ON PERMEABILITY AND ROUGHNESS EFFECTS IN FLOW PAST A …

E-proceedings of the 37th IAHR World Congress

August 13 – 18, 2017, Kuala Lumpur, Malaysia

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ON PERMEABILITY AND ROUGHNESS EFFECTS IN FLOW PAST A ROW OF

CIRCULAR CYLINDERS ARRANGED ALONG THE CENTERLINE OF A STRAIGHT FLUME

SATOSHI YOKOJIMA(1), RYOSUKE ASAOKA(2), KOUDAI YOSHINO(3), HIROSHI NODA(4) & TAKASHI MIYAHARA(5)

(1-3,5) Department of Mathematical and Systems Engineering, Shizuoka University, Hamamatsu, Japan.

e-mail: [email protected] (2) Department of Architecture, Kindai University, Higashi-Osaka, Japan.

ABSTRACT

The presence of a row of circular cylinders along the centerline in a flume gives a significant drag to the flow.

The row can be seen macroscopically as a porous flat plate having both permeability and roughness effects,

and the flow over semiannular stripe roughness elements is the system obtained by removing only the

permeability from the flow past the cylinders. In the present study, these flows have been studied comparatively

by a two-dimensional high-fidelity numerical experiment to elucidate the impacts of the permeability and

roughness on the flow resistance. The drag on the row of cylinders is found to be roughly one order of magnitude

higher than that over the roughness elements. In other words, permeability causes much more resistance to the

flow than roughness. The importance of the former tends to be decreased with decrease in the number of

cylinders (i.e., increase in the plate porosity). Visualization of the instantaneous flow fields reveals the presence

of large scale vortices in both types of the flows, and the intensity of the vortices is significantly enhanced by

the boundary permeability. This then promotes fluid mixing and associated momentum exchange over the whole

cross section, and increases the resistance to the flow.

Keywords: Permeability Effect, Roughness Effect, A Row of Circular Cylinders, Drag, Two-dimensional High-fidelity Numerical Experiment

1 INTRODUCTION

It is well known that adding permeability to a body immersed in a fluid flow can increase the resistance to

the flow. For example, Noymer et al. (1998) reported that, in a flow past an isolated porous circular cylinder, the

drag becomes about 30-50 percent higher than that of the corresponding solid cylinder when the porosity is only

about a few percent, at the stem Reynolds numbers of 100 and 1000. The drag increase due to permeability

was also found in flows parallel to a permeable wall (e.g., Jimenez et al., 2001; Yokojima, 2011). Since porous

bodies and permeable boundaries are quite ubiquitous in hydraulic and geophysical flows, it is of substantial

importance to gain a greater understanding of the mechanism of the drag enhancement.

In the present study, an attention is paid to a flow past a row of monodisperse circular cylinders of diameter

D arranged along the centerline of a straight flume of width 2, as shown in figure 1(a). It can be a simple model

for flows around vegetative canopies (Yokojima et al., 2015). The row can be seen macroscopically as a porous

flat plate of thickness D and hence has both the permeable and roughness effects. While the presence of such

cylinders may bring a substantial resistance to the flow, it is unclear how much the permeability and roughness

contribute to the drag. To elucidate the impacts of these basic properties of solid boundaries encountered in

practical applications on the flow resistance, a flow over semiannular stripe roughness elements (figure 1(b)),

which can be obtained by inserting a flat plate through the centers of the cylinders, is introduced. The

introduction of the plate removes the permeability from the row of cylinders but leaves the roughness element

unchanged. These flows are studied comparatively by two-dimensional high-fidelity direct simulations.

2 PHYSICAL AND NUMERICAL DETAILS

The details of the flows to be examined, i.e., the flow past a row of circular cylinders (figure 1(a)) and the

flow over semiannular stripe roughness elements (figure 1(b)), are described here. The former is established in

a straight channel of width 2, and a row of circular cylinders of diameter D is arranged along the centerline of

the flume with uniform surface-to-surface spacing S. The latter is obtained by installing a no-slip impermeable

plate along the flume centerline so that the flow domain size is halved in the transverse direction. The no-slip

and impermeable constraints are imposed also on the surface of the cylinders/roughness elements as well as

E-proceedings of the 37th IAHR World Congress August 13 – 18, 2017 - Kuala Lumpur, Malaysia

(a) permeable case. (b) impermeable case.

