Analysis of Blade Performance in Compressible Flows

P M V SubbaraoProfessor

Mechanical Engineering Department

Enhanced Effects due to Flow with Snestive Density…..

Tangency Condition in Potential Flow

A uniform flow is perturbed by an airfoil.

Tangency condition for an inviscid flow past an airfoil, y=f(x) is defined as:

tan

uV

v

dx

df

For small perturbations (thin airfoil at low AOA),

Vu

TC @ Small Pertrubations

V

v

dx

df

For potential flow :y

v

Tangency condition for linearized theory:

dx

dfV

yv

Subsonic Compressible Flow

02

2

2

2

yx

A Laplacian equation in (x,y) co-ordinates govern the incompressible potential flow in physical plane.

02

2

2

22

yx

This equation in (x,y) co-ordinates govern the subsonic compressible potential flow in physical plane.

A transformation function yxyx ,,&;

will convert 02

2

2

22

yx

in physical plane into

02

2

2

2

in transformed plane into

Aerofoil in Z& planes

02

2

2

2

This Laplacian equation will also govern

the incompressible potential.

Hence represents an incompressible flow in (,) space which is related to a compressible flow in the (x,y) space.

Shape of the airfoil: space. , in the yxxfy

space. , in the g

Tangency Condition in Transformed Plane

dx

dfV

yyv

1

Applying tangency condition in Transformed plane on the airfoil

d

dgV

Transform the TC of physical plane:

d

dg

dx

df

Similarity Nature of Thin Airfoil

• This equation tells that the airfoil in (x,y) space and the (,) space is the same.

• This confirms that the proposed transformations relate the compressible flow over an airfoil in the physical space to the incompressible flow over the same airfoil in transformed space.

d

dg

dx

df

Pressure Coefficient in incompressible flows

• The pressure p may be found from Bernoulli’s equation. For an incompressible flow it is written as

222

2

1

2

1 Vpvup

Cp p

V

u v

V

V

Vp

1

2

1 12

2 2

2

2

2

The velocity components in the flow influenced by the airfoil are represented in the form

vvuVu &

Linearized Pressure Coefficient

• For thin airfoils at low angle of attacks,

uVpp

Further, taking into account that the perturbation of the longitudinal velocity is related to the perturbation of the potential as

xu

'

The linearized pressure coefficient for incompressible flow past aThin airfoil at low angles of attacks is:

xVV

ppC p

'2

21 2

It was already shown that if the incompressible flow behavior is known, then there is no need to solve the compressible problem

Prandtl-Glauert rule

• Following up with linearized pressure coefficient :

xVV

ppC p

'2

21 2

xx

'1'

Transformation model states that

'1'

x

'12

VC p

V

uC p

21

The Final Outcome of Prandtl Glauert Rule

2

00

1

M

CCC pp

p

Thus, it can be claimed that the pressure coefficient Cp at any point on a thin aerofoil surface in an compressible flow is (1 − M

2)−1/2 times the pressure coefficient Cp0 at the same point on the same aerofoil in incompressible flow.

The thickness of the aerofoil in the subsonic compressible flow is times (1 − M

2)−1/2 the thickness of the incompressible aerofoil

These are called the final statements of Prandtl-Glauert rule.

The lift & Moment Coefficients

Validity of of Prandtl Glauert Rule : NACA4412

Improved Compressibility Corrections • The Prandtl-Glauert rule is based on the linearized velocity

potential equation.

• Other compressibility corrections do take the nonlinear terms into account.

• Examples are the Karman-Tsien rule, which states that

Laitone’s rule, stating that

Selection of Correct Formula for Cp

Cp

Variable Mach Number Effect

• The flow velocity is different on different positions on the wing.

• Let the Mach number of the flow over our wing at a given point A be MA.

• The corresponding pressure coefficient can then be found using

Loca Mach Number Variations