A variational multiscale method for particle-cloud ...

A. Corsini, F. Rispoli, A.G. Sheard, K. Takizawa, T.E. Tezduyar, and P. Venturini, “A variational multiscale method for particle-cloud tracking in turbomachinery flows,” Computational Mechanics, 54 (2014) 1191–1202, http://dx.doi.org/10.1007/s00466-014-1050-0

A variational multiscale method for particle-cloud tracking in turbomachinery flows A. Corsinia, F. Rispolia, A.G. Sheardb, K. Takizawac, T.E. Tezduyard and P. Venturinia

a

Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza University of Rome Via Eudossiana, 18, I-00184 Rome, Italy [email protected], [email protected] b

Fläkt Woods Ltd Axial Way, Colchester, Essex, C04 5ZD, UK [email protected] c

Department of Modern Mechanical Engineering and Waseda Institute for Advanced Study, Waseda University 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, JAPAN [email protected] d

Mechanical Engineering, Rice University - MS 321 6100 Main Street, Houston, TX 77005, USA [email protected]

ABSTRACT We present a computational method for simulation of particle-laden flows in turbomachinery. The method is based on a stabilized finite element fluid mechanics formulation and a finite element particle-cloud tracking method. We focus on induced-draft fans used in process industries to extract exhaust gases in the form of a two-phase fluid with a dispersed solid phase. The particle-laden flow causes material wear on the fan blades, degrading their aerodynamic performance, and therefore accurate simulation of the flow would be essential in reliable computational turbomachinery analysis and design. The turbulent-flow nature of the problem is dealt with a RANS model and SUPG/PSPG stabilization, the particle-cloud trajectories are calculated based on the flow field and closure models for the turbulence-particle interaction, and one-way dependence is assumed between the flow field and particle dynamics. We propose a closure model utilizing the scale separation feature of the variational multiscale (VMS) method, and compare that to the closure utilizing the eddy viscosity model (EVM). We present computations for axial- and centrifugal-fan configurations, and compare the computed data to those obtained from experiments, analytical approaches, and other computational methods.

1 Introduction Particle-laden flows are typically present in the operating environment of large axial and centrifugal fans used in process industries (such as cement, steel and power industries). In such applications, the flow, which is turbulent, carries solid or molten particles,

1

formed during the combustion process [1 - 3]. The particles are responsible for fouling and erosion phenomena, which drive the fan performance deterioration. The dynamics of the dispersed solid phase in turbulent flow is affected by the particle’s inertia and drift velocity. Those factors, in combination with the physical and chemical properties of the particles, influence the erosion in turbomachinery configurations [4, 5]. Therefore accurate simulation of the particle-laden flow is essential in reliable computational turbomachinery analysis and design that takes erosion into account. In the 1970s, motivated by in-service erosion in helicopter and other aircraft engines, several researchers started developing methods for computing the particle trajectories and related erosion in turbomachinery. Tabakoff and coworkers [6, 7] carried out particle-trajectory computations in axial and centrifugal turbomachinary, underpinning the interaction between the particle dynamics and the inertial forces in rotating cascades. Later numerical studies [8–10] focused on various modeling aspects of sand erosion. Turbulence-particle interaction is one of the main computational chellenges in simulation of particle-laden flows. In this paper, we present a computational method for simulation of particle-laden flows in turbomachinery, with

emphasis on the turbulence-particle interaction. We are targeting large, induced-draft fans, used in the process industries to extract exhaust gases in the form of a two-phase fluid with a dispersed solid phase. We consider both axial and centrifugal fans. The method is based on a stabilized finite element fluid mechanics formulation and a finite element particle-cloud tracking (PCT) method. The PCT method was formulated by Baxter [11], and then further developed [12 - 15] and improved to obtain statistically independent results [16]. One-way dependence is assumed between the flow field and particle dynamics. The trajectory of the particle-cloud center is calculated with a finite element method, where the discrete representation of the cloud is based on the elements of the particle mesh inside the cloud with trajectory-depenedent radius. The tracking method accounts for the drifting-velocity gradient in the near-wall regions [12, 13].

The turbulent-flow nature of the problem is dealt with a 3D Reynolds-Averaged Navier-Stokes (RANS) model and the Streamline-Upwind/Petrov-Galerkin (SUPG) [17] and Pressure-Stabilizing/Petrov-Galerkin (PSPG) [18, 19] stabilizations. These are complemented with the DRDJ stabilization [20 - 23]. The stabilization and discontinuitycapturing parameters to be used with the SUPG and PSPG formulations received much attention (see, for example, [24 50]). Here we use the ones given in [30]. The particle-cloud trajectories are calculated based on the flow field and closure models for the turbulence-particle interaction. We propose a closure model utilizing the scale separation feature of the variational multiscale (VMS) method [51]. We compare that to the classical closure approach utilizing the eddy viscosity model (EVM). We use an in-house parallel finite element (FEM) parallel solver [52] based on C++ libMesh libraries [53]. We carry out computations for two induced-draft fans for coal-fired power plant exhaust systems, with axial and centrifugal impellers and similar total pressure rise [14, 15]. We compare the computed data to those obtained from

2

experiments, analytical approaches, and other computational methods. In Section 2, we provide an overview of the mathematical model, including the RANS and PCT models. The SUPG/PSPG/DRDJ stabilized finite element formulations for the Navier-Stokes and RANS equations are described in Section 3. In Section 4, we describe the discretized particle equations, including the turbulence-particle interaction. The computations are presented in Section 5, and the concluding remarks are given in Section 6.

