Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 15, N

o
 3, 2017, pp. 479 - 493 

https://doi.org/10.22190/FUME170911026B  

© 2017 by University of Niš, Serbia | Creative Commons Licence: CC BY-NC-ND 

Original scientific paper 

 DETERMINATION OF AVERAGED AXISYMMETRIC FLOW 

SURFACES AND MERIDIAN STREAMLINES  

IN THE CENTRIFUGAL PUMP USING NUMERICAL 

SIMULATION RESULTS 

UDC 681.5 

Jasmina Bogdanović-Jovanović, Dragica Milenković, 

Živojin Stamenković, Živan Spasić 

University of Niš, Faculty of Mechanical Engineering, Serbia 

Abstract. One of the most important aims in the turbo pump design is to achieve an 
optimal design of the pump impeller. The basic assumption in the design procedure of 

the impeller is that of the axisymmetric fluid flow. It can be confirmed or disputed by 

using the method presented in the paper, which uses the results of numerical simulation 

of fluid flow in the pump impeller. The method is actually a procedure for determining 

averaged axisymmetric flow surfaces and meridian streamlines. Furthermore, according 

to the obtained streamlines, a correction of the impeller blade geometry can be made (if 

the streamlines deviate significantly from the assumed axisymmetric ones). It is also 

possible to calculate the specific works of the elementary stages and compare them with 

the previous assumptions. The pump impeller torque can be calculated as well. 

Key Words: Centrifugal Pump, Averaged Flow Surface, Meridian Streamlines, 

Numerical Simulations 

1. INTRODUCTION 

In view of the vast number of applications of centrifugal pumps in various systems, 

such as water supply, irrigation, drainage, water cooling systems, etc., it is crucial to 

ensure they are working in the most effective way possible. Turbo machine designing has 

always been a complex task, which is largely based on the designer’s experience, due to 

many assumptions made in the process of calculating the impeller geometry. Thus, the 

impeller calculation has to be followed by the prototype(s) testing and later, if needed, 

                                                           
Received September 11, 2017 / Accepted November 27, 2017 

Corresponding author: Jasmina Bogdanović-Jovanović  

University of Nis, Faculty of Mechanical Engineering, A. Medvedeva 14, 18000 Niš, Serbia  
E-mail: bminja@masfak.ni.ac.r 



480 J. BOGDANOVIĆ-JOVANOVIĆ, D. MILENKOVIĆ, Ž. STAMENKOVIĆ, Ž. SPASIĆ 

impeller blade correction (which is usually necessary for obtaining a more efficient 

impeller and higher pump efficiency). 

A special attention in the pump designing must be paid to the pump impeller. The 

basic assumption is that the fluid flow in the pump impeller is axisymmetric, which is the 

case of profile cascades with an infinite number of infinitely thin blades [1-4]. The 

continuity of the flow must be maintained, and the fluid flow energy must be increased 

by the required design value [5, 6]. By averaging the flow velocities according to the 

circular coordinate, the fluid flow in the real pump impeller can be derived into that of the 

fictive impeller with an infinite number of infinitely thin blades, which creates the equal 

flow declination as the real pump impeller [7, 8].  

An accurate determination of meridian streamlines in the centrifugal pump impeller is 

very important since they represent meridian traces of the axisymmetric flow surfaces in 

the pump impeller.  

On the other hand, numerical simulations of the flow in the radial pump impeller 

provide for obtaining all flow parameters in the flow domain (pump impeller). There are 

many attempts to use the CFD results of flow in turbo pump impellers in the process of 

developing their optimal design [9-12].  

Values of the numerically obtained flow parameters, especially of all the velocity 

components could be very useful, as will be presented in the method for determining 

averaged axisymmetric flow surfaces. This method for averaging flow parameters and 

calculating averaged flow surfaces according to the circular coordinate, in order to obtain a 

model of the fictive impeller with an infinite number of infinitely thin blades, is presented 

in the paper. Meridian streamlines are calculated using the integral continuity equation 

for the previously averaged flow parameters [8]. The comparison of the obtained meridian 

streamlines and the streamlines defined in the process of blade designing can be made.  

