1、1Lecture 4: Equations governing GeophysicalFlows with turbulence effects 第4章 考虑湍流效应的地流控制方程及边界定义和涡度方程24.1 时间平均的湍流模型时间平均的湍流模型3Laminar vs turbulent flow :Turbulence is flow dominated by recirculation, eddies, and apparent randomness. Flow in which turbulence is not exhibited is called laminar. 45Osborn
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中文名称：中文名称：雷诺数 英文名称：英文名称：Reynolds number 定义定义1：在流体运动中惯性力对黏滞力比值的无量纲数Re=UL/ 。其中U为速度特征尺度，L为长度特征尺度，为运动学黏性系数。 雷诺数小，意味着流体流动时各质点间的粘性力占主雷诺数小，意味着流体流动时各质点间的粘性力占主要地位，流体各质点平行于要地位，流体各质点平行于管路管路内壁有规则地流动，内壁有规则
6、地流动，呈呈层流层流流动状态。雷诺数大，意味着惯性力占主要地流动状态。雷诺数大，意味着惯性力占主要地位，流体呈位，流体呈紊流紊流流动状态，一般管道雷诺数流动状态，一般管道雷诺数Re2000为层流状态，为层流状态，Re4000为紊流状态，为紊流状态，Re20004000为过渡状态。为过渡状态。 7Laminar Flow is also referred to as streamline or viscous flow. These terms are descriptive of the flow because, in laminar flow, (1) layers of water fl
7、owing over one another at different speeds with virtually no mixing between layers, (2) fluid particles move in definite 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
8、characterized by the irregular movement of particles of the fluid. The particles travel in irregular paths with no observable pattern and no definite layers.Turbulence is flow dominated by recirculation, eddies, and apparent randomness.8Exercise : The steady Incompressible viscous laminar flow in a
9、channel . Assuming, v=0, w=0, impressible, steady 0 xp1) Write the continuity equation, momentum equation, 2) Write the no-slip boundary conditions for r=r0.3) Try to find the expression for the velocity u(r) .4) What is the maximum of velocity u ? (at r=r0)xr0RVpdtVdFg2/910Re=0.1 Re=50 Re=105 Dark
10、area is the area where viscous effects are important Turbulent flows are 3D, unsteady, rotational, viscous and chaotic fluid motions with 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 respon
11、sible for the enhanced diffusivity and stresses. Larger eddies carry small eddies, and transfer kinetic energy to smaller ones. Ultimately, the smallest eddies dissipate into heat by the action of molecular viscosity.11Momentum equation for turbulent flow 湍流流动湍流流动2ijdVpVgdt ;Vuiv jwkuuu vvvwww pppij
12、jiijkkijijijxuxuSxuS21);31(2For N-S equation in previous lectures, we didnt consider turbulence and didnt separate the variables into time-average part and fluctuation part. 12Color shades of the instantaneous streamwise velocity in LES of turbulent flow in plane diffuser Spatial and temporal fluctu
13、ations exist in turbulent flow. 时间和空间的脉动 13140u0v0vu00001:,Kolmogorov scaletFFFFFdttt;pppwwwvvvuuuEddies and DNS for turbulence computations The largest eddy size (l0) is comparable to the boundary-layer thickness. The smallest eddy size (h) is Kolmogorovs scales (much larger than molecular length s
14、cales) The larger Re, the greater l0/h 2141413tuEddy velocityDissipation of turbulent kinetic energyEddy size Direct numerical simulations (DNS) resolve the smallest eddies Wind u = 10 ms-1, = 1.5 x 10-5 m2s-1, h = 1.5 x 10-6 m 1 km3 volume: we need (1000/1.5 x 10-6)3 = 2.963 x 1026 grid points! A 2
