Understanding Turbomachinery Mechanics: Two-Dimensional Blade Cascades

1. 4thLecture Two-dimensional Cascades

2. Introduction The operation of any turbomachine is directly dependent upon changes in the working fluid’s angular momentum as it crosses individual blade rows. A deeper insight of turbomachinery mechanics may be gained from consideration of the flow changes and forces exerted within these individual blade rows. The flow past two-dimensional blade cascades will be examined. (a) Conventional low-speed (b) Transonic/supersonic

3. Cascade nomenclature A cascade blade profile can be conceived as a curved camber line upon which a profile thickness distribution is symmetrically superimposed. Referring to the figure the camber line y(x) and profile thickness t(x) are shown as functions of the distance x along the blade chord l. Blade camber and thickness distributions are generally presented as tables of y/l and t/l against x/l. Summarising, the useful parameters for describing a cascade blade are: camber line shape, b/l, a/l, type of thickness distribution and maximum thickness to chord ratio, tmax/l.

4. Two important additional geometric variables which define the cascade are the space-chord ratio s/l and the stagger angle  which is the angle between the chord line and a reference direction perpendicular to the cascade front.

5. Analysis of cascade forces Applying the principle of continuity to a unit depth of span and noting the assumption of incompressibility, yields The momentum equation applied in the x and y directions with constant axial velocity gives, Energy losses A real fluid crossing the cascade experiences a loss in total pressure p0 due to skin friction and related effects. Thus

6. where The total pressure-loss coefficient can be defined as:

7. Lift and drag A mean velocity cmis defined as Considering unit depth of a cascade blade, a lift force L acts in a direction perpendicular to cmand a drag force D in a direction parallel to cm. Rearranging

8. Lift and drag coefficients may be introduced as Alternatively,

9. Circulation and lift The lift of a single isolated aerofoil is given by the Kutta Joukowski theorem Kutta Joukowski theorem A body experiences a lift if the mean pressure above the body is lower than below. The condition of different mean velocities on the two sides of a body can be represented by superposing on a flow from left to right with constant velocity, a circulating flow in clockwise direction:

10. The circulation must be of such magnitude as to obtain a smooth wash of the flow at the trailing edge according to the so-called stagnation hypothesis of Jaukowiski: The following derivation gives us the dependence of the lift L upon the velocity difference w; the calculations are made for a span width b cut oit from the infinite long wing. The pressure at the upper side of the wing shall be denoted by puand at the lower side by pl. The lift L can be expressed by the pressure difference pl- pu: The pressure difference can be estimated by Bernoulli's equation: This leads to:

11. The velocity component w can be represented by a circulation around the airfoil. This circulation can be imagined to be induced by a vortex with its center in the airfoil. The magnitude of the vortex motion is designated by  The magnitude of the integration is independent from the chosen line and proportional to the velocity of circulation: The circulation velocity at the radius r=1 of a potential vortex may be w' = v. Thus the circulation velocity w' at any radius r amounts to w' = v/r. According to the given sketch we find This leads to Applying the magnitude of the circulation expressed by the term w as follows in the next slide.

12. The magnitude of circulation around the treated airfoil amounts to: Substituting this expression into the lift equation leads to Kutta- Joukowski' law For unit span width and velocity c=w the law can be rewritten as:

13. Efficiency of a compressor cascade The efficiency Dof a compressor blade cascade can be defined in the same way as diffuser efficiency; this is the ratio of the actual static pressure rise in the cascade to the maximum possible theoretical pressure rise (i.e. with p0= 0). Thus, Using the definitions of  and Cfin slide no. 6 Assuming a constant lift drag ratio, last eqn. can be differentiated with respect to mto give the optimum mean flow angle for maximum efficiency. Thus, so that