9
Canonical Turbulent Flows
2.1.3
The Reynolds Stress –u! v !
If you apply Reynolds’ decomposition to the governing equations for fluid mechanics
(the Navier-Stokes equations) and then average the equations over some time scale (which
could be an infinite time in an ergodic flow) you can develop an expression for the features
of the mean quantities (e.g., the mean velocity and pressure fields). The linear terms
in the Navier-Stokes equations just become the mean over the time scale of interest.
However, decomposing the non-linear terms and averaging yields new terms. Let’s take
a look at why. Consider the two instantaneous quantities A and B which we decompose
to A = A + A! where the overline indicates a temporal average over the time scale of
interest. Based on our definition of the Reynolds’ decomposition we have:
A! = 0
Proof ⇒ A = (A + A! ) = A + A! = A + A! ⇒ A! = 0
(2.4)
and importantly the nonlinear product yields:
AB = (A + A! )(B + B ! ) = (AB + AB ! + A! B + A! B ! )
(2.5)
= AB + A! B + AB ! + A! B ! = AB + A! B + A B ! + A! B !
= AB + A! B !
Where A! B ! is the correlation of fluctuating quantities. These types of terms are the
source of turbulent transport and stirring (and great modeling challenges!) in turbulent
flows.
Now, when you average the continuity equation and the Navier-Stokes equations you
have linear terms
∂
∂
A=
A
∂x
∂x
(2.6)
where the temporal mean just passes through to the variable, why? Since the averaging
process is a time integration we can move the spatial derivative outside the integral.
1
e.g.,
T
!
t0 +T
t0
∂ 1
∂A
dτ =
∂x
∂x T
!
t0 +T
t0
A dτ =
∂A
∂x
10
Now we continue down Osborne Reynolds’ path and apply our decomposition to the
Navier-Stokes equations under the assumption of ergotic incompressible flow. It is left
as an exercise for the student to show that continuity just looks like:
∂u ∂v ∂w
+
+
=0
∂x ∂y
∂z
(2.7)
And the Navier Stokes equations become:
1 ∂P
∂u
∂u
∂u
+v
+w
=
+ν
u
∂x
∂y
∂z
ρ ∂x
"
∂2u ∂2u ∂2u
+
+ 2
∂x2 ∂y 2
∂z
#
−
∂u! 2 ∂u! v ! ∂u! w !
−
−
(2.8)
∂x
∂y
∂z
" 2
#
∂v
∂v
∂v
1 ∂P
∂ v ∂2v ∂2v
∂u! v ! ∂v ! 2 ∂v ! w !
u
+v
+w
=
+ν
+
+
−
−
−
(2.9)
∂x
∂y
∂z
ρ ∂y
∂x2 ∂y 2 ∂z 2
∂x
∂y
∂z
" 2
#
∂w
∂w
∂w
1 ∂P
∂ w ∂2w ∂2w
u
+v
+w
=
+ν
+
+ 2
∂x
∂y
∂z
ρ ∂z
∂x2
∂y 2
∂z
−
∂u! w ! ∂v ! w ! ∂w ! 2
−
−
−g
∂x
∂y
∂z
(2.10)
These equations, known as the Reynolds Averaged Navier Stokes equations, look essentially the same as the original equation with two exceptions: The unsteady temporal
terms have been averaged out and a new set of terms emerges on the right-hand-side.
Looking more carefully at the new terms they actually arose from the nonlinear terms on
the left-hand side but it is customary to move them to the right hand side as they can
be viewed as the turbulent analog to the viscous stress terms, why? Let’s look at these
terms, known as the Reynolds’ stress terms, a bit more carefully.
These terms are of the form u! w !. Let’s try to get a physical sense for what these sort of
terms are doing.
Consider a turbulent shear flow with ∂u/∂z > 0. A shear flow is any flow with a mean
gradient. A picture of our flow might look like:
11
Canonical Turbulent Flows
If a parcel sitting at the point (xo , z o ) has a fluctuating vertical velocity w ! > 0 it has
high probability that u! < 0 since it is low momentum fluid moving to a region of high
momentum. Now consider hundreds and thousands of such parcels, on average we expect
⇒
u! w ! < 0.
If parcels have w ! < 0 then they have high probability that u! > 0 since high momentum
fluid is moving to a region of lower mean momentum
⇒
u! w ! < 0 again.
Therefore u! w ! < 0 in a turbulent shear flow with mean gradient ∂u/∂z > 0. What does
this mean? Consider that the w ! term is an advection term - meaning it is responsible for
transporting ‘stuff’ with it up or down (positive or negative). What is the stuff? It is u!
in our example – what is u! ? Well, what if we multiply by the density, ρ? We get that w !
is transporting ρu! – horizontal momentum! Thus we can think of u! w ! as the turbulent
vertical advection of streamwise turbulent momentum, or more simply the vertical flux of
streamwise momentum. In general terms like u! w ! are known as momentum flux terms.