2 Model 2.1 Fluid-phase RANS model for incompressible turbulent flows Let   R nsd be the spatial domain with boundary, and (0, T) be the time domain. The unsteady RANS equations of incompressible turbulent flows can be written on  and t   0,T  as  u  u u    t



      0 , 

(1)

 u  0 ,

(2)

   u   Bk    k  t



          νk   0 , 

(3)

T where  is the density, u the velocity vector,   k ,  is the vector of turbulence closure variables, and k and  are the turbulent

kinetic energy and homogeneous dissipation. The symbols  and  k represent the vector of external forces and the source vector of turbulence closure equations. As proposed in Corsini and Rispoli [54],  accounts for the volume sources related to the second- and third-order terms in the non-isotropic stress-strain relation [55]. The force vector reads as 

 

1  

     0.1 t  ε  u   ε  u   ε  u  : ε  u  I  3 



0.1 t   u   ε  u     u   ε  u  





1   0.26 t   u    u     u  :   u  I  3  



10c2 t 2 ε  u   ε  u    u    ε  u   ε  u    u  

T



5c2 t 2  ε  u  : ε  u   ε  u   5c2 t  2   u  :   u   ε  u   .

(4.1)

3



Here ε  u    u    u 

T

 is twice the strain-rate tensor,   u   u   u  is twice the vorticity tensor,  is the turbulent T

t

kinematic viscosity defined as t = c f k, and  = k/  is the turbulence time scale, with c and f and other closure coefficients for the turbulence model [55] used given in Table 1. In Table 1, Ret = k2 /(  ) is the turbulence Reynolds number, with being the molecular viscosity, and eˆ and ˆ are, respectively, the strain-rate and vorticity invariants defined as eˆ   0.5ε  u  : ε  u  and ˆ   0.5  u  :   u  . Table 1: Turbulence and chemistry closure coefficients

ˆ ˆ     0.3 1  exp -0.36/exp -0.75 max  e, 





1.5

c

ˆ ˆ   1  0.35  max  e,

f

0.5 2 1  exp    Ret / 90    Ret / 400  





c1

1.44

c2

1.92

f2

[1‐0.3exp(‐ Ret2 )]



1.3

k

1.0

The source vector  k  is defined as

 k

 Pk  D  ,   c 1 Pk   E    k

(4.2)

where Pk  R : u is the production of turbulent kinetic energy, with R being the Reynolds stress tensor, D  2 k   k , and E = 2

0.0022 eˆ k t    u  . The stress tensor is defined as 2   σ  p, u     p   k  I   u ε  u  , 3  

(5)

with u =  + t.

4

The diffusion terms in the turbulence closure equations are represented with diffusivity matrix defined as

 k

  t  0     k   ,      t    0         

(6)

with the values of the coefficients k and  given in Table 1. The reaction terms, absorption like in Eq. (3), account for the dissipation-destruction matrices and are defined as B Bk    k 

 

, B 

(7.1)

with

Bk 

 k

, B  c 2 f 2

 k

,

(7.2)

and the values of the coefficients c2 and f2 are given in Table 1. The essential and natural boundary conditions for Eqs. (1) and (3) are represented as

u = g on g,

and

 = gk on gk,

(8.1)

n   = h on h,

and

n       νk  = 0 on hk,

(8.2)

where g, gk, h and hk are the subsets of the boundary , n is the direction normal to the boundary, and g, gk and h are given functions representing the essential and natural boundary conditions.

2.2 Solid-phase model Particle trajectories are simulated in a Lagrangian reference frame. Since particle concentration in this kind of applications is very small (i.e. less than 10-6 in the particle volume fraction), a one-way dependence approach can be used [56]. That is, the flow field affects particle motion but particles do not affect the flow field. The concept of one-way dependence has been used in other computational engineering analyses. For example, in [57], the concept was used for computing the aerodynamic forces acting on the suspension lines of spacecraft parachutes, where the suspension lines are assumed to have no influence on the flow field. We used the PCT model [58] to simulate a large number of particles without tracking them individually. The PCT approach was used in turbulent particle dispersion [11, 16, 59–61] and validated in turbomachinery and biomass furnaces [62, 63]. In the PCT model, each trajectory is not related to a particle, but to a group of particles (a “cloud”), thus representing the evolution of the cloud position at time t: (9)

t

xc   v c dt '   xc 0 . 0

5

Here, subscript c refers to the cloud, v c is the velocity of the cloud, and  xc 0 is the initial position of the cloud, which is at the inflow boundary in our computations. The equation of motion for the cloud is given by the Basset-Boussinesque-Oseen formulation, which, with one-way dependence hypothesis according to Armenio and Fiorotto [64], reads as

 dvc    R1  u  vc   f  1   p dt 

(10)

 g ,  

where f is the centrifugal and Coriolis forces, p is the particle material density, and  R is the particle relaxation time, which, for spherical particles, reads as

 R1 

3  CD u  vc . 4d p p

(11)

Here, dp is the particle diameter and CD is the drag coefficient based on the particle Reynolds number Rep  introduced in [65]. The initial condition for vc is given as v c  0   u

t 0

u  vc d p



, first

.

The ensemble average for the dispersed solid phase within the cloud is defined according to the hypothesis of independent statistical events, and for any quantity  it reads as

 

  PDF  x, t  d 

c



PDF  x , t  d 

(12)

.

c

Here,  is ensemble-averaged quantity, c is the cloud domain, and PDF(x, t) is the multi-variate probability density function of the dispersed phase. This definition of the ensemble average is appropriate for stationary and non-stationary quantities, and also for both continuous and discontinuous quantities The PCT approach assumes that particle position distribution within a cloud is Gaussian, and the cloud size varies in time according to the properties of the flow. To this end, the PDF describing the particle distribution within the cloud reads as

PDF (x, t ) 

(13)

 1  x  x 2  c   exp  . 12 2 σ   2  σ     1

Here, σ is the square root of the variance of particle position, which accounts for the turbulent dispersion of particles. We will define it in Section 4. The cloud size (i.e. cloud radius) is taken as 3 , and that gives us c. Each cloud is assumed to consist of perfectly spherical particles with the same chemical and physical characteristics. Combining Eqs. (10) and (11), we obtain

 dv c   CD u  v c  u  v c   f  1   p dt 

(14)

 g ,  

where

6

3  CD . p 4d p

CD 

(15)

