Road to Turbulence in Stratified Flow

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Leon Jennings
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1 Road to Turbulence in Stratified Flow (Exaple: Boundary flow with f =0) Lainar Flow U H Turbulent Flow I II III IV V u u p u i i i u b t x x x x b g 0 i N 0 0 g z Turbulence occurs when: II UL R e R eyonlds N u ber = IV R i V N R ichardson N u ber = U ( ) z II 4

2 Advection u u x i and the Cascade Process D exaple u =A sin(k x) k u = A sin(k x) k.5

3 Kelvin-Helholtz Instabilities R i = N U ( ) z 4

5 The Path to Turbulent Fluctuations u u p u i i i u b t x x x x 0 i u u u ' i,, Let u U ( z), u u 0 i i i u ' u '; u ' v '; u ' w ' p p p ' P p ' b B ( z ) b ' N ote: b ' g w here B g 0 ' 0

6 U U P U t x x z 0 Average Equations P 0 z B Equations for turbulent fluctuations u ' u ' u ' u ' i i i i I U u ' w ' p ' u ' (i=, H orizontal) i t x x z w ' w ' w ' U p ' II U w ' w ' + w ' + b' V ertical t x z z z 0 0 Multiply (Dot Product) I by u i ', I I by w, Add and Average. w Use: u 0 z

7 TKE (Turbulent Kinetic Energy) Equation D ( TKE ) P B Tr Dt D U D t t x TK E T urbulent K inetic Energy = ( ( u ') ( v ') ( w ') u P Pr oduction T urbulent Eneregy by M ean Sh ear = u ' w ' z g B w ' ' Sink or Source of T urbulent Potential Energy s s D issipation of T K E s i i i u ' u ' i ( ) T urbulent Strain Rate T ensor x x i Tr T ransport of T E= [ u ' p ' u ' u ' u ' u ' s ] i i i x i

8 Steady State Hoogeneous TKE Equation D ( TKE ) 0 Tr 0 Dt P P B B du u ' w ' dz g ' w'

9 Turbulent Teperature Fluctuations T T T T 7 u v w T k ther al diffusivity.4 0 T T t x y z T ypical O cean C ase u U u ' U =[U (z),0,0] T T ( z ) T ' i i i i T ' T ' T T ' T ' T ' U w ' u ' v' w ' T ' T t x z z y z Multiply Above Equation by T and Average. Assue Stationarity and Hoogeneity in horizontal directions TV(Teperature Variance) Equation P T = dt P w ' T ' k ' ' T T dz P= Production of Teperature Variance Molecular Diffusion of Teperature (heat)

10 Steady State Hoogenous TKE and TV Equations : Linkage P P B T = du P u ' w ' dz g B ' w ' = g w ' T ' P T dt w ' T ' dz k ' ' T

11 Relating to Case of Density only a Function of Teperature, ( T ) g g N B ' w ' g w ' T ' dt dt ( ) dz dz 0-4 o ther al expansion coefficient =.4 0 ( C ) T B D efine M ixing Efficiency B g N dt dt ( ) dz dz Observations show in the ocean that.

12 Measuring Turbulence

13 Stratified Turbulence ( z ) Shear R i N N U 4 ( ) z g 0 z (B) Buoyancy Flux to Potential Energy Mixing Production (P) Fro Mean Shear Flow Turbulence Dissipation () to heat is rate of energy transfer (Watts/kg) fro larger scale to the sallest scales

14 Characteristic Scales of Turbulence Key Factor: Cascade of kinetic energy fro large to sall scales, 0 W atts W atts ( ) N ote =, pow er transfered to s aller scales V kg kg Energy Containing (Large) Scales of the Turbulent Field Scale with, L u, L, T ~ u u ~ or L ~ L T L u Dissipative (Sall) Scales of the Turbulent Field Scale with, v,, ~ ( ) 4 v ~ ( ) ~ ( ) 4 L

15 Ratio of Sallest (Dissipative) to Largest (Energy Containing) Turbulence Scales ul ~ [ ] L v ul ~ [ ] u 4 4 R ul ~ [ ] R T 4 4 R = Turbulent Reynolds nuber R L ul {0,0 }() R 0,0 6 0 Typical oceanic values 4 U [ ] ~., while ~.0,.00 z w atts 0 0 kg 6 9 L u ( l) ~ 0,0 w atts kg

16 Steady State, Hoogenous, No Stratification, B=0, P = U u ' u ' i i P u ' w ' s s D issipation of T K E i i z x x u ' u ' u i T urbulent Strain R ate s ( ) s ~ ( ) ( ) i i x x L U U u i M ean Strain R ate S ( ) S ~ ( ) i i x x L si U u R U z V W ~ N ote for U ( ), 0 S ( ) S z L i i i Page 65 T/L

17 Isotropy at Dissipation Scales u ' u ' i i x x where u ' u ' u ' x x x u ' u ' u ' x x x u ' u ' u ' x x x u ' u ' u ' x x x u ' u ' u ' u ' u ' u ' x x x x x x Only one coponent of turbulent shear is required to obtain u ' u ' u ' = = x x x u ' u ' u ' u ' u ' u ' = = x x x x x x

18 u ' ( u ') x ( u ') x 4 u ' ( u ') x u u 5 L L ul R 5 L R Taylor Microscale u ' using R 5 R L N ote: L w hen R >> 4

