ocean motion of L=103 km, 分子粘性m=10-6 m2/s, Ek10-14,湍流Ekman number,For geophysical flows, EH is small. For ocean, with an eddy viscosity AH as large as 102

23、m2/s(much larger than fluid viscosity 10-6 m2/s), = 7.3 105 s1 and L = 10 km, vertical EH = 1.4 106.,33,Ek=湍流粘性力/科氏力,Only consider friction (turbulent shear) in Ekman layers,The Ekman thickness, d, of a thin layer is such that the Ekman number is on the order of one at that scale, allowing friction

24、to be a dominant force:,34,H is the fluid height of the motion,For ocean at mid-latitude (104 s1), eddy viscosity AV 102 m2/s ，fluid thickness H,as H=10m,as H=100m,Ek=湍流粘性力/科氏力,Ekman number,Ekman depth/thickness (Ekman厚度) in atmosphere and sea flows,35,For ocean 海洋, with an eddy viscosity AV as larg

25、e as 102 m2/s ，= 7.3 105 s1 d10m H = 100 m,For atmosphere 大氣, with an eddy viscosity AV as large as 5 m2/s ，= 7.3 105 s1 d103m=1km H = 10 km,36,?？寺鼘樱‥kman layer）是流體中壓力梯度力、科氏力和湍流粘性力三力平衡的一層, 粘性力不可忽視。由瑞典海洋學(xué)家?？寺岢?。?？寺鼘永碚撨m用于許多地區(qū)，包括大氣層底部（接近地球表面和海洋），大洋底部（海床附近）和表層海水（海氣界面附近）,研究邊界層時(shí), 需保留湍流粘性項(xiàng) 大氣1km, 海洋10m量級(jí) A

26、lthough the Ekman number is small, indicating that the dissipative terms in the momentum equation may be negligible, these need to be retained when it is considered that vertical friction creates a very important boundary layer.,37,Richardson number,Ri10, 密度差(熱力因素)占主導(dǎo)地位 Ri0.1-10,密度差和水平慣性力都重要 Ri0.1,水

27、平慣性力主導(dǎo)，溫度差和密度差可忽略,38,4.3 邊界條件和初始條件,39,法向無滲透邊界條件 w=dz/dt=db/dt,運(yùn)動(dòng)邊界條件,40,法向無滲透邊界條件,41,自由表面邊界+底部邊界+側(cè)邊界,42,自由表面 邊界條件,43,動(dòng)力邊界條件 （壓力-pressure）,44,動(dòng)力邊界條件 （湍流剪應(yīng)力-turbulent shear stress）,看是否考慮邊界層粘性效應(yīng),或者用AV 表示,45,Open boundary,46,在海氣交界處最復(fù)雜 海溫異常，能影響氣候變化,47,Chapter 4.4 Vorticity equation 渦度方程,48,Vorticity(渦度)

28、and divergence (散度) are two properties of geophysical flows,渦度(Vorticity)是地球物理流體力學(xué)中的一個(gè)基本概念，它描述一個(gè)水質(zhì)點(diǎn)在空間中如何旋轉(zhuǎn)，并與各種海洋和大氣運(yùn)動(dòng)現(xiàn)象緊密聯(lián)系。從行星尺度的波動(dòng)到中小尺度渦旋，其垂直渦度分量尤為重要。,49,渦量與流體微團(tuán)自身的旋轉(zhuǎn)角速度成正比，而與流體微團(tuán)重心圍繞某一參考中心作圓周運(yùn)動(dòng)的角速度無關(guān)。流動(dòng)是否有旋與流體質(zhì)點(diǎn)的運(yùn)動(dòng)軌跡無關(guān)。一個(gè)作圓周運(yùn)動(dòng)的流體微團(tuán)可能渦量為零,速度的渦度（Vorticity)：描述速度的旋轉(zhuǎn)性,To check this flow field:V=ay i

29、+0 j +0 k, 有旋還是無旋 ?,50,夸父追日，渦度為0 胡旋舞，有渦度,渦度方程推導(dǎo)和物理意義,51,渦度的生成、發(fā)展和減弱均可用其垂直渦度的變化加以描述。垂直渦度方程（簡(jiǎn)稱渦度方程）描述流體質(zhì)點(diǎn)在運(yùn)動(dòng)中的渦度變化，它是地球物理流體力學(xué)中的一個(gè)基本方程，被廣泛應(yīng)用于各種現(xiàn)象的分析和理論研究。,渦度是流體微團(tuán)繞其內(nèi)部一瞬時(shí)軸作旋轉(zhuǎn)運(yùn)動(dòng)的角速度的二倍 (如何證明？),For earth rotation, the absolute vorticity,旋轉(zhuǎn)坐標(biāo)系中, 絕對(duì)渦度和相對(duì)渦度的關(guān)系,52,Vorticity equation (推導(dǎo)),53,梯度(gradient),壓力梯度,

30、 溫度梯度, 密度梯度: 其方向?yàn)閴毫ψ兓羁斓姆较颍?標(biāo)量(P, T) 的梯度為矢量,or,標(biāo)量梯度場(chǎng)為有(位)勢(shì)場(chǎng)(例壓力梯度場(chǎng)), 為無旋場(chǎng), 其渦度為0 或沿回線的環(huán)量(circulation)為0,55,56,The component of planetary vorticity normal to the earths surface is the Coriolis parameter, 科氏參數(shù) f = 2 sin , the latitude 羅斯貝數(shù) (Ro, Rossby number): The ratio of Ro=(U/L)/f; relative vorticity to planetary vorticity. Ro=(U2/L)/(f *U); inertial force to Coriolis force Ro=(1/f) / (L/U); time scales,57,For large scale fl