3 SUPG formulation of fluid mechanics equations of turbulent flows 3.1 Stabilized formulations

In describing the SUPG formulation of Eqs. (1), (2) and (3), we assume that we have constructed some suitably-defined finitedimensional trial solution and test function spaces  uh , ph ,  h and  uh , ph ,h . The SUPG/PSPG formulation reads as follows: find uh   uh , p h   ph , h   h such that w h  uh , q h  ph and  h h :  u h  h   uh u h   h  d   w     t 

 ε  w  : σ  p , u  d     q   u d   h

h

e 1

h

h

w h  hh d  

h

nel

 

h

e









h P stab w h ,q h   Ł p h , uh    d   0  

(16.1)

where  w h  Ł(qh, wh) =   + u h  w h       q h , w h  ,  t  

(16.2)

and

 ψ

h

  h   uh  h  Bk    kh  t 

 ψ nel

  e 1

e

nel

  e 1

e

h

 

 

  d  

:   h ν k  d  

 

 

Pkstab ψ h   Łk  h   kh  d    h h Κ kDC  ψ :  d   0,

(17.1)

where   h  Łk(h)=   + u h    h  Bk   h         h  ν k  .  t  





(17.2)

Notably, in the discretization of the source vector of the turbulence closure equation  kh , we compute the    u  in the E term by first calculating the nodal values of u by least-squares projection and then taking the divergence of the interpolated value of u .

7

DC In Eqs. (16.1), (16.2), (17.1), and (17.2) P stab , Pkstab  , and Κ k  are, respectively, the SUPG stabilization operators and the dissipation

matrix for the discontinuity-capturing (DC) scheme. The vectors P stab , Pkstab  take the following forms:

 





P stab w h   SUPG uh  w h 

0  SUPG-k Pkstab ψh     SUPG-  0

 PSPG 

q ,

(18.1)

 h  u   h . 

 





(18.2)

Here SUPG and PSPG are the SUPG and PSPG stabilization parameters, and the latter is for being able to use equal-order velocitypressure approximations. These are defined in Section 3.2. The DC dissipation terms for advection-diffusion-reaction equations are defined as  DRDJ  k Κ kDC     0



0

 DRDJ  

.

(19)

Here DRDJ-k and DRDJ- are the DRDJ diffusivities, (see [20 – 23]]).

3.2 Stabilization parameters We first define the element length [27] in the advection-dominated limit: 1

 nen  hUGN  2   s  N a  ,  a 

(20.1)

where s is the unit vector in direction of the velocity, nen is the number of element nodes, and Na is the interpolation function associated with node a. In the diffusion-dominated limit, the element lengths [30, 34] are defined as follows:  nen  hRGN  2   r  N a   a 

1

(20.2)

hRGN  k  2(  rk  N a )1 ,

(20.3)

hRGN   2(  r  N a )1 ,

(20.4)

a

a

where r, rk and r are the unit vectors in the direction of the solution gradient defined as r

 u  k   , rk  , r  .  u  k  

(21)

8

The components of SUPG corresponding to the advection-, transient-, and diffusion-dominated limits were defined in [30, 38] as follows:  nen



1

h

 SUGN 1    u  N a   UGN , 2 u  a 

(22.1)

 SUGN 2 

t 2

 SUGN 3 

2 hRGN h2 h2 ,  SUGN 3 k  RGN  k ,  SUGN 3  RGN  . 4 4 k 4 

(22.2) (22.3)

From these, the stabilization parameters are defined by using the r-switch [28]:

 SUPG

 1 1 1   rs  rs  rs   SUGN 1  SUGN  SUGN 3 2 

   



1 rs

,

(23.1)

 PSPG   SUPG ,

 SUPG 

(23.2)

 1 1 1   rs  rs  rs    SUGN 1  SUGN 2  SUGN 3 

   



1 rs

.

(23.3)

Here the subscript  generates the expressions corresponding to k and  . Typically, rs = 2. The SUPG “diffusivities” are defined as 2

2

 SUPG   SUPG u ,  SUPG    SUPG  u ,

(24)

with  indicating the variables modeled with advection-diffusion-reaction equations.

4 Discretized particle equations

In the discretized particle equations, ensemble averaging is carried out over the discretized cloud domain c =

 c e

nelc

  c e , where

e 1

is the cloud element, and nelc is the number of elements. In our computations, the cloud elements come from a fixed mesh,

which we call “particle mesh”, and consist of the elements of that fixed mesh within the radius  = 3. With that, the discretized version of ensemble averaging is written as

9

nelc



h



  e 1

 PDF  x, t  d 

  e c

,

nelc

  e 1

  e c

(25)

PDF  x, t  d 

where the element-level integration is performed by Gaussian quadrature.

4.1 Trajectory calculation

Spatially-discretized version of Eq. (14) is written as dv ch  ach , dt

(26)

where h

ach  CD u

 v ch  u 

h

 v ch   f 

h

   1   p 

 g .  

(27)

Temporal discretization of Eq. (26) is carried out with a predictor/multi-corrector algorithm. Predictor:

v 

h 0 c n 1

   a 

 v ch

h c

n

n

t .

(28)

Corrections:

v 

h i 1 c n 1

 

 v ch

n

   a 

  ach 

n

h i  c n 1  

t , 2

(29)

where the superscript i is the counter for the multiple corrections. We stop the corrections when

v 

h i 1 c n 1

 

 v ch

 

i 1 v ch n 1

i n 1

 2  102 .

