Tarbulent flow

Advanced Fluid Mechanics
Turbulent Flow
1-1

Introduction
1-2
 Turbulent motion is an irregular motion.
 Turbulent fluid motion can be considered as an irregular condition of flow in which
various quantities (such as velocity components and pressure) show a random
variation with time and space in such a way that the statistical average of those
quantities can be quantitatively expressed.
 It is postulated that the fluctuations inherently come from disturbances (such as
roughness of a solid surface) and they may be either dampened out due to viscous
damping or may grow by drawing energy from the free stream.
 At a Reynolds number less than the critical, the kinetic energy of flow is not
enough to sustain the random fluctuations against the viscous damping and in such
cases laminar flow continues to exist.
 At somewhat higher Reynolds number than the critical Reynolds number, the
kinetic energy of flow supports the growth of fluctuations and transition to
turbulence takes place.

Characteristics of Turbulent Flow
1-3
 The most important characteristic of turbulent motion is the fact that velocity and
pressure at a point fluctuate with time in a random manner.
Fig. Variation of horizontal components of velocity for laminar and turbulent flows at a point P
 The mixing in turbulent flow is more due to these fluctuations. As a result we can
see more uniform velocity distributions in turbulent pipe flows as compared to the
laminar flows.
Fig. Comparison of velocity profiles in a pipe for (a) laminar and (b) turbulent flows

1-4
 Turbulence can be generated by -
1. frictional forces at the confining solid walls
2. the flow of layers of fluids with different velocities over one another
The turbulence generated in these two ways are considered to be different
 Turbulence generated and continuously affected by fixed walls is designated as
wall turbulence , and turbulence generated by two adjacent layers of fluid in absence
of walls is termed as free turbulence .
 Turbulence can be categorized as below -
1. Homogeneous Turbulence: Turbulence has the same structure quantitatively in
all parts of the flow field.
2. Isotropic Turbulence: The statistical features have no directional preference and
perfect disorder persists.
3. Anisotropic Turbulence: The statistical features have directional preference and
the mean velocity has a gradient.

1-5
 Homogeneous Turbulence : The term homogeneous turbulence implies that the
velocity fluctuations in the system are random but the average turbulent characteristics
are independent of the position in the fluid, i.e., invariant to axis translation.
Consider the root mean square velocity fluctuations
In homogeneous turbulence, the rms values of u', v' and w' can all be
different, but each value must be constant over the entire turbulent field. Note that
even if the rms fluctuation of any component, say u' s are constant over the entire field
the instantaneous values of u necessarily differ from point to point at any instant.

1-6
Isotropic Turbulence : The velocity fluctuations are independent of the axis of
reference, i.e. invariant to axis rotation and reflection. Isotropic turbulence is by its
definition always homogeneous . In such a situation, the gradient of the mean velocity
does not exist, the mean velocity is either zero or constant throughout.
In isotropic turbulence fluctuations are independent of the direction of
reference and
It is reemphasized that even if the rms fluctuations at any point are same,
their instantaneous values necessarily differ from each other at any instant.

Laminar-Turbulent Transition
1-7
 For a turbulent flow over a flat plate,
 The turbulent boundary layer continues to grow in thickness, with a small region
below it called a viscous sub layer. In this sub layer, the flow is well behaved, just as
the laminar boundary layer
Fig. Laminar - turbulent transition

1-8
 Observe that at a certain axial location, the laminar boundary layer tends to
become unstable. Physically this means that the disturbances in the flow grow in
amplitude at this location.
 Free stream turbulence, wall roughness and acoustic signals may be among the
sources of such disturbances. Transition to turbulent flow is thus initiated with the
instability in laminar flow
 The possibility of instability in boundary layer was felt by Prandtl as early as
1912. The theoretical analysis of Tollmien and Schlichting showed that unstable
waves could exist if the Reynolds number was 575. The Reynolds number was
defined as

