RANS Modeling of a Particulate Turbulent Downward Jet

Alexander Kartushinsky Efstathios E. MichaelideS2

STallinn University of Technology, Akadeemia tee, 21E, 12618 Tallinn, Estonia,

fax: (+372) 6703601, ph. (+372) 6703610, e-mail: aleksander.kartusinski@ttu.ee;

2 University of Texas at San Antonio, One UTSA Circle, San Antonio TX, 78249, USA

Fax: 2 10-45 8-65 04, Ph. 2 10-45 8-5 580, e-mail : stathi s.michaelides@utsa. edu



Abstract

The Reynolds Average Navier-Stokes equations are used to perform a complete and accurate

modeling for the symmetric gas-solid turbulent round jet in three different orientations a)

downward flow, b) upward flow and c) zero gravity. The two-fluid model is applied to describe

the averaged characteristics of the gas and particulate phases, including the particle mass

concentration, the turbulent kinetic energy and its dissipation in the mixture of gas-solid

particles. The model includes effects such as particle-turbulence interaction; turbulence

production; and turbulence modulation. The drag and gravity forces are incorporated into the

system of equations using appropriate closure models that have been tested and validated in the

past. A finite difference numerical scheme for the solution of the set of the governing equations

and the closure relations is applied. The results of the model were validated by comparison with

experimental data. The computational results show the influence of the particulate phase on the

velocity and turbulence structure of the j et.

1. Introduction

The classical study by Abramovich [1] and the several subsequent studies on this subject make

the round, symmetric jet one of the most well-studied and well-calibrated problems in fluid

dynamics. The round, symmetrical multiphase jet (with particles, drops or bubbles) has also been

studied by several researchers both experimentally and computationally [2, 3]. Experimental

investigations of such jets have been performed by several research teams, including [4, 5] who








investigated how the fluid velocity and turbulent kinetic energy structures are affected by the

presence of particles and bubbles. The experimental investigation of the two-phase turbulent jet

[6] showed that there is a marked anisotropy in the fluctuations of the velocity of the dispersed

phase in both the axial and radial directions and that the anisotropy increased with heavier

particles.

On the theoretical front Abramovich [7] used Prandtl's mixing-length theory to take into

account the feedback of the particles on the turbulent structure and predicted attenuation of

turbulence by the particles. This effect was not confirmed experimentally or numerically until

almost two decades later and for Eine particles only [8, 9]. With the greater availability of

computational power, several numerical studies have been performed on the subject of round

turbulent particulate jets using the turbulent boundary layer approximation [10-12].

A Lagrangian-Eulerian approach was used in [13] for a two-phase jet laden with

Stokesian particles. The modeling of a nonisothermal turbulent round jet carrying small and

large particles was accomplished using a large eddy simulation (LES) approach coupled with a

Lagrangian method for the modeling of particles in the dispersed phase [14].

A full numerical simulation using the complete Reynolds Averaged Navier-Stokes

(RANS) equations is the subject of the current investigation. The RANS method and the two-

fluid approach for the gas and the dispersed phase are used to model the flow of the two phases

and their interactions. The system of equations for the mass and momentum conservation

equations and the turbulence closure equations (an extended k-E model) are derived and solved

using a Einite difference scheme. The dispersed phase is considered monodisperse and is dilute,

so particle-particle collisions do not affect significantly the flow structure. The particles are

subjected to the viscous drag and gravitation forces. The coefficient of the particles' turbulent

diffusion was used to close the mass conservation equation for the dispersed phase. Similarly,

the introduction of the particle turbulent viscosity coefficient was used as a closure equation for

the momentum equation of the dispersed phase. For the turbulence modeling, we have used the








original formulation of the k-E model with the addition of the particulate interactions through the

slip velocity of the particles supplemented with the hybrid length scale, which accounts for the

turbulence modulation. We refer to this model as the extended k-e model for particle-turbulence

interactions. Both the particle turbulent diffusion coefficient and the particle viscosity coefficient

were calculated from a PDF model, which is described in [15]. The model results are well tested

by comparison with experimental results obtained from single-phase turbulent round jet [1] and

also, from the existing experimental results of two-phase round jet [5, 11]. The results obtained

for the 2D two-phase round turbulent jet describe more accurately the development of the

particulate jet than those obtained using the two-phase turbulent boundary layer approach or the

quasi-2D approach [10, 11].

