Modelling of self-induced oscillations in the mixing head of a RIM machine.

INTRODUCTION

Reaction Injection Moulding (RIM) is widely used in industry for making polymer parts. In RIM systems, mixing is carried through directly opposed jet-to-jet impingement of the reactant streams and through the stretching and folding of the fluid elements, which promotes an almost direct contact of pure reactant streams. In practice, RIM is of importance for high viscosity fluids, such as polymer solutions, polymer melts, etc.

Many visualization studies have reported the flow characteristics in the mixing head of a RIM machine and have shown that self-induced and highly organized oscillations will be observed when the jet Reynolds number, [Re.sub.d], is beyond a threshold value (Malguarnera and Suh, 1977; Tucker and Suh, 1980; Lee et al., 1980; Kolodziej et al., 1982; Denshchikov et al., 1983; Sandell et al., 1985; Akaike et al., 1986; Sebastian et al., 1986; Wood et al., 1991; Roy et al., 1994; Johnson et al., 1996; Unger et al., 1998). A self-induced oscillation is excited by flow without any other external force. It was found that the injector velocity and the geometry of the mixing head have a great influence on the quantities of the oscillation. However, only few papers have paid attention to the frequency of the oscillations experimentally or numerically. Denshchikov et al. (1983) found that the frequency of the oscillations could be well correlated by [Re.sub.d] and the dimensionless distance between the jets. Wood et al. (1991) investigated the flow-induced oscillations in the mixing head by Laser Doppler Anemometry (LDA) and three-dimensional simulation and found that numerical simulations agree well with the experimental visualizations for the Strouhal numbers of the oscillations. Johnson and Wood (2000) evaluated the quantities of the oscillations in the mixing head experimentally and numerically and investigated the effect of [Re.sub.d] and velocity at the injectors, [u.sub.inj], on the frequency of the oscillations.

To understand the self-induced oscillations, it is necessary to study the mechanism of the excitation process. Several mechanisms for this phenomenon were proposed, with the emphasis on the pressure fluctuating on the impingement point (Rockwell and Naudascher, 1979; Rockwell, 1983; Yeo, 1993; Naudascher and Rockwell, 1994; Johnson and Wood, 2000; Ekmekci and Rockwell, 2003; Ozalp et al., 2003). Johnson and Wood (2000) suggested that the impingement surface plays a significant role in the onset and sustaining of the oscillations and the oscillating flow field can be looked as a class of self-sustaining oscillation where instabilities in the jet shear layer are amplified because of feedback from pressure disturbances in the impingement region. The simulation results by Yeo (1993) also showed that asymmetric distribution of the pressure at the impingement point decides the quantities of the oscillatory motion in the mixing head. Whereas, no corresponding model to these mechanisms appear to have been reported. This paper aims to model the motion of the self-sustained oscillations by the pressure fluctuation and further to verify this model by computing the frequency of the oscillations and comparing simulations results with experimental ones. Finally, the effect of [R.sub.ed] on the Strouhal number will be investigated.

MODELLING ON THE SELF-INDUCED OSCILLATIONS IN THE MIXING HEAD

In order to study the self-induced oscillations in the mixing head, consider a 2-D chamber and taking a control volume over the chamber as shown in Figure 1. For an incompressible, laminar flow, in x direction we have:

[rho][Du.sub.x]/Dt = [rho][a.sub.x] = [partial derivative][[tau].sub.xx]/[[partial derivative].sub.x] + [partial derivative][[tau].sub.xz]/[[partial derivative].sub.z] + [rho][g.sub.x] (1)

Since there is no gravity component in x direction, Equation (1) can be expressed as follows to the coordinate system used in this paper:

[rho][a.sub.x] = [partial derivative][[tau].sub.xx]/[partial derivative]x + [partial derivative][[tau].sub.xz]/[partial derivative]z (2)

where [[tau].sub.xx] and [[tau].sub.xz] denote the normal and shear stresses of the control volume, respectively, and can be expressed as:

