1. Turbomachinery Class 8a

2. Axial Flow Turbomachine Design
Three-dimensional flow through machine very complex
Decompose problem into series of two-dimensional problems
Blade-to-blade [Cascade] – (x, y=)
Plane cascade (no spanwise effect)
Stream surface (span effect modeled within 2D construct
Through flow [Meanline] – (r, x)
Blades modeled as thin or actuator discs
Secondary flow [normal to mainstream] – (r, )
Design problem largely treated as inviscid analysis with viscous effects accounted for through empirical and semi-empirical methods

3. Airfoil/Cascade Design
Cascade - array of airfoils providing forces that change flow vectors.
Requirements:
Produce required forces
Low total pressure loss
Wide range of low loss incidence: operate at off-design points
Stable exit angles
Turning should be produced so that losses are minimum
Airfoils [compressors] typically selected from family or “series” of airfoils

4. Geometry of a Rectilinear Cascade
Geometry Parameters
Solidity
Stagger angle: angle between chord line and leading edge front of cascade
Camber / Turning
Maximum thickness
Leading edge / trailing edge thickness / radius
Uncovered turning

5. Airfoil Nomenclature
Chord: c or b = xTE-xLE; straight line connecting leading edge and trailing edge
Camber line: locus of points halfway between upper and lower surface, as measured perpendicular to mean camber line itself
Camber: maximum distance between mean camber lineand chord line
Thickness t(x), tmax
Angle of attack: , angle between freestream velocity and chord line

6. Geometry of a Rectilinear Cascade

8. Geometry of a Rectilinear Cascade

9. Compressor Airfoil Design
Compressor Airfoil Series - geometric families
Circular arc (CA): for high Mach number flows
mean camber line is circular arc with max camber and max thickness at 50%c
65-series for moderate subsonic Mach numbers
mean camber line is a parabola with max camber at 50%c and max thickness at 40%c
400-series for low Mach numbers
max camber at 40%c and max thickness at 30%c
thickness distribution NACA SP-36 p.198 or Abbot and Von Doenhoff

10. Compressor Airfoil Design
Insert NACA [Nat’l Advisory Committee on Aeronautics] file here

11. Compressor Airfoil Design
Compressor Airfoil Series - geometric families
Initially developed from wing data
Used through 1970's
Large bodies of cascade data NASA, P&W, UTRC, DFVLR, NGTE, ONERA
Loss & flow turning = f (incidence, Mach no., area ratio & geometry)
Experimental Data Verifies Design Codes

12. Airfoil/Cascade Performance
Shear layer development over solid surface

13. Airfoil/Cascade Performance
Friction effects can adversely affect cascade performance

14. Airfoil/Cascade Performance
Boundary layer thickness assessment parameters:
* = displacement thickness
 = momentum thickness
H = shape factor = * /  (1< H < 2.2)
Equal mass =

15. Airfoil/Cascade Performance
Boundary layer thickness assessment parameters:
* = displacement thickness
Howell correlation

16. Airfoil - Cascade Comparisons
Isolated Wing Lift-Drag Curve
Observations
Cd dependence on Mach is small until critical Mcr
Cl dependence on Mach is strong
Shocks & shock boundary layer interaction lead to flow separation, lift loss, drag rise

17. Airfoil - Cascade Comparisons
Transonic flow shocks
M<1
M>1

18. Airfoil - Cascade Comparisons
Plane Cascade Lift-Drag Curve
Lift
 Circulation ( )
 turning
 deflection
Drag
 total pressure
change
 loss
Dixon – Howell [1942]
What observations can you make about these curves?

19. Loss “Bucket”

20. Minimum loss incidence range
Choke
Stall

21. Limits of Compressor Cascade Operation
Stall
Analogous to wing stall
High positive incidence, separation from suction side
Loss & Deviation (difference between flow and exit camber angles) rising rapidly, work & efficiency fall
Choke
Pressure surface separation due to negative incidence
Actual choke: not enough area at throat to pass mass flow
Choke margin decreases as flow becomes more axial (M=const)

22. Generic Airfoil Velocity Distribution

23. Airfoil/Cascade Design
Airfoil shapes constructed to have desirable surface pressure distributions and boundary layer characteristics
Airfoil shapes must also meet structural criteria
Low pressure, high Mach number [suction side]
Adverse
gradient
T.E. Kutta condition
High pressure, low Mach number [pressure side]
% chord

24. Design for Max Lift, Min Loss, Max Range &
Choke Margin
Avoid leading
edge sep. bubble
Ideal
Avoid separation
Avoid flow reversal
Can also view this in terms of ps/p0?

