Simulation of Unsteady Flow and Vortex Shedding for Narrow Spacer

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Ind. Eng. Chem. Res. 2003, 42, 4962-4977

Simulation of Unsteady Flow and Vortex Shedding for Narrow Spacer-Filled Channels J. Schwinge,† D. E. Wiley,*,† and D. F. Fletcher‡ UNESCO Centre for Membrane Science and Technology, School of Chemical Engineering and Industrial Chemistry, The University of New South Wales, Sydney, NSW 2052, Australia, and Department of Chemical Engineering, The University of Sydney, Sydney, NSW 2006, Australia

Insights into the effect of spacer filaments on the unsteady flow behavior in narrow spacerfilled channels were obtained using a computational fluid dynamics model. The flow was computed for different filament configurations for channel Reynolds numbers up to 2000. For the filament diameter, channel height, and spacer mesh length examined here, the transition to unsteady flow for channels obstructed with cylindrical spacer filaments occurs somewhere between Rech of 200 and 800, depending on the spacer configuration. When the flow is unsteady, vortex shedding occurs behind the spacer filaments and the shed vortices scour the channel walls as they move toward the next downstream filament, thereby increasing the wall shear stress. For each spacer configuration examined, a characteristic vortex formation pattern was found. 1. Introduction Steady flows in spacer-filled channels have been examined previously,1-3 and unsteady flows around cylinders placed in the middle of a wide “free-flow” channel are well documented in the literature.4-6 For such free-flow channels, interactions with the channel walls are negligible. For cylinder Reynolds numbers above 40, the recirculation region starts to show unsteady flow in the form of periodic movements with time. For a cylinder Reynolds number greater than 200, a vortex street develops behind the cylinder.4 The flow in obstructed channels can become unsteady for Reynolds numbers between 200 and 600.7-32 Flows have been examined for wide channels that had grooved channel walls,12-14,18-21 submerged rectangular bars in the center of the channel,23,24 and cylinders placed near the channel wall.33 Flow over a backward-facing rectangular step was found to be stable for higher Reynolds numbers than for grooved channels or channels with obstacles suspended in the channel. The flow directly behind the step can be time-independent for Reynolds numbers as high as 2000. However, small disturbances in the upstream region propagate unsteadiness downstream at lower Reynolds numbers.17 Practical application of spiral-wound membrane modules can require flow rates that lead to Reynolds numbers that exceed those at which steady flow occurs (Rech > 400). In this paper, the transition from steady to unsteady flow in narrow spacer-filled channels is examined in detail and flow patterns for unsteady flows at higher Reynolds numbers are discussed. This paper expands on our previous computational fluid dynamics (CFD) studies1-3 of the effects of spacer filament configurations on flow patterns, mass transfer, and pressure loss during steady flow. * To whom correspondence should be addressed. Tel.: +61 2 9385 4304. Fax: +61 2 9385 5966. E-mail: D.Wiley@ UNSW.edu.au. † The University of New South Wales. ‡ The University of Sydney.

Figure 1. Transverse filament configurations used in the channel to obstruct the flow.

2. Problem Description, Modeling Assumptions, and Numerical Methods Previously,1 flow patterns for Reynolds numbers that lead to steady-state flow in spacer-filled channels were examined, but flow patterns at higher Reynolds numbers were not discussed. However, the examination of unsteady flow in spacer-filled channels is equally important because unsteady flows may be present in spiral-wound membrane modules as a result of the presence of spacers. Unsteadiness may also be propagated into the membrane module because of unsteadiness caused by the pipe work and the flow distributor at the entrance of the module housing. To examine unsteady flows in this paper, channel geometries identical with those of Schwinge et al.1 were used (df/hch ) 0.5, lm/hch ) 4), but the Reynolds numbers were increased up to 2000, leading to unsteady flow with vortex shedding. The setup of the spacer-filled channel is shown in Figure 1. The channel has long entrance and exit lengths in order to (a) fully develop the flow along the channel entrance before the first upstream filament and

10.1021/ie030211n CCC: $25.00 © 2003 American Chemical Society Published on Web 08/29/2003

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4963

Figure 2. Vortex shedding behind a cylinder in the free-flow channel at Recyl ) 200 (the computed channel is much wider and longer than depicted).

Figure 3. Steady and unsteady flows caused by a single filament located in the center of the obstructed and narrow channel (df/hch ) 0.5) for Rech ranging from 200 to 2000.

