A three-dimensional method for the calculation of flow in turbomachines using finite elements on a blade-to-blade surface of revolution

fv o -^ o U1 o^

( - OD

A three-dimensional method

for the calculation of flow

in turbomachines using finite

elements on a blade-to-blade

surface of revolution

1 BIBLIOTHEEK TU Delft P 1836 7194

A three-dimensional method for the

calculation of flow in turbomachines

using finite elements on a

blade-to-blade surface of revolution

PROEFSCHRIFT ter verkrijging van

de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus

ir. H. B. Boerema, hoogleraar

in de afdeling der elektrotechniek,

voor een commissie aangewezen

door het college van dekanen

te verdedigen op

woensdag 30 oktober 1974

te 16.00 uur door

CORNELIS KORVING

werktuigkundig ingenieur,

geboren te 's-Gravenhage

/9h6 7/p(/

Dit proefschrift is goedgelceurd door de promotor

PROF. DR. R. TIMMAN

Aan mijn ouders Aan Marjan

CONTEMTS

CHAPTER 1 : Introduction

CHAPTER 2 : Fundamental flow relation

CHAPTER 3 •• Formulation of the flow problem 19

CHAPTER k : The method of solution 27

CHAPTER 5 : Computation methods and computer program 59

CHAPTER 6 : The two-dimensional case T3

CHAPTER 7 : Results of the three-dimensional calculations

83

CHAPTER 8 : Conclusions

CHAPTER 9 : Acknowledgements 91

APPENDIX 1 : A surface form of Green's theorem 93

APPENDIX 2 : Evaluation of the quantities for n = 2 97

APPENDIX 3 : The collocation method compared to the 101 variational method and the errors involved

References 111

A three-dimensional method

for the calculation of flow

in turbomachines using finite

elements on a blade-to-blade

surface of revolution

1. INTRODUCTION

The problem of analysing the three-dimensional flow in turbomachines of the axial-, radial-, and mixed-flow types is a formidable one, even if it is simplified by assuming the irrotational flow of an incompressible,

inviscid fluid. The various quasi-three-dimensional methods which are used to solve both direct and design problems in turbomachinery, are based on the assumption that the flow follows a known stream surface on which a two-dimensional solution for the velocity distributions is obtained. The methods presented in references 2 and 3, chiefly used for the design of pump impellers, consider the flow on a stream surface, which is generally assumed to be parallel to the mean blade siarface that extends from hub to shroud. When the streamlines in this surface are projected on the meridional plane - that is, a plane passing through the axis of the rotor - a velocity gradient equation can be derived which, combined with the condition of continuity of the flow, serves for the determination of the meridional flow pattern. A similar result is obtained by the methods, discussed in references 7,8 and 9, which are based on the assumption of axially symmetric flow, introducing a

"body force" in the equations of motion to allow for the fact that each element of blade surface exerts a

circumferential force on the fluid. Since this effectively linearizes the variation of pressure from the driving surface of the blade to the corresponding point on the trailing surface of the next blade, the pressure change can be used to determine the blade velocity distribution.

On the other hand, various approximations for the calculation of the blade velocity distribution are suggested in

references 2 and 3, where no axially symmetric flow of the fluid is assumed. The most accurate method, presented in references 5 and 6, assumes axisymmetric flow surfaces, which leads to the problem of a two-dimensional flow on a blade-to-blade surface of revolution. The knowledge of the

flow in the meridional plane is then used to select a streamtube between the hub and the shroud for the blade-to-blade analysis.

All these methods have the disadvantage that the accixracy of the results of the calculations depends on the geometry of the hub and shroud walls, the blade shape and the number of blades. Besides this, it is very difficult to estimate the

error involved. A method of solution for the direct problem has therefore been developed on the basis of a real

three-dimensional approach. Unfortunately, the general three-dimensional theory, presented in reference k, is long and complex. This theory is based on the investigation of a combination of flows on stream surfaces of both types, namely the hub-to-shroud plane and blade-to-blade plane. The equations obtained to describe the fluid flow on these surfaces lead to a solution in a mathematically

two-dimensional matter through an iterative process. However, even when a high-speed computer is available for the computation, the programming of the problem for the computer is a long and tedious process.

Instead of the above methods, which are characterized by the consideration of flows on stream surfaces, a method of calculation is presented on the basis of an approximation

leading to a system of two-dimensional equations in unknown quajitities which are defined on a regular set of coaxial surfaces of revolution. These surfaces are not considered as stream surfaces, but are expressed by their fixed position during the calculation. The three-dimensional approach of the method is shown by the fact that any desired accuracy in the results of the calculation can be achieved by a suitable choice of the parameters involved. The basic flow relation, described by the Laplace equation, is derived from the steady flow which occurs in the system of coordinates fixed to the rotor. This equation must be satisfied in a region between two successive blades, this section being extended upstream and downstream from the rotor. In contrast to the shape of the extensions from the blades in the inlet region, the shape of the corresponding surfaces in the outlet region cannot be chosen arbitrarily, but must coincide with the vortex sheets at the point where the vortices shed at the trailing edge of the blades are

transported downstream with the fluid particles. As outlined in chapter 3, a well-defined boundary value problem can be formulated if the position of these surfaces is fixed as well as possible beforehand. This particular surface is

subject to both the condition of continuity of flow through this surface and the condition of continuity of the pressure. The solution of the system of partial differential

equations, resulting from the collocation method presented in chapter h, is obtained by the Galerkin finite element approximation. The suitable choice of the system of coordi-nates enables the finite element division to be restricted to a bounded area on a blade-to-blade surface of revolution. The computer program developed during this study is described in chapter 5> and the results of the finite element method

are compared with exact solutions for the two-dimensional case in chapter 6. As an example of the three-dimensional method, the blade velocity distribution of two convenient impellers is calculated, in chapter 7. The results indicate that, in contrast to existing methods, a quick and accurate solution can be arrived at by this means.

2. FUNDAMENTAL FLOW RELATION

The fundamental flow relations in the theory of fluid flows are the equation of continuity and the equations of motion. The latter expresses the equilibrium condition which, in the Eulerian approach to the description of fluid motions, must be satisfied at each point in the flow field at every instant of time. For arbitrary flows the equation of motion is given by

where

p = the density

V = the velocity vector

f = the frictional force per unit mass p = the pressure in the flow

-gz = the potential of the gravity field with z vertically upward from any chosen horizontal plane of reference

•T:- = the substantial time derivative with local and dt

convective terms; for any vector u

dR^ ^ i-(V.v)u (2.2)

Two hypotheses are used to analyse fluid flows in turbo-machines j namely:

a) the fluid is incompressible b) the flow is steady.