Furthermore, the specific works of elementary stages in the meridian cross-section 

can be calculated and compared with the predefined specific works of elementary stages. 

 The pump impeller torque can be calculated as well, and also compared with the 

numerical simulation results. 

2. EQUATIONS FOR AVERAGING FLOW PARAMETERS OVER CIRCULAR COORDINATE 

AND FLOW EQUATIONS 

Observing the universal curvilinear orthogonal coordinate system (q1, q2, q3), there are 

following coordinate surfaces [7, 8]: meridian planes q3=const., axisymmetric surfaces 

q2=const. and axisymmetric flow surface q1=const. perpendicular to q3=const. (Fig. 1). 

Lame's coefficients are also defined as a function of curvilinear coordinates, L1(q1,q2), 

L2(q1,q2) and L3=r/ro, where r=r(q1,q2) is a radial distance from the origin of cylindrical 

coordinate system (r, , z) [7, 8]. 

Direction of circumferential coordinate q3( 3
o

e ) does not have to be the same as that of 

the impeller rotation ( 


); also, the right (positive) coordinate system is used (dashed 

blade lines in Fig.1 correspond to the left (negative) coordinate system).  

Using cylindrical coordinate system (r, , z): q1=z, q2=r and q3=ro (L1=1, L2=1, 
L3=r/ro), where ro is the radial distance in the pump inlet.  



 Determination of Averaged Axisymmetric Flow Surfaces and Meridian Streamlines in the Centrifugal… 481 

 

Fig. 1 Cross-section of the centrifugal pump impeller in curvilinear coordinate system 

If circumferential velocity u=-r<0 and absolute velocity c3=-cu<0, where  is the 

angular velocity and cu is the circumferential component of the absolute velocity, then 

relative velocity w3=c3+r=-cu+r>0 
for unit vectors 

3

o o
e u  , also, if circumferential 

velocity u=r>0 and c3=cu>0,  then w3=c3-r=cu-r<0 for 3
o o

e u . 

The suction side of the impeller blade is denoted by “l” and the pressure side is 

denoted by “g” (Fig. 1), indexes “a” and “b” are: a=l, b=g for 
3

o o
e u  , respectively 

a=g, b=l for 
3

o o
e u   [8]. 

Absolute (c) and relative (w) flow velocities are related, taking into account  
o

z
e  

(in the direction of the rotation axis),  

  ,    c w u w r , therefore,  rot rot 2  c w . (1) 

According to the cosines theorem, the velocity triangle exists: c
2
=w

2
+

2
r

2
-2cu, 

therefore, w
2
= c

2
-

2
r

2
+2cu. 

Scalar and vector functions can be considered as an averaged value over the circular 

coordinate and pulsation component. The averaged pulsation component is equal to zero 

[5].  

2.1. Averaging of flow parameters in turbo pumps 

Any scalar function f(q1,q2,q3) can be averaged over circumferential coordinate q3: 

 
3 1 2

3 1 2

( )

1 2 1 2 3 3

3 ( )

1
( ) ( )d

Δ

b

a

q q ,q

q q ,q

f q ,q f q ,q ,q q
q

  , (2) 

where the circumferential coordinate can be written as q3=rφ (Fig. 1). 



482 J. BOGDANOVIĆ-JOVANOVIĆ, D. MILENKOVIĆ, Ž. STAMENKOVIĆ, Ž. SPASIĆ 

Point M(q1,q2) in the meridian cross-section of the centrifugal pump impeller belongs to 

radial coordinate r=r(q1,q2). All the flow parameters, such as pressure p or vector 

components (relative wj
 
and absolute cj flow velocity), can be averaged over circumferential 

coordinate q3. The averaging interval is q3=q3b(q1,q2)q3b(q1,q2), where q3b(q1,q2) and 

q3a(q1,q2) are equations of two neighboring blade surfaces in the blade passage. 

