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Advanced Potential Flow 2

### Transcript of 4 Advanced PotentiaAdvanced Potential Flow 2l Flow 2

• HCMC University of Technology 29/09/200957:020 Fluid Mechanics 1

DWFS Tool CANICE 2D/3D

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

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

= 0 < 4piRuo = 4piRuo > 4piRuo

The Analytical Methods

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

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

= (s): source strength per unit length along s

An infinitesimal portion ds of sheet with strength of ds induces an infinitesimally small potential d 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 (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 (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 1, 2 n Find j, 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 j 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

Lets 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

= 0UL (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?

= 0UL

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

= (s): strength of the vortex sheet, per unit length along s.

An infinitesimal portion ds of sheet with strength of ds induces an infinitesimally small potential d 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

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

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