Figure 1. The panel (a) shows a flow past a row of circular cylinders and (b) presents a flow over semiannular

stripe roughness elements, which can be obtained by inserting a flat plate to the centers of the cylinders of (a).

at the side walls of the flume. Here periodicity of length L1=6 is assumed in the streamwise direction. It has

been confirmed that the size L1 does not affect the drag property of the flows significantly.

In all the cases considered, the driving force of the flow (the mean pressure gradient) is adjusted dynamically

so that the bulk mean velocity Ub is kept constant. The total resistance to the flow is therefore found by the

magnitude of the driving force. After the statistically steady state is reached in each case, the time series of the

flow driving force are sampled.

The flows are characterized by three parameters: the stem Reynolds number based on the diameter of the

cylinders/semiannular roughness elements ReD=UbD/, the bulk Reynolds number based on the channel half-

width Reb=Ub/, and the porosity of the row of circular cylinders/the stripe roughness pattern =(L1-ncD)/L1.

Here denotes the kinematic viscosity of the fluid and nc is the number of the cylinders/roughness elements

arranged over the periodic length L1. Hence the porosity shows the fraction of the fluid occupied on the flume

centerline. In the present study, the stem and bulk Reynolds numbers are fixed to be 720 and 14400 respectively,

and the porosity is changed systematically as =0.3, 0.4, 0.5, 0.6, 0.7, 0.75, 0.8, 0.85 and 0.9. The number of

the cylinders/roughness elements nc in the abovementioned nine cases are 84, 72, 60, 48, 36, 30, 24, 18 and

12, respectively. Note that ReD=720 and Reb=14400 are typical values found in laboratory experiments of flows

past arrays of circular cylinders (e.g. Yokojima et al., 2015).

The flows are reproduced by a high-resolution numerical experiment based on an immersed boundary

method (Uhlmann, 2005; Kempe and Frohlich, 2012). The grid spacing around each cylinder/roughness

element is set to be D/20 in the present study. It enables to reproduce the details of the fluid flows around the

immersed bodies and the hydrodynamic forces acting on each infinitesimal surface element of the bodies

accurately. The code used in the present study has been verified in a flow past an isolated circular cylinder

against experimental and numerical results obtained from past studies. The drag and lift coefficients and the

Strohal number are found to be well reproduced with the spatial resolution of =D/20.

3 SIMULATION RESULTS In the following, all the variables are normalized by the bulk-mean velocity Ub and the cylinder diameter D unless otherwise stated.

3.1 Macroscopic drag property

Figure 2(a) shows the statistics of the flow driving force required to keep the mass flow rate constant in each

case of the flow past a row of circular cylinders/the flow over semiannular stripe roughness elements. Note that

nc=0 denotes the case with no cylinders, and that nc=∞ is the case with infinite number of cylinders, which means

the presence of a no-slip and impermeable plate of width D along the flume centerline. It is clearly shown that

the presence of a row of cylinder increases the flow resistance significantly and that the influence of the friction

drag due to the side walls is negligible. The simulation results with =D/30 as well as with =D/20 are presented

for the case of =0.5 of the flow past the cylinders, where the drag due to the row of cylinders is maximized. No

significant difference is found between the results from the two computations, demonstrating that =D/20 is fine

enough to reproduce the drag property of the flow accurately.

The flow resistance is maximized at =0.5 in the flow past a row of cylinders. The same conclusion was

obtained also on other conditions such as ReD=180, Reb=3600 and ReD=270, Reb=5400. Figure 2(a) indicates

that the drag acting on the flow over semiannular stripe roughness elements is roughly one order of magnitude

lower than that on the flow past the cylinders. In other words, permeability causes much more resistance to the

flow than roughness. In the flow over the roughness elements, the drag is maximized at = 0.8. It is in good


August 13 – 18, 2017 - Kuala Lumpur, Malaysia

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(a) (b)

Figure 2. (a) Driving force required to keep the mass flow rate constant as a function of the porosity . The

symbols denote the time averaged values and the error bars are the standard deviations. The filled symbol at

=0.5 presents the results with the spatial resolution of =D/30. (b) Ratio of the driving force required in the

impermeable case to that in the permeable case.

(a) =0.9. (b) =0.7.

(c) =0.5. (d) =0.3.

