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MAE 5360: Hypersonic Airbreathing Engines

MAE 5360: Hypersonic Airbreathing Engines. Simplified Internal and External Flow Modeling Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk. Dynamic Pressure for Compressible Flows. Dynamic pressure, q = ½ r V 2

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MAE 5360: Hypersonic Airbreathing Engines

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  1. MAE 5360: Hypersonic Airbreathing Engines Simplified Internal and External Flow Modeling Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

  2. Dynamic Pressure for Compressible Flows • Dynamic pressure, q = ½rV2 • For high speed flows, where Mach number is used frequently, convenient to express q in terms of pressure p and Mach number, M, rather than r and V • Derive an equation for q = q(p,M)

  3. Summary of Total Conditions • If M > 0.3, flow is compressible (density changes are important) • Need to introduce energy equation and isentropic relations Must be isentropic Requires adiabatic, but does not have to be isentropic

  4. Review: Normal Shock Waves Upstream: 1 M1 > 1 V1 p1 r1 T1 s1 p0,1 h0,1 T0,1 Downstream: 2 M2 < 1 V2 < V1 P2 > p1 r2 > r1 T2 > T1 s2 > s1 p0,2 < p0,1 h0,2 = h0,1 T0,2 = T0,1 (if calorically perfect, h0=cpT0) Typical shock wave thickness 1/1,000 mm

  5. Summary of Normal Shock Relations • Normal shock is adiabatic but nonisentropic • Equations are functions of M1, only • Mach number behind a normal shock wave is always subsonic (M2 < 1) • Density, static pressure, and temperature increase across a normal shock wave • Velocity and total pressure decrease across a normal shock wave • Total temperature is constant across a stationary normal shock wave

  6. Tabulation of Normal Shock Properties

  7. Summary of Normal Shock Properties

  8. Normal Shock Total Pressure Loss Example: Supersonic Propulsion System • Engine thrust increases with higher incoming total pressure which enables higher pressure increase across compressor • Modern compressors desire entrance Mach numbers of around 0.5 to 0.8, so flow must be decelerated from supersonic flight speed • Process is accomplished much more efficiently (less total pressure loss) by using series of multiple oblique shocks, rather than a single normal shock wave • As M1 ↑ p02/p01 ↓ very rapidly • Total pressure is indicator of how much useful work can be done by a flow • Higher p0→ more useful work extracted from flow • Loss of total pressure are measure of efficiency of flow process

  9. Attached vs. Detached Shock Waves

  10. Detached Shock Wave Normal shock wave model still works well

  11. Examples of Schlieren Photographs

  12. Oblique Shock Wave Analysis Upstream: 1 M1 > 1 V1 p1 r1 T1 s1 p0,1 h0,1 T0,1 Downstream: 2 M2 < M1 (M2 > 1 or M2 < 1) V2 < V1 P2 > p1 r2 > r1 T2 > T1 s2 > s1 p0,2 < p0,1 h0,2 = h0,1 T0,2 = T0,1 (if calorically perfect, h0=cpT0) q b

  13. Oblique Shock Control Volume Analysis • Split velocity and Mach into tangential (w and Mt) and normal components (u and Mn) • V·dS = 0 for surfaces b, c, e and f • Faces b, c, e and f aligned with streamline • (pdS)tangential = 0 for surfaces a and d • pdS on faces b and f equal and opposite • Tangential component of flow velocity is constant across an oblique shock (w1 = w2)

  14. Summary of Shock Relations Normal Shocks Oblique Shocks

  15. q-b-M Relationship Strong M2 < 1 Weak M2 > 1 Shock Wave Angle, b Detached, Curved Shock Deflection Angle, q

  16. Some Key Points • For any given upstream M1, there is a maximum deflection angle qmax • If q > qmax, then no solution exists for a straight oblique shock, and a curved detached shock wave is formed ahead of body • Value of qmax increases with increasing M1 • At higher Mach numbers, straight oblique shock solution can exist at higher deflection angles (as M1→ ∞, qmax → 45.5 for g = 1.4) • For any given q less than qmax, there are two straight oblique shock solutions for a given upstream M1 • Smaller value of b is called the weak shock solution • For most cases downstream Mach number M2 > 1 • Very near qmax, downstream Mach number M2 < 1 • Larger value of b is called the strong shock solution • Downstream Mach number is always subsonic M2 < 1 • In nature usually weak solution prevails and downstream Mach number > 1 • If q =0, b equals either 90° or m

  17. Examples • Incoming flow is supersonic, M1 > 1 • If q is less than qmax, a straight oblique shock wave forms • If q is greater than qmax, no solution exists and a detached, curved shock wave forms • Now keep q fixed at 20° • M1=2.0, b=53.3° • M1=5, b=29.9° • Although shock is at lower wave angle, it is stronger shock than one on left. Although b is smaller, which decreases Mn,1, upstream Mach number M1 is larger, which increases Mn,1 by an amount which more than compensates for decreased b • Keep M1=constant, and increase deflection angle, q • M1=2.0, q=10°, b=39.2° • M1=2.0, q=20°, b=53° • Shock on right is stronger

  18. Oblique Shocks and Expansions • Prandtl-Meyer function, tabulated for g=1.4 in Appendix C (any compressible flow text book) • Highly useful in supersonic airfoil calculations