(30)

At each time step, the PCT model requires the computation of the cloud mean position and radius, and the identification of the elements contained within the cloud volume. This is done with the search algorithm described in [13]. 4.2 Turbulence–particle interaction parameters

The variance is taken to be dependent upon the Lagrangian time scale of the particle laden flow, L, and, according to Baxter [58], its Markovian approximation reads as (31)

t  2 σ 2  2  v c  L 2   1  e t / L   σ 02 .   L 





where L is defined as

 L  max  , p   max  , R 

(32)

with p given as p = R and  to be defined below. The fluctuating component of the particle velocity for the cloud, as driven by the turbulence–particle interaction [66], reads as

10

 v 'c   u 'c 1  e  2

2

p

.

(33)

We adopt two sets of definitions for  u ' c and . The first one uses the information given by the eddy viscosity model (EVM) 2

through the turbulence scale determining equations with the following definitions [58]:

 u 'c  EVM 2



2 k 3

C  

h 34



0.817 

h

k

,

(34)

h

.

h

(35)

The second set of definitions, on the other hand, is based on the variational multiscale (VMS) approach first proposed in [51] and further developed for RANS computations [21, 38, 67]. In this case  u ' c is obtained based on the VMS scale separation u  u h  u ' , 2

where u h is the resolved flow velocity and u ' is the fine-scale flow velocity modeled as





h 1 u '    SUPG Ł( p h , u h )   .



(36)

Then the definitions of the VMS turbulence–particle interaction parameters become

 u 'c VMS 2

 u'

2

h

,

(37)

   SUPG .

(38)

5 Computations Large industrial fans, used in power industry, are usually tailor-made. They have to meet the flow-rate and pressure rise requirements, the machinery spacing limits, and the structural requirements, often with a specific requirement about resistance to erosion. We perform our studies using two induced-draft fans, for coal-fired power plant exhaust systems, with axial and centrifugal impellers and similar total pressure rise. The equations are solved in the rotating reference frame of the fan rotor. Consequently, the computations are based on the version of Eq. (1) that includes the non-inertial terms, and in the implementation of the stabilized formulations these terms are just added to the other source terms. Alternatively, the ALE techniques as in [68 - 70], or the space-time technique as in [50, 51], can be employed to handle the rotating motion. We solve the Navier-Stokes and turbulence closure equations in a fully-coupled fashion. The linear solver uses 5 outer and 5 inner GMRES iterations, with SOR preconditioning.

5.1 Description of the axial fan The axial fan has a rotor with 24 unswept blades with blade sections belonging to the C4 profile family. Table 3 provides the fan geometrical specifications, together with the details of the blade geometry at the hub and tip sections [13, 14]. Two different meshes are used (see Figure 1). The fluid mechanics mesh is fully unstructured. It consists of about 4.5 million linear P1-P1 elements. The particle mesh is block-structured hexahedral. It consists of about 0.9 million tri-linear elements. The

11

meshes have sufficient refinement near solid boundaries, with the ratio of the first-layer spacing to mid-span blade chord set as 710-4 on the blade tip, casing wall and blade surfaces. Table 3 Axial-fan specifications Fan rotor data Blade number

24

Hub-to-casing diameter ratio

0.7 1767 mm

Tip diameter

6

Rotor tip clearance (% blade height)

890 rpm

Rated rotational frequency Blade geometry

Hub

Tip

Chord

487 mm

446 mm

Solidity

1.48

0.96

Stagger angle

38°

60°

Camber angle

45°

36°

Operating point 3

Volume flow rate

380 m /s

Total pressure rise

12000 Pa

The inflow boundary conditions for the fluid mechanics equations are defined according to recent experimental and numerical studies on ducted high-solidity fans [72]. Specifically, turbulence measurements in ducted industrial fans provide the inlet distribution of the turbulence variables [73]. Flow periodicity, upstream and downstream of the blade row, and Neumann outflow conditions complete the set of boundary data.

Figure 1 Axial fan a) blade, b) fluid mesh, c) particle mesh

5.2 Description of the centrifugal fan The centrifugal fan is a high-pressure fan unit with backward curved blades with performance spanning over a range of duties typical of modern power industry [15]. Table 4 lists the specifications for the fan.

12

The model consists of three parts (inlet bell-mouth, impeller and volute), which are discretised using a block-structured mesh to enable a simple handling of the rotor-stator interface (see Figure 2). The grid for the inlet bell-mouth, the impeller and the volute is hexahedral, made of about 6.7 million tri-linear Q1-Q1 elements. The particle mesh consists of only the inlet bell-mouth and impeller parts of the model. Table 4 Centrifugal-fan specifications Fan rotor and blade data Impeller inlet diameter

1804 mm

Impeller outlet diameter

3440 mm

Volute outlet diameter

5600 mm

Blade height at the impeller inlet

400 mm

Blade height at the impeller outlet

200 mm

Impeller exit angle

105.5°

Impeller blade count

11

Rotational frequency

900 rpm

Operating point 3

Volume flow rate

237 m /s

Total pressure rise

16000 Pa

The inflow velocity profile is derived from an earlier simulation in order to account for the presence of the inlet plenum [15]. The double-width/double-inlet centrifugal-fan configuration is approximated by using symmetry condition on the mid-plane of the fan. An inflow average turbulence intensity of 10% is used, while the dissipation length scale is set to be 10% of the mean blade chord.

Figure 2 Centrifugal fan a) blade, b) fluid and particle meshes, c) close up view.

5.3 Results 5.3.1 Initial conditions for the particle clouds Clouds enter the domain at three different locations on the inflow section orthogonal to the fan axis. The particle seeding radii are illustrated in Figure 3 and Figure 4, for the axial and the centrifugal fans respectively.

13

Figure 3 Axxial-fan particcle seeding poositions at thee inflow sectioon (r1 = 7% blaade span - squ uare, r2 = 70% % blade span n - triangle, r3 = 80% bladee span - circlee)

Figure 4 Centriifugal-fan particle seedingg positions at the inflow secction (r1 = 30% inflow w radius - circcle, r2 = 50% inflow radiuss - triangle, r3 = 75% infloow radius - sq quare) T The seeding ppositions are sselected in ordder to explore the dynamic response of tthe particles uunder differentt turbulence leevels as per thhe endwall booundary layerss. Table 5 show ws the diverse turbulence inttensity values at the seedingg radii.