2. The Governing equations

Because the flow of the two phases is in the vertical direction, the equations for the particulate

round jet flow are symmetric. The full-set of the Reynolds-Averaged Navier-Stokes (RANS)

equations (volume averaged) in two dimensions and in cylindrical coordinates have been used in

this study. A summary of the governing (conservation) equations is as follows:

A. For the particulate jet:

Al. Continuity

an a(ry)
- + = 0, (1)
8x rar

where x and r are the axial and radial coordinates; u and v are the axial and radial velocity

components of the gaseous phase in the two-phase round jet.

A2. Momentum equation in the longitudinal direction:

Bu mu 8 fu 8 u 8 (u- u,)
u- + v- 2vt + Wt +- a (2)
ax ar 8x \ xl rar r xl z

where u, is the axial velocity component of the dispersed phase; a is the particle mass


concentration; vt is the coefficient of the turbulent viscosity; r= c 18py/p di~ is the








particle viscous response time, which is derived from viscous drag considerations; and p and p,

are the gas and particle material densities, respectively; d is the particle diameter; cD, is a

correction factor for non-Stokesian drag [16] and is related to the particle drag coefficient:


c' +015e67.The particle Reynolds number is Res u-u)2+V-V Where vs


is the radial velocity component of the dispersed phase. The kinematic viscosity of the gas v is

neglected in equations of the gaseous phase because v << vt -

In the two-dimensional model of the particle laden turbulent round submerged j et there is

no effect of particle-particle collision. We have followed the model by [17] for the following two

reasons:

a) The jet expansion, that leads to increased inter-particle distance and by extend the time of the

particle collisions versus particle relaxation time and

b) The relatively low mass loadings with the corresponding very low volume concentration of

the particles.

The main components of the hydrodynamic force acting on the particles are: the drag, the

Saffman lift and the gravitational forces. These components along with particle's turbulent

dispersion, which is described by the turbulent diffusion coefficient of the particles and the

turbulent viscosity coefficient of the dispersed phase, determine the motion and dispersion of the

particles. It is assumed also that the particles enter the flow with no rotation in their motion and,

therefore there is no Magnus lift force.

A3. The turbulent kinetic energy and rate of dissipation of turbulent kinetic energy are given by

the following equations:

dk 8k 8 vt 8k 8 vt 8k
u-+ v- -- +-r+k+P- 3
dx dr 8x Gk kx sd ,d 3


dE dE 8 vt dE d vt dE E C,(I ) ," 4
u-+v- --- +-r +-c Pk+s-ss(4
8x dr ax o 8x rar a, dr k








where k and E are the kinetic turbulent energy and its dissipation rate, respectively. Pk is the

turbulence production by velocity gradients and Ps is the turbulence production by the average

velocity slip between two phases [18]:


Pk =t(Z +I +-+ ,(a



Ps = au-s2+VVs2(4b)


A4. The turbulent viscosity vt is given by the closure equation:


vt = c (5)


where the typical constants of the k-E model are: Gk s ,o = 1.3 c = 1.44, cs2 = 1.92 and

c, = 0.09, respectively.

A5. The radial velocity of the particulate jet may be obtained from the conservation of

momentum and, for the two dimensional jet flow is simply given by the following integral [1,

11]:

u 'rdro 8d au\ 8 fu 8v (uU, us
0 u2LX x a a x

B. The particulate, dispersed phase:

Bl. The continuity equation, which includes the diffusion of the solid particles in the j et is:

(3au, 8(rays) 8/ it 8 i
sI + D + D (7)
8x rar 8x s x rar sr

where Ds is the particle turbulent diffusion coefficient [15]:


Ds=2k(T+To, 1-x(T),z (7a)

where To is the integral time and length scales for the single phase turbulent jet.

B2. The momentum equation for the dispersed phase in the axial direction is:








As 8 l1n as x (- s d s (u -us
u h+ vs -D uy+-rays8 ++ +g, (8)



where vs=[ v+~ 1-exp(- ) is the viscosity of the dispersd phase due the part~ice

involvement into turbulent motion. The method to calculate this parameter is given in [15]. The

positive sign of the gravitational acceleration denotes that positive velocities are oriented in the

81n u
downward direction. The term D -is the contribution of the drift diffusion model.


