# Boundary layer thickness: Definition, Equation, Diagram, Pdf

In the case of flow of fluid over a solid surface, the boundary layer is the thin region near the solid surface in which the velocity gradient is developed in fluid in the direction perpendicular to the solid surface. In this article, we will go over one of the most important terms in this topic, boundary layer thickness.

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## What is Boundary layer thickness?

Boundary layer thickness is the perpendicular distance from the solid surface at which the velocity of the fluid becomes equal to 0.99 times the free stream velocity of the fluid approaching toward the solid surface.

OR

It is also known as a perpendicular distance between the boundary layer and the solid surface whereas the boundary layer is the locus of all the points where the velocity is 0.99 times the free stream velocity (u_{\infty} or U).

It is denoted by the symbol \delta and its value changes over the length of the solid surface.

The below figure shows the generation of the boundary layer over the flat solid surface. Over the length of the plate, the boundary layer region is divided into following different regions. i.e. Laminar, turbulent, and transition where laminar flow converts into turbulent.

Based on it, the respective boundary layer thickness in each region is known as, laminar boundary layer thickness, and turbulent boundary layer thickness.

The turbulent zone also has a smaller laminar region near the solid surface known as the laminar sublayer and the thickness of this layer is denoted by \delta’.

### Characteristics of boundary layer thickness:

• At x = 0, \delta = 0
• At x = L, \delta = \delta_{max}
• The boundary layer thickness increases with the increase in distance from the leading edge.
• At y = \delta, \frac{du}{dy}\approx0 and u = 0.99U

## Why boundary layer thickness always increases over the surface?

As shown in the figure, assume a laminar flow of fluid over a flat plate. Before reaching the stationary solid plate, the fluid particles are moving with the constant free stream velocity of ‘U’.

When the fluid particles in the bottom layer come in contact with the leading edge of the plate, due to the adhesion, the fluid particles in the bottom layer stick to the plate.

Because of the cohesive force between the fluid particles (viscosity), the particles in the bottom layer attract the fluid molecules in the adjacent upper layer. Thus it slightly retards the molecules in the first layer.

As the fluid advances over the plate, the molecules in the bottom layer undergo more retardation, therefore it causes the molecules in the first layer to undergo more retardation. Due to the increase in retardation, the molecules in the first layer slightly retards the molecules in the second layer.

Now as the molecules in the second layer advances further, it undergoes more retardation, and thus it affects the molecules in the layer above it.

In this way as the fluid advances on the solid surface, the motion of more and more fluid layers gets affected due to the viscosity. Thus the more fluid layers enter the region of the boundary layer.

Hence the thickness of the boundary layer continuously increases over the surface.

## Boundary layer thickness for laminar flow:

The concept of critical Reynolds number ignores the transition zone. Actually, the critical Reynolds number lies in between the transition zone. For the flow over flat plate, the value of the critical Reynolds number (Re_{cr}) used for the calculation is 5\times10^{5}.

If Re_{x} < Re_{cr} then the flow is laminar.
If Re_{x} > Re_{cr}, then the flow is turbulent.

For the laminar boundary layer (Re_{x} < Re_{cr}), the boundary layer thickness by the Blasius solution is given by the below formula,

\delta = \frac{4.91x}{\sqrt{R_{e_{x}}}}

## Boundary layer thickness for turbulent flow:

For the turbulent flow (Re_{x} > Re_{cr}), the value of boundary layer thickness based on the Blasius solution is given by the below formula,

\delta = \frac{0.37x}{Re_{x}^{1/5}}

Where,
Re_{x} = Reynolds number at length of x
x = Distance from the leading edge

## Thickness of Laminar sublayer:

The laminar sublayer is also called a viscous sublayer.

In the region of turbulent flow, the laminar sublayer is the smaller region near the solid surface, where no turbulence existed. In this region, the fluid velocity is only affected by the viscous effects, not by the inertial effects.

The thickness of the laminar sublayer (\delta’) is given by,

\delta’ = 11.6 \frac{\text{Kinematic viscosity} (\nu)}{\text{Shear velocity} (V\text{*})}

Where,
V\text{*} = \sqrt{\frac{\tau_{0}}{\rho}}
\tau_{0} = Wall shear stress

## Factors affecting boundary layer thickness:

The thickness of the boundary layer is affected by some of the following factors:-

The pressure gradient shows the rate of change of pressure along the x-direction (dP/dx).