144

Table 5 Turbulence intensity percentage at the fan inflow

Axial fan

Centrifugal fan

r1

10.3

4.9

r2

3.2

4.8

 





Table 6 provides the particle characteristics used in our studies in terms of the particle diameter, particle density, and the Stokes





number (Stk). The Stokes number in our studies is defined as Stk = ˆp / ˆ , where ˆ p   p d p2 /  

 and ˆ

= L0/U0. Here, L0 is the

blade-tip diameter and U0 is the average inflow velocity. In our studies, dp is in the range 50 to 70 m, and p is in the range 2000 to 3500 kg/m3, and, consequently, Stk is in the range 0.65 and 1.28.

Table 6 Particle characteristics

Axial fan dp (μm)

Centrifugal fan

50

50

70

50

62

p (kg/m )

2250

3500

2250

2000

2000

Stk

0.65

1.01

1.28

0.65

1.00

3

5.3.2 Turbulent dispersion of the particles To evaluate the long-term turbulent dispersion of particles, we have first computed the Eulerian and Lagrangian time scales, TmE and TL respectively. These time scales are defined as TmE = ˆ and TL =  L . The sensitivity study has been carried out by computing the time scales as mean values along the cloud trajectories. We assess the time scale behavior of the turbulence-particle interaction by using the algebraic equation given by Wang and Stock [73] for the dispersion of heavy particles as a function of their inertia. We evaluate that expression based on our computed data, and compare that to the evaluation based on the data [74] derived from direct numerical simulation (DNS) of homogenous and isotropic turbulent flows. Accordingly, the trajectory-averaged fluid mechanics time scale T seen by a particle as a function of its inertia (i.e. Stk number) reads as:

1  TL TmE  T ( Stk ) .  1 0.4 1 0.01Stk  TmE 1  Stk  

(39)

We perform a series of simulations computing TL and the ratio T/TmE while varying Stk as given in Table 6. Since TL changes at each time instant, the ratio T/TmE ratio also changes, thus a trajectory average is computed for each simulation.

15

In Figure 5, we show the ratio T/TmE as evaluated from Eq. (32) based on the VMS and EVM turbulence-particle interaction closure models and based on the DNS data. We also show in Figure 5 two curves bounding the experimental values from Sato and Yamamoto [75] for grid-generated turbulence.

1

0.8

T/TmE

0.6

0.4

0.2

0 -0.1

0.4

0.9

1.4

1.9

Stk (-)

Figure 5 Axial fan. Turbulence-particle time scale as a function of the particle inertia (VMS closure: empty symbols, EVM closure: filled symbols, Wang and Stock [73]: solid line, Sato and Yamamoto [75]: region bounded by the dashed lines)

The time scale behaviors given by the VMS and EVM closure models match well the behavior given by the DNS. Furthermore, independent of the turbulence-particle interaction closure model used, the computed data falls within the range of experimental values for grid-generated turbulence [74]. Examining the influence of particle seeding location, the clouds in the annulus boundary layer (represented by the square symbols) always show higher particle time scales, approaching TmE. This is due to the reduction of the inertia effect within the velocity-defect region. In a similar fashion, Figure 6 shows for the centrifugal fan the ratio T/TmE predicted with the VMS and EVM closure models compared to the behaviour given by Equation (39) [73], and the experimental data from Sato and Yamamoto [75]. Again, the time

16

scale values obtained with the two closure are quite close. Comparing the computed, DNS-based, and experimental data, it is evident that for all seeding locations the turbulence-particle time scale is less sensitive to Stk. Furthermore, for all seeding locations, the combination of the turbulence intensity level and the fluid velocity magnitude gives time scale values that are closer to the lower bound of the experimental data from Sato and Yamamoto [75].

1.00

0.80

T/TmE

0.60

0.40

0.20

0.00 -0.1

0.2

0.5

0.8

1.1

1.4

Stk (-)

Figure 6 Centrifugal fan. Turbulence-particle time scale as a function of the particle inertia (VMS closure: empty symbols, EVM closure: filled symbols, Wang and Stock [73]: solid line, Sato and Yamamoto [75]: region bounded by the dashed lines)

Figure 7 shows the square of the particle velocity-scale ratio computed with the VMS and EVM turbulence-particle interaction closure models. The square of the particle velocity-scale ratio, according to the Lagrangian velocity correlation in [73], is obtained

 

 

from the minimum v'1c and maximum v'3c particle fluctuating velocities, and reads as

 v'1c   v'3c

2

 0.5 TmE / ˆ  Stk  ,   1  2   2   1  TmE / ˆ  Stk 1   T / ˆ    

(40)

17

wherre    R g / U0 . Here, thee particle veloccity-scale is coomputed from m the cloud enssemble averagge quantities.

Figure 7 Square of the particle velocity-scale raatio as a functtion of particlle diameter

The aaxial-fan data is for particlees with 70 m m diameter, andd the centrifuggal-fan data is for particles w with 62 m diiameter (see Table T 6). In booth cases, the results plottedd are obtainedd by averagingg from the sim mulations carrieed out with diifferent seedinng locations. F Figure 7 also shows the com mputed data ffrom [73], [599] and [76], annd the Csanaddy limit for zeero-inertia partticles [77]. We note from F Figure 7 d reported in i literature. that, independent oof the turbulennce-particle closure model, tthe computed data is withinn the range of data

6 Co onclusions We hhave presentedd a computatioonal method foor simulation oof particle-ladden flows in tuurbomachineryy. The two maain componentts of the methhod are a stabiilized finite element fluid m mechanics form mulation and a finite elemennt PCT methood. Our focus w was on induceed-draft fans used in proceess industries to extract exhhaust gases inn the form of a two-phase ffluid with a diispersed solidd phase. The particlep ladenn flow causes material wearr on the fan blades, degradiing their aeroddynamic perfoormance. Accuurate simulatioon of the flow w would

188

be essential in reliable computational turbomachinery analysis and design, and this motivated the development of the method presented. The turbulent-flow nature of the problem is dealt with a 3D RANS model and the SUPG, PSPG and DRDJ stabilizations. The particle-cloud trajectories are calculated based on the flow field and closure models for the turbulence-particle interaction, and one-way dependence is assumed between the flow field and particle dynamics. We proposed a closure model utilizing the scale separation feature of the VMS method, and compared that to the closure utilizing the EVM. We presented computations for axial- and centrifugal-fan configurations, and compared the computed data to those obtained from experiments, analytical approaches, and other computational methods. With these computations and comparisons, we demonstrated that the method presented is quite effective in simulation of particle-laden turbomachinery flows.

Acknowledgements

The authors acknowledge MIUR support under the projects Ateneo and the Visiting Professor Programme at University of Rome “La Sapienza”. The authors also acknowledge FläktWoods Solyvent and FläktWoods AB.

REFERENCES

[1] Kurz, R., Brun, K., 2001, “Degradation in gas turbine systems”, ASME Journal of Engineering for Gas Turbine and Power, 123, pp. 70–77. [2] Hamed, A., Tabakoff, W., and Wenglarz, R., 2006, “Erosion and Deposition in Turbomachinery,” J. Propul. Power, 22, pp. 350– 360. [3] Tabakoff, W., Kotwal, R., and Hamed, A., 1979, “Erosion Study of Different Materials Affected by Coal Ash Particles,” Wear, 52(1), pp. 161–173. [4] Grant, G., and Tabakoff, W., 1975, “Erosion Prediction in Turbomachinery Resulting From Environmental Solid Particles”, J. Aircr., 12(5), pp. 471-478. [5] Richardson, J. H., Sallee, G. P., and Smakula, F. K., 1979, “Causes of High Pressure Compressor Deterioration in Service,” AIAA Paper No. 79-1234. [6] Hussein, M. F., and Tabakoff, W., 1974, “Computation and Plotting of Solid Particle Flow in Rotating Cascades,” Comput. Fluids, 2(1), pp. 1–15. [7] Elfeki, S., and Tabakoff, W., 1987, “Erosion Study of Radial Flow Compressor With Splitters,” J. Turbomach., 109(1), pp. 62–69.

19

[8] Ghenaiet, A., Tan, S. C., and Elder, R. L., 2004, “Experimental Investigation of Axial Fan Erosion and Performance Degradation,” Proc. Inst Mech. Eng., Part A, 218(6), pp. 437–446. [9] Ghenaiet, A., 2005, “Numerical Simulations of Flow and Particle Dynamics Within a Centrifugal Turbomachine,” Compressors and Their Systems 2005, IMechE Paper No. C639-52. [10] Ghenaiet, A., 2009, “Numerical Study of Sand Ingestion Through a Ventilating System,” Proceedings of the World Congress on Engineering, London, UK. [11] Baxter, L.L., and Smith, P.J., 1993, “Turbulent dispersion of particles: the STP model”, Energy and Fuels, 7, pp. 852–859. [12] Venturini, P., 2010, “Modelling of particle-wall deposition in two-phase gas-solid flows”, PhD thesis, Sapienza Università di Roma, Rome, Italy. [13] Corsini, A., Marchegiani, A., Rispoli, F., and Venturini, P., 2012, “Predicting blade leading edge erosion in an axial induced draft fan”, ASME Journal of Engineering for Gas Turbines and Power, 134(4), 042601 (9 pages). [14] Corsini, A., Rispoli, F., Sheard, A.G. and Venturini, P., 2013, “Numerical Simulation of Coal Fly-Ash Erosion in an Induced Draft Fan”, ASME, Journal of Fluids Engineering, 135, 081303 (12 pages). [15] Cardillo, L., Corsini, A., Delibra, G., Rispoli, F., Sheard, A.G., and Venturini, P., 2014, “Simulation of particle-laden flows in a large centrifugal fan for erosion prediction”,58th American Society of Mechanical Engineers Turbine and Aeroengine Congress, Düsseldorf, Germany, 16–20 June, Paper No. GT2014-25865. [16] Kær, S.K., 2001, “Numerical investigation of ash deposition in straw-fired furnaces”, PhD thesis, Aalborg University, Aalborg, Denmark. [17] A.N. Brooks, and T.J.R. Hughes, “Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations”, Computer Methods in Applied Mechanics and Engineering, 32 (1982) 199-259. [18] T.E. Tezduyar, “Stabilized finite element formulations for incompressible flow computations”, Advances in Applied Mechanics, 28 (1992) 1-44. [19] T.E. Tezduyar, S. Mittal, S.E. Ray and R. Shih, “Incompressible flow computations with stabilized bilinear and linear equal-orderinterpolation velocity-pressure elements”, Computer Methods in Applied Mechanics and Engineering, 95 (1992) 221-242. [20] A. Corsini, F. Rispoli, A. Santoriello, T.E. Tezduyar, “Improved discontinuity-capturing finite element techniques for reaction effects in turbulence computation”, Computational Mechanics 38 (2006) 356-364.

