# Computational Hydraulics Turbulence, Lecture Notes- Physics - , Study notes for Physics. The University of Manchester

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CFD 7 – 1 David Apsley

7. TURBULENCE SPRING 2012 7.1 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 Summary Examples PART (a) – THE NATURE OF TURBULENCE 7.1 What is Turbulence?

Instantaneous Mean

• A “random”, 3-d, time-dependent eddying motion with many scales, superposed on an

often drastically simpler mean flow. • A solution of the Navier-Stokes equations. • The natural state at high Reynolds numbers. • An efficient transporter and mixer ... of momentum, energy, constituents. • A major source of energy loss. • A significant influence on drag and boundary-layer separation. • “The last great unsolved problem of classical physics.” (variously attributed to

Sommerfeld, Einstein and Feynman)

CFD 7 – 2 David Apsley

7.2 Momentum Transfer in Laminar and Turbulent Flow In laminar flow adjacent layers of fluid slide past each other without mixing. Transfer of momentum occurs between layers moving at different speeds because of viscous stresses. In turbulent flow adjacent layers continually mix. A net transfer of momentum occurs because of the mixing of fluid elements from layers with different mean velocity. This mixing is a far more effective means of transferring momentum than viscous stresses. Consequently, the mean- velocity profile tends to be more uniform in turbulent flow. 7.3 Turbulence Notation The instantaneous value of any flow variable can be decomposed into mean + fluctuation. is decomposed into mean + fluctuation Mean and fluctuating parts are denoted by either: • an overbar and prime: uuu ′+= or • upper case and lower case: uU + The first is useful in deriving theoretical results but becomes cumbersome in general use. The notation being used is, hopefully, obvious from the context. By definition, the average fluctuation is zero:

0=′u In experimental work and in steady flow the “mean” is usually a time mean, whilst in theoretical work it is the probabilistic (or “ensemble”) mean. The process of taking the mean of a turbulent quantity or a product of turbulent quantities is called Reynolds averaging. The normal averaging rules for products apply:

222 uuu ′+= (variance) vuvuuv ′′+= (covariance) Thus, in turbulent flow the “mean of a product” is not equal to the “product of the means” but includes an (often significant) contribution from the net effect of turbulent fluctuations.

v

u

laminar

turbulent

CFD 7 – 3 David Apsley

7.4 Effect of Turbulence on the Mean Flow Engineers are usually only interested in the mean flow. However, turbulence must still be considered because, although the averages of individual fluctuations (e.g. u′ or v′ ) are zero, the average of a product (e.g. vu ′′ ) is not and may lead to a significant net flux. Consider mass and momentum fluxes in the y direction across surface A. For simplicity, assume constant density. 7.4.1 Continuity Mass flux: vA Average mass flux: Av The only change is that the instantaneous velocity is replaced by the mean velocity. The mean velocity satisfies the same continuity equation as the instantaneous velocity. 7.4.2 Momentum x-momentum flux: AuvuvA )()( = Average x-momentum flux: AvuvuuvA )()( ′′+= The average momentum flux has the same form as the instantaneous momentum flux … except for additional fluxes Avu ′′ due to the net effect of turbulent fluctuations. These additional terms arise because of the averaging of a product of fluctuating quantities.

A net rate of transport of momentum Avu ′′ from lower to upper side of an interface ... • is equivalent to a net rate of transport of momentum Avu ′′− from upper to lower; • has the same dynamic effect (i.e. same rate of transfer of momentum) as a stress (i.e.

force per unit area) vu ′′− . This apparent stress is called a Reynolds stress. In a fully-turbulent flow it is usually much larger than the viscous stress.

