Explicit Kutta Condition Correction for Rotary Wing Flows A. D'alascio, A

Explicit Kutta Condition Correction for Rotary Wing Flows

A. D'Alascio, A. Visingardi, and P. Renzoni CIRA, Centra Italiano Ricerche Aerospaziali

EMail: [email protected], [email protected],

p.renzoni@cira. it

Abstract

An explicit correction to the Kutta condition applied to rotary wing potential flows is described. Indeed, the simple application of the classical steady Kutta condition, which ensures the uniqueness of the solution of the potential flow problems [1,2], is unable to guarantee a zero pressure jump at the blade trailing edge. Several investigations performed on this problem have indicated the presence of considerable non-zero pressure jumps for highly 3D flows (blade root and tip) [3] and for unsteady flows characterized by high reduced frequencies [4]. The application of a Kutta condition correction in CIRA's BEM code RAMSYS [3] has shown the ability to ensure a continuous pressure across the blade trailing edge.

1 Introduction

The flow field around helicopter rotors is unsteady and highly three-dimensional. A low frequency unsteadiness is induced in the flow by the relative

motion between rotating and non-rotating components of the helicopter. The interaction between each main rotor blade with the wake of preceding blades is responsible for mid frequency unsteadiness whereas high frequency

unsteadiness is generated by blade/tip vortex encounters.

The rotor blades during their motion are subject to cross flows which make the flow field three-dimensional, expecially in the vicinity of the ro-

780 Boundary Elements

tor blade root and tip. In addition, considering the role of the fuselage, its displacement effect distorts the onset flow further increasing the three- dimensionality of the flow field in the rotor disk. The rotor wake is deformed due to the presence of the fuselage which in turn alters the interaction pro- cess between rotor blades and their wake.

In the numerical simulation of such complex flow fields the simple application of the classical steady Kutta condition has turned out to be unable to guarantee a zero pressure jump at the blade trailing edge. This pressure jump has shown to influence the whole pressure distribution on the blade, thus considerably altering the prediction of the aerodynamic loads.

An explicit unsteady Kutta correction has been introduced in the CIRA computer code RAMSYS [3], based on Morino's boundary integral formulation, for the unsteady, incompressible aerodynamics of multi-body configurations in arbitrary rigid motion. The aim of this paper is to il- lustrate an algorithm of an unsteady Kutta correction by providing the necessary details about the structure of the algorithm itself, and to present some results relative to the application of this correction to the aerodynamic analysis of an oscillating airfoil and a helicopter rotor.

2 BEM formulation

The formulation consists in the solution of Laplace's equation written in terms of the velocity potential

V^ = 0 Vx^^(f) (1)

such that v = V<£. S(t) is a surface outside of which the flow is potential and consists of a surface SB surrounding the body geometry and a surface Sw surrounding the wake geometry.

The boundary condition at infinity is that — 0. The surface of the body is assumed to be impermeable hence |^ = v# • n where MB is the velocity of a point on the body. The wake is a surface of discontinuity which is not penetrated by the fluid and across which there is no pressure jump. The second wake condition implies that A0 remains constant following a wake point x*y, and equal to the value it had when XVK left the trailing edge.

The value of A at the trailing edge is obtained by using the Kutta- Joukowski hypothesis that no vortex filament exists at the trailing edge; this implies that, at the trailing edge, the value of A on the wake and the value of A on the body are equal.

The application of Green's function method to Eq.(l), yields the following boundary-integral-representation for the velocity potential cf>

E(x*)

with

and

representing respectively the contribution of the body and the wake. E(x*) is a domain function defined as zero inside S and unity elsewhere.

The helicopter geometry and the wake are respectively discretised by M and N hyperboloidal quadrilateral panels on which the unknown velocity potential, the normal wash and the velocity potential jump are constant

(zeroth-order formulation). Using the collocation method and setting the collocation points at the centroids of each element on the body geometry, the integral equation, Eq.(2), is replaced by an algebraic linear system of equations for the velocity potential :

M M N Ekk(t) = £ BfenVm(t) + £ Ckm4>m(t] + ]T F^A^(<) (3) 771=1 m = l n=l

where 0, C, and F are respectively the body source, body doublet, and

wake doublet influence coefficients.

The system of linear Eqs.(3) can be written in matrix form as:

A • = B - <0 + F - A0 (4)

being Akm = EkSkm — Ckm with 8km representing the Kronecker function.

A Conjugate Gradient Method (GMRES solver) is applied for the numerical solution of the problem.

Once obtained the velocity potential <^(Z), the pressure coefficient Cpk(t) is computed, for each body panel k G [1,M], by applying the unsteady Bernoulli equation:

= _-- + V* • V^ + v& • Vfa (5) *>e/ \ Ut * )

where V^e/ is the reference velocity and V& is the entrainment velocity of the panel k.

