STUDY OF TURBULENCE MODELS AND ALGORITHMS FOR PRESSURE-VELOCITY COUPLING IN SIMULATION OF EMULSION...

09 a 12 de setembro de 2012

Búzios, RJ

STUDY OF TURBULENCE MODELS AND ALGORITHMS

FOR PRESSURE-VELOCITY COUPLING IN SIMULATION

OF EMULSION FLOW

A.K.N. VARGAS

1, J.L de PAIVA

1, R. GUARDANI

1

1 DEQ – Escola Politécnica, Universidade de São Paulo

ABSTRACT- In this study were carried out two-phase flow simulations where oil is

dispersed phase into water as microscopic drops with colloidal size (350 µm) and an

inlet average velocity equivalent to 0,6169 m/s. An Euler-Euler model was used to

simulate two phases flow system. In this model both phases are considered as

interpenetrating fluids. It was applied k- , k-ω models in fluid dynamics analysis and

its influence in interaction between phases. Finally, different algorithms were

compared for pressure-velocity coupling: PIMPLE, SIMPLEC, PIMPLEC, for testing

their performance for each method with regard to computational times and

convergence. Computational package used was OpenFOAM, free Software to

numerical solvers, and pre-/post-processing utilities for solving continuum

mechanical problems as CFD. This work is part of a research project in the program

BRAGECRIM, cooperation Brazil – Germany with CAPES, FINEP, CNPq e DFG

(Germany) funds.

1. INTRODUCTION

Several algorithms for pressure-velocity coupling have been proposed to resolve

Computational Fluid Dynamics (CFD) problems assuring convergence in process simulations.

Over last few years, some authors have compared these algorithms analyzing

efficiency especially with regard to computational time. Results vary depending of simulated

process. Zen and Tao, 2002, studied some variants of Semi-Implicit Method for Pressure

Linked Equations (SIMPLE): SIMPLE-revised (SIMPLER), SIMPLE-Consistent (SIMPLEC)

and SIMPLE-extrapolation (SIMPLEX). Their work showed that SIMPLEC and SIMPLE

were the most efficient in CPU time comparing with SIMPLEX and SIMPLER, likewise, it is

concluded that SIMPLEC algorithm was the most robustness method becoming the algorithm

recommend by the authors for solution of incompressible fluid flow and heat transfer

problems, mainly when the grid density is fine. Yin and Chow, 2003, compared four

algorithms: SIMPLEC, SIMPLER, SIMPLE and Pressure Implicit with Splitting of Operators

(PISO), these algorithms were analyzed for simulation of atrium fire. In this case PISO has

presented the best performance by its stability and low computational time.


Búzios, RJ

In this work were analyzed another solver considered more stable than PISO and

developed in OpenFoam, free software used in CFD problems (OpenFoam Documentation).

In its last version (version 2.1.0) includes a solver to resolve problems for two incompressible

fluid phases with one dispersed phase (twoPhaseEulerFoam). The algorithm for pressure-

velocity is PIMPLE, improved version of PISO where is possible to relax all variables.

Based in this, three algorithms were studied: PIMPLE, SIMPLEC, PIMPLE-consistent

(PIMPLEC), the latter being a PIMPLE and SIMPLEC combination, to observe the best

solver applied for dispersions system especially in emulsions. The simulations have been

done using two turbulence models: K-Epsilon (k- ), K-Omega (k-ω), which are commonly

used to evaluate flows in industrial process (Parvini and Mohtashami, 2010; Versteeg and

Malalasekera, 2007).

2. MATHEMATICAL MODEL

2.1. The Navier-Stokes equations

A first approximation to model flow emulsion behavior is to consider a transient state

process with two phases and Newtonian fluid. The effects like aggregation, breakage and

coalescence phenomena will not be analyzed in this study. The mathematical model is based

on solver proposed by Oliveira and Issa, (2003).

If a is dispersed phase and b is continuous phase, continuity and momentum equations

are solved for each phase in the system:

(1)

Where α means fraction of phase , means material density of same phase

constituent, and is phase velocity.

The phase momentum equation is expressed by (Rusche, 2002):

(2)

Here

is combined Reynolds (turbulent) and viscous stress tensor and is

averaged inter-phase momentum transfer term. The solution for these couple of equations is

not simple due to an explicit equation for pressure is not available. One of the most common

approaches is to derive an equation for pressure by taking momentum equation divergence

and by substituting it in continuity equation.


