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Shear stress in turbulent flow

The shear stress for laminar or viscous flow is given by Newton's law of viscosity as:-

$\tau_v=\mu\dfrac{du}{dy}$

Similarly for the expression of viscous shear, J. Bossinesq expressed the turbulent shear in mathematical form as,

$\tau_t=\eta\dfrac{d\overline{u}}{dy}.............................(1)$

Where,

$\tau_t$ = Shear stress due to turbulence

$\eta$ = Eddy viscosity

$\overline{u}$ = Average velocity at a distance y from boundary.

The ratio Eddy viscosity and $\rho$ (mass density) is known as kinematics Eddy viscosity and is denoted by $\varepsilon$ (epsilon).

$\varepsilon=\dfrac{\eta}{\rho}$

Now, if the shear stress due to laminar flow is also considered, then the total shear stress becomes,

$\tau=\tau_v+\tau_t$

$\tau=\mu\dfrac{du}{dy}+\eta\dfrac{d\overline{u}}{dy}$

$\eta=0$ (for laminar flow). For other cases the value of ‘$\eta$’ may be several thousand times the value of ‘$\mu$’.

To find shear stress in turbulent flow equation (1) is used. But since the value of ‘$\eta$’ cannot be predicted, this equation is having limited use.

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