3 The unsteady Kutta condition problem

The Kutta-Joukowski hypothesis that no vortex filament exists at the trailing edge is no longer sufficient for unsteady and/or highly three-dimensional fluidflows . Indeed the numerical validation of the RAMSYS code has shown that pressure jumps might occur at the blade trailing edge expecially for cases of helicopter rotors in forward flight and for highly unsteady flows on oscillating wings.

Taking the idea of Kinnas and Hsin [5] an iterative procedure has been developed with which for each trailing-edge wake panel the pressure jump, computed by applying Bernoulli's equation for unsteady flows Eq.(6), has been explicitly set equal to zero, in order to impose a continuous pressure across the trailing edge. By denoting with A^_/ the difference between the values computed on the upper and lower body panels at the T.E. it is obtained:

r^ 1 i = 0 (6)

Indicating by NTE the number of trailing edge wake panels, NTE linearised equations are coupled, through the velocity potentials at the blade trailing edges, to the M Eqs.(3) of the algebraic system. A linearised expression of the Eq. (6) is:

. • V(

Figure 1: Discretisation stencil

The time derivatives are evaluated with second-order accuracy. The right hand side of Eq.(7) is discretised, consistently with the expression of the pressure coefficient of Eq.(5), using a spline function chordwise and a

central second-order discretisation function spanwise. The left hand side can be discretised using several schemes without changing the result of the iterative procedure, being the Bernoulli's equation written in terms of vari- ations of the velocity potential jump at the T.E.. Therefore the stencil influences the rate of convergence of the algorithm.

In the RAMSYS code the term (V-f V<^'),^ is computed as the average of correspondent values on the upper and lower T.E. body panels, Fig. 1. A good rate of convergence has been obtained by imposing:

(v + v^ = \ (v,,_ + v<. + v,,, + v<)

where both V<^ and V^ are computed using the three neighbouring velocity potentials respectively in the panel centroids i^ and z&/, as shown in Fig. 1. Eq.(7) in the discretised form is written as:

being cj = c^ w representing the over-relaxation factor set equal to 1.5. In matrix form:

T' - 6 A^ = q^ (8)

By coupling Eqs.(4) and (8), a new set of equations is obtained which is solved for a certain number of iterations. Convergence is reached when the maximum value of the trailing edge pressure coefficient jump is of the order of 10~^. In the RAMSYS code a number of iterations not greater than

5 are necessary to reach convergence.

4 Numerical validation

The test cases performed are relative to an oscillating NACA 0012 airfoil and an isolated rotor in forward flight. In the first test case the pressure jump at the T.E. of the airfoil is due to the unsteadiness of the motion, while in the second case it is essentially due to the three-dimensional flow generated on the rotor blades. A comparison of the pressure coefficient distributions and the lift forces acting on both the airfoil and the rotor are computed by RAMSYS with and without the Kutta condition correction.

4.1 Oscillating airfoil

This test case deals with a NACA 0012 airfoil, discretised by 60 panels, oscillating about its quarter chord point at a reduced frequency k = ^j = 0.2

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and with a pitching amplitude c*o = 10°. The nondimensional time step has been set equal to A^ = ^^ = ?r/8.

In Fig. 2 the comparison of chordwise pressure distributions with and without the Kutta correction are presented for the nondimensional times of tn — 39.7 and tn = 48.3. In Fig. 3 the comparison of the lift coefficient time history with and without the Kutta correction is shown. A slight reduction in the C\ amplitude together with a slight phase shift can be observed by applying the Kutta correction.

For this 2D test case the convergence is reached in two or three iterations, reducing the pressure coefficient jump by 3 orders of magnitude.

4.2 Helicopter rotor in forward flight

This test case, HELINOISE Dpt 1637 [6], deals with a four-bladed isolated helicopter rotor at moderate-speed levelfligh t( p = 0.273), at CT = 0.00452 and MH = 0.644. Each blade, of rectangular planform and NACA 23012 airfoil sections, has been discretised by 60 panels in the chordwise direction and 10 panels in the spanwise direction. Four rotor revolutions have been performed, the first 2.5 with an azimuth step of 30° and the last 1.5 with an azimuth step of 15°. The Kutta correction has been applied after 1.5 rotor revolutions. Two spirals have been employed to model the wake, Fig. 4.

Fig. 5 illustrates a comparison of the thrust force time history without and with the Kutta condition correction. A considerable decrease in the thrust force can be observed by activating the correction thus highlight- ing the strong influence that this correction exerts on the evaluation of the aerodynamic loads.

