Download Computational Hydraulics Turbulence, Lecture Notes- Physics - and more Study notes Physics in PDF only on Docsity!
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 )
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: u = u + u ′ or
- upper case and lower case: U + u
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:
u ′= 0
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:
u^2 = u^2 + u ′^2 ( variance ) uv = uv + u ′ v ′ ( 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
In a simple shear flow the total stress is
stress turbulent stress viscous
u v y
u − ′′ ∂
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, − u ′ v ′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 : u ′^2 , v ′^2 , w ′^2
Shear stresses : v ′ w ′, w ′ u ′, u ′ v ′
(In slightly careless, but extremely common, usage both − u ′ v ′ and u ′ v^ ′ are referred to as
“stresses”.)
Most turbulent flows are anisotropic ; i.e. u ′^2 , v ′^2 , w ′^2 are different.
Turbulent kinetic energy : k = 21 ( u ′^2 + v ′^2 + w ′^2 )
Turbulence intensity : U
k U
u mean velocity
root-mean-square flu ctuation rms 32
y
U
v'
B
A
v u
y
U
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.
12 3 additionalflux
v φ = v φ + v ′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.
- The production terms for different Reynolds stresses involve different mean velocity
gradients; for example, the rate of production (per unit mass) of u 1 (^) u 1 ≡ u^2 and u (^) 1 u 2 ≡ uv are, respectively,
12
11
z
U
vw y
U
vv x
U
vu z
V
uw y
V
uv x
V
P uu
z
U
uw y
U
uv x
U
P uu
( 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. u^2 , v^2 , w^2 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 k = 12 ( u 2 + v^2 + w^2 ), relating the other stresses to this by an eddy-viscosity formula (see Section 8).
PRODUCTION ADVECTION (^) by mean flow
u
v
w REDISTRIBUTION
DISSIPATION by viscosity
by pressure fluctuations
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 u^2 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 v^2 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 ui j
u v^2
2
y
U
∆y
∆ (^) U
y
u ui j
uv u^2
y
U
∆y
∆U
y
u ui j
v^2 u^2 uv
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:
12 3 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 )
B
u y U = u + (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:
U y B
U y
= +
ln
(loglayer)
(viscoussublayer) (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.
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. turb =− u ′^ v ′
- 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).
