** Fluid Mechanics **3rd Year Mechanical Engineering Prof Brian Launder ## Lecture 10 **The Equations of Motion for Steady Turbulent Flows**
**Objectives** ## To obtain a form of the equations of motion designed for the analysis of flows that are turbulent. ## To understand the physical significance of **the Reynolds stresses.** ## To learn some of the important differences between laminar and turbulent flows. ## To understand why the turbulent kinetic energy has its peak close to the wall.
**The strategy followed** ## We adopt the strategy ad-vocated by Osborne Reynolds in which the instantaneous flow propert-ies are decomposed into a **mean** and a **turbulent** part. (for the latter, Reynolds used the term* ***sinuous**). ## We shall mainly use tensor notation for compactness. (Tensors hadn’t been invented in Reynolds’ time.)
**Preliminaries** ## We consider a turbulent flow that is **incompressible** and which is **steady** so far as the mean flow is concerned. ## For most practical purposes one is interested only in the **mean **flow properties which will be denoted *U*,* V*,* W *(or *Ui* in tensor notation). ## The instantaneous **total** velocity has components . (or ) ## So ## The difference between *Ui *and * * is denoted *ui*, the **turbulent** velocity: ## NB the time average of *ui *is zero, i.e.
**An important point to note** ## If a variable ** is a function of two independent variables, *x* and *y*, differential or integral operations on it with respect to *x* and *y* can be applied in any order. ## Thus ## So
**Averaging the equations of motion** ## First, note that the instantaneous static pressure is likewise written as the sum of a mean and turbulent part: ## The time average of , where the overbar denotes the time-averaging noted on the previous slide. ## Treating the viscosity as constant, the time averaged value of the viscous term in the Navier-Stokes equations may be written: ## But:
**The continuity equation in turbulent flow** ## For a uniform density flow: ## But …so ## ..or ## Thus, the fluctuating velocity also satisfies ## or -
**The averaged momentum equation** ## From the averaging on Slide 6: ** Convection Diffusion**
## Note that this is really three equations for *i* taking the value 1,2 and 3 in three orthogonal directions ## Recall also that because the *j* subscript appears twice in the convection and diffusion terms, this implies summation, again for *j=*1,2*, *and* *3*.* ## Thus:
**Boundary Layer form of the Reynolds Equation** ## The form of the Reynolds equation appropriate to a **steady 2D boundary layer** is taken directly from the laminar form with the inclusion of the same component of turbulent and viscous stress: i.e. ## The accuracy of this boundary layer model is, for some flows, rather less than for the laminar flow case (i.e. the neglected terms are less “negligible”). ## The form: ## is a higher level of approximation.
**Who was Osborne Reynolds?** ## Osborne Reynolds, born in Belfast - appointed in 1868 to the first full- time chair of engineering in England (Owens College, Manchester) at the age of 25. ## Initially explored a wide range of physical phenomena: the formation of hailstones, the effect of rain and oil in calming waves at sea, the refraction of sound by the atmosphere… ## …as well as various engineering works: the first multi-stage turbine, a laboratory-scale model of the Mersey estuary that mimicked tidal effects.
**Entry into the details of fluid motion** ## By 1880 he had become fascinated by the detailed mechanics of fluid motion….. ## ….especially the sudden transition between ** direct **and** sinuous **flow which he found occurred when: *UmD*/ 2000. ## Submitted ms in early 1883 – reviewed by Lord Rayleigh and Sir George Stokes and published with acclaim. Royal Society’s Royal Medal in 1888.
## In 1894 Reynolds presented orally his theoretical ideas to the Royal Society then submitted a written version. ## This paper included *“Reynolds averaging”*, *Reynolds stresses* and the first derivation of the turbulence energy equation. ## But this time his ideas only published after a long battle with the referees (George Stokes and Horace Lamb – Prof of Maths, U. Manchester)
**Some features of the Reynolds stresses** ## The stress tensor comprises nine elements but, since it is symmetric ( ), only six components are independent since etc. or in Cartesian coordinates . ## If turbulence is *isotropic *all the normal stresses (components where *i=j*) are equal and the shear stresses ( ) are zero. (Why??) ## The presence of mean velocity gradients (whether normal or shear) makes the turbulence *non-isotropic*. ## Non-isotropic turbulence leads to the transport of momentum usually orders of magnitude greater than that of molecular action.
