Higher Order Panel Method / BEM Current State Overview

Zeeshan,

Is there a paper or other reference which describes the method you used, and in particular the quadrature technique to avoid the singularities?

I assume your code models the thickness of the lifting surface with the panels distributed over the surface of the lifting surface. I assume by "non-linearity" you mean the effects of separation of the flow over the low pressure side of the lifting surface at high angle of attacks. Unfortunatley the effects of separation can not be modeled directly by panel methods. Panel methods are based on using potential flow assumptions which do not include viscous effects and separation.

Attempts have been made in the past to model separation using a combination of a panel method and a boundary layer code with separation prediction. The potential flow around the lifting surface is first solved for without including the separation and a set of streamlines along the surface are calculatied. Next the boundary layer code is run along the streamlines and the separation locations predicted. Then some sort of ad-hoc model of the separation zone is used to modify the potential flow results. If the separtion zone is small these types of models can sometimes provide reasonable approximations of experimental data. However they don't usually provide results for large separation such as with a stalled lifting surface which are in reasonable agreement with experimental data.

 

Bem

Thanks for the quick reply. Thanks for the suggestion on separation.

Infact I have two codes. One for linearily distributed singularities and using node as a collocation point and other one is simple low order panel method code (constant singularity strength over a panel and centroid as a collocation pt). These days I am using the low order one and I am using doublet only formulation. Can I ask u one more question which is different from the previous one. I am trying to achieve the manouevring characterstics of a submarine using low order panel method code. Unlike aircraft, submarine has low aspect ratio planes/rudders (wings). I am considering thick wing section. Toward the trailing edge of the rudder (NACA 0012 wing section), we have Morino Kutta condition (linear), which cater for the zero pressure jump across the wake but what about the chord tip and root tip of rudder spanwise. I mean how the panel method is catering for the sudden change in geometry at the tip chord and what abt the influence coefficient of panel at the two chord sections. Should we treat the normally or there must be condition at the tip of wing section.

 

Nonlinear aerodynamics means modeling flow separation. You will need a boundary layer calculation to find the separation point. Then you need to be able to model the wake shed by the separated region.

Here are some papers that may help with the wake problem:
Katz, Joseph, "Large-Scale Vortex-Lattice Model for the Locally Separated Flow over Wings," AIAA Journal, Vol.20 No.12 (1640-1646), 1982.

Daniel Levin, David and Katz, Joseph, "Vortex-Lattice Method for the Calculation of the Nonsteady Separated Flow over Delta Wings," Journal of Aircraft, Vol.18 No.12 (1032-1037), 1981.

Katz, Joseph and Maskew, Brian, "Unsteady low-speed aerodynamic model for complete aircraft configurations," Journal of Aircraft, Vol.25 No.4 (302-310), 1988.

Basically, Katz uses vortex rings that are shed from the separated portion of a partially stalled wing.

 

You can actually do a good job with a panel code by ignoring the tip. The CMARC panel code manual has a discussion of this issue.

A bigger problem is how to deal with the wake of the hull, as there will be separation on the hull and the wakes of the diving planes will intersect with the hull. This paper may help:

WARD, KENNETH C.; KATZ, JOSEPH, "Boundary layer separation and the vortex structures around an inclined body of revolution," AIAA-1987-2276. AIAA Applied Aerodynamics Conference, 5th, Monterey, CA, Aug 17-19, 1987, Technical Papers (A87-49051 21-02). New York, American Institute of Aeronautics and Astronautics, 1987, p. 83-93.

 

I don't understand how the inclusion of the free surface requires a so-called "iterative" solution, unless you mean an iterative matrix inversion? BYDE is your technique frequency domain, linearized about the wave amplitude, e.g. for generalized sea keeping analysis?

EDIT:
Here's a paper with some info on WAMIT's HO method

http://www.wamit.com/Reports/Report03.pdf

 

It depends on what version of a free surface condition is being used and where it is applied. If a linearized free surface condition is used and applied on the mean, flat surface of the water then the resulting set of equations are linear and in theory iteration is not required. This is equivalent to using Havelock sources and singularities derived from those sources. Whether iterative solution of a matrix is required depends on the other assumptions and boundary conditions Michel's thin ship theory as used in Michlet is an example where iteration is not used due to the assumed symmetry and application of boundary conditions on the centerplane of the vessel. Including asymmetry and/or applying the boundary conditions on the surface of the vessel away from the hull while using the linearized free surface boundary condition applied on the mean surface of the water generally requires interative solution of the resulting linear equations.