(b) to avoid interference between the recirculation region behind the last downstream filament and the channel exit. Here, similar to the previous work,1 the fluid used is water at a temperature of 293 K, which is assumed to be incompressible and isothermal and to have constant fluid properties. The fluid motion is described by the Navier-Stokes equations, which are valid for all Reynolds numbers, but under laminar conditions, they can be solved without the need to consider the resolution of turbulent eddies.4,34 The Navier-Stokes equations were solved using a commercial CFD code (CFX, version 4.4, AEA Technology). The pressure-velocity coupling is handled via the SIMPLEC algorithm, and the QUICK scheme was used for the velocity components.34 A second-order

accurate fixed time stepping method was chosen, with time steps small enough to resolve the unsteady flow. At each time step, convergence was achieved in less than 20 iterations. The sum of the mass residuals was less than 0.05% of the inlet mass flux, ensuring a high degree of convergence. In addition, a small enough grid size had to be chosen to obtain grid-independent unsteady flow (details about the choice of time steps and grid size are given in section 4.1). No artificial perturbations were introduced into the simulation, and any instability that occurred was triggered by numerical perturbations arising from round-off or truncation errors. The calculations required more than 2 GB of computer memory and usually required 30 000-100 000 time steps to calculate at least one transit time of the

4964 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

Figure 4. Unsteady flow caused by a single filament located in the center of the obstructed and narrow channel (df/hch ) 0.5) at Rech equal to 500.

geometry (defined as the channel length divided by the bulk velocity). This leads to computational times between 50 and 150 h for each simulation on a NEC SX5 supercomputer. 3. Simulating Unsteady Flows Caused by Single Filaments in Obstructed Free-Flow and Narrow Channels 3.1. Simulating Unsteady Flow and Vortex Shedding in an Obstructed Free-Flow Channel. To check the ability to produce grid-independent unsteady flows for a channel obstructed with a single filament, unsteady flows were first simulated for flow in a freeflow channel using a single cylinder with the same diameter as that of the filaments in the spacer-filled channel. The free-flow channel was created by placement of a cylinder in a wide channel. The channel was wide enough to minimize any effects of the walls on the flow around the cylinder and long enough to avoid any interference of the channel exit on the formation of vortices and on the vortex street formed behind the cylinder. Flow in such a free-flow channel is well documented in the literature.4-6 The results of the CFD simulations of the single cylinder in the free-flow channel agree with accepted literature findings reported earlier.4-6 The oscillation frequency of the vortex can be described by Strouhal’s equation.4

(

fdf 19.7 ) 0.198 1 u∞ Recyl

)

(1)

Table 1. Comparison of the Shedding Frequency of the Vortex for a Single Submerged Cylinder in the Narrow Channel and the Shedding Frequency of the Vortex for a Cylinder in a Free-Flow Channela uave

fa

fb

uave

fa

fb

0.2 0.25

204 250

69 84

0.8

833

303

a f : shedding frequency for the vortex in a narrow channel a obstructed with a single filament located in the center of the channel (submerged filament) [Hz]. fb: shedding frequency for the vortex in a “free-flow” channel obstructed with a single filament [Hz]. uave: average velocity in either channel [m/s].

where the cylinder Reynolds number is defined by

Recyl ) dfu∞/ν

(2)

A vortex street is formed for Recyl greater than 200, as shown in Figure 2. In addition, the shedding frequency of 139 Hz obtained here shows good agreement with a value of 143 Hz predicted by Strouhal’s equation. Having shown that the method of modeling unsteady flows with the CFD code can accurately reproduce vortex shedding, flows in narrow channels obstructed with single filaments were then modeled at Reynolds numbers, leading to unsteady flow. To characterize the flow in obstructed narrow channels, in contrast to the cylinder Reynolds number Recyl for the flow around a cylinder in a free-flow channel, a modified channel Reynolds number Rech was used,1 which accounts for changes in the hydraulic diameter dh and voidage due to the presence of spacer filaments. Simulations of


Figure 5. Unsteady flow for Rech ranging from 200 to 2000 when a filament is adjacent to the wall and small flow disturbances are convected downstream (df/hch ) 0.5).

Figure 6. Unsteady flow for Rech equal to 1000 when a filament is adjacent to the wall and small flow disturbances are convected downstream (df/hch ) 0.5).

unsteady flows were made for the Reynolds number Rech in the range from 40 to 2000.

Rech ) dhuave/ν

(3)

3.2. Investigation of Unsteady Flow Patterns for Single Filaments in a Narrow Channel. For a single

filament located in the center of a narrow channel, the flow does not show any transient movements for Rech below 200, despite using small time steps and a fine mesh. For such steady flows, the length of the recirculation region behind the filament increases with the Reynolds number as shown previously.1


Figure 7. Vortex shedding in a spacer-filled channel with multiple filaments adjacent to the bottom wall at Rech ) 1600 for various time steps (lm/hch ) 4; df/hch ) 0.5; the figure shows a section of the channel and depicts two sequential filaments).