If [dP/dx < 0], it means that the pressure is decreasing along the direction of flow, thus it indicates that the kinetic energy is increasing along the flow direction. It increases the velocity along the direction of flow which results in a lowering of the boundary layer thickness.

If [dP/dx<0], it means that, the pressure is increasing along the flow direction. It indicates that the kinetic energy of the fluid is decreasing in the flow direction. Thus it causes deceleration of the fluid along the x-direction.
We know that the layers of the fluid near the solid surface have minimum velocity and this adverse pressure effect can cause more deceleration of these bottom layers. Thus it results in an increase in the thickness of the boundary layer.

2] Viscosity:-

The thickness of the boundary layer increase with an increase in the viscosity of the fluid. The increase in viscosity causes the fluid to undergo more retardation over the solid surface which results in increased thickness of the boundary layer.

Thus the fluid with higher viscosity can generate a larger boundary layer thickness in comparison with fluid with lower viscosity.

3] Velocity of free stream:-

The thickness of the boundary layer increases with the decrease in the velocity of the free stream coming toward the solid surface.

4] Distance over solid surface:-

The thickness of the boundary layer increase with the increase in distance from the leading edge. Thus the thickness of the boundary layer is minimum at the leading edge and maximum at the trailing edge.

## Boundary layer thickness answered examples:

1] A free stream of water is flowing over the flat plate of length 4m at the velocity of 0.5 m/s. If the kinematic viscosity of the water is 1 centistoke, Calculate the boundary layer thickness at the length of 0.4 m from the leading edge of the plate.

Given:
L = 4m
x = 0.4 m
U = 0.5 m/s
\nu = 1 centistokes = 10^{-6} m²/s

Solution:-

The Reynolds number at the distance x from the leading edge is given by,

Re_{x} = \frac{U.x}{\nu}

Re_{x} = \frac{0.5\times0.4}{10^{-6}}

Re_{x} = 2\times10^{5}

As the Reynolds number is less than the critical Reynolds number, thus the flow is laminar.

The boundary layer thickness for the laminar flow is given by,

\delta = \frac{4.91x}{\sqrt{R_{e_{x}}}}

\delta = \frac{4.91\times0.4}{\sqrt{2\times10^{5}}}

\delta = 0.00439 m

2] The air is flowing over a flat plate at the velocity of 3 m/s. If the plate has a length of 3 m, calculate the boundary layer thickness at the trailing edge of the plate. (Assume,\nu_{air} = 1.5\times10^{-5} m²/s)

Given:
U = 3 m/s
L = 3 m
\nu_{air} = 1.5\times10^{-5} m²/s

Solution:-

At the trailing edge, x = L

Thus the Reynolds number at the trailing edge of the plate is given by,

Re_{L} = \frac{U\timesx}{\nu_{air}}

Re_{L} = \frac{3 times3}{1.5\times10^{-5}}

Re_{L} = 6 \times 10^{5}

As the Reynolds number is greater than the critical Reynolds number (Re > 5 \times 10^{5}), the boundary layer is of turbulent nature.

As per the Blasius solution, the thickness of the turbulent boundary layer is given by,

\delta = \frac{0.37x}{Re_{x}^{1/5}}

\delta = \frac{0.37 \times 3}{(6\times 10^{5})^{1/5}}

\delta = 0.077 m

## FAQs:

1. Why boundary layer thickness is significant?

The boundary layer is the region in the fluid that gets affected when it comes in contact with the solid surface at different momentum. Thus the thickness of the boundary layer has much importance in study of aerodynamics, fluid mechanics, heat transfer.

2. Is the laminar or turbulent boundary layer thicker?

The turbulent flow has a thicker boundary layer than the laminar.

3. Is the thickness of the boundary layer fixed?

No, the boundary layer thickness increases over the surface.

4. Why does boundary layer thickness decrease with increasing velocity?

The increase in velocity increases the Reynolds number. As the thickness of the boundary layer is inversely proportional to the Reynolds number, thus the boundary layer thickness decreases with the increase in velocity.