4 Advanced PotentiaAdvanced Potential Flow 2l Flow 2

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HCMC University of Technology 29/09/200957:020 Fluid Mechanics 1 Advanced Potential Flow DWFS Tool CANICE 2D/3D

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Advanced Potential Flow 2

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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

HCMC University of Technology 29/09/200957:020 Fluid Mechanics

B. Lifting Flows over Arbitrary Bodies:The Numerical Vortex Panel Method