25. Design for Max Lift, Min Loss, Max Range &
Choke Margin

26. Airfoil/Cascade Design
Cascade Testing:
Vital Requirements
Periodicity
Endwall boundary layer control
Uniform flow
Accurate pitch-wise Traverse data

27. Compressor Airfoil/Cascade Design
Additional Factors:
AVDR - Axial Velocity - Density Ratio - area ratio due to end wall boundary layer growth [Geometry 2D but flow 3D]– Effects deviation [ ]
Contraction of streamlines due to boundary layer thickening

28. Compressor Airfoil/Cascade Performance
Background: Boundary layer thickness parameters:
 = momentum thickness
* = displacement thickness
H = shape factor = * /  (1< H < 2.2)

29. Compressor Airfoil/Cascade Performance
Loss Analysis - Lieblein's Dfactor [Diffusion Factor]
Momentum Integral Equation describes the growth of boundary layer thickness along the suction surface:
where:
V = relative velocity at edge of boundary layer
x = distance along airfoil surface
- Boundary layer eqns.
- Integrate y: 0 to 
Boundary layer PDE integrated over y from 0 to 

30. Compressor Airfoil/Cascade Performance
Loss Analysis - Lieblein's Dfactor [Diffusion Factor]
Momentum Integral Equation describes the growth of boundary layer thickness along the suction surface:
where:
V = relative velocity at edge of boundary layer
x = distance along airfoil surface
- Boundary layer eqns.
- Integrate y: 0 to 
Boundary layer PDE integrated over y from 0 to 

31. Compressor Airfoil/Cascade Performance
Vmaximum
Lieblien's insight:
Correlation of cascade pressure distribution data for constant radius:
Velocities are in Relative Frame [V=W]. Solidity =b/s.
The 2 is empirical [from cascade data].
Now to connect Df to loss……...
V(x)
Vsurf V2

32. Wake Momentum Thickness vs. Calculated Diffusion
Empirical relation
[Leiblien] connecting
/c (or loss) to Df

33. Compressor Airfoil/Cascade Performance
Uses of Dfactor Today
- Preliminary design surge margin limit for known clearance & aspect ratio
- Low speed loss prediction in mean line systems
0 < Dfactor < 0.7
- Given loss [ /c ], now find loss coefficient [  ]

34. Example A compressor rotor with the following conditions:
U=200 mps
Cx1=Cx2=150 mps
2=35 degs.
 = 1
Calculate W1, W2, Df

35. Loss Coefficient Directly Related to Wake Momentum
Thickness Ratio, /c

36. Relation Between /c and  [Extra]
Empirical expression

37. What is the Impact of D-factor on Airfoil Shape
Carter’s Rule for compressor cascades
For HWK 6.3, since  =1
Metal angle decreases to achieve design exit angle goals

38. Carter’s Rule for Compressors

39. Compressor Airfoil/Cascade Performance
Compressor Airfoil Performance
Dfactor sets Wake Thickness
Wake Thickness and # Wakes sets Loss Coefficient
Loss Coefficient and Work Coefficient sets Efficiency

40. Turbomachinery Class 8b

41. Overview of Loss Analysis - Lieblein's Dfactor
Correlation of cascade data [Velocities in Relative Frame]
or de Haller [ 0.72<W2/W1<1]
Momentum thickness [ ] correlated to Dfactor [0 < Dfactor < .7]
Loss coefficient related to cascade wake
momentum thickness
Efficiency related to loss coefficient

42. Compressor Airfoil/Cascade Performance
High deflection compressor airfoil [reducing 2 with fixed 1] means reducing V2.
Early diffusion based design method by de Haller called for 0.72<V2/V1<1 to avoid high losses.
Lieblien [NACA] related diffusion to velocity gradient and solidity
Vmaximum
V(x)
Vsurf V2

43. Compressor Airfoil/Cascade Performance
Repeating stage, repeating row [mirror image airfoil]

44. Compressor Airfoil/Cascade Performance
Dfactor analysis for repeating row stage

45. Airfoil/Cascade Performance
Airfoil Sections Now Designed to Pressure Distributions
Compressible Potential flow code for local M<1.1 (no
shocks)
Integral Boundary Layer codes for performance prediction
(viscous effects)
Wake Mixing (Stewart’s Control Volume) Analysis to Obtain
Relative P0 Loss
Navier-Stokes codes for studies near separation (viscous &
inviscid flow solved at once)