Figure 3 shows that the flow is unsteady for Rech ) 400, and therefore the transition has occurred somewhere in the Reynolds number range of 200-400. When the transition to unsteady flow begins, the recirculation region oscillates up and down in a flapping motion behind the filament and flapping movements convect unsteadiness downstream, as depicted in detail in Figure 4. The flapping of the recirculation region is clearly identified by local changes in velocity observed at an array of monitoring points located downstream of the cylinder. With an increase in the Reynolds number to 800, the

flapping movements of the recirculation region become stronger and the recirculation region increases in length and size, as shown in Figure 3. With an increase in the Reynolds number to 1600 and above, the recirculation region does not increase further in length and size behind the filament but is broken down and vortex shedding occurs immediately behind the filament. The shed vortices move downstream adjacent to the top and bottom walls. This was evident by the periodic changes in velocity at a monitoring point in the downstream region of the channel. The periodic movements of the vortex behind a


Figure 8. Vortex formation pattern for the cavity spacer.

filament submerged in the center of the channel can be described by the shedding frequency of the vortex, as depicted in Figure 4 for the monitoring point. The shedding frequencies for the free-flow channel and for the narrow channel with a filament in the center of the channel are shown in Table 1. The comparison is based on the same average velocity in the channel. This comparison is used instead of a comparison based on Rech and Recyl because for unsteady flow in the narrow spacer-filled channel Rech does not consider the local velocity changes near the filament and the use of Recyl is also not appropriate because a uniform free-stream velocity cannot be defined. At the same average velocity uave in the channel, Table 1 shows that the shedding frequency of the vortex behind the filament in the narrow channel is almost 3 times higher than the frequency of a cylinder in free flow. For a single filament adjacent to the bottom wall, the flow in the channel is steady for Reynolds numbers up to 400, as shown in Figure 5. At a Reynolds number of

800, small disturbances caused by the flow around the filament and the large recirculation region behind the filament propagate unsteadiness downstream. Figure 6 shows details of the phenomenon. The large recirculation region behind the filament does not show any timedependent movement as would be identified by changes in the velocity as a function of time at a monitoring point adjacent to the bottom wall, but downstream of it, the velocity at a monitoring point changes with time in a complex manner. For a single filament adjacent to the bottom wall, vortex shedding immediately behind the filament occurs at a Reynolds number of approximately 2000, as depicted in Figure 5. Comparison of the results of the spacer-filled channel with the results of a free-flow channel suggests that the walls near a cylinder delay the transition to unsteadiness. However, once unsteadiness commences, it is at a higher oscillation frequency than that in a free-flow channel. In addition, location of a cylinder adjacent to the wall delays the transition to unsteadiness when compared with a channel with a cylinder in the center of the channel. 4. Investigation of Unsteady Flow Patterns in Narrow Channels Obstructed with Multiple Filaments 4.1. Resolving Vortex Shedding in the SpacerFilled Channel. It was relatively straightforward to achieve grid-independent solutions for a single filament in the channel, but when considering multiple filaments, an additional check was performed to ensure grid independence. As an initial test case, unsteady flow was simulated for a channel with multiple filaments adjacent to the bottom wall (cavity spacer in Figure 1). This initial test case is used to determine the grid size and time steps required to produce grid-independent results.

Figure 9. Temporal velocity variations at monitoring points in the channel containing a cavity spacer.


Figure 11. Unsteady flow for Rech ranging from 400 to 2000 when the filaments are adjacent to the wall (lm/hch ) 4; df/hch ) 0.5).

Figure 10. Shear stress patterns on the top and bottom channel walls for the cavity spacer at Rech ) 1600 (lm/hch ) 4; df/hch ) 0.5).

The grid size was chosen based on the comparison of results from a succession of finer meshes. When the grid is too coarse, unsteady flow is suppressed. For the simulations reported here, a numerical grid size of less than 0.005 mm in both the x and y directions is required to resolve a channel with a height of typically several millimeters. A channel with such a fine grid and long entrance and exit lengths can have more than a million cells despite a maximum total length of the channel of less than 100 mm. The time steps must also be small enough to resolve the unsteady flow. The time steps were chosen based on the comparison of results from a succession of smaller time steps. When the time steps are too large, either unsteady flow is suppressed or convergence at each time step is not achieved. The time steps used here needed to be smaller than 0.1 ms in order to achieve convergence at each time step for Reynolds numbers below 400-800. For Reynolds numbers of 800-2000, time steps had to be as small as 1 µs in order to achieve convergence. It is concluded that, for narrow obstructed channels, the time steps needed to be almost 3 orders of magnitude smaller than the period of oscillation in order to resolve vortex shedding in the channel, while in the free-flow channel time steps of only 50-100 times smaller than the period of oscillation are required to resolve vortex shedding. 4.2. Unsteady Flow Patterns in Spacer-Filled Channels. The figures presented here for the spacerfilled channels (cavity, submerged, and zigzag spacer) show a section of the flow channel with multiple filaments depicting two sequential filaments. A Reynolds number of 1600 was chosen to ensure unsteady flow in the channel. The figures depict a sequence of time steps showing the movement of the vortices. The actual modeling covered a real time of the fluid flow of up to 1 s, and the depicted sequences are at least 0.75 s into the total modeling time in order to avoid any effects of the flow development on the vortex shedding. The time tref used in the figures is a reference time for the illustrated sequences. 4.2.1. Cavity Spacer. Figure 7 shows the vortex shedding pattern for the cavity spacer at Rech ) 1600. As shown in Figure 7a, four large vortices are present at tref ) 0 s: three (1, 2, and 4) adjacent to the bottom wall spread out between the filaments and one (3) adjacent to the top wall near the downstream filament.