If a linear free surface condition is used and applied on the free surface with the location of the free surface being determined as part of the solution then the resulting set of equations is non-linear and requires iterative solution.

Likewise if a nonlinear free surface condition is used and applied on either the mean surface or on the free surface then the resulting set of equations is non-linear and requires iterative solution.

 

That's what I thought, thank you for clarifying.

I have a friend in Houston who has added the free surface condition to the aerospace panel code APAME for his Master's degree (I met him at an AQWA training seminar):

http://www.3dpanelmethod.com/

I don't think his approach is exactly what you're doing however (he's trying to make APAME work more like WAMIT, whereas you're more interested in making it work like Michlet, with the centerline Havelock sources?)

Are Havelock sources complex numbers like in the radiation/diffraction theory found in WAMIT/AQWA?

This is really interesting stuff.

EDIT:

What I've seen in AQWA is that the non-linear free surface condition is handled in a time domain response by calculating the Hydrostatics plus Froude-Krylov forces based on the instantaneous shape of the water and position of the vessel, with the linear frequency domain diffraction and radiation forces calculated by convolution of those frequency domain parameters to bring them into time domain.

 

First the steady state situation where the final results are real, not complex.

It's common to use complex number notation in the formulas for sources which satisfy the linearized free surface condition, with the final results being real, not complex. This is done because it can be considerably simpler to manipulate exponential notation then trig functions, ie sine and cosine. Two different approaches can be used. Probably the most common is to put a "Real" function in front of the formula as in
cosine(x) = Real[ e^ix ]
The other approach is more complicated (though what I usually prefer) with the form which doesn't use the "Real" function as in
cosine(x) = [ e^ix + e^-ix ] / 2

As for the formulas themselves Tuck, Scullen and Lazauskas's discuss Havelock sources with the older form and the one derived by Newman in the 1980's in http://www.cyberiad.net/library/pdf/tsl02b.pdf

For the unsteady situation complex notation can be used in a similar manner as for the steady situation, even with a time domain formulation.

For the frequency domain formulation complex notation is usually used with the real part being in phase and the imaginary part one-quarter cycle out of phase.

 

If the calculations are done in the time domain then a time marching solution method can be used rather than an iterative solution.

One common method for solving "steady" non-linear problems is to convert them to an equivalent "time-dependent" problem which will converge to the "steady" solution.

 

I'm not dead yet!

 

My apologies, spelling corrected.

Anything else you disagree with or wish to expand upon?

 

Not really. Only that the good fun is in trying to calculate various distributions
of the potential and derivatives accurately and efficiently.

 

I neglected to mention that a time domain solution needs to start with a valid solution at t=0. That solution may require iteration.

 

Thanks for the link to the paper and the explanations.

I'm still going through the derivation and use of Michell's integral to compare it to the theory in WAMIT/AQWA. The only conclusion I can really come up with is that the underlying theories are similar, but totally different! I mean, you use Michell's integral for resistance and powering, but WAMIT/AQWA for traditional RAO-based seakeeping calculations.

Do you use splines in Michlet to do this by any chance?

EDIT:

I've found that to be especially true when modeling cables.

 

I'm not familar with WAMIT/AQWA other than looking at the website and a brief look at a manual. So the following is based on some guesses about WAMIT/AQWA. Please correct me if my guesses are wrong. And I expect Leo will also speak up if I make some mistakes about Michlet.

Similarities

Both use potential flow assumptions.

Both use surface distributions of singularities.

Differences

Michlet is for steady state only. WAMIT/AQWA can treat unsteady.

Michlet uses a linearized free surface boundary condition applied on the mean surface. WAMIT/AQWA uses a non-linear free surface condition.

Michlet is based on thin ship assumptions and assumes a transversly symmetric hull. Sources are distributed along the centerplane of the hull (not the surface of the hull) and the source strengths are obtained from the freestream velocity multiplied by the local transverse slope of the hull surface. There is no interaction between source strengths. This results in the solution being an explicit integral, not an integral equation. The integral can be calculated directly as a quadrature. Multi-hull solutions are similar to mono-hulls with an integral for each multihull.

WAMIT/AQWA allows assymetric shapes and arbitrary geometry, and uses singularities on hull surfaces, etc. Singularity strengths interact with each other which results in integral equations rather than simple integrals. The integral equations require iteration or time marching for solution.