representations. The problem is governed by a field ...€¦ · method based on Treftz solution for...

An alternative approach to integral equation

method based on Treftz solution for inviscid

incompressible flow

Antonio C. Mendes, Jose C. Pascoa

Universidade da Beira Interior, Laboratory of Fluid

Mechanics, Covilhd 6200, Portugal

Jan A. Kolodziej

Poznan University of Technology, Institute of Applied

Mechanics, 60-965 Poznan, Poland

Abstract

The paper deals with the numerical modelling of inviscid, incompressible flowabout two-dimensional lifting surfaces at low speed. The problem is formulatedin terms of the Laplace equation. For its solution two different approaches arediscussed. The first one tackles the problem in terms of the velocity potentialwith normal derivative prescribed at the boundary. An integral equation methodbased on a distribution of vortices is utilised to solve the flow. As for the secondapproach the governing equation is written in terms of the stream function. Treftzmethod is then used to express the solution as a power series of the fundamentalsolutions of Laplace equation.

1 Introduction

Modelling of inviscid incompressible flow about lifting surfaces of anarbitrary geometry may be conducted by means of integral or differential

Transactions on Engineering Sciences vol 18, © 1998 WIT Press, www.witpress.com, ISSN 1743-3533

58 Advances in Fluid Mechanics II

representations. The problem is governed by a field equation which hasoriginally been solved by numerical methods applied to a grid extendedalong the fluid domain. By making use of Green's theorem the solution of

the problem may be reduced to solve an integral equation written for thephysical boundaries of the flow. In this context the classical approach bysurface singularity distributions has originated the well-known panelmethods in two and three-dimensions - Hess [1].

Another classical approach, which is not so commonly appliedto solve this class of problems, is the Treftz Method - Kolodziej [2],Herein we introduce this methodology as an alternative to the moretraditional integral equation approach. In this last approach the problemis formulated in terms of the velocity potential with normal derivativeprescribed at the boundary. This harmonic function is afterwardssearched for as the velocity field generated by a distribution of vorticesalong the body contour. The resulting Fredholm integral equation, of thefirst kind, is subsequently solved for the tangential velocity at a discretenumber of panels that are distributed along the body profile.

The numerical results that were obtained by modelling a NACA0012 airfoil have been subsequently compared to those experimentallyobtained by the authors in a low-speed wind tunnel.

2 Potential flow equations

Let us consider a two-dimensional lifting surface of an arbitrary shapedcross-section, like the one represented in Fig. 1. Its contour ( C ) is aclosed line having continuous derivative everywhere except at the

trailing edge, whose outward normal n points to the unbounded fluiddomain (D). Here OXYZ is a stationary co-ordinate system and oxyz is asystem of axes attached to the body.

Fig. 1 Wing section moving with speed U and angle of attack a in fluiddomain (D).


Advances in Fluid Mechanics II 59

The solid body advances with velocity U in air which is initially

at rest; a is the angle between the chord c and the direction of motion of

the wing. At low speed the flow induced in the vicinity of the body maybe assumed as incompressible. Moreover, if we assume inviscid,irrotational flow, the fluid velocity V may be derived from an absolutepotential. This function is the solution of the following system ofequations:

V 0(z) = 0 , (zeD) (1)

-•"•' m

where z stands for the complex co-ordinate z=x+iy. The velocitypotential is, therefore, the solution of Laplace equation (1) and theimpermeability condition (2), a condition to be applied on the solidboundaries of the flow. At infinity a regularity condition applies whichmay be written in terms of the relative potential:

= -U , (z-»±oo) (3)

3 Integral-equation approach

The harmonic function formulated above is herein searched for as thereal part of a complex velocity potential F(z). This velocity field will begenerated in every point z of the fluid domain (D) by a distribution of

Rankine vortices with density y(z') at a generic point along the contour (z'

At this stage the velocity field associated with this functionsatisfies automatically to the governing equation (1) and must equallyverify the boundary conditions stated above. As to condition (2) imposedon the surface of the body, this condition may be formulated in theintegral form as follows:



= 0.5 (5)

taking into account eq (3).In the previous equation the kernel of the integral may be easily

determined after differentiation of the function defined in eq (4). The

angle 0 represents here the argument of the unit-vector tangent to thecontour of the body. Equation (5) is a Fredholm integral equation, of the

first kind, which is to be solved in order to the vortex densities y, for aprescribed body motion with velocity U.

Fig. 2 Discretized contour of the wing section.

The solution of this equation for a body having a contour ofarbitrary geometry requires the use of a numerical approach. A numericaltreatment that is commonly adopted consists of sub-dividing the contour( C ) in N linear elements Sj (j=l..,N), as it is shown in Fig. 2. Each ofthese elements Sj is the support of a uniform distribution of vortices with

density yj. Sj is defined by its extremity points Zj and Zj+i, and its

argument 8,; Z=z/c are non-dimensional complex co-ordinates.

