On trapeze wing aerodynamics calculations based on improved vortex

lattice method

JACOB NAGLER

NIRC,

Haifa, Givat Downes, ISRAEL

Abstract: This paper presents, aerodynamics coefficients calculation (Lifting & drag coefficients, pressure central

location) of Trapeze wing shape configurations for different aspect ratios (ARs) values by using improved vortex

lattice method (VLM), compared with finite-wing and slender body theories. The planar wing was divided into

panels of the size: 6X6 with trapezoid shape panels. As expected, for high ARs the VLM solution for the lifting

coefficient is coincided with the finite wing theory whereas for small ARs (<1) it is coincided with the slender

body theory (~1). Afterwards, we obtained that the calculated VLM induced drag becomes closer to the finite-

wing theory as the AR value is increased.

Key-words: VLM; finite wing; slender body; leading edge suction force;

Received: June 14, 2021. Revised: January 15, 2022. Accepted: March 4, 2022. Published: April 13, 2022.

1 Introduction

In this current study, VLM (Vortex lattice

method) theory is applied on Trapeze wing

shape configurations for different aspect

ratios (ARs) values by using improved

vortex lattice method (VLM), compared

with finite-wing and slender body theories.

This study continues the previous author

study and VLM model [1] by improving it,

using the leading edge suction analogy that

considers the suction force and explains

analytically the vortex-lift theory. In the

past the leading edge suction analogy was

proposed by Polhamus [2, 3], applied on

Delta wings, and later extended by Traub

[4]. Advanced numerical analysis of

aerodynamics properties due to wing

geometry shape and different geometry

wing angles that includes different

geometry manipulations in order to achieve

the optimal aerodynamic flight have been

studied over the two decades. For instance,

studies concerning swept and semi-slender

wings for blunt leading edge shape [5], flap

and aileron deflection [6], flapping wings in

a hover [7], various leading edge shape [8,

9], morphing wings [10] and recently non-

slender delta wings [11] have been studied.

Finally, classic experimental and theoretical

work was performed over the years by [12-

16].

2 Improved VLM analytic model

The improved model [30] includes the

calculated suction coefficient (based on Fig.

2) as:

 

Cqs

  

  



(1)

while the wing perpendicular velocity

generates an axial force which is the leading

edge suction force. Note that according to

Katz & Plotkin [17], the calculation is only

considering the cells trailing vortices

without their bound vortex. Additionally,

according to Moran [18] for specific cell

along with the symmetric cell in the other

side – the calculation only considers the

trailing vortices, similarly to Katz & Plotkin

[17]. However, according to Moran [18],

the other cells should be considering all the

other vortices. Similar way, appears in

Margason & Lamar [19]. Since we concern

here a trapezoidal wing geometry shape;

then the attack and flow leading edges are

located on straight lines, therefore the

adjacent vortices of all the cells in the whole

framework are located along the projection

wingspan. In other words, an adjacent

vortex will contribute infinity induced

velocity on each cell located further along

the projection wingspan such as singularity

is obtained using methods [18, 19].

Although method [17] might be used (no-

DESIGN, CONSTRUCTION, MAINTENANCE

DOI: 10.37394/232022.2022.2.14

Jacob Nagler

E-ISSN: 2732-9984

Volume 2, 2022

singularity), however, inaccuracy might be

generated due to the negligence the adjacent

vortices. Hence, we will use a different

method, based on induced velocity

calculation on each cell, only the adjacent

vortices will be calculated without the

vortices located on the same straight line

such as accurate improved calculation will

be resulted for [17].

Next, measuring length chord and

calculating wing trapezoid surface area,

averaged aerodynamic chord together with

rearward sweep angle, we have obtained the

following geometrical parameters:

 

, , tan

2 / 2

root tip

root tip w

AR b

b c c S AR b





     



(2)

while

root tip

cc

and b are the root chord,

tip chord, sweep angle calculated and

spanwise length geometry parameters (see

Fig. 4 in [1]). Additionally, the total

rearward sweep angle is dependent on

coordinate only. As a result, the angle

calculation was based on simple triangle

calculation. From here, Finite wing theory

will be brought about by the current context.

Note that calculation was performed

according to Fig. 1 around the theoretical

point – wing apex (0, 0).

Fig. 1 Extension wing with vortex and

X Coupled vortices

O Collocation Points

Half infinity vortex from the left side

Half infinity vortex from the right side

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Jacob Nagler

E-ISSN: 2732-9984

Volume 2, 2022

collocation points partition

Fig. 2 Description of cross flow and

induced velocity including suction force.

