## Introduction to Boundary Layer Analysis

**Introduction: **The plane channel, which is also called plane Poiseuille ﬂow or duct ﬂow, is a canonical conﬁguration for studying internal ﬂows. Understanding the structure of channel ﬂow is obviously of great engineering interest since this can be applied in many applications. This ﬂow is obviously a Newtonian ﬂuid, so that the important boundary problems are raised. To study the plane channel ﬂow, we need to understand the behavior of ﬂow in boundary layers for both laminar and turbulence ﬂows.

For the laminar channel ﬂow, we know the solutions since we could calculate it analytically, but in turbulent case, we cannot get an analytical turbulent solution since turbulence is more complex, high Reynolds number is applied, and becomes unstable. Moreover, the reason why turbulence is more complex is the boundary layer starts off laminar, but at some critical Reynolds number, it becomes unstable to disturbance, e.g. noise, vibration, surface, and so on. Therefore, we will discuss the boundary.

**Boundary layer: **In physics and fluid mechanics, a boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface. On an aircraft wing the boundary layer is the part of the flow close to the wing, where viscous forces distort the surrounding non-viscous flow. See Reynolds number.

Laminar boundary layers can be loosely classified according to their structure and the circumstances under which they are created. The thin shear layer which develops on an oscillating body is an example of a Stokes boundary layer, while the Blasius boundary layer refers to the well-known similarity solution near an attached flat plate held in an oncoming unidirectional flow. When a fluid rotates and viscous forces are balanced by the Coriolis effect (rather than convective inertia), an Ekman layer forms. In the theory of heat transfer, a thermal boundary layer occurs. A surface can have multiple types of boundary layer simultaneously

**Aerodynamics:** The aerodynamic boundary layer was first defined by Ludwig Prandtl in a paper presented on August 12, 1904 at the third International Congress of Mathematicians in Heidelberg, Germany. It simplifies the equations of fluid flow by dividing the flow field into two areas: one inside the boundary layer, dominated by viscosity and creating the majority of drag experienced by the boundary body; and one outside the boundary layer, where viscosity can be neglected without significant effects on the solution. This allows a closed-form solution for the flow in both areas, a significant simplification of the full Navier–Stokes equations. The majority of the heat transfer to and from a body also takes place within the boundary layer, again allowing the equations to be simplified in the flow field outside the boundary layer. The pressure distribution throughout the boundary layer in the direction normal to the surface (such as an airfoil) remains constant throughout the boundary layer, and is the same as on the surface itself.

**Formulation: **We begin with the equations for two dimensional steady continuity and Navier-Stokes equations: