![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](https://image.slidesharecdn.com/presentationmefte09-201028172913/85/Leading-Edge-Vortex-Flow-Modelling-Around-Delta-Wings-Using-a-Boundary-Element-Method-2-320.jpg)
![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](https://image.slidesharecdn.com/presentationmefte09-201028172913/85/Leading-Edge-Vortex-Flow-Modelling-Around-Delta-Wings-Using-a-Boundary-Element-Method-14-320.jpg)
Leading-Edge Vortex Flow Modelling Around Delta Wings Using a Boundary Element Method
- 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