Figure 12. Propagation of large eddies between sequential cavity spacer filaments caused by small upstream disturbances at Rech ) 800 (lm/hch ) 4; df/hch ) 0.5).

Figure 13. Vortex shedding for the submerged spacer at Rech ) 1600 at two time intervals (lm/hch ) 4; df/hch ) 0.5).

As shown in Figure 7b, at tref ) 1.5 ms, vortices 1 and 2 have moved downstream adjacent to the bottom wall and vortex 1 is fully detached from the upstream filament. A new vortex 5 has formed behind the upstream filament. Another new vortex 6 has developed adjacent to the top wall halfway between vortices 1 and 2, while vortex 3 has moved along the channel closer to the downstream filament and has reduced in size and vortex 4 has disappeared because it has been squeezed over the top of the filament. As shown in Figure 7c, at tref ) 2.5 ms, vortices 5 and 6 have increased in size. Vortex 5 is still attached to the upstream filament, while vortex 6 has moved downstream. Vortex 1 has also moved further down-

stream, while vortex 2 disappears because it is squeezed over the top of the filament. As shown in Figure 7d, at tref ) 3.5 ms, vortices 1 and 6 have moved downstream while a new small vortex 7 has formed adjacent to the top wall halfway between vortices 1 and 5. Vortex 5 is still adjacent to the upstream filament and has again increased in size. As shown in Figure 7e, at tref ) 5 ms, vortex 6 no longer appears in the figure because it has moved past the downstream filament. Vortex 5 has fully detached from the upstream filament and moved along the channel. Vortex 1 has reached the downstream filament and disappears because it is squeezed over the top of the filament. A new vortex 8 has formed behind the


upstream filament. Vortex 7 has moved along the channel adjacent to the top wall and has increased in size. As shown in Figure 7f, at tref ) 6 ms, vortex 7 has moved closer to the downstream filament, vortex 5 has moved downstream, and vortex 8 has increased in size behind the upstream filament. A new vortex 9 has started to form adjacent to the top wall halfway between vortices 5 and 8. As shown in Figure 7g, at tref ) 7.5 ms, vortex 7 has moved past the downstream filament while vortex 5 has reached the downstream filament. Vortex 9, being adjacent to the top wall, has moved downstream and increased in size. Vortex 8 is fully detached from the upstream filament, and a new vortex 10 is forming behind the upstream filament. Thus, at a Reynolds number of 1600, there is a regular pattern to the vortex formation, as shown schematically in Figure 8. After vortex I is formed and has detached from the upstream filament, a new vortex II immediately forms behind the filament and adjacent to vortex I (step a). In step b, both vortices I and II increase in size, with vortex I moving along the channel while vortex II remains attached to the filament. When vortex I has moved a sufficient distance along the channel, vortex III forms adjacent to the top wall between vortices I and II. As vortices I and III move downstream, in step c, they first increase in size but then decrease when they get closer to the downstream filament. Vortex III passes above the filament, while vortex I disappears because it is squeezed over the top of the filament. As shown in Figure 9, monitoring points were placed behind the upstream filament to help study the vortex shedding (MPI) and halfway between the downstream filament and the top wall to further elucidate the movements of the flow passing the filament (MPII). The velocity data at monitoring point MPI shows that a vortex passes the monitoring point every 3 ms for a Reynolds number of 1600, but the intensity of the vortex changes each time, as indicated by the figure. For the fluid that passes the downstream filament, the velocity variation at monitoring point MPII shows complex behavior. The temporal variations are more complex at MPII than at MPI because the vortices at both the top and bottom wall are squeezed together in front of the filament and then pass over the filament. This causes a highly unsteady flow, as shown by the time-dependent velocity at MPII. The unsteady shear stress pattern for the cavity spacer is shown in Figure 10 for three time steps. The maximum shear stress values occur where the shed vortices are adjacent to the wall. The shear stress increases more than an order of magnitude because of the presence of a shed vortex. With the movement of the vortices, the high shear stress region moves along the channel, as depicted for regions 1 and 2 in Figure 10. The high shear stress region due to the formation of the vortex is observed immediately behind the upstream filament, and when the vortex has been shed, this region moves along the channel toward the downstream filament. The shed vortices and the high shear stress region are likely to have a great impact on reducing fouling in membrane systems by breaking up the concentration boundary layer.35 The boundary layer breakup is repeated frequently along the channel as vortices pass. Thus, foulants are less likely to deposit

Figure 14. Vortex formation pattern for the submerged spacer.