15、 GHz single-processor computer: 4.695 x 109 years! (exactly the age of the Earth!) 15Geophysical flow is large-Reynolds-number turbulent flow.Reynolds-averaged Navier-Stokes equations for impressible flow雷诺平均方程iiiuuuLarge-scale time-averaged flowTurbulent fluctuationsTotal instantaneous 0iu Substitu
16、ting into the governing equations22()iijuuvvwwuuf vvtxyzuuppxx 2 uuuupu uu vu wuvwfvutxyzxxyz 00001:,Kolmogorov scaletFFFFFdttt16 Reynolds stress represents an average flux of x-momentum due to small-scale motions across the y-surface2*22*()()()xyxxxzxyyyyzyzxzzzuuuupuvwf wfvutxyzxxyzvvvvpuvwfuvtxyz
17、yxyzwwwwpuvwf uwtxyzzxyz gxyy1y2uvReynolds stress tensor (雷诺应力雷诺应力)How to model them to close the equations? jiijuu17182222*2220222222202222*222001()1()1()HVHVHVdupuuuf wfvAAudtxxyzdvpvvvfuAAvdtyxyzdwpwwwf uAAwgdtzxyz duuuuuuvwdttxyzNew governing equation for geophysical flows with turbulent stress今
18、后简化方程, 默认为是雷诺时间平均项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.ijjiMiMjjiuuAAxxAM105 m2s-1 for free atmosphere compared to 1.5 x 10-5 m2s-1 for kinematic viscosity19 ;/iiijMjjj
19、Mjuuu uAxxA / fluid viscositykinematic viscosityBoussinesqs eddy viscosity modelEffect of turbulence for momentum/scalar/energy transport.湍流中湍流输运粘性输运 (AM is turbulent kinematic viscosity);iiijM jjjjpEjuuu uAxxTu c TKx20Turbulent momentum transportTurbulent energy transportWater vapor or pollutant tr
20、ansport湍流交换系数湍流动量交换系数湍流热量输运通量密度湍流水汽通量密度湍流动量输运通量密度wjwqjqu qKx湍流动量交换系数AM的各向异性(变化剧烈) 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 ; AH105KV21xyy1y2uv Huu vAyu Vuu wAzxzz1z2uwuHorizontalReynolds stressVerticalReynold
21、sstress垂直方向受下面刚体边界和重力场的作用, 都抑制垂直方向的湍流发展22Six components of Reynolds stresses2322222202222220222222001()1()1()HVHVHVuuuupuuuuvwfvAAtxyzxxyzvvvvpvvvuvwfuAAtxyzyxyzwwwwpwwwuvwgAAtxyzzxyzgoverning equation for geophysical flows with turbulent stress and Boussinesq approximation (2)今后简化方程, 默认为是雷诺时间平均项 00
22、;uvwpgxyzz 244.2 量纲分析量纲分析Scale analysis in geophysical flows (z) HL; WU, V25WTUWLUWLWWH0PH2HA WL2VAWH222222001()HVwwwwpwwwuvwgAAtxyzzxyz10pgz 2001HA WPPLUULUHHLFor large scaleHydrostatic balanceScale analysis in geophysical flows (x ,y) HL; WU, V; For large scale WL/(UH)1 or 1, Coriolis force can be
23、 neglected;Ro1，both are important; Ro1，inertial force can be neglectedScales when rotation effect is important大尺度 地转平衡关系式 (二力平衡)00011pfvxpfuypgzGeostropic flow (地转流2) between Coriolis term and pressure gradient term11hVkpkff 31WTUWLUWLWWH0PH2HA WL2VAWH222222001()HVwwwwpwwwuvwgAAtxyzzxyz10Ekman numbe
24、r 埃克曼数埃克曼数Ek=/(2f H2 ) = ( 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, is the kinematic viscosity运动粘性系数 f = 2 sin is the Coriolis frequency, the latitudeEk1 viscosity can be neglectedEk1 vis
25、cosity is importantFor ocean motion of L=103 km, 分子粘性=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 m2/s(much larger than fluid viscosity 10-6 m2/s), = 7.3 105 s1 and L = 10 km, vertical EH = 1.4 106. 3322;VHVHAAEEHLEk=湍流粘性