Búzios, RJ

2.2. The pressure equation

The phase momentum correction and pressure equation will be derived using a semi-

discretized equation (Equation 3) from momentum equation (Equation 2):

(3)

Where “H” operator ()H and diagonal ()D operator are defined in Rusche, 2002;

denotes Turbulent drag and corresponds to system of linear algebraic equations

arising from the discretization of phase momentum equations without terms which are

proportional to or :

(4)

Rearranging Equation 3 yields phase momentum correction equations:

(5)

Equation 5 is used to correct velocities after an updated pressure field is obtained by

solving pressure equation.

Pressure equation is obtained from volumetric mixture continuity equation:

(6)

To calculate flux and are derived by interpolating the momentum correction

Equation 5 to face centers. Using central differencing, they are given by:

for (7)

Where the flux predictions are given by:

(8)

The flux predictions are composed of two parts: the first contribution originates the

“H”- operator, which is evaluated by re-substituting the velocity estimates obtained earlier

into their discretised equations and the second one is due to the turbulent drag term. (Rusche,

2002)


Búzios, RJ

Finally, by replace Equation 8 into Equation 6, it is obtained the equation to solve

pressure field:

(9)

2.3. Algorithms for pressure-velocity coupling

The PIMPLE algorithm is a modification of PISO described by Versteeg and

Malalasekera, (2007). This algorithm allows equation under-relaxation and ensures the

convergence of all equations at each time-step. The loop is:

(a) Under-relax the Equation 5 for velocity.

(b) Predict fluxes using Equation 8.

(c) Construct and solve the pressure Equation 9.

(d) Correct fluxes, Equation 7.

(e) Correct velocities, Equation 5.

The SIMPLEC (SIMPLE-Consistent) follows same steps as SIMPLE (Versteeg and

Malalasekera, 2007), however momentum equations are manipulated so that SIMPLEC

velocity correction equation omit terms which are less significant than those in SIMPLE.

Finally the PIMPLEC algorithm follows same steps as PIMPLE but handling velocity

correction equation as in SIMPLEC.

2.4. Turbulence Models The k- . Standard model is available to solve turbulence problems in

twoPhaseEulerFoam (Open Foam solver to simulate dispersions), which is derived to the

continuous phase fraction b. For this study, a new solver was implemented; this includes model

k- to analyze turbulence effect in simulation.

k- Turbulence Model: It has been used in most general purpose CFD codes and it is

considered a standard model. This model of turbulence is expressed by transport equations

“k” and “ ” described in Equations 10 and 11:

(10)

(11)


Búzios, RJ

First part of Equations 10 and 11 represents change rate of k or and transport by

convection for the continuous phase b. Second part corresponds to transport by diffusion,

production rate and destruction rate of k or . The turbulent viscosity of b is calculated by:

(12)

And production rate as:

(13)

Where:

(14)

The k- model employs values for constants which are arrived by comprehensive data

fitting for a wide range of turbulent flows. These values are presented in Table 1.

k-ω TurbulenceModel: It is a model proposed by Wilcox (1988), which uses

turbulence frequency (dimensions s-1

) as second variable. Transport equation for k

and for turbulent flows at high Reynolds is as follows:

(15)

(16)

Table 1 –Constants present in k- equations

Model Parameter Default value

1,44

1,92

0,09

1,00

1,30

The model constants are as follows in table 2.

One of main features of k- model are the equations implemented near to wall for

measuring turbulence in that region. The boundary conditions imposed at a solid boundary

are:

(17)


Búzios, RJ

where is normal to boundary.

Table 2 –Constants present in k-ω equations

Model Parameter Default value

5/9

3/40

9/100

1/2

1/2

Moreover, the centroid values in cells adjacent to solid wall are specified as:

(18)

(19)

In an alternative approach, k production terms is modified.

Automatic wall: The purpose of automatic wall treatments is to achieve results

insensitive with regard to wall mesh refinement. Many blending approaches have been

proposed. The one by Menter, (2001) takes advantage of that solution to equations is

known for both viscous and log layer (FLUENT Documentation):

(20)

(21)

Where is cell centroid distance from the wall. Using this blending can take

following form:

(22)

Subsequently Menter, 2001 proposes also blending for friction velocity. Friction

velocities for viscous and logarithmic region are:

(23)

(24)


Búzios, RJ

And the blending suggested:

(25)

3. RESULTS

3.1. Model Validation

Parvini and Mohtashami, 2010, carried out simulations to evaluate the effect of

various significant forces in turbulent flow of liquid dispersions. Results were compared with

Farrar and Bruun, 1996, experiments which were made in a vertical pipe with diameter D and

length L. In the present paper, the experimental data have been used to validate the model

employed in this work.