[[tau].sub.xx] = -p + 2[mu] [partial derivative][u.sub.x]/[partial derivative], [[tau].sup.xz] = [mu]([partial derivative][u.sub.x]/[partial derivative]z + [partial derivative][u.sup.z]/[partial derivative]x) (3)

where [u.sub.x] denotes the velocity component in x direction and [u.sub.z] in z direction. Substituting Equation (3) to Equation (2), we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[FIGURE 1 OMITTED]

The first term on the right-hand side denotes the pressure gradient along x direction, and the remaining terms on the right-hand side denote the viscosity dissipation along x direction, which can be ignored in our case (verified by a simulation case in Result and Discussion section). So Equation (4) can be simplified and rearranged as follows:

[rho][a.sub.x] + [partial derivative]p/[partial derivative]x = 0 (5)

Equation (5) is similar to and can be expressed as the formulation of a linear spring in the undamped system:

[rho][??] + kx = 0 (6)

where k is the spring coefficient per volume:

kx = 1/x [partial derivative]p/[partial derivative]x

So the vortex-induced oscillations in our case can be simplified as a discrete mass per volume, ?, free to vibrate in one direction, which is depicted as a body oscillator in Figure 2. The solution to Equation (6) is:

x = [x.sub.0]cos([[omega].sub.n]t -[phi]) (7)

with

[[omega].sub.n] = 2[pi][f.sub.n] = [square root of k/[rho]]

where [[omega].sub.n] is the natural circular frequency and fn is the natural frequency of the discrete-mass system. Over the half domain of the chamber, [[omega].sub.n] can be expressed as follows:

[[omega].sub.n] = 2[pi][f.sub.n] = [square root of ([partial derivative]p/[partial derivative]x)/[rho]x] = [square root of [DELTA][p.sub.x]/l/[rho]l)] (8)

where l is the amplitude of the self-induced oscillation in the chamber and [DELTA][p.sub.x] the pressure fluctuation along l in x direction. The amplitude will be discussed in detail in the Result and Discussion section. In this paper, only the dimensionless pressure fluctuation, [DELTA][p'.sub.x], is available from CFD results (details can be found in the next section). You can refer to the next section for the definition of [DELTA][p'.sub.x], so Equation (8) can be further represented as:

[[omega].sub.n] = 2[pi][f.sub.n] = [square root of ([rho][u.sup.2.sub.inj][DELTA][p'.sub.x]/l)/[rho]l] = [u.sub.inj]/l[square root of [DELTA][p'.sub.x]] (9)

In our simulation, [DELTA][p'.sub.x] is varied from time to time, so it is useful to calculate the average frequency, <[[omega].sub.n]> based on the averaged pressure fluctuation over a time period, <[DELTA][p'.sub.x]>, which should be long enough to secure the computational accuracy. Note that <[DELTA][p'.sub.x]> is the time-average of the absolute value of the pressure fluctuations over a time period, since the sum of the pressure fluctuations approaches to zero, which can be easily seen from the simulation results in the Result and Discussion section. The dimensionless frequency, Strouhal number, St, can be expressed as:

St = [[omega].sub.n] x d/[u.sub.inj] = d[square root of <[DELTA]p'x>/l] = [square root of <[DELTA][p'.sub.x]>/l'] (10)

where l' is the dimensionless amplitude of the oscillation in the chamber, l' = l/d. d is the diameter of the injectors of the mixing head. More details on the geometry of the mixing head can be found in the next section.