1-9
 Taylor developed an alternate theory, which assumed that the transition is caused
by a momentary separation at the boundary layer associated with the free stream
turbulence. In a pipe flow the initiation of turbulence is usually observed at Reynolds
numbers in the range of 2000 to 2700.
 The development starts with a laminar profile, undergoes a transition, changes over
to turbulent profile and then stays turbulent thereafter (Fig.). The length of
development is of the order of 25 to 40 diameters of the pipe.
Fig. Development of turbulent flow in a circular duct

Correlation Functions
1-10
 A statistical correlation can be applied to fluctuating velocity terms in turbulence.
Turbulent motion is by definition eddying motion. A high degree of correlation exists
between the velocities at two points in space, if the distance between the points is
smaller than the diameter of the eddy. Conversely, if the points are so far apart that the
space, in between, corresponds to many eddy diameters (Figure), little correlation can
be expected
Fig. Velocity Correlation

1-11
Consider a statistical property of a random variable (velocity) at two points
separated by a distance r. An Eulerian correlation tensor (nine terms) at the two points
can be defined by
 In other words, the dependence between the two velocities at two points is
measured by the correlations, i.e. the time averages of the products of the quantities
measured at two points. The correlation of the components of the turbulent velocity of
these two points is defined as
 It is conventional to work with the non-dimensional form of the correlation, such as

1-12
A value of R(r) of unity signifies a perfect correlation of the two quantities
involved and their motion is in phase. Negative value of the correlation function
implies that the time averages of the velocities in the two correlated points have
different signs. Figure shows typical variations of the correlation R with increasing
separation r .
 The positive correlation indicates that the fluid can be modeled as travelling in
lumps. Since swirling motion is an essential feature of turbulent motion, these lumps
are viewed as eddies of various sizes. The correlation R(r) is a measure of the strength
of the eddies of size larger than r. Essentially the velocities at two points are correlated
if they are located on the same eddy.

1-13
 To describe the evolution of a fluctuating function u'(t), we need to know the
manner in which the value of u' at different times are related. For this purpose the
correlation function
Between the values of u' at different times is chosen and is called
autocorrelation function.
 The correlation studies reveal that the turbulent motion is composed of eddies
which are convected by the mean motion . The eddies have a wide range variation in
their size. The size of the large eddies is comparable with the dimensions of the flow
passage.
 The size of the smallest eddies can be of the order of 1 mm or less. However, the
smallest eddies are much larger than the molecular mean free paths.

Reynolds decomposition of turbulent flow
1-14
 The Experiment: In 1883, O. Reynolds conducted experiments with pipe flow by
feeding into the stream a thin thread of liquid dye. For low Reynolds numbers, the dye
traced a straight line and did not disperse. With increasing velocity, the dye thread got
mixed in all directions and the flowing fluid appeared to be uniformly colored in the
downstream flow.
 The Inference: It was conjectured that on the main motion in the direction of the
pipe axis, there existed a superimposed motion all along the main motion at right
angles to it. The superimposed motion causes exchange of momentum in transverse
direction and the velocity distribution over the cross-section is more uniform than in
laminar flow. This description of turbulent flow which consists of superimposed
streaming and fluctuating (eddying) motion is well known as Reynolds decomposition
of turbulent flow.
 Here, we shall discuss different descriptions of mean motion. Generally, for
velocity u , the following two methods of averaging could be obtained.

1-15
 Time average for a stationary turbulence:.
 Space average for a homogeneous turbulence:
 For a stationary and homogeneous turbulence, it is assumed that the two averages
lead to the same result: and the assumption is known as the ergodic hypothesis.
 In our analysis, average of any quantity will be evaluated as a time average . Take a
finite time interval t1. This interval must be larger than the time scale of turbulence.
Needless to say that it must be small compared with the period t2 of any slow variation
(such as periodicity of the mean flow) in the flow field that we do not consider to be
chaotic or turbulent .