**More features of the Reynolds stresses** ## Turbulent flows unaffected by walls (jets, wakes) show little if any effect of Reynolds number on their growth rate (i.e. they are independent of **). ## Turbulent flows (like laminar flows) obey the no-slip boundary condition at a rigid surface. This means that all the velocity fluctuations have to vanish at the wall. ## So, right next to a wall we have to have a viscous sublayer where momentum transfer is by molecular action alone; ## The presence of this sublayer means that growth rates of turbulent boundary layers *will* depend on Reynolds number.
**Comparison of laminar and turbulent boundary layers** ## **Laminar B.L.** ## Recall: The very steep near-wall velocity gradient in a turbulent b.l. reflects the damping of turbulence as the wall is approached ## But why do turbulent velocity fluctuations peak so very close to the wall?
**The mean kinetic energy equation** ## By multiplying each term in the Reynolds equation by *Ui* we create an equation for the mean kinetic energy: ## The left side is evidently: ## or, with *K**Ui*2 /2, ## Re-organize the right hand side as: ## ## A B C D E ## See next slide for physical meaning of terms
**The “source” terms in the mean k.e eqn** ## B: Viscous diffusion of kinetic energy ## C: Viscous dissipation of kinetic energy ## D: Reversible working on fluid by turbulent stresses ## E: Loss of mean kinetic energy by conversion to turbulence energy
**A Query and a Fact** ## Question: How do we know that term E represents a loss of mean kinetic energy to turbulence? ## Answer: Because the same term (but with an opposite sign) appears in the **turbulent** kinetic energy equation! ## The mean and turbulent kinetic energy equations were first derived by Osborne Reynolds.
**Boundary-layer form of mean energy equation ** ## For a thin shear flow (*U*(*y*)) the mean k.e. equation becomes: ## Consider a fully developed flow where the total (i.e. viscous + turbulent) shear stress varies so slowly with *y* that it can be neglected. ## In this case, where does the conversion rate of kinetic energy reach a maximum?
**Where is the conversion rate of mean energy to turbulence energy greatest?** ## This occurs where: ## or where ## or: ## or, finally: ## Thus, the turbulence energy creation rate is a maximum where the viscous and turbulent shear stresses are equal
**The near wall peak in turbulence explained** ## The peak in turbulence energy occurs very close to the point where the transfer rate of mean energy to turbulence is greatest ## This occurs where viscous and turbulent stresses are equal – i,e. within the viscosity affected sublayer! ## Why the turbulent velocity fluctuations are so different in different directions will be examined in a later lecture.
**Why is the normal stress perpendicular to the wall so much smaller than the other two?** ## Continuity for turbulent flow: ## Apply this at *y *=0 (the wall) ## But on this plane *u=w=*0 for all *x* and *z* ## So, ; but *u* and *w* deriv’s w.r.t. *y* **** 0 ## Expand fluctuating velocities in a series: ## But *b*1 must be zero (if ) ## So, while ## Q: How does the shear stress vary for small *y*?
**Extra slides** ## The following slides provide a derivation of the kinetic energy budget from the point of view of the turbulence. ## They confirm the assertion made earlier that the term represents the energy source of turbulence. ## We do not work through the slides in the lecture (Dr Craft will provide a derivation later) but the path parallels that for obtaining the mean kinetic energy.
**The turbulence energy equation-1** ## Subtract the Reynolds equation from the Navier Stokes equation for a steady turbulent flow ## This leads to: ## Note the above makes use of since by continuity
**The turbulence energy equation - 2** ## Multiply the boxed equation from the previous slide by and time average. ## Note: where *k* is the **turbulent ** **kinetic energy:**
## The viscous term is transformed as follows: ** turbulence energy **dissipation rate**
**The turbulence energy equation - 3** ## After collecting terms and making other minor manipulations we obtain: ## viscous turbulent diffusion generation dissipation ## Note this is a scalar equation and each term has to have two tensor subscripts for each letter. ## Repeat Q & A: How do we know that represents the generation rate of turbulence? **Ans**: The same term but with opposite sign appears in the **mean **kinetic energy equation.
**A question for you** ## Compile a sketch of the mean kinetic energy budget for fully developed **laminar** flow between parallel planes.
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