1. MAE 3241: AERODYNAMICS AND FLIGHT MECHANICS Compressible Flow Over Airfoils: Linearized Subsonic Flow Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. 2 WHAT ARE WE DOING NOW? Goal: Examine and understand behavior of 2-D airfoils at Mach numbers in range 0.3 < M ∞ < 1 Think of study of Chapter 11 (compressible regime) as an extension of Chapter 4 (incompressible regime) Why do we care? –Most airplanes fly in Mach 0.7 – 0.85 range –Will continue to fly in this range for foreseeable future –“Miscalculation of fuel future pricing of $0.01 can lead to $30M loss on bottom line revenue” – American Airlines Most useful answers / relations will be ‘compressibility corrections’: Example: 1. Find incompressible c l,0 from data plot NACA 23012,  = 8º, c l,0 ~ 0.8 2. Correct for flight Mach number M ∞ = 0.65 c l = 1.05 Easy to do!

3. 3 For M ∞ < 0.3,  ~ const C p = C p,0 = 0.5 = const Effect of compressibility (M ∞ > 0.3) is to increase absolute magnitude of C p and M ∞ increases Called: Prandtl-Glauert Rule Prandtl-Glauert rule applies for 0.3 < M ∞ < 0.7 (Why not M ∞ = 0.99?) PREVIEW: COMPRESSIBILITY CORRECTION EFFECT OF M ∞ ON C P Sound Barrier ? M∞M∞

4. 4 OTHER IMPLICATIONS Subsonic Wing SweepArea Rule

5. 5 REVIEW Continuity Equation True for all flows: Steady or Unsteady, Viscous or Inviscid, Rotational or Irrotational 2-D Incompressible Flows (Steady, Inviscid and Irrotational) 2-D Compressible Flows (Steady, Inviscid and Irrotational) steady irrotational Laplace’s Equation (linear equation) Does a similar expression exist for compressible flows? Yes, but it is non-linear

6. 6 STEP 1: VELOCITY POTENTIAL → CONTINUITY Flow is irrotational x-component y-component Continuity for 2-D compressible flow Substitute velocity into continuity equation Grouping like terms Expressions for d  ?

7. 7 STEP 2: MOMENTUM + ENERGY Euler’s (Momentum) Equation Substitute velocity potential Flow is isentropic: Change in pressure, dp, is related to change in density, d , via a 2 Substitute into momentum equation Changes in x-direction Changes in y-direction

8. 8 RESULT Velocity Potential Equation: Nonlinear Equation Compressible, Steady, Inviscid and Irrotational Flows Note: This is one equation, with one unknown,  a 0 (as well as T 0, P 0,  0, h 0 ) are known constants of the flow Velocity Potential Equation: Linear Equation Incompressible, Steady, Inviscid and Irrotational Flows

9. 9 HOW DO WE USE THIS RESULTS? Velocity potential equation is single PDE equation with one unknown,  Equation represents a combination of: 1.Continuity Equation 2.Momentum Equation 3.Energy Equation May be solved to obtain  for fluid flow field around any two-dimensional shape, subject to boundary conditions at: 1.Infinity 2.Along surface of body (flow tangency) Solution procedure (a 0, T 0, P 0,  0, h 0 are known quantities) 1.Obtain  2.Calculate u and v 3.Calculate a 4.Calculate M 5.Calculate T, p, and  from isentropic relations

10. 10 WHAT DOES THIS MEAN, WHAT DO WE DO NOW? Linearity: PDE’s are either linear or nonlinear –Linear PDE’s: The dependent variable, , and all its derivatives appear in a linear fashion, for example they are not multiplied together or squared No general analytical solution of compressible flow velocity potential is known –Resort to finite-difference numerical techniques Can we explore this equation for a special set of circumstances where it may simplify to a linear behavior (easy to solve)? 1.Slender bodies 2.Small angles of attack –Both are relevant for many airfoil applications and provide qualitative and quantitative physical insight into subsonic, compressible flow behavior Next steps: –Introduce perturbation theory (finite and small) –Linearize PDE subject to (1) and (2) and solve for , u, v, etc.

11. 11 HOW TO LINEARIZE: PERTURBATIONS

12. 12 INTRODUCE PERTURBATION VELOCITIES Perturbation velocity potential: same equation, still nonlinear Re-write equation in terms of perturbation velocities: Substitution from energy equation (see Equation 8.32, §8.4): Combine these results…

13. 13 RESULT Equation is still exact for irrotational, isentropic flow Perturbations may be large or small in this representation Linear Non-Linear

14. 14 HOW TO LINEARIZE Limit considerations to small perturbations: –Slender body –Small angle of attack

15. 15 HOW TO LINEARIZE Compare terms (coefficients of like derivatives) across equal sign Compare C and A: –If 0 ≤ M ∞ ≤ 0.8 or M ∞ ≥ 1.2 –C << A –Neglect C Compare D and B: –If M ∞ ≤ 5 –D << B –Neglect D Examine E –E ~ 0 –Neglect E Note that if M ∞ > 5 (or so) terms C, D and E may be large even if perturbations are small A B C D E

16. 16 RESULT After order of magnitude analysis, we have following results May also be written in terms of perturbation velocity potential Equation is a linear PDE and is rather easy to solve (see slides 19-22 for technique) Recall: –Equation is no longer exact –Valid for small perturbations: Slender bodies Small angles of attack –Subsonic and Supersonic Mach numbers –Keeping in mind these assumptions equation is good approximation

17. 17 BOUNDARY CONDITIONS 1.Perturbations go to zero at infinity 2.Flow tangency Solution must satisfy same boundary conditions as in Chapter 4

18. 18 IMPLICATION: PRESSURE COEFFICIENT, C P Definition of pressure coefficient C P in terms of Mach number (more useful compressible form) Introduce energy equation (§7.5) and isentropic relations (§7.2.5) Write V in terms of perturbation velocities Substitute into expression for p/p ∞ and insert into definition of C P Linearize equation Linearized form of pressure coefficient, valid for small perturbations

19. 19 HOW DO WE SOLVE EQUATION (§11.4) Note behavior of sign of leading term for subsonic and supersonic flows Equation is almost Laplace’s equation, if we could get rid of  coefficient Strategy –Coordinate transformation –Transform into new space governed by  and  In transformed space, new velocity potential may be written

20. 20 TRANSFORMED VARIABLES (1/2) Definition of new variables (determining a useful transformation is done by trail and error, experience) Perform chain rule to express in terms of transformed variables

21. 21 TRANSFORMED VARIABLES (2/2) Differentiate with respect to x a second time Differentiate with respect to y a second time Substitute in results and arrive at a Laplace equation for transformed variables Recall that Laplace’s equation governs behavior of incompressible flows Shape of airfoil is same in transformed space as in physical space Transformation relates compressible flow over an airfoil in (x, y) space to incompressible flow in ( ,  ) space over same airfoil

22. 22 FINAL RESULTS Insert transformation results into linearized C P Prandtl-Glauert rule: If we know the incompressible pressure distribution over an airfoil, the compressible pressure distribution over the same airfoil may be obtained Lift and moment coefficients are integrals of pressure distribution (inviscid flows only) Perturbation velocity potential for incompressible flow in transformed space

23. 23 OBTAINING LIFT COEFFICIENT FROM C P

24. 24 IMPROVED COMPRESSIBILITY CORRECTIONS Prandtl-Glauret –Shortest expression –Tends to under-predict experimental results Account for some of nonlinear aspects of flow field Two other formulas which show excellent agreement 1.Karman-Tsien –Most widely used 2.Laitone –Most recent