Lund Institute of Technology Department of Energy Sciences Division of Fluid Mechanics Robert Szasz, tel 046-0480 Johan Revstedt, tel 046-43 0 Turbulence - Theory and Modelling GROUP-STUDIES: Turbulence Theory During this course the theory behind transition, turbulence and turbulence modelling will be discussed and explained. This exercise will cover the turbulence theory part of the course, but the focus will be on two classes of flow, free shear flow (Ch. 5, Pope) and wall bounded shear flow (Ch. 7, Pope). You should preferably work in groups of -4 persons. The answers to the questions should be handed in latest 15 November to Johan Revstedt, one report per group! Note! Questions in italics are optional for undergraduate students. Preparation, read: Pope, S.B., Turbulent flows, Ch. -7 Suggestions for extracurricular reading: Tennekes, H. and Lumley, J.L., A first course in turbulence, MIT Press, 197. Landahl, M. and Mollo-Christensen, E., Turbulence and random processes in fluid mechanics, Cambridge University Press, 199 Mathieu, J. and Scott, J., An introduction to turbulent flow, Cambridge University Press, 000. Governing equations and averaging 1. Do exercise 4.4 (Pope, p. 90). Do exercise 4.9 (pope, p. 95) 3. Define: a. Isotropic turbulence b. homogeneous turbulence c. statistical stationarity Free shear flow 1. Give some examples where the following phenomena appear: a) far wake b) mixing layer c) jet. What does self-similarity mean? 3. Define turbulent kinetic energy (k). How large are the fluctuations in velocity, as a function of k, in isotropic turbulence?
4. Define the spreading rate. Also, do Exercise 5.1 (Pope, p. 103) 5. Do exercise 5.3 on page 13 (Pope). 6. Do exercise 5.5 on page 133 (Pope). Scales of turbulent motion 1. Sketch an energy spectrum for a turbulent flow. Mark the different regions and indicate the locations of the relevant length scales.. Do exercise 6.6 on page 00 (Pope). Wall bounded shear flow 1. Give at least three examples of technical applications where wall bounded shear flow occurs.. Sketch the velocity profiles for both fully developed laminar and turbulent pipe flow. Give the physical background to the differences between the two velocity profiles. 3. The boundary layer can be divided into different layers. Name these layers and which length scales are valid in the different layers. 4. Derive the velocity profile in the inertial sub-layer, state clearly the assumptions that you make. 5. Figure 7.34 on page 314 show the energy budget for turbulent kinetic energy. Now consider Figure 7.34b. At which y + -value does the peak value of gain occur? Give a physical explanation. 6. Do exercise 7.7 on page 87 (Pope) 7. Do exercise 7.4 on page 319 (Pope)
Answers to the Case-Study Wall free shear flow. Examples of far wake, mixing layer and jet: Far wake: δ A wake is created behind objects moving in a fluid, for example cars, trains, ships, submarines, missiles without engines, bullets, footballs, wingprofiles etc. If the fluid is moving, then the object may be steady or move in another velocity than the fluid. For example the flow around radio masts, bridges, flag poles, chimneys etc. Jet: δ Jet flow is found in rocket engines, jet airplanes, water hoses, engines of direct injection etc. Mixing layer: δ Mixing layer flow (layers of different velocities) is for example found in the oceans, in the atmosphere, at the wind blowing past a roof, in the close region behind bodies moving in a fluid etc. Note that both jets and wakes contain mixing layers. Further, some flow fields will be some kind of mix of the above mentioned flow cases. For example rockets creates a mix of far wake and jet flow. 3. Self-similarity? Self-similarity means that velocity profiles at different places downstream will have the same shape when scaled by some velocity. When the flow fulfils self-similarity, then the details of the geometry and the flow situation far enough upstream has negligible effect on the flow downstream. For example, far enough
downstream of an object, the behaviour of a wake flow will be similar independent of what kind of object that created the wake. For these kinds of flow, the approximate velocity can be expressed as a function: y U( x, y ) = U0( x ) F = U0( x ) F( η ) δ ( x ) By plotting U( x, y ) / U0 ( x ) versus η, the scaled velocity profile in a cross section to x will be similar independent of x. A flow of this behaviour is also called "self-preserved". An advantage of a self-similar flow is that you can calibrate your model to a cheap smallscale experiment instead of an expensive large-scale experiment, since the scaling is straightforward. 4. Physical meaning of l mix : Compare l mix with l mean free path : l mean free path = the average distance a molecule keeps its momentum (then it will collide and exchange momentum) l mix can be seen as a "turbulent mean free path", i.e. an equivalent length that is describing the momentum exchange (the distance that an equivalent eddy is keeping its momentum). Wall free and wall bounded shear flow: For both wall free shear flows and wall bounded shear flows, l mix is a length comparable to the thickness of the disturbed region, since the thickness of the disturbed region is an estimate of how big an eddy may be and how far it may travel before it is breaked down. For wall free shear flows, l mix has a simple expression: l mix = α δ(x) For wall bounded shear flows, the boundary layer is divided into several regions, namely: viscous sublayer: l mix = υ/u τ inertial subrange (log layer): l mix = κ y defect layer: l mix = α δ(x) outside the boundary layer: l mix = α y 5. Turbulent kinetic energy Definition: k = 1 uiui = 1 u + v + w where prime denotes the fluctuations.