In spite of the simplification resulting from the above hypotheses,the solution of the complex flow relations with the boundary conditions imposed on viscous flows is actually impossible. We have to consider two aspects of flow:

1) an assumed non-viscous flow governing the largest part of the flow region

2) a wall flow strongly influenced by the viscous effects.

The analysis of the flow, however, will be based on the assumption that the first part constitutes the whole flow field. Accordingly, the boundary conditions at the solid walls are represented by the requirement that the velocity vectors must be at a tangent to all solid boundaries. Thus, starting from the Euler equation for incompressible flows

we now consider the equilibrium of a mass unit of fluid under the effects of the forces of inertia, pressure and gravity. Using the relation

V vv= 1. Vfy-v)-VKfvxv) (2.^)

the equation of motion can be rewritten as:

The working conditions - namely the rotational speed and the total axisymmetric inlet and outlet flow - being fixed, the fluid motion in a rotor will be unsteady when it is viewed from a standing frame of reference. The blades pass across the point of observation, clearly indicating the unsuitabi-lity of using eq. (2.5). On the other hand, the flow is steady with reference to the turning rotor. In order to obtain the equation of motion in this rotating coordinate system, the absolute velocity vector V is replaced by the relative velocity vector w , satisfying the relation

V= i^y- u5xr (2.6)

to = the angular velocity vector directed along the axis of the rotor

r = the radius vector

The substitution of eq. (2.6) into eq. (2.5) leads to the equation of motion for relative flows containing centrifugal and Coriolis terms ; thus

£i!r' _

WK(7x1^^- 2Zo)

y-

j - v(lv. ^)

-f-2i

--^ V(rM><'f').(o3xF)) - - vfR-f-^z)

'2.7)

f

In this equation the gravitational force will be considered to be only dependent on the position z of the axis of

El

rotation of the machine, expressed in the replacement of the pressure p by the hydrostatic pressure P = p + pgz . This means that the effect of the force of gravity in tangential direction is neglected.

Since the flow is steady in the system of coordinates fixed to the rotor, the term involving the local time derivative

can be dropped. The equation of motion (2.7) then takes the simple form:

t 7 ^ _ iv

X (T'x ^ -f- 2 uS)

.

(2.8)

The quantity P_ represents the relative stagnation pressure, n

which can be defined as

It is now assumed that the absolute flow far ahead of the rotor has constant total energy, for instance, in the plane perpendicular to the axis at station 1. (Fig. 2.1)

Since the absolute flow at station 1 must be axisymmetric and steady to produce a steady relative flow in the rotor, the total pressure P. = P... defined by

P^=p.^J.fV' f2.to)

must be constant at station 1.

The absolute velocity V. at station 1 can be expressed by means of the components of the relative velocity W . In the cylindrical systems of coordinates of Fig. 2.1 which turns with the rotor

"K^ = 4

Wu^ •i-T^^'Wa, -hT^Wti , (2.n)

where W , W and W , respectively,denote tangential, axial and radial components of the relative velocity. The components of the velocity vectors V.. and W are only different in relation to the tangential component, so

Yu, = Wui i- oot-j . (2.12)

Hence,

V;^= h// f^ Z CO r^ V^, - oo^r, ^ (2.13)

With this relation, the total pressxire is

P^,=P, +±f>(h/;^-f2u3r,Vu,-uj^r,'-J (z./z,)

Thus, with the definition of the relative stagnation pressure of eq. (2.9), we have at station 1:

PA, = P„, + fuo rjYu, (z./S)

Now, in addition to being axisymmetric, the absolute

velocity V ahead of the rotor is assumed to have tangential velocity components V . which vary inversely with the

radius, or

r, Vuf - consjiant = A> (2./6J

Then, by eq. (2.15), the relative total pressure P^. is con-stant at station 1, since uniform values of P. .. were assumed at the outset. A flow satisfying eq. (2.16) is called a vortex-free flow. The vector w x ( V x w + 2 to) of eq. (2.8) is perpendicular to w. Therefore, the dot product of eq.(2.8) with w is zero and the term on the right-hand side of

eq. (2.8) vanishes. For steady motions, we simply have

^

^y ^o , (2;y)

indicating that the relative stagnation pressure is constant along particular streamlines. However, since all streamlines

originate at station 1, where, with the stipulated

assumption, the value of P ., is constant, the total relative

HI

pressure P„ must be an absolute constant equal to

P„-everywhere in the relative flow field. Thus the equation of motion (2.8) establishes the condition

Vx W = - 2 oo - constant (^J^)

The other possibility of satisfying eq. (2.8) is provided by the case where the curl vector \ =Vxw + 2to is parallel to the velocity vector w-that is, if the fluid particles rotate like solid bodies with the velocity vector as the axis of rotation. This situation only occurs in the vortex sheets where the vortices shed from the trailing edge of the blades are transported downstream with the fluid particles . Assuming the validity of condition (2.18), which specifies that the curl is not zero for steady relative flows, it does not seem possible to express these flows by the

gradients of a potential function. It is possible, however, to change the relative flow field into another flow,

obtained by vectorial addition of the tangential velocity vector i, ur to w. The resultant flow is not the actual absolute flow, but a hypothetical flow, say,

Vfl

= iv -^

ij u> r (2.ig)

where the symbol V. is used to indicate that the flow field V , called the absolute rotor flow, does not correspond to the absolute velocity field V. If w is steady, the velocity V. will also be steady. On the other hand, the velocity V appears to an absolute observer as non-steady, in spite of the fact that a steady relative flow can occur in a rotor

only, if the absolute motion at the rotor entrance is

axisymmetric with V. = V, The insertion of eq. (2.19) into

eq. (2.18), yields

V X V^ = o (^-^o)

This can be seen by expressing eq. (2.20) in an axisymmetric

orthogonal system of coordinates fixed to the rotor.

Figure 2.2 shows the position of the coordinates in the

meridian plane (Fig. 2.1). If

1 2 3

where V , V and V.^, respectively, denote x = 9 , x and x

^ - 1 . 2

components of V , and k = :r— is the curvature of the x

A 2 H2

coordinate curves in the meridian plajie, then the three

relations, one for each unit vector, obtained from (2.20)

are:

' K

0-0.^^.-o

(Z.2O

^iLK)-dyL=o (2.Z1)

^9.2.2

W-'-^-° ('-"^

If the relative flow is assumed to be axisymmetric again

downstream from the rotor - particularly far downstream

from it, say at station 2 - then relations(2.22) and (2.23)

indicate that neither V nor V will change in the direction

of 9 and the value rV must be constant, say

u

r^Ki = Comiant = /Tj . (2.2^)

As i n e q . ( 2 . 1 5 ) , t h e t o t a l p r e s s u r e P. a t s t a t i o n 2 i s

^ 2 = ^ 2 ^ F^r^Vuz . (2-2s)

The value of P„„, however, was found to be equal to the

H2

relative total pressure P_.. Therefore, from eqs. (2.15),

K I

( 2 . 1 6 ) , (2.2U) and (2.25) we f i n d :

'Pai-hi = f ^ (n Vuz - f, yu,) =

(2.26)

= foo fKz - f<i) = con^iiant.

