A boundary layer is a thin viscous layer adjacent to a solid surface, where the fluid viscosity predominates. Due to viscous effects, fluid sticks to the solid surface such that the velocity of fluid layer adjacent to the surface is equal to the velocity of the surface itself. In other words, the relative velocity of the fluid with respect to a solid surface is zero. This condition is commonly referred to as no-slip condition.

The extent of viscous effects on a surface is measured in terms of boundary layer thickness

\(\mathrm {\left( \delta \right) }\), defined as the distance normal to the surface such that the local flow velocity

\(\mathrm {\left( u\right) }\) at that distance is 99% of the freestream velocity

\(\mathrm {\mathrm {\left( U_{a}\right) }}\). That is

$$\begin{aligned} \text {at}&\mathrm {y=0;}&\mathrm {u\left( y\right) =0}\\ \text {at}&\mathrm {y=\delta ;}&\mathrm {u(y)=0.99U_{a}} \end{aligned}$$

The displacement thickness

\(\mathrm {\left( \delta ^{*}\right) }\) is defined as the distance perpendicular to the boundary, by which the freestream is displaced due to the formation of boundary layer.

$$\begin{aligned} \mathrm {\delta ^{*}}&= \mathrm {\intop _{0}^{\delta }\left( 1-\frac{u}{U_{a}}\right) dy} \end{aligned}$$

The momentum thickness

\(\mathrm {\left( \theta \right) }\) is defined as the distance through which the boundary layer must be displaced to compensate the reduction in momentum of the flowing fluid due to boundary layer formation.

$$\begin{aligned} \mathrm {\theta }&= \mathrm {\intop _{0}^{\delta }\frac{u}{U_{a}}\left( 1-\frac{u}{U_{a}}\right) dy} \end{aligned}$$

Kinetic energy thickness

\(\mathrm {\left( \delta ^{**}\right) }\) is the distance measured perpendicular to surface of the solid body through, which the boundary should be displaced to compensate for the reduction in kinetic energy of the flowing fluid due to boundary layer formation.

$$\begin{aligned} \mathrm {\delta ^{**}}&= \mathrm {\intop _{0}^{\delta }\left( \frac{u}{U_{a}}\right) \left[ 1-\left( \frac{u}{U_{a}}\right) ^{2}\right] } \end{aligned}$$

The Reynolds number is an important similarity parameter for viscous flows, defined as the ratio of inertia force to viscous force.

$$ \mathrm {Re=\frac{\rho vL}{\mu }} $$

The ratio of inertia force to elastic force is known as the Mach number. From order of magnitude of analysis, the Mach number is defined as

$$ \mathrm {M=\frac{v}{a}} $$

That is, the Mach number can be defined as the ratio of local flow speed to the speed of sound.

For very thin boundary layers, the Prandtl boundary layer equations are

$$\begin{aligned} \mathrm {\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}}&= \mathrm {0}\\ \mathrm {u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}}&= \mathrm {-\frac{\partial p}{\partial x}+\mu \frac{\partial ^{2}u}{\partial y{}^{2}}}\\ \mathrm {\frac{\partial p}{\partial y}}&= \mathrm {0} \end{aligned}$$

The boundary conditions that satisfies the above equations are

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {at}&\mathrm {y=0:}\end{array}\right. }&\mathrm {u=0,}&\mathrm {v=0}\\ {\left\{ \begin{array}{ll} \text {at}&\mathrm {y=\delta :}\end{array}\right. }&\mathrm {u=U_{a},}&\mathrm {v=0} \end{aligned}$$

The momentum integral equation suggested by von Karman is

$$ \mathrm {\frac{d\theta }{dx}+\frac{\theta }{U_{a}}\frac{dU_{a}}{dx}\left( H+2-M_{a}^{2}\right) =\frac{C_{f}}{2}} $$

where

$$\begin{aligned} \mathrm {C_{f}}&= \mathrm {\frac{\tau _{w}}{\frac{1}{2}\rho _{a}U_{a}^{2}}\equiv \text { coeffcient of skin friction}}\\ \mathrm {H}&= \mathrm {\frac{\delta ^{*}}{\theta }\equiv \text { shape factor}}\\ \mathrm {\frac{d\theta }{dx}}&\equiv \mathrm {\text { inertia term}}\\ \mathrm {\frac{\theta }{U_{a}}\frac{dU_{a}}{dx}\left( H+2-M_{a}^{2}\right) }&\equiv \mathrm {\text { pressure gradient term}}\\ \mathrm {\frac{C_{f}}{2}}&\equiv \mathrm {\text { wall skin friction term}} \end{aligned}$$

