Figure 1 Development of the hydrodynamics boundary layer over a
flat plate [2]
Boundary layer theory is a subject connected with the study of velocity
gradient, shear stress, forces and energy loss in the boundary layer.
For laminar flow, the shear stress can be calculated from Newton’s law:
\(\tau_{w}=\mu\frac{\text{du}}{\text{dy}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(1.1\right)\)
While for turbulent flow, the shear stress can be obtained from the
equation inserted below:
\(\tau_{w}=0.0233\rho U^{\frac{7}{4}}\left(\frac{\nu}{y}\right)^{0.25}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(1.2\right)\)
Reynolds number
The Reynolds number is a criterion which defines the nature of flow if
it is laminar, transitional or turbulent by measuring its inertial and
viscous forces are given by the equation inserted below[1, 3, 4]:
\(Re=\frac{\rho\text{\ U}_{\infty}\text{\ x}}{\mu}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(2.1\right)\)
Where Re is the Reynolds number; \(\rho\) is the density of
air; \(\text{\ U}_{\infty}\) is the free stream velocity; \(x\) is the
typical length scale of the system; \(\mu\) is the dynamics viscosity
As illustrated before, the Reynolds number is used to categorize the
nature of the flow type in three regions as illustrated below in Figure
2.