Other Reynolds stresses ( uu ′′− , vv ′′− , etc.) emerge when considering the average flux of the different momentum components in different directions. The mean velocity satisfies the same momentum equation as the instantaneous velocity,

except for additional apparent stresses: the Reynolds stressesjiuu ′′−

vAρ

u

vAρ

CFD 7 – 4 David Apsley

In a simple shear flow the total stress is

{

321

stress turbulent

stress viscous

vu y

u ′′− ∂ ∂= (1)

In fully-turbulent flow turbulent stress is usually substantially bigger than viscous stress. can be interpreted as either:

• the apparent force (per unit area) exerted by the upper fluid on the lower, or • the rate of transport of momentum (per unit area) from upper fluid to lower. The dynamic effect – a transfer of momentum – is the same. The nature of the turbulent stress can be illustrated by considering the motion of particles whose fluctuating velocities allow them to cross an interface. 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).

In both cases, vu ′′− is positive and, on average, tends to reduce the momentum in the upper fluid or increase the momentum in the lower fluid. Hence there is a net transfer of momentum from upper to lower fluid, equivalent to the effect of an additional mean stress. Velocity Fluctuations

Normalstresses: 222 ,, wvu ′′′ Shear stresses: vuuwwv ′′′′′′ ,, (In slightly careless, but extremely common, usage both vu ′′− and vu ′′ are referred to as “stresses”.)

Most turbulent flows are anisotropic; i.e. 222 ,, wvu ′′′ are different.

Turbulent kinetic energy: )( 2222 1 wvuk ′+′+′=

Turbulence intensity: U

k

U

u

itymean veloc

ctuationsquare fluroot-mean- rms 3 2

= ′

==

y

U

v' B

A

v u

y

U

τ τ

CFD 7 – 5 David Apsley

7.4.3 General Scalar In general, the advection of any scalar quantity φ gives rise to an additional scalar flux in the mean-flow equations; e.g.

321

vvv φ′′+φ=φ (2)

Again, the extra term is the result of averaging a product of fluctuating quantities. 7.4.4 Turbulence Modelling At high Reynolds numbers, turbulent fluctuations cause a much greater net momentum transfer than viscous forces throughout most of the flow. Thus, accurate modelling of the Reynolds stresses is vital. A turbulence model or turbulence closure is a means of approximating the Reynolds 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.

CFD 7 – 6 David Apsley

7.5 Turbulence Generation and Transport 7.5.1 Production and Dissipation Turbulence is initially generated by instabilities in the flow caused by mean velocity gradients (e.g. ∂U/∂y). These eddies in their turn breed new instabilities and hence smaller eddies. The process continues until the eddies become sufficiently small (and fluctuating velocitygradients ∂ui/∂xj sufficiently large) that viscous effects become significant and dissipate turbulence energy as heat.

This process – the continual creation of turbulence energy at large scales, transfer of energy to smaller and smaller eddies and the ultimate dissipation of turbulence energy by viscosity – is called the turbulent energy cascade.

7.5.2 Turbulent Transport Equations 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). It can be proved mathematically (Section 10) that:

• Each Reynolds stress jiuu satisfies its own scalar-transport equation.

• The source term for an individual Reynolds stress jiuu transport equation has the form: ndissipatiotionredistribuproductionsourcenet −+=

where: – productionPij is determined by meanvelocity gradients; – redistributionij transfers energy between stresses via pressure fluctuations; – dissipationij involves viscosity acting on fluctuating velocity gradients.

There are also “advection” terms (turbulence carried with the flow) and “diffusion” terms

(if turbulent stresses vary from point to point).

large eddies

small eddies

PRODUCTION

by mean flow

DISSIPATION

by viscosity

CFD 7 – 7 David Apsley

• The production terms for different Reynolds stresses involve different mean velocity

gradients; for example, the rate of production (per unit mass) of 211 uuu ≡ and uvuu ≡21 are, respectively,

)()(

)(2

12

11

z

U vw

y

U vv

x

U vu

z

V uw

y

V uv

x

V uuP

z

U uw

y

U uv

x

U uuP

∂ ∂+

∂ ∂+

∂ ∂−

∂ ∂+

∂ ∂+

∂ ∂−=

∂ ∂+

∂ ∂+

∂ ∂−=

(3)

(Exercise: by “pattern-matching” write production terms for the other stresses). • Because: (i) mean velocity gradients are greater in some directions than others, (ii) motions in certain directions are selectively damped (e.g. by buoyancy forces or

rigid boundaries),

turbulence is usually anisotropic, i.e. 222 ,, wvu are all different.

