Computational Simulation of Supersonic Deltawings

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Computational Simulation of Supersonic Delta-wings

• Mentor Dr. Sritharan, Chairman of Department of
  Mathematics
• Presenter Jingling Guan
• Department of Mathematics

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Introduction

• What are Delta-wings?

• A delta-wing is a wing whose shape looks like a
  triangle when viewed from above. Delta-wings are
  very efficient in high-speed flight.

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Introduction

• Several Popular Delta-wings

Delta wings were used on several military
aircrafts that had a need for speed.

• YF 102

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Introduction

• Several Popular Delta-wings
• Concorde

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Introduction

• Several Popular Delta-wings

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Introduction

• What is Supersonic Speed?
• Supersonic Speed is the speed faster than the
  speed of the sound.

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Introduction

• What are Shock Waves?A shock wave is a very
  strong pressure wave in air, produced by
  supersonic crafts, that creates sudden, huge
  changes in pressure.
• Shock waves are very important in designing an
  aircraft, because the change of pressure will
  influence the performance of aircrafts.

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Introduction

• Design Requirements
• - Transonic leading edge ---cross flow
  shocks
• - Attached flow ---shock free or contains
  only weak shocks

Hence, it would be very helpful if we can find a
numerical method to study the behavior of the
cross-flow around the wing.
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Coordinate Systems
To find such a numerical method, first, the
intersect of the wing is modeled in the spherical
coordinate system.
Intersect of an elliptic wing
Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Coordinate Systems
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Coordinate Systems
A 3D plot of the grids in the spherical
coordinate system.
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Coordinate Systems

• Stereographic Projection

O
A
B
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Coordinate Systems
A plot of the grids on a plane after
stereographic projection
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Transformation of the Coordinate Systems

• Joukowskis transformation

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Coordinate Systems
By a simple shearing, the circular computational
domain will be transformed into a rectangular
region.
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Transformation of the Coordinate Systems

• The first Fundamental Form

Cartesian system
Any coordinate system
a,ß1, 2
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Transformation of the Coordinate Systems

• The first Fundamental Form (continued)Metric
  Tensor

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Governing PDE

• The Governing Partial Differential Equation

Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Governing PDE

• The density

Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Governing PDE

• Also, we have the velocity defined on the unit
  sphere

Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Governing PDE

• By substituting the previous equations, we get

Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Sonic Line

• Sonic Line

On the sonic line, the governing PDE changes type
between elliptic and hyperbolic.
Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Numerical Method

• The geometric quantities at the center of primary
  cells1. metric tensor in the spherical
  coordinate system

Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Numerical Method

• The geometric quantities at the center of primary
  cells2. metric tensor in the mapped coordinate
  system

Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Numerical Method

• The geometric quantities at the center of primary
  cells3.

3
2
1
4
Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Numerical Method

• The flow quantities

Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Numerical Method

• The flow quantities (continued)

Source Nonlinear Aerodynamics of Conical Delta
Wings, Dr. S. S. Sritharan
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Computational Results

• 3-D Plot of the Sonic Line
• angle of attack10o

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Computational Results

• 3-D Plot of the Sonic Line
• angle of attack10o

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Computational Results

• 3-D Plot of the Sonic Line angle of attack20o

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Computational Results

• 3-D Plot of the Sonic Line
• angle of attack20o

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Computational Results

• 3-D Plot of the Sonic Line
• angle of attack10o, elliptic wing

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Computational Results

• 3-D Plot of the Sonic Line
• angle of attack10o, elliptic wing

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Summary
This research project is based on Nonlinear
Aerodynamics of Conical Delta Wings, by Dr. S. S.
Sritharan

• Transformation of Coordinate Systems
• Governing PDE
• Numerical Method
• 3-D Numerical Results
• Many things can be done based on the research
  results.

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Special Thanks to

• Dr. Sritharan
• Wyoming EPSCoR
• Dr. Lynne Ipina
• Mr. John Spitler
• Regent
• Thanks everybody for coming!

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Questions?Comments?