Concept of Reynold Nuber, Re Ignore Corioli and Buoyancy and forcing Acceleration Advection Preure Gradient Friction I II III IV u u 1 p i i u ( f u ) b + u t x x x j i i i i i i U U U? U L L If IV < < I, II R e = UL 1 T u rb u le n c e O c c u r U c h a ra c te ri tic flo w fie ld v e lo c ity L c h a ra c te ri tic flo w fie ld le n g th c a le ti e c a le = L U
Flow pat a circular cylinder a a function of Reynold nuber Fro Richardon (1961). Re<1 Lainar Cae Re = Re =174 Exaple : a toothpick oving at 1/ Exaple: a finger oving at c/ Turbulent Cae Re = 5, Re = 14,48 Note: All flow at the ae Reynold nuber have the ae trealine. Flow pat a 1c diaeter cylinder at 1c/ look the ae a 1c/ flow pat a cylinder 1c in diaeter becaue in both cae Re = 1. Re = 8 Re = 1,, Exaple: hand out of a car window oving at 6ph.
Wind Ocean Turbulence (3D, Microtructure) Air Mixed Layer Turbulence Water Therocline Turbulence Botto Boundary Layer turbulence
Turbulent Frictional Effect: The Vertical Reynold Stre Air du d t i No Turbulence p ( f u ) g F r x i or in coponent for i i i d u d t 1 p ( ) f v + x F r x Water d v d t 1 p ( ) fu y F r y d w d t 1 p ( ) g F r
Mean and Fluctuating Quantitie u u ' u u u e a n + flu c tu a tin g ' u u ' u Three Type of Average Eneble Tie Space Ergodic Hypothei: Replace eneble average by either a pace or tie average tie N o ta tio n q < q >
How doe the turbulence affect the ean flow? Mean Flow 3D turbulence u u ', v ', w ' u u u ' v = v v ' w w w ' u ' w ' < u w C o n c e p t o f R e yn o ld S tre u ' w '
Moentu Equation with Molecular Friction Approach for Turbulence d u 1 p fv ( ) + F rx d t x d v 1 p fu ( ) F r d t y d w 1 p ( ) g F r d t w h e re d u v w d t x y Fr x, y, But (u,v,w ) o le c u la r v i c o i ty = y e c d u fv 1 p ( ) d t x d v fu 1 p ( ) d t y d w 1 p ( ) g d t w h e re d u v w d t x y B u t u u u ' u u u ' v v + v ' w w + w ' i i i
Exaple Unifor unidirectional wind blowing over ocean urface Dienional Analyi Boundary Layer Flow Gradient in x direction aller than in direction Exaple:Mean velocity unidirectional, no gradient in y direction d u d t d v d t fv 1 p ( ) x fu 1 p ( ) y u u u u v w u fv 1 p ( ) t x y x u in g u v w x y d w d t 1 p ( ) g u u u ( ) ( v u ) ( w u ) ) fv 1 p ( ) t x y x w h e re d u v w d t t x y
u u u ( u w ) fv 1 p ( ) t x x B u t u u u ' u u u ' v v + v ' i i i ( u u') ( u u ') ( u u ')( w w ') x w w + w ' Now we average the oentu equation u u ( u ' w ' ) 1 p u fv t x x u u 1 p 1 u fv t x x x w h e re x = u ' w ' " x " c o p o n e n t o f R e y n o ld S tre
U n tratfied flo w co n tan t Exaple: Tidal flow over a ound U H Lainar Flow Turbulent Flow R e y n o ld N u b e r R e ul ; u, L c h a ra c te ri tic v a lu e o f th e e a n f lo w, 1 F o r u n tra tfie d flo w c o n ta n t 6 e c T u rb u le n c e o c c u r w h e n R e R e ~ 3 c
3 D Turbulence: Navier Stoke Equation (no gravity, no corioli effect) Exaple: tidal channel flow, pipe flow, river flow, botto boundary layer) I II III IV u u u u 1 p u u u i i i i i i i i { u v w } ( ) t x y x x y j I. Acceleration II. Advection (non-linear) III. Dynaic Preure IV. Vicou Diipation R e y n o ld N u b e r = II IV