By enforcing the impermeability condition on N control points TJtaken as the middle point of each element of the contour, eq (5) assumesthe discretized form:

, = M,,U = i..,N) (6)

The influence coefficients of the velocity By may be calculated after

integration of the left-hand side of eq (5), taking yj constant along eachlinear boundary element.



B,, =0 (i- j);

As to the right-hand side of the linear system, n, is the scalar product ofthe unit-vector in the direction of the body motion to the normal unit-

vector at the middle point ij=0.5(Zi+Zi+i).

The system of N equations (6) is to be solved for the N unknown

normalised densities (yj/U). In order to get a finite value for the velocityat the wing trailing edge and the system to be determined it is, however,necessary to impose an additional condition at this point. Although Kuttacondition may be imposed in few different ways, a convenient form fornon-cusped airfoils is to ensure equal velocities on either side of the

trailing edge, as suggested by Anderson [3]: yi+y#=0. This is

achieved by simply replacing the last equation of system (6) by theformer relationship.

Finally the pressure coefficient is derived from Bernoulli's law:

Following Katz & Plotkin [4], the tangential velocity within the fluidadjacent to the profile may be obtained directly from the vortices densitythat were calculated above, on each panel of the body. The lift coefficientof the airfoil may then be derived from Joukowski theorem as a functionof the circulation F:

C -— F^L- (9)

4 Treftz solution

The governing equation is here written in terms of the stream function Yof the flow and the value of this function is prescribed at the boundary. Intwo-dimensional potential flow, Cauchy-Riemann relationships appliedtoeq(l) yell the following Laplace equation:



(10)

The solution for this flow field is now assumed to be the superposition of

a uniform stream of constant velocity U and incidence a, with aperturbation of the flow which is due to the presence of the body.

(11)k=\ i=\

where A^ are unknown weighting coefficients of the truncated series that

may be determined by means of a simple collocation procedure - Collatz

[5] - or by using a Galerkin approximation; y\(x,y,Xk,yd are trialfunctions that exactly fulfil eq (10). These fundamental solutions aresources forz = l, vortices for i=2, doublets for i=3 and multipoles for

(12)

where

n -, X X i

The co-ordinates (x .y ) represent here the positions of these

singularities placed inside the contour of the airfoil.Solution (11) fulfils exactly the governing equation (10) and

must also satisfy the boundary condition required on the surface of the

airfoil, where the stream function must be zero: (*¥ \ -0). Keeping thisin mind, normal and tangential velocities may be obtained afterdifferentiation of equation (11),



03)

(14)

k = \ /=!

Here cosG and sinG are direction cosines of the unit-vector tangent to the

contour of the body, and:

g\

82=-

§3 =

X XL,

X XL,

/2 ~ ~

/3 =

/, = -(z - 2)!r ' cos[(/ -

i -4 5i — -t, ,

i = 4, 5,...

(15)

In the numerical procedure used to determine the NxMcoefficients A^ in eq (11), the boundary conditions which are to beimposed on the stream function and normal velocity will be fulfilled onlyapproximately, at a finite number of collocation points along the bodyprofile. In this way we are led to an over-determined system of equations



which must be solved by the least-squares method. Collocation of thestream function and Neumann conditions for (P-l) control points

Tp(Xp,yp) defined along the boundary of the airfoil, yell the followingsystem of equations:

TV

k = \ 7 = 1

AT M(16)

Moreover, Kutta condition states that the backward stagnation pointcoincides with the trailing edge:

TV

k=\ 7=1

k=\ i=\(18)

k = \ 7 = 1

Two different collocation procedures, both in least squares sense,have been used. In the first one only the stream function condition isimposed along with Kutta condition; (P+2) linear equations are to besolved in this case. As for the second one, the collocation involvesstream function and its tangential derivative (normal velocity), and Kuttacondition is enforced as well. Equations (16), (17) and (18) constitute alinear system of (2P+1) equations to be solved for NxM A^j unknowns.Eq (14) is then used to calculate Cp from eq (8) and eq (9) delivers thelift coefficient.



5 Numerical results and discussion

The numerical results presented here have been obtained for a standard

lifting surface (NACA 0012). Fig. 3 gives the pressure coefficient alongthe wing section, calculated by the integral equation method. The valuesof Cp are compared with the ones obtained by Moran [6] for the sameprofile.

The lift coefficient is shown in Fig. 4 for different angles of

attack a. The computational results are here compared to the results ofAbbott and Doenhoff [7] for a Reynolds number equal to 6x10 , and toour own experiments at Re =2.5x10 . We have used for that purpose a

low-speed wind tunnel Flint TE44 (0.45x0.45x0.572m), in which the

velocity of air can be continuously varied till 30 m/sec. The model ismounted on an electronic three-component balance TE81/E, equipped

with piezo-resistive load cells. A preliminary analysis of the uncertainty

of the measured forces points out to a precision of 1% during thecalibration procedure. The data acquisition is controlled by an A/D LabPC card from National Instruments working in Lab VIEW environment.The signals are afterwards processed in order to evaluate the lift and dragcoefficients, as well as the aerodynamic moment. The speed of theapproaching stream is controlled at the main section of the tunnel by aPrandtl probe.

