Environmental Flow Physics& Computation Group

Undisturbed Fluid Flow for Improved Macroscopic Estimation of Drag Forces Acting on Circular Cylinders
S. Yokojima1, T. Uchida2, Y. Kazehaya1, and Y. Kawahara2 (1Shizuoka University, Japan; 2Hiroshima University)
Abstract
A drag-force model is essential in practical predictions of flows aroundvegetation/urban canopies, since it is too prohibitive to resolve details of
the canopy elements and associated fluid motions in practical applications.
The model, however, has a serious difficulty inherent in its formulation: how
one can specify the representative velocity? In ideal situations such as flows
past an isolated obstacle, it is the velocity of the inflow and is easily
available. In flows past multiple obstacles, on the other hand, astraightforward extension of the idea behind the ideal situations leads tointroduce undisturbed flow, which is the flow that would exist at an obstacle
location in the absence of that obstacle but with all other obstacles present.
Here the undisturbed flow has been evaluated directly in fully resolved
computations of a two-dimensional flow past circular cylinders and thefundamental properties of the flow are discussed.

Introduction•

Macroscopic drag force model(1)✓ No standard way to find a proper CD even for simple array ofcylinders – a bottleneck in practical applications
˗ CD,global by Yokojima & Kawahara (2015)
˗ CD,local by Yokojima & Kawahara (2016)✓ Here particular attention to representative velocity urep

Figure 1. In ideal situations like (a), urep must be the inflow U0 (see (b)). In flows pastmultiple obstacles like (c), a straightforward extension of the idea behind (b) introduces
undisturbed flow, which would exist at an obstacle location in the absence of thatobstacle but with all other obstacles present. Clearly the undisturbed flow is unavailable
in general (see (d)), which causes serious difficulties in application of the model.

Figure 3. Computational domain and arrangement of circular cylinders: (a) Case 1;(b) Case 2. Periodicity is imposed in x1 direction. Each of the cylinders colored with
red is removed to compute the undisturbed flow uun at the cylinder location. For
clarity, the cylinders are depicted larger than the actual size.

Figure 5. Mean streamwise velocity profiles along streamwise rows passing center of eachcylinder. The curve colored by black, red, blue, magenta, and cyan represent velocity profiles
along r1, r2, r3, r4, and r5.

••

Numerical details✓ Introduce horizontally two-dimensional system, as zerothapproximation to target flow
✓ Stem Reynolds number ReD = 700✓ Immersed boundary method, Dx = D/12
• Undisturbed flow uun & drag coefficient CD,un✓ Direct calculation of undisturbed flow uun˗ Refer cylinder i-th from the top (row) and j-th from the left(column) to as ‘ricj’˗ Remove a cylinder, simulate the new system, and calculate
uun at the cylinder location
˗ Calculate uun at five different locations (r1c1, r2c1, r3c1,r4c1, r5c1) in Case 1 and at 36 locations (r1, r3, r5 in thetransverse direction, and c1, c2, c3, c4, c5, c9, c17, c25,c30, c31, c32, c33) in Case 2 (see figure 3)
˗ Number of cylinders that are statistically independent is fivein Case 1, and 165 (= 5*33) in Case 2
˗ Obtain uun by both spatial averaging over entire volumeoccupied by the target cylinder and temporal averaging✓ Evaluation of drag coefficient CD,un based on eq. (1)˗ Assume that eq. (1) holds at time-averaged flow level˗ Obtain drag to the target cylinder FD from simulation resultsof the original system where no cylinders are removed
˗ Calculate CD,un = 2FD/(rDuun2)(a)•Figure 2. (a) Case 1; (b) Case 2. Dashed rectangle indicates the computational domain
for the present, microscopic, fully resolved numerical simulations (see figure 3).

(b)

Figure 6. (a) Distributions of undisturbed velocity uun and approaching velocity ua. (b) CD-ReD
relation based on uun. In (b), solid line - an isolated circular cylinder in 3-D space (Finnemore andFranzini (2001)); open symbols - an isolated circular cylinder in 2-D space (realized by CFDsimulations); filled symbols - array of circular cylinders.

•Comparison of CD,global, CD,local, and CD,unFigure 7. Estimated drag coefficient profiles. Since uun is available only at 36 locations in Case 2, alinear interpolation was used to obtain uun elsewhere.

•(c)Case 13mm
Physical details (see figure 2 and Yokojima et3cmal. (2015))✓ Fully-developed open-channel flow through
27cm
80cm
emergent vegetation✓ Vegetation model consists of circular cylinders of diameter 3 mmwith in-line arrangement of center-to-center distance 3 cm
✓ Flow discharge is 9000 cm3/s, and resulting water depth H andbulk Reynolds number Reb = UbH/n are about 4.7 cm and 11250,
respectively, in both cases

(a)

(b)

✓ Objective – to examine fundamental properties of undisturbed
flow in a two-dimensional flow past circular cylinders

Physical and numerical details

Profiles of uun and ua, and the drag coefficient CD,un(d)Figure 4. Profiles of mean streamwise velocity u/Ub ((a), (b)) and mean drag acting
on each cylinder FD/(rUb2D) ((c), (d)), obtained from the original system (i.e., nocylinders are removed).

Results and discussion•Approaching velocity ua (see Yokojima & Kawahara (2016))✓ ua to the cylinder j-th from the left (upstream) is defined by themaximum value of u1 between the (j-1)-th and j-th cylinders.
✓ uun is generally unavailable (as can be seen in figure 1(d)), andhence ua can be an alternative to uun in practical applications

Discussion✓ CD-ReD relation for each cylinder in array of cylinders is deviated from thatfor an isolated cylinder even when CD and ReD are evaluated based on uun.
✓ Most CD,un are in between 1.5 and 2.5.
✓ ua captures basic characteristics of uun qualitatively but not quantitatively.
✓ CD,global tends to underpredict vegetation drag (Yokojima & Kawahara (2015)),and CD,local basically overestimate flow resistance (Yokojima & Kawahara(2016)). Since CD,un falls between CD,global and CD,local, CD,un is expected toprovide a proper resistance to the flow.

References•••

Yokojima et al. (2015) “Impacts of vegetation configuration on flow structure and resistance in arectangular open channel” JHER 9(2) 295.
Yokojima and Kawahara (2015) “Influence of model drag coefficient on LES prediction
accuracy of vegetated open-channel flows” 36th IAHR World Congress.
Yokojima and Kawahara (2016) “Drag coefficient distribution in LES of vegetated open channelflows” River Flow.