Formula (2) becomes [7, 8]: 

 
3

3

( ) ( )

1 2 1 2 3 3 1 2

3 ( ) ( )

1 1
( , ) ( , , ( ))d ( ) ( , , ( ))d ( )

Δ Δ ( )

b b

a a

q r r

q r r

f q q f q q q r q r f q q r r
q r





 


   , (3) 

where, dq3=dq3(r)/L3 and q3=q3(r)/L3=r(r)/L3, for (r)=b(r)–a(r) [deg]. 
Since coordinate q3 is circular, the averaging is obtained over a circular arc in the 

blade passage area. Depending on the number of numerically calculated flow parameters 

the integral in Eq. (3) can be resolved numerically using the trapezoidal rule.  

In the Ansys CFX software, even only one blade with half of the blade passages on 

each side can be numerically simulated (in order to reduce the computational time due to 

impeller symmetry). The same strategy applies to the Fluent software [10]. Anyway, 

regardless of whether the whole impeller or just a single blade is numerically simulated, 

only one blade with half of the blade passages around it will be enough for analyzing 

numerical simulation results, as shown in Fig. 2. 

 

Fig. 2 Averaging over circular coordinate q3 (around the blade) 

Averaging of the scalar function for the values as given in Fig. 2 can be written: 

      
1

1 2

( ) ( )

1 2 1 2 1 2

( ) ( )

1 1
, , , ( ) d ( ) , , ( ) d ( )

Δ ( ) Δ ( )

m n

m

r r

r r

f q q f q q r r f q q r r
r r

 

 

   
 

 
 

  , (3a) 

where, 
1 1

Δ ( ) ( ) ( )
m

r r r     , 
2

Δ ( ) ( ) ( )
n m

r r r    
 
[deg].

The application of the trapezoidal rule for Eq. (4) yields: 

1

2

1 2 1 1 1 1
2 1

1 1
( , ) ( )( ) ( )( )

2Δ ( ) 2Δ ( )

m n

j j j j j j j j
j j m

f q q f f f f
r r

   
 

   
  

  
       

    
  , (4) 



 Determination of Averaged Axisymmetric Flow Surfaces and Meridian Streamlines in the Centrifugal… 483 

where: 

1 1
Δ ( ) ( ) ( )

m
r r r     , 

2
Δ ( ) ( ) ( )

n m
r r r     ,  [deg], fj=fj(r),  j=1,2,...m1,m2,...,n. 

2.2. Averaging of derivatives of flow parameters, gradient, divergence and curl 

The corresponding derivatives of the flow parameters can be averaged as follows [7]: 

 
1 2 1 2

3 3

1
Δ ,   where Δ ( , ) ( , )

Δ
b a

f
f f f q q f q q

q q

 
     

 and (5) 

 

 3 3 3 3 3
1,2 3 1,2 3 1,2 1,2 1,2 1,2

Δ1 1
Δ ,  where: Δ

Δ Δ
b a

q f q q q qf
f f f f

q q q q q q q q

            
            

                       

 (6) 

Inclination angles of blade profiles, a,b and a,b, are measured in the negative 

direction of the circumferential flow velocity, as shown in Fig. 3. Also, for a radial pump 

impeller, the vectors perpendicular to blade surfaces, 1 , 2 , 3 ,, ,a b a b a bn n n , are given in Fig. 3. 

Thus, the averaged derivative (6) can be transformed into the next equation: 

1 2 1 23

1 2 3 1 2 3 3 3

(Δ )1 1
Δ

Δ Δ

, ,

, ,

L nq ff
f

q q q q L n

    
         

, for  
1,2 1,2 1,2

3 3 3

Δ

b a

n n n
f f f

n n n

     
      

     
 (7) 

where, 
3 1 1 1

,

1 3 3 3, ,

a b

a b a b

q L L n
ctg

q L L n

    
       

     
and 

3 2 2 2
,

2 3 3 3, ,

a b

a b a b

q L L n
ctg

q L L n

    
       

     
 

for a point M(q1, q2). 

 

Fig. 3 Cross-sections of the centrifugal pump impeller where 
3

o o
e u   (a=l, b=g) 



484 J. BOGDANOVIĆ-JOVANOVIĆ, D. MILENKOVIĆ, Ž. STAMENKOVIĆ, Ž. SPASIĆ 

Gradient of a scalar function f, divergence and curl of any vector v  become [7, 8]: 

 

3

3 3 3 3 3 3 3

3

3 3 3 3 3 3 3

3

3 3 3 3 3 3

1 1 1
grad grad(Δ ) Δ grad Δ    

Δ Δ Δ

1 1 1
div div(Δ , ) Δ , =div Δ ,  

Δ Δ Δ

1 1 1
rot rot(Δ , ) Δ , rot Δ

Δ Δ Δ

n n
f q f f f f

q L q n L q n

n n
v q v v v v

q L q n L q n

n n
v q v v v

q L q n L q n

   
        

   

   
     

   

 
    

  3
, v










 
 
  

 (8) 

Also, the following relations are obtained: 

 
3 3 3 3 3 3 3 3 3

1 1 1
grad Δ   div Δ ,  rot Δ

Δ Δ Δ

     
            

     

n n n
f f , , v ,v v ,v

L q n L q n L q n
. (9) 

2.3. Averaging of fluid flow equations 

There are two equations used for describing a turbulent incompressible fluid flow for 

the stationary operation of turbo machine, and the first one is continuity equation: 

 div 0w . (10) 

Neglecting gravity, the flow equation is given by the formulation [13]: 

 [ , rot ] gradw c E , (11) 

where E is the energy per unit mass of the fluid of density ρ, according to the Bernoulli’s 

integral for relative fluid flow: 

 
2 2

1 2 3

( ω)
,   ( , , )

2 2

p w r
E E E q q q


    . (12) 

Averaging the continuity Eq. (10) over the circular coordinate, it becomes: 

 
3

3 3

1
div(Δ , ) Δ , =0

n
q w w

L n

 
  

 
. (13) 

Equation (13) can be transformed due to the requirement of perpendicularity of vectors 

n  and w , then the second term of Eq. (13) is equal to zero. The remaining part of the 

equation has been developed and then multiplied with the number of blades zl = 2/, 

where:  = 2/zl is an angular blade pitch, k = φ/τ = zlφ/2 is the blockage factor, which 
represents the cross-section reduction due to the real thickness of the impeller blades, and 

q3 = ro [7, 8]. Thus, Eq. (13) becomes: 

 
3 2 1 3 1 2

1 2

(2 ) (2 )
o o

r kL L w r kL L w
q q

 
 

 
 

. (14) 



 Determination of Averaged Axisymmetric Flow Surfaces and Meridian Streamlines in the Centrifugal… 485 

Equation (14) is a necessary and sufficient condition for the existence of meridian 

stream function, 
1 2

( , )
m m

q q  , where the meridian velocity is 
1 1 2 2

o o

m
w w e w e  , thus 

 
3 2 2 3 1 1

1 2

2    and    2
m m

o o
r kL L w r kL L w

q q

 
 

 
  

 
. (15) 

Equation (15) can be used to calculate the averaged meridian stream function (meridian 

streamlines) 
m

 , for known averaged velocities and blockage factor. It determines volume 

flow rate (Q) through axisymmetric cross-sections,
 
d d

m
Q   

Second term in Eq. (13) can be neglected only in the case of a very thin boundary 

layer (a fully developed primary turbulent flow with very high Reynolds numbers).  

In practice, the fluid flow in turbo pumps is highly turbulent but often showing 

separations of the boundary layers, when the second term in Eq. (13) should not be 

neglected and the meridian streamlines should be determined by using the requirement of 

equal flow rate passing between the axisymmetric flow surface and the hub surface (not 

using Eq. (15)). 