on the membrane surface and will be convected away from the membrane wall by the scouring vortices. Figure 11 shows further details of the fluid flow pattern for the cavity spacer at a fixed mesh length for a series of Reynolds numbers. For Reynolds numbers below 400, a large and steady “fully formed” recirculation region, as discussed previously,1 exists between sequential filaments. The flow is unsteady at a Reynolds numbers of 800. At Rech ) 800, a large fully formed but unsteady recirculation region appears only between the first two sequential filaments. However, as shown in Figure 12, a large fully formed recirculation region is no longer observed between other downstream filaments and vortex shedding occurs. At a Reynolds number of 1600, vortex shedding commences after the first filament in the channel, as shown in Figure 11. 4.2.2. Submerged Spacer. Figure 13 shows the vortex shedding pattern for the submerged spacer at Rech ) 1600. In Figure 13a, at tref ) 0 s, the recirculation region 1 behind the upstream filament oscillates periodically and, because of the flapping movements of the recirculation region, vortices 2 and 3 have been shed previously. The shed vortices 4-6 have moved along the channel adjacent to the wall, with vortex 6 having almost reached the downstream filament. In Figure 13b, at tref ) 0.5 ms, it is evident that the recirculation region 1 continues to move periodically behind the upstream filament and a new vortex 7 has shed adjacent to the bottom wall. Vortices 2-5 have moved along the channel, while vortex 6 has passed the downstream filament. Thus, at a Reynolds number of 1600 a vortex formation pattern is observed, as shown in Figure 14. The recirculation region behind the upstream filament oscillates periodically shedding vortices at the top and bottom walls. The shed vortices move along the channel adjacent to the wall at which they were shed. When the shed vortices reach the downstream filament, the vortices reduce in size and disappear as they squeeze past the downstream filament. As shown in Figure 15, monitoring points were placed behind the upstream filament (MPI), near the top wall halfway between the sequential filaments (MPII) and halfway between the downstream filament and the top wall (MPIII). The transient velocity data at monitoring point MPI shows that the periodic movements of the recirculation region behind the filament have a period of oscillation of 0.7 ms for a Reynolds number of 1600. The data at monitoring point MPII show a period of oscillation of 1.4 ms, which is exactly twice the period of oscillation of the recirculation region behind the filament. An identical period of oscillation is observed


Figure 15. Temporal velocity variations at monitoring points in the channel (lm/hch ) 4; df/hch ) 0.5). Note the highly regular nondissipative variations.

for a monitoring point at the bottom wall. It is concluded that every 0.7 ms a vortex is shed in the channel alternately adjacent to the top and bottom walls. The data at monitoring point MPIII show that the fluid passing the downstream filament still shows a period of oscillation of 1.4 ms even though the vortex itself is suppressed when passing the downstream filament.

The unsteady shear stress pattern for the submerged spacer is shown in Figure 16 for two times. Similar to the cavity spacer, the maximum shear stress values are achieved where the vortices are moving adjacent to the walls. The shear stress increases more than an order of magnitude because of the arrival of a shed vortex. The high shear stress region moves along the channel,


Figure 16. Shear stress patterns on the top and bottom channel walls for the submerged spacer at Rech ) 1600 (lm/hch ) 4; df/hch ) 0.5).

as depicted for regions 1 and 2 in Figure 16. The highest shear stress regions due to the formation of the vortex are observed immediately behind the upstream filament (regions 3 and 4), and when the vortex has shed, the high shear stress regions move along the channel toward the downstream filament. Further details of the unsteady flow patterns for the submerged spacer are shown in Figure 17. When Rech is increased to 400, the recirculation region behind each filament starts to move periodically, which is indicated by changes in the velocity as a function of time at a monitoring point behind the filament. At Rech ) 800 and above, vortices are shed behind each filament and the shed vortices move toward the downstream filament. When the flow passes the downstream filament, the vortices are suppressed as a result of the flow around the filament, but immediately behind the filament, vortex shedding occurs again. 4.2.3. Zigzag Spacer. Figure 18 shows the vortex shedding pattern in the zigzag spacer at Rech ) 1600. In Figure 18a, at tref ) 0 s, five vortices are present. Vortex 1 is being formed behind the upstream filament and is adjacent to vortex 2, which has been previously formed behind the upstream filament. Vortex 3 has formed previously adjacent to the top wall and moved along the channel reaching the downstream filament. Vortex 4 is adjacent to the bottom wall and has reached the downstream filament. Near the downstream filament, vortex 4 decreases in size. Vortex 5 has started to form adjacent to the top wall. Figure 18b at tref ) 1 ms shows that vortex 2 has moved along the channel and vortex 1 has detached from the upstream filament. Vortex 3 has decreased in size in front of the downstream filament. A new vortex 6 is formed behind the upstream filament similar to vortex 1 previously. Vortices 1 and 6 are adjacent to each other. Figures 18c at tref ) 2 ms shows that vortices 1, 2, and 5 have moved further along the channel and vortex 3 has disappeared in front of the downstream filament. Vortex 6 has increased in size but is still adjacent to the upstream filament. A squeezed vortex 7 passes the

Figure 17. Unsteady flow for Rech ranging from 100 to 2000 for multiple filaments located in the center of the channel (lm/hch ) 4; df/hch ) 0.5). The figure shows a truncation of the channel and depicts two sequential filaments.

