Characteristics Of Turbulent Flow
Fig. 32.1 Variation of horizontal components of velocity for laminar and turbulent flows at a point P
Fig. 32.2 Comparison of velocity profiles in a pipe for (a) laminar and (b) turbulent flows
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 . One of the effects of viscosity on turbulence is to make the flow more homogeneous and less dependent on direction.
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.
In isotropic turbulence fluctuations are independent of the direction of reference and
It is re-emphasised that even if the rms fluctuations at any point are same, their instantaneous values necessarily differ from each other at any instant.
The turbulent boundary layer continues to grow in thickness, with a small region below it called a viscous sublayer. In this sub layer, the flow is well behaved,just as the laminar boundary layer (Fig. 32.3)
Fig. 32.3 Laminar - turbulent transition
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 Reynolds number was defined as
where U∞ is the free stream velocity , δ* is the displacement thickness and ν is the kinematic viscosity .
The development starts with a laminar profile, undergoes a transition, changes over to turbulent profile and then stays turbulent thereafter (Fig. 32.4). The length of development is of the order of 25 to 40 diameters of the pipe.
A statistical correlation can be applied to fluctuating velocity terms in turbulence. Turbulent motion is by definition eddying motion. Not withstanding the circulation strength of the individual eddies, 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 32.5), little correlation can be expected.
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
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 32.6 shows typical variations of the correlation R with increasing separation r .
The positive correlation indicates that the fluid can be modelled 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
between the values of u' at different times is chosen and is called autocorrelation function.
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 and the turbulent motion does obey the principles of continuum mechanics.
Reynolds decomposition of turbulent 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.
(i) Time average for a stationary turbulence:
(ii) 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.
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 signifies any of the space variables.
Invoking Eq.(32.1) in the above expression, we ge
Since , Eq.(32.2) depicts that y and z components of velocity exist even for the parallel flow if the flow is turbulent. We have-
Contd. from Previous slide
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
The time averages of the spatial gradients of the fluctuating components also follow the same laws, and they can be written as
The degree of turbulence in a wind tunnel can be brought down by introducing screens of fine mesh at the bell mouth entry. In general, at a certain distance from the screens, the turbulence in a wind tunnel becomes isotropic, i.e. the mean oscillation in the three components are equal,
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 component of velocity by and fluctuating component as u' we can write down the following,
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 .
If f and g are two dependent variables and if s denotes anyone of the independent variables x, y