##### Document information

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 stresses* *jiuu *′′−

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

*Normal* *stresses*: 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

*fluxadditional
*

*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
**velocity* *gradients ∂*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:
– *production* *Pij* is determined by **mean velocity gradients**;
–

*redistribution*

*ij*transfers energy between stresses via

**pressure fluctuations**; –

*dissipation*

*ij*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

ENERGY CASCADE

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).

**PRODUCTION ADVECTION **by mean flow

2*u *2*v *2*w ***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 *P*11 > 0 but that

*P*22 = *P*33 = 0, and hence
2*u * 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 2*v * 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 stress* *w* (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 velocity* *u*τ (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 *, 2*u*′ , 2*v*′ , *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 2*u * 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 *P*22, *P*33, *P*23, *P*31, the rates of

production of 2*v *, 2*w *, *vw * and *wu * respectively.
(b) Write down expressions for *P*11, *P*22, *P*33 and *P*12, *P*23, *P*31 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*.