The Reynolds averaged approach is used to decompose mean quantities  and  fluctuations  for  all  of  the  physical  variables  to

where n0 = total number of vapor bubbles per unit volume; and R = radius of a bubble。 Therefore, the physical properties of fluid can be attained from the linear interpolation as shown in following formulas

ρ ¼ Vf ρv  þ ð1 − Vf Þρl ð7Þ

μ ¼ Vf μv  þ ð1 − Vf Þμl ð8Þ

where the subscripts l and v refer to liquid and vapor, respectively。

The formula of Vf becomes

n0 4 πR3

handle turbulent flows。 Therefore, 3D governing equations can be denoted as

Solving Vf  in the flow field can find out the regions  where

1 cavitation occurs。 From the mass conservation, the equation    that

is used to predict the distribution of Vf

is derived as

where U = mean velocity component and the subscript j refers

to three directions in the Cartesian coordinate system。 In addition,

momentum conservation can be described by the Reynolds   aver-

aged Navier-Stokes (RANS) equations, which can be written   as

Because Vf is a function of n0

and R, it requires another equa-

tion to solve n0  and R。 Therefore, the Rayleigh-Plesset equation

can be utilized to predict the variation of R in a flow field。 It isdenoted as

where psatðTÞ = saturated vapor pressure of liquids at a specific temperature; μl  = dynamic viscosity of liquids; ρl  = density of

= ambient pressure。 One can obtain the distribution

ij ¼ −ρuiuj ð4Þ

of vapors in the flows inside a globe valve by solving Eqs。 (10)

and (11)。

where  τ ij  =  viscous  shear  stress  of  the  mean  flow;  and  τ 0 =

Reynolds stress attributable to velocity fluctuations。 A turbulence model is needed to simulate the Reynolds stress in turbulent flows。 The standard k—ε turbulence model mentioned in Launder (1989) is employed in this study。 Coefficients appeared in the stan- dard k—ε turbulence model are Cμ ¼ 0。09; σk ¼ 1; σε ¼ 1。29; Cε1  ¼ 1。44; and Cε2  ¼ 1。92。

Cavitation Model

To predict the distribution of vapors due to cavitation in a globe valve, a cavitation model is required。 The adopted cavitation model described by Hympendahl (2003) is based on some assumptions。 First, bubbles evaporate from liquids。 Dissolved gas in liquids is not considered。 In addition, bubbles are assumed to be spherical。 Although bubbles may interact and coalesce with each other, the proposed cavitation model does not consider interaction and coalescence among bubbles。 Presumably, bubbles form and are immersed in liquids in the low pressure region。 To predict the distribution of vapors, a new parameter called the fraction of vapors Vf  is defined as

Vf ¼ Vb þ Vl ð5Þ

Numerical Methods and Parameters

The governing equations for flow and cavitation predictions are solved by the CFD software STAR-CCM+, which is based on the finite-volume method。 The resultant linear simultaneous algebraic equations are solved by the algebraic multigrid (AMG) method。 Furthermore, the SIMPLE algorithm is adopted to iterate velocity and pressure at each time  step。