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# Water Science and Engineering, 2010, 3(3): 321-330

doi:10.3882/j.issn.1674-2370.2010.03.008
http://www.waterjournal.cn
e-mail: wse2008@vip.163.com

## Orifice plate cavitation mechanism and

its influencing factors
Wan-zheng AI*, Tian-ming DING
Marine College of Zhejiang Ocean University, Zhoushan 316000, P. R. China

Abstract: The orifice plate energy dissipater is an economic and highly efficient dissipater.
However, there is a risk of cavitaion around the orifice plate flow. In order to provide references for
engineering practice, we examined the cavitation mechanism around the orifice plate and its
influencing factors by utilizing mathematical analysis methods to analyze the flow conditions
around the orifice plate in view of gas bubble dynamics. Through the research presented in this
paper, the following can be observed: The critical radius and the critical pressure of the gas nucleus
in orifice plate flow increase with its initial state parameter 0 ; the development speed of bubbles
stabilizes at a certain value after experiencing a peak value and a small valley value; and the orifice
plate cavitation is closely related to the distribution of the gas nucleus in flow. For computing the
orifice plate cavitation number, we ought to take into account the effects of pressure fluctuation.
The development time of the gas nucleus from the initial radius to the critical radius is about
10-7-10-5 s; therefore, the gas nucleus has sufficient time to develop into bubbles in the negative
half-cycle of flow fluctuation. The orifice critical cavitation number is closely related to the orifice
plate size, and especially closely related with the ratio of the orifice plate radius to the tunnel radius.
The approximate formula for the critical cavitation number of the square orifice plate that only
considers the main influencing factor was obtained by model experiments.
Key words: orifice plate cavitation; gas nucleus; critical radius; critical pressure; pressure
fluctuation; critical cavitation number

1 Introduction
The orifice plate energy dissipater has the advantages of a simple layout, economy, and
high energy efficiency (He and Zhao 2010). The energy dissipation efficiency of the orifice
plate and its cavitations are a pair of contradictions. Orifice plates with high efficiency of
energy dissipation often have a high risk of cavitation damage. Cavitation can lead to
structural damage of buildings and reduce spillway tunnel discharge capacity (Rahmeyer 1988;
Plesst and Zwick 1977; Zhang and Chai 2001; Neppiras 1984). Cavitation can be divided into
four categories: wavering cavitation, fixed cavitation, vortex cavitation, and oscillation
cavitation (Knapp et al. 1970). Oscillation cavitation and vortex cavitation are the main forms
of orifice plate cavitation (Xu et al. 1988). The study of cavitation has been mainly focused on
model tests in the past and some useful conclusions have been obtained. Many studies have

This work was supported by the National Natural Science Foundation of China (Grant No. 50879021).
*Corresponding author (e-mail: aiwanzheng@126.com)
Received Nov. 22, 2008; accepted Feb. 27, 2009
concluded that the orifice plate critical cavitation number decreases with the increase of the
contraction ratio (Xu et al. 1988; Russell and Ball 1967). The successful application of the
orifice plate energy dissipater in the Xiaolangdi Project in China has proved that installing
orifice plates in tunnels, as shown in Fig. 1, where D and d are the diameters of the
channel and orifice plate, T and b are the thicknesses of the orifice plate and anti-vortex
ring, r is the radius of the round corner of the orifice plate, is the slope angle at the top
of the orifice plate, and is the slope angle of the anti-vortex ring, is an effective way to
dissipate flow energy (Chen et al. 2006). However, orifice plate cavitation is a very complex
problem, and there has been little research on the orifice plate cavitation mechanism and its
influencing factors, especially on the principle of cavitation caused by pressure fluctuation.
Based on a summary of previous results, this paper attempts to further reveal the mechanism
of the orifice plate cavitation and to analyze factors that affect its cavitation in view of gas
bubble dynamics.