B3. Momentum equation in the radial direction:


u ~+ v- v + 2ry


Fs (u- us), (9)

where the Saffman (shear) lift force is determined by the expression:

3.0759p uv [u\
F, Iu- s (9a)
s p,d Iu a


and the correction coefficient uy is obtained from [19] as:

r1-0.3314 pexp(-0.1Res)+ 0.3314Jj Res <; 40
0.0524JPe Res > 40 (b



8x aBr 8x aBr
with the shear rate p =
2lv 2 (u-us)2 + V-Vs 2

C. Transformation to similarity coordinate system:

In order to facilitate the computations and obtain more comprehensible results, it is used the jet

similarity transformation and define the following new variables, which are related to the

similarity conditions that are met in round submerged jets [20, 21]:

r r/R 8 8 8 8 yz 8
z =, a = 1 + y(x / R), ,,(10)
R +yx 1 + y(x /R)' Br ac~z 8x 8x a 8z2

where the coeffieient of transformation is y=0.09 and R is the radius of the orifice.








3. Boundary conditions

The jet emanates from an orifice facing the downward direction. Since the main flow direction of

the jet is in the direction of gravity, the symmetry conditions at the jet centerline are expected to

be valid. In the first few diameters, the velocity profie of the jet evolves from the "top hat"

profile or uniform velocity to the developed jet profie. This evolution typically takes less than

20 orifice diameters and in our computations it was completed between 13 and 16 orifice

diameters. At the outer boundary of the jet, the longitudinal velocity vanishes asymptotically. In

the computations of this study, the outer boundary of the jet was defined as the location where

the longitudinal velocity was 2% of the centerline velocity. Thus, the boundary conditions used

in the study are as follows:

A. The entrance conditions for the particulate jet, at x=0 for r < /R, are:

u(0, z)= u(0,0)- (1- z)! ", us = u(0, z); v(0, z)= vs (0,) = 0; k(0, z)= 2.5 -10-3

s(0, z)= 10 -k3/2(0, z), u(0, z)= m*, (18)

where m* is the initial mass loading of the flow.

B. At the jet axis, the symmetry conditions are applied:

Bu 8k 88 au, 82
---------, v =v, =0 (19)
8z 8z 8z 8z 8zs

C. The outer boundary in the radial direction of the j et is defined as the radial distance where the

axial velocity of the jet is equal to 2% of its orifice velocity. At this distance the following

boundary conditions are prescribed [10, 13, 21]:


us =0.0u(0,0),lr km=t.= Oam =0.0a(0,0)~a, svZava = 0 (20)


D. The exit boundary conditions at the forward end of the computational domain:

Bu 8k 88 av au 8v 8 2
- - - - 0 (2 1)
8z 8z 8z 8z 8z 8z 8zX=eXlt


4. Numerical computations and results








A finite difference tridiagonal matrix algorithm has been used for the computation of the

jet parameters. The discretized governing equations are cast along both directions. We use an up-

wind difference scheme for the convective terms, which results in a first-order scheme, using an

equidistant mesh size grid. The number of iterations is determined by the minimum value of the

residual (less than 0.1%) for the axial velocity of the gaseous phase and the turbulent energy.

This minimum value is determined by comparison between the previous and the current

parameters throughout the flow domain (jet domain). The dimensions of the domain are

(width)x(length)=1 50x9,000. These conditions for the iteration procedure are typical in jet flows

and allow the spatial development of the 2D turbulent single phase and particulate j ets.

The first row of nodes in the radial direction contains 50 points corresponding to a mesh

size of Ar=0.02. The number of grid points was increased downstream using the condition,


- <; 0.01, which has been recommended in [22]. Therefore, the jet expansion was determined


and conditioned by the smooth behavior of the axial component of the carrier velocity. In the

overall computations, the area of the flow increased up to 2.5 times as the flow progressed

downward, that is up to 150 radial nodes from the initial set of 50 nodes. For the actual

computations we used the similarity transformation of the jet described by Eq. (10) and used the

variables x and z. However, the results are reported in the figures that follow in terms of the

actual coordinates x and r.