Figure 3. Time-averaged streamwise velocity distributions in the flow past a row of circular cylinders. The region

of 0≤x1/D≤60, x2≥0 is enlarged.

agreement with the findings from past studies on turbulent flows over stripe roughness. To gain an insight into

the impacts of the permeability and roughness on the flow resistance, the time-averaged driving force required

in the flow over the roughness elements normalized by that in the flow past the cylinders is plotted in figure 2(b).

It is just around 10 percent when the porosity is low, but it reaches up to 30 percent or larger at =0.8-0.9.

3.2 Time-averaged velocity and pressure fields

The time-averaged streamwise velocity profiles for some representative cases are presented in figures 3

and 4. In the flow past a row of circular cylinders of figure 3, the flow is strongly accelerated for =0.9 in the

vicinity of the side edges of the cylinders, and the fluid between the cylinders has a relatively high velocity of

O(Ub). As the porosity is decreased, the row of the cylinders is being immersed in the boundary layer. On the

other hand, the roughness elements are found to be involved in the low-speed layer of O(0.1Ub) in all the cases.

Comparison between figures 2 and 3 indicates that an active momentum exchange in the transverse direction


(a) =0.9. (b) =0.7.

(c) =0.5. (d) =0.3.

Figure 4. Time-averaged streamwise velocity distributions in the flow over semiannular stripe roughness

elements. The region of 0≤x1/D≤60 is enlarged.

(a) =0.9. (b) =0.7.

(c) =0.5. (d) =0.3.

Figure 5. Time-averaged pressure distributions in the flow past a row of circular cylinders. The region of

0≤x1/D≤60, x2≥0 is enlarged.

due to the boundary permeability increases the fluid velocity near the cylinders and the resistance to the flow

also.

Figures 5 and 6 show the time-averaged pressure distributions. While a pressure difference between the

fore and aft of each circular cylinder can be observed clearly for ≥0.5 in the flow past a row of circular cylinders,

the pressure difference is not substantial in the flow over semiannular stripe roughness elements even for =0.7.

The mean pressure becomes low in the flume core region for both types of the flows. It will be presented in

section 3.3 that this pressure depression is due to the presence of large-scale vortices at the instantaneous flow

level. The extent of the pressure depression for the flow past the cylinders is deeper than that for the flow over

the roughness elements. It suggests the presence of more intense vortices in the former case.

To examine more details of the flow structures around the cylinders/roughness elements, one-dimensional

streamwise profiles of the time-averaged streamwise velocity and pressure at x2=D/40, i.e., in the immediate

vicinity of the boundaries, are presented in figure 7. A recirculation region is observed behind of each

semiannular roughness element. For =0.9, where the surface-to-surface distance between the elements is



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(a) =0.9. (b) =0.7.

(c) =0.5. (d) =0.3.

Figure 6. Time-averaged pressure distributions in the flow over semiannular stripe roughness elements. The

region of 0≤x1/D≤60 is enlarged.

(a) streamwise velocity, permeable case. (b) pressure, permeable case.

(c) streamwise velocity, impermeable case. (d) pressure, impermeable case.

Figure 7. Time-averaged streamwise velocity and pressure profiles at x2=D/40 (immediate vicinity of x2=0). In

each panel, the data from =0.9, 0.8, 0.7, 0.6, 0.5, 0.4, and 0.3 are presented from bottom to top.


(a) =0.9. (b) =0.8.

(c) =0.7. (d) =0.6.

(e) =0.5. (f) =0.4.

(g) =0.3.

Figure 8. An instantaneous snapshot of the flow past a row of circular cylinders, where the pressure (color

contours) and the fluctuating velocity (vector, obtained by subtracting the bulk-mean velocity Ub) are depicted.

4.5D, the separated streamline is reattached before the fluid encounters the next element. At =0.8 or lower,

however, the reattachment is no more observed, and a roughness element is involved in the separation/

recirculation region developed behind the immediate upstream neighbor. The pressure difference between the

fore and aft of each element is therefore substantially decreased with decrease in . For the flow past a row of

cylinders, on the other hand, such a reverse flow does not occur even at =0.5 where the surface-to-surface

distance is 1.0D, and the fluid encounters the next cylinder with a fairly high velocity. The drag acting on the

roughness elements is therefore decreased more substantially than that on the row of cylinders with decrease

in the porosity . It should be borne in mind that the pressure range covered in figure 7(b) is five times wider

than that in figure 7(d). The difference in the macroscopic drag property between the flow past the cylinders and

the flow over the roughness elements presented in figure 2(a) can be attributed to the differences in the time-

averaged streamwise velocity profiles and also in the associated shape drag between the cylinders and the

roughness elements.