The uniform energy changes of the fluid particles passing

the rotor depend only on the tangential velocity component

at the discharge station. This component can be found by

satisfying the Kutta-Joukowski condition at the trailing

edge of the blades, a condition obviously responsible for

the change in the energy of the fluid particles. Since

the quantity r V must change in the direction of the flow

from r V = K to r V = K equations (2.22) and (2.23)

clearly show that the flow cannot remain axisymmetric

everywhere in the flow field. Hence, V does not correspond

to the absolute velocity field V.

The absolute rotor flow V is "irrotational" (eq. 2.20) if

w satisfies condition (2.18). Obviously this condition must

be satisfied at all stations in the flow field, because the

assumed axisymmetric flow ahead of the rotor involves V. = V and the absolute incoming flow is irrotational (eq. 2.16). For such flows it must therefore be possible to establish a potential function (f> from which V. can be obtained by

Y = V ^ . (227)

relative flow of an incompressible fluid takes the simple form:

V. W = o (2.za)

v.V^ =0, (2.29)

namely the Laplace equation

V^<l> ^ o . (2.30)

However, the simplicity of eq, (2.30) and its similarity to the equation for incompressible absolute flows are

misleading since the flow equation for absolute rotor flows must be solved for more difficult boundary conditions than the flow equation for absolute motions. At the solid walls of a relative flow, the relative velocity vector w must be parallel to the surface and not the velocity V.. Let n be the normal unit vector of an element of surface of a solid wall. Then

n . 17 ^

= n. ij oo r

(2.iz)

In order to formulate a boundary value problem,it is necessary to constitute a closed flow domain, partly bounded by the solid walls under the above-mentioned boundary conditions, and partly bounded by surfaces which must be indicated more precisely. The position of these sxirfaces, and the boundary conditions to be prescribed, are discussed in the next chapter.

3. FORMULATION OF THE FLOW PROBLEM

For steady relative flows of an incompressible fluid in turbomachines, satisfying the condition (2.18), it has been established that the flow equation is represented by the Laplace equation. It contains the absolute velocity potential which is a function of the space coordinates in the rotating relative system only. The flow will be described for the general case of a rotor of the mixed flow type with an arbitrary geometry and an arbitrary blade shape. The metric tensor quantities are introduced,using coordinates

determined by this geometry. The Laplace equation then reads as follows:

Since the absolute flow is assumed to be known at the

stations upstream and downstream from the rotor - the latter as far as the meridian component of the velocity is concerned - it is necessary to consider a flow region consisting of three parts: an inlet, a part with rotating blades and an outlet. Because of the periodicity in the geometry, we shall confine ourselves to a region between two successive blades, and extensions from these blades in the inlet and outlet parts. The shape of the surface lying between the leading edge of the blade and the entrance plane may be arbitrary. The boundary conditions at this surface are only subject to the condition of periodicity,' specified by the equality of the normal and tangential velocity components at

corresponding points of the two successive surfaces. The corresponding surface in the outlet part, however, must actually coincide with the vortex sheet containing the trailing vortices generated at the trailing edge of the blades because of the change in the circulation round the blades from the hub to the shroud walls of the rotor. In addition, since the vortex sheet is a stream surface whose position is unknown, the shape must be assumed prior to performing the calculations, and the resulting shape must be checked with the assumed shape to see whether improved approximations are necessary. Thus, if the shape of the surface between the trailing edge of the blade and the discharge plane is assumed to be known, it is initially only possible to impose boundary conditions specified by both the condition of continuity of the pressure over the assumed surface and by the condition of identical normal derivatives at corresponding points of the two successive surfaces. The Kutta condition must be satisfied at the trailing edge of the blade. This practical condition is fulfilled by the requirement that the stream surface or vortex sheet at the trailing edge must correspond to the smooth extension from the mid surface of the blade in the outlet region.

The boundary conditions can be summed up successively as follws:

1) At the blade surfaces (Fig. 3.1) .'

•n'' (jijl =- U) f cos oc

.

f-3-z)

Ihis condition comes from eq. (2.32) where the inner product of the unit vectors normal to the blade and normal to the meridian plane is replaced by the cosine of the angle between the two normals.

2) At the hub and the shroud of the rotor:

n'-<j>,L - 0 . (3.3)

since these surfaces are coaxial surfaces of revolution this boundary condition is not influenced by the

rotation of the rotor (eq. 2.32).

3) At the entrance and discharge stations 1 and 2 (Fig. 3.1)

^'<^,i = f. A ^ ;

particularly the component perpendicular to the entrance and discharge planes is,assumed to be known. As explained in chapter 2, since the flow ahead of the

rotor must be axisymmetric and the absolute flow is assumed to be irrotational, the function f cannot be given arbitrarily. From eq. 2.21, which represents a vanishing component of the curl of V in an

axi-symmetric coordinate system, it follows that f must be a constant at the entrance station (1) if there is a parallel flow between the straight cylinders in the region ahead of the inlet. If the term V_k_ is

neglected, a similar result can also be obtained at the discharge station (2) as a result of the small

curvature kp of "the walls. In addition, it is

necessary to satisfy the condition of continuity of the total flow passing through the rotor which is expressed by

\7

./•

/

/ '

h) At the surfaces S. formed by the extensions from the blade surfaces in the inlet region:

These conditions represent the requirement of periodicity: the velocities are identical at

corresponding points of the two successive surface S., denoted by the "trailing" surface (t) and the "driving" surface (d). This equality can be transformed into the condition that the normal derivative of i^ (eq. 3.5) and the derivative of ((> with respect to an arbitrary curve s at these surfaces are equal (Fig. 3.2). Since the tangential velocity component at station 1 is known, condition (3.6) can be simplified. When this component is equal to zero, which is usual, it is expressed by identical values of the potential at the corresponding points P and P' at the lines of inter-section of the surfaces S. and the entrance plaxie.

