This document summarizes research on modeling leading-edge vortex flows around delta wings using a boundary element method. The study aims to implement a partial wake relaxation model to simulate vortex sheet separation at the wing's leading edge. Results show the boundary element method can realistically capture leading-edge vortex behavior by using a prescribed wake geometry based on previous research. Comparisons to prior work show good agreement in pressure distributions, vortex positions, and lift/drag coefficients at varying angles of attack. The method is deemed suitable for modeling separated vortex flows where the general flow pattern is known.

1. Leading-Edge Vortex Flow Modelling
Around Delta Wings
Using a Boundary Element Method
J. Baltazar
Marine Environment and Technology Center (MARETEC)
Department of Mechanical Engineering
Instituto Superior T´ecnico (IST)
Lisbon, Portugal
MARETEC
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 1 / 15

2. Motivations
U∞
α [º]
0 10 20 30 40
0.0
0.5
1.0
1.5
2.0
CL
with leading-edge vortex sheet separation
vorticity is shed from the trailing-edge only
Although viscosity is responsible for the formation of the free
shear layer, potential flow methods can be used to model
the vortex sheet separation.
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 2 / 15

3. Objectives
Application of a boundary element method for delta wings
with leading-edge vortex sheet separation.
Implementation of a partial wake relaxation model.
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 3 / 15

4. Mathematical Formulation
Potential Flow Problem
Undisturbed onset velocity:
U∞ = U∞ cos αi + U∞ sin αj
Velocity field: V = U∞ + φ
Laplace equation: 2
φ = 0
Boundary conditions:
∂φ
∂n = −n · U∞ on SB
V + · n = V − · n and
p+ = p− on SW
φ → 0 if |r| → ∞
Kutta condition: | φ| < ∞
x
y
z
U∞
α
SB
SW
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 4 / 15

5. Mathematical Formulation
Integral Equation
Fredholm integral equation for Morino formulation:
2πφ (p) =
SB
G ∂φ
∂nq
− φ (q) ∂G
∂nq
dS −
SW
∆φ (q) ∂G
∂nq
dS,
where p ∈ SB.
Green’s function: G(p, q) = −1/R(p, q).
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 5 / 15

6. Numerical Method
Potential Flow Model With Leading-Edge Vortex Sheet Separation
x
y
z
U∞
α
SB
STW
SFW
SLW
Far Wake
Trailing-edge Wake
Delta Wing
Leading-edge Wake
Vortex Filament
Feeding Sheet
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 6 / 15

7. Numerical Method
Vortex Filament/Feeding Sheet Vortex Core Model
µ µ
0
A
A
t t
t
Γ
Γ
∞t
Feeding Sheet
Vortex
Filament
Continuous Model
Vortex Filament/Feeding Sheet
Vortex Core Model
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 7 / 15

8. Numerical Method
Partial Wake Relaxation Model
Prescribed wake geometry defined based on the numerical results
of Hoeijmakers (1989).
ri
0
ri
1
θi
0
θi
1
(yi
VF,zi
VF)
z
y
SB
SLW
Feeding
Sheet
Vortex lines are allowed to move freely on the prescribed wake
surface in order to have zero-pressure-jump.
The solution of equation ∆p = −ρ(Vm · ∆V ) is obtained by
the method of Newton-Raphson, where ∆V = S (∆φ).
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 8 / 15

9. Panel Arrangement
Discretisation: 20×20 Wing, 15×30 LE Wake, 5×20 TE Wake
Y
Z
X
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 9 / 15

10. Results for the 76 deg. Swept Delta Wing
at 20 deg. Incidence
Pressure Side
0.150
0.250
0.200
0.275
0.325
0.300
0.350
Suction Side
0.00
-0.50
-1.00
-0.50-1.00-1.50
-2.00
Cp:
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 10 / 15

11. Results for the 76 deg. Swept Delta Wing
at 20 deg. Incidence
t
0.0 1.0 2.0 3.0
0.0
0.5
1.0
1.5
2.0
Leading-Edge
x/C0
=0.15
x/C0
=0.55
x/C0
=0.35
x/C0=0.75
x/C0
=0.95
x/C0=1.15
x/C0=1.35
∆φ/(U∞
S)
y/S
0.0 0.2 0.4 0.6 0.8 1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
x/C0=0.15
x/C0=0.35
x/C0=0.55
x/C0=0.75
x/C0
=0.95
-Cp
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 11 / 15

12. Results for the 76 deg. Swept Delta Wing
at 20 deg. Incidence
y/S
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Present Method
Hoeijmakers (1989)
x/C0=0.45
z/S
x/C0
=0.95y/S
z/S
0.00 0.02 0.04 0.06
0.00
0.02
0.04
0.06
x/C0=0.05
y/S
0.0 0.2 0.4 0.6 0.8 1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Present Method
Hoeijmakers (1989)
x/C0
=0.95
-Cp
x/C0=0.45
x/C0=0.05
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 12 / 15

13. Results for the 76 deg. Swept Delta Wing
at Different Incidences
y/S
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Present Method
Hoeijmakers (1989)
α=10º
z/S
α=20º
α=30º
α=40º
y/S0.0 0.2 0.4 0.6 0.8 1.0
-0.5
0.0
0.5
1.0
1.5
-Cp
α=20º
y/S0.0 0.2 0.4 0.6 0.8 1.0
-0.5
0.0
0.5
1.0
1.5
2.0
-Cp
α=40º
y/S0.0 0.2 0.4 0.6 0.8 1.0
-0.5
0.0
0.5
1.0
1.5
α=10º
-Cp
y/S0.0 0.2 0.4 0.6 0.8 1.0
-0.5
0.0
0.5
1.0
1.5
2.0 Present Method
Hoeijmakers (1989)
-Cp
α=30º
x/C0 = 0.95
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 13 / 15

14. Results for the 76 deg. Swept Delta Wing
at Different Incidences
α [º]
0 10 20 30 40
0.0
0.5
1.0
1.5
2.0
Present Method
Hoeijmakers (1989)
CL
α [º]
0 10 20 30 40
0.0
0.5
1.0
1.5
Present Method
Hoeijmakers (1989)
CDi
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 14 / 15

15. Conclusions
Potential flow computations about delta wings using a BEM
with a partial wake relaxation model are feasible.
By providing an appropriate shape of the vortex sheet geometry
the method is able to realistic capture the behaviour of the
leading-edge vortex flow.
The calculations are in good agreement with the results of
Hoeijmakers (1989).
It is believed that the method may be used for the modelling
of separated vortex flows in the cases where the flow pattern
is known in advance.
MEFTE 2009 Bragan¸ca, Portugal 17-18 September 15 / 15