Lecture 7. Turbulence

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1 Lecture 7

2 Content Basic features of turbulence Energy cascade theory scales mixing

3 Basic features of turbulence What is turbulence?

4 spiral galaxies NGC 2207 and IC 2163

5 Turbulent jet flow Volcano jet flow

6 Clouds: turbulence in atmosphere Large eddies and small eddies

7 Water flow Motor oil floating on water runoff from a street after a snowstorm Large eddies and small eddies

8 wake flow

9 Jet flow in gas turbine combustor

10 Turbulent flames

11 Origin of turbulence Shear layer instability Kelvin Helmholtz instability Transition from laminar to turbulent

12

13 Clouds: turbulence in atmosphere Large eddies and small eddies Kelvin-Helmholtz instability

14 A turbulent jet flow 3D eddies

15 Turbulent jet flow K-H instability Direct numerical simulations

16 Turbulent jet flow laminar turbulent K-H instability PLIF imaging

17 Turbulent jet flow turbulent laminar K-H instability Direct numerical simulations

18 Wall generates turbulence

19 Near wall turbulence

20 Basic features of turbulence Irregularity Statistical approaches, not deterministic approaches unsteady, random

21 O. Reynolds experiment (1880s) Osborne Reynolds Born: 23 Aug 1842 in Belfast, Ireland Died: 21 Feb 1912 in Watchet, Somerset, England

22 Reynolds experiment Reynolds number

23 Reynolds number and turbulence scales Large scales: independent of Re Small scales: dependent of Re Lower Re Small eddy Large eddy Higher Re

24 Basic features of turbulence The nature of turbulence: what is turbulence? High Reynolds numbers Instability of laminar flows Non-linearity and randomness make the problem nearly intractable Three dimensional vorticity fluctuations The large scale structure in tornado is not mainly turbulent! Vortex stretching mechanism Dissipation Energy transfer from large scales to small scale and dissipated to heat is continuum flow!

25 What is turbulence? is flow is three dimensional high Reynolds number flow is a random, chaotic, and irregular flow is made up of 3D eddies of various size is dissipative is effective in transferring of mass and heat

26 Basic features of turbulence Diffusivity Enhanced mixing & heat and momentum transfer Wind/ocean current momentum transfer Coffee/milk mixing Fuel/air mixing Prevent separation of golf ball, airfoil

27 Other examples

28 7.2 Energy cascade of turbulence

29 eddies Richardson s definition of turbulence eddies (1922) consists of different eddies An eddy is a localized flow structure Large eddies consists of small eddies Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity. Lewis Fry Richardson

30 Breakdown of large eddies Due to the non-linearity of the N-S eqaution (convective transport) large eddies tend to breakdown to smaller eddies If the viscous term is very large then the breakdown process is prevented: viscous terms tend to stabilize the flow (think about a very sticky fluid)

31 Small scale in turbulence Kolmogorov eddies the smallest eddy must be at the length and velocity scale at which viscous term is at least as important as the non-linear convective term Physically, viscous damping should be so strong that any velocity gradient would be quickly smooth away Andrey Kolmogorov

32 Energy Cascade inlet, other boundaries Energy transfer at a constant rate e heat

33 Energy cascade theory Kolmogorov s universal equilibrium theory Large eddies are not affected by viscosity Large eddies transfer energy to small eddies The rate of energy transfer from large eddies must be in the order of energy dissipation from the small eddies to heat, since otherwise the energy at small eddies will be accumulating or dying out! Since no accumulation of energy at any of the scales, one may assume the energy transfer rate is the same at all scales

34 7.2 eddy scales

35 Taylor s estimation of the energy transfer rate (1935) Large eddy size Large eddy speed l 0 u 0 one eddy turnover time t 0 µ l 0 u 0 Large eddy kinetic energy Large eddy loses a significant fraction of their kinetic energy within one eddy turnover time - Energy transfer rate u 0 2 e µ u 2 0 µ u 0 t 0 3 l 0 Sir Geoffrey Ingram Taylor

36 Energy cascade Integral scale length: l 0 velocity: u 0 =IU, I=turbulent intensity, U=mean velocity time: t 0 =l 0 /u 0 Reynolds number: Re l =u 0 l 0 /n Kolmogrov scale length: l k velocity: u k time: t k =l k /u k Reynolds number: Re k =u k l k /n=1!!!

37 Energy cascade Dissipation rate of turbulent kinetic energy e is constant at all scales (Cascade assumption). At smallest scale, it is estimated as e u t 2 k k, t or k t l u k k k, 2 k l, n e u l 3 k k u e n l eµ u 2 0, t t 0 µ l 0, eµ u u 0 l 0 2 k 2 k \ u æ 0 µ l ö 0 u ç k è ø l k 1/3, t 0 µ l u æ 0 k µ l 0 t k u 0 l ç k è l k ö ø 2/3

38 Scale relations At the viscous scale (Kolmogorov scale) convection term is in the same order as diffusion term, i.e., Re k =1, so... eµ n u 2 k 2 l k µ n u 2 k 2 l k n 2 u 2 k l µ n3 2 4 k l k or eµ u 3 0 µ u 3 0 l 0 l 0 æ n ç è u 0 l 0 ö ø 3 æ ç è u 0 l 0 n ö ø 3 µ n3 l Re 3 4 l 0 \ l 0 µ Re 3/4 l l, k u 0 µ Re 1/4 u l, k t 0 1/2 µ Re t l k

39 Turbulent jet flows l l 0 k Re l 3 / 4 Lower Re Small eddy Large eddy Higher Re

40 7.2 mixing

41 mixing u 0 u k, l k l 0

42 mixing u 0 u k, l k l 0

43 mixing u 0 Large eddy passing time = l 0 /u 0 l 0

44 mixing u 0 Large eddy passing time = l 0 /u 0 Molecular mixing time = l 0 /(D/l 0 ) = l 02 /D l 0

45 mixing rate u 0 Large eddy passing time = l 0 /u 0 u k, l k Kolmogorov eddy turnaround time = l k /u k <<l 0 /u 0 = large eddy passing time Molecular mixing time = l k2 /D = l k2 /(u k l k ) = Kolmogorov eddy turnaround time l 0 It implies that as soon as the entire large eddy has past the surface it would be mixed with the local material on the molecular scale

46 mixing u 0 Mixing time of An entire large eddy = l 0 /u 0 u k, l k Kolmogorov eddy turnaround time = l k /u k << large eddy passing time Molecular mixing time = l k2 /D = l k2 /(u k l k ) = Kolmogorov eddy turnaround time l 0 mixing time = l 0 /u 0 = l 02 /(u 0 l 0 ) = l 02 /D t eddy diffusion coefficient: D t >>D

47 Estimation of turbulence mixing

48 Mixing by molecule motion Mixing of milk and coffee H Molecular diffusion coefficient nµaxµ10-6 m 2 / s Coffee height H=0.06 m Mixing time t=h 2 /n = 1 hour

49 Mixing by turbulent motion Mixing of milk and coffee Molecular diffusion coefficient n t µ u 0 l 0 µ10-3 m 2 / s Coffee height H=0.06 m H Mixing time t=h 2 /n t = 3.6 s

50 Summary of key issues Concepts turbulent eddies Energy cascade Key parameters integral scales Kolmogrov scales Reynolds number Damköhler number Karlovitz number eddy viscosity

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