Duan C.G., Karelin V.Y. Abrasive Erosion and Corrosion of Hydraulic machinery

Подождите немного. Документ загружается.

124

Abrasive Erosion and Corrosion of Hydraulic Machinery

The much more simple model to close the turbulent flow governing equations

(3.54 ~ 3.55) of particle phase is to use the algebraic turbulence model - A

model, that is, to use the Hinze-Tchen formula for particle phase eddy

viscosity. The k - e - A

turbulence model of two-phase flow is the

combination of the standard k - s turbulence model for liquid phase and the

algebraic turbulence model - A

for particle phase, in which the ratio of the

turbulent kinetic energy k

for particle phase to this energy k for liquid phase

equals to the ratio of the particle phase eddy viscosity v to the liquid eddy

viscosity v

, that is,

k D

l+Iz.

(3.56)

where r is particle response time, according to the Stokes drag formula, it

is indicated in the equation (3.57).

^=%r

(357)

The less the time r is the closer the particle follows the fluid motion.

is the fluctuation time of liquid flow. According to the isotropic

assumption of liquid turbulent flow and the assumption that the distribution of

liquid flow velocity fluctuation accords with the Gaussian statistic

distribution, it is obtained as follows:

=- =

4V2-

(3.58)

from which it is clear that the fluctuation of particles is less than that of liquid

based on the present turbulence model. In equation (3.55) neglecting the

productive and the diffusive terms, as well as the second part of the^

Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow 125

expression, and assuming that C

equals to one, then the following equation

can be obtained:

" -

-{^dT

-k

)

(3.59)

dt r.

Integrating equation (3.59) in the period of t,, and at the same time letting

the turbulent kinetic energy k and k

equal to constants, then we can get:

f O

-2

k y 2r

t y

which is much similar to the Hinze-Tchen expression. Wu et al (1994) applied

this k - s- A

turbulence model to calculate the liquid-particle two-phase flow

at dilute concentration conditions through a centrifugal pump impeller

[3.21].

3.2.4 Lagrangian-Eulerian Model for Liquid-Particle Two-Phase

Flow

Lagrangian-Eulerian model for particle-laden two-phase flow is increasingly

becoming a powerful tool to predict dilute turbulent two-phase flow of

engineering significance. This hybrid model consists of the Eulerian

formulation of

the

liquid phase and the Lagrangian formulation of

the

particle

phase together with a stochastic dispersion model to account for the influence

of liquid-phase turbulence on the particle phase. Most of existing Lagrangian

stochastic models, in which the particles are individually treated and a

representative number of particle trajectories are calculated and have adopted

the fundamental method pioneered by Peskin and Kau (1976), Yuu et al

(1978) and Gosman and loannides (1981) [3.22] to account for the effects of

the liquid phase turbulence on the particle phase dispersion. Many authors

have conducted in common that detailed inlet conditions play an important

role in accurate Lagrangian predictions, and that a large number of particle

trajectories are required to achieve a stochastically significant solution

[3.23].

In Lagrangian stochastic computations, particle properties are determined

using an ensemble averaging of many individual particle trajectories, such as,

126

Abrasive Erosion and Corrosion of Hydraulic Machinery

particle trajectories, which requires consuming

great amount

computer CPU

time.

The stochastic dispersed-width transport (SDWT) model

was presented to provide reduced sampling requirements.

Lagrangian Equation for Particles

In the Eulerian-Lagrangian model, the liquid flow

simulated by using the

same Eulerian conservative equations as indicated in last session. But in the

Lagrangian approach, the equation of motion for each of the particle parcel

can be written as

dllpk

Upk

(3.60)

where

Uk + Uk

the instantaneous liquid velocity, F

is an

external

force. The response time of particle,

r ,

is defined as

P~~ P

18///,

^=l+0.15Re

0687

(0<Re

;

, <1000)

The particle trajectories are computed by

dx.

