Investigation of flow through centrifugal pump impellers 2

THEORETICAL MODELS OF SINGLE-PHASE FLOW 2.1 Governing Equations Flow field within centrifugal pump impellers can usually be described by the three-dimensional Reynolds-averaged Navier-S

THEORETICAL MODELS OF SINGLE-PHASE FLOW

2.1 Governing Equations

Flow field within centrifugal pump impellers can usually be described by the three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations with a suitable turbulence model Assuming a constant property flow, the RANS equations are as follows:

0

=

•

M

T

k p U

•

3

2 ( { )

where “×” is a vector cross-product, U is the velocity, ρ the density, δ the identity matrix, S M the source term, µeff is the effective viscosity accounting for turbulence (µeff =µ+µt; µ and µt are the molecular and turbulent viscosities respectively), p

is the static pressure, k is turbulence kinetic energy

For flows in a rotating frame of reference, rotating at the constant rotation speed, Ω , the effects of the Coriolis forces are modeled in the code In this case,

)]

( 2

× Ω

× Ω +

× Ω

−

where r

is the location vector

2.2 Turbulence Models

The mathematical models used in the present investigation employ the usual continuity and momentum equations together with a suitable turbulence model

Calculations have been carried out with four selected turbulence models: the standard

ε

−

k model, RNG k−ε model, the Wilcox k−ω model and the shear stress transport (SST) model The details regarding these turbulence models are discussed below

2.2.1 k−ε Turbulence Model

In Eq (2.2), µeff is effective viscosity coefficient, which is equal to molecular viscosity coefficient µ plus turbulent eddy viscosity coefficient µt:

t

The turbulent viscosity µt is modelled as the product of a turbulent velocity scale, V , and a turbulent length scale, t l , as proposed by Kolmogorov (1941) t

Introducing proportionality constant gives

t t

t ρcµl V

Both one and two equation models take the velocity scale V to be the square t

root of the turbulent kinetic energy

k

The turbulent kinetic energy, k, is determined from the solution of a

semi-empirical transport equation

In the standard k−ε two-equation model it is assumed that the length scale is

a dissipation length scale, and when the turbulent dissipation scales are isotropic, Kolmogorov determined that

where ε is the turbulent dissipation rate

Therefore, the turbulence viscosity, µt, can be derived from Eqs (2.5), (2.6) and (2.7) to link to the turbulence kinetic energy and dissipation via the relation:

ε ρ

µ µ

2

k C

where C is a constant, the value is 0.09 (Launder and Sharma, 1974) µ

The values of k, ε come directly from the differential transport equations for the turbulence kinetic energy and turbulence dissipation rate:

ρε σ

µ µ

ρ −∇⋅ + ∇ = −

⋅

k

) (

] ) [(

)

σ

µ µ ε

ε

C P C k

⋅

where C =1.44, ε1 C =1.92, ε2 σk=1.0 and σε =1.3 are empirical constants (Launder and Sharma, 1974)

term, which for incompressible flow is:

)

t

Eqs (2.1), (2.2), (2.9) and (2.10) form a closed set of non-linear partial differential equations governing the fluid motion

2.2.2 The RNG k−ε Model

The RNG k−ε model (Yokhot and Orszag, 1986) is an alternative to the

standard k−ε model In this model, the transport equations for k and ε are the same

as those for the standard k−ε model except that the model constants are different

The transport equation for turbulence dissipation becomes

) (

) (

)

σ

µ ε

eff

C P C k

•

where,

Cε1RNG = 421 − fη

) 1

(

) 38 4 1 (

3

η β

η η

η

RNG

f

+

−

=

ε ρ

η

k C

P

=

and Cε2RNG=1.0, σεRNG =0.7179, βRNG =0.012, CµRNG =0.085

2.2.3 The Wilcox k−ω Model

Turbulence modelling is one of the most important aspects in numerical

simulations of fluid flow and heat transfer In conjuction with empirical wall functions, the conventional k −ε model has been widely used in enginnering

practice, and has turned out a success in many applications Nevertheless, some problems exist when using the wall-fuction method (Patel et al., 1984) The lack of universality of the wall functions has been frequently criticized Another major deficiency with the k−ε based models is the uncertainty of specifying ε at the wall

(Peng et al 1997) In addition, the k −ε models have other drawback that they are

very inaccurate for flows with adverse pressure gradient (Wilcox, 1993) Therefore, it

is necessary to investigate on some other turbulence models to find out if they could

provide more accurate numerical results when simulating water flow between the impellers under pump off-design operating condition