1-16
Thus, for a parallel flow, it can be written that the axial velocity component is
As such, the time mean component determines whether the turbulent
motion is steady or not. The symbol signify any of the space variables
 While the motion described by Fig.(a) is for a turbulent flow with steady mean
velocity the Fig.(b) shows an example of turbulent flow with unsteady mean velocity.
The time period of the high frequency fluctuating component is t1 whereas the time
period for the unsteady mean motion is t2 and for obvious reason t2>>t1. Even if the
bulk motion is parallel, the fluctuation u ' being random varies in all directions.
Fig. Steady and unsteady mean motions in a turbulent flow

1-17
 The continuity equation, gives us
Invoking Eq.(1) in the above expression, we get
Since, Eq.(2) depicts that y and z components of velocity exist even for the
parallel flow if the flow is turbulent. We have-

1-18
 However, the fluctuating components do not bring about the bulk displacement of a
fluid element. The instantaneous displacement is u’dt, and that is not responsible for
the bulk motion. We can conclude from the above
 Due to the interaction of fluctuating components, macroscopic momentum transport
takes place. Therefore, interaction effect between two fluctuating components over a
long period is non-zero and this can be expressed as
 Taking time average of these two integrals and write

1-19
 Now, we can make a general statement with any two fluctuating parameters, say,
with f ' and g' as
 The time averages of the spatial gradients of the fluctuating components also follow
the same laws, and they can be written as
 The intensity of turbulence or degree of turbulence in a flow is described by the
relative magnitude of the root mean square value of the fluctuating components with
respect to the time averaged main velocity. The mathematical expression is given by

1-20
 For isotropic turbulence,
In this case, it is sufficient to consider the oscillation u' in the direction of flow and to
put
 This simpler definition of turbulence intensity is often used in practice even in
cases when turbulence is not isotropic.
Following Reynolds decomposition, it is suggested to separate the motion into a
mean motion and a fluctuating or eddying motion. Denoting the time average of the u
component of velocity by and fluctuating component as , we can write down
the following,

1-21
 By definition, the time averages of all quantities describing fluctuations are equal to
zero.
 The fluctuations u', v' , and w' influence the mean motion, in such a way
that the mean motion exhibits an apparent increase in the resistance to deformation. In
other words, the effect of fluctuations is an apparent increase in viscosity or
macroscopic momentum diffusivity .
 Rules of mean time - averages :
If f and g are two dependent variables and if s denotes anyone of the independent
variables x, y

Governing Equations for Turbulent Flow
1-22
 For incompressible flows, the Navier-Stokes equations can be rearranged in the
form
 Express the velocity components and pressure in terms of time-mean values and
corresponding fluctuations. In continuity equation, this substitution and subsequent
time averaging will lead to

1-23
 From Eqs (3a) and (2), we obtain

1-24
 It is evident that the time-averaged velocity components and the fluctuating
velocity components, each satisfy the continuity equation for incompressible flow.
 Imagine a two-dimensional flow in which the turbulent components are
independent of the z -direction. Eventually, Eq.(3b) tends to
 On the basis of condition (4), it is postulated that if at an instant there is an increase
in u' in the x -direction, it will be followed by an increase in v' in the negative y -
direction. In other words, is non-zero and negative. (Figure 1)

1-25
Fig 1 Each dot represents uν pair at an instant
 Invoking the concepts of into the equations of motion (eqn
1a,b,c), we obtain expressions in terms of mean and fluctuating components. Now,
forming time averages and considering the rules of averaging we discern the following

1-26
 The terms which are linear, such as and vanish when they are averaged
(Refer Equation 6)
 The same is true for the mixed like
 But the quadratic terms in the fluctuating components remain in the equations.
After averaging, they form
 Perform the aforesaid exercise on the x-momentum equation, we obtain

1-27

1-28
 Introducing simplifications arising out of continuity Eq. (3a), we shall obtain.
 Performing a similar treatment on y and z momentum equations, finally we obtain
the momentum equations in the form

1-29
 Comments on the governing equation
I. The left hand side of Eqs (5a)-(5c) are essentially similar to the steady-state
Navier-Stokes equations if the velocity components u, v and w are replaced by
II. The same argument holds good for the first two terms on the right hand side of
Eqs (5a)-(5c).
III. However, the equations contain some additional terms which depend on turbulent
fluctuations of the stream. These additional terms can be interpreted as
components of a stress tensor.

