2. Panel methods to Innovate a Turbine Blade-1 P M V Subbarao Professor Mechanical Engineering Department A Linear Mathematics for Invention of Blade Shape…..

3. Panel methods Panel methods are numerical models based on simplifying assumptions about the physics and properties of the flow of air over an aircraft. The viscosity of fluid in the flow field is neglected. However, the net effect of viscosity on a blade is considered by forcing the flow to smoothly leave the sharp trailing edge of the blade. The compressibility of fluid is neglected, and the curl of the velocity field is assumed to be zero. The vector velocity describing the flow field can be represented as the gradient of a scalar velocity potential. The resulting flow is referred to as potential flow.

4. Basic Approach in panel methods 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. Vortex sheets give rise to circulation, and hence lift. Vortex sheet/panel is a collection of miniature vortices of strength  ds, where ds is the length of a panel. Vortex sheets mimic the boundary layer around airfoils.

5. The Source Consider a point source

6. The Source Panel Imagine spreading the source along a line. Distribute the source as certain strength per unit length q(s) that could vary with distance s along the line. Each elemental length of the line ds would behave like a miniature source and so would produce a velocity field: Where z 1 (s), gives the coordinate of the line source at s. The total flow field produced by the line is the strength of the panel.

7. Uniform Strength Source Panel If strength q is constant with s then The velocity in terms of components aligned with the panel: The panel is not a solid boundary. To make it behave like a solid boundary in a flow you would have to set the strength q so that the total V n (due to the panel and the flow) is zero

8. Uniform Strength Vortex Panel The integral vortex panel is used to perturb the uniform flow to create a possible lifting body. This is the stream function at any point in 2-D space. Panel methods are evaluated only at the closed streamline. This may be the surface of the would be airfoil as the body surface is a streamline. Therefore, the stream function value will be some constant value C.

9. Analogy between boundary layer and vortices Upper surface boundary layer contains, in general, clockwise rotating vorticity. Lower surface boundary layer contains, in general, counter clockwise vorticity. Because there is more clockwise vorticity than counter clockwise Vorticity, there is net clockwise circulation around the airfoil. In panel methods, this boundary layer, which has a small but finite thickness is replaced with a thin sheet of vorticity placed just outside the airfoil.

10. Panel method treats the airfoil as a series of line segments On each panel, there is vortex sheet of strength  =  0 ds 0. Where ds 0 is the panel length. Each panel is defined by its two end points (panel joints) and by the control point, located at the panel center, where we will apply the boundary condition  = Constant=C. The more the number of panels, the more accurate the solution,since we are representing a continuous curve by a series of broken straight lines

11. 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.

12. Stream Function due to freestream The free stream is given by Recall This solution satisfies conservation of mass And irrotationality It also satisfies the Laplace’s equation. Check!

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

14. Stream function Vortex Panel Pay attention to the signs. A counter-clockwise vortex is considered “positive” In our case, the vortex of strength  0 ds 0 had been placed on a panel with location (x 0 and y 0 ). Then the stream function at a point (x, y) will be Panel whose center point is (x 0,y 0 )