058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 1Chapter 6: Viscous Flow in Ducts6.3 Turbulent FlowMost flows in engineering are turbulent: flows overvehicles (airplane, ship, train, car), internal flows (heatingand ventilation, turbo-machinery), and geophysical flows(atmosphere, ocean).V(x, t) and p(x, t) are random functions of space and time,but statistically stationary flows such as steady and forcedor dominant frequency unsteady flows display coherentfeatures and are amendable to statistical analysis, i.e. timeand place (conditional) averaging. RMS and other low-order statistical quantities can be modeled and used inconjunction with the averaged equations for solvingpractical engineering problems.Turbulent motions range in size from the width in theflow δ to much smaller scales, which becomeprogressively smaller as the Re = Uδ/υ increases.058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 2058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 3Physical description:(1) Randomness and fluctuations:Turbulence is irregular, chaotic, and unpredictable.However, for statistically stationary flows, such as steadyflows, can be analyzed using Reynolds decomposition.u=u+u'u=1T∫t0t0+Tu dTu '=0u '2=1T∫t0t0+Tu '2dTetc.u = mean motionu '= superimposed random fluctuationu '2= Reynolds stresses; RMS = √u'2Triple decomposition is used for forced or dominantfrequency flowsu=u+u ''+u 'Where u '' = organized oscillation(2) Nonlinearity Reynolds stresses and 3D vortex stretching are directresult of nonlinear nature of turbulence. In fact, Reynoldsstresses arise from nonlinear convection term after058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 4substitution of Reynolds decomposition into NS equationsand time averaging.(3) DiffusionLarge scale mixing of fluid particles greatly enhancesdiffusion of momentum (and heat), i.e.,Reynolds Stresses: −ρu 'iu 'j>> τij=μεij⏞viscous stressIsotropic eddy viscosity:−u 'iu 'j= νtεij−23δijk(4) Vorticity/eddies/energy cascadeTurbulence is characterized by flow visualization aseddies, which varies in size from the largest Lδ (width offlow) to the smallest. The largest eddies have velocityscale U and time scale Lδ/U. The orders of magnitude ofthe smallest eddies (Kolmogorov scale or inner scale) are:LK = Kolmogorov micro-scale = [υ3δU3]14LK = O(mm) >> Lmean free path = 6 x 10-8 mVelocity scale = (νε)1/4= O(10-2m/s)Time scale = (ν/ε)1/2= O(10-2s)Largest eddies contain most of energy, which break upinto successively smaller eddies with energy transfer to058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 5yet smaller eddies until LK is reached and energy isdissipated by molecular viscosity.Richardson (1922):LδBig whorls have little whorlsWhich feed on their velocity;And little whorls have lesser whorls,LKAnd so on to viscosity (in the molecular sense).(5) Dissipationℓ0=Lδu0=√k k =u '2+v '2+w '2=0 (U )Reδ=u0ℓ0/υ=bigε = rate of dissipation = energy/time=u02τoτo=ℓ0u0=u03l0independent υLK=[υ3ε]14Energy comes from largest scales and fed by mean motionDissipationoccurs at smallest scalesDissipation rate isdetermined by theinviscid large scaledynamics.Decrease in decreasesscale of dissipation LK notrate of dissipation .058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 6Fig. below shows measurements of turbulence forRex=107.Note the following mean-flow features:(1) Fluctuations are large ~ 11% U∞(2) Presence of wall causes anisotropy, i.e., thefluctuations differ in magnitude due to geometric andphysical reasons. u '2 is largest, v '2 is smallest andreaches its maximum much further out than u '2 or w '2.w '2 is intermediate in value.(3)u ' v '≠0 and, as will be discussed, plays a veryimportant role in the analysis of turbulent shear flows.(4) Although uiuj=0 at the wall, it maintains largevalues right up to the wall(5) Turbulence extends to y > δ due to intermittency. Theinterface at the edge of the boundary layer is called thesuperlayer. This interface undulates randomly between058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 7fully turbulent and non-turbulent flow regions. The meanposition is at y ~ 0.78 δ.(6) Near wall turbulent wave number spectra have moreenergy, i.e. small λ, whereas near δ large eddies dominate.058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 8Averages:058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 9For turbulent flow V (x, t), p(x, t) are random functions oftime and must be evaluated statistically using averagingtechniques: time, ensemble, phase, or conditional.Time AveragingFor stationary flow, the mean is not a function of time andwe can use time averaging.u=1T∫t0t0+tu (t ) dtT > any significant period of u '=u−u (e.g. 1 sec. for wind tunnel and 20 min. for ocean)Ensemble AveragingFor non-stationary flow, the mean is a function of timeand ensemble averaging is usedu(t )=1N∑i=1Nui(t )N is large enough that u independent ui(t) = collection of experiments performed under identical conditions (also can be phase alignedfor same t=o).058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 10058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 11Phase and Conditional AveragingSimilar to ensemble averaging, but for flows withdominant frequency content or other condition, which isused to align time series for some phase/condition. In thiscase triple velocity decomposition is used: u=u+u ''+u 'where u΄΄ is called organized oscillation.Phase/conditional averaging extracts all threecomponents.058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 12Averaging Rules:f =f +f 'g=g+g 's = x or tf '=0f =ff g=f gf ' g=0f +g=f +gf fs s� �=� �fg=f g+f ' g '∫f ds=∫f ds058:0160 Chapter 6-part3Professor Fred Stern Fall 2010 13Reynolds-Averaged Navier-Stokes EquationsFor convenience of notation use uppercase for mean andlowercase for fluctuation in Reynolds decomposition.u~i=Ui+uip~=P+ p23~0~ ~ ~ ~~1iii i iiii i j juxu u p uu gt x x x xu dr�=�� � � �+ =- + -� � � � �Mean Continuity Equation∂∂ xi(Ui+ui)=∂Ui∂ xi+∂ui∂ xi=∂Ui∂ xi=0∂u~∂ xi=∂Ui∂ xi+∂ui∂ xi=0 →∂ui∂ xi=0Both mean and
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