CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 61, 2017 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš 
Copyright © 2017, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-51-8; ISSN 2283-9216 

Theory Analysis and CFD Simulation of the Pressure Wave 

Generator 

Bing Yang, Jianqiang Deng*, Kai Liu, Fanghua Ye 

School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China 

dengjq@mail.xjtu.edu.cn 

In a pressure wave generator mainly consisting of a stator and a rotor with several valve blocks, pressure 

waves are generated by the continuous shear perturbation of fluids. In this paper, theory analysis of pressure 

wave generated in water hammer principle was conducted. Then a simplified CFD model of a pressure wave 

generator connecting the upstream and downstream pipes was established, and the grid and the time step 

independence tests were carried out. Generating process of controllable pressure waves in pipe flows was 

simulated, and the frequency and amplitude characteristics were obtained. The results suggest that pressure 

waves in upstream and downstream pipes have the similar shapes of approximate sinusoidal waveform and 

opposite phase. Pressure is inversely proportional to the first order derivative of x-velocity at a certain position, 

which is in good agreement with theoretical analysis; pressure wave amplitude increases with an increment of 

the rotational frequency of the rotor at the beginning of the pressure wave generator operation, but reaches 

the maximum amplitude in an optimal frequency when the generator is in stable operation, which means there 

exists a transition process for the generator. The critical frequency related to the first characteristics time can 

be applied to avoid the waveform distortion. 

1. Introduction 

Pressure waves in transient flow are generally induced by flow disturbances which may be caused by planned 

or accidental changes in a system (Chaudhry, 2014). Pressure waves with high propagation speed and 

energy concentration properties can be utilized in transmission signal in measurement while drilling system 

during drilling process (Patton et al., 1977). Another significant application of the pressure waves is the early 

detection of gas influxes because of encountering gas-bearing fractures in gas drilling process to prevent the 

accidents and ensure security and continuity of gas drilling (Meng et al., 2015). Pipe leakage may cause 

wastes for water and energy resources and can produce contaminants in urban water supply systems. 

Pressure waves can also be applied in identifying and detecting potential defects in water supply pipe systems 

(Duan, 2017). 

Fang and Su (2004) summarized the fundamental types of the pressure wave generator in a measurement 

while drilling system in petroleum drilling engineering, and classified pressure wave generators into two main 

categories including the depressurization type and throttle type. The principle of the pressure wave generator 

based on the thin-walled cutting edge flow characteristics was analyzed and the curve orifice generating the 

continuous sinusoidal pressure wave was designed and optimized (Jia et al., 2010). Xue et al. (2011) 

designed a new type pressure wave generator applied in vibration cementing based on water hammer theory. 

In order to obtain an optimized valve orifice satisfying the requirements for the continuous sinusoidal pressure 

output, an improved arc-fillet-line triangular valve orifice was designed based on a general line triangular valve 

orifice (Yan et al., 2015). Hansen and Kjellander (2016) employed the CFD method to explore the 

consequences of a sudden opening of a storage tank due to an external aggression and analyzed 

characteristics and attributions of the pressure waves.  

Although many studies have been published concerning the applications of the pressure wave and generator 

and qualitative analysis of pressure wave characteristics, a little of research has been done on generating 

controllable pressure wave and theory analysis of pressure wave generator are inconsistent in different 

literatures. In the present paper, theoretical analysis is carried out by extending classic water hammer theory 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1761078

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Yang B., Deng J., Liu K., Ye F., 2017, Theory analysis and cfd simulation of the pressure wave generator, Chemical 
Engineering Transactions, 61, 481-486  DOI:10.3303/CET1761078  

481



in continuous transient process and CFD simulations of pressure wave generated in the generator are 

conducted. The simulation results reveal the amplitude and frequency characteristics which are discussed 

quantitatively. 

2. Mechanism of a pressure wave generator 

2.1 Theory analysis 
Water hammer with spatial and temporal changes in velocity and pressure in pipe systems is caused by rapid 

flow disturbances; such transient flows are assumed to be unidirectional because the axial fluxes of mass, 

momentum and energy are far greater than their radial counterparts (Ghidaoui et al., 2005).  