All the results in the following figures are presented in dimensionless form. We followed

the usual method for rendering the spatial and flow variables dimensionless [1, 20] and used the

following characteristic quantities to render the pertinent variables dimensionless:

a) For the radial dimensions the half-radius of the jet, R1/2. This is the radial distance where the

velocity of the j et is equal to one-half of the value of the centerline velocity.

b) For the longitudinal distances the initial orifice radius, R, is used.








c) For the longitudinal velocities and turbulence parameters, the pertinent centerline velocity,

U,, either at the entrance of the jet or at the corresponding axial distance, u(x,0) or us(x,0), as

stated in the text.

d) For all the particles parameters, their distributions are normalized with their numerical values

at the centerline of the j et.

The numerical modeling of the two-phase turbulent round jet is performed for the

following flow conditions: the particle sizes of glass beads (p~ = 2500kg/m ), d=0.2 mm with

mass loading: m*=1 are used. The Stokesian response time, z corresponding for these flow

conditions has the following values: z =0.307 for glass beads of d=0.2 mm.

For the validation of the numerical results we compared the single-phase results with the now

classical experimental data of Abramovitch [1], which has become the yardstick for experimental

and computational studies.

The flow conditions in the simulation results are as follows: jet diameter at the orifice

exit, D=2R=30.5mm and mean flow velocity at the jet exit 10m/s. The exit Reynolds number

corresponding to this velocity is Re = 2.2-104. The exit velocity and the Reynolds number are

two parameters that vary in some of the computations as indicated in the text and the captions of

the figures. This is dilute enough to result to negligible inter-particle collisions.

Figure 1 shows the distribution of the axial velocity component of the single phase jet as

well as the comparison of the numerical data with the experimental results from [1]. The exit

cross-section is at x/R=60 and the mid section is at x/R=30. It is observed that the numerical

results agree very well with the experiments, a fact that validates the numerical method used

here. Also confirmed is the self-similarity of the jet, which is expected to occur at downstream

distances x/R greater than 20. The self-similarity of the jet follows the quasi-one-dimensional

approach, such as the turbulent boundary layer approach. The self-similarity is not so obvious in

the case of a 2D approach with the diffusive terms in both the axial and the radial directions.

However, the results of this investigation show that self-similarity is preserved in the two-








dimensional case, most probably because the effects of the inertia of the jet dominate any

diffusion effects.


- x/R=30
....... x/R= 60

x experimental


1




O 0.75
o






a0.25


o


.5


Figure 1. Normalized axial velocity of the j et and experimental data. The experimental data are

from [1].

Figure 2 shows the development of the normalized radius determined by the half magnitude of

the gaseous phase velocity, R1/2 for single phase and dispersed phase. The experimental data are

taken from [1]. It is observed that the half radius is reduced by the addition of the particles in the

jet which corresponds to experimental study of [5].


0.5 1 1

Radial distance, r/R1/2









7.5







o






~o "-single ph, R1/2

.2 two ph,R1/2

at I single ph,R1/2,exp

0 15 30 45 60

Axial distance, 2x/D


Figure 2. Normalized half radius of the j et velocity. The experimental data, denoted by o, are

from [1].

Figure 3 depicts the distribution of the parameter d/Lo, the ratio of the particle diameter to the

integral turbulence length scale, which is very important in turbulence modulation. The length Lo

is defined as: La = k /2 0E, Where ko and so are turbulent energy and its rate of dissipation in

the single phase jet. The particles are glass beads with sizes d=0.2 and 0.5mm. According to this

distribution and following Crowe's criteria [8, 9, 23] the particles are expected to generate

turbulence in the initial part of the flow where d/Lo>0. 1 and attenuate the turbulence at distances

greater than 2x/D=20, where d/Lo<0.1.










-- d/Lo, d=0.2mm

.5r -e- d/Lo, d=0.5mm










0




0 20 40 60

Axial d ista nce, 2x/D


Figure 3. Normalized distribution of parameter d/Lo along the jet.