[FIGURE 2 OMITTED]

MATHEMATICAL FORMULATION FOR THE FLOW CONDITIONS IN THE MIXING HEAD

It is useful to use CFD (Computational Fluid Dynamics) to study the flow-induced oscillations in the mixing head of a RIM machine, which is sketched in Figure 3. In formulating flow field inside the chamber, the following assumptions are made: (1) the flow is laminar and incompressible; and (2) the fluid is viscous and Newtonian with constant physical properties. Based on the above assumptions, the flow field in the mixing head can be described by the continuity and full incompressible fluid Navier-Stokes equations:

[nabla] x u = 0 (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [rho] and [mu] are the density and viscosity of the reactant, respectively, g the gravity acceleration. The governing equations above can be written in the dimensionless form as follows:

[nabla] x u' = 0 (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[FIGURE 3 OMITTED]

In Equations (13) and (14), the relevant dimensionless physical variables are defined by:

t' = [tu.sub.inj]/d, u' = u/[u.sub.inj], [nabla]' = d[nabla], p' = p/[[rho].sup.2.sub.inj]

The relationship between the dimensionless time, t', and the time expressed as a function of the residence time, [tau], is:

t/[tau] = 2[d.sup.2]/HD t' = 9 x [10.sup.-3]t'

and the two dimensionless groups are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [u.sub.inj] is the value of the velocity at the injector, d the diameter of the injector. [Re.sub.d] is the jet Reynolds number and Fr the Froude number.

For the governing equations, the boundary conditions are defined as follows: velocities at all solid walls taken to be zero in accordance with no-slip condition, a zero derivative boundary condition applied at the outlet, which can be expressed specifically as follows:

[u.sub.x] = 0 (15)

[partial derivative][u.sub.z]/[partial derivative]z = 0 (16)

Based on the governing equations and boundary conditions above, a first symmetry steady-state solution will be obtained with initial guess set to zero for all zones. Then the dynamic behaviour of the RIM impingement mixing head can be obtained numerically by inputting the symmetry steady solution as the initial condition and by imposing a small perturbation to the right injector. The velocity at right injector can be described by:

[u.sub.inj](t') = [u.sub.inj](t') (17)

where f(t') is the perturbation time function affecting the steady velocity profile at one of the injectors:

f(t') = 1 + a/2(1 - H(t' - b))(1 - cos(2[pi]t'/b)) (18)

In Equation (18), H(t'-b) is the Heaviside function, which is equal to zero if t'<b, and to one otherwise; a is the maximum amplitude of the perturbation and equal to 0.1 in this paper; b represents the time period of the perturbation occurring and is set to be 11.11, a tenth of the fluid passage time, [tau]. The perturbation curve of Equation (18) is shown in Figure 4.

NUMERICAL SOLUTION

The steady and hydrodynamics simulations of the mixing head are carried out on the commercial software package Fluent. In this paper, some experimental cases by Santos (2003) will be simulated. The physical parameters, as well as the values of the dimensionless groups, [Re.sub.d], Fr, for those cases are given in Table 1. Note that the critical Reynolds number for fully developed oscillations in this paper is 250. Only the cases with the jet Reynolds number beyond this threshold value were simulated to study the quantities of oscillations. In those cases, the geometric model and the space coordinate are put in dimensionless form by the injector diameter, d:

x' = x/d, x = z/d, H' = H/d, D' = D/d, L' = L/d

and the key dimensions of the geometry of the mixing head are given in Table 2.

[FIGURE 4 OMITTED]

The symmetric, 2-D simulation domain is represented by 44 167 quadrilateral cells with non-uniform spacing with a particularly fine grid in injectors. Santos et al. (2001) have demonstrated numerically that the computation is sufficiently accurate when [DELTA]x equals to [DELTA]z [less than or equal to] 0.1.

Previous mixing simulations have demonstrated that false prediction of mixing may result from numerical errors, especially in the chaotic regime. To minimize such errors, a coupled implicit solver will be employed in this simulation to solve the momentum in x and z directions simultaneously, which can decrease the numerical error efficiently. Note that a segregated solver will be used first for the steady simulation. The default Courant number (CFL) for the coupled implicit solver is 5.0, which is the main control over the time-stepping scheme. Whereas, a lower Courant number (CFL = 2) is required during start-up for our case and will be increased to 4, 8, 16, 32, 64 as the solution progresses. A steady solution will be reached as the CFL gives rise to 64. When the guesses for all the zones are taken to be zero, a symmetric steady-state will be obtained.