Averaging of the flow Eq. (11), where the flow values can be treated as the sum of 

mean and fluctuation values, w w w  and rot rot (rot )c c c   , it becomes [7, 8]: 

  rot rot grad     w, c w ,( c ) E . (16) 

Equation (16) can be finally written in the next form [8]: 

 
(1) (2) (3)

, rot gradw c F F F E       , (17) 

 

 (1) (2) (3)
3 3 3 3 3 3

2 2

1 1
, Δ , ,   Δ , ,  , rot  where,    

Δ Δ

( ω)
and  

2 2

 
n n

F w w F E F w w
L q n L q n

p w r
E



    
         

     

  

 (18) 

Equation (18) represents the forces obtained as the result of averaging, which should 

be treated as mass forces of the blades acting on the averaged fluid flow. Since 
   3 1

F F , especially if w  is changing linearly over circumferential coordinate q3, 
 3

F  

can be neglected [13]. Also, for axisymmetric flow and model of inviscid fluid, 
 2

0F  . 

But if the fluid flow is not axisymmetric, 
 2

F  has to be taken into account. Thus, 
   1 2

 F F F : 

 

 

 

 

(1)

1 2 2 3 3 2 1 3 1

3 3

(1)

2 3 3 1 1 1 2 3 2

3 3

(1) 1
3 3 2 3 1 1 2 2

3 3

1
Δ( )ctg Δ( ctg ) Δ

Δ

1
Δ( )ctg Δ( ctg ) Δ    

Δ

Δ( ctg ) Δ( ctg ) Δ Δ
Δ

F w w w w w w w w
L q

F w w w w w w w w
L q

w
F w w w w w w w

L q


        




         



         


 (19) 



486 J. BOGDANOVIĆ-JOVANOVIĆ, D. MILENKOVIĆ, Ž. STAMENKOVIĆ, Ž. SPASIĆ 

and

 

     2 2 2
1 2 3

3 3 3 3 3 3

1 1 1
Δ( , ctg ),   Δ( , ctg ),   Δ ,

Δ Δ Δ
F E F E F E

L q L q L q
          (20) 

where, 
1 3 1 3 2 3 2 3

/ / ctg   and   / / ctgn n F F n n F F      . 

Assuming axisymmetric flow surfaces (when 
 2

0F  ) the general equation of the 

averaged axisymmetric flow in centrifugal pump can be obtained. The surfaces are 

perpendicular to 
 1

F  and vector 
)1(

n


 is in the direction of 
 1

F ( 0],[
)1()1(
Fn


), then 

0),(
)1(

wn


. Since 
(1)

 and 
(1)

 are corresponding blade angles and considering 1 1c w , 

2 2
c w , 

3 3
c w r  , for 

3

o o
e u   and 3 3c w r  , for 3

o o
e u , Eq. (11), i.e. Eq. (17) 

transforms into the general equation of the averaged fluid flow in pump impellers: 

 (1) (1)3 3 2 2 1 1

2 2 1 1 1 2 1 2 1 2 2

( ) ( ) ( ) ( )1 1 1 1 1
0

rc rc L c L c E
ctg ctg

rL q rL q L L q q c L q

     
       

     
 (21) 

 

(1) (1)3 3

2 2 2 1 1 1

2 2 1 1

1 2 1 2 1 2 2

( ) ( )1 1
or,   2 2

( ) ( )1 1 1
         0.

rw rwr r
ctg r ctg r

rL q q rL q q

L w L w E

L L q q w L q

     
        

      

   
    

   

 (22) 

Real and viscous fluids differ from non-viscous ones; and, according to the energy 

losses, E , the averaged Bernoulli’s integral for relative and absolute flow can be written [8]: 

 ( ) ( ) Δ ( ) ( ) ( ) ( ) Δ ( )
m o m g m o m u o m g m

E E E G rc E            (23) 

where ( )o mE   is averaged mechanical energy of the relative fluid flow in inlet control 

surface “o”, Δ ( )g mE   is averaged flow energy loss from the inlet control surface to the 

arbitrary circular cross-section, when the meridian trace is line const.
m
   

In Eq. 23, ( )oG   relates to averaged absolute flow, 
2

( ) / / 2
o o o

G p c   , where 

3u
c c   for 

3

o o
e u  (for “+”) and 3uc c  for 3

o o
e u (for “‒“).  

In vaneless parts of the pump, the differential equation of the averaged flow is 

obtained for 0F  . 