20

[21] A. Corsini, F. Menichini, F. Rispoli, A. Santoriello, T.E. Tezduyar, “A multiscale finite element formulation with discontinuity capturing for turbulence models with dominant reaction like terms”, Journal of Applied Mechanics,76 (2009) 021211. [22] A. Corsini, C. Iossa, F. Rispoli and T.E. Tezduyar, “A DRD Finite Element Formulation for Computing Turbulent Reacting Flows in Gas Turbine Combustors”, Computational Mechanics, 46 (2010) 159-167. [23] A. Corsini, F. Rispoli and T.E. Tezduyar, “Stabilized finite element computation of NOx emission in aero-engine combustors”, International Journal for Numerical Methods in Fluids, 65 (2011) 254-270. [24] T.E. Tezduyar, Y.J. Park, “Discontinuity capturing finite element formulations for nonlinear convection–diffusion–reaction equations”, Computer Methods in Applied Mechanics and Engineering, 59 (1986) 307-325. [25] T.E. Tezduyar, Y.J. Park and H.A. Deans, “Finite Element Procedures for Time-dependent Convection-Diffusion-Reaction Systems”, International Journal for Numerical Methods in Fluids, 7 (1987) 1013-1033. [26] T.J.R. Hughes and T.E. Tezduyar, “Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler 3quations”, Computer Methods in Applied Mechanics and Engineering, 45 (1984) 217-284. [27] G.J. Le Beau, S.E. Ray, S.K. Aliabadi and T.E. Tezduyar, “SUPG finite element computation of compressible flows with the entropy and conservation variables formulations”, Computer Methods in Applied Mechanics and Engineering, 104 (1993) 397-422. [28] T.E. Tezduyar and Y. Osawa, “Finite element stabilization parameters computed from element matrices and vectors”, Computer Methods in Applied Mechanics and Engineering, 190 (2000) 411-430. [29] J.E. Akin, T. Tezduyar, M. Ungor and S. Mittal, “Stabilization parameters and Smagorinsky turbulence model”, Journal of Applied Mechanics,70 (2003) 2-9. [30] T.E. Tezduyar, “Computation of moving boundaries and interfaces and stabilization parameters”, International Journal for Numerical Methods in Fluids, 43 (2003) 555-575. [31] J.E. Akin and T.E. Tezduyar, “Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements”, Computer Methods in Applied Mechanics and Engineering, 193 (2004) 1909-1922. [32] T.E. Tezduyar and M. Senga, “Stabilization and shock-capturing parameters in SUPG formulation of compressible flows”, Computer Methods in Applied Mechanics and Engineering, 195 (2006) 1621-1632. [33] T.E. Tezduyar, “Finite elements in fluids: stabilized formulations and moving boundaries and interfaces”, Computers & Fluids, 36 (2007) 191-206. [34] L. Catabriga, A.L.G.A. Coutinho and T.E. Tezduyar, “Compressible Flow SUPG parameters computed from element matrices”, Communications in Numerical Methods in Engineering, 21 (2005) 465-476.

21

[35] T.E. Tezduyar and M. Senga, “SUPG finite element computation of inviscid supersonic flows with YZ shock-capturing”, Computers & Fluids, 36 (2007) 147-159. [36] L. Catabriga, A.L.G.A. Coutinho and T.E. Tezduyar, "Compressible Flow SUPG parameters computed from degree-of-freedom submatrices”, Computational Mechanics, 38 (2006) 334-343. [37] T.E. Tezduyar, M. Senga and D. Vicker, “Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZ shock-capturing", Computational Mechanics, 38 (2006) 469-481. [38] F. Rispoli, A. Corsini, T.E. Tezduyar, “Finite element computation of turbulent flows with the Discontinuity-Capturing Directional Dissipation (DCDD)”, Computers & Fluids, 36 (2007) 121-126. [39] Y. Bazilevs, V.M. Calo, T.E. Tezduyar and T.J.R. Hughes, “YZ discontinuity-capturing for advection-dominated processes with application to arterial drug delivery”, International Journal for Numerical Methods in Fluids, 54 (2007) 593-608. [40] Y. Bazilevs, V.M. Calo, J.A. Cottrel, T.J.R. Hughes, A. Reali, G. Scovazzi, “Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows”, Computer Methods in Applied Mechanics and Engineering, 197 (2007) 173-201. [41] T.J.R. Hughes, G. Scovazzi and T.E. Tezduyar, “Stabilized methods for compressible flows”, Journal of Scientific Computing, 43 (2010) 343-368. [42] L. Catabriga, D.A.F. de Souza, A.L.G.A. Coutinho and T.E. Tezduyar, “Three-dimensional edge-based SUPG computation of inviscid compressible flows with YZ shock-capturing”, Journal of Applied Mechanics,76 (2009) 021208. [43] M.-C. Hsu, Y. Bazilevs, V.M. Calo, T.E. Tezduyar and T.J.R. Hughes, “Improving stability of stabilized and multiscale formulations in flow simulations at small time steps”, Computer Methods in Applied Mechanics and Engineering, 199 (2010) 828840. [44] T.E. Tezduyar, “Comments on `Adiabatic shock capturing in perfect gas hypersonic flows'”, International Journal for Numerical Methods in Fluids, 66 (2011) 935-938. [45] S. Takase, K. Kashiyama, S. Tanaka and T.E. Tezduyar, “Space-time SUPG Formulation of the shallow-water equations”, International Journal for Numerical Methods in Fluids, 64 (2010) 1379-1394. [46] K. Takizawa and T.E. Tezduyar, “Multiscale space-time fluid-structure interaction techniques”, Computational Mechanics, published online, 48 (2011) 247-267. [47] K. Takizawa, B. Henicke, T.E. Tezduyar, M.-C. Hsu and Y. Bazilevs, “Stabilized space-time computation of wind-turbine rotor aerodynamics”, Computational Mechanics, 48 (2011) 333-344.

