Drag and lift forces on flat plate

[JimK]

I'm trying to calculate the aerodynamic forces on a flat plate as a function of angle of attack.

The drag force is calculated from F = ρ*V^2*A*Cd/2

Where ρ is the air density, V is the wind velocity, A is the area, and Cd is the drag constant.

Simple question:

For A, should I use the Frontal area facing the wind, or the total Surface area of the plate? I've searched, and I can't find which assumption the calculation of Cd is based on.

Second question:

Can you give me the formula for calculating the lift vector for this same plate as a function of the angle of attack?

Thank you for your help.

Jim

 

[boneh3ad]

You typically use the frontal area that "sees" the wind. That said, in this situation where your angle of attack is changing, C_D will be changing as well. Lift uses the same formula except using C_L instead of C_D.

Lift always points up and drag always points backward, so if you know the forces, you know the vectors.

Keep in mind that these formulas are empirical and are not based on the physics governing lift and drag. There are no analytical, physics-based formulas for lift and drag. There are some good approximations, at least for lift, but they are substantially more complicated.

 

[JimK]

Thank you for your quick reply. I appreciate it.

I am already taking into account the variable Cd and Cl as a function of angle of attack. I've fitted curves to the published table data.

I'm using the Frontal area for my drag calculations.

However, I'm using the total Surface area for the lift calculations. This is based on reading several aerodynamic papers. It somewhat makes sense, based on how the lift is generated for a normal wing cross section. I really don't know whether this is right or wrong. The explanations I've seen are somewhat vague. It seems wrong to use the Frontal area for the drag calculations and the Surface area for the lift calculations. You'd think they would be consistent. It makes a big difference, depending on how the C coefficients are calculated.

Do you know which area I should be using for the lift calculations, using the standard Cl coefficient?

Thank you for your help.

Jim

 

[boneh3ad]

It's hard to say without knowing the source of your coefficients. The correct reference area is generally the one that was used when generating the coefficient in the first place. For drag that is usually frontal area, for lift, wing area. It would theoretically work either way though depending on the particular reference area used by whoever came up with the coefficient.

 

[JimK]

I agree. The problem seems to be that the references to the coefficients don't cite the definition of the area used to derive them.

For now, I'm going to assume: Frontal area for drag. Surface area for lift.

Thank you for your help.

Jim

 

[256bits]

However, I'm using the total Surface area for the lift calculations. This is based on reading several aerodynamic papers. It somewhat makes sense, based on how the lift is generated for a normal wing cross section. I really don't know whether this is right or wrong. The explanations I've seen are somewhat vague. It seems wrong to use the Frontal area for the drag calculations and the Surface area for the lift calculations. You'd think they would be consistent. It makes a big difference, depending on how the C coefficients are calculated.

Cd has a pressure component and a skin component. A flat parallel to the fluid velocity has overiding skin friction, whereas perpendicular to the fluid flow an overdiding pressure friction component. At both these conditions it seems sensible to use the area of the plate as the characteristic for the total drag. It does then not seem reasonable to move from one characteristic area ie chord length and span, in determining the drag of an object, and then moving to another characteristic area ( frontal area ) with changing angle of attack for the same body.

Also, induced drag, a result of induced vortex at the span edges may be something to look at. Depending upon the aspect ratio, ie slenderness, and fluid velocity this can become important in calculating the total drag.

 

[JimK]

Thanks for the feedback. I completely agree with your logic.

It would be really helpful if they would define the area they use to get the two coefficients. One argument I've heard is that they are usually used for small angles of attack. For this case, the drag is predominantly due to frontal area. And the lift is predominantly due to the lift produced from the surface area of the airfoil.

Would the real answer please step forward. :) I'm actually designing a blade using these formulae, so it would help to have the best data...

 

[JimK]

I've given into using the Surface area for both CD and CL. I ran across a post that said the surface area used to calculate CD is SPAN*CHORD, which is the Surface area.

After doing the new calculation, it turns out the difference between using the Frontal area and the Surface area is very small (< 1%) for my application. I'm calculating the effect around a whole revolution with a varying blade pitch.

For future reference, it looks like the Surface area is the correct area to use for both the Drag and Lift force equations.

Thank you for all your help!

Jim

 

[256bits]

It just seems to make more sense for comparison.
A listings of Cd using the same characteristic area ie span and chord , would be easier to make inferences as the angle changes from 0 degrees ( parallel ) to 90 degrees ( perpendicular).

The value of Cd would suffer a discontinuity from that of a parallel plate based on total area. A change to frontal area with an angle of attack lowers the characteristic area and Cd would have to abrupty increase.

Frontal area is a function of area of the plate mutliplied by the angle of attack anyways, so why not just use one characteristic area for all angles of attack, and the user can determine his own frontal area if he/she needs that calculation.

Having said that, the tabulation of Cd should give some indication of how the data was determined, in reference to boneh3ad statements.

This may give you some more insight on wings.
http://www.history.nasa.gov/SP-367/chapt4.htm

 

[JimK]

Interesting discussion, all based on people not defining their assumptions when they present a result.

Thanks for the reference.

 

[boneh3ad]

That's a very valid point, 256bits.

This, of course, highlights the pitfalls of using formulas based on empirically-determined coefficients and reference dimensions.