• In practice, most turbulence models do not actually solve transport equations for all turbulent stresses, but only for the turbulent kinetic energy )( 2222

1 wvuk ++= , relating the other stresses to this by an eddy-viscosity formula (see Section 8).

2u 2v 2w REDISTRIBUTION

DISSIPATION by viscosity

by pressure fluctuations

CFD 7 – 8 David Apsley

7.6 Simple Shear Flows A flow for which there is only one non-zero mean velocity gradient, ∂U/∂y, is called a simple shear flow. Because they form a good approximation to many real flows, have been extensively researched in the laboratory and are amenable to basic theory they are an important starting point for many turbulence models. For such a flow, the first of (3) and similar expressions show that P11 > 0 but that

P22 = P33 = 0, and hence 2u tends to be the largest of the normal stresses because it is the

only one with a non-zero production term. On the other hand, if there is a rigid boundary on

y = 0 then it will selectively damp wall-normal fluctuations; hence 2v is the smallest of the normal stresses. If there are density gradients (for example in atmospheric or oceanic flows, in fires or near heated surfaces) then buoyancy forces will either damp (stable density gradient) or enhance (unstable density gradient) vertical fluctuations. 7.6.1 Free Flows Mixing layer Wake (plane or axisymmetric) Jet (plane or axisymmetric)

y

U

∆ y

∆U

y

u u i j

u v2

2

y

U

∆ y

∆U

y

u u i j

u2uv

y

U

∆ y

∆U

y

u u i j

uv2 2

uv

CFD 7 – 9 David Apsley

For these simple flows: • Maximum turbulence occurs where yU ∂∂ / is largest, because this is where turbulence

production occurs. Note, however, that in the case of wake or jet, some turbulence must have diffused into the central core, where 0/ =∂∂ yU .

uv has the opposite sign to ∂U/∂y and vanishes when this derivative vanishes. • These turbulent flows are anisotropic: 22 vu > . This is because, for these simple shear

flows, only the streamwise component has a production term:

0,2 2211 =∂ ∂−= P

y

U uvP

7.6.2 Wall-Bounded Flows Pipe or channel flow Flat-plate boundary layer Even though the overall Reynolds number /Re LU ∞= may be large, and hence viscous transport much smaller than turbulent transport in the majority of the flow, there must be a thin layer very close to the wall where the local Reynolds number based on distance from the wall, /Re yuy = , is small and hence molecular viscosity is important. Wall Units An important parameter is the wall shear stressw (drag per unit area). Like any other stress this has dimensions of [density] × [velocity]2 and hence it is possible to define an important velocity scale called the friction velocityuτ (also written u*):

/� wu = (4) From u and it is possible to form a viscous length scale l = /u. Hence, we may define non-dimensional velocity and distance from the wall in so-called wall units:

u

U U =+ ,

yu y =+ (5)

UD

y

U

y

u ui j

u2

-uv v2 constant-stress layer

CFD 7 – 10 David Apsley

The direct effects of molecular viscosity are usually only important when y+ is O(1). The total mean shear stress is made up of viscous and turbulent parts:

321 321 turbulent

viscous

uv y

U − ∂ ∂=

When there is no streamwise pressure gradient is approximately constant over a significant depth and is equal to the wall stress w. This assumption of constant shear stress allows us to establish the velocity profile in regions where either viscous or turbulent stresses dominate. Viscous Sublayer Very close to a smooth wall, turbulence is damped out by the presence of the boundary. In this region the shear stress is predominantly viscous. Assuming constant shear stress,

y

U w

∂=

y

U w= (6)

i.e. the mean velocity profile in the viscous sublayer is linear. This is generally a good approximation in the range y+ < 5. Log-Law Region At large Reynolds numbers, the turbulent part of the shear stress dominates throughout most of the boundary layer so that on dimensional grounds, since u and y are the only possible velocity and length scales,

y

u

y

U �∝ ∂ ∂

Integrating, and putting part of the constant of integration inside the logarithm (to make its argument dimensionless):