With the unidirectional assumption, the one-dimensional water hammer governing equations in transient pipe 

flows can be derived by applying the principles of mass and momentum conservation to a control volume in an 

oblique tapered pipe. The continuity equation for a control volume is expressed as follows: 

2
sin 0

P P V
V c V

x t x
 

  
   

  
 (1) 

where v is the x-direction velocity, P is the pressure, α is the angle between pipe centreline and the horizontal 

line, g is the gravitational acceleration, c is the wave speed, ρ is the density of fluid. Considering gravitational, 

wall shear and pressure gradient forces as externally imposed, the momentum equation can be given by: 

1
0

2

fV VP V V
V

x x t D

  
   

  
 (2) 

where D is the inner diameters of the pipe, f is the friction coefficient. The wave speed can be calculated by: 

  
2

1

/

1 / /

K
c

K E D e c




   

 
(3) 

where K is the bulk modulus of elasticity of the fluid, E is the Young’s modulus of elasticity of pipe materials, e 
is the thickness of the pipe, c1 is the parameters related to the constraint conditions of the pipes. Eq(1-2) can 
be used in 1D compressible pipe flow. 
Compared to other terms in Eq(1-2), the convective terms are negligible in magnitude because the wave 
speed is considerably larger than x-direction velocity of fluid in most of the engineering applications 
(Chaudhry, 2014). To analyze the pressure wave propagation characteristics, the fluid is supposed to be 
inviscid. Based on the above assumptions and simplifying the equations, the pressure wave equation can be 
written as: 

2 2
2

2 2

P P
c

t x

 


 
 (4) 

which is a one-dimensional second-order wave equation.  
For the case of a rapid closure of the downstream valve in a piping system with a constant level reservoir at 
the upstream end and a valve at the downstream end, the general solution of Eq(4) is represented as: 

P c v    (5) 

It can be noticed from Eq(5) that the pressure wave amplitude is proportional to change in x-direction velocity. 
The disturbance of the case for the rapid closure of the valve is instantaneous and discontinuous. For a 
continuous disturbance case, the process can be viewed as a linear superposition of instantaneous closure 
process at each moment. Assuming that ρ and c are constant, the partial derivative of time on right side of 
Eq(5) can be calculated to obtain the relationship between the pressure and the x-direction velocity which can 
be written as: 

   P t k c v t     (6) 

where k is the correction coefficient. The Eq(6) can describe the quantitative relation between the pressure 
and the velocity accurately, which may validate the simulation results in Part 4. 

2.2 Construction and working principle 
A pressure wave generator shown in Figure 1(a) mainly consists of a stator, a rotor, a shaft, an upstream 

pipeline and a downstream pipeline. Both the stator and rotor are fan-shaped and have four valve blocks. 

When the generator works, the stator remains stable and the rotor driven by a variable frequency motor 

rotates at a certain speed to disturb the fluid flow in both upstream and downstream pipes continuously. As 

mentioned for continuous disturbances, variation in pressure is closely related to change rate of velocity. 

482



Therefore, the periodic variation in velocity field which is induced by the continuous shear disturbance will 

produce the pressure waves in the generator and waves will propagate along the pipelines. 

 

Inlet
Stator

Outlet

Rotor

Shaft

(a)

Upstream pipeline

Downstream pipeline

Gap
Upstream pipe Downstream pipe

Stator Rator
Gap

InterfaceInlet OutletWall Wall

Flow 

A

A

(b)

X

o 

A-A

Valve 

blocks

Flow 

Flow 

Flow 

Flow 

(c)
 

Figure 1: (a) The 3D model of a pressure wave generator, (b) The Main parts and boundary conditions of the 

CFD model, (c) Left view of the rotor 

3. CFD method 

3.1 CFD modelling  
In order to investigate the generating process and characteristics of pressure waves, the effect of the bend 

pipe on pressure wave is neglected and the pressure wave generator system can be reduced to five parts, 

including an upstream pipe (length: Lu = 5,000 mm, diameter: Du = 80 mm), a stator (inner diameter: Dsi = 40 

mm, outer diameter: Dso = 80 mm), a rotor (Dri = 40 mm, Dro = 80 mm), a downstream pipe (Ld1 = 5,000 mm, Dd 

= 80 mm) and a gap (thickness: δg = 3 mm, Dg = 80 mm) between the stator and the rotor which are shown in 

Figure 1(b). Figure 1(c) shows the left view of the rotor to illustrate the valve block shape. The origin of 

coordinate system is at the centre of the left surface of the gap. Grids of each part is generated by using the 

ICEM CFD 14.5 and merged together. The junction surfaces of each part are defined as interface boundaries 

to accomplish the data communication. Attributed to the complexity of each part, the grids of the gap, the 

upstream and downstream pipes are structured; the stator and the rotor are meshed by tetrahedral 

unstructured grids. ANSYS Fluent 14.5 with 3D mode was applied to perform CFD calculations. Simulations in 

this paper mainly focused on the pressure wave characteristics caused by velocity fluctuations in a single 

phase flow and involved strong unsteadiness occurring in the flow. The realizable k–ε turbulence model was 

chosen to simulate the pressure wave better (Yang et al., 2015). The motion of the rotor was achieved by 

setting in the cell zone motion dialog box. All types of boundary conditions and corresponding values are 

presented in Table 1. Considering slight compressibility of water as the fluid in cases, the density of water was 

determined by a user defined function(UDF) of pressure (ANSYS., 2012). The density of water was calculated 

by the following equation: 

  1ref opP P K     (7) 

Where ρref  is the reference density, ρref = 998.2 kg/m3; Pop is the operating pressure, Pop = 101,325 Pa; K is 

the bulk modulus of elasticity of the fluid, K = 2,180 MPa. All the details regarding the chosen solver settings 

are given in Table 2. In all cases, the steady results remaining the rotor stationary were obtained to be the 

initial conditions for unsteady calculations. Monitoring surfaces along the axis positioned by the different x-

coordinates were created to achieve the pressure and velocity at certain positions, which were named by the 

x-coordinate and the measurement units in m. 

Table 1: Boundary conditions 

Part of geometry  Inlet All walls All interfaces Outlet 

Boundary condition type Pressure inlet Stationary wall, no slip Interface Pressure outlet 

Value 1.005 MPa - - 1 MPa 

Table 2: ANSYS Fluent solver settings 

Solve type Density Wave speed Turbulence models 
Pressure-velocity 

coupling 
Convergence criteria 

Pressure-

based 
UDF 1,000 m/s Realizable k–ε SIMPLE 

10-6 for energy 
10-3 for the rest 

483



3.2 Independence test 
Grid independence test was performed on three meshes, consisting of the following number of nodes on per 

meter in axial direction of the pipe: 200 (coarse mesh), 300 (medium mesh) and 320 (fine mesh). Due to the 

periodic unsteady flow, time step independence test was conducted on several time steps determined by: 

 1/ rt N n f     (8) 

where fr is the rotational frequency of the rotor, N is the number of valve blocks, n is a specific integer. fr was 

set to 30 Hz. Variations in x-direction velocity of the monitoring surface at x = -1 m in different mesh types and 

time steps are illustrated in Figure 2(a) and (b). It can be observed from Figure 2(a) that velocity variation with 

time between medium mesh and fine mesh is closely related. Figure 2(b) shows the velocity changes with 

time in different time steps cases. The medium mesh and n = 24 were chosen to conduct the CFD calculation. 

0.00 0.02 0.04 0.06
0.32

0.34

0.36

0.38

0.40

0.42

0.44

V
x
(m

/s
)

t(s)

 coarse mesh

 medium mesh

 fine mesh

(a)
 

0.00 0.02 0.04 0.06
0.46

0.48

0.50

0.52

0.54

0.56

0.58

0.60

0.62

V
x
(m

/s
)

t(s)

 n=70  n=40

 n=24  n=21

 n=17

(b)
 

Figure 2: (a) Grid independence test, (b) Time step independence test 

4. Results and discussion 

Figure 3 shows the pressure variation with time in different positions at fr = 30 Hz. Time was 

nondimensionalized by multiplying the frequency of the rotor and N. As shown in Figure 3, the pressure waves 

in upstream and downstream pipes have the similar shapes of approximate sinusoidal wave and opposite 

phase, which means that the compression wave in upstream and expansion wave in downstream are 

generated simultaneously when the rotor rotates and disturbs the flow. In different observation positions, there 

exists obvious phase difference because of the finite wave speed for compressible fluids. 

  

0.0 0.5 1.0 1.5 2.0
980

990

1,000

1,010

1,020

1,030

1,040
 -0.5m  -1m  -2m  -3m

 -4m  0.5m  1m  2m

 3m  4m

 

P
re

s
s
u
re

(k
P

a
)

t
*

 

20.0 20.5 21.0 21.5 22.0
970

980

990

1,000

1,010

1,020

1,030

1,040

1,050

1,060
 -0.5m  -1m  -2m  -3m

 -4m  0.5m  1m  2m

 3m  4m

 

P
re

s
s
u
re

(k
P

a
)

t
*

 

Figure 3: Pressure variation with time in different positions at fr = 30 Hz 

Figure 4(a) shows changes in the pressure and x-direction velocity of the monitoring position at x = -3 m and 

the opening of rotary valve with time. From Figure 4(a), it can be seen that the pressure reaches a maximum 

when the valve was fully closed, and reaches a minimum when the valve were totally open. In order to validate 

the simulation result, the first order derivative of velocity and the coefficient k in Eq(6) was calculated, and the 

mean value of pressure were determined. Pressure of the monitoring position at x = -3 m can be computed by 

using the following theoretical relation between pressure and the first order derivative of velocity: 

484



   P t P k c v t     (9) 

where P  is the mean pressure at a certain position in a time series. The pressure variations of the simulation 

result and the theoretical calculation result with time are shown in Figure 4(b). It can be noted from Figure 4(b) 

that CFD results are in good agreement with theoretical results.  

120 121 122 123 124 125 126
970

975

980

985

990

995

1,000

1,005

1,010

1,015

1,020

1,025

P
re

s
s
u
re

(k
P

a
)

t
*

 -3m(P)

open

(a)

0.200

0.208

0.216

0.224

0.232

0.240
 -3m(v

x
)

v
x
(m

/s
)

 

120 122 124 126 128 130
990

995

1,000

1,005

1,010

1,015

1,020

1,025

 

P
re

s
s
u
re

(k
P

a
)

t
*

 -3m(CFD)  -3m(Th)

(b)
 

Figure 4: (a) Pressure and velocity variations with time, (b) Pressure variation of both the simulation result and 

the theoretical calculation result with time 

Effects of the rotational frequency of the rotor on pressure waves in the pipes are shown in Figure 5(a). In 

Figure 5(a), it can be found that the pressure wave amplitude increased with an increment of rotational 

frequency at the early stage of the pressure wave generator operation. The reason for this could be that the 

shear disturbance of rotor to flow might increase when frequency went up in the same initial conditions. Figure 

5(b) shows the frequency characteristics of pressure wave more accurately by employing the fast Fourier 

transform method. As shown in Figure 5(b), the amplitude of pressure increased when the rotational frequency 

was from 0 Hz to 30 Hz; when rotational frequency was higher than 30 Hz, the amplitude decreased as the 

rotational frequency went up. These phenomena may be attributed to the shear disturbance of rotor and the 

flow response in different rotational frequency. The waveform distortion induced by the short pipes with low 

rotational speed can be noticed from Figure 5(a). For a certain pipe system, there exists a critical frequency 

which could be estimated by the first characteristics time Tic of pressure wave. Tic can be computed by: 

2 /
ic p

T L c  (10) 

where Lp is the length of pipe. For a given number blocks of rotor, the critical frequency can be determined by: 

 1cr icf N T   (11) 

where fcr is the critical frequency. To avoid the distortion, the rotational frequency should be higher than fcr , 

which is helpful to regulate the pipe system and rotational frequency to obtain a better waveform. 

 

0.00 0.01 0.02 0.03 0.04
980

990

1,000

1,010

1,020

1,030

1,040

1,050

1,060

 

P
re

s
s
u
re

(k
P

a
)

t(s)

 15Hz  25Hz

 30Hz  40Hz

(a)
 

0 50 100 150 200 250 300 350 400 450 500
0

2

4

6

8

10

12

14

16

 

30Hz

40Hz

25Hz

15Hz

A
m

p
it
u

d
e

(k
P

a
)

f(Hz)

 15Hz

 25Hz

 30Hz

 40Hz

(b)
 

Figure5: (a) Pressure variation with time in different rotational frequency, (b) Frequency characteristics of 

pressure wave  

485



5. Conclusions 

In the present paper, theoretical analysis and numerical experiments on pressure wave generated in pipelines 

have been conducted. Extending approximate solution of water hammer theory was obtained to reveal 

correlation between pressure field and velocity field concisely, which may explain the generation of pressure 

wave in continuous disturbance flow. The simulation results were validated by theory results and showed both 

amplitude and frequency characteristics. Pressure wave amplitude increased with the rotational frequency at 

the early stage of the pressure wave generator operation but existed an optimal frequency to reach the 

maximum amplitude when the generator was in stable operation, indicating that there existed a transition 

process in the generator. And the waveform distortion will appear when the rotational frequency is lower than 

the critical frequency which can be avoided by regulating the pipe system and rotatory frequency. Further 

research is needed to investigate practical operating conditions and the flow transition phenomenon in high 

rotational frequency. The study on pressure wave characteristics can help to apply the pressure wave in 

practice and achieve the velocity fluctuations in scientific research.  

Acknowledgments  

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of 

China under Grant No 21376187. 

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