7. TURBULENCE SPRING 2019
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1 7. TRBLENCE SPRING What is turbulence? 7.2 Momentum transfer in laminar and turbulent flow 7.3 Turbulence notation 7.4 Effect of turbulence on the mean flow 7.5 Turbulence generation and transport 7.6 Important shear flows Summar Examples 7.1 What is Turbulence? A random, 3-d, time-dependent edding motion with man scales, superposed on an often drasticall simpler mean flow. A solution of the Navier-Stokes equations. The natural state at high Renolds numbers. An efficient transporter and mixer... of momentum, energ, constituents, A major source of energ loss. A significant influence on drag and boundar-laer separation. The last great unsolved problem of classical phsics ; (variousl attributed to Sommerfeld, Einstein and Fenman). Instantaneous Mean Jet Boundar laer () 7.2 Momentum Transfer in Laminar and Turbulent Flow In laminar flow, adjacent laers of fluid slide past each other without mixing. Transfer of momentum occurs between laers moving at different speeds because of viscous stresses. CFD 7 1 David Apsle
2 In turbulent flow adjacent laers continuall intermingle. A net transfer of momentum occurs because of the mixing of fluid elements from laers with different mean velocit. This mixing is a far more effective means of transferring momentum than viscous stresses. Consequentl, the mean-velocit profile tends to be more uniform in turbulent flow. Which regime the flow adopts depends on the Renolds number with low Renolds numbers favouring laminar flow, high Renolds numbers turbulent. What constitutes low or high depends on (a) the geometr of the flow; (b) what scales have been chosen for and L. There is not a single dividing value. 7.3 Turbulence Notation The instantaneous value of an flow variable can be decomposed into mean + fluctuation. v u is decomposed into Mean and fluctuating parts are denoted b either: an overbar and prime: upper case and lower case: mean + fluctuation The first is useful in deriving theoretical results, but cumbersome in general use. The notation being used should be obvious from the context. B definition, the average fluctuation is zero: In experimental work and in stead flow the mean is usuall a time mean, whilst in theoretical work it is the probabilistic (or ensemble ) mean. The process of taking the mean of a fluctuating quantit or product of fluctuating quantities is called Renolds averaging. The normal rules for averages of products appl: (variance) (1) (covariance) (2) Thus, in turbulent flow the mean of a product is not equal to the product of the means but includes an (often ver substantial) contribution from the net effect of turbulent fluctuations. CFD 7 2 David Apsle
3 7.4 Effect of Turbulence on the Mean Flow Engineers are usuall onl interested in the mean flow. However, turbulence must still be considered because although the averages of individual fluctuations (e.g. or ) are zero the average of a product ( ) is not and ma lead to a significant net flux. Consider mass and x-momentum fluxes in the direction across surface area A. For simplicit, assume constant densit Continuit Mass flux: Average mass flux: va The onl change is that the instantaneous velocit is replaced b the mean velocit. The mean velocit satisfies the same continuit equation as the instantaneous velocit Momentum va x-momentum flux: Average x-momentum flux: u The average momentum flux has the same form as the instantaneous momentum flux except for additional fluxes due to the net effect of turbulent fluctuations. These additional terms arise from averaging a product of fluctuating quantities; i.e. the are a consequence of the fluid-flow equations being non-linear. A net rate of transport of momentum from lower to upper side of an interface... is equivalent to a net rate of transport of momentum from upper to lower; has the same dnamic effect (rate of transfer of momentum) as a stress (force per unit area) of. This apparent stress is called a Renolds stress. Other Renolds stresses (,, etc.) emerge when considering the average flux of the different momentum components in different directions. The mean velocit satisfies the same momentum equation as the instantaneous velocit, except for additional apparent stresses: the Renolds stresses CFD 7 3 David Apsle
4 Total stress reall means the net rate of transfer of momentum (per unit area) and is given in simple shear flow b (3) In full-turbulent flow the turbulent contribution is usuall substantiall larger than the viscous stress. τ can be interpreted as either: the apparent force (per unit area) exerted b the upper fluid on the lower; or the rate of transport of momentum (per unit area) from upper fluid to lower. The dnamic effect a net transfer of momentum is the same. The nature of the turbulent stress can be illustrated b considering the fluctuations of fluid particles across some arbitrar surface. If particle A migrates upward (v > 0) then it tends to retain its original momentum, which is now lower than its surrounds (u < 0). If particle B migrates downward (v < 0) it tends to retain its original momentum which is now higher than its surrounds (u > 0). B A v' In both cases, is positive and, on average, reduces momentum in the upper fluid and increases momentum in the lower fluid. On its own this would give rise to a net transfer of momentum from upper to lower fluid, equivalent to the effect of an additional mean stress. Velocit Fluctuations Normal stresses: Shear stresses: In careless, but widespread, use, both and are referred to as stresses. The latter, with or without a minus sign(!), should reall be called kinematic turbulent stress. v Most turbulent flows are anisotropic; i.e. are different. u The level of turbulence ma be quantified b either its absolute or relative size: turbulent kinetic energ: (4) turbulence intensit: (5) CFD 7 4 David Apsle
