The instantaneous stream-wise velocity of a turbulent flow is given as follows:

\(u\left( {x,\;y,\;z,\;t} \right) = \bar u\;\left( {x,\;y,\;z} \right) + u'\left( {x,\;y,\;z,\;t} \right)\)

The time-average of the fluctuating velocity u’ (x, y, z, t) is

Concept:

For turbulent flow

\(u = \bar u + u'\)

\(\bar{u}=\frac{1}{T} \int_0^T u d t\)

where,

u = velocity in x-direction at any instant.

\(\bar u\) = Average velocity in x-direction.

u’ = Fluctuating component of velocity in x-direction.

Time average of fluctuating component of velocity,

\(\overline{u^{\prime}}=\frac{1}{T} \int_0^T u^{\prime} d t=\frac{1}{T} \int_0^T(u-\bar{u}) d t\)

\(\overline{u^{\prime}}=\frac{1}{T} \int_0^T u d t-\frac{1}{T} \int_0^T \bar{u} d t=\bar{u}-\left[\frac{1}{T} \times(\bar{u} T)\right]=0\)

Important Points

Interaction effect between two fluctuating components over a long period is non-zero.

\(\mathop \smallint \limits_{ - T}^T u'v'dt \ne 0\)

where v’ → fluctuating component of velocity in y-direction