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. Normal shock wave model still works well
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 Shock Wave Angle, b Detached, Curved Shock
Strong
M2 < 1
Weak
Shock Wave Angle, b
M2 > 1
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

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

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 Relationships for Lift and Drag Coefficients

34. http://www. hasdeu. bz. edu
CASE 1: a=0°
Shock waves
Expansion

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 Compare with your solution
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 Linear Non-Linear
Equation is still exact for irrotational, isentropic flow
Perturbations may be large or small in this representation

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

52. 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 B A C D E

53. 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
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

54. BOUNDARY CONDITIONS Solution must satisfy same boundary conditions
Perturbations go to zero at infinity
Flow tangency

55. IMPLICATION: PRESSURE COEFFICIENT, CP
Definition of pressure coefficient
CP 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 CP
Linearize equation
Linearized form of pressure coefficient, valid for small perturbations

56. HOW DO WE SOLVE EQUATION
Note behavior of sign of leading term for subsonic and supersonic flows
Equation is almost Laplace’s equation, if we could get rid of b coefficient
Strategy
Coordinate transformation
Transform into new space governed by x and h
In transformed space, new velocity potential may be written

57. 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

58. 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 (x, h) space over same airfoil

59. FINAL RESULTS Insert transformation results into linearized CP
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

60. OBTAINING LIFT COEFFICIENT FROM CP

61. 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
Karman-Tsien
Most widely used
Laitone
Most recent

62. Compressible Flow Over Airfoils: Linearized Flow, Supersonic Case

63. SMALL PERTURBATION VELOCITY POTENTIAL EQUATION
Equation is a linear PDE and easy to solve
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
Nature of PDE:
Subsonic: (1 - M∞2) > 0 (elliptic)
Supersonic: (1 - M∞2) < 0 (hyperbolic)

64. SUPERSONIC APPLICATION
Linearized small perturbation equation
Re-write for supersonic flow
Solution has functional relation
May be any function of (x - ly)
Perturbation potential is constant along lines of x – ly = constant

65. DERIVATION OF PRESSURE COEFFICIENT, CP
Solutions to hyperbolic wave equation
Velocity perturbations
Eliminate f’
Linearized flow tangency condition at surface
Linearized definition of pressure coefficient
Combined result
Positive q: measured above horizontal
Negative q: measured below horizontal

66. KEY RESULTS: SUPERSONIC FLOWS
Linearized supersonic pressure coefficient
Expression for lift coefficient
Thin airfoil or arbitrary shape at small angles of attack
Expression for drag coefficient