5 7.4.3 General Scalar In general, the advection of an scalar quantit gives rise to an additional scalar flux in the mean-flow equations; e.g. va (6) Again, the extra term is the result of averaging a product of fluctuating quantities Turbulence Modelling At high Renolds numbers, turbulent fluctuations cause a much greater net momentum transfer than viscous forces throughout most of the flow. Thus, accurate modelling of the Renolds stresses is vital. A turbulence model or turbulence closure is a means of approximating the Renolds stresses (and other turbulent fluxes) in order to close the mean-flow equations. Section 8 will describe some of the commoner turbulence models used in engineering. 7.5 Turbulence Generation and Transport Production and Dissipation Turbulence is initiall generated b instabilities in the flow caused b mean velocit gradients. These eddies in their turn breed new instabilities and hence smaller eddies. The process continues until the eddies become sufficientl small (so that fluctuating velocit gradients become sufficientl large) that viscous effects become significant and dissipate turbulence energ as heat. This process the continual creation of turbulence energ at large scales, transfer of energ to smaller and smaller eddies, and the ultimate dissipation of turbulence energ b viscosit is called the turbulent energ cascade. PRODCTION b mean flow ENERGY CASCADE large eddies, scale L (low frequenc) Turbulence Transport DISSIPATION b viscosit small eddies, scale (high frequenc) l It is common experience that turbulence can be transported with the flow. (Think of the turbulent wake behind a vehicle or downwind of a large building.) The following can be proved mathematicall (see the appendix to Section 8 and the optional Section 10). Each Renolds stress satisfies its own scalar-transport equation. These equations contain the usual time-dependence, advection and diffusion terms. CFD 7 5 David Apsle
6 The source term for an individual Renolds stress has the form: where: production P ij is determined b mean velocit gradients; redistribution Φ ij transfers energ between stresses via pressure fluctuations; dissipation ε ij involves viscosit acting on fluctuating velocit gradients. The k equation just contains production (P (k) ) and dissipation (ε) terms because it doesn t distinguish individual stresses. The production terms for different Renolds stresses involve different mean velocit gradients; for example, it ma be shown that the rates of production (per unit mass) of and are, respectivel, (7) (Exercise: b pattern-matching write production terms for the other stresses). PRODCTION ADVECTION b mean flow 2 2 u v 2 w REDISTRIBTION b pressure fluctuations DISSIPATION b viscosit Turbulence is usuall anisotropic; i.e. are all different. This is because: (i) mean velocit gradients are greater in some directions than others; (ii) bod forces (particularl buoanc) onl act in one direction; (iii) motions in certain directions are selectivel damped (b boundaries). In practice, most turbulence models in regular use do not solve transport equations for all turbulent stresses, but onl for the turbulent kinetic energ, relating the other stresses to this b an edd-viscosit formula (see Section 8). CFD 7 6 David Apsle
7 7.6 Simple Shear Flows Flows for which there is onl one non-zero mean velocit gradient, /, are called simple shear flows. Because the form a good approximation to man real flows, have been extensivel researched in the laborator, and are amenable to basic theor, the are an important starting point for man turbulence models. For such a flow, the first of (7) and similar expressions show that P 11 > 0 but that P 22 = P 33 = 0, and hence tends to be the largest of the normal stresses. If there is a rigid boundar on = 0 then it will selectivel damp wall-normal fluctuations; hence is the smallest of the normal stresses. If there are densit gradients (for example in atmospheric or oceanic flows, in fires, or near heated or cooled surfaces) then buoanc forces will either damp (stable densit gradient) or enhance (unstable densit gradient) vertical fluctuations Free Flows Mixing laer v 2 u 2 u i u j Wake (plane or axismmetric) uv u 2 u i u j Jet (plane or axismmetric) uv 2 u v 2 u i u j For these simple flows: maximum turbulence occurs where (hence turbulence production) is largest; has the opposite sign to / and vanishes when this derivative vanishes; these turbulent flows are anisotropic:. CFD 7 7 David Apsle
8 7.6.2 Wall-Bounded Flows Pipe or channel flow D Flat-plate boundar laer constant-stress laer -uv v 2 u 2 u i u j Wall nits Even if the overall Renolds number,, is large, and hence viscous stresses much smaller than turbulent stresses in most of the boundar laer, there must ultimatel be a thin laer ver close to the wall where velocit fluctuations are damped and molecular viscosit is important. An important parameter is the wall shear stress τ w. Like an other stress this has dimensions of [densit] [velocit] 2 and hence it is possible to define an important velocit scale called the friction velocit u (sometimes written u * ): (8) From u τ and ν it is possible to form a viscous length scale ν/u τ. Hence, we ma define nondimensional velocit and distance from the wall in wall units: (9) + is a measure of proximit to a solid boundar and how important molecular viscosit is compared with turbulent transport. When + is small ( + < 5) viscous stresses dominate. When + is large ( + > 30) turbulent stresses dominate. In between there is a buffer laer. The total mean shear stress is made up of viscous and turbulent parts: When there is no streamwise pressure gradient, τ is approximatel constant over a significant depth and is equal to the wall stress τ w. CFD 7 8 David Apsle