What would u! c! be (where c! is a scalar concentration fluctuation)? A scalar flux - the
turbulent streamwise flux of mass!
If you make the same back-of-the-envelope analysis for ∂u/∂z < 0 you find that u! w ! > 0.
Thus the sign of the momentum flux terms tell whether the flux is inducing a net increase
or decrease in momentum (positive and negative, respectively).
12
The momentum flux terms are frequently referred to as the Reynolds stress terms as
they play an analogous role in the governing equations to the viscous stress. The sign
convention is such that the Reynolds stress is most properly written in the form −ρu! v !
but frequently, the terms of the form u! v ! will be referred to as the Reynolds stress.
Looking at our Reynolds Averaged Navier Stokes (RANS) equations (equations 2.8-2.10)
we see that there are six independent Reynolds stress terms, three tangential stress terms
−ρu! v ! , −ρu! w !, and −ρv ! w ! and three normal stress terms, −ρu! 2 , −ρv ! 2 , and −ρw ! 2 . We
should note that for modelers this is a problem, we have introduced no new equations yet
we have six new terms, i.e., six new unknowns! This is referred to as a closure problem
as the RANS equations are not a closed set of equations. It is solved by developing
models for the new terms based on resolved quantities. The best known of these models
is perhaps the k − ε model which developed the models for the Reynolds stress terms
based on k and ε and includes two new equations to track k and ε. This is part of a
broad class of solutions to the closure problem known as 2-equation models since two
new equations are introduced. Signficantly, experimentalist have no closure problem we can directly measure the Reynolds stresses!
2.2
The Turbulent Round Jet
Jet flows are a subset of the general class of flows known as free shear flows where free
indicates that the shear arises in the absence of a boundary (wall) and instead results
from a maintained velocity difference. The scale of round jet flows ranges from the
nozzle on an ink jet printer to a volcano. For a perfectly round jet the flow is essentially
completely characterized by the Reynolds number
ReD =
UJ D
ν
(2.11)
where D is the diameter of the jet orifice and UJ is the velocity of the jet at the orifice.
Ideally the velocity at the orifice will be constant across the exit plane, but if not we
take UJ = Q/A where Q is the flow rate and A is the jet cross-sectional area. At low
Canonical Turbulent Flows
13
ReD jets are laminar and at high ReD jets are turbulent. It is a bit difficult to define
exactly where the transition occurs as it depends on the exact conditions of the flow at
the orifice. Jets begin to exhibit turbulent behavior at relatively low ReD , as low as 300,
but the fundamental parameters that characterize the turbulent round jet (to be defined
in a moment) are weakly a function of ReD and the jet exit velocity profile (Zarruk &
Cowen, 2008. Simultaneous velocity and passive scalar concentration measurements in
low Reynolds number neutrally buoyant turbulent round jets. Experiments in Fluids
44, 865 - 872). For an essentially uniform jet exit velocity profile (known as a top-hat
profile), the fundamental parameters of the turbulent jet become independent of Re for
ReD > 4000, thus the canonical turbulent round jet problem is for ReD > 4000.
Round jets are one of many flows that exhibit what is known as self-similar behavior.
Self-similarity indicates that a flow will look similar when scaled appropriately. This
appropriate scaling involves defining similarity variables that reduce the overall dimensionality of the problem. In the case of the turbulent round jet, after a transition region
that occurs roughly from 0 < x/D < 30 (where x is the coordinate along the jet axis –
coordinate system shown on next page), the jet cross-sectional properties are independent of x when properly scaled. The similarity variable in the round jet can be chosen
in multiple ways but basically comes down to a statement that the radial coordinate
depends on the axial coordinate in a fixed way. We will choose as our similarity variable
the ratio r/r1/2 where r1/2 is the half-width of the jet defined as the radial point where
the velocity has dropped to half the centerline velocity. Several other normalizations
have been developed including r/(x − x0 ) where x0 is a virtual origin to be defined in a
√
moment and r/lq where lq = A. Our choice of definition has the advantage that a single cross-sectional profile can be normalized without exact knowledge of the streamwise
14
position of the profile relative to the jet orifice.
For the round jet we define
U0 (x) = u(x, 0, 0)
U0 (x)
u(x, r 1 , 0) =
2
2
(2.12)
(2.13)
where U0 is our nomenclature for the jet centerline velocity. If the centerline velocity
is plotted versus the non-dimensional distance downstream an inverse relationship with
downstream distance is found (this can actually be analytically predicted)
"
#
B
1 x − x0
U0 (x)
UJ
# or
="
=
x − x0
UJ
U0 (x)
B
D
D
(2.14)
where B (the slope term of the centerline velocity decay which Hussein, Capp & George
(1994) define as Bu , see below) and x0 (the virtual origin of the jet) are constants of the
expected linear fit that must be determine empirically – for ReD > 4000 B ∼ 5.8 while
x0 /D is generally ∼ 4 but each are a function of Reynolds number and jet exit profile
(Zarruk & Cowen, 2008).
