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# Why we study boundary layer theory ?

• We all know that according to Newton’s law of viscosity

τ ∝ du/dy

Where u = Average velocity.
y = Vertical distance measure from the solid surface of plate.

• According to Newton’s law of viscosity, if y = 0, τ is maximum. Due to this τ max, the fluid generates opposite force on a solid surface that force is known as drag force.
• To properly study of drag force, it is necessary to have knowledge of boundary layer theory.
• There are lots of things in engineering applications where fluids come in contact with the solid body. So, there will be drag force produced on their solid surface. To design such solid bodies like the body of an aeroplane, aircraft, vehicle body etc the understanding of the theory of boundary layer is required to find out how much drag force will be produced.

# Concept of boundary layer

• When fluid flows over a solid surface, the very first layer of fluid is stuck with the solid surface due to attraction force. If the velocity at the very first layer is zero, it is known as a no-slip condition. Now, at a solid surface velocity is zero. As moves toward the y-direction or perpendicular to the plate, velocity is increasing (Parabolic velocity profile is generated). So, we can observe from the velocity profile that at every layer of fluid the velocity is different. So, I can say there is a zone in which a relative velocity is existing.

• Now, As more towards the y-direction or perpendicular to the plate, the velocity is increasing. There is a one-point will come at which fluid velocity becomes 99 % of free stream velocity. which means the velocity of the fluid is almost equal to free stream velocity. Now, To perpendicular to every point of solid surface, we can find a number of points at which the velocity of the fluid is 99.99 % of free stream velocity. If we join all points we get one layer that is known as the boundary layer.

## Definition of boundary layer

• The boundary layer is a region in a flow field, inside which fluid flows with higher relative velocity or the layer inside which the relative motion between fluid particles may exist or in another word, A boundary layer beyond which there is no relative velocity.
• Outside the boundary layer, there is no relative velocity along y-direction hence du/dy = 0 but inside the boundary layer there is relative velocity so du/dy ≠ 0. du/dy also known as the rate of shears strain.

• If we observe through this equation, beyond the boundary layer du/dy = 0. So, τ = 0 and for the region inside the boundary layer, the shear strain is not equal to zero (τ ≠ 0). A viscous effect will be created inside the boundary layer. We all know that there is a relative velocity inside the Boundary layer means the adjacent layer of fluids are sliding along with each other. That’s why the fluid particles will spin or rotate about their axis so that inside the boundary layer the flow is rotational flow and outside the boundary layer, there is no relative motion so flow is irrotational flow.

## The thickness of boundary layer

• As we know, as move towards the X direction, the boundary layer thickness is increased. let considered one point X. At a point X, the velocity u is equal to zero. As the move vertically towards the y-direction velocity is keeping increase and one point S is coming where fluid velocity become 99 % of free stream velocity. The gap or the distance between these two points X and S are known as boundary layer thickness. So the definition of the boundary layer thickness can be written as….
• Distance of point at which velocity of fluid particles equals 99 % of free stream velocity and this distance is measured from the solid surface of the plate. Boundary layer thickness is not a constant throughout the layer.

Thickness of boundary layer

• it has a different thickness at a different location (See above fig). if we measure thickness at a point X2, δ will be different. if we measure thickness at a point X3, δ will be different. So, the boundary layer thickness is a function of X (distance from the leading edge) different values of the δ will get at different points from the leading edge.
• As the fluid moves ahead, the nature of the fluid will change. Up to some particular length, the nature of the fluid is laminar. After one particular point, this fluid will convert into a turbulent flow. The small zone in which the fluid flow is converting from laminar flow to turbulent flow that zone is called the transition zone (flow).
• The nature of flow is identified by Reynolds number. The value of Reynolds number at which fluid is converting from laminar to turbulent is known as critical Reynolds number. For the case of a flat plate, the critical Reynolds number is 5 × 10⁵. On a solid surface, a point for which the Reynolds number is greater than 5 × 10⁵ is called transition point.

## Different types of the thickness of boundary layer

1. Displacement thickness
2. Momentum thickness
3. Energy thickness

### Displacement thickness

• During the formation of the boundary layer, the mass outside the boundary layer continuously comes in the boundary layer. Hence the reduction occurred in mass flow rate outside the boundary layer. To compensate for this reduction, the plate needs to travel a little distance in the y-direction(upward). This distance is known as displacement thickness.
• In a similar manner remaining two thicknesses can be defined. Distance measure perpendicular to the solid surface of the plate by which the solid surface or plate must be displaced to compensate the reduction in mass flow rate, that distance is called “Displacement thickness”.

### Momentum thickness

• Due to the reduction in mass flow rate, it is obvious there is some reduction in momentum too. So the distance measure perpendicular to the solid boundary by which the solid boundary must be displaced in order to compensate for the reduction in momentum or change in momentum. That distance is called “Momentum thickness”.

### Energy thickness

• The distance measure perpendicular to the boundary by which the solid boundary should be displaced in order to compensate for the reduction in kinetic energy. That distance is called “Energy thickness”. Note that, in boundary layer theory energy thickness represent only kinetic energy.
• The thickness of different boundary layers is calculated by a mathematically derived formula. We can’t discuss the derivation here. Here, I can just explain the concept of derivation to derive these 3 thicknesses (displacement thickness, momentum thickness, energy thickness).
• To derive the equation of displacement thickness in the first step, we will calculate the mass flow rate at particular section XX in the boundary layer. In the second step, if there is no boundary layer, then how much mass flow rate will occur at a section XX that we will check then we will compare this two-step and calculate the total reduction in mass flow rate by integrating the equation.
• Similarly, the equation of momentum thickness and energy thickness can be derived. The only difference is, in the equation of momentum thickness, momentum difference is calculated and in the equation of energy thickness, the energy difference is calculated.
• You can check the complete step-by-step procedure of derivation of these 3 thicknesses (displacement thickness, momentum thickness, energy thickness).

## Equation of Displacement thickness, Momentum thickness & Energy thickness.

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