Then, the steady solution will be used as initial guess for the dynamics simulation. Note that a coupled implicit solver will still be employed in this simulation. Numerical tests were conducted by Santos et al. (2005) to achieve a time step not affecting the numerical simulation and showed that [DELTA]t' = 0.01 was satisfactory. The simulation takes the total number of iterations to reach the final time of 10[tau]. Because of memory and time requirements for the iteration process, the Fluent and the Gambit software were run on Dell Precision workstation (Intel Xeon 2.2 GHz, 1.5GB SDRAM). Relevant parameters for all dynamic simulations are shown in Table 3, namely: grid elements, Reynolds number and CPU time for the simulation from 0 to 10[tau].

RESULTS AND DISCUSSION

This paper aims to study the quantities of the oscillations in the mixing head of a RIM machine, especially the frequencies of the oscillations. For this purpose, Equation (10) will be applied to calculate the Strouhal numbers and the effect of [Re.sub.d] and Fr on Strouhal numbers will be investigated in this paper. First, the contribution of the pressure fluctuation and viscosity dissipation to the self-sustained oscillations in the mixing head were investigated in this paper and the numerical results are shown in Figure 6. It was found that the viscosity dissipation is only ([10.sup.-2]) to pressure fluctuation, so it was confirmed that contribution of dissipation term to the oscillation can be ignored as stated in the second section.