The energy integral equation derived by K. Wieghardt (1948) for laminar boundary layer is given by

$$\begin{aligned} \mathrm {U_{a}^{2}\frac{\partial \delta ^{*}}{\partial t}+\frac{\partial }{\partial t}\left( U_{a}^{2}\theta \right) +\frac{\partial }{\partial x}\left( U_{a}^{3}\delta ^{**}\right) }&= \mathrm {\frac{2\varepsilon }{\rho }} \end{aligned}$$

where

\(\mathrm {\varepsilon =\mu \intop _{0}^{\infty }\left( \frac{\partial u}{\partial y}\right) ^{2}dy}\);

\(\mathrm {\varepsilon }\) is the energy dissipation rate due to viscosity across the boundary layer.

\(\mathrm {\delta ^{*}}\),

\(\mathrm {\theta }\) and

\(\mathrm {\delta ^{**}}\), respectively, are the displacement thickness, the momentum thickness and the energy thickness of the boundary layer.

A laminar boundary layer is the one where the fluid layers are assumed to slide over one another smoothly. That is, in a laminar boundary layer the fluid layers, also called laminas, slide over adjacent layers such that there exists only the microscopic exchange of matter (mass, momentum, and energy) among the layers. However, if we try to observe the laminar flows on macroscopic scale, we will not feel any exchange of matter taking place between the fluid layers. Moreover, any small disturbances or instabilities are quickly damped out by the viscous forces that resist the relative motion of adjoining layers. Thus, a laminar boundary layer flow is an orderly flow in which fluid elements move in an orderly manner such that the transverse exchange of momentum is negligibly small, however, the axial exchange of momentum may be considerable. In contrast, the flow in a turbulent boundary layer is three-dimensional random phenomena, exhibiting multiplicity of scales, possessing vorticity, and showing very high dissipation.

For the laminar flow past a flat plate, the Blasius solutions are

$$ \mathrm {\frac{\delta }{x}=\frac{5.0}{\sqrt{Re_{x}}}} $$

$$\begin{aligned} \mathrm {\delta ^{*}}&= \mathrm {1.7208\sqrt{\frac{\nu x}{U_{a}}}} \end{aligned}$$

$$\begin{aligned} \mathrm {\theta }&= \mathrm {0.664\sqrt{\frac{\nu x}{U_{a}}}} \end{aligned}$$

$$\begin{aligned} \mathrm {\delta ^{**}}&= \mathrm {1.0444\sqrt{\frac{\nu x}{U_{a}}}} \end{aligned}$$

$$ \mathrm {\mathrm {C_{f}}=\frac{\tau _{w}}{\frac{1}{2}\rho U_{a}^{2}}=\frac{0.664}{\sqrt{Re_{x}}}} $$

For compressible flow over the flat plate, the boundary layer thickness

\(\mathrm {\left( \delta _{\text {comp}}\right) }\) can be defined as

$$ \mathrm {\delta _{\text {comp}}=\frac{5.0x}{\sqrt{Re_{x}}}G\left( M_{e}, Pr,\frac{T_{w}}{T_{e}}\right) } $$

The boundary conditions are:

$$\begin{aligned} \mathrm {{\left\{ \begin{array}{ll} \text {at}&y=0;\end{array}\right. }}&\mathrm {T=T_{w}} \end{aligned}$$

(11.237)

$$\begin{aligned} \mathrm {{\left\{ \begin{array}{ll} \text {at}&y=d;\end{array}\right. }}&\mathrm {T=T_{e}} \end{aligned}$$

(11.238)

where

\(\mathrm {T_{w}} = \) wall temperature,

\(\mathrm {T_{e}} = \) temperature at the boundary layer edge,

\(\mathrm {d} = \) local distance from the wall, and function

\(\mathrm {G}\) can be obtained through the numerical simulation of the problem. Furthermore, the skin friction coefficient

\(\mathrm {\left( C_{f,\text {comp}}\right) }\) can be written as

$$ \mathrm {C_{f}=\frac{1.328}{\sqrt{Re_{c}}}F\left( M_{e}, Pr,\frac{T_{w}}{T_{e}}\right) } $$

For a uniform flow past the flat plate, boundary layer starts growing as laminar flow beginning at the leading edge of the flat plate. However, due to instabilities this laminar flow turns into transition flow which subsequently turns into turbulent flow. The instabilities introduced in a laminar flow are amplified and result in flows which are orderly in nature in some parts, but also shows temporarily (and/or spatially) irregular fluctuations of all the flow quantities in other parts, one may say of a transitional flow state. In this state, intermittent laminar and turbulent behavior is observed; some phases occur in which the flow behaves as laminar, and in other phases the flow exhibits turbulent characteristics. Since the transition phenomena is exceedingly complex and thus there is no accurate theory available to predict the process. It is the area of active research.

For the boundary layer over the flat plate, transition is usually predicted in terms of critical Reynolds number.

$$\begin{aligned} \mathrm {Re_{cr}}&= \mathrm {\frac{\rho vx_{cr}}{\mu }} \end{aligned}$$

where

\(\mathrm {x_{cr}}\) is called the critical length. It is the axial length measured from the leading edge of the plate at which transition begins. For the flow over a flat plate, the value of critical Reynolds number is

\(\mathrm {\text {5}{\times }\text {10}^{5}}\) whereas, for pipe flows the critical Reynolds number based on the inner diameter of the pipe is 2300.

The turbulence or turbulent flows can be described as a random three-dimensional phenomenon in which the perturbation components are superimposed over mean flow, i.e., it is the flow which has irregular fluctuations. By definition, a turbulent flow can be described as the flow which is three-dimensional, random, exhibiting multiplicity of scales, possessing vorticity and shows very high dissipation. Most of the naturally occurring flows are turbulent in nature.

In a turbulent flow, the instantaneous flow parameters can be expressed as

$$\begin{aligned} \mathrm {u}&= \mathrm {\overline{u}+u'}\\ \mathrm {v}&= \mathrm {\overline{v}+v'}\\ \mathrm {w}&= \mathrm {\overline{w}+w'}\\ \mathrm {p}&= \mathrm {\overline{p}+p'} \end{aligned}$$

where

\(\mathrm {u}\),

\(\mathrm {v}\),

\(\mathrm {w}\) and

\(\mathrm {p}\) are instantaneous values;

\(\mathrm {\overline{u}}\),

\(\mathrm {\overline{v}}\),

\(\mathrm {\overline{w}}\) and

\(\mathrm {\overline{p}}\) are time-averaged values; and

\(\mathrm {u'}\),

\(\mathrm {v'}\),

\(\mathrm {w'}\) and

\(\mathrm {p'}\) are fluctuating components.

The turbulence is said to be isotropic if the statistical averaging of flow parameters are direction independent. That is, they remain invariant with respect to reflection or rotation of axes. Mathematically,

$$\begin{aligned} \mathrm {\overline{u'^{2}}}&= \mathrm {\overline{v'^{2}}=\overline{w'^{2}}} \end{aligned}$$

In those situations where mean velocity shows the gradient, turbulence will be non-isotropic or anisotropic. In this case,

$$\begin{aligned} \mathrm {\overline{u'^{2}}}&\mathrm {\ne \overline{v'^{2}}\ne \overline{w'^{2}}} \end{aligned}$$

For two-dimensional turbulent boundary layers, the time-averaged boundary layer equations are

$$\begin{aligned} \mathrm {\frac{\partial \overline{u}}{\partial x}+\frac{\partial \overline{v}}{\partial y}}&= \mathrm {0}\\ \mathrm {\rho \left( \frac{\partial \overline{u}}{\partial t}+\overline{u}\frac{\partial \overline{u}}{\partial x}+\overline{v}\frac{\partial \overline{u}}{\partial y}\right) }&= \mathrm {-\frac{\partial \overline{p}}{\partial x}+\frac{\partial \overline{\tau }}{\partial y}} \end{aligned}$$

Turbulent boundary layer can be classified into three different regimes. The bottommost layer adjacent to wall shows the flow characteristics similar to the laminar flow and thus known as laminar sub-layer. The layer beyond it is the transition or buffer layer, where the magnitudes of viscous stresses and Reynolds stresses are almost equal. The outermost part of boundary layer blended with the buffer layer has the characteristics similar to that of the free shear layer, and thus this outer layer is known as the outer boundary layer or the fully turbulent layer.