Cascade biplane theory

[willfox]

I am currently working on a design of a yawl and wondered if anyone knew of any papers in relation to cascade biplane theory. Any help would be great.

Will Fox

[daiquiri]

A broad question... Is there something particular that interests you?

To make it very concise, a theory for multiple lifting surfaces uses the same math used the theory for a single lifting surface.

In it's simplest 2D form, a single lifting surface can be rappresented by a flat plate at an angle of attack to the flow. It can be modeled mathematically as a vortex placed at 1/4 chord and a control point (where tangential flow to the surface is imposed) placed at 3/4 chord of the plate. By resolving the resulting equation at the control point, you obtain the vortex circulation and hence the lift at that given angle of attack.

A biplane model is the same thing, you just add another flat plate somewhere in the space, with its vortex and control point at the 1/4 and 3/4 chord position, respectively. Now you have a system of 2 equations with 2 unknown vortex circulations, which you resolve to obtain the total lift (the sum of individual surfaces), or to analyze the contribution of each surface to the total lift.

Though very simplistic, this vortex model gives you a fundamental insight to what is happening in a system of two lifting surfaces. You will, for example, discover that the surface which "works" (produces most of the lift) is the one
placed forward, the other one makes it work more by inducing an upwards-directed velocity component, which increase the angle of attack of the forward surface.

Once you have understood how the effects of two vortices sum up to give a resultant velocity at the control points (we can do that superimposition because we assume a potential flow around the foil), you can extend the analysis to an arbitrary number of vortices. By placing a number of vortices along an airfoil's chord (or along several airfoils), for example, you obtain a vortex sheet, which will result in an even more detailed flow around the body (or the bodies).

It can all be extended to 3D surfaces, by implementing a vortex panel method

You can learn more about it by googling the keywords written in bold letters above, there are many resources available in internet on that topic.
A qualitative 2D analysis of a system of two lifting surfaces can be done in JavaFoil too, as long as you use airfoils (and not sails, which have a sharp leading edge) for the analysis.

See this link for more info on vortex theory of lift, for example:
http://www.adl.gatech.edu/classes/lo...5/lospd5n.html
Something on Vortex Panel Method:
http://www.fluid.mech.ntua.gr/wind/b....html#num_meth
Also see this discussion, from the post #9 on, and the Tspeer's excellent explanation on how single elements in a multi-element lifting surface influence each other (post #13):
Reynolds number and JAVAfoil

[willfox]

Thank you so much for your help and the useful links. I will be working on this much of next week so I may need to pick your brains more if thats ok. Thank you again,

Will

[tspeer]

Originally Posted by willfox

I am currently working on a design of a yawl and wondered if anyone knew of any papers in relation to cascade biplane theory. Any help would be great.

I suspect the sails on a yawl are so widely separated that there's not much interaction with regard to their section characteristics.

The interaction of their wakes and impact on spanload and induced drag is another matter, however. A vortex lattice program would be the simplest CFD method capable of dealing with the 3D interaction between the sails.