Source-Doublet panel potential derivatives

lWiNdY · July 11, 2022, 10:06

In a 3D panel method one computes the panel tangential velocities using the doublet strengths $\mu$ as

$q_l = \frac{d\mu}{d\l}$ ,
[LaTeX Error: Syntax error] in a local panel reference frame (see figure). However, how does one compute the derivative using finite differences?

That is in the case of a sphere there are triangular panels (see second and third figure) for which each panel is in a direction not parallel or orthogonal to the l-m-n coordinate system. What would be the tangential velocities in the l-m-n coordinate system?

1st figure : Katz & Plotkin, Low Speed Aerodynamics
2nd figure : Kirian Dixon, The Near Wake Structure of a
Vertical Axis Wind Turbine

sbaffini · July 11, 2022, 10:09

You need to compute the superficial gradient using methods for unstructured grids. It is a vector that you can later project along any direction you might need

lWiNdY · July 11, 2022, 10:21

Do you have any pointers to literature on that topic?

sbaffini · July 11, 2022, 10:27

Nope, but I have seen a similar treatment for vortex rings. You can also give a look at apame to see how it does that

ichamail · August 3, 2022, 16:35

Check the book COMPUTATIONAL FLUID DYNAMICS Principles and Applications by JIRI LAZEK.
Read the chapter 5.3.4 evaluation of the gradients (least squares approach).
At page 151
check eq 5.54 and 5.55
(rij · ∇)μi = μj − μi, where rij is the vector from i-th panel's control point to j-th panel's control point and μj, μi, are j-th and i-th panel's doublet strengths.

July 11, 2022, 10:27		#4
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Nope, but I have seen a similar treatment for vortex rings. You can also give a look at apame to see how it does that Last edited by sbaffini; July 13, 2022 at 05:35.

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July 11, 2022, 10:09		#2
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	You need to compute the superficial gradient using methods for unstructured grids. It is a vector that you can later project along any direction you might need

July 11, 2022, 10:21		#3
lWiNdY New Member L. Join Date: Jun 2022 Posts: 12 Rep Power: 4	Do you have any pointers to literature on that topic?

August 3, 2022, 16:35		#5
ichamail New Member Join Date: Jul 2022 Posts: 20 Rep Power: 4	Check the book COMPUTATIONAL FLUID DYNAMICS Principles and Applications by JIRI LAZEK. Read the chapter 5.3.4 evaluation of the gradients (least squares approach). At page 151 check eq 5.54 and 5.55 (rij · ∇)μi = μj − μi, where rij is the vector from i-th panel's control point to j-th panel's control point and μj, μi, are j-th and i-th panel's doublet strengths.