Fig. 6 shows the chordwise pressure distributions at two radial stations r/R = 0.40 and r/R — 0.94 for three different azimuth angles. The azimuth station at ^ = 0° is characterized by the free stream velocity vector being aligned with the blade spanwise direction and pointing outward. In this case the pressure jumps at the trailing edge are caused by the strong cross flows acting on the blade, giving rise to gaps in the root region of the blade and crossings in the tip region. At the azimuth station # = 90° the free stream velocity vector is normal to the blade thus not giving rise to cross flows. The pressure distribution presents a zero pressure jump even without applying the Kutta correction. Finally, an interesting result can be observed by considering the azimuth station at # = 180°. As for the station at ^ — 0° the free stream velocity vector is aligned with the blade spanwise direction but pointing inward. In this case the presence of cross flows on the blade gives rise to gaps in the tip region and to crossings in the root region. In

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all the cases the Kutta correction algorithm is able to guarantee continuous pressure at the trailing edge.

5 Conclusions and further developments

It has been shown that the Kutta correction method applied in the RAM- SYS code modifies a non-zero pressure jump at the trailing-edge and that this correction influences the solution along the whole blade, thus cosider- ably modifying the global aerodynamic loads.

The algorithm in both cases reaches convergence in no more than five iterations. The use of GMRES solver is fundamental to contain the compu- tational time per iteration.

A development of the method will deal with the description of the velocity potential jump on the T.E. wake panels with a linear distribution. This feature, as highlighted by Kinnas and Hsin [5], is important to have a solution independent of the time step for a larger range of time steps.

References

1. Katz, J. and Plotkin, A. "LOW-SPEED AERODYNAMICS - From wing theory to panel method," McGraw-Hill, International Editors, 1991.

2. Morino, L. and Tseng, K. "A general Theory of Unsteady Compressible Potential Flows with Applications to Airplanes and Rotors," Eds.: P.K.

Banerjee and L. Morino, Developments in Boundary Element Theory, Vol.6: Boundary Element Methods in Nonlinear Fluid Dynamics, 1990.

3. Visingardi, A., D'Alascio, A., Pagano, A. and Renzoni, P. "Validation of CIRA's Rotorcraft Aerodynamic Modelling SYStem with DNW experimental data," 22nd European Rotorcraft Forum, 17-19 September, Brighton, UK, 1996.

4. Bose, N., "Explicit Kutta Condition for an Unsteady Two-Dimensional Constant Potential Panel Method," AIAA Journal, Vol. 32, No. 5,

1994, pp. 1078-1080. 5. Kinnas, S.A. and Hsin, C.-Y., "Boundary Element Method for the

Analysis of the Unsteady Flow Around Extreme Propeller Geometries," .4MA JowrW, Vol. 30, No. 3, 1992, pp. 688-696. 6. Splettstosser, W.R., Niesl, G., Cenedese, F., Nitti, F., and Papanikas,

D.G., "Experimental Results of the European HELINOISE Aeroacous- tic Rotor Test in the DNW," Proceedings of the Nineteenth European Rotorcraft Forum, Cernobbio, Italy, Sept. 1993.

Nondimensional time =39.7

WITHOUT KUTTA CORRECTION WITH KUTTA CORRECTION

0.4 0.6 x/c Nondimensional time =48.3 WITHOUT KUTTA CORRECTION WITH KUTTA CORRECTION

0.2 0.4 0.6 0.8 x/c Figure 2: Chordwise pressure distribution at two nondimensional times

(NACA 0012: k = 0.2, Af* = Tr/8, ao = 10°)

2. 5 WITHOUT KUTTA CORRECTION 2 WITH KUTTA CORRECTION •

1.5 1

0.5 0 -0.5

-1

10 20 30 Nondimensional time

Figure 3: Lift coefficient time history (NACA 0012: k = 0.2, A^ = Tr/8, ao = 10°)

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Figure 4: Development of the wake about an isolated rotor (DPT 1637: moderate-speed level flight, ^ = 0.273, C? = 0.00452, M* = 0.644)

0.008 WITHOUT KUTTA CORRECTION WITH KUTTA CORRECTION

0.007

0.006

0.004

0.003 360 720 1080 1440 Azimuth Angle

Figure 5: Thrust force time history (DPT 1637: moderate-speed level flight, ,4 = 0.273, CY = 0.00452, M* = 0.644)

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r/R = 0.40 r/R = 0.94 WITHOUT KUTTA CORRECTION WITHOUT KUTTA CORRECTION - WITH KUTTA CORRECTION - WITH KUTTA CORRECTION -

0 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c x/c

r/R = 0.40 r/R = 0.94 WITHOUT KUTTA CORRECTION WITHOUT KUTTA CORRECTION WITH KUTTA CORRECTION WITH KUTTA CORRECTION

0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c

r/R = 0.40 W=180 r/R = 0.94 -2 WITHOUT KUTTA CORRECTION WITHOUT KUTTA CORRECTION - WITH KUTTA CORRECTION - WITH KUTTA CORRECTION - -1.5 -1

-0.5

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/c x/c

Figure 6: Chordwise pressure distribution at radial stations r/R=0.40 and r/R-0.94 for three azimuth angles (# = 0° - 90° - 180°) (Dpt 1637: moderate-speed level flight, ^ = 0.273, CT = 0.00452, MH = 0.644)