In this lesson, we will be discussing the following topics:
1. Momentum Thickness (θ or δ**), Drag (D)
2. Thickness (δ) Estimate
3. Shear Stress (Tw) Estimate
4. Displacement Thickness (δ*)
5. Example – Is the Boundary Layer Thin?
video coming soon
In 1921, Von Kármán proposed a momentum integral theory for solving boundary layer equations. We will talk more on boundary layer equations in upcoming lessons, but this momentum integral theory has a few equations that can be used to solve for variables, such as drag.
Momentum Thickness, Drag
Consider the following flow along the flat plate:
The drag force D on the plate is given by the following equation, where b is the plate width and the integral is along the boundary layer thickness (δ):
The equation above was then written in a form that relates the total drag D to the momentum thickness (can be defined as θ or δ**). The equation for the momentum thickness, which is used in the drag equation, is given below too.
Momentum thickness is the distance by which the boundary should be displaced to compensate for the reduction in the momentum of the fluid due to the boundary layer formation. We will talk about the boundary layer and why the fluid gets displaced later in this lesson. Momentum thickness is defined from the displacement thickness, so we will talk more on this below.
This equation for drag can also be rewritten as either one of the following two equations, where Tw is the wall shear stress along the plate:
Then, the derivative of the drag dD/dx, can also be written as:
With some mathematical manipulation of the equations above, Karman determined the momentum integral relation for flat-plate boundary layer flow (for either laminar or turbulent flow):
Thickness Estimate
Another equation that was found was the thickness estimate (δ). The thickness estimate comes into play with the boundary layer. The thickness estimate is a measurement from the surface of the fluid (zero velocity) to where the velocity becomes the free stream velocity – it is the height of the boundary layer.
A boundary layer can be considered to be thin if δ/x in the equation below is smaller than about 0.1, which corresponds to a Reynolds Number of greater than about 2500.
Shear Stress (Tw) Estimate
The shear stress estimate along the flat plate, where Cf is the skin friction coefficient and is similar to the friction factor f that we’ve used in previous lessons, is given below:
Displacement Thickness
When there is a boundary layer around a flat plate, it actually displaces the flow around the plate (aka, the streamlines). δ* is called the displacement thickness of the boundary layer. Displacement thickness is the distance by which the flow around the plate is displaced because of the velocity decrease in the boundary layer. The flow is displaced outward, so it looks like a thickening of the flow. To find this value, we can use the following equations that relate the displacement thickness with other values:
This is what the displacement thickness and thickness estimate looks like:
Momentum thickness, although we defined it above, needs to be mentioned here again because it is actually defined from the displacement thickness. Another way to think of it, now that we know what the boundary layer is, is the following:
Momentum thickness is the loss of momentum in the boundary layer as compared with that of freestream flow.
More information on the differences between the thickness estimate, displacement thickness and momentum thickness (respectively δ, δ*, δ**) can be found at this link.
Example – Is the Boundary Layer Thin?
(a) We are trying to figure out whether air and water boundary layers at specific conditions are considered thin. Recall that I have written above if δ/x is smaller than 0.1 or if the Reynolds Number is greater than 2500 , the boundary layer is considered thin. We can find both δ/x and the Reynolds Number to prove whether the boundary layer is thin or not in both ways.
(b) Let’s just do this for air instead of water, since you can repeat the same calculations to do it for part (b). Let’s find the Reynolds Number with the equation we always use, based on the information given and the kinematic viscosity. The kinematic viscosity of air at 68F is 1.61*10^-4 ft^2/s, as found in textbook tables.
(c) The Reynolds Number calculated above is greater than 2500, so the boundary layer is considered thin. Now let’s find δ/x to confirm this again, if the value of δ/x is smaller than 0.1. We will use the equation below, which was discussed in a previous lesson plan, to find δ/x. Since the Reynolds number is less than 10^6, we have laminar flow and can use the appropriate equation:
(d) Now δ/x is 0.0635, which is less than 0.1. So this confirms that the boundary layer is thin.