Boundary Layer Correction of Viscous Flow Through 2 D Turbine Cascades

P M V SubbaraoProfessor

Mechanical Engineering Department

A Classical Method Recommended by Schlichting .……

Validity of Inviscid Flow Theory

Cascade Boundary Layer

Losses in 2D Cascades

• For flow of viscous fluids through two-dimensional cascades, the main object of the investigations has been to find a way to calculate theoretically the loss coefficients of the cascade.

• They depend on the geometrical and aerodynamic parameters of the cascade.

• A turbulent boundary layer computation has been incorporated into the inviscid design.

• This method solves three ordinary differential equations for three independent parameters, the momentum thickness, the shape factor and the entrainment coefficient.

The Displacement Thickness• Conservation of mass is applied to this Engineering CV

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CS

Adv

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H

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Y

Udyudy

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dyU

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* 1

Momentum Thickness of BL

Conservation of Integral x momentum

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x AdvuDF

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Physical Interpretation of Momentum and displacement thicknesses

Their ratio of displacement thickness to momentum thickness is called the shape factor, is often used in estimation of boundary-layer impact on cascade peformance

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BL Correction Equations

dS

dU

UMH

C

ds

d f 222

dS

dU

UH

CHC

dH

Hd

ds

Hd fE

121

1

2

1

2/12/1

1

18.2

MfdS

dU

UdS

dU

UdH

HdCC

HHF

ds

dC

EQEQ

E

Y

e

dyU

uH

0

11

Y

eE udy

ds

d

UC

0

1

Y

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dyU

uH

0

1

2

2

UC

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wf

Comparison of optimum solidities

Non-Dimensional Variables for Cascade Analysis

• Non dimensional deflection:f

ru

V

V

• Loss Coefficient: 2

21

fV

h

• Solidity ratio:c

s

• Reynolds Number:

cVr 2Re

Non-Dimensional Variables for Cascades : Design

• Pressure Coefficient:2

21 U

p

• Flow Coefficient:22fVR

Q

f

ru

V

V

f

ru

V

V

Wind Tunnel Testing of Cascades

COMPRESSIBLE CASCADE FLOW

• This is regarded as very important for basic research on cascades, because the aerodynamic coefficients in most cases depend considerably on both the Mach number and the Reynolds number of the blade.

• The independent variation of Mach number and Reynolds number is achieved by installing the cascade wind tunnel in a tank which can be evacuated from 1 atm down to 0.1 atm.

• The Mach number range is from M = 0.2 to about 1.1.

• The blade length L = 300 mm and the blade chord c = 60 mm.

• For one year an extensive programme of pressure distribution measurements has been carried out on cascade blades at high subsonic speeds.

High Speed Wind Tunnel Testing of Cascades

Effect of Mach Number

Effect of Mach Number

Effect of Mach Number