3 Isotropic turbulence: u = v = w => k = u => u RMS = u = k 3 6. Intermittency factor The intermittency factor is a factor that takes into account that the flow is sometimes turbulent and sometimes laminar. The intermittency factor in a specified point is in Tennekes/Lumley defined as the relative fraction of time that the flow in that the point is turbulent. An example of both laminar and turbulent flow is the turbulent wake behind a body in a laminar free stream, but if the free stream is turbulent there may be different levels of turbulence that the factor may be used for as well. For example in a turbulent boundary layer, the flow is laminar close to the wall (since the velocity approaches zero at the surface of a non-moving body, viscous forces always dominates there). Another example is the laminar free stream past a flat plate where the boundary layer is turbulent. The flow is as well laminar close to the wall in the viscous sublayer (since the velocity approaches zero at the surface of an unmoving body, viscous forces always dominate there), but close to that region the Van Driest damping factor may be used instead. In the Baldwin-Lomax model, the "intermittency" is a correction factor for outer viscosity. 7. Spreading rate The definition may differ for different authors, but it means how fast the disturbed region is extended downstream. In Wilcox (p. 99), the definition for jet flow is Δy/Δx where y is the coordinate where U(x,y)=U max (x)/ (if the flow fulfils self-similarity, then the spreading rate is equal to δ/ x). 8. Reason for success of algebraic models for these types of flows: 1. Used for simple flow cases with only one characteristic length and with slowly varying properties.. Calibration of constants to experimental data for each case can be done rather easily (a lot of data available). 3. The turbulence can be described by a local model, i.e. the development of the flow is only depending on local time and length scales. 4. Characteristic turbulent time scales << characteristic time scales of the mean flow. 5. Dimension analysis is fulfilled (of course). 6. Only one length and no extra equations => easy to use.
9. Asymptotics of wall free shear layers Assumptions: 1. The flow is almost parallel, -dimensional and the mean flow is steady.. The turbulence is slowly developing in the streamwise direction. 3. The Reynolds number is high. 4. Small terms in Navier-Stokes equations are negligible. 5. Incompressible, Newtonian fluid 6. Molecular viscosity neglected 7. dp/dx = 0 (x is the flow direction) 8. The turbulence is fully developed. Basic results: 1. The flow is "self-similar".. Only one characteristic length scale of the flow. 3. A mixing-length model with one coefficient (based upon Boussinesq eddy-viscosity approximation). 4. The model yields good results if the coefficient is calibrated to experimental data for each flow case (see page 44 in Wilcox): α far wake = 0.180 α mixing layer = 0.071 α plane jet = 0.098 α round jet = 0.080 Our formulation is therefore NOT universal (is not valid for all flow cases without calibration). Why is the value of α far wake bigger than the others? If comparison is made to the jet values, where the velocity in the turbulent region is higher than in the free stream compared to the wake where the relation is the opposite, then the time scales of the jet flow should be smaller compared to the wake flow, yielding smaller eddies and less values of the mixing length, Jet flow: If the free stream value of the turbulent kinetic energy is high, then there is much turbulence. Turbulence increases mixing of momentum yielding higher spreading rate. If the dissipation rate is high, then the turbulence is "killed" fast. If the turbulence is decreasing, the mixing will be less and the spreading rate will be reduced.
