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

2. Progress in Steam Power Plants : Need for New Turbine Designs

3. Blade Design : Industrial Practice
The process of blade design proceeds from a knowledge of the boundary layer properties and the relation between geometry and pressure distribution.
The goal of a blade design varies.
Some blades are designed to produce low drag (and may not be required to generate lift at all.)
Some sections may need to produce low drag while producing a given amount of lift.
In some cases, the drag doesn't really matter -- it is maximum lift that is important.
One approach to airfoil design is to use an airfoil that was already designed by someone.
This "design by authority" works well when the goals of a particular design problem happen to coincide with the goals of the original airfoil design.

4. Case Study A company is in the business of building turbines.
They decide to use a version of one of Bob Liebeck's very high lift airfoils.
The pressure distribution at a lift coefficient of 1.4 is shown above.
Note that only a small amount of trailing edge separation is predicted.
Actually, the airfoil works quite well, achieving a Clmax of almost 1.9 at a Reynolds number of one million.

5. Off-design Performance
At lower angles of attack, the turbine seemed to drop out the load.
The plot above the pressure distribution with a CL of

6. Need for High Lift Aerofoil Shapes for Blades
“A 1.5% increase in maximum lift coefficient is equivalent to an increase in torque of 29,360N-m a fixed blade speed”
• A 1% increase in L/D leads to not only increase in torque, but also isentropic efficiency of turbine.

7. High Lift Aerofoils for Aviation Turbines

8. Inverse Design of Turbine Blades - 1
Another type of objective function is the target pressure distribution.
It is sometimes possible to specify a desired Cp distribution and use the least squares difference between the actual and target Cp's as the objective.
This is the basic idea behind a variety of methods for inverse design.
Airfoil theory can be used to solve for the shape of the camber line that produces a specified pressure difference on an airfoil in potential flow.

9. Inverse Design of Turbine Blades - 2
The second part of the design problem starts when one has somehow defined an objective for the airfoil design.
This stage of the design involves changing the airfoil shape to improve the performance.
This may be done in several ways:
1. By hand, using knowledge of the effects of geometry changes on Cp and Cp changes on performance.
2. By numerical optimization, using shape functions to represent the airfoil geometry and letting the computer decide on the sequence of modifications needed to improve the design.

10. Invention of Blade Shape
The first step in blade shape invention is choosing a method.
A variety of computational methods are available to the airfoil designer.
Linear methods -- concerned with solving the velocity potential equation
Complicated methods that involve solving the Euler (inviscid) equations at various locations around a known geometry.
The linear methods are much faster but much more limiting in application.
Panel methods are most popular of the linear methods.

11. 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.
These line segments are called as panels.
Either soure sheets of strength m are vortex sheets of strength g are placed on each panel.
Vortex sheets mimic the boundary layer around airfoils.

12. The Point Source
Consider a point source

13. 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 z1(s), gives the coordinate of the line source at s.
The total flow field produced by the line is the strength of the panel.

14. Uniform Strength Source Panel
If strength q is constant with s then
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 Vn (due to the panel and the flow) is zero
The velocity in terms of components aligned with the panel:

15. Linear Source Panels

16. Constant-Strength Doublet Distribution

17. 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.

18. 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.

19. Panel method treats the airfoil as a series of line segments
On each panel, there is vortex sheet of strength DG = g0 ds0.
Where ds0 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 y= 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

20. 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) y= C
The stream function is due to superposition of the effects of the free stream and the effects of the vortices g0 ds0 on each of the panel.

21. Stream function due to a Counterclockwise Vortex of Strengh G

22. Stream function Vortex Panel
Pay attention to the signs.
A counter-clockwise vortex is considered “positive”
In our case, the vortex of strength g0ds0 had been placed on a panel with location (x0 and y0).
Then the stream function at a point (x, y) will be
Panel whose center
point is (x0,y0)

23. 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 g0 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.

24. Adding the free stream and vortex effects..
The unknowns are the vortex strength 0 on each panel, and the value of the Stream function C.
Before we go to the trouble of solving for  0, we ask what is the purpose..

25. Physical meaning of g0
V = Velocity of the flow just outside the boundary layer
Sides of our contour
have zero height.
Bottom side has zero Tangential velocity
Because of viscosity
Panel of length ds0 on the airfoil.
If we know g0 on each panel, then we know the velocity of the flow outside the boundary layer for that panel, and hence pressure over that panel.

26. Pressure distribution and Loads
Since V = -g0

27. 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.
g2upper = V2upper
g2lower = V2lower
From this sketch above, we see that pressure will be equal, and the flow
will leave the trailing edge smoothly, only if the voritcity on each panel
is equal in magnitude above and below, but spinning in opposite
Directions relative to each other.

28. The Closure
We need to solve the integral equation derived earlier
And, satisfy Kutta condition.

29. 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 g0 is a piecewise constant. Thus, on a panel numbered j, the unknown strength is g0,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

30. Numerical procedure
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

31. Numerical procedure
We thus have N+1 equations for the unknowns g0,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.
This linear set of equations may be solved easily, and g0 found.
Once go is known, we can find pressure, and pressure coefficient Cp.

32. An Useful Aerofoil

33. PABLO
A powerful panel code is found on the web.
It is called PABLO: Potential flow around Airfoils with Boundary Layer coupled One-way
See
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.

34. Innovative Mathematics close to the Current Reality
It is time to modify the theory to model advanced practice.
The potential theory learnt during few past lectures, is it truly a realistic invention?
Can we see another important positive characteristic of flow is being missed in developing the theory(earlier) of designing a lifting body ?!?!?!?
Is it correct to assume 2=0 for all potential flows???
The real behavior of flow in steam and gas turbines is consider compressible subsonic flow through flow path and hence past a blade.

35. Lifting Bodies in Subsonic Compressible Flows
The Velocity Potential function is also valid for compressible isentropic subsonic flows.
From this velocity potential we can find the velocity components