3. NUMERICAL SIMULATION OF FLUID FLOW IN RADIAL PUMP IMPELLER 

The given methodology for obtaining averaged flow surfaces and averaged streamlines 
in the radial pump impeller is illustrated in the case of a centrifugal pump, which is 
constructed and tested as the part of the previous project of the Department of 
Hydroenergetics, Faculty of Mechanical Engineering Niš (“Improving the constructive 
solution of a centrifugal pump in order to increase its operating area and to improve 
cavitation characteristics”).  

It is a centrifugal norm pump (Fig. 4), used mainly for water supply or irrigation 
purposes, which operates with the following operating parameters: flow rate Q=30 l/s, 

pump head H=14,9 m, number of revolutions n=1490 min
-1

, efficiency =0,78, inlet 



 Determination of Averaged Axisymmetric Flow Surfaces and Meridian Streamlines in the Centrifugal… 487 

diameter is D1=110 mm, outlet diameter D2=250 mm (cut to 220 mm) and 6 impeller 

blades. Specific speed is nq=nQ
1/2

H
-3/4

=33,3. Detailed impeller geometry is given in [8].   

 

Fig. 4 Geometric model of the centrifugal pump impeller  

and a detail of the discretization mesh used in CFD 

A discretization mesh is generated using the Ansys Workbench BladeGen and it 

consists of 166998 nodes and 700506 elements, mostly tetrahedral, 589332, pyramidal 

463 and wedges 110711, (Fig. 4). Spiral casing is deliberately omitted since in some 

studies [14] it has been shown that for such models of centrifugal pumps the spiral does 

not affect too much pump performance characteristics.  

Numerical simulations of the centrifugal pump are performed using the Ansys CFX 

14.0, which solves RANS equations, with standard k- turbulent model [15, 16].  Specified 

conditions are: mass flow at the impeller inlet and static pressure at the outlet. High 

resolution advection scheme is used. Numerical solving of governing equations continues 

until the root mean square values of the equation residuals become smaller than 10
-5

. 

  

 

 



488 J. BOGDANOVIĆ-JOVANOVIĆ, D. MILENKOVIĆ, Ž. STAMENKOVIĆ, Ž. SPASIĆ 

 

Fig. 5 Numerical and experimental results of the pump head characteristic 

The volume flow rate is changed from 18 l/s to 36 l/s, giving the following operating 

characteristic chart. Validation of obtained results is presented in Fig. 5; the numerical 

and experimental curves show a very small difference in the values obtained (1,7%). 

4. DETERMINATION OF MERIDIAN STREAMLINES OF AVERAGED FLOW  

USING THE INTEGRAL CONTINUITY EQUATION 

First of all, it is necessary to determine a series of control cross-sections across 

meridian lines (Lk, k=1,2,…), as shown in Fig. 6 [8].  

 

Fig. 6 Control cross-sections 



 Determination of Averaged Axisymmetric Flow Surfaces and Meridian Streamlines in the Centrifugal… 489 

Each of these lines contains an arbitrary number of calculation points (n) and for 

every point an averaged flow parameter (such as wr, wz, w, cr, cz, c, w
2
, c

2
, p and pt , Eq. 

(4)) is calculated, using the values obtained by numerical simulations. 

Using the integral continuity equation, for each meridian trace represented with a line 

l, the volume flow rate through an axisymmetric flow surface, for k=zlΔφ/2π is: 

 

o

( ) ( )

( ) ( )

2 d d
n n n

o o

r l z r

l z r

r l z r

Q kc r r kc r z
 

  
  
  . (24) 

In the centrifugal pump impeller, 6 different cross-sections are selected, two of which 

are outside of the blade area (a-a in front of the blades and f-f behind the blades) and four 

cross-sections are in the impeller blade area (b-b, c-c, d-d and e-e), as shown in Fig. 7. 

These streamlines represent the initial (or the first) approximation. There are nine points 

on each meridian control line, which are used for averaging all the flow parameters. 

Along every circular arc passing through these points, there are ten evenly distributed 

points in which all the flow parameters are obtained by the CFD. 

