In turbulent flow, the velocity and pressure at a given point are not constant with time but shows irregular fluctuations of velocity with high frequency as shown in. This fluctuation is due to random. haphazard and irregular nature of fluid motion.


The velocity in the turbulent flow is consisting of a time average portion and a fluctuating part u. The velocity can be written by
-
u = u + u'

Where, u = Instantaneous velocity.

u  = Temporal mean velocity or time average velocity

u = Velocity fluctuation.

Instantaneous Velocity:

In turbulent flow, the velocity changes continuously with respect to time. The velocity of any point at any instant is called as Instantaneous velocity. It is denoted by u.

Temporal Mean Velocity:

Temporal mean velocity is found out by taking average of velocity at a point over a certain period of time. It is denoted by u.

Fluctuating Velocity:

Fluctuating velocity is difference between the instantaneous velocity u and temporal mean velocity u at any point in a given instant.

The fluctuating velocity is the component of velocity which may be positive, negative or zero. It is denoted by u.

By definition, temporal mean velocity component can be expressed as,

-            T                     T                     T
u = 1/T ∫ u dt, v = 1/T ∫ vdt, w = 1/T ∫ w dt
            0                     0                     0

And, by definition fluctuating velocity

-            T                     T                     T
u = 1/T ∫ u dt, v = 1/T ∫ vdt, w = 1/T ∫ w dt
            0                     0                     0
                T         -
= u = 1/T ∫  (u - u) dt
                0
-         T                     T
 = 1/T ∫ u dt, v = 1/T ∫ u dt = u
          0                     0
-     -     -
u = v = w = 0

Where T = Time inverval over which average is taken.

Since these quantities are zero, the intensity of turbulence is estimate by root mean square value of u.
                                                                                       -
Intensity of turbulence = √1/3 ( u-2 + v-2 + w-2) / u

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