1

The identity of the potential for all corresponding points at the surfaces S. then follows immediately from condition (3.6). Thus,condition (3.6) can be replaced by

5) At the surface S formed by the extensions from the blade surfaces in the outlet region:

aJ n'<j>,Jt = -n'i>,cld (3.8)

4/ Pi ^ Pu . P-9)

As already mentioned, the shape of the surfaces S is fixed from the outset, so that it is impossible to prescribe conditions which ensure that the surfaces S act as vortex sheets and stream surfaces at the

o

same time. Now, a well-defined problem can be formula-ted when the condition of continuity of the pressure over these surfaces (3.9) is combined with condition

(3.8). These conditions are necessary, but not

sufficient to permit the solution of the flow problem to be found. This actually means that the method of calculation on the basis of above boundary value problem will produce values for the normal derivative of (f) at S which generally differ from the values belonging to a stream surface. In effect, an iteration process has to be carried out, improving the shape of the surface S from the previous calculation until the

o

position of the surface S coincides with a stream •^ o surface.

Condition (3.9) can be transformed into a condition containing ^ . Using the result, obtained in the preceding chapter, that the relative total pressure P_

H is an absolute constant everywhere in the flow field, condition (3.9) combined with eq. (2.9) can be

transposed into the condition of identicaJ. magnitudes of the velocities at both sides of the vortex sheet S

In spite of the fact that the direction of the velocities is different, condition (3.6) can also be used if the curve s is formed by the bisector of curves s, and s, , representing the direction of the velocities

d u

at the driving and trailing surfaces respectively (Fig. 3.3).

Since the difference of the potential at points R and R' - i.e. the points at which the curves s intersect with the discharge plane - is determined by the tangential component of the velocity, condition (3.9) can be finally transformed into a condition along s:

^^ (^i = <i>d + % , (3. to)

indicating an unknown quantity q which follows from the Kutta condition at the trailing edge of the blade. Generally speaking, the quantity q will be a function

3

of the coordinate x at station 2.

A boundary value problem in terms of eq. (3.1) subject to the boundary conditions (3.2), (3-3), (3.it), (3.5), (3.7), (3.8) and (3.10), has now been formulated. The uniqueness of the solution of this problem can be proved by means of Green's theorem which reads as follows:

ft.n)

jj fn.'-<^,idS

.

Let (j). and (j)_ be two solutions of the problem. Then the boundary conditions for (jt =({> -ij) are:

1) n V * = o ,

replacing the conditions in(3.2), (3.3) and (3-^);

2) ^^Vf,/^ = _ ^ . > * ^ ^

representing the unchanged conditions (3-5) and (3.8);

3)

(^.f. =.(^^ ,

stating the conditions for both (3.7) and (3.10).

Substituting in (3.11) both ^i/ = i^ = ^ and the conditions derived above, we finally obtain

Iff f^</>*^ </>,* d V = o

2)

The integrand is everywhere positive, except when it is zero. Thus, a positive integral is proved to be equal to zero, which is only possible if at every point inside S

V <f>* = o ,

k. THE METHOD OF SOLUTION

As pointed out in chapters 2 and 3, the solution must be sought for the boundary value problem governed by the Laplace equation in general curvilinear coordinates x'(i = 1(1)3)

subject to the boundary conditions (see Fig. k,^

?i'p,l=u:>r cos oc (e<}. 3.z) ai S/^

-n^'^.l = O

r o.

^

4-\- ^^ X

'^i-^ ^d-^

^(^-YV A ? -

3-7O)^

f^.z)

(e^.3.ij

a / ^ . ^

(^.3)

{^•^)

(^•7)

On the one hand, the solution of this boundary value problem can be based on a direct method of numerical integration of the Laplace equation and, on the other, it can be based on

V \

<

I / I . A

cO

the method of solving the equivalent variational problem. When all boundary conditions at S are given by relations of the nature (U.U), the variational problem is formulated by the problem of minimizing the functional:

where D is the domain enclosed by all boundaries S (Fig.i+.l). In order to find a solution to the above problem, the

extremal principle may be used with the Ritz finite element approximation. A set of linear equations in the model values can then be derived.However, the solution of this enormous set of equations requires too much computation time, especially when an iteration procedure is used in order to satisfy the Kutta condition.

This difficulty can be overcome by adopting a suitable approximation for the potential function (ji beforehand and, in addition, by making a suitable choice of the system of coordinates. This method will be outlined below.

A regular set of coaxial surfaces of revolution is located in the space between the hub and the shroud of the rotor.

3 These are denoted as coordinate surfaces of constant x

3 3

at which x = x . (j = l(l)n), where j = 1,n corresponds to J

the hub and the shroud respectively. The class of functions <!). at x = X. is defined, representing the potential if- and

J J o J its derivatives with respect to x up to and including order

p. The functions w- are only dependent on the surface coordinates x , a=1,2. The potential cj) at an arbitrary

3 3 point in a subdomain (j) lying between the surfaces x = x.

and X

3

^J-1

x^.. (Fig. k.2) is assumed to be a polynomial in x with coefficients containing the defined quantities V- and

i(r , . The coefficients of the Hermitian interpolation polynomials in x are calculated from the condi-tion that the approximated

shroud

Fig.4.2

potential and its deriva-3 tives with respect to x up to and including order p correspond exactly to the quantities ^. and

P "^

i). , on the boundaries J + 1

3 3 * 3 3 X = X. and x = x . .

J J + 1 respectively. I f p is equal to 1, the expression of t h e potential in the region (j) takes the form:

•^/V./

-

- /^^^..^

^tJ^'(x^)</>t''(x'y)

.

A',

fo)

H e r e , the quantity i^ represents the derivative of (j) with

3 ^ 3 3

respect to x at the surface x = x.. The polynomials 1. and

J 1

m. (i=1,2) are determined by the four conditions:

k-/v

The calculations produce polynomials of the third degree,

expressed by:

1^"^^ = 2(u-^)2(u+l)

l[^^

=-2(u+J)2(u-l)

{h.^^)

m\^^

= 5(u-i)(u+J)^

where (see Fig.

k.2)

^ 3 3

0 = x.^, - X.

J + 1 J

, - X . + X .

and u = 1 (x3 - - ^ ^ - ^ ± 1 )

In order to use the approximation U.10, which is

inten-ded to reduce the dimension of the problem, the metric

quantity v^g

^

and f (eq. ^.9) must be approximated by

expressions of the same kind as (U.IO) when the

variational method is used. The integrands of (U.9) then

3

.

.

.

contain known functions m x and the integration with

3

respect to x can be performed for each subdomain (j).

This operation is very laborious, especially when the

position of the blade surfaces and their extensions in

3

the inlet and outlet regions depend on the coordinate x .

In this respect an important improvement is achieved if

the system of coordinates x is completely defined by the

geometry of the rotor and the blade shape: the x , a=1,2

coordinate curves with respect to

zhe

geometry of the

3

rotor and the x coordinate curves with respect to the

blade shape. This will be clarified with the help of

Fig. U.3, which shows the axisymmetric system of

coordinates, drawn at a particular surface of revolution

3 3 1 .

x = X.. The surfaces x is constant will be half planes

J

through the axis of rotation of the rotor. They are called

meridian planes, determined by x = 9 is constant, where 0

is the angle measured from a fixed x plane. As defined

earlier, the coaxial surfaces of revolution, denoted as

3 3 3

X = X., are surfaces of constant x . Consequently, the

.'^

2 . . .

coordinate curve x , for instance, through R, is fixed by

1 3 3

the intersection of the surfaces x = 9 and x = x., whereas

3 o J

the coordinate curve x , determined by the intersection of the

1 2

surfaces x = 9 and x is constant, may still be positioned

arbitrarily in the meridian plane. However, in order to

satisfy the requirement that the intersections of the

3 .

boundaries of the flow region D with surfaces x is constant

1 2

are given by the same values of the x , x coordinates, the

3 . . .

X coordinate curve through a particular point R must

coincide with the intersection line RS of the blade surface

and the meridian plane. In addition, this is connected with

the easiest way to represent an arbitrary blade surface

-that is, to intersect the blade surface with planes 9 is

con-stant and to show these curves in a particular meridian plane

X = 9 . These curves are then labelled 9 is constant, now

.°

. .

3 .

also indicated as the coordinate curves x . As illustrated

in Fig. ^ . 3 , the line along which the blade intersects with

3

the plane 6. is constant, denoted as PQ, generates the x

coordinate curve P'Q', labelled 9 is constant, if PQ is

rotated into the meridian plane under consideration. Moreover

2 1

, if it is stated that x = x on the blades, the position of

the point P' is then completely designated by the values of

1 2 3 1 2

the the parameters x , x and x , namely x = 6 , x = S-IJ

3 3

°

4

Accordingly, the integration of eq. (U.9) with respect to x need only be performed for one subdomain (j). As a

consequence of the identical integration limits with respect 1 2

to X and x for each subdomain ( j ) , the result for the complete region D is obtained by a simple addition of all contributions under the integral signs. This result, now related to the two-dimensional domain S at a particular

3 3 ^ . . surface x = x., enclosed by the curve r , can be written m

J the general form:

J^*. (<^',^]—.(^^j ^L, J4.^-•'$4 / =

Cjf K^ ^u",

^ «;:

i>f.

^ ^ - -

r^.'Z)

-—+^y^i>^^''fd,'d.^^

r

If the Ritz finite element method is used to solve the variational problem stated in the form (i+.12) it can be concluded that some progress has been made in comparison with the three-dimensional version of solving eq. k.9' Now, it is only necessary to divide the two-dimensional domain S into finite elements, resulting in a considerable reduction of the number of nodes. But as far as the size of the linear system of equations is concerned, the system remains large, for one nodal point k contains the 2n unknowns (j>>i and

(f) (j = l(l)n). If it is possible, to transform this large Uk

system of equations into a diagonal form, however, a process of successive iteration could be used, thus considerably reducing the computation time. Since the structure of the expression (U.12) is too complicated and, moreover, the

calculation of the coefficients a., etc., is too laborious, Ji^

the collocation method may be considered as an alternative to the variational method. Starting from the same

approximation (it. 10) and the same system of coordinates, this proves to be more successful. The collocation method implies that the Laplace equation has to be satisfied on the

3 3 defined set of coaxial surfaces of revolution x = x.. In

J order to obtain these partial differential equations, the expression (ii.lO) is substituted into eq. (U.l), without the factor 1/>^, producing the equations at the boundaries

3 3 3 3

X = x. and x = x. ., . Since the subdomains (j-l) and (j)

J J ' o o

are separated by the intersurface x = x. (Fig. k.2) two

o

partial differential equations are formed for each surface 3 3

X = X. (j=2(l)n-l), which are only different in relation J

to the term concerning the second derivative of (ji with

3 . . . . . .

respect to x . If the continuity of this derivative is 3 3

guaranteed at the intersurfaces x = x. (j=2(l)n-1), J

leading to the n-2 relations

^

r/''- ^^•')= ^f'. ^ ^^ ^ ^^''

,

(^. ;3)

then the two partial differential equations belonging to one intersurface will be interdependent. Subsequently n

independent partial differential equations are obtained when the equations from the boundary surfaces j=1 and j=n are combined with the equations collected by the addition of the two equations for each intersurface.

where, with ( /gg ) = G,

|--k

U^^ = G^^s!

v.. = (

G^^). 6^

8x-

^ij = ^( - ^ ^''^i

1 0 -1

2 1+

3x

2 G. } 5.

Next, two equations are added to the system of equations

following from the boundary conditions at the hub and the

shroud (eq. U.3). These are written as:

(^•7S)

If the X coordinate is perpendicular to both surfaces of

revolution we simply have:

The equations (U.13),

(k.lk)

and (U.I5) represent a complete

system of 2n equations in the 2n unknowns

(

j

j and

(\i ,

j = l(l)n.

The boundary conditions, which have to be prescribed on the

3 3

bounding curves enclosing the domains at each surface x =x.,

J

follow from the unused conditions (U.2), (U.U), (U.5), (U.6),

(U.7) and (U.8). Since the bounding curves are given by the

1

2

.

same values of the x and x coordinates, the suitable choice

3 .

of the x coordinate curves, enables the boundary conditions

to be imposed on one closed curve r . Fig.

k.k

shows the

domain S enclosed by the

curve r , and consisting

of three main parts:

AA'BB' the inlet, BB'CC

the part bounded by the

blade curves BC and B'C

and CC'D'D the outlet.

The conditions can be

defined as follows:

Fig.^.i

1 ) BC and B'C (eq.

k.2):

f'n'<^,J =corQ) cos cc^yj ;

_f^-^7)

2)

AA' and DD' (eq.

k.k):

f^'S)

f^.zo)

3) AB and A'B' (eqs. (U.5) and (^4.6))

V

fib:

r^V>^/=

/«>^')

^^ =^'(xy)

k)

CD and C D ' (eqs. (U.7) and (i+.8))

In these conditions the unknown functions g and y are

introduced to indicate the relationship between the boundary

values at those points of the curves in question at distances

of the pitch t, which is equal to 2Tr divided by the number

of the blades, in the same way as P and P' in Fig.

k.k.

It should be noted that the boundary condition, containing

q (U.20) does, in fact, hold good if the surfaces x =x.

J

coincide with the surfaces found by turning the curves s from

condition 3.10 round the axis of the rotor. Thus, by

prescribing condition U.20, the position of the curves s is

assumed to be known initially, but the possibility of

altering the shape of the vortex sheet after a previous

calculation can at the same time be combined with an

3 3

improvement m the shape of the surfaces x =x. m the outlet

region. The unknown constants q must be iteratively

determined by the requirement that the Kutta condition now 3 3

has to be satisfied at each surface x = x. (j=l(l)n). J

The solution of the system of partial differential equations - under the condition that the constants q are assumed to be known initially - is obtained by applying the Galerkin method. An approximate solution is then expressed in the

form:

^%U5^ = ^;/'?^>V

and

s^/(f)N

/ = t(f)n

(^.2j)

^if^uv=^sp'f^y),

where p (x ,x ) is a certain system of functions, in the form of polynomials in the finite element representation, and

i i . . . . s a , b are undetermined coefficients. The functions p are

s' s

considered to be linearly independent and to represent the 1 1 2 first N functions of some system of functions p (x ,x ) ,

{ l = 1 , ..., N,...| which is complete in the given region. The approximate solution is found by the requirement that

Lc(f>'>—-J^: <^:,—-J:) = 0, Ut(f)n (^.2ZJ

where L. is the operator belonging to equation i of the system (U.lU), combined with equations (U.13) and (U.l5) under the boundary conditions (U.I7 - U.20). According to the method of Galerkin , by putting conditions (U.17-U.20) into the general form

the above boundary value problem is tranformed into the

equivalent problem based on the requirement of the

ortho-gonality of expressions (U.22) and (U.23) to the N functions

s 1 2

of the system p (x ,x ). Stating these conditions, we arrive

at the system of equations:

ff L, /d.'d.^-foc,^ f (^-i,)'- f^J /d^ =o

^c r

s=r{0/V (4^2^)

C.J. = rffjfi

which, combined with equations (U.13) and (U.I5), serves for

the determination of the coefficients a and b .

s s

The system of equations

{k.2k)

is transformed by means of

Green's formula

i-///i;jG'U..xjdx^cix^d^^=.

{^.25)

In order to obtain the two-dimensional representation, this

3 3

IS applied to a region bounded by the surfaces S : x = x.

3 3 c 1

and X = x- + e and the surfaces S spanned by the bounding

curves r of the surfaces of revolution between x = x. and

3 3 . . I

•

"

"

X = X. + e. Taking the limit lim — (see Appendix 1) we

e^O ^

arrive at the result:

where m dn is connected to the formula defining an element of area do of the surface S :

d(r =.'\/a dv dx^. (^2?)

I n p a r t i c u l a r , i f G/. % (jj, ^ i s r e p l a c e d by t h e q u a n t i t y

a6 J s .

• <J>„ P J which occurs ] ij ^'6

U.26) can be rewritten as;

U ? 4„ p , which occurs in eq. {k.2k) then the formula

(^28)

^ 3= iCr)N oc,jS =f,z with

Fig.i.5

where g is the angle

between the unit vectors

in the x direction and

the tangent to the

curve r (Fig. U.5), then

the substitution of the

relations (U.28) and

(U.29) into the system

of equations (U.2U)

leads to a system of

n X N equations:

f^.^o)

r

ss= ?Ci) N

indicating that only terms up to and including first

derivatives occur. Now, the complete system of 2n x N

equations in the unknowns a and b is obtained by

substitu-ting the expressions (U.21) into equations (U.13), (U.I5)

and (U.30). In contrast to the result obtained by the

variational method (U.12), the result of the collocation method proves to be very attractive for developing a procedure of solution based on a process of successive iteration. The system (U.30) then has to be transformed into a system of dominant diagonal subsystems. In effect, such a subsystem i(i=l(l)n) consisting of N equations can be solved independently of the remaining n - 1 subsystems, but only on condition that this operation is part of a process of

successive iteration. The transformation to the new dependent variables \p , outlined below, has the advantage that the quantities ^ can be eliminated at the same time. Moreover, since the size of one subsystem just corresponds to the size of the system of equations when two-dimensional rotor flows - similar to n = 1 in the above problem - are considered, it is only necessary to determine one constant in relation to the Kutta condition. This is done by an interior process of

iteration, which is a simple operation as compared with the exterior process of iteration required to calculate the n constants q belonging to the complete system of equations. In order to obtain the desired transformation the first two terms of eq. (U.30), which are the dominant terms, are considered and the eigenvalues and eigenfunctions are determined from

combined with equations (U.13):

C,^ fj

=

j - B,j j>^ i^ Z(r) n-r (^.iZ)

and the homogeneous boundary conditions (eq. U.l6):

^u"'=a yn=1.n (^-33)

where C. • = A. . i 1^ j

and

= k

i = j.

The relationship between * and

i,

follows from the

^ ^u ^ u

condition (see eqs. (U.13) and (I4.32)):

r 1^ r

^ V

L^zfrJ-n-i

/ = ///;

n

the homogeneous boundary conditions for

^

(eq. U.33) and

the inhomogeneous conditions for

^

(eq. U.I5). This means

that

r !'< r I*'' V 'dp L,k^z(i)TX'i

where

R.. = 0, except for

^21 = «l''/S''

R , = G ^^/G 23

n-1,n n n s

or

k'=4u'.Ky^. X,^ = i(l)n (^.3^J

with R. = C R. .,

'^.

^•^

i,k,m=2(1)n-1

5-^ 0 , ^ = ? 0-1(1 h

and 22

* 1

^i 1

R. = - - - L ^ 5^ 1 = 1 ,n .

J 1^ 33 J

j

~ j * j

If the eigenfunctions for (j) and (ji are expressed in the form

£/

= (^Z

CO^Cy-O/i^ (Z/.3S)

and

^ 4 = ^/

^^'^(J--')N {'^•^^)

respectively, where Eu!: satisfies the homogeneous boundary

conditions if B ='"• , then the eigenvalues can be calculated

after substituting eqs. (U.35) and (U.36) into eqs. (U.31)

and (U.32). If follows that

X^ = -2 ^-jo^ft^

/V.37)

-^ Z -f- cos /3^ ' '/

1 2

and the relation between the constants C and C becomes

In addition,

/3, = JLfJlll . r^.,s}

-1 *-i 1

By performing the transformation from cj) and (j) to

^

according to:

the system of equations (U.30) can be reduced to a system with dominant terms of tp for 1 = m in the subsystem m after multiplying the system with the transposed matrix of

eigen-=^)"

m * m

Then, after inserting expression (U.3U) in eq. (U.30) as well functions (E^) which is equal to E^ because of its symmetry. Then, after , we obtain:

-2(u-,r'^Q^^^')l^.}dx'd.^,

s =. f(t) N

fn,i= / {')n

IV, iA : U-^ ~ G^.j Jl^^j (fcoj. E(f)

TT-'

_ r ' 7/" r '^'

Q., = E:u-iR*;Ei

F^

=

E^

DC,y

f^

If expressions (U.21) are used, combined with eq. (U.Uo), so that

f'(x',x')= ^f p^(x'y) i=rO)n (I^.^z)

we then arrive at the system of equations:

which serves for the determination of the n x N coefficients

1 rs s ip . The system tensor S ., and the load tensor T are derived

r ml m from eq. U.Ul, thus

in which

r

and

r

r s

Now, the system tensor S ^ contains dominant terms for 1 = m, ml

as can be seen from the different terms which occur under the integral sign. The terms of the form E X. . E!; are found

m i j 1

to produce the l a r g e s t values for 1 = m. This i s shown for

the special case X.. is constant. Then

i j

n.