 

Fig. 7 Designed meridian cross-sections  

The following values are needed: the blockage factor due to blade thickness, k = zl/2 

number of blades zl = 6, and angle Δφ  60
o
. Note that in blade area k < 1, and in vaneless 

space k=1.  

Volume flow rates in j cross-section of the pump impeller can be calculated: 

 
1 1 1 1 1

( )( ) ( ( )( )),  for 1,2,...,
j j j j j j j j j j

Q Q f f r r F F z z j n 
    

           (25)
 

where, Qo = 0, and f and F are functions calculated knowing the values of averaged axial 

(
z

w ) and radial relative velocity (
r

w ) in the pump impeller: 

 

.

.

( ) ( , ) ( , ) ( , ) ( , ),  for 1,2,...,

( ) ( , ) ( , ) ( , ) ( , ),  for 1,2,...,

k j

k j

j z L j j j j j j z j j

j r L j j j j j j r j j

f krw f z r k z r r z r w z r j n

F krw F z r k z r r z r w z r j n

     

     
 (26) 



490 J. BOGDANOVIĆ-JOVANOVIĆ, D. MILENKOVIĆ, Ž. STAMENKOVIĆ, Ž. SPASIĆ 

Flow parameters are obtained by numerical simulation of the flow in the pump 
impeller, for the nominal operating pump regime. The volume flow rates are calculated 
for all cross-sections (a, b, c, d, e, f) and the average value is Q=0,0282 m

3
/s=28,2 l/s, 

obtaining an error of 6% mostly due to blade thickness and flow separation from the blade 
surfaces and reverse fluid flow. In cross-sections a-a and f-f, which are outside the blade 
area (in front and behind the impeller blades), the volume flow rates are close to the 
designed nominal value. Thus, the values obtained are 29,436 l/s and 29,945 l/s, respectively. 

4.1. Correction of meridian streamlines in radial pump impeller 

Considering that in the observed control cross-sections the flow rates between the hub 
and these surfaces are not equal, it is clear that the assumed axisymmetric flow surfaces in 
the initial approximation are not averaged meridian streamlines. Therefore, the next step is 
the correction of streamlines. Fig. 8 shows the initial meridian line (j), the corrected 
meridian line (j’), and the distances between these lines and the next (or neighboring) line, 
where k-k is the flow cross-section perpendicular to the meridian streamlines. 

 

Fig. 8 An illustration for the correction methodology 

Volume flow rates between the streamlines can be calculated using the mean values 

of meridian velocity (
)(

.

j

avgm
c ): 

 

( ) ( )

1 . . . 1 .

( ) ( )

1 . . . 1 .

1
2 ,   for ( )

2

1
2 ,   for ( )

2

j j

j j j m avg m avg m j m j

j j

x j x m avg m avg m j m x

Q r n c c c c

Q r n c c c c





 

 

 

    

    

 (27) 

Assuming a linear change of the meridian velocity along meridian traces nj and nx, 

the relation can be obtained: 

 

2

.

. 1

2( / ) A( / )
,     where  A 1

(2 A)

x j x j m jx

j m j

n n n n cQ

Q c


    
  

 
. (28) 

Since  

 
j

jx

j

x

jxjx
Q

QQ

Q

Q
QQQQ









 1

 
    then  , . (29) 

and the quadratic equation is obtained: 

 
























































j

jx

j

x

j

x

Q

QQ

n

n

n

n
1)A2(Bfor       ,0B2A

2

. (30) 



 Determination of Averaged Axisymmetric Flow Surfaces and Meridian Streamlines in the Centrifugal… 491 

There are two possible solutions of Eq. (30): 

 
. . 1

1
if A 0, ,   then ( 1 1 AB)

A

x
m j m j

j

n
c c

n



     


   and (31) 

 
j

jx

j

x

jmjm
Q

QQ

n

n
cc












1  then , 0,A if

1..
. (32) 

Note that as a result of Eq. (31) one should take only a real and positive solution. 

In Fig. 9 the initial approximation of the meridian streamlines (dashed lines) and the 

corrected meridian streamlines (full lines) are presented. A noticeable difference is clearly 

visible but in some parts of the impeller there is a greater difference of the corrected 

streamlines, compared to the initially assumed.    