The Wilcox k−ω model is chosen for two reasons First, it gives superior results for the adverse pressure gradient flows computed by Wilcox (1988) when compared with other two-equation models Second, because of its mathematical simplicity, it does not need damping functions in the low Reynolds number region close to the wall, and the ω equation has an exact boundary condition at the wall Of specific interest is the behavior of this model in the case of separated flow, since no detailed comparison with sperated flow data in the centrifugal pump impeller has yet been reported

For the Wilcox k−ω model (Wilcox, 1988), the turbulence viscosity µt is given by µt =α*ρk/ω , where k is turbulence kinetic energy, ω is specific dissipation rate, and α* is a closure coefficient The equations for k and ω are as follows:

] ) [(

) ( Uk = P k − * k+∇• + * t ∇k

•

] ) [(

) (ρ ω =αω −βρω2 +∇• µ+σµ ∇ω

•

k

where P is the turbulent kinetic energy production, its definition is given in the k

relation (2.11)

The closure coefficients for the Wilcox k−ω turbulence model are given as

follows:

*

α =1, α=5/9, β*=9/100, β=3/40, σ*=1/2, σ =1/2

2.2.4 Shear Stress Transport (SST) Model

While standard two-equation models provide good predictions for many flows

of engineering interest, there are still applications for which these models will fail Among these are:

Flows with sudden changes in the mean strain rate

Flows over curved surfaces

Flows in rotaing fluids

Flows with boundary layer separation

The weaknesses of standard two-equation models are well known and have resulted in a number of modifications and model enhancements A major improvement in terms of flow separation predictions has been achieved by the k−ω

based SST model It accounts for the transport of the turbulence shear stress and gives highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients

The Shear Stress Transport (SST) model developed by Menter (1994) uses the

ω

−

k model in the sublayer as well as in the logarithmic part of the boundary layer

At the boundary layer edge and outside boundary layer, the standard k−ε turbulence model is then recovered The equations of the SST model are given by

] ) [(

) ( Uk = P k − * k+∇• + k t ∇k

•

and

ω ω σ ρ

ω µ σ µ βρω

υ

γ ω ρ

ω

ω

∇

∇

− +

∇ +

•

∇ +

−

=

•

∇

k F

P

t

1 ) 1 ( 2

] ) [(

) (

2 1 2

(2.16)

where ω is the specific dissipation rate, υt is the eddy kinematic viscosity (=

ω

k

)

Also, the blending function F1 is such that it is unity near the surface and is zero away from it, resulting in the k−ω model in the near-wall region and the k−ε in the remainder of the flow field The formulation of F1 is defined as

) tanh(arg4

1

1 =

F

]

4 );

500

; 09 0 min[max(

2 1

y CD

k y

y

k

kω ω

ρσ ω

υ ω

=

where y is the distance to the next surface, υ is the molecular kinematic viscosity

and CD is the positive portion of the cross-diffusion term of Eq (2.16): kω

) 10 , 1

2

2 ∇ ∇ −

ω

ρσω

CD k

The values of the constants σk, σω, σω2, β*, β, γ in Eqs (2.15) and (2.16) are derived from the constants of k−ω and k−ε models If we let φ1 represent any constant in the k−ω model (σk1, …), φ2 any constant in the transformed k −ε

model (σk2 , …) and φ the corresponding constant of the SST model (σk, …), then the relation between them is:

2 1 1

1φ (1 )φ

φ = F + −F

The constants of set 1 (φ1) are (Wilcox):

85 0

1 =

k

σ , σω1 =0.5, β1 =0.0750

09 0

* =

β , κ =0.41, 2 *

1

* 1

1 β /β σ κ / β

γ = − ω The constants of set 2 (φ2) are (standard k−ε ):

0 1

2 =

k

σ , σω2 =0.856, β2 =0.0828

09 0

* =

β , κ =0.41, 2 *

2

* 2

2 β /β σ κ / β

γ = − ω

The eddy viscosity υt in Eq (2.16) is defined as:

)

; max( 1 2

1

SF a

k a

υ =

where S is the absolute value of the vorticity, α1 is constant, its value is 0.31 F2 is

given by tanh(arg2)

2

2 =

09 0 2 max(

ω

υ

ωy y

k

=

2.3 Log-Law Wall Functions

There are very large gradients in the dependent variables near the wall It is

very costly to fully resolve the solution in this near-wall region, as the required number of nodes would be quite large Thus a common approach known as “wall functions” is applied to model this region