= C2^ cas(m-l)/ii coi {•£-l)/ii -O m^ i , i^-^-^ odd

— C m =^ ^, ^77,'/ ei/eh

-•^CZ^ I Cos(nn-f-z) (3- + cos(m-.i)pi_j ~ Cn Z^m + t,n

" ' =iC(y,.l)

With : f3i = Ti ELL.

following from

n .

21

COSrts ELL = ± {1 4-(-l)'\ O <S< z(»-l) .

In view of the fact that the coefficients of the diagonal elements m = 1 exceed the other coefficients, the solution to the system can be found by the method of successive approximations. This involves the successive solution of n subsystems of each set of N equations of the form:

(m) rf' S

S m(m) ft -

'«•

(^°

-S"*^ '"'

"^) f^-^^)

and

T: ^J(F^ . Q^e % n^^v^V^ ^

'''

JJ{(Z^z*p'p'- lE^* p'^plfi -f.

r.s =f (,) N

To simplify the calculations, the subsystem has been arranged —rs —sr

in a symmetrical form, indicating that S = s . The mm mm

tensor are therefore transposed to the right-hand side of the equations. These terms do not disturb the diagonal configuration of the equations because they are connected to the skew system of coordinates with |g„_| < /g 800> and hence are small in comparison with the principal terms which are represented by -Z and U

mm mm

By virtue of the fact that the proper values X are always non-positive, it is evident that the equal sign of both terms strengthens the diagonal form of the equations. Moreover, these terms enable the behaviour of the system t be investigated for large values of n.

1 . 3 3

Since 6 = — r , if x. = 0 and x = 1, the dominant quantit

n-1 ' 1 n ^ in the equations is represented by the term containing Z

mm knowing that the eigenvalue X (eq. U.37) tends to -k for

m a large value of m, we conclude that

/n-^-n

'

From eq. (U.UU) it immediately follows that:

clearly indicating that for a definite number n the values of the unknowns ^ are continually reduced with increasing values of m.

On the other hand^the behaviour of the system in the neighbourhood of m = 1 is governed by:

^^m ^J^ =-n^(m-j)^^/y(<^') ^

now showing the finite value of the term relating to Z ° "= mm If m is equal to 1 this term even vanishes, in this case transposing the dominant values of the diagonal terms to the term relating to IT; . Considering the terms in eq. (U.Ul), we conclude that only terms containing - ^ remain,

1 3x"'

because Eu": = 0. This indicates that ij;! can be determined except for one arbitrary constant,which is in agreement with the conclusion already arrived at when the uniqueness of the solution came up for discussion. There it was proved that the solution of the three-dimensional potential (|) could be determined for all but one constant. In the approximated solution it is obviously expressed in ipl- , resulting in the dependency of the equations in subsystem 1. If ij; . is put equal to zero, then the last equation can simply be cancelled from the first subsystem of equations. It must be remarked that the second term in V.. (eq. U.Ul) requires

1J

some attention, because it is divided by 5 . However, a closer investigation of this term verifies that B. R. is

iq J of the order of fi which means that no difficulties can arise with respect to this term.

Until now the term relating to F in eq.(U.UU)has been assumed to be known. The quantity F , however, expressed by

(eqs (U.23) and (U.Ul)):

/ v „ - C ^ v

(^^V.-^/

in which

is known on a part of the curve f . Fof instance, on the blade surfaces we know that

(v''(t),k)^ =^ U)

f^J

COS

<V^J

= LO ]/jJ,(^) COi o(^) .

Fig. U.6, showing the blade surface, will next be considered in order to find an expression for F which contains known

m 2 quantities. Now, the quantity -^. /g . cos eu dx'dx dn

J ' IJ J

embodies the volume of the parallelepiped spanned by the

. . . 1 3 . .

infinitesimal elements along the x , x and n axis, since the surface element ^a". dx dn on the blade is multiplied by the

1

height of the parallelepiped v€"< < • cos a.dx . Since dn cose.=

p ' ' J J o o J

dx , in order to produce the same surface elements in x =x . , the volume element dV becomes:

/ / " / COS 6. '

which is equal to the value of the volume element expressed in the curvilinear coordinates x , generally written as: dV = >-§". dx''dx^dx-^.

J Hence

Err, = U)E^(\/^ COSe)l '

/VW

A similar result can be obtained for the entrance and dis-charge curves. Then

or

At the remaining parts of the curve (see Fig. U . 7 ) , however, the value of F is unknown. To avoid the difficulty arising from the complexity of the system of equations (U.UU) when the unknowns on the right-hand side of the equations are transposed to the left-hand side, the relationship between the quantities at corresponding points of the boundary at distances of the pitch is used initially. Since the normal derivatives at these points are opposite each other - for instance r. and r. of Fig. U.7 - the term F of T (eq. U.UU)

1 2 m m ^

is eliminated if the elements of row s = r. are added to the r r

elements of row s = rp, on condition that p 1 = p 2. Moreover , it is possible to maintain the symmetry of the system matrix if at the same time account is taken of the other condition, expressing the equality of the unknowns \b ^ and

m r o . . . ifi ^ except f o r a constant in t h e outlet r e g i o n . T h i s a c t u a l

-elements of column r = r,, resulting in a symmetrical system of equations in which row T„ and column r^ are cancelled. It is evident that the final system of equations for each subdomain m consists of N equations reduced by a set of equations in accordance with the nimiber of nodal points which are located on the curves AB and CD (Fig. U.7). We shall now consider the region S , shown in Fig. U.7, divided into triangular elements. Within a finite element e, the approximated function i//" of eq. k,k2 is defined by a linear function which is only dependent on the nodal values at the nodes r = i,j,k belonging to the element e. The "shape" function p is then described by

/3^= (a^ +• a, X ^ aj_ x J/ZA

_"=

_{''E'^ ,}

=

=

t

with the other coefficients a. etc. obtained by cyclic permutation of the subscripts in the order i,j,k and where

2 A - dei

1 Xi z

Xk

= 2 (area of itiatio'e. iik)

0 and s r . r The coefficients of p satisfy the condition p

1 . ^ . therefore the nodal values of -^ now define the function uniquely and continuously throughout the region. The

integration of the relations in eq. (U.UU) can be carried out with the known function p as soon as the quantities Z

U'^'^etc are taken as constant within the elements. mm

Using the results

JJ p^dx'dK^

= f 4

e

JJ f>'f>'dx'dx^=jL(r^r')A

—r s

the contribution to the system matrix S of an element e is mm

S^m

= -

:JJ^Z^„

(f-i-6 Ji-^ lE^I al, a^ ^jL Q„^ a^ ^a^.