3 Finite wing and slender body theories

In the first step, finite-wing theory

assumptions which are based on lifting line

theory are presented in [1]. However, lift-

curve slope in case of swept wing is given

by Kuchemann [20]:

Finite-Wing

2 cos

2cos











(3)

According to the slender-body theory [21]:

Slender-Body 2

LAR







(4)

We will also define the errors, such as:

finite slender

L L L L

C C C C

   









(5)

which present the lifting coefficient errors

for the finite wing (



) and slender body (



) theories, respectively.

The center of pressure of the finite wing

and slender body theories will be calculated

by:

 

finite slender

;

Cp Cp root tip

X X c c  

(6)

Here, the difference value

root tip

cc

for all

wing configurations will be equal to 0.75.

Thus, the center of pressure will be

calculated in relative to the average

aerodynamic chord (

1Cp

) and the wing

root chord (

2Cp

The induced drag coefficient will be

calculated for the finite wing theory by the

following form:

 

finite

Finite-Wing

Elliptic distribution

Di L i

CC AR







  

(7)

In similar way, the slender body induced

drag will be:

 

slender

Elliptic distribution

Di L

CC AR









  

(8)

while the induced drag errors for both

theories will be defined as:

finite slender

D Di D Di

Di Di

C C C C







(9)

Next section, final results and comparison

to finite-wing and slender body theories

will be presented and discussed.

4 Results & Discussion

Comparisons between VLM specific

aerodynamics parameters for different aspect ratio

values, Finite-Wing and Slender body theories are

presented in Table 1 alongside Figs. 3-5.

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Jacob Nagler

E-ISSN: 2732-9984

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Fig. 3 Lifting coefficients comparison for different AR values: VLM calculation vs. Finite wing and

Slender Body theories.

Fig. 4 Center of pressure location comparison for different AR values: VLM calculation vs. Finite

wing and Slender Body theories.

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Jacob Nagler

E-ISSN: 2732-9984

Volume 2, 2022

Fig. 5 Induced drag comparison for different AR values: VLM calculation vs. Finite wing and Slender

Body theories.

Table 1. Aerodynamics parameter comparisons

between VLM, Finite-Wing and Slender body

theories.

Examining Table 1 alongside with Figs.

3-5 leads to the following comprehensions

about VLM obtained results compared to

the finite-wing and slender body theories.

One might observe that for AR ~1, the

VLM lifting coefficient calculated value

becomes closer to the slender body theory,

whereas for exceeding ARs value (>1)

becomes closer to finite wing theory (Fig.

3). In addition, for relatively high ARs, the

VLM center of pressure location value

becomes closer to the Finite-Wing theory,

whereas the comparison with the slender

body theory is unclear (Fig. 4). Finally, it

was revealed that for relatively high ARs,

the VLM induced drag calculated value

becomes closer to the Finite-Wing theory

parameter value, whereas the error between

the slender body theory and VLM is

relatively increasingly high (Fig. 5).

5 Conclusion

In this study we have calculated

numerically (improved VLM model) the

aerodynamics properties (Lifting & drag

coefficients, pressure central location) of

Trapeze wing shape configurations for

different aspect ratio values, compared with

finite-wing and slender body theories. The

wing was divided into

planar trapezoid

shape panels of the size: 6X6. As expected,

for high ARs the VLM solution for the

lifting coefficient was coincided with the

finite wing theory whereas for small ARs

(<1) it was coincided with the slender body

DESIGN, CONSTRUCTION, MAINTENANCE

DOI: 10.37394/232022.2022.2.14

Jacob Nagler

E-ISSN: 2732-9984

Volume 2, 2022

theory (~1). Afterwards, we obtained that

the calculated VLM induced drag had

become closer to the finite-wing theory as

the AR value was increasing.

Acknowledgements

This study was performed using Dr. Jose

Meyer Aerodynamics of wings and bodies

course notes from Technion – Israel

Institute of Technology and RAFAEL

Company. Also, I would like to thank my

colleague Mr. Nitai Stein from RAFAEL

Company for his assistant the numerical

and analytic method presented in this study.

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[18] Moran J., (1984) An Introduction to

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All data, models, and code generated or used during

the study appear in the submitted article.

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Nagler) states that there is no conflict

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DESIGN, CONSTRUCTION, MAINTENANCE

DOI: 10.37394/232022.2022.2.14

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