Intro. Turbulent Flows INTRODUCTION TO TURBULENT FLOWS What is turbulence? How is turbulence created? How do we solve turbulent flows? General Purposes - u and U Mass diffussion and concentration statistics Details of turbulent motion and how they interact Three Types of turbulence - Grid turbulence -- not self-sustaining Wall shear layers --- self sustaining - wall effect as a turbulence source Free shear layers -- mixing layers -- two fluid at different speeds -- jets, wakes Characteristics of turbulence - Velocity fluctuates in a random manner -- Statisitically can be studied High levels of vorticity fluctuations High Reynolds numbers Described by the Navier-Stokes (N-S) equations Higher levels of momentum and energy transfers Dissivative Continuum level Certain spatial structures - eddies –vortices - mushroom like, etc, all are distributed continuesly. Reynolds Decompositions (RANS = Reynolds Average of N-S) u~ i U i ui Ui Lim 1 T T t 0 T t0 u~ i dt 1 Intro. Turbulent Flows ui u~ (t), m/s Ui Fig. 1. Typical of velocity fluctuation in turbulent flows. The mean value of a fluctuation components is zero, ui Lim 1 T T t 0 T t0 (u~ i U i)dt 0 For time averages to make sense, the integral have to be independent of t0. U i 0 t u~i U i ui ui ; 0 x j x j x j x j u~i u~ j (U i ui )(U j u j ) =0 =0 U iU j U i u j u iU j u i u j U iU j ui u j 2 Intro. Turbulent Flows u i u j 0 if ui and uj are correlated. = 0 if ui and uj are uncorrelated. ~ p P if ~p = P + p. xi xi N-S equations can be written as u~i ~ u~i 2 u~i 1 ~ p uj t x j xi x j x j u~i 0 xi u~ i and u~ j are the instantaneous velocities. (1) (2) Equation for the mean flow for a turbulent flow The momentum equation is obtained by substituting u~ i U i ui and ~p = P + p and taking a time average ( U i 0 ). t U i ui 2U i 1 P Uj uj x j x j xi x j x j (3) The continuity equation becomes: U i u 0 and i 0 . xi xi (U i u i ) u~i 0 0 xi xi uj u j u i u j u j u i u u j i ui 0. , since u i x j x j x j x j x j U i 2U i 1 P Uj ui u j x j xi x j x j x j Uj U i 1 P 1 U i ui u j x j xi x j x j or 3 Intro. Turbulent Flows P ij U i u i u j x j where ij is Kronecker delta. ij = i if i = j, and 0 if i j. Uj U i x j x j (4) Equation (4) is the momentum equation for the mean flow. For turbulent flows, it is not enough equations to solve the problem, because of the attendance of the Reynolds stress tensors u i u j . This leads to a closure problem. In general, one can write: ij ui u j (5) 11 12 13 where ij 21 22 23 . If ij is symetric, then ij ji and there are six independent 31 32 33 components, instead of nine. The diagonal components of ij are normal stresses: u12 ; u 22 ; u 32 . The off-diagonal components of ij are shear stresses, and they play an important role in the transport of mean momentum by turbulent motion. One of the methods to solve the closure problem is the use of turbulence models. How do we estimate or model u i u j ? Length Scales in Turbulent Flows Turbulent flows are characterized by the existence of several lengths. Consider a laminar boundary layer flow over a flat plate: u = u(y) U y x L Fig. 2. Laminar boundary layer over a flat plate. L = convective length scale U = convective velocity scale = diffusive length scale 4 Intro. Turbulent Flows The time scale is then: L . U We can also estimate that L ~ U UL 1/ 2 1 . Re 1L/ 2 (6) Furthermore, L is related to the convection of momentum, and relates to the molecular diffusion of momentum deficit across the flow, away from the surface. Suppose we have a turbulent boundary layer: U largest eddy size y u u smallest eddy size L x Fig. 3. Turbulent boundary layer over a flat plate. L convective time scale. U ~ (u) and L U u ~ L U where u is the characteristic velocity fluctuation. L ~ turbulent diffusion time scale matches the convective time scale. U u What is the smallest eddy size? or 5 Intro. Turbulent Flows What is the smallest length scale in the turbulent flow? Large scales are as big as the width of the flow Interact with the mean flow. The smallest scales? Kolmogorov length scale. Energy transfer among eddies Large scale smallest eddies dissipation to heat Fig. 4. Energy cascade in turbulent motion. Turbulence generates new (smaller) length scales till the local Reynolds number becomes small for viscous dissipation to become significant. How can we estimate the Kolmogorov length scale ()? Kolmogorov theory: ~ f(,) (7) where = Kolmogorov length scale, = dissipation rate, and = kinematic viscosity For an equilibrium turbulent boundary layer, dissipation = energy input The dissipation () and can be used to obtain the smallest scales of length (), time (), frequency (fK), and velocity (), in the flow, and are referred to as the Kolmogorov scales (Tennekes and Lumley, 1972) = (3/)1/4 (8) = (/)1/2 (9) fK = U/(2) (10) 6 Intro. Turbulent Flows (11) The turbulent kinetic energy dissipation rate plays an important role in turbulent flow analysis. In an isotropic turbulence, the dissipation rate, , is given by 15 u / x 2 (12) Values of u / x can be obtained experimentally using Taylor’s frozen hypothesis. The 2 hypothesis states that if the turbulent velocity fluctuations are small compared to the mean velocity, then the autocorrelation of the fluctuating velocity with time delay will be the same as the spatial correlation with separation U in the streamwise direction (Bradshaw, 1971). In a mathematical form, the Taylor’s hypothesis can be expressed as u 1 u x U ( y ) t (13) where u / t can usually be obtained from instantaneous velocity measurements. In addition to the Kolmogorov length scale, the Taylor microscale ( Taylor) and integral length scale () are often used in the analysis of turbulent flows. The Taylor microscale, with dimension of length, is defined as (after Tennekes and Lumley, 1972) Taylor u2 u / x 1/ 2 2 (14) By recalling Taylor’s frozen hypothesis, based on Eq. (14), we can obtain () = ( ) (15) where ( ) is the spatial correlation with separation ( = U. Figure 5 shows a typical spatial correlation curve. The area under the spatial correlation curve is the integral length scale, so that 7 Intro. Turbulent Flows ( ) d (16) 0 The Taylor microscale (Taylor) and the integral length scale ( ) are far larger than the Kolmogorov length scale (). 1.0 ( ) 0 Fig. 5. A typical spatial correlation curve. 8