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 (y)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 get
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
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
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