— s

The matrix T can be foiind in a similar way. In particular, "^ . . -23

the terms containing H ., and U , become ml ml

f: ^SH^.aEff.iU'Zafey/.--- (e^.l).

since the system of equations contains integrals taken along the boundary r , it is necessary to add to the above terms the contributions of the elements which actually form the boundary, like p,q,k illustrated in Fig. U.7.Consider, for

-23 example, the face p-q of the element.If the quantities U ,

mm F and Q are taken as constant along that face and

al

7>»

then the extra contributions to S and T (see eq. U.UU) are,

mm m ^ ' respectively:

and

iE[^„d d (x^-xp)

i^-<^'J%-V.)-iz Q^i<fJ^''(H-Xp)'

In this, the term containing F may be written as:

m

io^E:\f,d yxl-x',) Uj(,)n

for the blade curves (eq. U.U5), and as

i Fn, d^ fx^ - Xp )

for the entrance and discharge curves (eq. U.U6).

The final system of equations is obtained by the assembled

contributions of all elements. As pointed out earlier, one

constant q^ with respect to the Kutta condition has to be

determined for each subsystem m. Then the quantity

AF^ = F^ -LoE^ (\/fcose)i f^-^j)

is considered in the node located immediately behind the

trailing edge of the blade, for instance r = r_ in Fig.

U.7. The operation to be carried out by the computer

Fig.4.8

until the value of AF is near zero, depending on the m

desired accuracy. This is done by a numerical equivalent of Newton's approximation in order to obtain better solutions from early solutions. The errors between the required value of AF = 0 and the predicted values of AF at q^., a and q^ are obtained as the result of a simple linearizing

technique (see Fig. k.8), but the error and value of q^ are stored before computing q^ ,and its corresponding error in AF . Then the value of q^, is found by linearizing between the results for q^ and q.^_ and so on. The calcula-tion method was now applied to two impellers, particularly for the case n = 2. The quantities used in the computer program are derived in Appendix 2. Next, a comparison of the results of the collocation method and the variational

method is breefly presented in Appendix 3 for a special case. Nevertheless it is again emphasized that the collocation method is preferred to the variational method.

Briefly, the reasons for this preference are as follows: 1 ) The collocation method has a wider range of application,

problems, like rotational and compressible flows. 2 ) The variational method is more laborious than the

collocation method.

The difference in accuracy of both methods, however, shows a slight advantage for the variational method (see App. 3 ) , but this is not important, since the minimum value of n

(is equal to 2) gives quite accurate results when the collocation method is used.

5. COMPUTATION METHODS AND COMPUTER PROGRAM

This chapter describes the computer program with the various computation methods used to calculate the velocity

distribution in an impeller of a given geometry and blade shape. The data for the program, a flow chart of which is shown in Fig. 5.1., consists of the total flow, the speed in r.p.m., the number of blades, the average blade thickness and the metric quantities g-.. Since the geometry of the

1 J

hub and shroud locations and the mid-surface of the blade for various values of 6 is constant are given by discrete values of the radius r and the axial coordinate z, initial calculations must be carried out to obtain the metric quantities g.. in the x coordinate system introduced. If the coordinate surfaces x is constant are known as

-1 -2 functions of the cylindrical coordinates x = 6, x = z and

-3 . . .

x = r ,then it is possible to calculate the components of the metric tensor g from the relation

dx' dx'^

-^ dx' dx' -

/V ,1

where g^ . = 0 for i ?i j , g^ .^ = r and "i^^ ^ ^ 3 3 = 1-In practice, however, it is impossible to specify the relationship between both coordinate systems without the help of a graphical method. To demonstrate the scarcity of data with respect to the curvilinear coordinates x an impeller is considered in Fig. 5.2 of which the hub and

3 3 shroud walls can be denoted as surfaces x = 0 and x = 1 respectively. The blade of the impeller is represented by

START

i

I

INPUT HftTft

COf^FI GUfiA TlOf^ \oF FiNirt eL EM BAITS

f \CAlCUlflrrOA/ Of THE 3}£Fr/^xr£ QUANnriEs FOft £fiC» £L£M£/jr

'

\

ave/^Aii sysTsr^

\

(SIfVV) OF EE At^D STOUE TH£ MATfiJCFS

E/<-TA^^

/E;'c,f

'

_ _

\

'

f: = Pi. XI

t

SocuTiof^ OF Tf/a

£(fUArZO/VS OF

5^asysT£F1 ^fvE')

'

liF„i>e

'

r

' "

lA4l>e

OUTPUT DATA

,

fiq. ST./

curves S = constant and, as pointed out in the previous

3

chapter, these curves are indicated as x coordinate curves

2 . 2

for which X can be defined as x = c. + c_9 , but m the

2

inlet and outlet regions the curves x = constant are

unknown initially. A graphical method is therefore used to

obtain the quantities g... Then, for instance, the metric

quantity v^pp is obtained from:

where ds represents the length of the element of arc along

the coordinate line dx . If the v^pp/x relationship is

considered (Fig. 5-3), only the curves between the leading

edge and the trailing edge of the blade can be drawn; for

2 2 2 2

the hub, these are between x = x,, and x = x , , and for

2 2

^\ 2

* .

the shroud between x = x ., and x = x ^. To obtain the

2 Is t

set of curves x is constant in the inlet region, the

following conditions are imposed

0 J vj:, dx^

=

5,-^^ £^ ^ L. =- •

X sX Q

In the first condition, s. represents the length of arc

along the hub and shroud locations between the entrance

2 2 . 2 2

station X = X and the leading edges x = x ,, and

2

2

^

.

. ^^ .

x = X respectively. Since the shape of the extension

from the blade in the inlet region may be arbitrary, the

curve may be chosen as smooth as possible in that region.

2

x^'^x^.

\ \ / \ x = 0 . 2

Fig.5.2