Surface Wind Stre (Untratified Boundary Layer Flow) Definition: Stre = force per unit area on a parallel urface wind a ir a ir u a ir ' w a ir ' Air w u ' w ' Water What i the relationhip between and? a ir w w a ir
Epirical Forula for Surface Wind Stre Drag C D Coefficient C U w h e re U D 1 1 i th e w in d p e e d 1 a b o v e t h e w a te r C D 3 1 U < 5 3 1.5 1 U > 5 1 e c e c Concept of Friction Velocity u* Definition u * u ' w ' ( u * ) u* Characteritic velocity of the turbulent eddie
Exaple. If the wind at height of 1 over the ocean urface i 1 /ec, calculate the tre at the urface on the air ide and on the water ide. Etiate the turbulent velocity on the air ide and the water ide., u*=?, u*=? Since U > 5 C.5 1 1 D e c kg C ( U ) (1. )(.5 1 )(1 ) a ir D 1 3 e c N. 5 a ir 3 3 w a te r N. 5 * a ir u = =.5 a ir kg e c a ir 1. 3 N. 5 * w u = =. 1 6 w kg e c w 1 3
General Cae of Vertical Turbulent Friction u u 1 p 1 i i i u ( f u ) g t x x w h e r e j i i j = u ' w ' f o r u ' u ' u ', u ' u ' v ' i i 1 x y = f o r i = 3 ( ) Note that we oetie ue 1,,3 in place x, y, a ubcript Convention: When we deal with typical ean equation we drop the ean Notation! i u u 1 p 1 i i i u ( f u ) g t x x j i i j i
Coponent for of Equation of Motion with Turbulent Vertical Friction u u u u v u w fv 1 p ( ) + 1 t x y x v v v v 1 p 1 u v w fu ( ) d t x y y w u w w v w w 1 p ( ) g t x y x y Note: in any cae the ean vertical velocity i all and we can aue w = which lead to the hydrotatic approxiation and u (1) u u u v fv 1 p ( ) + 1 t x y x v v v 1 p 1 ( ) u v fu ( ) d t x y y x y (3 ) p g
Role of Botto Stre Exaple : Steady State Channel flow with a contant urface lope, a. (No wind) = D a Surface Surface Stre Stre Flow Direction Why? = Botto p g D ( x Botto Stre ga D = 1 p 1 x b u t p x { g ( D } x g a ga ( D ) N o te a x
= D Surface Flow Direction Why? = Botto p g D ( x Botto Stre ga D Typical Value g a ( D ) a x 5 6 a 1 c /(1 k to 1 k ) 1 to 1 & fo r D = 1 F ric tio n v e lo c ity o n th e b o tto i u * g a D u * (1 3 ) c / e c
Relating Stre to Velocity Vicou (olecular) tre in boundary layer flow Low Reynold Nuber Flow Note: Vicou Stre i proportional to hear. u e c o le c u la r v i c o i ty Turbulence Cae: Eddy Vicoity Auption u e e eddy vicoity Note. At a fixed boundary u becaue of olecular friction. In general = (). Mixing Length Theory: Modeling e ul l a characteritic length, u a characteritic velocity of the turbulence
Back to contant urface lope exaple where we found that ga ( D ) = D a = If we ue the eddy vicoity auption with contant k u k g a ( D e ) u g a ( D k ) ( u* ) k (1 ) D u * = b o tto f r ic tio n v e lo c ity u u( D ) ga D k e 6 5 D = 1, a 1, k 1 u u( D ) ga D k.5 e c Exaple Value e e e c ( 9.8 1 * 1 e c * 1 6 5 e c
Log Layer Note: in the previou exaple near the botto, independent of Botto Boundary Layer co n tan t E d d y i e to d i ta n c e fro b o tto k u * v u ( u * ) u v u * v.4, V o n K a r a n ' c o n ta n t Note we have ued the fact that ln ( ) 1 u u * ln ( ) th e ro u g h n e p a ra e te r v
