4 advanced potentiaadvanced potential flow 2l flow 2
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Advanced Potential Flow 2TRANSCRIPT
HCMC University of Technology 29/09/200957:020 Fluid Mechanics 1
Advanced Potential Flow
DWFS Tool CANICE 2D/3D
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
Advanced Potential Flow
Flows over Arbitrary Bodies: The Numerical Method
1. The Source Panel Method
2. The Vortex Panel Method
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
Γ = 0 Γ < 4πRuoΓ = 4πRuo
Γ > 4πRuo
The Analytical Methods
Advanced Potential Flow
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
Advanced Potential Flow
The Numerical Methods
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
Source sheet
• Side by side line sources form a source sheet
• λ = λ(s): source strength per unit length along s
• An infinitesimal portion ds of sheet with strength of λds induces an infinitesimally small potential dΦ at point P
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
Superposition of a uniform flow and a source sheet on a body of given shape, to produce the flow over the body
• Cover the surface of the prescribed body with a source sheet
• Strength λ(s) varies in such a fashion that the combine action of the uniform flow and the source sheet makes the airfoil surface a streamline of the flow.
• Find λ(s) -> numerically method!
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
Source panel distribution over the surface of a body
• n source panels with strengths per unit length are λ1, λ2… λn
• Find λj, j=1:n such that the body surface becomes a streamline of flow
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
• Define mid-point of each panel to be a control point (Mj)
• Determine λj such that normal component of flow velocity is zero at Mj
• Velocity potential at P:
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
• Let’s put P at the control point of ith panel. Velocity potential at P(xi,yi):
• Normal component of velocity induced at P(xi,yi):
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
• Normal component of flow velocity at ith control point is the sum of that due to the frees stream and that due to the source panels
• The boundary conditions states that this sum must be zero:
i = 1,2,…,n
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
• Tangential component of flow velocity at ith control point:
• The total surface velocity at the ith control point:
• The pressure coefficient at the ith control point:
• The body itself do not add or absorb mass from the flow, so:
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
Example1: Calculate the pressure distribution around a circular cylinder using the source panel method
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
i = 1,2,…,n
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
Crux of the source panel method:
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
Pressure distribution around a circular cylinder
A. Non-lifting Flows over Arbitrary Bodies:The Numerical Source Panel Method
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
Streamlines Pressure Coefficient
Flow around a cylinder with circulation Г
Γ= 0UL ρ(Kutta-Joukowsky Theorem)
B. Lifting Flows over Arbitrary Bodies:The Numerical Vortex Panel Method
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
• Equation gives the relation between the surface pressure distribution (which produces lift L) and circulation.
• In the theory of incompressible, potential flow, it is generally much easier to determine the circulation around the body rather than calculate the detailed surface pressure distribution
• How can we calculate the circulation for a given body in a given incompressible, inviscid flow?
Γ= 0UL ρ
B. Lifting Flows over Arbitrary Bodies:The Numerical Vortex Panel Method
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
B. Lifting Flows over Arbitrary Bodies:The Numerical Vortex Panel Method
• γ = γ(s): strength of the vortex sheet, per unit length along s.
• An infinitesimal portion ds of sheet with strength of γds induces an infinitesimally small potential dΦ at point P
• Circulation of the vortex sheet:
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
B. Lifting Flows over Arbitrary Bodies:The Numerical Vortex Panel Method
Simulation of an arbitrary airfoil by distributing a vortex sheet over the airfoil surface
Thin airfoil approximation
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
B. Lifting Flows over Arbitrary Bodies:The Numerical Vortex Panel Method
Tangential velocity jump across a vortex sheet
Kutta Condition:
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
B. Lifting Flows over Arbitrary Bodies:The Numerical Vortex Panel Method
For the camber to be streamline:
Crux of Numerical Vortex Panel Method!
HCMC University of Technology 29/09/200957:020 Fluid Mechanics
B. Lifting Flows over Arbitrary Bodies:The Numerical Vortex Panel Method
Example 2: Flow over a flat plate at angle of attack α