9 Viscous Sublaer (tpicall, + < 5) Ver close to a smooth wall, turbulence is damped b the presence of the boundar. In this region the shear stress is predominantl viscous. Assuming constant shear stress, (10) i.e. the mean velocit profile in the viscous sublaer is linear. Log-Law Region (tpicall, + > 30) At large Renolds numbers, the turbulent shear stress greatl exceeds the viscous stress throughout most of the boundar laer so that, on dimensional grounds, since u τ and are the onl possible velocit and length scales, or where κ is a constant ( Ká á constant). Integrating, and putting part of the constant of integration inside the logarithm, gives either of the forms: or (11) κ, B and E are universal constants. Experiments give values 0.41, 5.0 and 7.8 respectivel. sing wall units (equation (9)) velocit profiles are often written in non-dimensional form: Viscous sublaer: (12) Log laer (smooth wall): or (13) Experiments indicate that, in full-developed flow, the log law is a good approximation over much of the boundar laer. (This is where the logarithm originates in friction-factor formulae such as the Colebrook-White formula for pipe flow). Consistenc with the log law is probabl the single most important consideration in the construction of turbulence models. For rough walls the logarithmic velocit profile is often written as or (with the wall-normal distance written as z rather than in the latter to reflect its common application in environmental flow. k s is the Nikuradse roughness; z 0 is just called the CFD 7 9 David Apsle
10 roughness length and varies tpicall from about 0.1 m in rural settings to 1 m in cities.) Summar Turbulence is a 3-d, time-dependent, edding motion with man scales, causing continuous mixing. An instantaneous flow variable ma be decomposed as mean + fluctuation. The process of averaging turbulent variables or their products is called Renolds averaging and leads to the Renolds-averaged Navier-Stokes (RANS) equations. Turbulent fluctuations make a net contribution to the transport of momentum and other quantities. Turbulence enters the mean momentum equations via the Renolds stresses; e.g. A means of specifing the Renolds stresses (and hence solving the mean flow equations) is called a turbulence model or turbulence closure. Turbulence energ is generated at large scales b mean-velocit gradients (and, sometimes, bod forces such as buoanc). Turbulence is dissipated (as heat) at small scales b viscosit. Due to the directional nature of the generating process (mean shear and/or bod forces) turbulence is initiall anisotropic. Energ is subsequentl redistributed amongst the different components b the action of pressure fluctuations. Turbulence modelling is guided b experiment and theor for simple free shear flows (mixing laer, jet, wake) and wall-bounded flows (pipe, flat plate). Theor and experiment suggest that throughout much of a turbulent boundar laer the shear stress is constant: and the velocit profile is logarithmic: Wall units determine regions of a turbulent boundar laer. In the viscosit-dominated region ( + < 5) the velocit profile is linear. In the full-turbulent region ( + > 30) an equilibrium-boundar-laer velocit profile has a universal logarithmic form. CFD 7 10 David Apsle
11 Examples Q1. Which is more viscous, air or water? Air: ρ = 1.20 kg m 3 μ = kg m 1 s 1 Water: ρ = 1000 kg m 3 μ = kg m 1 s 1 Q2. The accepted critical Renolds number in a round pipe (based on bulk velocit and diameter) is At what speed is this attained in a 5 cm diameter pipe for (a) air; (b) water? Q3. Sketch the mean velocit profile in a pipe at Renolds numbers of (a) 500; (b) What is the shear stress along the pipe axis in either case? Q4. Explain the process of flow separation. How does deliberatel tripping a developing boundar laer help to prevent or dela separation on a convex curved surface? Q5. The following couplets are measured values of (u,v) in an idealised 2-d turbulent flow. Calculate,,,, from this set of numbers. (3.6,0.2) (4.1, 0.4) (5.2, 0.2) (4.6, 0.4) (3.4,0.0) (3.8, 0.4) (4.4,0.2) (3.9,0.4) (3.0,0.4) (4.4, 0.3) (4.0, 0.1) (3.4,0.1) (4.6,-0.2) (3.6,0.4) (4.0,0.3) Q6. The rate of production (per unit mass of fluid) of and are, respectivel, (a) (b) (c) B inspection, write down similar expressions for P 22, P 33, P 23, P 31, the rates of production of,, and respectivel. Write down expressions for P 11, P 22, P 33 and P 12, P 23, P 31 in simple shear flow (where / is the onl non-zero mean velocit gradient). What does this indicate about the relative distribution of turbulence energ amongst the various Renolds-stress components? Write down also an expression for P (k), the rate of production of turbulence kinetic energ, in this case. The different production terms can be summarised b a single compact formula using the Einstein summation convention implied summation over a repeated index (in this case, k). See if ou can relate this to the above expressions for the P ij. CFD 7 11 David Apsle
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