All simulations were performed in a computer Intel core i5 CPU 650 Processor 4M

Cache, 3.20 GHz. The associated Computational Time (CPU time) required for numerical

experiments were recorded.

Summary of liquid-liquid data points and value of parameters used in experiments and

simulations are presented in Table 3.

Table 3 –Flow conditions for mathematical model validation (Parvini and Dabir, et al. 2010)

Velocity

(m/s)

Dispersed

fluid diameter

(m)

L(m) D(mm) Drag Model Dispersed

Phase fraction

Turbulence

model

0,69 0,05 1,50 78 Schiller and

Naumann 0,1912 k-

The simulation results can be seen in Figure 1. Simulation data predict the profile of

dispersed phase fraction mainly in center of pipe, the highest difference is close to wall. This

performance depends on assumptions taken in model which should be improved in futures

works.

3.2. Study Case: Sudden Enlargement in a Circular Pipe

Computational analysis: To evaluate the effect in a flow with similar characteristics in

emulsions was carried out different simulations with the parameters defined in Table 4 using

as geometry the classic case of sudden enlargement in a circular pipe (Figure 2). The problem

is considered two–dimensional and the calculations are made on a two dimensional axi-

symmetric geometry. The mesh used is hexahedral with 11441 cells, 9 cell on inlet and 18

cells on outlet.


Búzios, RJ

The profiles of dispersed phase fraction and velocities were analyzed in three different

planes across the pipe as is showing in Figure 2.

Figure 1 – Comparison between simulation using PIMPLE and k- and experimental data

The boundary conditions and the parameters to carry out the simulations are presented

in table 4:

Table 4 – Summary of conditions to run simulations.

Velocity

(m/s)

Dispersed fluid

diameter (m) Drag Model

Dispersed

Phase fraction

Relaxation

Factor

Real Time

(seg)

0,69 3,5e -04 Schiller and

Naumann 0,10 0,9 60

As can be seen from Table 5, all of algorithms used in this work are stable. Residuals

were between 10-3

and 10-6

. On the one hand, the highest values correspond to the PIMPLE

case and simulations using k-ω. Conversely, simulation using PIMPLE k- guaranteed the

lowest residuals so the highest precision.

With regard to computational time (Table 6), the SIMPLEC cases expend less CPU

time comparing with the other algorithms because PIMPLE and PIMPLEC are based in

algorithm PISO which one involve one predictor step and two corrector steps whereas

SIMPLEC do not need to do this procedure.


Búzios, RJ

Figure 2 – Sections considered to evaluate velocity and dispersed phase fraction.

Table 5 – Residuals of main parameters calculated in all simulations

Algorithm Turbulence

model

Residuals

Continuity Pressure Alpha k /ω

PIMPLEC k- 5,61e-12 6,79e-04 3,98e-06 1,80e-05 1,02e-05

PIMPLEC k-ω 4,12e-12 6,31e-04 2,45e-06 1,44e-05 1,78e-03

PIMPLE k- 4,53e-11 7,73e-05 8,67e-06 1,00e-05 1,00e-05

PIMPLE k-ω 4,11e-11 7,03e-05 2,45e-06 8,99e-06 2,06e-03

SIMPLEC k- 1,31e-07 2,54e-03 1,07e-05 1,03e-05 1,03e-05

SIMPLEC k-ω 9,91e-08 2,49e-03 8,67e-06 1,06e-05 1,84e-03

Table 6 – Computational times necessary to carry out each simulation

Algorithm Turbulence

model

Execution

Time Clock Time

PIMPLEC k- 3047,35 3123

PIMPLEC k-ω 3044,34 3116

PIMPLE k- 2911,91 2986

PIMPLE k-ω 2878,88 2949

SIMPLEC k- 1972,54 2027

SIMPLEC k-ω 1925,95 1974


Búzios, RJ

Velocities and Dispersed phase fraction Analysis: For particles with mean diameter

350 µm it is observed that dispersed phase fraction is practically constant across the area 1

and velocities profiles have been not development completely (Figures 3, 4, 5).

Figure 3 – Dispersed Phase Fraction (Alpha) Profile across sectional area 1.

Figure 4 – Continue Phase Velocity (Ub) Profile across sectional area 1.

