Addition of a Second Blade Row

5.10.1 Modifications to the Computational Procedure

The inclusion of a second (downstream) blade row in the computational procedure involves only a small increase in computational time, and involves the following modifications to the design process:-

a) Specification of the shaft speed, number of blades, axial chord, location, and

thickness and rV^ distributions for the additional blade row.

b) For there to be a unique solution for blade shape, it is necessary for the

downstream blade row to encounter axisymmetric onset flow. Therefore, a "mixing" plane, similar to that used by Hawthorne and Tan [11], is defined on one quasi-orthogonal between the trailing edge of the upstream blade row and the leading edge of the downstream row (see Section 5.10.3). At the mixing plane both the flow velocity and the flow vorticity are circumferentially- averaged, this being achieved as outlined in Section 5.10.2.

c) At the mixing plane the relative co-ordinate system is changed from that of the

upstream blade row to that of the downstream row. Thus the values of

rothalpy here are reset in accordance with

H " ^ H " + rV,{(S>, - © ,) 5 3 0

where subscripts 1 and 2 refer to the upstream and downstream blade rows respectively.

d) Downstream of the mixing plane the number of blades is taken to be that for the

downstream blade row.

e) It is necessary to march rV^ downstream between the two blade rows using

equation 5.21. The distribution on the downstream blade row is then

scaled in such a way as to match the updated values of kVq at its leading edge,

whilst maintaining the specified values of kVq at its trailing edge.

f) To give an extra degree of control over blade shapes, an independent stacking

condition is specified for the downstream blade row, this being used to determine all blade/wake shapes downstream of the mixing plane.

g) In addition to the Kutta condition at the trailing edge of the upstream blade row

it is necessary to satisfy a zero-incidence condition at the leading edge of the

downstream blade row. This essentially involves marching rV^ one quasi­

streamline downstream from the leading edge of the downstream row (using

equation 5.21), and then scaling and smoothing the new kVq distribution using

similar techniques to those employed for the blade Kutta condition (see Section 5.7). Additionally, if a non-uniform swirl distribution is chosen at the trailing edge of the downstream row, it is further necessary to satisfy the Kutta condition here explicitly.

h) Since rothalpy is reset at the mixing plane in accordance with c) above, there is

generally a jump in rothalpy across the duct wake. Therefore the duct Kutta condition (equation 5.26) for multiple blade row propulsors is modified to

From the procedure outlined above it should be apparent that it would be a relatively simple matter to include even more blade rows in the computational procedure if required.

5.10.2 Circumferentially-Averaging Velocity at the Mixing Plane

As stated in section 5.10.1, for the twin blade row problem the velocity field is circumferentially averaged at the mixing plane using the following procedure:-

1) From equations 4.16 and 5.18a..c it can be seen that the shear component of the

periodic velocity and vorticity fields can be circumferentially-averaged at the mixing plane by circumferentially-averaging rothalpy and drift function here i.e. setting

//" = 1" = 0 : 5.32

2) A downstream potential function is defined which replaces the upstream

potential 0 within the core flow downstream of the mixing plane. This

dissociates the periodic potential function downstream of the mixing plane from that upstream. The upstream potential function 0 is modified so that it satisfies a downstream boundary condition of 0^^ = 0 at the mixing plane.

3) From equations 4.16 and 4.22 it can be seen that the only general means of

circumferentially-averaging the remainder of the velocity and vorticity fields is if

0 2 satisfies

V02 = 5(a)VÂ% 5.33

at the mixing plane.

Unfortunately, not only would such a potential function require Von Neuman conditions on each boundary (highly undesirable from the point of view of numerical stability), but worse still it would seem that no such function that satisfies this relationship exists.

The solution adopted therefore was for to satisfy the Dirichtlet condition d>2 = 0 at the mixing plane (its far upstream boundary), and the Von Neuman

condition 0 2^ = 0 on the far downstream boundary.

From equations 4.16 and 4.22 it can be seen that this approach is sufficient for both the velocity and vorticity fields to be purely axisymmetric immediately downstream of the mixing plane, provided that:-

(a) the upstream blade row is uniformly loaded (i.e. VrVg = 0 at the mixing plane);

(b) the mixing plane is sufficiently far (typically half a blade chord) upstream of the

downstream blade row leading edge (see Section 5.10.3).

However, for cases of a non-uniformly loaded upstream blade row (VrVg at the

mixing plane), the resulting velocity at the mixing plane is clearly non-axisymmetric (equation 4.22 gives V = -S'(a)VrV^). As a result the velocity and vorticity downstream of the mixing plane were formulated as:-

V = V<I>J - 5 (a ) V(rVe - rV,„) + H Vx -I- Tv{h + h)x + xV H Q = V rV ^xV a + v(rV, - rV,„)x V a ( 5 ^ ( a ) - l ) + V //x V x 5.34a 5.34b

where rV^ = at the mixing plane, and is assumed to be convected along mean

streamlines to points downstream of this plane, satisfying

= 0 5.35

Whilst this "induced swirl" formulation ((^V^-rVg^) is, in effect, the mean swirl induced by the downstream blade row) is not strictly mathematically rigorous, it does ensure that the velocity and vorticity fields are axisymmetric at the mixing plane, and prevents the periodic component of the shed vorticity from being transferred across

the mixing plane. Furthermore, for cases of uniform blade loading this approach yields identical solutions to those obtained from steps 1...3 above.

Considerable time and thought was devoted to the problem of the non-uniformly loaded upstream blade row, and this solution was the best that could be found bearing in mind the limitations imposed by the concept of the mixing plane.

5.10.3 Positioning of the Mixing Plane

It is generally desirable to position the mixing plane as far downstream of the upstream blade row as possible to prevent the downstream boundary condition of O (which is only approximate - see Section 5.3.2) from significantly affecting the three-

dimensional velocity field around the upstream blade row. However, it is also

desirable for the mixing plane to lie far upstream of the downstream blade row so that

the upstream boundary condition on the downstream potential function » does not

inhibit the development of three-dimensional velocities around the downstream blade row.

Results of studies on the positioning of the mixing plane are too lengthy to present in detail here. However, it was found that for loading distributions such as those used in Chapter 6 the errors (as reflected in blade shape predictions) associated with the proximity of the mixing plane to the blading are small, provided that the mixing plane is no closer than one quarter of a blade chord to either blade row.