Grid-generated turbulence, drag, internal waves and mixing in stratified fluids

Transcription

1 Grid-generated turbulence, drag, internal waves and mixing in stratified fluids Not all mixing is the same! Stuart Dalziel, Roland Higginson* & Joanne Holford Introduction DAMTP, University of Cambridge *Now at USC Numerical models are not yet very good at this. Experiments leave many open questions. MOLECULAR DIFFUSION Essential for mixing at a molecular level Advection/diffusion of scalar S S 2 t + u. S = κ For heat (air) κ ~ 10 5 m 2 s 1 ν/κ ~ 0.7 heat (water) κ ~ 10 7 m 2 s 1 ν/κ ~ 7 salt κ ~ 10 9 m 2 s 1 ν/κ ~ 600 Diffusion slow: t ~ l 2 /κ Sc = ν/κ density diffusion slower than momentum at Kolmogorov scale Stuart Dalziel 1 Euromech 428, Torino, September 2001 S

2 Mixing = Stirring + Diffusion Stirring: intermingles fluid parcels of different properties and produces large gradients in those properties Diffusion: drives a flux that reduces gradients between adjacent fluid parcels Diffusion may be slow, but it need only act over small distances. Source of energy Stirring Turbulence Stratification Internal waves Nonlinearities + Advection Dissipation Molecular diffusion Gradients at small scales Mixing ENERGY BUDGET d dt ( PE + KE) = W D Total Work Dissipation Energy Can decompose PE into Background PE and Available PE. Stuart Dalziel 2 Euromech 428, Torino, September 2001

3 + + PE back PE avail KE! E back "" ""! E avail Lorenz (1955), Thorpe (1977), Winters et al. (1995) [For compressible fluid need to include internal energy] PE back is the minimum energy state that is achieved by adiabatic rearrangement of fluid parcels. Mixing increases PE back it cannot decrease it! PE avail is the component of PE that can be converted into KE, heat (through dissipation) and, if mixing occurs, into PE back. In the absence of external work: Stuart Dalziel 3 Euromech 428, Torino, September 2001

4 D D KE KE KE E avail PE avail PE avail PE avail PE PE back PE back PE back time Rewrite energy budget d dt ( E + E ) = W D back avail Total mechanical energy changes due to dissipation. Mixing efficiency definition Background energy E back changes due to molecular mixing. E η = E back avail PEBack = PEBack + ε dt = PE Back ( KE + PE ) Avail Stuart Dalziel 4 Euromech 428, Torino, September 2001

5 Previous work First experiments by Rouse & Dodu (1955) Houille Blanche 10. Most follows from work by Turner (1968) JFM 33. ω E = u e = f t u, ( Ri, Re Pe) Temperature Salt Turner, 1973: Buoyancy Effects in Fluids, CUP Stuart Dalziel 5 Euromech 428, Torino, September 2001

6 Linden (1979) Geophys. Astrophys. Fluid Dyn. 13. Mixing efficiency Stability z Low flux z High flux High flux Low flux Low flux High flux ρ ρ Stuart Dalziel 6 Euromech 428, Torino, September 2001

7 Depth (mm) Time (hours) Also lots of work on stratified shear flows Joanne Holford, DAMTP. Strang & Fernando (2001) JFM 428. Stuart Dalziel 7 Euromech 428, Torino, September 2001

8 New experiments SET-UP Ri = 0 N 2 W M 2 2 DRAG ON GRID Required to determine energy input Model as F = F f + F D + F S = ½ ρ 0 C f W 2 A f + ½ ρ 0 (C D + C S ) W 2 A g Stuart Dalziel 8 Euromech 428, Torino, September 2001

9 Drag on float Drag Coefficient of Float Drag Coefficient, C float Reynolds Number = WL float /ν close to that on a single sphere Stuart Dalziel 9 Euromech 428, Torino, September 2001

10 Drag on grid homogeneous 5.0 Drag Coefficient of Small Grid. 4.0 Drag Coefficient, C D Reynolds Number = W g M /ν C D < ~ 2.76 from PTV measurements for main grid Comte-Bellot & Corrsin (1966): C D = 4.53 c.f. Naudascher & Farell (1970): 2.12 C D 2.18 for similar Re. In stratified lower layer, need to account for Varying buoyancy Acceleration Added mass Inertial density (F = ρ 0 or F = ρ(z)?) Stuart Dalziel 10 Euromech 428, Torino, September 2001

11 The Drag Force on a Grid Shown With the Added Drag Due to a Density Gradient D grid, D strat (N = 1.1s -1, 1.69s -1 ) (Newtons) D strat, N=1.69s -1 D strat, N=1.1s -1 D grid, N=0s Reynolds Number = W g M /ν 40.0 The Coefficient of Added Drag Due to a Linear Density Gradient N=1.69s -1 N=1.1s -1 C S = D strat / (1/2)ρ 0 W g 2 Agrid Ri o = N 2 M 2 2 /W g C S = Ri o Stuart Dalziel 11 Euromech 428, Torino, September 2001

