058 0160 Professor Fred Stern Fall 2009 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 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 to much smaller scales which become progressively smaller as the Re U increases 058 0160 Professor Fred Stern Fall 2009 Chapter 6 part3 2 058 0160 Professor Fred Stern Chapter 6 part3 3 Fall 2009 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 u u u 1 u u dT T t 0 T 1 u u dT T u 0 2 t0 t 0 T 2 etc t0 u mean motion u superimposed random fluctuation u Reynolds stresses RMS u 2 2 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 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 Professor Fred Stern Chapter 6 part3 4 Fall 2009 3 Diffusion Large scale mixing of fluid particles greatly enhances diffusion of momentum and heat i e viscous stress Reynolds Stresses 6 474 8 u u Isotropic eddy viscosity 2 u i u j t ij ij k 3 i j ij ij 4 Vorticity eddies energy cascade Turbulence is characterized by flow visualization as eddies which vary 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 1 4 3 LK Kolmogorov micro scale 3 U 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 yet smaller eddies until LK is reached and energy is dissipated by molecular viscosity i e viscous diffusion 058 0160 Professor Fred Stern Chapter 6 part3 5 Fall 2009 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 l 0 L 2 u0 k 2 k u v w Energy comes from largest scales and fed by mean motion 2 0 U Re u 0 l 0 big rate of dissipation energy time u02 u03 o l0 o l0 Dissipation occurs at smallest scales u0 independent 1 3 4 LK 058 0160 Professor Fred Stern Chapter 6 part3 6 Fall 2009 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 u is largest v is smallest and reaches its maximum much further out than u or w w is intermediate in value 2 2 2 2 2 3 u v 0 and as will be discussed plays a very important role in the analysis of turbulent shear flows 4 Although u u 0 at the wall it maintains large values right up to the wall i j 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 058 0160 Professor Fred Stern Fall 2009 Chapter 6 part3 7 between 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 Fall 2009 Chapter 6 part3 8 058 0160 Professor Fred Stern Fall 2009 Chapter 6 part3 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 1 t0 t u u t dt T any significant period of u u u T t0 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 1 N i u t u t N is large enough that u independent N i 1 ui t collection of experiments performed under identical conditions also can be phase aligned for same t o 058 0160 Professor Fred Stern Fall 2009 Chapter 6 part3 10 058 0160 Professor Fred Stern Chapter 6 part3 11 Fall 2009 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 Averaging Rules f f f f 0 f g f g f ds f ds g g g f f f f s s s x or t fg fg f g 0 fg f g f g 058 0160 Professor Fred Stern Chapter 6 part3 12 Fall 2009 Reynolds Averaged Navier Stokes Equations For convenience of notation use uppercase for mean and lowercase for fluctuation in Reynolds decomposition u i U i ui p P p ui 0 xi ui ui 2 u i 1 p ui g i 3 xi t xi x j x j NS equation Mean Continuity Equation U u U U u 0 x x x x u U u u 0 0 x x x x i i i i i i i i i i i i i i i i Both mean and fluctuation satisfy divergence 0 condition 058 0160 Professor Fred Stern Chapter 6 part3 13 Fall 2009 Mean Momentum Equation 1 P p U i ui U j u j U i ui xi t x j 2 U i ui g i 3 x j x j U u U U u t t t t i i i U u j i i j U u U u U u U U u u x x x x x U U uu x x i i i i i j j j …
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