Lift Coefficient

Hello everyone.

Lift in various (aeronautical) literature is defined as

L = CL*0.5*ρ*v^2 * S

where CL is the lift coefficient
ρ is the fluid's density
v is velocity
and S is the reference area

However in marine design literature lift is shown as:

L = CL*0.5*ρ*v^2 * b^2

where b is the beam of the planning area.

Could someone point a rough guideline how b^2 is related to reference planning area as shown in first relation?

Thank you.

 

I suppose b^2 will be an (rough) approximation of the planing area, because it is difficult, to measure the actual lenght of immersed part of the bottom while planing.

 

The general definition of lift coefficient is

CL = L / (0.5 * rho * Vref^2 * Sref)

where Vref is a reference velocity and Sref is a reference area. These are completely arbitrary, but some choices make more physical sense than others.

For external flows around bodies, it makes most sense to choose Vref to be the freestream velocity, Vref = Vinf, which is what is usually done. (But note that for a sail in a wind profile, Vinf is not quite unique).

For a wing it's traditional to define Sref = S, which is the planform wing area projected along the lift direction, which also makes physical sense. Another defensible choice, although not traditional, is the span squared, Sref = b^2, as in your second example. In this case the resulting modified wing lift coefficient (call it CL') is actually equal to the induced-drag/lift ratio times pi times the span efficiency:

CL' = L / (0.5 * rho * Vinf^2 * b^2) = CL * S/b^2 = CL^2/(pi e b^2/S) * pi * e / CL = pi * e * CDi/CL


The bottom line is that whenever anyone posts CL or CD data, one should always specify what the reference quantities are, unless they are "traditionally obvious" like for a wing. If the reference quantities are not known, then the data is not usable.

 

Deadrise angle must have some effect on CL, is that factored into the equation?

 

Of course it does. It modifies the CL of the flat-plate, according to formulae by Savitsky shown in the attached pdf (page 82).

 

The relation as stated at my question is the general relation for lift regardless the shape of the plate. Stavisky has derived an empirical relation for determining the lift coefficient for plates with deadrise.

 

Welcome to fluid dynamics - the science of the non-constant constant!

I think it's most useful to think of coefficients as a bookkeeping system. There are different conventions, depending on what you know and what your objectives are. Often the largest effects are removed so the remainder can model the more subtle aspects. In the case of lift coefficient, the lift is divided by dynamic pressure, removing the effects of fluid density and speed. That leaves a lift area.

The aeronautical approach is to divide the lift area by the planform area to get a lift coefficient. You could do the much the same thing for a planing boat, but the problem is knowing the chord.

The wetted length of the bottom will depend on the boat's operating condition, so it's not very useful for nondimensionalizing the lift data. So naval architects use the square of the beam instead. You could think of this as adopting the beam as the reference length of the wetted patch. The actual variation in the wetted area then gets cooked into the lift coefficient. A typical planing hull has hard chines where the flow separates cleanly, so the beam is a reliable reference for scaling the forces.

It really is more bookkeeping than physics. If other naval architects use beam-squared in their definition of lift coefficient, then it makes sense for you to do the same if you want to be able to compare your data to theirs. But you may have some other objective and different information. For example, you might have underwater photographs from a tow tank that show the actual wetted surface. If so, and you were doing some research that was sensitive to the wetted area, you might want to define your lift coefficient based on the observed wetted area. That would be just as legitimate. However, you'd need to make very clear just what your definition of the lift coefficient was, and how to convert your data to the convention of others (such as multiplying by wetted area/beam^2).

 

Tom Speer,

thank you for your clear answer. I fully understand the meaning of the coefficient which is not constant at all. I just dont really grasp the idea of substituting the reference plan area with its beam (span) squared as an alternative reference area. On the other hand, if I see it just as a common way of expressing the relation then it gets more sense, especially if the definition is clearly stated.

I appreciate all for your answers, fluid dynamics is indeed a strange but beautiful subject!

 

I think the bottom width and the distance between the stern and the center of the weight should be the best. Because they are constant under all conditions.

js

 

Width between chines is logical since flow separates at chines. Longitudinal center of pressure is the chord distance but since it depends on dynamic trim angle, which is unknown until one finds equilibrium, you cannot reference it.

 

I wrote the weight center of gravity, not the planing surface pressure point. There is an important difference to me. Weight's center of gravity is constant as the bottom width on my boat. The lifting force consists of two forces, buoyancy and the planing surface pressure lift. The resultant of these two forces is at the weight's center of gravity when it is in equilibrium.


js

 

My fault, not giving enough attention on your previous reply. You are right about CG.

 

Unfortunately the publication of CL/CD values used for planing hulls don't always make the reference Area clear. Results using CL/CD values can only be helpful when the same reference Area is used. Sometimes CL/CD coefficients are derived from theory or often by scale experimental verification. Unfortunately the difference in reference Area's that is sometimes used makes the comparison of CL/CD values difficult (inaccurate) unless the reference Area's are normalized.


Tom's explanation is a good one.

 

But the CoG of a given boat can vary. And in effect, it is a variable in the calculations.
The chine width - well, varying that is much more difficult on existing boats. At the design stage too the chine width is often pretty much fixed value due to road transportability or marina-berthing cost considerations.