🗊Презентация Panel.Methods

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Слайд 1


Panel Methods
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Panel Methods

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What are panel methods? Panel methods are techniques for solving incompressible potential flow over thick 2-D and 3-D geometries. In 2-D, the airfoil surface is divided into piecewise straight line segments or panels or “boundary elements” and vortex sheets of strength  are placed on each panel. We use vortex sheets (miniature vortices of strength ds, where ds is the length of a panel) since vortices give rise to circulation, and hence lift. Vortex sheets mimic the boundary layer around airfoils.
Описание слайда:
What are panel methods? Panel methods are techniques for solving incompressible potential flow over thick 2-D and 3-D geometries. In 2-D, the airfoil surface is divided into piecewise straight line segments or panels or “boundary elements” and vortex sheets of strength  are placed on each panel. We use vortex sheets (miniature vortices of strength ds, where ds is the length of a panel) since vortices give rise to circulation, and hence lift. Vortex sheets mimic the boundary layer around airfoils.

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Analogy between boundary layer and vortices
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Analogy between boundary layer and vortices

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Panel method treats the airfoil as a series of line segments
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Panel method treats the airfoil as a series of line segments

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Boundary Condition We treat the airfoil surface as a streamline. This ensures that the velocity is tangential to the airfoil surface, and no fluid can penetrate the surface. We require that at all control points (middle points of each panel) = C The stream function is due to superposition of the effects of the free stream and the effects of the vortices  ds on each of the panel.
Описание слайда:
Boundary Condition We treat the airfoil surface as a streamline. This ensures that the velocity is tangential to the airfoil surface, and no fluid can penetrate the surface. We require that at all control points (middle points of each panel) = C The stream function is due to superposition of the effects of the free stream and the effects of the vortices  ds on each of the panel.

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Stream Function due to freestream The free stream is given by
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Stream Function due to freestream The free stream is given by

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Stream function due to a Counterclockwise Vortex of Strengh 
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Stream function due to a Counterclockwise Vortex of Strengh 

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Stream function Vortex, continued.. Pay attention to the signs. A counter-clockwise vortex is considered “positive” In our case, the vortex of strength 0ds0 had been placed on a panel with location (x0 and y0). Then the stream function at a point (x,y) will be
Описание слайда:
Stream function Vortex, continued.. Pay attention to the signs. A counter-clockwise vortex is considered “positive” In our case, the vortex of strength 0ds0 had been placed on a panel with location (x0 and y0). Then the stream function at a point (x,y) will be

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Superposition of All Vortices on all Panels In the panel method we use here, ds0 is the length of a small segment of the airfoil, and 0 is the vortex strength per unit length. Then, the stream function due to all such infinitesimal vortices at the control point (located in the middle of each panel) may be written as the interval below, where the integral is done over all the vortex elements on the airfoil surface.
Описание слайда:
Superposition of All Vortices on all Panels In the panel method we use here, ds0 is the length of a small segment of the airfoil, and 0 is the vortex strength per unit length. Then, the stream function due to all such infinitesimal vortices at the control point (located in the middle of each panel) may be written as the interval below, where the integral is done over all the vortex elements on the airfoil surface.

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Adding the freestream and vortex effects..
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Adding the freestream and vortex effects..

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Physical meaning of 
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Physical meaning of 

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Pressure distribution and Loads
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Pressure distribution and Loads

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Kutta Condition Kutta condition states that the pressure above and below the airfoil trailing edge must be equal, and that the flow must smoothly leave the trailing edge in the same direction at the upper and lower edge.
Описание слайда:
Kutta Condition Kutta condition states that the pressure above and below the airfoil trailing edge must be equal, and that the flow must smoothly leave the trailing edge in the same direction at the upper and lower edge.

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Summing up.. We need to solve the integral equation derived earlier And, satisfy Kutta condition.
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Summing up.. We need to solve the integral equation derived earlier And, satisfy Kutta condition.

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Numerical Procedure We divide the airfoil into N panels. A typical panel is given the number j, where J varies from 1 to N. On each panel, we assume that 0 is a piecewise constant. Thus, on a panel numbered j, the unknown strength is j We number the control points at the centers of each panel as well. Each control point is given the symbol “i”, where i varies from 1 to N. The integral equation becomes
Описание слайда:
Numerical Procedure We divide the airfoil into N panels. A typical panel is given the number j, where J varies from 1 to N. On each panel, we assume that 0 is a piecewise constant. Thus, on a panel numbered j, the unknown strength is j We number the control points at the centers of each panel as well. Each control point is given the symbol “i”, where i varies from 1 to N. The integral equation becomes

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Numerical procedure, continued Notice that we use two indices ‘i’ and ‘j’. The index ‘I’ refers to the control point where equation is applied. The index ‘j’ refers to the panel over which the line integral is evaluated. The integrals over the individual panels depends only on the panel shape (straight line segment), its end points and the control point í’. Therefore this integral may be computed analytically. We refer to the resulting quantity as
Описание слайда:
Numerical procedure, continued Notice that we use two indices ‘i’ and ‘j’. The index ‘I’ refers to the control point where equation is applied. The index ‘j’ refers to the panel over which the line integral is evaluated. The integrals over the individual panels depends only on the panel shape (straight line segment), its end points and the control point í’. Therefore this integral may be computed analytically. We refer to the resulting quantity as

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Numerical procedure, continued.. We thus have N+1 equations for the unknowns 0,j (j=1…N) and C. We assume that the first panel (j=1) and last panel (j=N) are on the lower and upper surface trailing edges.
Описание слайда:
Numerical procedure, continued.. We thus have N+1 equations for the unknowns 0,j (j=1…N) and C. We assume that the first panel (j=1) and last panel (j=N) are on the lower and upper surface trailing edges.

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Panel code Our web site contains a Matlab code I have written, if you wish to see how to program this approach in Matlab. See http://www.ae.gatech.edu/people/lsankar/AE3903/Panel.m And, sample input file http://www.ae.gatech.edu/people/lsankar/AE3903/panel.data.txt An annotated file telling you what the avrious numbers in the input means is found at http://www.ae.gatech.edu/people/lsankar/AE3903/Panel.Code.Input.txt
Описание слайда:
Panel code Our web site contains a Matlab code I have written, if you wish to see how to program this approach in Matlab. See http://www.ae.gatech.edu/people/lsankar/AE3903/Panel.m And, sample input file http://www.ae.gatech.edu/people/lsankar/AE3903/panel.data.txt An annotated file telling you what the avrious numbers in the input means is found at http://www.ae.gatech.edu/people/lsankar/AE3903/Panel.Code.Input.txt

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PABLO A more powerful panel code is found on the web. It is called PABLO: Potential flow around Airfoils with Boundary Layer coupled One-way See http://www.nada.kth.se/~chris/pablo/pablo.html It also computes the boundary layer growth on the airfoil, and skin friction drag. Learn to use it! We will later on show how to compute the boundary layer characteristics and drag.
Описание слайда:
PABLO A more powerful panel code is found on the web. It is called PABLO: Potential flow around Airfoils with Boundary Layer coupled One-way See http://www.nada.kth.se/~chris/pablo/pablo.html It also computes the boundary layer growth on the airfoil, and skin friction drag. Learn to use it! We will later on show how to compute the boundary layer characteristics and drag.

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