  19. Prandtl-Meyer Function and Mach Angle

  20. Swept Wings in Supersonic Flight • If leading edge of swept wing is outside Mach cone, component of Mach number normal to leading edge is supersonic → Large Wave Drag • If leading edge of swept wing is inside Mach cone, component of Mach number normal to leading edge is subsonic → Reduced Wave Drag • For supersonic flight, swept wings reduce wave drag

  21. Wing Sweep Comparison F-100D English Lightning

  22. Swept Wings Example M∞ < 1 SU-27 q M∞ > 1 • ~ 26º m(M=1.2) ~ 56º m(M=2.2) ~ 27º

  23. Supersonic Inlets Normal Shock Diffuser Oblique Shock Diffuser

  24. EFFECT OF MASS FLOW ON THRUST VARIATION • Mass flow into compressor = mass flow entering engine • Re-write to eliminate density and velocity • Connect to stagnation conditions at station 2 • Connect to ambient conditions • Resulting expression for thrust • Shows dependence on atmospheric pressure and cross-sectional area at compressor or fan entrance • Valid for any gas turbine

  25. NON-DIMENSIONAL THRUST FOR A2 AND P0 • Thrust at fixed altitude is nearly constant up to Mach 1 • Thrust then increases rapidly, need A2 to get smaller

  26. Supersonic and Hypersonic Vehicles

  27. SUPERSONIC INLETS • At supersonic cruise, large pressure and temperature rise within inlet • Compressor (and burner) still requires subsonic conditions • For best hthermal, desire as reversible (isentropic) inlet as possible • Some losses are inevitable

  28. REPRESENTATIVE VALUES OF INLET/DIFFUSER STAGNATION PRESSURE RECOVERY AS A FUNCTION OF FLIGHT MACH NUMBER

  29. C-D NOZZLE IN REVERSE OPERATION (AS A DIFFUSER) Not a practical approach!

  30. C-D NOZZLE IN REVERSE OPERATION (AS A DIFFUSER)

  31. Example of Supersonic Airfoils http://odin.prohosting.com/~evgenik1/wing.htm

  32. Supersonic Airfoil Models • Supersonic airfoil modeled as a flat plate • Combination of oblique shock waves and expansion fans acting at leading and trailing edges • R’=(p3-p2)c • L’=(p3-p2)c(cosa) • D’=(p3-p2)c(sina) • Supersonic airfoil modeled as double diamond • Combination of oblique shock waves and expansion fans acting at leading and trailing edge, and at turning corner • D’=(p2-p3)t

  33. Approximate Relationshipsfor Lift and Drag Coefficients

  34. http://www.hasdeu.bz.edu.ro/softuri/fizica/mariana/Mecanica/Supersonic/home.htmCASE 1: a=0° Expansion Shock waves

  35. CASE 1: a=0°

  36. CASE 2: a=4° Aerodynamic Force Vector Note large L/D=5.57 at a=4°

  37. CASE 3: a=8°

  38. CASE 5: a=20° At around a=30°, a detached shock begins to form before bottom leading edge

  39. CASE 6: a=30°

  40. Example Question • Consider a diamond-wedge airfoil as shown below, with half angle q=10° • Airfoil is at an angle of attack a=15° in a Mach 3 flow. • Calculate the lift and wave-drag coefficients for the airfoil. Compare with your solution

  41. Compressible Flow Over Airfoils:Linearized Flow, Subsonic Case

  42. Review True for all flows: Steady or Unsteady, Viscous or Inviscid, Rotational or Irrotational Continuity Equation 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

  43. 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 dr?

  44. STEP 2: MOMENTUM + ENERGY Euler’s (Momentum) Equation Substitute velocity potential Flow is isentropic: Change in pressure, dp, is related to change in density, dr, via a2 Substitute into momentum equation Changes in x-direction Changes in y-direction

  45. RESULT Velocity Potential Equation: Nonlinear Equation Compressible, Steady, Inviscid and Irrotational Flows Note: This is one equation, with one unknown, f a0 (as well as T0, P0, r0, h0) are known constants of the flow Velocity Potential Equation: Linear Equation Incompressible, Steady, Inviscid and Irrotational Flows

  46. HOW DO WE USE THIS RESULTS? • Velocity potential equation is single PDE equation with one unknown, f • Equation represents a combination of: • Continuity Equation • Momentum Equation • Energy Equation • May be solved to obtain f for fluid flow field around any two-dimensional shape, subject to boundary conditions at: • Infinity • Along surface of body (flow tangency) • Solution procedure (a0, T0, P0, r0, h0 are known quantities) • Obtain f • Calculate u and v • Calculate a • Calculate M • Calculate T, p, and r from isentropic relations

  47. WHAT DOES THIS MEAN, WHAT DO WE DO NOW? • Linearity: PDE’s are either linear or nonlinear • Linear PDE’s: The dependent variable, f, 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)? • Slender bodies • 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 f, u, v, etc.

  48. HOW TO LINEARIZE: PERTURBATIONS

  49. INTRODUCE PERTURBATION VELOCITIES Perturbation velocity potential: same equation, still nonlinear Re-write equation in terms of perturbation velocities: Substitution from energy equation: Combine these results…

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

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