Typical Ocean Profile of teperature (T), denity ( - 1 1k Mixed Layer therocline pycnocline T 4k But ( T, S, p )
Stratified Flow Horiontal Equation Du 1 1 f u p h D t D w h e re u ' w ' & u D t t Vertical Equation: Hydrotatic condition No tratification Vertical Equation: Hydrotatic condition Stratification Dw D t 1 p g Dw D t 1 p ( ) g p p g
Buoyancy If W > If W < F F B B b lo c k in k b lo c k ri e Archiede Principle Buoyancy Force FB g d e n ity o f th e w a te r Weight W g d e n ity o f th e b lo c k
Concept of Buoyancy frequency N F ( d ) g B +d W ( ) g F F W V { ( d ) ( ) } g n e t B b u t { ( d ) ( ) } d g g g F V d g V N d w h e r e N ( ) { } n e t c g c 5 4.4 1 e c
Turbulence in the Pycnocline Velocity Shear u N g Gradient Richardon Nuber Ri g N u ( ) Turbulence occur when Rig 1 4
Billow cloud howing a Kelvin-Helholt intability at the top of a table atopheric boundary layer. Photography copyright Brook Martner, NOAA Environental Technology Laboratory. 1 ( Ri ) g 4
Depth() Ditance () Turbulence Oberved in an internal olitary wave reulting in Goodan and Wang (JMS, 8) 1 ( Ri ) g 4
Teperature (Heat)Equation with Molecular Diffuion dt dt w h e r e T T Approach for Turbulence dt dt B u t T T T ' & w w w ' d u v w d t x y T o le c u la r d if f u iv ity o f T = 1.4 1 7 e c T T u v T w T t x y T u T v T w T w ' T ' t x y Cae of Vertical Advection and Turbulent Flux Eddy Diffuivity Model Note: Heat Flux i given by T w T w ' T ' t w ' T ' k T H k c T
T T T w T t Advection Diffuion Equation drop bar notation T S te a d y S ta te C a e t T T w w u p w e llin g v e lo c ity T S o lu tio n : T T w ( ) e x p [ ( ) ] D T T D ( ) exp( ) w here w T T T ( ) = T ( ) {1 e x p ( ( D )} D T T ( ) = T T {1 e x p ( ) } w h e r e T ( )
H e a t T r a n f e r e d Z=D Surface () T 3 J H = c ( ) w h e r e c p e c if ic h e a t o f w a te r 4. 1 T S o C k g T T() Z= u w u Exaple: Suppoe the heat input i H = 5, T o C in water of depth 5. 3 The turbulent diffuivity i k 1 (a) For the cae of no upwelling what i e c the heat tranferred, H, at the urface, id depth, and the botto? What i the water teperature at the urface id depth and the botto? (b) Suppoe there wa an upwelling velocity of.1 /ec how would the reult in part (a) change? W a tt Note change in nuber fro note!!
(a) No Upwelling w= W a tt H = 5, T o C Z=D Surface () T Z= T T w T w ( ) e x p [ ( D )] = ( ) W a tt H H 5 a t a ll d e p th! W a tt 5 ( T H ) ( T ) c T 3 k g 3 J 3 1 4. * 1 * 1 3 o k g C e c o.1 C a t a ll d e p th! T D T ( ) = T T {1 e x p ( ) } k T D A T ( ) = li [ T ( ) {1 e x p ( ) } ] w T T ( ) = T ( ) ( D ) a t: = 5, T = T o = 5, T = C.1 C o ( 5 ) 1 7 C o =, T = C.1 C o ( 5 ) 1 4 C o o C o
(a)upwelling w=.1 /ec W a tt H = 5, T o C Z=D Surface () T 3 1 T e c w 4 1 1 e c T T D H ( )= c k ( ) c k ( ) e x p ( ) D H ( )= H e x p ( ) u w u W a tt 5 ; H = H 5 W a tt 5 W a tt 5 ; H = 5 e x p ( ) 4 1 1 W a tt 5 W a tt ; H = 5 e x p ( ) 3.3 1 Z= D T ( ) = T T {1 e x p ( ) } T C T ( ) 1 *.1 1. a t: = 5, T = o C o o 5 o = 5, T = C 1. C [1 e x p ( ) ] 1 8.9 C 1 o o 5 o =, T = C 1. C [1 e x p ( ) ] 1 8.8 C 1 o o 1 o a t: = 1,T = C 1. C [1 e x p ( ) ] 1 9. C 1 o