22

[48] K. Takizawa, B. Henicke, D. Montes, T.E. Tezduyar, M.-C. Hsu and Y. Bazilevs, “Numerical-performance studies for the stabilized space-time computation of wind-turbine rotor aerodynamics”, Computational Mechanics, 48 (2011) 647-657. [49] S. Takase, K. Kashiyama, S. Tanaka and T.E. Tezduyar, “Space-time SUPG finite element computation of shallow-water flows with moving shorelines”, Computational Mechanics, 48 (2011) 293-306. [50] P.A. Kler, L.D. Dalcin, R.R. Paz and T.E. Tezduyar, “SUPG and discontinuity-capturing methods for coupled fluid mechanics and electrochemical transport problems”, Computational Mechanics, (2012) published online, DOI: 10.1007/s00466-012-0712-z. [51] T.J.R. Hughes, “Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods”, Computer Methods in Applied Mechanics and Engineering, 127 (1995) 387-401. [52] Borello, D., Corsini, A., and Rispoli, F., 2003, “A finite element overlapping scheme for turbomachinery flows on parallel platforms”, Computers and Fluids, 32, pp. 1017–1047. [53] Kirk, B.S., Peterson, J.W., Stogner, R.H., and Carey, G.F., 2006, “libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations”, Engineering with Computers, 22, pp. 237–254. [54] A. Corsini, F. Rispoli, “Flow analyses in a high-pressure axial ventilation fan with a non-linear eddy viscosity closure”, Int. J. of Heat and Fluid Flow, 17 (2005) 108-155. [55] T.J. Craft, B.E. Launder, K. Suga, “Development and application of a cubic eddy-viscosity model of turbulence”, Int. J. of Heat and Fluid Flow, 17 (1996) 108-155. [56] Laín, S. and Sommerfeld, M., Turbulence Modulation in Dispersed Two-phase Flow Laden with Solids from a

Lagrangian Perspective, Int. J. Heat and Fluid Flow, vol. 24, pp. 616-625 (2003). [57] T.E. Tezduyar, K. Takizawa, C. Moorman, S. Wright and J. Christopher, "Space-Time Finite Element Computation of Complex Fluid-Structure Interactions", International Journal for Numerical Methods in Fluids, 64 (2010) 1201-1218. [58] Baxter, L.L., 1989, “Turbulent transport of particles”, PhD thesis, Brigham Young University, Provo, UT, USA. [59] Wang, L.P., 1990, “On the dispersion of heavy particles by turbulent motion”, PhD thesis, Washington State University, Pullman, WA, USA. [60] Litchford, L.J., Jeng, S.M., 1991, “Efficient statistical transport model for turbulent particle dispersion in sprays”, AIAA Journal, 29, p. 1443-1451. [61] Jain, S., 1995, “Three-dimensional simulation of turbulent particle dispersion”, PhD thesis, University of Utah, Salt Lake City, UT, USA.

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[62] Borello, D., Venturini, P., Rispoli, F., and Saavedra, G.Z.R., 2013, “Prediction of multiphase combustion and ash deposition within a biomass furnace”, Applied Energy, 101, pp. 413-422. [63] Venturini, P., Borello, D., Iossa, C.V., Lentini, D., and Rispoli, F., 2010, “Modelling of multiphase combustion and deposit formation and deposit formation in a biomass-fed boiler”, Energy, 35, pp. 3008–3021. [64] Armenio V, Fiorotto V., (2001) The importance of the forces acting on particles in turbulent flows. Physics of Fluids 13 (8): 24372440. [65] Schiller, L., and Naumann, A. (1933). "Uber die grundlegenden Berechnungen bei der Schwekraftaubereitung." Zeitschrift des Vereines Deutscher Ingenieure, 77(12), 318-320. [66] Smith, P.J., 1991, “3-D Turbulent particle dispersion submodel development”, Quarterly Progress Report #1, Department of Energy, Pittsburgh Energy Technology Center, Pittsburgh, PA. [67] A. Corsini, F. Rispoli, and A. Santoriello, “A variational multiscale high-order finite element formulation for turbomachinery flow computations”, Computer Methods in Applied Mechanics and Engineering, 194 (2005) 4797-4823. [68] Y. Bazilevs, M.-C. Hsu, I. Akkerman, S. Wright, K. Takizawa, B. Henicke, T. Spielman, and T.E. Tezduyar, “3D Simulation of wind turbine rotors at full scale. Part I: geometry modeling and aerodynamics”, International Journal of Numerical Methods in Fluids, 65 (2011) 207-235. [69] M.-C. Hsu, I. Akkerman, and Y. Bazilevs, “Wind turbine aerodynamics using ALE-VMS: Validation and the role of weakly enforced boundary conditions”, Computational Mechanics, (2012), published online, DOI: 10.1007/s00466-012-0686-x. [70] M.-C. Hsu and Y. Bazilevs, “Fluid--structure interaction modeling of wind turbines: Simulating the full machine”, Computational Mechanics, (2012), published online, DOI: 10.1007/s00466-012-0772-0. [71] Sheard, A.G., Corsini, A., Minotti, S., and Sciulli, F., 2009, “The role of computational methods in the development of an aeroacoustic design methodology: application to a family of large industrial fans”, 14th Conference on Modelling Fluid Flows, Budapest, Hungary, 9–12 September. [72] Corsini, A., and Rispoli, F., 2004, “Using sweep to extend stall-free operational range in axial fan rotors”, Journal of Power and Energy, 218, pp. 129–139. [73] L.P. Wang, D.E. Stock, 1993, “Dispersion of heavy particles in turbulent motion”, Journal of the Atmospheric Science, 50, 18971913. [74] L.P. Wang, D.E. Stock, 1992, “Numerical simulation of heavy particle dispersion: time step and non-linear drag considerations”, Journal of Fluids Engineering, 114, 100-106.

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[75] Y. Sato, K. Yamamoto, 1987, “Lagrangian measurement of fluid-particle motion in an isotropic turbulent field”, J. Fluid Mech., 175, 183-199. [76] M.W. Reeks, 1977, “On the dispersion of small particles suspended in an isotropic turbulent field”, J. Fluid Mech., 83, 529-546. [77] G.T. Csanady, 1963, “Turbulent diffusion of heavy particles in the atmosphere”, J. Atmos. Sci., 20, 201-208.

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