## 2 Orifice plate cavitation mechanism

In low-pressure conditions, the motion equation of a gas nucleus developing into a gas
bubble is as follows (Huang and Han 1992):
3
dU 3 2 R 2 1
R + U = C02 0 ( p pv ) = f ( R , t ) (1)
dt 2 R R
where C02 = p1 ; p1 is the gas nucleus initial pressure; is the flow density; U is the
gas nucleus development velocity, which can be expressed as U = dR dt ; R is the radius of
the gas nucleus; R0 is the initial radius of the gas nucleus; t is time; is the air insulation
coefficient; is the surface tension coefficient; pv is the saturated steam pressure; and
p is the pressure on the non-disturbed sections, usually located at 0.5D in front of the orifice
plate or at 3D behind the orifice plate, shown in section 1 and section 2, respectively, in Fig. 2
(Xu et al. 1988), where Lb is the length of the backflow region behind the orifice plate. p
is also affected by the orifice plate geometry.
If f R = 0 , we can obtain the critical radius of the gas nucleus:
1
13

2
Rc = (2)
3 C0 R0
2 3

322 Wan-zheng AI et al. Water Science and Engineering, Sep. 2010, Vol. 3, No. 3, 321-330
Fig. 2 Flow around orifice plate
If R = Rc and f ( R, t ) = 0 , we can obtain the critical pressure of the gas nucleus:
2 (1 3 )
pc = + pv (3)
3Rc
If 0 = p1 R03 ( 0 represents the initial state parameters of the gas nucleus), we can obtain the
following equations:
1
13

2
Rc = (4)
3 0
2 (1 3 )
pc = 1
+ pv (5)
13

3 2 ( 3 0 )
Figs. 3 and 4 can be obtained at the normal temperature when = 4 / 3 ,
0.072 8 N m , and pv 2 452 Pa , and the following can be observed: Rc and pc
increase with 0 , that is to say, Rc and pc increase with p1 and R0 ; and if 0 ,
pc pv . Therefore, many researchers regard pv as the critical pressure of incipient
cavitation. In fact, if 0 is large enough, there is a little deviation between pv and pc , but
if 0 is small, there is a large deviation between pv and pc , and it is not accurate to regard
pv as the critical pressure of incipient cavitation under this condition. Integrating Eq. (1), the
following equation can be given:
1

dR 2C02 R0
3
R 2 R0
3 2
2 ( p pv ) R0 3 2
U= = 0 + 1 + 1 (6)
dt 3 (1 ) R R R R 3 R

## Fig. 3 Rc - 0 curve Fig. 4 pc - 0 curve

Wan-zheng AI et al. Water Science and Engineering, Sep. 2010, Vol. 3, No. 3, 321-330 323
The simulation results of Eq. (6) are shown in Fig. 5, which reflect the relationship
between U and R at the normal temperature when p = 600 Pa , and p1 = 4 500 Pa or
3 500 Pa .

Fig. 5 U -R curve
Fig. 5 shows that, in the initial phase, the development velocity of the gas nucleus
increases with its radius. When the gas nucleus radius develops into the critical radius, the
development velocity of the gas nucleus has a small peak value. At this time, cavitation
begins to occur. The critical radii of the gas nucleus, when p1 = 4 500 Pa , are 9.18 105 m,
7.97 105 m, and 6.82 105 m, respectively corresponding to initial radii of 5 105 m, 4.5
105 m, and 4 105 m; and the critical radii of the gas nucleus, when p1 = 3 500 Pa , are
8.44 105 m, 7.23 105 m, and 6.27 105 m, respectively corresponding to initial radii of 5
105 m, 4.5 105 m, and 4 105 m. If the gas nucleus inflation continues, the development
velocity of the gas nucleus reaches a small valley value. After that, the development velocity
of the gas nucleus increases with the radius and at last stabilizes at a certain value. For the gas
nucleus with the same initial radius, if the initial pressure of the gas nucleus is larger, its
inflation velocity at a critical radius is also larger. For the gas nucleus with the same initial
pressure, if the initial radius of the gas nucleus is larger, its inflation velocity at a critical
radius is also larger. If F ( R, p pv ) = U , we obtain the development time of the gas
nucleus from the initial radius to the critical radius:
Rc dR
t = (7)
( c v)
R0 F R, p p

The range of the initial radius of the gas nucleus is 2 106 m to 50 106 m in normal
flow (Ni 1999). At this range, if gas nuclei can develop into gas bubbles, the following
conditions must be met: p pc and p1 > p pv + 2 R0 . Table 1 shows the development
time of the gas nucleus under the conditions of 0.072 8 N m , pv 2 452 Pa , = 4 / 3 ,
and R0 ranging from 0.05 106 m to 50 106 m .