In this step, algorithms presented the same response with minimal difference that it

could be negligible to flow analysis. Moreover, on section 2, simulations show higher

differences than in area 1 in dispersed phase fraction (Alpha, Figure 6), probably due the flow

recirculation and presence of eddies (Figures 7 and 8). This difference is visible for Alpha

where simulations using PIMPLEC and SIMPLEC k-ω indicated high concentration near to


Búzios, RJ

wall whereas simulations using PIMPLE presented the lowest values over this section (Figure

6).

Figure 5 – Disperse Phase Velocity (Ua) Profile across sectional area 1.

Figure 6 – Dispersed Phase Fraction (Alpha) Profile across sectional area 2.

As can be seen in Figures 7 and 8, the simulations do not change when are illustrated

velocities of disperse and continuous phase.


Búzios, RJ

Figure 7 – Dispersed Phase Velocity (Ua) Profile across sectional area 2.

Variations among simulations with respect to Alpha are minimized in third section

where there is no recirculation (Figure 9). In this case, simulations carried out with PIMPLE

presented lower values close to wall than SIMPLEC and PIMPLEC, in some cases, below

average concentration.

Figure 8 – Continuous Phase Velocity (Ub) Profile across sectional area 2.


Búzios, RJ

Figure 9 – Dispersed Phase Fraction (Alpha) Profile across section 3.

Figure 10 – Continuous Phase Velocity (Ub) Profile across section 3.

In third section the velocity results corroborate the first assumption that the drops flow

with the fluid because dispersed phase velocity has the same profile than the continuous phase

(Figure 11) with a small difference (< 0,06 m/s) among methods.


Búzios, RJ

Figure 11 – Dispersed Phase velocity (Ua) Profile across section 3.

4. CONCLUSIONS

Simulations in circular pipe with sudden enlargement were performed to test three

algorithms for pressure-velocity (SIMPLEC, PIMPLE and PIMPLEC) coupling and two

turbulence models (k- , k-ω). The main conclusions were the following:

The dispersed phase profile calculated showed an approximation to real process

where the most variations in results are especially close to the wall.

For all cases computed, PIMPLEC needs the largest computational time, PIMPLE

comes next and SIMPLEC need the least computational time. When k- and k-ω

are analyzed in each algorithm it is observed that k-ω expends less time than k-

despite k-ω uses additional equations to evaluated turbulence effect near wall.

Under same conditions all algorithms showed appropriate convergence with

maximum residual of 1,84e-03 for calculate ω in PIMPLE k-ω case.

The highest difference among simulations was observed in the dispersed phase

fraction profile where variations occur in the recirculation region mainly when

PIMPLE algorithm is implemented.

The SIMPLEC algorithm is recommended for the solution of with one dispersed

phase by its low computational time and good convergence.


Búzios, RJ

5. REFERENCE

FARRAR B., BRUUN H.H. A computer based hot-film technique used for flow

measurement in vertical kerosene–water pipe flow. Int. J. Multiphase Flow, v. 22, p. 733–751,

1996.

MENTER, F., ESCH, T. Elements of industrial heat transfer predictions, 'COBEM

2001, 16th Brazilian Congress of Mechanical Engineering.' 2001.

OLIVEIRA, P. J., ISSA, R. I. Numerical aspects of an algorithm for the Eulerian

simulation of two-phase flows Int J Numer Meth Fl, v. 43, p. 1177-1198, 2003.

PARVINI, M.; DABIR, B et al. Numerical Simulation of Oil Dispersions in Vertical Pipe

Flow J Jpn Petrol Inst, v. 53 (1), p 42-54, 2010.

RUSCHE, H. Computational Fluid Dynamics of Dispersed Two Phase Flows at High

Phase Fractions. 2002. 343 (Doctor of Philosophy of the University of London and Diploma

of Imperial College). Departament of Mechanical Engineering, Imperial College of Science,

Technology & Medicine, ExhibitionRoad, London SW7 2BX.

VERSTEEG. H. K., MALALASEKERA, W. An introduction to computational fluid

dynamics : the finite volume method. 2nd ed. Harlow, England; New York: Pearson

Education, 2007. xii, 503 p.

YIN, R.; CHOW, W.K. Comparison of four algorithms for solving pressure-velocity

linked equations in simulating atrium fire. Int J Arch Sci, v. 4 (1), p 24-35, 2003

ZEN, M.; TAO, W.Q. A comparison study of the convergence characteristics and

robustness for four variants of SIMPLE-family at fine grids. Eng Computation, v. 20 p. 320-

340, 2003

STUDY OF TURBULENCE MODELS AND ALGORITHMS FOR PRESSURE-VELOCITY COUPLING IN SIMULATION OF EMULSION...

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