 

Fig. 9 Meridian streamlines of averaged flow and 
u

rc  distribution in the impeller outlet 

4.2. Calculation of specific work of pump elementary stages and torque 

The specific work of elementary stages in the pump impeller can be calculated: 

 
( ) ω[( ) ( ) ]

k j u f u a
y S rc rc   (33) 

In Fig. 9 a change of the specific work in an outlet cross-section (f-f), calculated for the 

given example of centrifugal pump is presented. It clearly shows an unequal distribution of 

u
rc  from the hub to the shroud of the pump impeller. First three cross-sections have similar 

values of specific work (15% smaller than design value). In the next (fourth) cross-section 



492 J. BOGDANOVIĆ-JOVANOVIĆ, D. MILENKOVIĆ, Ž. STAMENKOVIĆ, Ž. SPASIĆ 

this value increases, obtaining the design value in the next 3 cross-sections. In the last two 

cross-sections the specific work exceeds design value up to even 35%. 

The common practice in the design of all types of turbomachinery is the assumption 

of equality of the specific work of elementary stages.  

There are some earlier suggestions proposing the use of an unequal distribution of the 

specific work of elementary stages in turbomachinery designing [17, 18]. The justification 

of such an assumption (unequal distribution of 
u

rc ) is shown in Fig. 9. 

The pump impeller torque can be calculated using the formula: 

 
2 1

(1, 2) ω ( , d ) ω ( , d ) (2) (1)
(P)

k k u m u m k k

A A

M M r c c A r c c A M M       ,  (34) 

where A1 and A2 are control cross-sections on inlet (1) and outlet (2) of the pump 

impeller. Therefore, 

 
 

1,2 1,2

2 2

1,2 1,2
(1, 2) 2 2 ,  ( ( , ))

k u z u r

L L

M r c c dr r c c dz L L z r       (35) 

If the calculation points are uniformly and densely distributed along control lines L1 

and L2 (n1,n210), then the integrals of Eq. (34) can be determined, with good accuracy, 

using the trapesoidal rule. Using the notation, for j=0,1,2,...n1,2: 

 

(1,2) (1,2)

1,2. 1,2.

(1,2) 2 (1,2) (1,2) (1,2) (1,2) 2 (1,2) (1,2) (1,2)

,    ,  

( , ),    ( , )

j j j j

j u z j j j j u r j j j

z z r r

g r c c g z r G r c c G z r

 

   
  (36) 

Equation (34), i.e. Eq. (35), for inlet and outlet cross-sections, becomes: 

 

1,2 1,2

(1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2)

1 1 1 1
1 1

(1, 2) ( )( ) ( )( )
n n

k j j j j j j j j
j j

M g g r r G G z z 
   

 

         (37) 

In the given example of a centrifugal pump, it is obtained Mk=29,785 Nm, which 

differs for only 5% from the value obtained by numerical simulation. 

5. CONCLUSION 

The principle and method of averaging flow parameters and flow equations, using the 

results of numerical simulations of flow in the centrifugal pump impeller, are presented in 

the paper. Meridian streamlines of the averaged flow are obtained, as well as the distribution 

of specific work in the impeller. The specific work distribution in the impeller outlet shows 

unequal distribution, which indicates it deviates more or less from the pre-designed values. 

The impeller torque was estimated as well. Such information is of the greatest importance 

in the design and calculation of the pump impeller. The meridian streamlines in the 

centrifugal pump impeller should be corrected during the design process. Using the 

numerical simulation data, according to the procedure described in the paper, much more 

relevant information is available to the pump designer. At the same time, this procedure 

shortens the time required for pump model developing and testing. 



 Determination of Averaged Axisymmetric Flow Surfaces and Meridian Streamlines in the Centrifugal… 493 

Acknowledgements: The paper is a part of the research done within the project TR33040, 

“Revitalization of existing and designing new micro and mini hydropower plants (from 100 kW to 

1000 kW) in the territory of South and Southeast Serbia”, supported by the Ministry of Science 

and1 Technological Development of the Republic of Serbia.  

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