The profiles of the longitudinal velocity components of gas-phase in two-phase j et system

for downward, upward and zero gravity jets versus the longitudinal velocity of the single-phase

are presented in Figure 4 for glass particles of 0.2 mm and mass loading of 1. The profiles are at

the downstream distance x/R=30. It is observed that in such coordinate all profiles are almost

self-similar up to r/R1/2 =1. After this distance the velocity profiles of the two-phase jets

deviate significantly from that of the single-phase jet. At this location, the concentration of the

solids drops significantly, as it may be seen in Figure 8.









u: single-ph ase
\l~ uu: nog ravity
u:downwa rd
u:upward
a 0.75 i\











0.






0 0.5 1 1.5

Radial distance, r/R1/2


Figure 4. Normalized axial velocity of gas-phase for a j et with glass beads, 0.2 mm and mass

flow ratio m*=1 at x/R=30 and Re = 2.2 -104

The normalized distribution of axial velocity component of the dispersed phase along the jet

cross-section at x/R=30 are shown in Figure 5 for the three cases: downward flow, upward flow

and neutrally buoyancy jets. As in the previous figure, the velocity profies drop significantly in

the vicinity of r/R1 2=1.3 where it appears that the two-phase jet ends. Figure 5 also shows that

the profies of the axial velocity of the dispersed phase are fuller in the cases of the downward

and zero-gravity jets.














,o 0 75











E 0.25
o us:solid:nog ravity

us:sol id: downwa rd

us :so lid: upwa rd

0 0.5 1 1.5

Rad ial d ista nce, r/R1/2


Figure 5. Normalized axial velocity of dispersed phase for a jet with glass beads, 0.2 mm and

mass flow ratio m*=1 at x/R=30 and Re = 2.2 -104

The normalized distribution of the radial velocity components of gas- and the dispersed phases

along the jet cross-section at x/R=30 are also shown in Figure 6 for downward flow, upward

flow and neutrally buoyancy jets as well as for the single-phase jet. Figure 6 shows that the

magnitudes of the radial velocity of the dispersed phase are negative for all the two-phase jet

configurations. This implies that that two-phase jet becomes narrower than the single-phase, jet,

which is corroborated by the axial velocity results of the previous figures. This effect to make the

jet narrower and is more pronounced for the downward flowing jet, where the particle axial drag

forces tend to accelerate the surrounding gas.









0.005






o -------, /
0 -0.0 --





-00 -, -- vIigl-hse'


-cr- v: nog ray ity \ ,
o -v: down wa rd *

= -0.015vupad ?
E ------- vs:solid: nog ravity l
o .. .vs:solid: downward L

-00 vs:solid:upward

Rad ial d ista nce, r/R1/2


Figure 6. Normalized radial velocities of gas- and dispersed phases for a j et with glass beads, 0.2

mm and mass flow ratio m*=1 at x/R=30 and Re = 2.2 -104

Figure 7 shows the distribution of turbulent kinetic energy along the jet cross-section at x/R=30

for downward, upward and zero gravity flow of a jet carrying glass particles of 0.2 mm with

loading equal to 1. The observed distribution of the kinetic energy profiles indicates that: a) the

particles attenuate the turbulence, and b) the downward directed jet has significantly less

turbulence than those of upward and neutrally buoyancy j ets.








0.04
k: single-p hase
z~- kknogravity
I~ k: downwa rd
a0.03 k:upward







= 0.01




0.0



0 0.5 1 1.5

Radial distance, r/R1/2


Figure 7. Normalized turbulent kinetic energy profiles for a jet with glass particles 0.2 mm and

mass flow ratio: m*=1 for downward, upward and no gravity. The profiles are at x/R=30 and for

Re = 2.2 -104

Figure 8 shows the radial distribution of the particle mass concentration at the cross-section

x/R=30 for three downward flow, upward flow and zero gravity flow with glass beads of 0.2 mm

diameter and flow mass ratio m*=1. The distribution of mass concentration is in agreement with

the velocity profiles. The mass distribution appears to be more uniform in the upward case

where the particles exert a restraining hydrodynamic force to the fluid.