12 Simple model: M b W ρ(z) V Drift 2πb 2 γw/n γw g N ρ 2 2πb M = 2πγρ 0b z F S 2 4πγb 2 NMW C = Ri γ ( 0.35) Ri S A g 0 0 Stuart Dalziel 12 Euromech 428, Torino, September 2001

13 Homogeneous turbulence Between bars Temporal Average of the Flow Around the Bars W g t/m cms -1 Vorticity: s cm Stuart Dalziel 13 Euromech 428, Torino, September 2001

14 Behind bars Temporal Average of the Flow Behind a Bar W g t/m cms -1 Vorticity: s cm Stuart Dalziel 14 Euromech 428, Torino, September 2001

15 Mean velocity 1.00 Time Evolution of Mean Velocity Components 0.50 U /W g W /W g U /W g, W /W g W g t /M Fluctuations Time Evolution of rms Velocity Fluctuations u /W g w /W g u /W g, w /W g W g t /M Stuart Dalziel 15 Euromech 428, Torino, September 2001

16 Stratified turbulence Rio = 135 Temporal Average of the Stratified Flow Behind a Grid, with N = 1.63s -1 Vorticity: s Wg t/m cm 0 Stuart Dalziel 16 Euromech 428, Torino, September 2001

17 Rio = 29 Temporal Average of the Stratified Flow Behind a Grid, with N = 0.76s -1 Vorticity: s Wg t/m cm 0 Clear evidence of internal waves Stuart Dalziel 17 Euromech 428, Torino, September 2001

18 Typical Example of a Density Fluctuation Time Series Density Perturbation, ρ (g/cm 3 ) e e Nt sinθ = 2 cos θ 1 NM 2π W = 1 Ri o 2π Phase Angle of Predominant Internal Waves θ = cos -1 (ω/n) Ri o = N 2 M 2 2 /W g Stuart Dalziel 18 Euromech 428, Torino, September 2001

19 Velocity fluctuations Time Evolution of rms Horizontal Velocity Fluctuations N = 0s -1 N = 0.40s -1 N = 0.76s -1 N = 1.19s -1 N = 1.63s u /W g W g t /M Time Evolution of rms Vertical Velocity Fluctuations N = 0s -1 N = 0.40s -1 N = 0.76s -1 N = 1.19s -1 N = 1.63s w /W g W g t /M Stuart Dalziel 19 Euromech 428, Torino, September 2001

20 Adjustment of Vertical Velocity Fluctuations N = 0.40s -1 N = 0.76s -1 N = 1.19s -1 N = 1.63s w /W g Nt Mixing efficiency Integrated Flux Richardson Number, Rf The Integrated Flux Richardson Number Plotted as a Function of Ri o W g =2.47cm s -1 W g =1.37cm s -1 W g =0.67cm s -1 W g =0.41cm s Overall Richardson Number, Ri o = N 2 M 2 2 /W g Rf = 0.2 Ri o 0.53 Stuart Dalziel 20 Euromech 428, Torino, September 2001

21 For Ri o = 3842 still saw no reduction in η! Numerical: Flux Richardson Number for Numerical Simulations Integrated Flux Richardson Number, Rf Ri = (Nλ peak /q) 2 Stuart Dalziel 21 Euromech 428, Torino, September 2001

22 500 mm Statically unstable flows Top view 200 mm End view 400 mm Mixing driven by Available Potential Energy Maximum mixing efficiency: η = ½ (horizontal tank). Stuart Dalziel 22 Euromech 428, Torino, September 2001

23 Stuart Dalziel 23 Euromech 428, Torino, September 2001

24 Overall mixing efficiency η Angle of tank α o This is the cumulative mixing efficiency over a mixing event. The final state is nearly well mixed. Stuart Dalziel 24 Euromech 428, Torino, September 2001

25 1.00 η instantaneous q = q = = q + ε dpe dt back δpe δe back avail 0.80 η instantaneous Time t/τ Stuart Dalziel 25 Euromech 428, Torino, September 2001

26 Conclusions Drag on grid increased due to direct generation of internal waves. Model for additional drag Model for internal waves Mixing efficiency for vertically towed grid does not decrease at high stabilities Greater contribution to internal wave drag Ignoring wave drag leads to continued increase in η Added mass contribution always raising fluid elements Static instability very efficient Density contributes to growth of fine scales Numerics over-predict mixing efficiency It matters how you get the energy to small scales and whether the density helps or hinders this process. Stuart Dalziel 26 Euromech 428, Torino, September 2001