The two-way coupling sources between the liquid- and particle-phases are

given by

SS^^PPDINA!

P J

s£

»-

Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow

127

=—

s^-

UJW

where the summation is carried out over all droplet trajectories crossing the

control volume.

Dispersion-Width Transport Model for Particle Phase

To obtain the instantaneous liquid velocity in equation (3.60), an improved

model dispersion-width transport model can be used to determine the

fluctuating liquid velocity as follows:

",-=V«?£^

(

)

where £■ is a Gaussian random variable having zero mean and unity

deviation. In order to account for the anisotropic effects of liquid turbulence

on particle dispersion, the normal stresses, uf , have been used here. The

transverse component in equation (3. 62) is determined by

where £

is a Gaussian variable having zero meaning and unity deviation, r

the particle radial position, and a

a controlling parameter, being either zero

or unity, to switch on or off this modification. In the dispersion-width

transport model, both particle mean position and its variance are computed

for each particle trajectory. Particle means position can be obtained by

Lagrangian tracking of a computational parcel through a sequence of eddy.

Simultaneously, the particle dispersion-width induced by liquid turbulence is

determined and used as the standard deviation of the computational parcel's

PDF after each interaction. The overall mean square dispersion corresponding

to the variance of the parcel PDF after «-th interaction is determined by

128

Abrasive Erosion and Corrosion of Hydraulic Machinery

where

k-Tpk

i=k+\

T^Exp

j=k+]

<■

i>k

k <n

i = k

k-n

with A

defined by

=l-Exp

At,.

In addition, the above-computed variance for each PDF should be

normalized by the total number of computational parcels; that is,

(3.65)

where k is introduced as a correction factor account for undersampling, and

N^ is the number of computational parcels. It is clear that the normalization

of equation (3.65) can recover the conventional discrete delta-function

dispersion model as the number of computational parcels increases. However,

according to this normalization, use of more computational parcels does not

pay off as regard to the smoothing of Lagrangian computations. Obviously,

k = JN corresponds to the situation that no normalization is used. The

probability distribution function for a simplified isosceles triangle PDF can be

obtained by

Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow 129

P(r)

0 < r

w - r

™~r

<r<r

w-r

<r<w

r>w

+ r„

(

<w)

P(r)

0 < r

r„

<r<w-r

w-r

<r <w

1 r>w

r„

(o<r,<y

P(r) =

0 0 <r <r

-w

c r

-w<r<r

<r<w

where w =

2yf3cr

is the PDF half-width, and the terms

, P

, PD and

are defined by

PA =

-r\r

-2r

+(w

)

+\w-r

-4r

+6wr

-2r]

+(w

y+\w-r

130

Abrasive Erosion and Corrosion of Hydraulic Machinery

+3\w-r

+\w-r

-2r

+(w

)

+\w-r

-2r

+3(w

-2r

\w-r

-2r

+(w

)

+\w-r

~-3(w

-2r

+\w-r

-2r

+(w

)

+\w-r

Particle Motion Equation Disscussion

The equation of motion for each particle written in equation (3.60), like

equations for most of engineering practice only includes the main force, such

as,

the drag force. But the drag coefficient C

includes the effects of the drag

force, the Basset force and the added mass force. The following equation is

the general one of motion of

the

unsteady acceleration particle in liquid flow,

called as the Basset - Boussinesque - Oseen equation, as follows:

nd),

, ,/ v

?td\

~i:

d^py

-u\{u

-u)+-^p—

7id:

-p,dl^

--C/^p\u-u\iu-u)+'-^-p—

-^ """

Hp p

dt 2

' 6

du du

Tvd

" 4

ndlt-

+ ~f(p

-p)s-c,-f/>M"

M.,

-rj=

(3.66)

where

dt *&x

and

—

— + 11.—

and C

is the drag coefficient at the steady motion, C

the mass coefficient and

is the Basset term coefficient. The Basset term can be calculated by the

following integration:

Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow

131

du„ du

t-M

dr dr

, dr

dr y/t-r

4t-r

i— du

2VA/

h-h =

*i-\j

t-Al

Many theoretical and experimental studies have been undertaken on the

transient effects on drag coefficient

[3.28],

Odar and Hamilton (1964)

proposed an equation for the unsteady drag force exerted by a viscous fluid in

a sphere which was accelerating arbitrarily and moving rectilinear in an

otherwise quiet flow:

c +c

±UL\

u\\\ np \ yft - r

dr (3.67)

where C

is the drag coefficient at unsteady motion.

Karanfilian and Kotas (1978) conducted a series of similar experiments

with the condition of 10

< R

< 10

and the acceleration number 0 < A„ <

10.5.

They reformed all the terms in the right hand side of equation (3.67) in

to a single empirical term as follows:

=(i

where /? = 1.2 ± 0.03, U is the velocity of water flow and the acceleration

number is:

Sano (1981) considered the problem of unsteady viscous incompressible

flow past a sphere. The unsteady drag coefficient of a sphere for different

time domains is expressed as:

132 Abrasive Erosion and Corrosion of Hydraulic Machinery

(1) small time domain:

C - C

^-D ^d

H(t)+

-8(t)

(Kt)

-1/2

(2) large time domain:

C - C

D W

-R.

erf

-t

4rtR

v Kj

f ,r> 2\

exp

)

The similar relationships for steady drag coefficient C

, like that in

equation (3.14) are as follows:

c,=—(I

O.IS*,

687

)

d n \ <>P / ep

800

= — (l +

0.\25RJ

)

R < 1

d r> V ep ! ep

000

3.3 Numerical Simulation of Liquid-Particle Two-Phase

Flow through Hydraulic Machinery by Two-Fluid Model

3.3.1 Numerical Method for Simulating Liquid-Particle Two-

Phase Flow

Several numerical methods have been developed for solving the

incompressible N-S equations by using the primitive variables, i.e., velocities

and pressure. The main difference between these methods lies in the way of

Analysis and Numerical Simulation of Liquid-Solid Two-Phase Flow

133

finding a pressure field such that the flow field can be as close to divergence-

free as possible (in order to satisfy the mass conservation equation). This is

the main feature and difficulty for solving the incompressible

viscous/turbulent flow problem, in which there is no obvious equation to

obtain the unknown pressure. The pressure gradients only form a part of

source terms of momentum equations. The main numerical approaches for

solving this problem include:

1) The pseudo-compressibility method;

2) The approximate factorization scheme (the fractional step scheme);

3) The block-implicit finite difference method and the block-implicit finite

element method;

4) The successive pressure-velocity correction scheme. The original

version of the Semi-Implicit Method for Pressure Linked Equation

(SIMPLE) was proposed by Patanker and Spalding (1972), which

overcome the difficulty of solving pressure field by using the continuity

equation and new variables, i.e., velocity and pressure corrections. This

procedure has been widely used to solve many incompressible flow

problems. A version of SIMPLE methods improved by Y.S. Chen

(1986),

SIMPLEC, will introduce to solve incompressible two-phase

turbulent flow in this section

[3.24].

Numerical Approach of SIMPLEC in Body-Fitted Coordinates

* Transformation of equations

In many flow problems, the geometry's boundaries are very complex,

especially for internal flow problems with complicated boundaries, such as

those of centrifugal impeller and hydraulic turbine. So the use of non-

orthogonal body-fitted coordinates (BFC) can be beneficial in many aspects.

It is not only why the boundary geometry can be represented closely using

BFC systems, but also why the grid-refined solution can be easily obtained.

The governing equations of two-phase turbulent flow, expressed in the

BFC system \E,,r],C,), can be represented by the following transport model

equation (3.68) in the conservative form, in which $ denotes all the dependent

variables respectively and /"is the diffusion coefficient: