Derivation of the Governing Equations

Adopting the cylindrical polar co-ordinate system shown in Figure 4.2, a relative velocity W may be defined relative to the blading, viz:-

W = V - (orëg 4.3

Defining the blade surfaces by

a = Q - f ( r , z ) = : m = 0 , 1 , 2 , . . - 1 4.4

^ b

the condition of no flow normal to the blade may then be expressed

•Va = 0 4.5

pW X Q = p V //+ pFV a 4.6

where H is rothalpy and the term pFV a represents the blade force on the fluid which

must act normal to the blade on the blade surface.

Taking the tangential component of equation 4.6 at the blade surface, it can be seen that

Now, as it can be shown that

30

circumferentially averaging equation 4.7 over one blade pitch (i.e. over the range

0 < 0 < 2n/Nf^ ) yields

F = -W „ -V rV ;

As the blade force is exerted at the blade surfaces, it follows that

f = F 5 ,(a ) = - ( w „ V ^ ) 5 , ( a ) 4.8

with equation 4.6 becoming

W x Û = V / /- ( w „ V 7 v ; ) 5 ,( a ) V a 4.9

where 5 ^(a) is the periodic delta function given in Appendix II.

Now, taking the dot product of equation 4.9 with W and applying condition 4.5 it follows that

The flow vorticity is now split into components as follows :-

Q = 4.11

where represents the blade bound/shed vorticity, and is the additional

component of vorticity associated with the presence of shear flow. Decomposing

each component of vorticity as in equation 4.1, and noting that is only non-zero

on the blade/blade wake surface, equation 4.11 may be written

o = ( VI» X Vn ( a ) + X Vn, 4.12

provided A,,, and p., are chosen from considerations of the equations of

motion. Since the blade bound/shed vorticity must lie on the blade surfaces/wakes, it

is immediately apparent that A.^ = a may be chosen. Additionally, setting X^ = H ,

equation 4.9 may be written

f2 = ( V a x V / / X V|i^ 4.13

Substituting equation 4.13 into 4.9 and applying condition 4.10 now yields

F = (W „ -V n J § ^ (a ) 4.14

W*Vx = l 4.15

where x = |i^

Comparing equation 4.14 with equation 4.8 it can be seen that

Likewise from equation 4.15 it can be seen that x is the "Drift Function" given by Hawthorne [7]. This quantity represents the time between fluid particles passing some reference plane and reaching the point of concern in the fluid, as can be seen by expressing equation 4.15 in the integral form

_ f ^

' i w

where s is distance along the particle path.

The full Clebsch representation of vorticity is therefore

n = (VrV0 X V a)ô ^(a) + V //x Vx 4.16

At this stage it should be stressed that swirl is assumed to be absent from the onset flow. As a result the first term on the right hand side of this equation is zero upstream of the blading and in the bypass flow.

Furthermore, it should be noted that equation 4.16 is almost identical to the Clebsch formulation used (without derivation) by Dang and McCune [5], the only difference being that they use a tangential velocity term relevant to the design of rectilinear

cascades in place of rV^ . Additionally, the first term on the right hand side of

equation 4.16 is identical to the expression for vorticity derived by Borges [3] which has been widely applied to the design of turbomachinery blading in the absence of shear flow ([2], [3], [4]).

Now, whilst rV^ on the blading is specified as an input to the design process, rV^

downstream of the blading is determined by the condition that the blade wakes cannot

sustain a pressure jump. Thus substituting F=0 into equation 4.8, rV^ downstream

of the blading can be determined from

W ,;-V 5 % = 0 4.17

the implication of this being that the trailing vortex lines he along the wake streamlines.

At this stage it is convenient to split the velocity field into circumferential mean and periodic components along similar lines to Tan et al [2], viz

For computational convenience the velocity field V is formulated in terms of Stokes stream function, \j/, as follows:-

" ' - i f

" ■ - i f

where is a mean tangential blockage function introduced to account for blade

thickness:-

2tc r

7] = tangential blade thickness

= blade thickness normal to mean camberline (determined from structural considerations)

N^, = number of blades

This model for blade thickness, previously employed by Dang and McCune [8], and

Zangeneh [9], represents the blade thickness as a circumferentially-distributed

blockage. As a consequence blade thickness is only accounted for in the

circumferential mean component of velocity.

Now, the periodic velocity field V is formulated in terms of a periodic potential function 0 ( r,8 , z), as follows. Firstly, using the fundamental definition of vorticity

a = V x V 4.20

it can be shown that

Q = V x V 4.21a

Therefore, from equations 4.21b and 4.16 it can be seen that one formulation for the periodic velocity field is

V = VO - 5(a)VrVe + H V z - z V { H + H ) + xVH 4.222

where 5 (a )^ is the sawtooth function defined in Appendix II, and O (r,0 ,z ) is a

purely periodic potential function representing the irrotational component of the flow field.

Now, since the mean velocity V is represented using Stokes stream function and therefore automatically satisfies the condition

V.V = 0 4.23

the condition of Continuity reduces to

V.V = 0 4.24

which when combined with of equation 4.22 gives

V^O= 5(a)V^rVe + (V rV ^.V a)5'(a) - //V^x - V //-V x

+ xV^(T7+h) + Vx v(77+h) - xv*// + v x - v //

4.25

The one remaining condition for the formulation to be valid is that the mean velocities as given in equations 4.19a,b must satisfy equation 4.21a. Since the radial and axial components of equation 4.21a are independent of Stokes Stream function, this condition may be stated as

(V x v )-êe =[VrVe X V a +V / / X Vx]-Cq 4.26

At this stage it is convenient to introduce the Discrete Fourier Transform (DPT) to represent the tangential variations in the flow quantities (see Appendix I). This approach, previously used by Borges [3] and Zangeneh [4], allows the equations to

2 Equation 4.22 can easily be proved, since taking the curl o f both sides yields equation 4.16, as