2. Analysis of Blade Performance in Compressible Flows P M V Subbarao Professor Mechanical Engineering Department Enhanced Effects due to Flow with Snestive Density…..

3. Tangency Condition in Potential Flow A uniform flow is perturbed by an airfoil. Tangency condition for an inviscid flow past an airfoil, y=f(x) is defined as: For small perturbations (thin airfoil at low AOA),

4. TC @ Small Pertrubations For potential flow : Tangency condition for linearized theory:

5. 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

6. 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:

7. Tangency Condition in Transformed Plane Applying tangency condition in Transformed plane on the airfoil Transform the TC of physical plane:

8. Similarity Nature of Thin Airfoil This equation tells that the airfoil in (x,y) space and the ( ,  ) space is the same. This confirms that the proposed transformations relate the compressible flow over an airfoil in the physical space to the incompressible flow over the same airfoil in transformed space.

9. Pressure Coefficient in incompressible flows The pressure p may be found from Bernoulli’s equation. For an incompressible flow it is written as The velocity components in the flow influenced by the airfoil are represented in the form

10. Linearized Pressure Coefficient For thin airfoils at low angle of attacks, Further, taking into account that the perturbation of the longitudinal velocity is related to the perturbation of the potential as

11. The linearized pressure coefficient for incompressible flow past a Thin airfoil at low angles of attacks is: It was already shown that if the incompressible flow behavior is known, then there is no need to solve the compressible problem

12. Prandtl-Glauert rule Following up with linearized pressure coefficient : Transformation model states that

13. 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.

14. The lift & Moment Coefficients

15. Validity of of Prandtl Glauert Rule : NACA4412

16. 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

17. Selection of Correct Formula for C p CpCp

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

19. Loca Mach Number Variations