Mesh less Euler Computation for Cropped Delta Wing ...

11th Annual CFD Symposium 11th – 12th August 2009, Bangalore

Mesh less Euler Computation for Cropped Delta Wing with Aileron Deflection Manish K. Singh*, V. Ramesh* and S. M. Deshpande+ * CTFD Division, National Aerospace Laboratories, CSIR, Bangalore + Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore The determination of the aerodynamic characteristics of a wing with the control surface deflection is one of the most important and challenging task in aircraft design and development. Many military aircraft uses some form of the delta wing. Euler computation has been done for simulation of transonic flow over a cropped delta wing with the deflected aileron, using in-house developed mesh less solver. Results are presented for the flow past the wing with the aileron deflection d=6°, and a free-stream Mach number M∞=0.9, at angle of attack a=1° and 3°. The computed results are compared with available experimental data and Euler computation of finite volume solvers (NAL and Beijing University of Aeronautics and Astronautics, China).

Key words: Mesh less solver, cropped delta wing, aileron deflection, control surface, aerodynamics

Nomenclature C CM CP X/C α

= chord = Quarter chord pitching moment coefficient = Pressure Coefficient = fraction of chord length = angle of attack

I.

Introduction

The delta wing is planform in the form of a triangle. The main advantage of the delta wing is that its leading edge remains behind the shock generated by the aircraft nose while flying at the supersonic speed. Although same advantage can be obtained from the highly swept wing but delta wing are generally structurally stronger and having much more internal volume for fuel and other storage because delta’s platform carries across the entire aircraft. Another advantage is that at high angle of attack leading edge generates high velocity vortex on the upper surface causing lift augmentation. Most of the modern fighter aircraft uses some form of the delta wing which includes Euro-fighter Typhoon, JAS 39 Gripen, F-16, Dassault Mirage 2000, Tejas, and Sukhoi-11 etc. Pure delta wings have some unwanted characteristics like flow separation at high angle of attack and high drag at low altitude. F-16 uses cropped delta wing with horizontal tail surfaces. Nowadays high lift systems are used extensively to improve the take-off and landing performance. The determination of aerodynamic characteristics at high Mach number with deflected control surface is one of the most important and also challenging tasks. Any aircraft design during the preliminary and optimization phase requires a lot of studies involving elemental changes in the configuration. Use of the CFD codes to predict aerodynamic behaviour for such different changes in the configuration would require regenerating the grid every time. In case of

1

multi-block structured grid, multi-blocking strategy itself would require changes. Further with every new multiblock grid one will need to regenerate the inter-block connectivity data also. This inevitably requires huge efforts towards repeated grid generation process. Use of the unstructured grids does reduce the turn around time, but however they also require new grids to be generated for every delta change in configuration. Hence there is a great interest in developing the computational tool for faster computation. Mesh-less solvers are such tool which operates on the cloud of points. For a complex configuration, with such a solver, one can generate a separate cloud of points around each component, which adequately resolves the geometric features, and then combine all the individual clouds to get one set of points on which the solver directly operates. An obvious advantage of this approach is that any incremental changes in geometry will require only regeneration of the small cloud of points where changes have occurred. This feature eliminates the necessity of regeneration of the complete grid for incremental changes. This quality of Mesh-less solvers makes it very attractive for problems involving moving boundaries. The objective of the present study is to perform Euler computation of cropped delta wing using in-house developed mesh-less solver.

II. Mesh-less Solver Mesh-less solver uses kinetic flux vector splitting (KFVS) method to update the solution. The KFVS method involves two levels; the Boltzmann level and the Euler level (at which state update operates). The update at Euler level is then obtained by taking suitable moments of the update equation at Boltzmann level. Spatial derivative of flux vector is obtained by least squares kinetic upwind method. A brief methodology is presented here. More details can be obtained from references [1-15]. The Boltzmann equation for a Maxwellian velocity distribution function F is given by

¶F ¶F ¶F ¶F + v1 + v2 + v3 =0 ¶t ¶x ¶y ¶Z

(1)

The Maxwellian distribution function, F, is given by 3

r æ b ö 2 [ - b (v1 - u1 ) F = ç ÷ .e I0 è p ø

2

- b ( v 2 - u 2 ) 2 - b ( v3 - u 3 ) 2 -

I ] I0

(2)

Where r is density, b=1/(2RT), and I0 is the internal energy due to non-translational degrees of freedom. I0 = (53g)RT/2(g-1) and u1, u2 and u3 are fluid velocity Cartesian components, v1, v2 and v3 are Cartesian components of molecular velocity. R is the gas constant and T is absolute temperature of the fluid. We define 2

v + v22 + v32 T Moment Vector, y º [1, v1 , v2 , v3 , I + 1 ] 2 y Moment,

¥

¥

¥

¥

0

-¥

-¥

-¥

< y , F >º ò dI ò dv1 ò dv2 ò dv3yF

(3) (4)

The Euler equation can be written as

áy ,

¶F ¶F ¶F ¶F ¶U ¶ ¶ ¶ + v1 + v2 + v3 ñ= + (GX ) + (GY ) + (GZ ) = 0 ¶t ¶x ¶y ¶z ¶t ¶x ¶y ¶z

(5)

Where U represents the conserved vector, GX is x-component of the flux vector, GY is y-component of the flux vector and GZ is z-component of the flux vector. The expressions for these are given bellow.

