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

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 )

Superposition of All Vortices on all Panels In the panel method we use here, ds 0 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.

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..

Physical meaning of   Panel of length ds 0 on the airfoil. V = Velocity of the flow just outside the boundary layer If we know  0 on each panel, then we know the velocity of the flow outside the boundary layer for that panel, and hence pressure over that panel. Sides of our contour have zero height. Bottom side has zero tangential velocity because of viscosity

Pressure distribution and Loads Since V = -  0

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.  2 upper = V 2 upper  2 lower = V 2 lower 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.

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

Numerical Evaluation of Integral 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

Simplification of Numerical Integral 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

Closure of Numerical Integration 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. This linear set of equations may be solved easily, and g 0 found. Once  0 is known, we can find pressure, and pressure coefficient C p.

An Useful Aerofoil

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! Learn how to compute the boundary layer characteristics and drag.

Innovative Mathematics close to the Current Reality It is time to modify the theory to model current/advanced practice. The potential theory learnt during few past lectures, is it truly a realistic theory (predicting all positive features)? 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.

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

2-D Continuity Equation for Steady Compressible Flow

Mach Number Mach number of a flight For an ideal and calorically perfect gas:

The Linearized Velocity Potential Equation for Subsonic Compressible Flows For thin bodies at small angles of attack, based on few assumptions, the complicated equation reduces to This is the linearized perturbation velocity potential equation

Prandtl-Glauert Concern The hard work done for incompressible flow should not go waste. It must be possible to apply with few modifications to incompressible potential flow theory to the true subsonic compressible flows. How to estimate the factor of modification? Any non-dimensional parameter, which can guide …???

Subsonic Compressible Flow A Laplacian equation in (x,y) co-ordinates govern the incompressible potential flow in physical plane. This equation in (x,y) co-ordinates govern the subsonic compressible potential flow in physical plane. A transformation function will convert in physical plane into in transformed plane into

Aerofoil in Z&  planes This Laplacian equation will also govern the incompressible potential. Hence represents an incompressible flow in ( ,  ) space which is related to a compressible flow  in the (x,y) space. Shape of the airfoil:

Prandtl-Glauert Rule For subsonic flow past a blade at low angle of attack, the shape of streamlines in incompressible flow are similar to the streamlines in compressible flow. The shape of zero streamline remain unchanged in incompressible to compressible flow past a blade. What about effect? Will the coefficient of pressure reamain unchanged? Will C L and C D remain unchanged?

The Final Outcome of Prandtl Glauert Rule Thus, it can be claimed that the pressure coefficient C p at any point on a thin aerofoil surface in an compressible flow is (1 − M  2 ) −1/2 times the pressure coefficient C p0 at the same point on the same aerofoil in incompressible flow. The thickness of the aerofoil in the subsonic compressible flow is times (1 − M  2 ) −1/2 the thickness of the incompressible aerofoil These are called the final statements of Prandtl-Glauert rule.

The lift & Moment Coefficients

Validity of of Prandtl Glauert Rule : NACA4412

Improved Compressibility Corrections The Prandtl-Glauert rule is based on the linearized velocity potential equation. Other compressibility corrections do take the nonlinear terms into account. Examples are the Karman-Tsien rule, which states that Laitone’s rule, stating that

Selection of Correct Formula for C p CpCp

Variable Mach Number Effect The flow velocity is different on different positions on the blade. Let the Mach number of the flow over our blade at a given point A be M A. The corresponding pressure coefficient can then be found using

Local Mach Number Variations

Critical Mach Number The velocity of the flow on top of our wing is generally bigger than the free stream velocity V . It may be possible to get sonic flow (M = 1) over the blade, while the upstream flow still at M  < 1. The critical Mach number M cr is defined as the free stream Mach number M  at which sonic flow (M = 1) is first achieved on the airfoil surface. This is a very important value. Define the critical pressure coefficient C p,cr. The relation between M cr and C p,cr can be found from the same equation.

Flow above Critical Mach Numbers

1930’s Flying Story Cruising at High Altitudes ?!?!?! Aircraft were trying to approach high altitudes for a better fuel economy. This led to numerous crashes for unknown reasons. These included: The rapidly increasing forces on the various surfaces, which led to the aircraft becoming difficult to control to the point where many suffered from powered flight into terrain when the pilot was unable to overcome the force on the control stick. The Mitsubishi Zero was infamous for this problem, and several attempts to fix it only made the problem worse. In the case of the Super-marine Spitfire, the wings suffered from low torsional stiffness.

The P-38 Lightning suffered from a particularly dangerous interaction of the airflow between the wings and tail surfaces in the dive that made it difficult to "pull out“. Flutter due to the formation of thin high pressure line on curved surfaces was another major problem, which led most famously to the breakup of de Havilland Swallow and death of its pilot, Geoffrey de Havilland, Jr.