324 Wan-zheng AI et al. Water Science and Engineering, Sep. 2010, Vol. 3, No. 3, 321-330
Table 1 Development time of gas nucleus from initial radius to critical radius for different initial parameters
R0 p1 Rc pc p p Development t
( 105 m ) ( Pa ) ( 105 m ) ( Pa ) ( Pa ) ( Pa ) state ( 105 s )
4.50 4 500.00 7.97 1 082.61 3 716.44 500.00 Development 4.36
4.00 4 500.00 6.82 849.75 3 312.00 600.00 Development 4.13
1.00 4 500.00 1.07 7 721.47 7 608.00 1 000.00 No development
4.00 3 500.00 6.27 709.75 2 312.00 600.00 Development 4.38
0.10 156 350.00 0.18 59 590.00 143 000.00 59 590.00 Development 0.03
4.50 4 500.00 7.97 1 082.62 3 716.44 400.00 Development 4.22
4.50 4 500.00 7.97 1 082.61 3 716.44 600.00 Development 4.51

Note: p = p pv + 2 R0 .

From Table 1 we can observe the following: The development time of the gas nucleus from
the initial radius to the critical radius is about 107 - 105 s. Under the condition of the same
p and R0 , the gas nucleus with a larger p1 will need less time to develop into the critical
radius. Under the condition of the same p1 and R0 , the gas nucleus with a larger p will
need a longer time to develop into the critical radius. Under the condition of the same p1 and
p , the gas nucleus with a larger R0 will need a longer time to develop into the critical radius.
Physical factors that affect the gas nucleus development include R0 , p1 , and p . R0
and p1 determine the initial state parameter 0 , that is to say, if R0 and p1 are determined,
pc and Rc are also determined. If R0 , p1 , and p are determined, the development time
of the gas nucleus from the initial radius to the critical radius is also determined.
Two conditions determine whether cavitations happen in orifice plate flow or not. One
is whether there are many gas nuclei and whether gas nuclei meet this condition:
p1 = pout pv + 2 R0 , where pout is the pressure outside the gas nucleus. That is to say, the
orifice plate cavitation is closely related with the distribution of gas nuclei in flow. The other is
whether there is a low pressure condition for gas nucleus development, i.e., p pc and
p1 > p pv + 2 R0 . When flow passes through the orifice plate, streamlines suddenly
shrink, and then suddenly enlarge, which causes boundary layer separation. Therefore, the
boundary layer is closely related with cavitation. Meanwhile, whirlpool areas occur in front
of and behind the orifice plate. At the center of a whirlpool area, there often form low
pressure areas. At a distance about 0.5D behind the orifice plate, there is a contraction
section. At fringes of the contraction section, there also form low pressure areas. In addition,
because of turbulent shear, local low pressure also exists in the orifice plate flow. All these
low pressure areas create favorable conditions for gas nucleus development. Thus, the
orifice plate size, especially the ratio of the orifice plate radius to the tunnel radius, is
closely related with the orifice plate cavitation.

## 3 Factors affecting orifice plate critical cavitation number

The formula for the critical cavitation number of the orifice plate is as follows:

Wan-zheng AI et al. Water Science and Engineering, Sep. 2010, Vol. 3, No. 3, 321-330 325
p pv
c = (8)
0.5 u02
where u0 is the average flow velocity in the tunnel. Cavitation is mainly affected by the flow
velocity, viscosity, boundary conditions, and pressure. Therefore, the minimum pressure on
the border can also be used to determine whether cavitation has occurred. The minimum
pressure coefficient is defined as follows:
p p
Cp = min 2 (9)
0.5 u0
where pmin is the minimum pressure on the border. If cavitation occurs, the gas bubble is full
of steam, and the pressure in the gas bubble is equal to saturated steam pressure pv . Also,
cavitation begins to occur at the position of the minimum pressure; at this time, pmin is equal
to pv , providing the following equation:
p pmin p
c = Cp = = (10)
0.5 u0
2
0.5 u02
For an orifice plate shown in Fig. 1, the main geometry parameters that determine
boundary conditions are the tunnel diameter D , the orifice plate diameter d , the thickness
of the orifice plate T, and the radius for the round corner of the orifice plate r . Besides these
geometry parameters, the other hydraulic parameters that affect critical cavitation number are
the dynamic viscosity parameter , the average flow velocity in the tunnel u0 , the deviation
between the average pressure on the non-disturbed section and the minimum pressure on the
boundary p . All the above parameters are written into a formula as follows:
f1 ( D, d , , , u0 , p, r , T ) = 0 (11)
According to the dimensional analysis, D , u0 , and are three basic parameters of
the above eight parameters. A non-dimensional equation can be obtained using the
theorem as follows:
d T r p
f2 , , , , 2 =0 (12)
D u0 D D D u0
The above equation is changed as follows:
d T r
p = u02 f3 , , , (13)
D u0 D D D
Because ( u0 D ) = 1 Re , where Re is the Reynolds number, the above equation can be
written as
d 1 T r
p = u02 f3 , , , (14)
D Re D D
Combing Eq. (10) with Eq. (14), Eq. (15) can be rewritten as
1 d 1 T r
c = Cp = f 3 , , , (15)
2 D Re D D
In the above equation, d D is the contraction ratio, and T D and r D are