2

U = [r , ru1 , ru2 , ru3 , re]

T

[ ] GY = [ru , ru u , p + ru , ru u , u ( p + re )] GZ = [ru , ru u , ru u , p + ru , u ( p + re )] RT 1 e= + (u + u + u ) g -1 2 GX = ru1 , p + ru12 , ru1u2 , ru1u3 , u1 ( p + re ) 2

1 2

3

1 3

T

2 2

2 3

2

2 3

3

2 3

2 1

2 2

T

(6)

T

2 3

p = rRT Using MCIR splitting for all three components of the molecular velocity, the Boltzmann equation can be written as

¶F v1 + v1 f1 ¶F v1 - v1 f1 ¶F v2 + v2 f2 ¶F v2 - v2 f2 ¶F + + + + + ¶t ¶x ¶x ¶y ¶y 2 2 2 2

(7)

v3 + v3 f3 ¶F v3 - v3 f3 ¶F + =0 2 2 ¶Z ¶Z Where

f1 ,f2 and f3 represents

the dissipation control parameters corresponding to three components of the

molecular velocity. The y moment of the above equation will lead to the Modified Kinetic Flux split Euler equation

¶U ¶ ¶ ¶ ¶ ¶ ¶ + GX M+ + GX M- + GYM+ + GYM- + GZ M+ + GZ M- = 0 ¶t ¶x ¶x ¶y ¶y ¶z ¶z

(

±

)

(

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)

(

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(

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(

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(

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(8)

±

Where GX M , GYM and GZ M are modified KFVS split fluxes. These can be represented in terms of the usual KFVS fluxes as

{

}

{

}

{

}

1 GX ± f1 (GX + - GX - ) 2 1 GYM± = GY ± f2 (GY + - GY - ) 2 1 GZ M± = GZ ± f3 (GZ + - GZ - ) 2

GX M± =

(9)

GX ± , GY ± and GZ ± are x, y and z component of split flux. Space derivatives, in LSKUM, are evaluated using least squares approximation. Consider any point O shown in figure 1. Assume F to be function whose values are available at O and its immediate surrounding nodes, referred to as connectivity of point O. The first order approximations to the space derivatives, FxO , FyO and FzO using least squares approach [16] is given by the following formula

FxO D1

=

FyO D2

=

FzO D3

=

1 D

(10)

3

Where

å Dx Dy D = å Dy å Dy Dz å Dx D = - å Dx Dy å Dx Dz å Dx D = å Dx Dy å Dz Dx å Dx D = å Dx Dy å Dz Dx i

i

2 i

1

i

i

2 i

2

i

i

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i

2 i

3

i

i

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i

2 i

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Where

å

i

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å Dx Dz å Dy Dz å Dz å Dx Dz å Dy Dz å Dz å Dx Dy å Dy å Dy Dz å Dx Dy å Dy å Dy Dz i

i

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2 i

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å Dx DF å Dy DF å Dz DF å Dx DF å Dy DF å Dz DF å Dx DF å Dy DF å Dz DF å Dx Dz å Dy Dz å Dz i

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(11)

i

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i

2 i

represent the summation over all points in the connectivity N (Po) of point O. Discretising the time

derivative to the first order and using least squares formula to approximate the derivatives of the various spilt fluxes, an update scheme for 3D Modified KFVS Euler equations can be written as

n n éæ ¶GX + ö n ù æ ¶GX M- ö æ ¶GYM+ ö M ÷÷ ÷÷ ÷÷ êçç + çç + çç +ú êè ¶x ø N1 ( Po ) è ¶x ø N 2 ( Po ) è ¶y ø N3 ( Po ) ú U n+1 = U n - Dt.LS .ê ú n n n êæ ¶GYM- ö ú æ ¶GZ M+ ö æ ¶GZ M- ö ÷÷ ÷÷ ÷÷ + çç + çç êçç ú ¶y ø N ( P ) è ¶z ø N ( P ) è ¶z ø N ( P ) ú 4 o 5 o 6 o ëêè û

(12)

Following the stability analysis of Ghosh[16], the time step Dt used in state update is given by

ìï üï Dsi Dt = min í ý ïî q + 3. RT ïþ

(13)

Where Dsi is the distance between any point i and O as shown in figure 1, q is the magnitude of velocity at point O. In the above update formula, we choose a suitable subset of points in the connectivity (referred as sub-stencil) to ensure that the signal propagation property is not violated. The subscripts to the various flux derivative approximation in the equation indicates the sub-stencil chosen from the full connectivity set. The dissipation control parameter f1 , f2 and f3 are chosen as f1 = f2 = f3 = Dr , where 0