Then, the stream contours of the flow field in the mixing head will be investigated and two typical contours at t' = 645.36, 661.13 for Case 2 are given in Figure 7. It was found that at t' = 645.36, the vortices flow toward the opposite direction, to which they flow at t' = 661.13 in x direction. Therefore, the stream contours at t' = 645.36, 661.13 represent two different states of a circle for the self-sustained oscillations in the chamber, one for the former half and the other for the later half. From the stream contours, the flow field in the mixing chamber is very heterogeneous, presenting remarkably different features at different locations. At the impingement point, the two jets impinge approximately at the chamber axis. Immediately downstream the jets, two vortices are formed, each one occupying approximately half the chamber width. The amplitude of oscillation in the impingement region can be characterized by the measurement of the pressure fluctuation (Rockwell and Naudascher, 1979). The dependence of such pressure fluctuations on impingement length and flow speed has been studied for a variety of geometries (Powell, 1961; Chanaud and Powell, 1965). In this paper, the amplitudes in the impingement region can be measured approximately from the maps of turbulence intensity calculated from PIV data as l = 0.36D at z = 5 mm, l= 0.50D at z = 7.5 mm and l = 0.60D at z = 10 mm at [Re.sub.d] = 300, since the intensity of turbulence is scaled by velocity component fluctuations (Santos, 2003). Then, the two vortices evolve to fully developed vortices extending throughout the whole chamber width, as can be seen from the contours in Figure 7. The fully developed vortices flow from approximately a distance downstream the jets equal to the chamber width (Santos et al., 2002). Therefore, the amplitude of oscillation, l, can be considered as the whole D when z [greater than or equal to] 15mm for the chamber in this paper. From Equation (10), it was concluded that the Strouhal number of the fully developed oscillation is only dependent on the values of the dimensionless diameter of the chamber and the jet Reynolds number. This agrees well with the conclusion proposed by Denshchikov et al. (1983) that the frequency of the oscillations could be well correlated by the jet Reynolds number, [Re.sub.d], and the dimensionless distance between the jets. Further, the independence of the oscillation frequency on the time interval was evaluated. From Equation (9) or (10), it shows that the frequency or Strouhal number is mainly dependent on that of pressure fluctuations. Therefore, the independence of the pressure fluctuation on time interval will be studied and the values of <[DELTA][p'.sub.x]> on the different time intervals varied from t = 0.2[tau] to t = 4[tau] will be investigated for Case 2 at impingement point. The calculated pressure fluctuations versus time intervals are shown in Figure 5, and the calculated frequencies by Equation (10) are listed in Table 4. As discussed above, at impingement point l = 0.36D, we have l'=2.5. Figure 5 shows that when t [greater than or equal to] [tau], the pressure fluctuation will be independent of time interval. From Table 4, it was found that the frequencies at t = [tau] and t = 2t are subjected to a relative error of 0.2%. This proves that t = [tau] is satisfactory and will be applied in our simulation.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Then this model will be applied to calculate the Strouhal number of the self-sustained oscillations on the points at the chamber axis. Figure 8 gives the time history samples of pressure fluctuations and [u'.sub.x] for z = 5, 12.5 and 44 mm for Case 2, where [u'.sub.x] is the velocity component in x direction of those points. Comparing the time series of [u'.sub.x] with that of pressure drop at the impingement point in Figure 8, it was found that every peak in the series of pressure fluctuation corresponds exactly to one in the series of [u'.sub.x] in the same direction. This shows that the pressure drop along D is the driving force of the self-induced oscillations in the mixing head, and the values of pressure disturbance decide the frequency of the oscillation in the mixing head. According to the properties of the spring force of an undamped system, the sum of the pressure fluctuations equals to zero in a long time period, which can be easily seen from the time histories of pressure fluctuations in Figure 8. However, in the downstream, the peaks of [u'.sub.x] do not correspond to the peaks of pressure fluctuation as well as in the upstream, since [u'.sub.x] in the downstream is more affected by the oscillation in z direction than in the upstream. It was verified by Santos et al. (2005) that St in z direction downstream equals to 0.015 and is vanishingly small in the impingement regime. From this aspect, the time history of pressure fluctuation can provide more correct information on the quantities of oscillation than that of [u'.sub.x], due to the disturbance of measuring of [u'.sub.x] by the oscillations in z direction. The Strouhal number can be calculated by applying Equation (10) to the pressure fluctuation series in Figure 8. From the simulation results, the average Strouhal number, St, equals to 0.1 at the jet impingement regime, z = 5, and 7.5 mm. The pressure disturbance at the impingement point is very important in deciding the quantities of the oscillatory motion in the mixing head (Yeo, 1993). For z [greater than or equal to] 22 mm, the flow is characterized with a smaller typical frequency of St = 0.029. In this zone, the typical frequency is generated by the passage of fully developed vortices extending throughout the whole chamber width and keeping their diameter while evolving through the chamber. The calculated results were compared with the dominant frequency from time series of [u'.sub.x] by Santos (2003), and found in good agreement with them in high and low dominant frequency as shown in Figure 9. In addition, a transition zone of 11 mm [less than or equal to] z [less than or equal to] 22 mm was shown in Figure 9 on the curve of St from the pressure fluctuation. This zone provides direct communication between processes near the impingement surface and the separation region, to ensure highly organized oscillations occurred in the mixing head. Rockwell (1983) concluded that highly organized oscillations of an impingement flow are sustained through a series of interacting events: feedback, or upstream propagation, of disturbances from the impingement region to the sensitive area of the free shear layer near separation; inducement of localized vorticity fluctuations in this region by the arriving perturbations; amplification of these vorticity fluctuations in the shear layer between separation and impingement, and production of organized disturbances at impingement. In this zone, the Strouhal number drops abruptly from 0.1 to 0.03. Spectra taken at a single location in the shear layer for the case of a wedge flow (Hussain and Zaman, 1978) strikingly shows that the energy concentration gradually shifts from one dominant frequency to another. This shift may occur over a much shorter distance. Santos (2003) also noticed this phenomenon by the fact that two dominant frequencies were observed from the power spectra of the time series of u ' x in the transition zone, one at St = 0.1, the other at a lower frequency whereas the averaged frequency cannot be available, due to the unknown of their portion. In this paper, the averaged frequency can be calculated by Equation (10), which is of importance in describing the averaged oscillation quantities in the mixing chamber.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Further, the effect of [Re.sub.d] on the Strouhal number in the impingement regime was also evaluated. From Equation (10), it was found that [Re.sub.d] plays its role through [DELTA][p'.sub.x] on the Strouhal number. Figure 10 gives the time history samples of pressure fluctuations and [u'.sub.x] for the cases at [Re.sub.d] = 400, 500 in Table 1. From the time history of [Re.sub.d] = 500, a second oscillatory behaviour at the impingement point with a lower frequency is observed. The lower frequency oscillation is probably induced by the fully developed vortices formed downstream in the mixing head, which will affect the jets oscillatory behaviour. The Strouhal numbers were also calculated by applying Equation (10) to the pressure fluctuation samples in Figure 10. The amplitudes in the impingement region can also be measured from the maps of turbulence intensity calculated from PIV data and showed an increase with the Reynolds numbers. This is due to the increasing interaction of the jets with the surrounding fluid, that is, the jets are more disturbed in their path at higher Reynolds numbers (Santos, 2003). Johnson and Wood (2000) also reported the difficulties in measuring the typical frequencies at [Re.sub.d] = 150, due to the increasing randomness of the flow field. The simulation results were compared with the dominant frequency from the time series of [u'.sub.x] and LDA data by Santos (2003) and a considerable agreement was found as shown in Figure 11. Due to the 3-D geometry in experiments and 2-D model in simulations, Santos (2003) deduced that the frequencies generated by the 2-D flow structures in the simulations and 3-D flow structures in the experiments relate as [St.sub.2D] = 1.5[St.sub.3D], and that the Reynolds numbers relate as [Re.sub.2D] = 2[Re.sub.3D]. In Figure 11, a decrease was observed in the Strouhal numbers with Reynolds numbers when [Re.sub.d] > 250. It was concluded that the increase of the Reynolds number is not beneficial for improving the Strouhal number after the Reynolds number exceeds the threshold value for the onset of fully developed oscillations in the mixing head. Since our simulations concentrate on the frequency of the self-induced oscillation in the mixing head and the Reynolds numbers are required to be larger than 250, the calculated frequencies from our cases cannot be compared with the experimental data by Johnson and Wood (2000) directly.