3.3 Instantaneous flow property

Figure 8 shows an instantaneous snap shot of the pressure and velocity fields obtained from the flow past

a row of cylinders. The low pressure regions indicate the presence of large-scale vortices. More intense vortices

are observed in the cases with higher drag to the flow such as =0.3-0.6. The presence of such vortices whose

size is O() enhances fluid mixing and associated momentum exchange over the whole cross section.



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(a) =0.9. (b) =0.8.

(c) =0.7. (d) =0.6.

(e) =0.5. (f) =0.4.

(g) =0.3.

Figure 9. An instantaneous snapshot of the flow over semiannular stripe roughness elements, where the

pressure (color contours) and the fluctuating velocity (vector, obtained by subtracting the bulk-mean velocity Ub)

are depicted.

Figure 9 is the counterpart of figure 8 obtained from the flow over stripe roughness elements. Note that the

range of the colorbar for the instantaneous pressure is the same employed in figure 8. While vortices with

various sizes interact actively with each other in the flow past the cylinders, it is not in the flow over the roughness

elements. The vortices are found to be simply advected at almost a constant speed without any deformation for

=0.4, and 0.3. Merging and splitting of the vortices are observed at =0.5 or larger, but the irregularity of the

flow is not as high as in the flow past the cylinders. Comparison between figures 8 and 9 clearly shows that the

pressure depression at the central core of each vortex is more substantial in the flow past the cylinders. It is in

accordance with the properties of the time-averaged pressure field depicted in figures 5 and 6.

4 SUMMARY AND CONCLUSIONS To deepen the understanding of the macroscopic drag of the flow past a row of cylinders, another flow over semiannular stripe roughness elements, which can be obtained by inserting a flat plate through the centers of the cylinders, is introduced. These flows are studied comparatively by two-dimensional high-fidelity direct simulations based on an immersed boundary method, and the impacts of the permeability and roughness, the basic properties of solid boundaries encountered in practical applications, on the flow resistance are critically examined.

The numerical experiment where the porosity is systematically changed between 0.3 and 0.9, has

revealed that the drag on the row of cylinders is roughly one order of magnitude higher than that over the

roughness elements. In other words, permeability causes much more resistance to the flow than roughness.

While the drag on the row of cylinders is maximized at =0.5, the drag over the roughness elements has the

largest value at =0.8. The impact of the permeability on the flow resistance is therefore decreased with increase

in .

In the flow over semiannular stripe roughness elements, an element is involved in the separation/

recirculation region developed behind the immediate upstream neighbor at =0.8 or lower. The shape drag acting on the element is therefore substantially decreased. For the flow past a row of circular cylinders, on the

other hand, such a reverse flow does not occur even at =0.5 and the fluid encounters the next cylinder with a fairly high velocity. It causes a high resistance to the flow.

Vortices whose sizes are comparable to the flume width are frequently observed in both types of the flows.

The vortices are found to have various sizes and interact actively with each other in the flow past the cylinders.


In the flow over the roughness element, on the other hand, there exist some cases where the vortices are simply

advected at almost a constant speed without any deformation. The presence of intense vortices in the flow past

the cylinders enhances fluid mixing and associated momentum exchange over the whole cross section, and

increases the resistance to the flow.

REFERENCES Jimenez, J., Uhlmann, M., Pinelli, A., and Kawahara, G. (2001). Turbulent shear flow over active and passive porous surfaces. J. Fluid Mech., 442, 89-117. Kempe, T., and Frohlich, J. (2012). An improved immersed boundary method with direct forcing for the simulation of particle laden flows. J. Comput. Phys., 231, 2012. Noymer, P.D., Glicksman, L.R., and Devendran, A. (1998). Drag on a permeable cylinder in steady flow at moderate Reynolds numbers. Chem. Engng. Sci., 53, 2859-2869. Uhlmann, M. (2005). An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys., 209, 448-476. Yokojima, S. (2011). Effect of wall permeability on wall-bounded turbulent flows. J. Phys. Soc. Jpn., 80, 033401. Yokojima, S., Kawahara, Y., and Yamamoto, T. (2015). Impacts of vegetation configuration on flow structure and resistance in a rectangular open channel. J. Hydro-environ. Res., 9 (2), 295-303.

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