upstream filament. This illustrates how the passage of a previously fully developed vortex convects unsteadiness along the channel. Thus, it is most important to compute velocity profiles for a series of multiple filaments in order to fully understand the formation and propagation of unsteady flow patterns. At a Reynolds number of 1600, a vortex formation pattern is observed, as shown in Figure 19. When a


Figure 19. Vortex formation pattern for the zigzag spacer.

vortex is formed behind the upstream filament and shed, a new vortex is immediately formed behind the upstream filament adjacent to the previously shed vortex. The shed vortex moves along the channel toward the downstream filament. Near the downstream filament, the vortex decreases in size and is squeezed when passing the downstream filament. Vortices are also formed adjacent to the top wall near the upstream filament and move along the channel. These vortices move along the channel, decrease in size, and disappear when passing the downstream filament. The pattern of vortex formation and shedding behind the upstream filament of the zigzag spacer is similar to the vortex formation behind the upstream filament of the cavity spacer. As shown in Figure 20, monitoring points were placed behind the upstream filament (MPI), near the top wall halfway between the sequential filaments (MPII), and halfway between the downstream filament and the top wall (MPIII). The transient data at monitoring point MPI illustrates the vortex shedding, while that at MPII shows the movement of the shed vortices along the channel and that at MPIII shows the fluid motion past the upstream filament. The data from MPIII show that the complex unsteady flow motions convect along the channel past the filaments; thus, the downstream fluid motion is affected by the upstream fluid motion. The unsteady shear stress pattern for the zigzag spacer is shown in Figure 21 for three time steps. Similar to the cavity and submerged spacers, maximum wall shear stress values are achieved where the vortices scour along the channel adjacent to the walls. The shear stress increases more than an order of magnitude because of the motion of the shed vortex. The high wall shear stress regions move along the channel, as depicted by regions 1-3 in Figure 21. The high shear stress region due to the formation of the vortex is observed immediately behind the upstream filament. Further details for the unsteady flow of the zigzag spacer at a fixed mesh length are shown in Figure 22. At Rech above 800, vortices are shed behind each filament and move toward the downstream filament. 5. Pressure Loss for the Flow around the Spacer Filaments

Figure 18. Vortex shedding for the zigzag spacer at Rech ) 1600 for various time steps (lm/hch ) 4; df/hch ) 0.5).

The obstruction of the narrow flow channel with spacer filaments increases the pressure loss due to the additional form drag caused by each spacer filament. Figures 23 and 24 show the pressure loss for the steady region, the region when the flow is unsteady but large-


Figure 20. Temporal velocity variations at monitoring points in the channel (lm/hch ) 4; df/hch ) 0.5).

scale vortex shedding does not occur, and for the vortex shedding region. For an obstruction of the channel with a single filament, Figure 23 shows that pressure loss increases with the Reynolds number for the steady region, as discussed previously.1 When the flow becomes unsteady, pressure loss increases further, and when vortex shedding starts, the pressure increases even more steeply.

For multiple filament configurations, the pressure loss is shown in Figure 24. For the steady flow region, the submerged spacer has the highest pressure loss when compared with the cavity and zigzag spacers. The cavity and zigzag spacers have similar pressure losses for the mesh lengths chosen here, as discussed previously.1 For Reynolds numbers between 400 and 800, when the flow becomes unsteady, the pressure loss increases further


Figure 22. Unsteady flow for Rech ranging from 200 to 1600 for multiple filaments alternately adjacent to the top and bottom walls (zigzag spacer) at a fixed mesh length (df/hch ) 0.5). The figure shows a truncation of the channel and depicts two sequential filaments.

Figure 23. Pressure loss along the modeled channel for the single-filament configurations (df/hch ) 0.5).

for all spacers. The submerged spacer continues to have the highest pressure loss, followed by the cavity and zigzag spacers. For Reynolds numbers greater than 800, when vortex shedding occurs for all spacer configurations, pressure loss increases drastically for all spacers. It must be noted that, for practical applications, a pressure loss greater than 1 bar/m is not desirable in a spiral-wound module because the pumping costs and the loss of driving force become too large in an array. In these simulations, the pressure loss exceeds this desired value for a Reynolds number greater than 1200. 6. Conclusions

Figure 21. Shear stress patterns on the top and bottom channel walls for the zigzag spacer at Rech ) 1600.

