Defining the Target AP Distribution

The relationship between the pressure jump across the blade and the swirl velocity is given in equation (5.3), section 5.1.2. It is then clear that the local rate of change of the

mass-averaged swirl, rV^ is directly related to the blade pressure loading.

AP oc (5.46)

dm

or

_L

_{L ]}

which states that the area under the AP versus m curve is related to the specific work,

quantified by the overall change in swirl between the inlet and outlet of the cascade, i.e.

- ’■V. and K is the constant of proportionality.

Since the constant of proportionality in equation (5.47) is not known in a viscous,

compressible flow, the correct value of AP is obtained iteratively. This is carried out by

first defining a AP distribution (for example, using the three-segment method, see

Appendix VII) which, if necessary, is varied against the resulting overall change in swirl

may be performed manually or automatically using an outer loop attached to the proposed method. Implementation of the latter is given in the next section.

5.2.5.1 Secant Loop: Design for Specific Work

The objective of having an outer loop system is to automatically adjust the target pressure loading distribution such that the required specific work, quantified by the change in the mass-averaged swirl velocity from the inlet to the outlet of the final converged cascade (i.e. the bracketed terms on the r.h.s. of equation 5.48), is satisfied at every stage of the design. To illustrate the principle of this outer loop, details are presented here in two dimensions.

In two dimensions, the proportional relation between AP and the change in swirl is

given by

f TE

J AP dx = K

LE (5.48)

The idea is to vary the unknown constant of proportionality in equation (5.48) such that

the required specific work (i.e. F - K ) is attained.

• '7 E ^LE J

In the proposed method, the outer iterative loop is executed using a fast converging,

root-seeking Secant Method (Press et a l, 1992) at each blade update interval where the

pressure loading distribution is also updated.

Although this method may be used with any AP distribution, for the purpose of

illustration, the problem is simplified by assuming a parabolic AP distribution so that its

/• TE

T h e p a r a b o lic ta r g e t p r e s su r e lo a d in g f o l l o w s th e r e la tio n ,

AP* oc x(y - x ) L.E. < x< T.E. (5.49)

where the Kutta condition is satisfied at the trailing-edge, with x denoting the axial co­

ordinate from the leading-edge to the trailing-edge of the blade section whose axial

chord is set at 1.

Denoting the peak pressure load at the mid-axial chord as A PL , equation (5.49) can be written as,

(5^0)

^ ' = 4AP:„x( I -x)

To find the required peak pressure load, AP \ the algorithm starts with an arbitrary solution (point 1) on the first iteration (see figure 5.8) and the next improvement is taken as the point where the approximating line crosses the solution axis (point 2). As the iteration proceeds, the solution continually improves. Finally, the required value of the peak pressure load, AP^,^^ is obtained which, when used to define the target pressure loading using equation (5,50), gives a specific work sufficiently close to that specified. Typical tolerance is set at 1% of the required change in swirl between the leading- and trailing-edges.

A P R e q u ir e d APJJ', P t 1 : initial G u e s s P t 2 , 3 , 4 : i n te r m e d ia te Im p r o v e m e n ts O : R e q u ir e d S o lu tio n P o in t R e q u ire d AVy,^ A V y

Mathematically, the Secant algorithm is expressed as

AP’^ * '= A P " — — .(aP " - A P ’^~‘ ) (5.51)

- - A S 1 - A S 2 ^ ^ )

where A SI = A V ’ - z l F " and AS2 = A V ' - A V ^ , with AV,

y P E ^ L E y i E ^ L E y p E ^ L E y P E ^ L E ’ y T E _ L E

representing the change in the mass-averaged tangential velocity between the inlet and

outlet o f the cascade, and superscripts, N, N+1 and * denoting the previous, current

predicted and target values respectively.

The working of this automatic looping system is demonstrated in Chapter 7, section 7.I.2.I.

5.2.6 Predicting an Initial Blade Camber Geometry

In the case when the initial blade or blade section is not known, the geometry has to be

approximated. The approximation is carried out in the same way as that used in Method

5.2.7 Design Convergence Criteria

As described earlier, the main design modules consist of the modified Euler solver and the blade update algorithm. Every time the transpiring solver outputs its flow solution, the numerical procedure initiates a design iteration to change the blade shape. In the current implementation, the "design" mode flow calculation is allowed to converge before its solution is used for the blade modification. Numerical experimentation has shown that it is not necessary to achieve the same level of convergence as that required by the "direct" mode flow computation (see chapter 3, section 3.9) throughout the entire design process. Instead, it is found to be sufficient and computationally faster to relax the convergence requirements of the flow computation at the early stages of the design, when large geometrical modifications may be made quickly without affecting the final design.

In the current implementation, two convergence settings. Set 1 & Set 2 are used to

determine when the blade geometry is updated. When the number of iterations, Nueration

exceeds the given maximum number of time steps, N„,ax, the less stringent criteria. Set 2

are used so as to reduce computational time. This is given as follows.

S et the M inim um N um ber o f tim e ste p s in the "design m ode" an alysis: N„,i„

S et the M axim um N um ber o f tim e ste p s in the "design m ode" an alysis: N,„ax S et C on vergen ce C riteria: M ass E rror a n d Vr,„s E rror