326 Wan-zheng AI et al. Water Science and Engineering, Sep. 2010, Vol. 3, No. 3, 321-330
parameters of the orifice plate geometry. From Eq. (15), the following conclusion can be
obtained: the orifice plate critical cavitation number is closely related with the orifice plate
geometry, contraction ratio, and Reynolds number. With the assumption that
=d D (16)
= f (T D , r D ) (17)
where is a parameter regarding the orifice plate geometry, Eq. (15) can be rewritten as
c = f 4 ( , Re, ) (18)
Li et al. (1997) regarded as the main factor affecting the critical cavitation number of the
orifice plate. They also provided an empirical formula for the orifice plate critical cavitation
number according to hydraulic model experiments:
c = a b exp ( cRe ) (19)
where a , b , and c are determined by the orifice plate geometry. Because the flow passing
through the orifice plate is turbulent flow, the boundary layer has fully developed in most
cases, and the critical cavitation number changes little with the change of the Reynolds
number. At this time, the critical cavitation number can be expressed as
c = f ( , ) (20)
Most studies (Li et al. 1997; Takahashi et al. 2001) have regarded the critical cavitation
number as varying with by index relations. Thus, the critical cavitation number can be
expressed as
c = m n (21)
where m and n are determined by the orifice plate geometry parameters, excluding . Eq. (21)
shows that the larger the contraction ratio is, the smaller the orifice critical cavitation number is.

## 4 Pressure fluctuation effect on orifice plate cavitation

If we consider the effect of the flow fluctuation, we can obtain another equation about the
cavitation number:
p + p pv
t = (22)
0.5 u02
where p is the pressure fluctuation. Under the condition of low pressure, the gas nucleus
must have enough time tc to develop into a bubble. As the flows passing through the orifice
plate have the characteristics of a large low-frequency fluctuation, the spectrum scope of the
pressure fluctuation is about 0 to 5 Hz. Peak frequency, in general, is about 0 to 0.5 Hz. Thus,
the time that flow fluctuation stays in the negative half-cycle of the fluctuation cycle will be
longer than 102 s. From Table 1 we can observe that gas nucleus development time from the
gas nucleus to the gas bubble is about 107 - 105 s. Therefore, the gas nucleus has sufficient
time to develop into bubbles in the negative half-cycle of the fluctuation cycle.
Characteristics of the flow fluctuation have important effects on the orifice plate
cavitation. Under normal circumstances, cavitation will not occur when the cavitation number

