Autodidact
Well-Known Member
My project is to learn how to do the calculations of the circulation/lifting line theory of lift on a wing, and if I'm successful, it will turn into a design and build log for an ultralight aircraft. Why go to all that trouble? Because I want to learn this mathematical process, and because I think I can design a better aircraft through this knowledge. Here's an outline of what I (think) I know so far:
a) the value circulation around an (infinitely long) spinning cylinder can be calculated according to the magnus effect, and the stagnation points can be located.
b) the circular section of the cylinder can be transformed, via a geometric or "conformal" transform process, into an airfoil section, and the calculated streamlines around the circle are transformed as well; actually, you can transform an arbitrary airfoil section into a circle or a close approximation of a circle. Therefore, there is a way to relate the calculated "circulation" around a circular section to the circulation ("gamma") around an airfoil section and the lift is, V*rho*gamma, where V=velocity, rho=air density, and gamma=circulation.
c) for 3d wings the 2d circulation is related mathematically to the wing planform and the wingtip (tip vortices) effects by considering an infinite number of "U" shaped vortices of infinitesimal vorticity integrated across the span and the total vorticity of the trailing vortex sheet is equal to the maximum vorticity of the circulation about the wing (the "bound vortex"). Looking at the trailing vortex sheet (and the tip vortices which I understand so far to be a "rolling up" of the vortex sheet) from directly aft, the right side rolls up counter-clockwise, and the left side rolls up in a clock-wise direction so that they meet in the middle in a downward direction. The resultant of this downward velocity component added to the free stream velocity (aft of the wing) is what causes the downwash on the tail. This downwash, as well as the induced upwash (related to "tip-stall"), and the tilting of the lift vector (which neglects, at this point, the friction drag) which represents the induced drag, can be calculated via a process that takes into account something called the Biot-Savart law which defines the velocity of the streamlines of a vortex in terms of their distance from the origin of the vortex.
This will take some months, possible a year or two, to accomplish, and that's if I can get around the math. And, as stated, if I can get that far then I will continue with the design of a small single seater which will hopefully be Pt 103 legal and relatively sophisticated structurally and aerodynamically, so that it can operate efficiently with a PPG motor for propulsion.
First I have to review a quicky-book (Fluid Mechanics DeMystified by Merle C. Potter), which I have previously worked through, and then read a more detailed text (SCHAUM'S ouTline's FLUID DYNAMICS by Hughes and Brighton) and then Either Theoretical Aerodynamics by Milne-Thomson or Low Speed Aerodynamics by Katz and Plotkin (I'm currently leaning toward the latter).
This is a long term project, so the next post will be weeks or months from now. I hope I can make this interesting (at least to those interested in the subject matter) as well as informative and helpful - we'll see...
This is pushing the boundaries of the definition of this forum's Project Logs section, but I hope the moderators will be tolerant of it.
a) the value circulation around an (infinitely long) spinning cylinder can be calculated according to the magnus effect, and the stagnation points can be located.
b) the circular section of the cylinder can be transformed, via a geometric or "conformal" transform process, into an airfoil section, and the calculated streamlines around the circle are transformed as well; actually, you can transform an arbitrary airfoil section into a circle or a close approximation of a circle. Therefore, there is a way to relate the calculated "circulation" around a circular section to the circulation ("gamma") around an airfoil section and the lift is, V*rho*gamma, where V=velocity, rho=air density, and gamma=circulation.
c) for 3d wings the 2d circulation is related mathematically to the wing planform and the wingtip (tip vortices) effects by considering an infinite number of "U" shaped vortices of infinitesimal vorticity integrated across the span and the total vorticity of the trailing vortex sheet is equal to the maximum vorticity of the circulation about the wing (the "bound vortex"). Looking at the trailing vortex sheet (and the tip vortices which I understand so far to be a "rolling up" of the vortex sheet) from directly aft, the right side rolls up counter-clockwise, and the left side rolls up in a clock-wise direction so that they meet in the middle in a downward direction. The resultant of this downward velocity component added to the free stream velocity (aft of the wing) is what causes the downwash on the tail. This downwash, as well as the induced upwash (related to "tip-stall"), and the tilting of the lift vector (which neglects, at this point, the friction drag) which represents the induced drag, can be calculated via a process that takes into account something called the Biot-Savart law which defines the velocity of the streamlines of a vortex in terms of their distance from the origin of the vortex.
This will take some months, possible a year or two, to accomplish, and that's if I can get around the math. And, as stated, if I can get that far then I will continue with the design of a small single seater which will hopefully be Pt 103 legal and relatively sophisticated structurally and aerodynamically, so that it can operate efficiently with a PPG motor for propulsion.
First I have to review a quicky-book (Fluid Mechanics DeMystified by Merle C. Potter), which I have previously worked through, and then read a more detailed text (SCHAUM'S ouTline's FLUID DYNAMICS by Hughes and Brighton) and then Either Theoretical Aerodynamics by Milne-Thomson or Low Speed Aerodynamics by Katz and Plotkin (I'm currently leaning toward the latter).
This is a long term project, so the next post will be weeks or months from now. I hope I can make this interesting (at least to those interested in the subject matter) as well as informative and helpful - we'll see...
This is pushing the boundaries of the definition of this forum's Project Logs section, but I hope the moderators will be tolerant of it.
