Intro. Turbulent Flows
INTRODUCTION TO TURBULENT FLOWS
What is turbulence?
How is turbulence created?
How do we solve turbulent flows?
General Purposes
-
u and U
Mass diffussion and concentration statistics
Details of turbulent motion and how they interact
Three Types of turbulence
-
Grid turbulence -- not self-sustaining
Wall shear layers --- self sustaining
- wall effect as a turbulence source
Free shear layers -- mixing layers
-- two fluid at different speeds
-- jets, wakes
Characteristics of turbulence
-
Velocity fluctuates in a random manner -- Statisitically can be studied
High levels of vorticity fluctuations
High Reynolds numbers
Described by the Navier-Stokes (N-S) equations
Higher levels of momentum and energy transfers
Dissivative
Continuum level
Certain spatial structures - eddies –vortices
- mushroom like, etc, all are distributed continuesly.
Reynolds Decompositions (RANS = Reynolds Average of N-S)
u~ i  U i  ui
Ui 
Lim 1
T  T

t 0 T
t0
u~ i dt
1
Intro. Turbulent Flows
ui
u~ (t),
m/s
Ui
Fig. 1. Typical of velocity fluctuation in turbulent flows.
The mean value of a fluctuation components is zero,
ui 
Lim 1
T  T

t 0 T
t0
(u~ i U i)dt  0
For time averages to make sense, the integral have to be independent of t0.
U i
0
t
u~i U i ui ui
;

0

x j x j x j x j
u~i u~ j  (U i  ui )(U j u j )
=0 =0
 U iU j  U i u j  u iU j  u i u j
 U iU j  ui u j
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Intro. Turbulent Flows
u i u j  0 if ui and uj are correlated.
= 0 if ui and uj are uncorrelated.
~
p P

if ~p = P + p.
xi xi
N-S equations can be written as
u~i ~ u~i
 2 u~i
1 ~
p
uj


t
x j
 xi
x j x j
u~i
0
xi
u~ i and u~ j are the instantaneous velocities.
(1)
(2)
Equation for the mean flow for a turbulent flow
The momentum equation is obtained by substituting u~ i  U i  ui and ~p = P + p and taking a
time average (
U i
 0 ).
t
U i
ui
 2U i
1 P
Uj
uj


x j
x j
 xi
x j x j
(3)
The continuity equation becomes:
U i
u
 0 and i  0 .
xi
xi
 (U i  u i )
u~i
0
 0
xi
xi
uj
u j u i u j
u j
u i
u
 u j i  ui

 0.
, since u i
x j
x j
x j
x j
x j
U i
 2U i
1 P

Uj



ui u j
x j
 xi
x j x j x j
Uj

U i
1 P 1   U i



  ui u j 

x j
 xi  x j  x j

or
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Intro. Turbulent Flows
 P

   ij   U i  u i u j 
 

x j


where ij is Kronecker delta. ij = i if i = j, and 0 if i  j.
Uj
U i


x j
x j
(4)
Equation (4) is the momentum equation for the mean flow. For turbulent flows, it is not
enough equations to solve the problem, because of the attendance of the Reynolds stress
tensors   u i u j . This leads to a closure problem.
In general, one can write:
 ij    ui u j
(5)
  11  12  13 


where  ij   21  22  23  . If  ij is symetric, then  ij   ji and there are six independent


 31  32  33 
components, instead of nine. The diagonal components of  ij are normal stresses:
  u12 ;   u 22 ;   u 32 .
The off-diagonal components of  ij are shear stresses, and they play an important role in the
transport of mean momentum by turbulent motion. One of the methods to solve the closure
problem is the use of turbulence models.
How do we estimate or model   u i u j ?
Length Scales in Turbulent Flows
Turbulent flows are characterized by the existence of several lengths. Consider a laminar
boundary layer flow over a flat plate:
u = u(y)
U
y

x
L
Fig. 2. Laminar boundary layer over a flat plate.
L = convective length scale
U = convective velocity scale
 = diffusive length scale
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Intro. Turbulent Flows
The time scale is then:
L
.
U
We can also estimate that
L  
~

U  UL 
1/ 2

1
.
Re 1L/ 2
(6)
Furthermore, L is related to the convection of momentum, and  relates to the molecular
diffusion of momentum deficit across the flow, away from the surface.
Suppose we have a turbulent boundary layer:
U
largest eddy
size
y
u

u
smallest
eddy size
L
x
Fig. 3. Turbulent boundary layer over a flat plate.
L
 convective time scale.
U
~ (u)
and
L
U

u
~
L U
where u is the characteristic velocity fluctuation.
L

~
 turbulent diffusion time scale matches the convective time scale.
U
u
What is the smallest eddy size?
or
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Intro. Turbulent Flows
What is the smallest length scale in the turbulent flow?
 Large scales are as big as the width of the flow  Interact with the mean flow.
 The smallest scales?  Kolmogorov length scale.

Energy
transfer
among
eddies
Large
scale
smallest
eddies
dissipation
to heat
Fig. 4. Energy cascade in turbulent motion.

Turbulence generates new (smaller) length scales till the local Reynolds number becomes
small for viscous dissipation to become significant.
How can we estimate the Kolmogorov length scale ()?
Kolmogorov theory:
 ~ f(,)
(7)
where  = Kolmogorov length scale,  = dissipation rate, and  = kinematic viscosity
For an equilibrium turbulent boundary layer,
dissipation = energy input
The dissipation () and  can be used to obtain the smallest scales of length (), time (),
frequency (fK), and velocity (), in the flow, and are referred to as the Kolmogorov scales
(Tennekes and Lumley, 1972)
 = (3/)1/4
(8)
 = (/)1/2
(9)
fK = U/(2)
(10)
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Intro. Turbulent Flows


(11)
The turbulent kinetic energy dissipation rate plays an important role in turbulent flow
analysis. In an isotropic turbulence, the dissipation rate, , is given by
  15 u / x 
2
(12)
Values of u / x  can be obtained experimentally using Taylor’s frozen hypothesis. The
2
hypothesis states that if the turbulent velocity fluctuations are small compared to the mean
velocity, then the autocorrelation of the fluctuating velocity with time delay  will be the
same as the spatial correlation with separation U in the streamwise direction (Bradshaw,
1971). In a mathematical form, the Taylor’s hypothesis can be expressed as
u
1 u

x
U ( y ) t
(13)
where u / t can usually be obtained from instantaneous velocity measurements.
In addition to the Kolmogorov length scale, the Taylor microscale ( Taylor) and
integral length scale () are often used in the analysis of turbulent flows. The Taylor
microscale, with dimension of length, is defined as (after Tennekes and Lumley, 1972)
Taylor

u2


 u / x

1/ 2



2

(14)
By recalling Taylor’s frozen hypothesis, based on Eq. (14), we can obtain
() = (  )
(15)
where (  ) is the spatial correlation with separation  (  = U. Figure 5 shows a typical
spatial correlation curve.
The area under the spatial correlation curve is the integral length scale,  so that
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Intro. Turbulent Flows

   (  ) d 
(16)
0
The Taylor microscale (Taylor) and the integral length scale (  ) are far larger than the
Kolmogorov length scale ().
1.0
( )
0

 
Fig. 5. A typical spatial correlation curve.
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