In the wall function approach (Launder and Spalding, 1974), the near wall tangential velocity is related to the wall shear stress by means of a logarithmic relation, which can be written as follows:

C y

u+ = 1ln( +)+

τ

u

u

µ

ρ yuτ

2 / 1

) (

ρ

τ

where τw is the wall shear stress, u is the known velocity tangent to the wall at a t

distance of y∆ from the wall, and κ is the Von Karman constant for smooth walls κ

and C are constants depending on wall roughness

However, this form of wall function equations has the problem of becoming singular at separation points where the near wall velocity, u , approaches zero In the t

logarithmic region, an alternative velocity scale u can be used instead of * u : +

k c

µ

This scale has the useful property that it does not go to zero if u goes to zero t

(and in turbulent flow k is never completely zero) Based on this definition, the

following explicit equation for the wall shear stress is obtained:

+

=

u

y visc

w

*

τ

where τvisc =µu t/∆y,

µ

ρ * /

* u y

C y

u+ = 1ln( *)+

κ

The recommended practice is to locate near-wall nodes such that y is in the *

range of 20 to 50, for smooth walls

In the near wall region, an estimate of the dissipation consistent with the log-law can be presented as

y

k c

∆

=

κ

The dissipation at the first interior node is set equal to this value The

boundary nodal value for k is estimated via an extrapolation boundary condition

The near-wall production of turbulent kinetic energy is derived to be

2

k

visc

p

µ

τ

where * ( *)2 *

dy

du u

y

2.4 Discretisation Method

The present numerical study is based upon a Finite Volume Approach Method This approach involves discretisation of the integral form of the governing equations, which are solved over a number of finite volumes within the fluid domain

Consider the general scalar equation for the variable φ with no sources:

0 ) (

) ( )

(

=

∂

∂

∂

∂

−

∂

∂ +

∂

∂

j

eff j j

j

x x

x

u t

φ µ φ

ρ ρφ

This can be integrated over the control volume to give

0 ˆ

∂

∂

− +

∂

∂

∫

∫

j

A eff A

j

x d

n u d

t

φ µ φ

ρ ρφ

where nˆ is surface outward normal vector and A and V are outer surface area and j

volume respectively

The first step in solving these equations numerically is to approximate them using discrete function For example, the advection term can be approximated by,

ip

ip ip A

j

Aρu j n φd ∑m φ

where m ip = ρu j nˆj∆A

is the discrete mass flow through a surface of the finite

volume, A∆ is the surface area, φip is the discrete value of φ at the integration point, and the sum is over all the surfaces of the finite volume

To complete the discretisation of the advection term, the variable φip must be related to the dependent variables stored at the nodes of the element, φn As transported variables move with the flow, it is physically reasonable to approximate

the variable at ip by the upstream nodal variable at n, to give φip = φn

This is called the Upwind Difference Scheme (UDS) and it is first-order accurate

2.5 Boundary Conditions

The boundary conditions of single-phase flow are specified as follows:

Inlet Boundary A constant mass flow rate is specified at the inlet of

calculation domain Different mass flow rates are specified to study pump design and

off-design conditions The turbulent kinetic energy k, diffusion rate ε and the specific dissipation rate ω in the inlet of calculation domain are defined as:

2

005

) / ( 2

3 4

3

m

ε =

m

in

u

) 10 1

( →

=

ω

where u is inlet velocity, in C is an experimental constant and is 0.09 and µ L is the m

Prandtl’s mixing length scale and is assumed to be 0.5 inlet hydraulic diameters (Wang et al., 2002)

Outlet Boundary In the outlet of calculation domain, the gradients of the

velocity components and k, ε, ω are assumed to be zero respectively by following the suggestion of Keimasi and Taeibi (2001)

0

=

∂

∂

n

u j

(j=1,2,3)

0

=

∂

∂

=

∂

∂

=

∂

∂

n n n

Solid Walls For surface of blade, hub and casing, relative velocity

components are set as zero Also the standard wall function is used near the walls when the standard k−ε model is implemented For the k−ω and SST models, the

turbulence kinetic energy k was set to zero at the walls and ω is obtained by the following relation suggested by Menter (1994):

2 1

/

60υ β y p

where y is the distance to the next point away from the wall, p υ is the molecular kinematic viscosity, and β1 is a constant with the value of 0.075

Periodic Condition In the pairs of the boundaries of the computational domain before and after impeller, the periodic conditions are applied:

right left |

ϕ =

where ϕ = u,v,w,k,ε,ω,p