Wan-zheng AI et al. Water Science and Engineering, Sep. 2010, Vol. 3, No. 3, 321-330 327
is larger than the critical cavitation number, i.e., when t > c . However, the critical
cavitation number of the orifice plate computed through Eq. (8) is often larger. For example,
if the flow fluctuation is in the negative half-cycle of a fluctuation cycle, the pressure
fluctuation is a negative value in Eq. (22). At this time, the actual cavitation number
computed through Eq. (22) is smaller than the critical cavitation number computed through
Eq. (8). Only when the flow fluctuation is in the positive half-cycle of a fluctuation cycle can
we find that the cavitation number calculated with Eq. (22) is larger than the critical cavitation
number calculated with Eq. (8). Thus we assume there is no cavitation in the orifice plate flow.
Therefore, if we consider the fluctuation effect on cavitation, we will find that cavitation
occurs locally.
Hu (1994) showed that, in flow fluctuation, the gas nucleus expands in a negative
half-cycle at first and then is compressed in a positive half-cycle; compression and expansion
of the gas nucleus appear alternately, which cause saturated gas in the flow to permeate into
the gas nuclei through their side walls. In the turbulent shear flow, when cavitation begins,
high-frequency components in the power spectrum of pressure fluctuation increase. In general,
there is a predominant frequency. The fluctuation at the predominant frequency leads to the
bubble oscillation. In the course of bubble rebounding, bubbles have a resonance movement
because of the appearance and disappearance of compression in the bubble. In turn, resonance
changes the pressure fluctuation. This phenomenon helps to produce a series of large and
small fluctuations. These large and small fluctuations cannot be ignored; they will shorten the
time the gas nucleus takes to develop into the gas bubble and cause cavitations to occur.
The intense flow fluctuations through the orifice plate are caused by the orifice plates
sudden enlargement and contraction and dramatic changes of streamlines. Fluctuation
amplitude is closely related with orifice plate geometry and contraction ratio. Russell and Ball
(1967) arrived at the following conclusions: when d D 0.65 , the pressure fluctuation
coefficient is almost unchanged around 0.5; when 0.85 d D > 0.65 , the pressure fluctuation
coefficient increases with the contraction ratio of the orifice plate, with the peak pressure
fluctuation coefficient reaching the value of 3 when d D = 0.85 ; and when d D > 0.85 , the
pressure fluctuation coefficient decreases with the increase of the contraction ratio. Taking
into account the effects of pressure fluctuation, the critical cavitation number can be written as
2
p
c = Cpmin +K (23)
0.5 u02
where p is the average value of pressure fluctuation, and K is a parameter closely related
with the probability density and properties of the spectrum of pressure fluctuation. Yang et al.
(2004) determined that if we consider 95% of pressure fluctuation as the characteristic value
of cavitation occurrence, it is reasonable to assume a K value of 1.6.

328 Wan-zheng AI et al. Water Science and Engineering, Sep. 2010, Vol. 3, No. 3, 321-330
5 Model experiments on critical cavitation number of square
orifice plate
Model experiments were conducted in a decompression cabinet. The decompression cabinet
was designed based on the principles of gravity similarity and cavitation number similarity. As
presented in Tullis and Govindarajan (1973), the contraction ratio is the most important factor
influencing the critical cavitation number of the orifice plate. Therefore, only the contraction
ratio was considered in the model experiments. The models were designed as in Fig. 2. The
orifice plate thickness T of each model was 0.01D. The other parameters of the models were
arranged as in Table 2. The critical cavitation number of the orifice plate was calculated by
Eq. (8), but p was divided into two parts as in Eq. (24):
p = pu + pa (24)
where pu is the relative pressure in section 1, and pa is the air pressure in the
decompression cabinet. All experimental results are shown in Table 3. Fig. 6 can be drawn
using the data in Table 3.
Table 2 Model parameters
Model number D (m) d (m)
M1 0.21 0.60 0.126 0
M2 0.21 0.70 0.147 0
M3 0.21 0.78 0.163 8
M4 0.21 0.80 0.168 0

## Table 3 Experimental results

Model number pu (g) (m) pa ( g ) (m) pv ( g ) (m) u 2 ( 2 g ) (m) c
M1 0.651 0.311 0.192 0.031 22.621
M2 0.098 0.372 0.211 0.073 12.312
M3 0.711 0.310 0.211 0.073 7.963
M4 2.070 0.311 0.201 0.282 6.960

Fig. 6 shows that the critical cavitation number of the orifice plate inversely varies with
the contraction ratio. An approximate formula for the critical cavitation number of the square
orifice plate can be obtained:
c = 2.1 4.6 (25)
In Eq. (25), the effects of Reynolds
number, pressure fluctuation and orifice
plate thickness are not considered. This
expression is valid for T = 0.01D ,
Re >105 , and values from 0.6 to
0.8 . Fig. 6 c - curve

6 Conclusions
(1) Critical radius and critical pressure increase with the initial state parameter 0 . That

Wan-zheng AI et al. Water Science and Engineering, Sep. 2010, Vol. 3, No. 3, 321-330 329
is to say, critical radius and critical pressure increase with the gas nucleus initial radius and its
initial pressure.
(2) The orifice plate cavitation is closely related with the gas nucleus distribution, the
contraction ratio and geometry of the orifice plate. The larger the contraction ratio is, the
smaller the critical cavitation number of the orifice plate is.
(3) The gas nucleus has sufficient time to develop into bubbles in the negative half-cycle
of flow fluctuation, so the characteristics of the flow fluctuation have a certain effect on
cavitation. It is necessary to take into account the effect of pressure fluctuation when
computing the orifice plate cavitation number.

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