058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 1 Chapter 6: Viscous Flow in Ducts 6.3 Turbulent Flow Most flows in engineering are turbulent: flows over vehicles (airplane, ship, train, car), internal flows (heating and 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 forced or dominant frequency unsteady flows display coherent features and are amendable to statistical analysis, i.e. time and place (conditional) averaging. RMS and other low-order statistical quantities can be modeled and used in conjunction with averaged equations for solving practical engineering problems. Turbulent motions range in size from the width in the flow δ to much smaller scales, which become progressively smaller as the Re = Uδ/υ increases.058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 2058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 3 Physical description: (1) Randomness and fluctuations: Turbulence is irregular, chaotic, and unpredictable. However, for statistically stationary flows, such as steady flows, can be analyzed using Reynolds decomposition. 'uuu += ∫=+TttdTuTu001 0'=u dTuTuTtt∫=+0022'1' etc. u = mean motion 'u = superimposed random fluctuation 2'u = Reynolds stresses; RMS = 2'u Triple decomposition is used for forced or dominant frequency flows ''' uuuu ++= Where ''u = organized oscillation (2) Nonlinearity Reynolds stresses and 3D vortex stretching are direct results of nonlinear nature of turbulence. In fact, Reynolds stresses arise from nonlinear convection term after substitution of Reynolds decomposition into NS equations and time averaging.058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 4 (3) Diffusion Large scale mixing of fluid particles greatly enhances diffusion of momentum (and heat), i.e., Reynolds Stresses: 48476stressviscousijijjiuuμετρ=>>− '' Isotropic eddy viscosity: kuuijijtjiδευ32'' −>>− (4) Vorticity/eddies/energy cascade Turbulence is characterized by flow visualization as eddies, which vary in size from the largest Lδ (width of flow) to the smallest. The largest eddies have velocity scale U and time scale Lδ/U. The orders of magnitude of the smallest eddies (Kolmogorov scale or inner scale) are: LK = Kolmogorov micro-scale = 4133⎥⎦⎤⎢⎣⎡Uδυ LK = O(mm) >> Lmean free path = 6 x 10-8 m Velocity 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 up into successively smaller eddies with energy transfer to yet smaller eddies until LK is reached and energy is dissipated by molecular viscosity (i.e. viscous diffusion).058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 5 Richardson (1922): Lδ Big whorls have little whorls Which feed on their velocity; And little whorls have lesser whorls, LK And so on to viscosity (in the molecular sense). (5) Dissipation biguUwvukkuL===++===υδδ/Re)(0'''0022200ll ε = rate of dissipation = energy/time ouτ20= 00uol=τ =030lu independent υ 413⎥⎦⎤⎢⎣⎡=ευKL Energy comes from largest scales and fed by mean motion Dissipation occurs at smallest scales058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 6 Fig. below shows measurements of turbulence for Rex=107. Note the following mean-flow features: (1) Fluctuations are large ~ 11% U∞ (2) Presence of wall cause anisotropy, i.e., the fluctuations differ in magnitude due to geometric and physical reasons. 2'u is largest, 2'v is smallest and reaches its maximum much further out than 2'u or 2'w . 2'w is intermediate in value. (3) 0'' ≠vu and, as will be discussed, plays a very important role in the analysis of turbulent shear flows. (4) Although 0=jiuu at the wall, it maintains large values right up to the wall (5) Turbulence extends to y > δ due to intermittency. The interface at the edge of the boundary layer is called the superlayer. This interface undulates randomly058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 7 between fully turbulent and non-turbulent flow regions. The mean position is at y ~ 0.78 δ. (6) Near wall turbulent wave number spectra have more energy, i.e. small λ, whereas near δ large eddies dominate.058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 8058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 9 Averages: For turbulent flow V (x, t), p(x, t) are random functions of time and must be evaluated statistically using averaging techniques: time, ensemble, phase, or conditional. Time Averaging For stationary flow, the mean is not a function of time and we can use time averaging. ∫+=tttdttuTu00)(1 T > any significant period of uuu−=' (e.g. 1 sec. for wind tunnel and 20 min. for ocean) Ensemble Averaging For non-stationary flow, the mean is a function of time and ensemble averaging is used ∑==NiituNtu1)(1)( N is large enough that u independent ui(t) = collection of experiments performed under identical conditions (also can be phase aligned for same t=o).058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 10058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 11 Phase and Conditional Averaging Similar to ensemble averaging, but for flows with dominant frequency content or other condition, which is used to align time series for some phase/condition. In this case triple velocity decomposition is used: ''' uuuu ++= where u΄΄ is called organized oscillation. Phase/conditional averaging extracts all three components. Averaging Rules: 'fff += 'ggg += s = x or t 0'=f ff = gfgf = 0' =gf gfgf +=+ ffss∂∂=∂∂ '' gfgffg += ∫=∫dsfdsf058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 12 Reynolds-Averaged Navier-Stokes Equations For convenience of notation use uppercase for mean and lowercase for fluctuation in Reynolds decomposition. pPpuUuiii+=+=~~ 23~0~~ ~ ~~1iiii iiiiijjuxuu puugtx xxxυδρ∂=∂∂∂ ∂∂+=−+ −∂∂ ∂∂∂ Mean Continuity Equation 00~0)(=∂∂→=∂∂+∂∂=∂∂=∂∂=∂∂+∂∂=+∂∂iiiiiiiiiiiiiiiixuxuxUxuxUxuxUuUx Both mean and fluctuation satisfy divergence = 0 condition. NS equation058:0160 Chapter 6-part3 Professor Fred Stern Fall 2009 13 Mean Momentum
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