058 0160 Professor Fred Stern Fall 2010 Chapter 6 part3 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 loworder statistical quantities can be modeled and used in conjunction with the averaged equations for solving practical engineering problems Turbulent motions range in size from the width in the flow to much smaller scales which become progressively smaller as the Re U increases 058 0160 Professor Fred Stern Fall 2010 Chapter 6 part3 2 058 0160 Professor Fred Stern Chapter 6 part3 3 Fall 2010 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 1 u T u u u t 0 T t0 u dT u 0 1 u 2 T t 0 T u 2 dT t0 etc u mean motion u superimposed random fluctuation 2 u 2 Reynolds stresses RMS u Triple decomposition is used for forced or dominant frequency flows u u u u Where u organized oscillation 2 Nonlinearity Reynolds stresses and 3D vortex stretching are direct result of nonlinear nature of turbulence In fact Reynolds stresses arise from nonlinear convection term after 058 0160 Professor Fred Stern Chapter 6 part3 4 Fall 2010 substitution of Reynolds decomposition into NS equations and time averaging 3 Diffusion Large scale mixing of fluid particles greatly enhances diffusion of momentum and heat i e viscous stress Reynolds Stresses Isotropic eddy viscosity u i u j ij ij 2 u i u j t ij ij k 3 4 Vorticity eddies energy cascade Turbulence is characterized by flow visualization as eddies which varies in size from the largest L width of flow to the smallest The largest eddies have velocity scale U and time scale L U The orders of magnitude of the smallest eddies Kolmogorov scale or inner scale are 3 3 U LK Kolmogorov micro scale 1 4 LK O mm Lmean free path 6 x 10 8 m Velocity scale 1 4 O 10 2m s Time scale 1 2 O 10 2s Largest eddies contain most of energy which break up into successively smaller eddies with energy transfer to 058 0160 Professor Fred Stern Chapter 6 part3 5 Fall 2010 yet smaller eddies until LK is reached and energy is dissipated by molecular viscosity Richardson 1922 L 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 0 L u0 k k u v w 0 U Re u 0 0 big 2 2 Energy comes from largest scales and fed by mean motion 2 rate of dissipation energy time u 20 o o u0 u30 l0 independent Dissipation rate is determined by the inviscid large scale dynamics Dissipation occurs at smallest scales 0 LK 3 1 4 Decrease in decreases scale of dissipation LK not rate of dissipation 058 0160 Professor Fred Stern Fall 2010 Chapter 6 part3 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 causes anisotropy i e fluctuations differ in magnitude due to geometric 2 2 physical reasons u is largest v is smallest 2 reaches its maximum much further out than u or w 2 is intermediate in value the and and w 2 3 u v 0 and as will be discussed plays a very important role in the analysis of turbulent shear flows 4 Although ui u j 0 at the wall it maintains large values right up to the wall 5 Turbulence extends to y due to intermittency The interface at the edge of the boundary layer is called the superlayer This interface undulates randomly between 058 0160 Professor Fred Stern Fall 2010 Chapter 6 part3 7 fully turbulent and non turbulent flow regions The mean position is at y 0 78 6 Near wall turbulent wave number spectra have more energy i e small whereas near large eddies dominate 058 0160 Professor Fred Stern Averages Fall 2010 Chapter 6 part3 8 058 0160 Professor Fred Stern Fall 2010 Chapter 6 part3 9 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 u 1 T t 0 t u t dt T any significant period of u u u e g 1 sec for wind tunnel and 20 min for ocean t0 Ensemble Averaging For non stationary flow the mean is a function of time and ensemble averaging is used N 1 u t u i t N i 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 Professor Fred Stern Fall 2010 Chapter 6 part3 10 058 0160 Professor Fred Stern Fall 2010 Chapter 6 part3 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 u u u u where u is called organized oscillation Phase conditional averaging extracts all three components 058 0160 Professor Fred Stern Chapter 6 part3 12 Fall 2010 Averaging Rules f f f f 0 f g f g f ds f ds g g g f f s x or t f g f g f f s s f g 0 fg f g f g 058 0160 Professor Fred Stern Fall 2010 Chapter 6 part3 13 Reynolds Averaged Navier Stokes Equations For convenience of notation use uppercase for mean and lowercase for fluctuation in Reynolds decomposition ui U i ui p P p u i 0 xi 2 u i u i 1 p u i u i u gdi 3 t xi r xi xj xj NS equation Mean Continuity Equation U u U i ui U i 0 xi i i xi xi xi ui u U i ui 0 0 xi xi xi xi Both mean and fluctuation satisfy divergence 0 condition 058 0160 Professor Fred Stern Chapter 6 part3 14 Fall 2010 Mean Momentum Equation 1 P p U i ui U j u j U i ui t xj r xi 2 U i ui gdi 3 xj xj U u U i u i U i t i i t t t U i ui U i ui U j u j U u U j U j u u …
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