)ln 1

( 

 B yu

uU += (7)

(von Kármán’s constant) and B are universal constants with experimentally-determined values of about 0.41 and 5 respectively. Using the definition of wall units (equation (5)) these velocity profiles are often written in non-dimensional form:

ByU

yU

+=

= ++

++

ln 1

layer) (log

sublayer)(viscous (8)

Experimental measurements indicate that the log law actually holds to a good approximation over a substantial proportion of the boundary layer. (This is where the logarithm originates in common friction-factor formulae such as the Colebrook-White formula for pipe flow). Consistency with the log law is probably the single most important consideration in the construction of turbulence models.

CFD 7 – 11 David Apsley

Summary • Turbulence is a 3-d, time-dependent, eddying motion with many scales, causing

continuous mixing of fluid. • Each flow variable may be decomposed as mean + fluctuation. • The process of averaging turbulent variables or their products is called Reynolds

averaging and leads to the Reynolds-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 Reynolds stresses, e.g.

vuturb ′′−= • A means of specifying the Reynolds stresses (and hence solving the mean flow

equations) is called a turbulence model or turbulence closure. • Turbulence energy is generated at large scales by mean-velocity gradients (and,

sometimes, body forces such as buoyancy). Turbulence is dissipated (as heat) at small scales by viscosity.

• Because of the directional nature of the generating process (i.e. mean-velocity

gradients and/or body forces) turbulence is initially anisotropic. Energy is subsequently redistributed amongst the different stress components by the action of pressure fluctuations and ultimately dissipated by the action of viscosity on the smallest scales.

• Turbulence modelling is, to a large extent, guided by experimental observations and

theoretical considerations for simple shear flows which may be free (e.g. mixing layer; jet; wake) or wall-bounded (e.g. pipe or channel flow; flat-plate boundary layer).

CFD 7 – 12 David Apsley

Examples Q1. Which is more viscous, air or water? Air: = 1.20 kg m–3 = 1.80×10–5 kg m–1 s–1 Water: = 1000 kg m–3 = 1.00×10–3 kg m–1 s–1 Q2. The accepted critical Reynolds number in a round pipe (based on bulk velocity and diameter) is 2300. At what speed is this attained in 5-cm-diameter pipe for (a) air; (b) water? Q3. Sketch the mean velocity profile in a pipe at Reynolds numbers of (a) 500; (b) 50 000. What is the shear stress along the pipe axis in either case? Q4. Explain the process of flow separation. How does deliberately “tripping” a developing boundary layer help to prevent or delay separation on a convex curved surface? Q5. The following couplets are measured values of (u,v) in an idealised 2-d turbulent flow.

Calculate u , v , 2u′ , 2v′ , vu ′′ 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 2u and uv are, respectively,

)(211 z

U uw

y

U uv

x

U uuP

∂ ∂+

∂ ∂+

∂ ∂−=

)()(12 z

U vw

y

U vv

x

U vu

z

V uw

y

V uv

x

V uuP

∂ ∂+

∂ ∂+

∂ ∂−

∂ ∂+

∂ ∂+

∂ ∂−=

(a) By inspection, write down similar expressions for P22, P33, P23, P31, the rates of

production of 2v , 2w , vw and wu respectively. (b) Write down expressions for P11, P22, P33 and P12, P23, P31 in simple shear flow (where

U/∂y is the only non-zero mean velocity gradient). What does this indicate about the relative distribution of turbulence energy amongst the various Reynolds-stress components? Write down also an expression for P(k), the rate of production of turbulence kinetic energy.

(c) A mathematician would summarise the different production terms by a compact

formula

)( k

i

kj k

j

kiij x

U uu

x

U uuP

∂ ∂

+ ∂ ∂

−=

using the Einstein summation convention – implied summation over a repeated index (in this case, k). See if you can relate this to the above expressions for the Pij.