I have handed out Hussein, Capp & George (1994). Velocity measurements in a highReynolds-number, momentum-conserving, axisymmetric, turbulent jet. J. Fluid Mech.
258, 31-75. Look over this article for plots of Eq 2.14 and all of the plots below for
jets. This is the current ‘standard’ article on turbulent round jets. The ‘standard’ paper
prior to this paper (and still a benchmark!) was (also handed out) Wyzgnanski & Fiedler
15
Canonical Turbulent Flows
(1969). Some measurements in the self-preserving jet. J. Fluid Mech. 38, 577-612. The
same plots exist in this paper. Note both papers are excellent examples of experimental
design and technology and should be regarded as exemplary experimental papers. Each
reviews hot-wire anemometry and Hussein et al. reviews laser Doppler velocimetry (LDV)
as well.
We can define the jets spreading rate, S, based on r 1 as
2
S=
d r 1 (x)
(2.15)
2
dx
The conservation of momentum can be used to show that in a round jet the product
r 1 (x)U0 (x) is a constant. Eq 2.14 indicates that U0 (x) ∼ x−1 hence r 1 (x) must go as x1 .
2
2
Integration of Eq 2.15 and incorporation of the virtual origin results in
r 1 (x) = S(x − x0 )
2
(2.16)
A typical value for S is 0.094 (Hussein et al.). Note based on the definition arctan S gives
the angle of spread of the r 21 (x) contour. For S = 0.094 the angle with respect to the x
axis is about 5.4◦ . A more physical sense of the jet width is based on 2S (greater than
90% of the momentum is contained within this contour). The included angle of the jet
based on 2S is about 21◦ (e.g., the angle between the ±2S contours).
The above analysis is all based on mean quantities – how does the turbulence in a round
$
jet behave? It turns out that it too is self-similar. If we consider u! 2 and normalize it by
the jet centerline velocity, U0 (x), we find that it also decays as x−1 and in the self-similar
$
region has a constant value of about 27% (Hussein et al.). If we consider v ! 2 we find
the same dependence however the constant value is a bit lower, about 22% (Hussein et
al.). Note that the round jet is axially symmetric (there is no dependence on the exact
direction we choose perpendicular to the jet axis). Hence if we continue our analysis of
$
the turbulence in Cartesian coordinates we find that w ! 2 has the same dependence and
the constant is 22%.
Some general points about the shape of the turbulence profiles in the radial direction are
worth noting. If you consider the shape of u it decays monotonically (but in a Gaussian
16
like manner) from the peak centerline value to zero at an infinite radius. There is an
inflection point in the profile where the maximum radial gradient occurs (at around
r/r 1 = 0.5). This is the point of maximum shear and it is the point where turbulence
2
production is a maximum. Since the shear is in the u component of the velocity field it
$
should not be surprising that u! 2 shows a peak value above that of the centerline value
at this point in the profile and then decays. As there is minimal shear driven production
$
$
in the plane perpendicular to the jet axis the v ! 2 and w !2 profiles show no such offaxis peak but instead decay in a way very similar to u. See Hussein et al.’s Figure 9 - 11
for examples.
The final turbulence parameter we need to discuss is the Reynolds stress component
u! w ! (which is the same as the u! v ! in the case of the round jet). Recall that this is the
turbulent transport of momentum term. Physically the jet has its momentum focused
along the jet axis and we expect that the turbulence is trying to spread the momentum
radially away from the axis. Thus if we consider the ‘upper half’ of the jet profile where
∂u/∂z < 0 a positive w ! fluctuation will carry high momentum fluid away from the jet
axis and hence u! > 0. Thus in this region we expect u!w ! > 0. An easy way to get a
sense for the Reynolds stress behavior is to use what is called a scatter plot – a plot of
u! vs w ! in the case just described. Here is an example of a scatter plot from ADV data
near the region of maximum shear in a turbulent round jet (e.g., lab #1!).
6
4
w’ (cm/s
2
0
−2
−4
−6
−6
−4
−2
0
2
u’ (cm/s
4
6
Canonical Turbulent Flows
17
Similarly if we now consider the ‘lower half’ of the jet profile where ∂u/∂z > 0 a negative
w ! fluctuation will carry high momentum fluid away from the jet axis and hence u! > 0.
Thus in this region we expect u! w ! < 0. There will not be a discontinuity at the jet
centerline so the value at jet centerline must be u! w ! = 0. The Reynolds stress terms
play a pivotal role in turbulence production and hence we expect their value to be a peak
where the mean velocity shear is a maximum - at around r/r 1 = 0.5 as indicated above.
2
%
2
Further, as u(x)! /U0 (x) has a self-similar region we expect that u! w !/U02 will also be
self-similar — and it is. See Hussein et al.’s Figure 12 for the profiles exact shape.
That’s it for now for the turbulent round jet. For more details see the two articles handed
out and the excellent chapter in Pope (2001), sections 5.1 – 5.3, on the reference list. We
will investigate the mass flux in a turbulent round jet in the final laboratory exercise so
the story is not quite over!