Incompressible Potential Flow Analysis Using Panel Method

Incompressible Potential Flow Analysis Using Panel Method

ShahNor Basri, Norzelawati Asmuin & Aznijar Ahmad Yazid

Universiti Putra Malaysia

Jabatan Kejuruteraan Aeroangkasa

Fakulti Kejuruteraan,

Universiti Putra Malaysia, 43400 UPM SERDANG, Selangor D E, Malaysia.

kaa@eng.upm.edu.my

ABSTRACT

Incompressible potential flow problems are governed by LaplaceÐŽ¦s equation. In solving linear, inviscid, irrotational flow about a body moving at subsonic or supersonic speeds, panel methods can be used. Panel methods are numerical schemes for the solution of the problem. The tools at the panel-method user's disposal are the representation of nearly arbitrary geometry using surface panels of source-doublet-vorticity distributions, and extremely versatile boundary condition capabilities that can frequently be used for creative modeling. Panel-method capabilities and limitations, basic concepts common to all panel-method codes and different choices that were made in the implementation of these concepts into working computer programs are discussed.

Keywords

Panel method (fluid dynamics), incompressible potential flow, application programs (computer), computational fluid dynamics.

INTRODUCTION

Incompressible inviscid flow is governed by LaplaceÐŽ¦s equation. An extremely general method to solve this equation is the panel method. The flow may be about a body of any shape or past any boundary. Almost any boundary conditions, not just due to the fluid flow, can be solved. For 2-dimensional problems, the profile is approximated by a many-sided inscribed polygons. For 3-dimensional cases, a flat quadrilateral elements are used instead. The name ÐŽ§panel methodЎЁ derived from these treatments of the body shape.

Proper design of an airfoil requires an accurate prediction of the pressure distribution. Initially, thin-airfoil theory is used to analyse or design airfoils. However, due to its deficiencies for multi-element airfoils, this theory is not much used these days. Among the shortcomings of this theory is the inability to take into account the effect of thickness distribution on the lift and moment coefficients as well as the results at areas nearer to the stagnation points are not good.

After TheodorsenÐŽ¦s work on single-element airfoil problems using a semi-analytic method in the 1930s, more work was performed to produce even more accurate prediction of pressure distributions on airfoils. These works are based on the distributions of sources and vortices or doublets. In order to avoid the inaccuracies of the thin-airfoil theory, the body surface must satisfy the flow-tangency conditions without approximations. Additionally, the singularities are distributed on the body surfaces rather than any other line within the body or the chord line. This technique could be used to treat any airfoils, including multi-elements airfoil to any desired accuracy and could also be extended to handle three-dimensional flows.

The methods of approximating the body surfaces by a collection of ÐŽ§panelsЎЁ, are aptly named ЎҐpanel methodsÐŽ¦. There are various ways in setting-up this method. Using any combinations of sources, vortices or doublets the types of singularities used can be varied. The pioneering work using sources and vortices were conducted by Hess and Smith.

Computational work on fluid dynamics requires the calculation for the entire 3-dimensional field about the body. However, the panel method requires only calculation over the surface of the body when handling 3-dimensional field. This reduces the calculation, hence lowering computing time.

INCOMPRESSIBLE POTENTIAL FLOW

An inviscid, incompressible fluid is also sometimes called an ideal fluid, or perfect fluid. The Laplace equation is the governing equation for the solution of the problems of this inviscid, incompressible fluid.

The assumption made in solving LaplaceÐŽ¦s equation is that the flows satisfy the equations for continuity and irrotationality.

The equation for continuity, that is, the conservation of mass is given by

(1)

While the vector equation expressing irrotationality is

(2)

For two-dimensional flows, the velocity components, u and v can be expressed as,

(3)

Substituting the velocity components, u and v, for two dimensional flows, the equations for continuity and irrotationality would then become,

(4)

In the equation 4 above, the operator,

(5)

is termed the Laplacian operator, after Pierre de Laplace (1749-1827).

LaplaceÐŽ¦s equation is normally recognised just by the equation 5.

PANEL METHOD

There are various ways how the panel method can be implemented, and not only limited to the classical technique described by Hess and Smith.

Various techniques have been used by lots of different groups, producing different, competing codes. The variations in the implementation of the techniques are by the use of:

1. various singularities,

2. various distributions of the singularity strength over each panels, as well as,

3. different panel geometry.

The Hess-Smith code, developed in 1962 used a Neumann velocity boundary condition based on flat constant source panels. Initially meant for non-lifting potential flow, this code was added with constant doublet elements to handle flow with lift. Woodward then introduced a supersonic option code later in 1966 in Seattle. The code was aptly named Woodward. Using the same flat, linear sources but with linear