CFD is a powerful tool for revealing the unsteady flow patterns and vortex shedding that occurs in narrow spacer-filled channels. The transition to time-dependent flow in an obstructed channel occurs at Reynolds


Acknowledgment The authors acknowledge Paul Ryan and Russell Standish at the Centre for Advanced Computations and Communications for help with access to the highperformance computers used in this work. List of Symbols

Figure 24. Pressure loss along the channel for the three spacer configurations with the pressure loss scaled up from the modeled channel to a pressure loss given in bar/m (df/hch ) 0.5; lm/hch ) 4).

numbers between 200 and 800 (depending on the filament configuration). In the spacer-filled channel, multiple vortices form at the top and bottom wall behind a filament and convect downstream toward the sequential filament. The vortex shedding between spacer filaments can only be resolved for time steps that are 3 orders of magnitude smaller than the period of oscillation; otherwise, unsteadiness is suppressed. Also, the grid size must be up to 3 orders of magnitude smaller than the distance between sequential spacer filaments in order to resolve vortex shedding. Even when vortex shedding does not occur between the first upstream filaments, it may still occur between filaments further downstream. Thus, it is important to develop the flow patterns for a number of filament sequences. In addition, the upstream region requires fine numerical grid resolution in order not to suppress any unsteadiness. It was found that a minimum of six filaments in the channel is necessary to develop the unsteady flow patterns. Modeling such a large computational domain leads to large computational times. Because large computational times and high-speed computing facilities are required, the modeling cases must be chosen carefully. Variations in spacer mesh lengths and filament diameters must be carefully considered and cannot be completed as quickly as those for steady-state computations. Therefore, effort was concentrated on a fundamental demonstration of how vortices are formed and shed in the spacer-filled channel and not on parameter variations. An extension of this work to three dimensions will require careful and systematic identification of appropriate modeling procedures because time-dependent movements of three-dimensional recirculation regions are likely to be more complex and will require huge numbers of grid nodes. The scouring of the shed vortices in the channel and thus the frequent local changes in velocity and in wall shear stress are likely to have a great impact on reducing fouling in membrane systems. Pressure loss increases for unsteady flows, which leads to higher energy costs, and a careful economic analysis would be necessary when attempting to estimate the appropriate mass transfer and pressure loss that achieve optimized operating conditions for spiral-wound modules.

df ) filament diameter [m] dh ) hydraulic diameter [m] fi ) oscillation frequency [Hz] hch ) channel height [m] lm ) mesh length [m] Recyl ) cylinder Reynolds number as defined in eq 2 Rech ) channel Reynolds number as defined in eq 3 t ) time [s] tref ) reference time [s] uave ) average velocity in the channel [m/s] u∞ ) uniform free-stream velocity [m/s] ) voidage (flow volume of a channel containing spacers divided by the volume of the same channel with no spacers present) ν ) kinematic viscosity [m2/s]

Literature Cited (1) Schwinge, J.; Wiley, D. E.; Fletcher, D. F. Simulation of Flow Around Spacer Filaments Between Channel Walls. Part I: Hydrodynamics. Ind. Eng. Chem. Res. 2002, 41, 2977. (2) Schwinge, J.; Wiley, D. E.; Fletcher, D. F. Simulation of Flow Around Spacer Filaments Between Channel Walls. Part II: Mass Transfer. Ind. Eng. Chem. Res. 2002, 41, 4879. (3) Schwinge, J.; Wiley, D. E.; Fletcher, D. F. A CFD Study of Unsteady Flow in Narrow Spacer-filled Channels for Spiral Wound Membrane Modules. Desalination 2002, 146, 195. (4) Massay, B. S. Mechanics of Fluids; Chapman & Hall: London, 1989. (5) Fox, R. W.; McDonald, A. T. Introduction to Fluid Mechanics; John Wiley & Sons Inc.: New York, 1994. (6) van Dyke, M. An Album of Fluid Motion; The Parabolic Press: Stanford, CA, 1982. (7) Karniadakis, G. E.; Mikic, B. B.; Patera, A. T. Minimumdissipation Transport Enhancement by Flow Destabilization: Reynolds’ Analogy Revisited. J. Fluid Mech. 1988, 192, 365. (8) Karniadakis, G. E.; Beskok, A. Simulation of Heat and Momentum Transfer in Complex Microgeometries. J. Thermophys. Heat Transfer 1994, 8, 647. (9) Karniadakis, G. E. Simulating Turbulence in Complex Geometries. Fluid Dyn. Res. 1999, 24, 343. (10) Zagarola, M. V.; Smits, A. J.; Karniadakis, G. E. Heat Transfer Enhancement in a Transitional Channel Flow. J. Wind Eng. Ind. Aerodyn. 1993, 49, 257. (11) Crawford, C. H.; Evangelinos, C.; Newman, D.; Karniadakis, G. E. Parallel Benchmarks of Turbulence in Complex Geometries. Comput. Fluids 1996, 25, 677. (12) Ghaddar, N. K.; Korczak, K. Z.; Mikic, B. B.; Pater, A. T. Numerical Investigation of Incompressible Flow in Grooved Channels. Part 1. Stability and Self-sustained Oscillations. J. Fluid Mech. 1986, 163, 99. (13) Ghaddar, N. K.; Magen, M.; Mikic, B. B.; Pater, A. T. Numerical Investigation of Incompressible Flow in Grooved Channels. Part 2. Resonance and Oscillatory Heat-transfer Enhancement. J. Fluid Mech. 1986, 168, 541. (14) Patera, A. T.; Mikic, B. B. Exploiting Hydrodynamic Instabilities. Resonant Heat Transfer Enhancement. Int. J. Heat Mass Transfer 1986, 29, 1127. (15) Zdravkovich, M. M. Smoke Observations of the Formation of a Karman Vortex street. J. Fluid Mech. 1969, 37, 491. (16) Aref, H.; Siggia, E. D. Evaluation and Breakdown of a Vortex street in Two Dimensions. J. Fluid Mech. 1981, 109, 435. (17) Kaiktsis, L.; Karniadakis, G. E.; Orszag, S. A. Unsteadiness and Convective Iinstabilities in Two-dimensional Flow over a Backward Facing Step. J. Fluid Mech. 1996, 321, 157.