26、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 to be a dominant force:34221(2 sin )VVVkAAAEddHdff H is the fluid height of the motion For ocean at mid-latitude
27、 (104 s1), eddy viscosity AV 102 m2/s ，fluid thickness H222110VkVkAEHAEHas H=10mas H=100mEk=湍流粘性力/科氏力Ekman numberEkman depth/thickness (Ekman厚度) in atmosphere and sea flows35 For ocean 海洋, with an eddy viscosity AV as large as 102 m2/s ，= 7.3 105 s1 d10m H = 100 m For atmosphere 大气, with an eddy vis
28、cosity AV as large as 5 m2/s ，= 7.3 105 s1 d103m=1km 10, 密度差(热力因素)占主导地位Ri0.1-10,密度差和水平惯性力都重要Ri0.1,水平惯性力主导，温度差和密度差可忽略384.3 边界条件和初始条件边界条件和初始条件39法向无渗透边界条件法向无渗透边界条件 w=dz/dt=db/dt运动边界条件运动边界条件40u db=w dxu db=w dybbbwuvtxy法向无渗透边界条件法向无渗透边界条件41自由表面边界自由表面边界+底部边界底部边界+侧边界侧边界42自由表面自由表面 边界条件边界条件dzdwdtdt43动力边界条件动力
29、边界条件 （压力（压力-pressure）44动力边界条件动力边界条件 （湍流剪应力（湍流剪应力-turbulent shear stress）看是否考虑边界层粘性效应看是否考虑边界层粘性效应或者用AV 表示45Open boundary 46在海气交界处最复杂在海气交界处最复杂海温异常，能影响气候变化海温异常，能影响气候变化47Chapter 4.4 Vorticity equation 涡度方程48Vorticity(涡度) and divergence (散度) are two properties of geophysical flows 涡度(Vorticity)是地球物理流体力学中
30、的一个基本概念，它描述一个水质点在空间中如何旋转，并与各种海洋和大气运动现象紧密联系。从行星尺度的波动到中小尺度涡旋，其垂直涡度分量尤为重要。49 涡量与流体微团自身的旋转角速度成正比，而与流体微团重心围绕某一参考中心作圆周运动的角速度无关。流动是否有旋与流体质点的运动轨迹无关。一个作圆周运动的流体微团可能涡量为零速度的涡度（速度的涡度（Vorticity)：描述速度的旋转性 To check this flow field:V=ay i+0 j +0 k, 有旋还是无旋 ?jjkjkijkikjkxyzuuuVeeexxxwvyzuwzxvuxy 50夸父追日，涡度为0胡旋舞，有涡度涡度方程
31、推导和物理意义涡度方程推导和物理意义51涡度的生成、发展和减弱均可用其垂直涡度的变化加以描述。垂直涡度方程（简称涡度方程）描述流体质点在运动中的涡度变化，它是地球物理流体力学中的一个基本方程，被广泛应用于各种现象的分析和理论研究。22()()2Vd rV dArrVrVr 涡度是流体微团绕其内部一瞬时轴作旋转运动的角速度的二倍 (如何证明？)xyzrOFor earth rotation, the absolute vorticity()2aRVr 旋转坐标系中, 绝对涡度和相对涡度的关系52Vorticity equation (推导推导)211(2)(/2)ijVVpVt 22/2()/2
32、VVVVVVV 112;ijVVVVpt gg 为重力211(2)(/2)ijVVpVt 21(2)()ijpVt is relative vorticity 2ais absolute vorticity 53梯度梯度(gradient) 压力梯度, 温度梯度, 密度梯度: 其方向为压力变化最快的方向）nnkzjyixxexeiiii)(标量(P, T) 的梯度为矢量P=p1P=p2n0zyxzyxkji0dlor 标量梯度场为有(位)势场(例压力梯度场), 为无旋场, 其涡度为0 或沿回线的环量(circulation)为0222()(2 )1()()aaaaijddVdtdttdpVVd
33、t ()()()()ABABBABAAB(2)(2)(2)()(2)(2)VVVVV is relative vorticity 2ais absolute vorticity 5556()()();xyzijkVxyzuvwwvuwvuijkyzzxxywvuwvuyzzxxy =V 尝试证明：（）0 有旋场无源222222()()()()Vwvuwvuxyzyzxzxywvuwvux yx zy zx yx zy z 21()()(2cos )(2sin)aaaijaxyzdpVVdtijk The component of planetary vorticity normal to th
34、e earths surface is the Coriolis parameter, 科氏参数 f = 2 sin , the latitude罗斯贝数罗斯贝数 (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 scales57 kzyFor large scale flows (Ro1):Inertial force c
35、oriolis forceRelative vorticity 0,0,zxxwwxxxw若则使开始时平行于 轴的作逆时针旋转的涡管由于 的不均匀分布而发生倾斜，从而产生 方向的垂直涡度分量xxyz6364()zyxdfwwdtyx垂直涡度1xxyz642xxyz64w1w2210;0 xwwwxx2z65()zyxdfwwdtyx垂直涡度xxyz652xxyz65w1w2210;0 xwwwxx2z科氏力，粘性，斜压科氏力，粘性，斜压 与与 外力无势外力无势是引起相对速度环量和相对涡通量发生变化的四大因素.66Equation of Circulation环量方程的解释环量方程的解释(2)ijCCCCV drdpVdrdrdrdrdt 作业：描述涡度方程各项的意义6721()()(2cos )(2sin)aaaijaxyzdpVVdtijk