Answers to the Case Study Wall bounded shear flow. Examples of wall bounded shear flows: a. around vehicles such as aeroplanes, cars, trains, rockets, boats b. around turbine blades or propellers c. in channels, pipes, heat exchangers, ventilation systems 3. Assumptions a. Incompressible flow b. Newtonian fluid (continuum, constant viscosity) c. The mean flow is steady and two dimensional d. The velocity is zero at the wall due to viscous forces e. The flow is parallel to the wall close to the surface (u >> v) f. The wall is smooth (roughness is small) g. The Reynolds number is high h. The development of the flow in the flow direction is slow i. The pressure gradient in the x direction is limited j. Only the local scales δ, ν/u * and u * are relevant in the dimension analysis. δ is the thickness of the boundary layer, ν/u * is the length scale in the viscous sublayer and u * is the friction velocity. k. δ/l<<1 where L is a length scale in the flow direction l. The velocity is equal to the free stream velocity at y = δ (in case there is a free stream) 4. Important results The analysis result in several different length scales in different parts of the boundary layer. There is a viscous region where the velocity profile is linear. If δ u * /ν is large enough, there exist an "inertial sublayer" where y (the distance to the wall) is the only relevant length scale. The turbulence models must reflect the different layers. For example, the Baldwin-Lomax model (Wilcox page 5) has an inner and outer eddy viscosity. The inner viscosity is damped close to the wall by the exponential expression below, due to the viscous sublayer. µ T,inner = ρ mix l w + + where y / A0 = + y lmix κ y 1 e and y = υ If y approaches zero, then μ T,inner approaches zero as well. The damping factor of l mix makes μ T,inner approaches zero even faster. For two-equation models based on Boussinesq hypothesis integration through the viscous sublayer to the wall poses a problem. The problem is more severe for k-ε models compared to k-ω models. Without corrections the standard k-ε model fails to give the right value for B in the law of the wall * u
+ ( y ) + B u 1 = ln * u κ while k-ω performs better. Furthermore, the models have problems for the wall bounded flows with adverse pressure gradients and/or separation. This problem is more severe for k-ε formulations. k-ω gives reasonable results when integrated all the way to the wall without viscous corrections while this is not the case for the k-ε model. 5. Pipe flow Turbulent flow is mixing both mass and momentum. The mixing of momentum makes the velocity profile more flat. If the mass flow rate is assumed constant for a turbulent case compared to a laminar case, the gradient at a solid boundary will be bigger and the maximum velocity will be less. mean velocity laminar turbulent Figure. Laminar and turbulent velocity profile. 6. Different layers of the boundary layer For wall bounded shear flows, the boundary layer is divided into several regions, namely: Viscous sublayer: Inertial subrange (log layer): Defect layer: l mix ~ υ/u τ l mix ~ κ y l mix ~ δ(x) The range between the viscous sublayer and inertial subrange is called buffer layer (T&L, page 160). 7. Velocity profile in the inertial sublayer For a steady, two-dimensional incompressible flow of a Newtonian fluid, assume that convection terms and pressure terms are much smaller than the viscous terms and the Reynolds stresses. Assume as well that the development in the x-direction is negligible. Then the mass and momentum equations may be simplified as: U U µ + τ xy = 0 => µ + τ xy = C Since this expression is valid for all y close to the surface, then C = τ w = ρ u τ. In the inertial sublayer, the viscous part is negligible compared to the turbulent one. The Reynolds stress term is approximated by
τ xy µ T U U where µ T = ρ lmix and l mix = κ y. This results in U ρ κ y = ρ uτ U If > 0 (i.e. no separation), this yields U u τ U 1 y = => = ln( y ) + D κ uτ κ where D is constant. Using y + = y u * /ν, the formula can be rewritten into U 1 = ln( y + ) + B * u κ A fully derivation can be found in T&L, chapter 5.. B is approximately 5 for a smooth wall (small roughness) and sufficiently high Reynolds number. In the viscous sublayer, the turbulent part is negligible compared to the viscous one. Since the derivative of the velocity is constant for a Newtonian fluid (constant viscosity), the velocity profile is linear in the viscous sublayer. 8. Gain and loss of turbulent kinetic energy Maximum gain: y + 10 Gain = Turbulent production τ ij du i /dx j where τ = ρu u is the Reynolds stress tensor. ij i j The turbulent production is to be maximised: 1. τ ij is decreasing when the distance to the wall is decreasing. du i /dx j is decreasing when the distance to the wall is increasing Since one of the two parameters is small when the distance to the wall is small/big, there must be a maximum somewhere in between. The maximum is where both parameters are large, i.e. close to the wall but outside of the viscous sublayer.