[FIGURE 11 OMITTED]

Finally, the effect of Froude number, Fr, on the oscillations in the mixing head was investigated. Since Fr is a relation between the inertial and body forces, the values of Fr can be adjusted by changing inertial forces (i.e. [u.sub.inj]) or by changing the body forces (i.e. gravity) keeping [u.sub.inj] and viscosity constant. In this paper, four cases with lower values of Fr = 1, 10, 500, 1000 will be simulated with Reynolds number fixed at 300. The Strouhal numbers at the impingement point versus the Froude number are presented in Figure 12. It was found that no oscillation was formed in the chamber when Fr [less than or equal to] 10, which can be seen from Figure 13 and St remains constant at 0.1 when Fr [greater than or equal to] 1000. A transition zone with lower frequency was observed between Fr = 1 and Fr = 1000. This result indicates that it is invalid to use lower viscosity fluids, i.e., gases, in the RIM machine at intermediate Reynolds number. The Froude number is a critical parameter in the RIM process since the operation at lower values of Fr presents an increasing stability up to the point where the system is unable to present dynamic evolution. Thus, even if the ratios of the inertial and viscous forces are kept at constant Reynolds number, if the inertial forces are decreased, the system dynamics is dumped. A probable reason is that a greater body forces increasingly imposes its application direction to the flow and thus compromise its ability to present recirculation patterns. It seems that the oscillation in z direction is of importance in feedback the instability from the downstream to upstream.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