1-30
 Now, the resultant surface force per unit area due to these terms may be considered
as
Comparing Eqs (5) and (6), we can write

1-31
 It can be said that the mean velocity components of turbulent flow satisfy the same
Navier-Stokes equations of laminar flow. However, for the turbulent flow, the
laminar stresses must be increased by additional stresses which are given by the
stress tensor (Eq. 7)
 These additional stresses are known as apparent stresses of turbulent flow or
Reynolds stresses
 Since turbulence is considered as eddying motion and the aforesaid additional
stresses are added to the viscous stresses due to mean motion in order to explain the
complete stress field, it is often said that the apparent stresses are caused by eddy
viscosity . The total stresses are now
and so on
 The apparent stresses are much larger than the viscous components, and the viscous
stresses can even be dropped in many actual calculations

Turbulent Boundary Layer Equations
1-32
 For a two-dimensional flow (w = 0)over a flat plate, the thickness of turbulent
boundary layer is assumed to be much smaller than the axial length and the order of
magnitude analysis may be applied. As a consequence, the following inferences are
drawn:

Turbulent Boundary Layer Equations
1-33
 The turbulent boundary layer equation together with the equation of continuity
becomes
 A comparison of Eq. (10) with laminar boundary layer Eq. depicts that: u, v and p
are replaced by the time average values
 Laminar viscous force per unit volume is replace by

Prandtl’s Mixing Length Theory
1-34
 Consider a fully developed turbulent boundary layer, quite away from leading edge
so that it may be considered s fully developed.
 Such flow are classified as one dimension parallel flow. The velocity varies only
from one stream line to next. The main flow direction is assumed to be parallel to X
axis.

1-35
 The time average component of the velocities are given by
 The fluctuating component of transverse velocity v’ transports mass and
momentum across a plane at a distance y1 from the wall. The shear stress due to
fluctuation is equal to the rate of x-momentum transport in y direction per unit area,
and so.
 The time mean value of the turbulent shear stress is
 In the above equation so the resulting equation is
 Considering the direction of force associated with the shear stress, we can write,

1-36
 A lump of fluid, which comes to the layer y1 from a layer (y1 -l) has a positive value
of v’.
 If a lump of fluid retain its original value of momentum then its velocity at its
current location y1 is smaller then the velocity prevailing there. The difference in
velocity is then,
The above expression is obtained by expanding the function
in a Taylor series and neglecting all higher order terms and higher ordered derivatives.
 The term l is small length scale and called Prandt’s mixing length. Prandtl’s
proposed that transverse displacement of any fluid particle is on an average equal to l.
 Let’s consider another lump of fluid with a negative value of v’. This arrives at y1
from a layer (y1 +l).
 If this lump of fluid retain its original value of momentum then its velocity at its
current location y1 will be somewhat more than the original velocity. This difference is
given by,

1-37
 The velocity differences caused by the transverse motion can be regarded as the
turbulent velocity component at y1.
 The time average of the absolute value of this fluctuation is given by,
 Based on the experiments and simplification Prandlt’s observe that
 Along with the condition that the moment at which u’ is positive, v’ is more likely
to be negative, and conversely when u’ is negative. Based on that we can write,

1-38
 Where C1 and C2 are different proportionality constant. If C2 is included in the
unknown mixing length then the above equation can simplified as
 From the expression of turbulent shear stress τt, we may write
 Where µt is the turbulent viscosity or eddy viscosity and it is given by
The velocity gradient was express in modulus to adjust its sign according to sign of
shear stress.

1-39
 The turbulent viscosity and consequently the mixing length are not properties of
fluid.
 They are purely local functions and dependent on the nature of the turbulent flow
field.
 The problem of determining the mixing length in terms of the velocity field has not
been resolved and it is a topic of continuous research. Several correlation using
experimental results for τt have been proposed to determine l.
 Variation of mixing length with the boundary layer is shown in figure. It is zero in
the sublayer, equal to χy over about 20 percent of δ and proportional to δ elsewhere.
Parameter χ, the von Karman constant has a value of 0.4.

1-40
 Where y is the distance from the wall, λ is a mixing length constant (=0.09) and δ is
the layer width.

Tarbulent flow

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