5

-4

-3

Cp -2

-1

0

10 10.2 0.4 0.6 0.8

x/c

Fig. 3 Pressure coefficient along the upper and lower surfaces of theNACA 0012 airfoil.

For an angle of attack till about 5° we have obtained a goodcorrelation between the numerical predictions and the experimental

results. For oc=10° the lift coefficient as predicted by the integralequation model, differs from the experimental value by about 25%.However, this fact is due to the separation of the flow and the



development of a wake behind the wing. Results comparable to the onesprovided above are attained by using the concept of boundary collocation

with adaptation - Golik & Kolodziej [8].

Fig. 4 Lift coefficient for the NACA 0012 wing section as a function ofthe angle of attack.

The previous results were calculated for an average of N=200panels distributed over the body surface. Fig. 5 shows the evolution ofthe values obtained for CL as we refine the discretization of the

boundary. For the NACA 0012, a=8.3° is a particular angle of attack for

which CL=!. By applying Joukowski theorem we have obtained a liftcoefficient 2% close to this value (0 =0.98), using an average of N=100panels. From direct integration of the pressure around the wing a morerough estimate of this coefficient is obtained (C[j=0.93), and this iscertainly due to the numerical oscillations on the tangential velocity nearthe trailing edge. The effect of these unwanted oscillations on pressure atthe vicinity of this point is visible in Fig. 3. Under these theoreticalconditions the wing experiences a small drag Co=0.15, that can be seenas a residual error in fulfilling Kutta condition at the trailing edge of thewing section. This numerical difficulty may be overcome by reasoning

on the field induced inside the body - Katz & Plotkin [4] - or byconsidering higher-order singularity distributions.

In general we may conclude that the Integral Equation methodreproduces correctly the flow properties. If vortices are used, aformulation based on the normal velocity leads inevitably to an integralequation of the first kind. Consequently, the matrix of the influencecoefficients does not exhibit a dominant diagonal; the number of panelsthat we need to input in order to get an accurate solution is in fact high, ifa direct elimination solver with full pivoting is used - Press et al [9]. The



improvement gained by using a singular value decomposition had nosignificance in this case.

Condition number based on the Euclidean norm is situated in ourcase between 10* and 10^, for N=10 to 200 boundary elements.Therefore, double precision is required in order to obtain increasedaccuracy of the numerical results. As it has been pointed out by Nathman[10] the evaluation of the relevant parameters using higher precision may

have some impact on the final results for the case of very thin wings.However, the logarithmic nature of the velocity coefficients in eq (7) willcertainly be an important limitation, with respect to the simulation ofwings with small thickness.

0 20 40 60 80 100 120 140N

Fig. 5 Convergence of the lift coefficient with number of panels.

As to the CPU time surveyed on calculations performed byintegral equation method on a PC double Pentium processor at 200 MHz,with 64 Mbyte RAM and symmetrical multi-processing, it is of the orderof msec for N up to 900 and between 1 and 2 sec for N up to 3000panels. After this value CPU time starts from 22 sec, as a result oflimitations in memory space.

Acknowledgements

The research work that is presented in this paper was carried outwithin the framework of PRAXIS grant 3/3.1/CTAE/1940/95. The workreflects a close collaboration between Universidade da Beira Interior(Portugal) and The Technical University of Poznan, in Poland. Thenumerical and experimental results have been obtained at the Laboratoryof Fluid Mechanics of Universidade da Beira Interior.



References

[1] Hess, J. L., Panel methods in computational fluid dynamics, Ann.

Rev. Fluid Mechanics, 22, pp. 255-274, 1990.[2] Kolodziej, J. A., Review of application of boundary collocationmethod in mechanics of continuous media, Solid Mechanics Archives,

12, pp. 187-231, 1987.[3] Anderson, J. D. Jr., Fundamentals of Aerodynamics, McGraw-Hill

International, New York, pp. 258-263, 1991.[4] Katz, J. & Plotkin, A., Low-Speed Aerodynamics: From Wing Theory

to Panel Methods, McGraw Hill, New York, 1991.[5] Collatz, L., The Numerical Treatment of Differential Equations,Springer-Verlag, Berlin, 1966.

[6] Moran, J., An Introduction to Theoretical and ComputationalAerodynamics, John Wiley & Sons, New York, pp 241, 1984.[7] Abbott, I. & Doenhoff, A., Theory of Wing Sections, Dover

Publications, pp 462-463, New York, 1959.[8] Golik W. L. & Kolodziej, J. A., An Adaptive Boundary CollocationMethod for Linear PDEs, Numerical Methods for Partial DifferentialEquations, 11, pp 555-560, 1995.[9] Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T.,Numerical Recipes, Cambridge University Press, Cambridge, pp 26-29,1992.[10] Nathman, J., Precision Requirements for Potential Based Panel

Methods, AIAA Journal, Vol. 32, 5, pp. 1089-1090, 1993.


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