19 Exaple: Wind Tunnel Decay of Turbulence U x U t u 0 d dt ( TK E ) u d dt ( u ) u l x U t U se l kt L kt k= u L for a ti e t such that l L du u dt kt u 0 u, w here t ( ) 9k ( u ) 0 0

20 0 = P Tr - TKE (x, z, y) Coponent Equations Fro T/L page 7 Red: Dissipation (Isotropic) Production Brown: Redistribution by Pressure fluctuations

21 Role of Vorticity in Turbulent Cascade {Navier Stokes Eqaution} = V orticity Equ ation u { u u} { p u} t u u t Inviscid C ase (no Friction)] u u t x x i i i u u u u u i i i ( ) ( ) x x x x x i i = s U sing r i i r i ik d i i i u s dt t x i

22 V orticity E quation u u t Inviscid C ase (no Friction)] d i i i u s dt t x i S i U 0 0 x V 0 0 y W 0 0 z S S S d dt d dt d dt Wind Tunnel contraction s = exp(s t ) 0 s = exp(- s t ) 0 s = exp(- s t) 0

23 Steady State Vorticity Variance Equation d ' ' i i ' ' (...) (...)... ' ' s i i i i dt x x Order of Magnitude scaling results in (Pages 87-9 T/L) ' ' i i ' ' s ' i i x x Key Points. Unlike TKE Equation no Source (production) ter.. Right Hand Side (RHS) represents decay and is always positive; thus LHS is positive.. LHS arises represents vortex stretching by strain rate. RHS > 0 iplies ore vortex stretching than squeezing. 4. Vorticity is then transferred to saller scales!

24 Cascade Mechanis Role of Vorticity Stretching u ' 0 0 x s ' 0 0 v ' s' s' 0 Let s ' s ' s ' & ' ' i y 0 0 s ' w ' 0 0 z ' ' i i ' ' s ' i i x x { ( ') ( ') } { s ' } > 0 if s ' 0, {< ( ') } { ( ') } if s ' 0, {< ( ') } { ( ') } Both stretching and squeezing results in an ibalance in vorticity variance Ibalances are dissipated Vortices in direction of larger variances are reduced in size (wind tunnel exaple)

25 Wind Convective Boundary Layer (CBL) -H watts/. z u* w* L= Monin- Oboukhov (MO) depth T(z) H= Depth of CBL L= Monin- Oboukhov (MO) depth z < L Doinated by Wind Stress Turbulence, u* z >L Doinated by Buoyancy Driven Turbulence, w* H = depth of CBL fro balancing rotation and buoyancy

26 Wind Convective Boundary Layer (CBL) -H watts/. Surface (s) z u* w* T(z) L h L= Monin Obhukov (MO) depth h= Depth of CBL P P B B<0 u* z' for a log layer U = log( ).4 z P u ' w ' ( u*) u *.4 z g B= < ' w ' du dz 0 R f L= g < ' w ' B = = = P ( u*) ( u*).4 z g < ' w '.4 ( u*) B.4L z L B P ( z L ) & ( B ) ( u*) ( w*) w * z B ( ).4 L z u * L z at depth z w* u * ( ) L k u* h = using k = u*h h f f

27 Wind Exaple Proble -H watts/. z =0 Surface (s) Convective Boundary Layer u* w* T(z) L h z z = D In winter tie the wind blows steadily over a region of the continental shelf at U 0 0 and loses heat at a rate of H 400 watts. The depth of water is found to be D = 400; 4 Take f 0. (a) How deep do you estiate the convective boundary layer to be? (b) Estiate the turbulent velocity at z = 0,50, 00, 50? (c ) If the wind U 0 decreases by 50% how does this change these answers? kg ( ).5 0 (0 ) C ( U ) 0 ( ) ( *) a D a u u*.06 kg w w * u * h 0 4 f 0

28 Wind Exaple Proble (Continued) -H watts/. z=0 Surface (s) Convective Boundary Layer u* w* T(z) L h z=d ( u*) ( u*) ( u*) L= g g.4<t ' w' H.4< ' w' g.4 c (.06 ) w atts o 9.8 *.4 0 ( C ).4 kg J o kg ( C ) p 78 z 0,50, z L 78 u u*.06 turb & 00 0 * * ( ).0 z z h w u 78 z 50 h w* u* 0

29 ( b) U 5 0 w kg ( ).5 0 (5 ) C ( U ) a D 0 ( *) u u*.008 kg w u * h 60 4 f 0 ( u*) ( u*) ( u*) L= g g.4<t ' w' H.4< ' w' g.4 c (.008 ) w atts o 9.8 *.4 0 ( C ).4 kg J o kg ( C ) p 9.5

30 z 0, z L 9.5 u u*.008 turb 50 z 50 L & h 60 w* u * ( ) z 00, 50 h w* u* 0

31 Concept of Buoyancy frequency N F ( z z) g B z-z z W ( z) g Block of fluid oscillates at N z-z N g g P ( ) { } S ubtract out effect of pressure changes z z P z c the speed of sound c 500 P c P g z

32 N g z g c g c g g T T If ( T ) z T z z.40 ( C ) T N T g z 4 o T g C If.0 z c N T g z g c o

Concept of Reynolds Number, Re

Concept of Reynolds Number, Re 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 <