46. Compressor Airfoil/Cascade Design
Controlled Diffusion Airfoils GE)
Higher peak Mach no., tapered dV/dx
More camber in front half of chord
Elliptical Leading edges
Stall range assured by gradual initial acceleration
Not optimum at end walls

47. Turbine Airfoil/Cascade Design
Turbine Airfoil Design
Historically more Analytical than Compressors
Accelerating Flow...Probably the Reason
Boundary layer separation easier to avoid
Design to Pressure Distributions
Correct for deviation effects
Zweifel Load Coefficient & Convergence
Desirable Pressure Distributions

48. Turbine Rotor 2  3 Constant Cx, no exit swirl [3=0]
High loading stage [E<<0, turbine]
Lower weight, lower stage efficiency  lower R

49. Turbine Rotor 2  3
For repeating row design and no exit swirl E=2[R-1]

50. Cascade Design Problem

51. Cascade Design Problem
Desirable pressure distributions
Pressure side
Exit
Pressure
Suction side
Good
Poor- too
much diffusion

52. What is the Impact of Deviation on Airfoil Shape
Carter’s Rule for turbine cascades
Metal angle decreased to achieve design exit angle goals
Effect for turbine is much smaller for turbines due to thinner boundary layers

53. Zweifel Coefficient Derivation
Solidity play important role in turbine efficiency: (1) spacing small, fluid gets
maximum turning force with large wall friction forces; (2) spacing large, fluid
gets small turning force with small wall friction losses.
p01
Area Fideal
Area F p2
Solidity issue: Enough airfoils must be used so that F = change in tangential
momentum of the fluid.

54. Zweifel Coefficient Derivation

55. Turbine Airfoil/Cascade Design [Details]
Zweifel Load Coefficient:
where bx is the axial chord and h is the streamtube spanwise width and x = bx / s
Treating the pressure difference incompressibly:

56. Turbine Airfoil/Cascade Design [Details]
Simplifying:
Zweifel Coefficient: 1st estimate of solidity.

57. Turbine Airfoil/Cascade Design
Zweifel Coefficient: 1st estimate of solidity.
Zweifel noted that turbine min.
loss for z 0.8. Optimum
spacing estimated from given
inlet and outlet angles.

58. Turbine Airfoil/Cascade Design
Zweifel Coefficient: 1st estimate of solidity.
Other Factors:
Pressure Distributions
Cooling
Resonances
Disk Stress
Weight
Cost
Some of this covered after
meanline analysis

59. Turbine Airfoil/Cascade Design
Turbine Airfoil Design Approach [covered in following charts]
Balance Acceleration & Diffusion
Use Laminar Boundary layer where possible
Diffuse with Turbulent Boundary layer
Control Maximum Mach No. & Shocks
Control Leading Edge Overspeed
Manage Uncovered Turning

60. Turbine Airfoil/Cascade Design
Balance Front End Acceleration With Rear Diffusion
Laminar boundary layers on suction surface reduces losses on front,
turbulent boundary layers on rear more tolerant to flow separation.

61. More Blades
Increases Solidity [ = b/s]
Reduces Force per Blade

62. Aft Curvature Moves Loading Aft

63. Turbine Airfoil/Cascade Design
High Convergence Eases Cascade Design
Like an area ratio: Aan is an annulus area
Convergence is Function of Velocity Diagrams, not airfoil shape
High Reaction = higher blade convergence

64. Turbine Airfoil/Cascade Design

65. Turbine Airfoil/Cascade Design
Thin Edges are best for Aerodynamic Performance
Trailing Edge Blockage Drives Base Drag
TET = Trailing Edge Thickness
Stewart’s mixing loss correlation is function of TET and *s.s. , *p.s.
Casting capability, stress & cooling set minimum edge thickness
High wedge angle eases effect at leading edge
Elliptical LE is better
Pressure Distribution Improved;
Overspeed reduced in both surfaces

66. Turbine Airfoil/Cascade Design
Manage Uncovered Turning
Transonic Flow:
More uncovered turning
lowers base pressure loss
(<pex) but higher boundary
layer loss.
Subsonic Flow:
Base pressure  pex
Uncovered
turning
Throat

67. Consequences of Stagnation Point Location

68. Turbine Airfoil/Cascade Design
Manage Uncovered Turning
Uncovered turning is less of an issue for subsonic turbine airfoils

75. Turbine Airfoil Active Cooling Methods

76. Thermo & Kinematic View of Compressor Stage
Note: U>C

77. Thermo & Kinematic View of Turbine Stage
Note: U<C
Cu3
Cu2

78. Thermo & Kinematic View of Turbine Stage