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4977 (18) Greiner, M.; Chen, R. F.; Wirtz, R. A. Heat Transfer Augmentation Through Wall-shape-induced Flow Destabilization. J. Heat Transfer 1990, 112, 336. (19) Greiner, M.; Chen, R. F.; Wirtz, R. A. Enhanced Heat Transfer/Pressure Drop Measured from a Flat Surface in a Grooved Channel. J. Heat Transfer 1991, 113, 498. (20) Greiner, M. An Experimental Investigation of Resonant Heat Transfer Enhancement in Grooved Channels. Int. J. Heat Mass Transfer 1991, 34, 1383. (21) Greiner, M.; Faulkner, R. J.; Van, V. T.; Tufo, H. M.; Fischer, P. F. Simulations of the Three-dimensional Flow and Augmented Heat Transfer in a Symmetrically Grooved Channel. J. Heat Transfer 2000, 122, 653. (22) Wirtz, R. A.; Huang, F.; Greiner, M. Correlation of Fully Developed Heat Transfer and Pressure Drop in a Symmetrically Grooved Channel. J. Heat Transfer 1999, 121, 236. (23) Majumdar, D.; Amon, C. H. Heat and Momentum Transport in Self-sustained Oscillatory Viscous Flow. J. Heat Transfer 1992, 111, 866. (24) Amon, C. H.; Majumdar, D.; Herman, C. V.; Mayinger, F.; Mikic, B. B.; Sekulic, D. P. Numerical and Experimental Studies of Self-sustained Oscillatory Flows in Communicating Channels. Int. J. Heat Mass Transfer 1992, 35, 3115. (25) Kang, I. S.; Chang, H. N. The Effect of Turbulence Promoters on Mass TransfersNumerical Analysis and Flow Visualization. Int. J. Heat Mass Transfer 1982, 25, 1167. (26) Kim, D. H.; Kim, I. H.; Chang, H. N. Experimental Study of Mass Transfer Around a Turbulence Promoter by the Limiting Current Method. Int. J. Heat Mass Transfer 1983, 26, 1007. (27) Thomas, D. G. Forced Convection Mass Transfer: Part II. Effect of Wires Located Near the Edge of the Laminar Boundary Layer on the Rate of Forced Convection from a Flat Plate. AIChE J. 1965, 11, 848.

(28) Thomas, D. G. Forced Convection Mass Transfer: Part III. Increased Mass Transfer from a Flat Plate Caused by the Wake from Cylinders Located Near the Edge of the Boundary Layer. AIChE J. 1966, 12, 124. (29) Thomas, D. G.; Griffith, W. L.; Keller, R. M. The Role of Turbulence Promoters in Hyperfiltration Plant Optimization. Desalination 1971, 9, 33. (30) Thomas, D. G.; Mixon, W. R. Effect of Axial Velocity and Initial Flux on Flux Decline of Cellulose Acetate Membranes in Hyperfiltration of Primary Sewage Effluents. Ind. Eng. Chem. Process Des. Dev. 1972, 11, 339. (31) Thomas, D. G. Forced Convection Mass Transfer in Hyperfiltration at High Fluxes. Ind. Eng. Chem. Fundam. 1996, 12, 396. (32) Watson, J. S.; Thomas, D. G. Forced Convection Mass Transfer: Part IV. Increased Mass Transfer in an Aqueous Medium Caused by Detached Cylindrical Turbulence Promoters in a Rectangular Channel. AIChE J. 1967, 13, 676. (33) Chang, H. N.; Park, J. K. Effect of Turbulence Promoters on Mass Transfer. In Handbook of Heat and Mass Transfer; Cheremisinoff, N. P., Ed.; Gulf Publishers: Houston, TX, 1986; Vol. 2. (34) CFX 4.4 User Manual; CFX International, AEA Technology plc: Oxfordshire, U.K., 2002. (35) Li, H.; Bertram, C. D.; Wiley, D. E. Mechanisms by Which Pulsatile Flow Affects Cross-flow Microfiltration. AIChE J. 1998, 44, 1950.

Received for review March 3, 2003 Revised manuscript received July 7, 2003 Accepted July 23, 2003 IE030211N

Simulation of Unsteady Flow and Vortex Shedding for Narrow Spacer

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