CONCLUSIONS

In this paper, the quantities of the self-sustaining oscillations in the mixing head of a RIM machine were modelled. The dynamic simulations of the experimental cases by Santos (2003) were carried out by commercial Fluent package successfully. By the comparison of the time series of pressure fluctuations and [u'.sub.x] in the impingement region, it was found that pressure fluctuation is the driving force of the self-excited oscillations in the mixing head. The calculated Strouhal numbers are in good agreement with the dominant frequency from the power spectral of measured velocity component [u.sub.x] by Santos (2003). This shows that the Strouhal numbers can be well calculated from the averaged pressure drop of self-sustained oscillations along its amplitude in the mixing head. Finally, the effect of [Re.sub.d] and Fr on the Strouhal number was also investigated in the impingement region. The calculated Strouhal numbers showed a good agreement with the numerical results of Santos (2003). The simulation results at lower Froude numbers showed the invalidity to use lower viscosity fluids in the RIM machine, mainly due to the increase of body forces, which imposes its application direction to the flow and thus compromise its ability to present recirculation patterns. This model and numerical work in this paper give some insights in understanding the quantities of the oscillatory flow in the mixing head of a RIM machine, as well as provides a numerical approach to evaluate the dominant frequency in the mixing chamber.

ACKNOWLEDGMENT

The authors gratefully acknowledge the financial support from FCT (No. SFRH/BPD/10076/2002).
NOMENCLATURE

a acceleration (m x [s.sub.-2] )
b time period of perturbation occurring, dimensionless
d injectors' diameter/width (m)
D chamber diameter/width (m)
f nature frequency (Hz)
Fr Froude number
g gravity acceleration (m x [s.sub.-2])
H injectors centre distance from the mixing chamber
top (m)
k spring coefficient (N x [m.sub.-1])
l amplitude of oscillation (m)
L chamber height (m)
H(t) Heaviside function
p pressure (Pa)
[Re.sub.d] jet Reynolds number
St Strouhal number
t time (s)
[u.sub.inj] velocity at the injector (m x [s.sub.-1])
u velocity vector (m x [s.sub.-1])
x space coordinate
z space coordinate

Greek Symbols

[mu] viscosity (Pa x s)
[rho] density (kg x [m.sub.-3])
[tau] passage time in the mixing chamber (s)
[tau] stress tensor (Pa)
[omega] circular frequency

Manuscript received October 25, 2005; revised manuscript received May 26, 2006; accepted for publication July 28, 2006.

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Xiaojin Li (1), Ricardo J. Santos (2) and Jose Carlos B. Lopes (2) *

(1.) Dalian Institute of Physical Chemistry, Dalian, Liaoning Province, China 116023

(2.) Laboratory of Separation and Reaction Engineering, Chemical Engineering Department, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

* Author to whom correspondence may be addressed. E-mail address: lopes@fe.up.pt
Table 1. Physical parameters of the experiments by Santos (2003)
and the corresponding values of computational groups for simulation

Case no. Experimental

[u.sub.inj'] [rho], [micro],
m/s Kg/[m.sup.3] Pa.s

1 3.8 1187 0.0273

2 4.5 1187 0.0273

3 6.0 1187 0.0273

4 7.6 1187 0.0273

Case no. Computational

[Re.sub.d] Fr

1 250 981.2

2 300 1376

3 400 2446

4 500 3924

Table 2. Scale of geometric parameters

Experimental (mm) Computational

Jet diameter, d 1.5 d

Chamber diameter, D 10 6.667d

Piston position, H 5 3.333d

Chamber length, L 50 33.333d

Table 3. List of parameters for the several simulations
at different Reynolds numbers

[Re.sub.d] 250 300 400 500

Grid elements 44 167 44 167 44 167 44 167

CPU Time, h 29.8 24.9 26.2 28.9

Table 4. Investigation of time interval on the numerical
results of averaged pressure fluctuation and frequency of
self-induced oscillation at impingement point in the mixing
head for Case 2

t 0.2[tau] 0.4[tau] 0.6[tau]

([DELTA][p.sub.x.sup.'] 0.046199 0.05142 0.05359
St 0.09119 0.096207 0.098213

t 0.8[tau] [tau] 2[tau]

([DELTA][p.sub.x. sup.'] 0.055683 0.055803 0.055819
St 0.101148 0.102179 0.102348
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