Cascade theory The theory in this lecture comes from: Fluid Mechanics of Turbomachinery by George F. Wislicenus Dover Publications, INC. 1965

c = c ∞ +  c c∞c∞ FYFY FXFX ds X Y Contour The contour is large compared to the dimensions of the vane  c is the change of velocity due to the vane

Decompose the velocity in the normal and the tangential direction of the contour

Bernoulli’s equation

Forces in the x-direction The forces in the x-direction acting on the element ds can be calculated as a force coming from pressure and impulse. Flow Rate, QVelocity in x-direction, c x

Forces in the x-direction We insert the equation for the pressure, p from Bernoulli’s equation.

Forces in the x-direction We insert the equation for the pressure, p from Bernoulli’s equation.

Forces in the x-direction

The change of velocity,  c is very small because the large distance from the airfoil to the contour. We neglect the terms that has the second order of  c.

Forces in the x-direction This is the force acting in the x-direction on a small element, ds of the contour.

Forces in the x-direction By integrating around the contour, we will find the total force acting in the x-direction. =0

d’Alembert paradox The term  c n ·ds is the flow rate through the contour. If the flow is incompressible, the integral of the term  c n ·ds around the contour will be zero. A body in a two-dimensional and non- viscous flow with constant energy will not exert a force in the direction parallel undisturbed flow, c ∞

Forces in the y-direction The forces in the y-direction acting on the element ds can be calculated as a force coming from pressure and impulse.

Forces in the y-direction This is the force acting in the y-direction on a small element, ds of the contour.

Forces in the y-direction By integrating around the contour, we will find the total force acting in the y-direction. =0

Lift

Circulation

Lift

The law of the circulatory flow about a deflecting body In the absence of any deflecting body inside the hatched area of the contour the force in y- direction must necessarily be zero. This leads to the theorem that: For a flow of constant energy, the circulation around any closed contour not enclosing any force-transmitting body must be zero.

The law of the circulatory flow about a deflecting body Let the circulation around the outer contour in the figure be: cscs Let the circulation around the inner contour in the figure be:

The law of the circulatory flow about a deflecting body Let the circulation around the inner and outer contour be connected along the line A-B. The circulation around the hatched area can now be written as: cscs

The law of the circulatory flow about a deflecting body From the figure we can see that: The circulation around the hatched area can now be written as: cscs

The law of the circulatory flow about a deflecting body Since we do not have any body inside the hatched area: Which gives: cscs

The law of the circulatory flow about a deflecting body cscs This leads to the theorem: For a given flow condition (with constant energy), the circulation around the deflecting body is independent of the size and shape of the contour along which the circulation is measured.

The law of the circulatory flow about a deflecting body cscs The mean velocity for the circulation around a contour having the length s is: For a constant value of the circulation, the mean velocity, c sm has to decrease if the length s increases. The circulation is in inverse ratio to the distance of the contour

Circulation about several deflecting bodies We have 3 wing profiles in a two- dimensional cascade and makes a contour around the whole cascade. This contour is marked ABGDEF.

Circulation about several deflecting bodies From the figure we can see that:

Circulation about several deflecting bodies Circulation around 3 wing profiles in a cascade becomes:

Cascade in an axial flow turbine Let us look at the cylindrical section AB through the axial flow turbine.

Cascade in an axial flow turbine By unfolding the cylindrical section AB from the last slide, we can look at the blades in a cascade

Cascade in an axial flow turbine Circulation around the blades is: (where Z is the number of blades)

Cascade in an axial flow turbine From the figure we can see that:

Cascade in an axial flow turbine

From the figure we can see that:

Cascade in an axial flow turbine The circulation becomes:

Cascade in an axial flow turbine The change of angular momentum is related to the vane circulation by the equation:

Cascade in an axial flow turbine By multiplying the change of angular momentum from the upstream to the downstream side of a turbine runner is the torque acting on the turbine shaft with the angular velocity of the runner we will recognize Euler’s turbine equation.