S0.75










-m 0.25 -1 alfa:nogravity

o alfa: downwa rd

alfa: upwa rd

0 0.25 0.5 0.75 1 1.25 1.5

Radial distance, r/R1/2


Figure 8. Normalized mass concentration profiles of the dispersed phases at x/R=30 for glass

beads of 0.2 mm and m*=1 at Re = 2.2 -104



The Figure 9 shows the distribution of the gas-phase center-line velocity for the same three cases

of upward, downward and zero-gravity flow. As one can see the particles attenuate the

turbulence kinetic energy and this results to slower rate of decrease of velocity in comparison

with the single-phase distribution (dashed line).









single-ph

- gas,u pwa rd

- gas,downwa rd

gas,nog ravity


o
a, 0.75
m






S0.25
O

0)


30 45
Axial distance, x/R


Figure 9. Normalized centerline velocity of gas-phase for two-phase j et loaded with glass beads

of 0.2 mm and m*=1 for Re = 2.2 104



The normalized axial velocity of the dispersed phase us is shown in Figure 10 along the entire

jet length for the three cases considered here. In all cases the axial velocity of the dispersed

phase is higher that that of the gas-phase. The rate of decrease is less pronounced for the

downward j et configuration.










gas,nog rav ity

o ~~ so lids,u pwa rd

a -solids,downward
> 0.75
I solids,nogravity









0.






0 15 30 45 60
Axial distance, x/R


Figure 10. Normalized velocities of the gasp- and dispersed phases, u, us along the centerline of

the jet for glass beads, 0.2 mm and m*=1 at Re = 2.2 -104 (D=30.5 mm).

The axial distribution of the normalized turbulent kinetic energy, made dimensionless by its

value at the orifice exit is presented in Figure 11. The jet in this case has glass beads of d=0.2

and with mass loading m*=1. The single-phase results are also included for comparison. It is

observed that, after an initial decrease, which is due to the jet development and which is

independent of the particle loading, the turbulent kinetic energy increases rapidly. It

subsequently falls to a level that is close to its initial level. In general, the small glass particles of

0.2 mm attenuate the turbulence in the developed part of the jet. This tendency is more

pronounced in the case of the downward flow of the two-phase jet. This is explained by the

significant increase of the characteristic length of the jet downstream as shown in Figure 3, that








is the increase of integral turbulent length scale, and is in agreement with several experimental

and analytical studies [23-25].


3.5


3


2.5


2


1.5


1


0.5


O


k:single-phase

Sk: gas, upwa rd

- k:gas,downwa rd

k:gas,nogravity


30 45
Axial distance, x/R


Figure 11i. Normalized profiles of turbulent energy for a single phase j et and a j et loaded with

glass beads.



4. Conclusions

A complete RANS method with its pertinent closure equations for particle viscous drag, gravity,

turbulence modulation has been developed to describe the flow of a particulate jet. A finite

difference method with the aid of a tridiagonal matrix solver has been used for the solution of the

governing and closure equations. The results obtained for the gaseous and solids velocities were

compared with sets of experimental data. The results showed good agreement with the data and

the general trends expected from experimental data and the pertinent theory.








The results of the computations show that the particulate phase, even at low loadings and

low concentrations, has a significant effect on the flow parameters of the gaseous jet. The jet

becomes narrower with the heavier particles and higher loading because of the drag the solids

exert on it. As a result of this interaction, the solids velocity profile becomes flatter and the

concentration of the solids narrower. The higher radial velocity of the jet at higher loadings

indicates that more gas is entrained from the sides when the particle-laden jet has higher inertia.

Finally, the effects of the particles on the turbulence of the carrier phase have been computed.

The general trends for the attenuation and enhancement of the turbulence by particles, which

have been observed by several experimental and analytical studies, have also been observed in

the numerical results of the RANS computations.

Acknowledgements. The work was performed within the frame of target financing from Proj ect

SF0140070s08 (Estonia) and partially supported by ETF grant Project ETF7620. The research of

the second author, EEM, has been partly supported by a grant from the DOE, through the

National Energy Technology Laboratory, (DE-NT0008064), Mr. Steven Seachman project

manager; and by a grant from the National Science Foundation (HRD-0932339) Drs. Demetris

Kazakos and Richard Smith, project managers.. The authors are grateful for the technical support

of the computers of the University of Texas at San Antonio.

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