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Effect of side walls on flow around prisms
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Effect of side walls on flow around prisms Ramnarayan Mondal a, Md. Mahbub Alam a, b, *, Rajesh Bhatt a, c a
Institute for Turbulence-Noise-Vibration Interaction and Control, Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China Digital Engineering Laboratory of Offshore Equipment, Shenzhen, China c Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, India b
A R T I C L E I N F O
A B S T R A C T
Keywords: Wall effect Flow structure Forces Strouhal number
Two-dimensional laminar flow around three side-by-side rectangular prisms placed between two parallel walls is numerically investigated at a Reynolds number Re ¼ 100. The cross-sectional aspect ratio of each prism is b/h ¼ 3, where b and h are the prism width and height, respectively. The gap spacing (g) between the two adjacent prisms and between the wall and the outer prism is varied from g* (¼ g/h) ¼ 0.8 to 3.4. The focus is given on the influence of g* on flow structures, Strouhal number (St), and fluid forces. Three distinct flow regimes are identified, namely single vortex street (0.8 � g* < 1.1), twin vortex streets (1.1 < g* < 2.1), and triple vortex streets (2.1 < g* � 3.4). Lift forces on the outer prisms are attractive for single vortex street and repulsive for twin and triple vortex streets. The St being identical for the three prisms exponentially declines from 0.39 to 0.18 with g* increasing from 0.8 to 3.4. Fluctuating lift and time-mean drag also shrink with increasing g*, time-mean drag becoming higher for the middle prism than for the outer prisms. Empirical relations of St and forces are obtained as a function of g*.
1. Introduction The problem of fluid flow around slender structures is of interest among the researchers and scientists for their enormous applications in the fields of aerodynamics (e.g. wind flow around high rise buildings, chimney stacks, overhead power line bundles), offshore engineering (e. g. flow past bridge piers, offshore platform, drilling rigs and riser sys tem), and subsea engineering (e.g. flow around sub-sea cables and pipelines). These structures very often appear in a group, making the flow physics more complicated than a single structure. For the closelyspaced structures, the complicated flow arises due to the mutual inter action between shear layers and between wake vortices. Because of everything getting compact in the modern engineering, applications of flow around multiple structures in wall proximity get broader, such as the flow around prisms/cylinders in a channel to enhance flow mixing and heat transfer, the flow around drive bays in CPUs, etc. It is therefore essential to assimilate the complicated flow around closely spaced multiple structures with a view to their practical applications. Due to the geometric simplicity, the structures are mostly found in circular, square or rectangular shapes. A comprehensive review of the wakes behind different bluff body shapes (including square, circular, and triangular cylinders) is can be found in Derakhshandeh and Alam, 2019.
Flow around multiple cylindrical structures of circular or square shape have been extensively investigated in the literature through ex periments (e.g. Okajima, 1982; LeGal et al., 1990; Sumner et al., 1999; Alam and Sakamoto, 2005; Alam et al., 2003; Alam and Zhou, 2007; Alam, 2014; Bai and Alam, 2018) and numerical simulations (e.g. Sohankar, 2006, 2008, 2014; Ding et al., 2007; Carini et al., 2014; Maiti, 2012; Maiti and Bhatt, 2014; Alam, 2016; Alam et al., 2017; Chen et al., 2018; Zheng and Alam, 2019; Zhou et al., 2019). One of the key features of the flow around a circular cylinder is that the flow separation point oscillates following the vortex shedding and also depends on the Rey nolds number. On the other hand, for the flow around sharp edge bodies like square/rectangular prisms, the flow separation point is fixed at the corners and less sensitive to Reynolds number. The effect of gap spacing between the two side-by-side square prisms on the wake formation and vortex shedding was reported in the litera ture (e.g. Agrawal et al., 2006; Alam and Zhou, 2013; Ma et al., 2017). Agrawal et al. (2006) studied the vortex shedding and wake at a low Reynolds number Re ¼ 73 for the gap spacing g* (¼ g/h) ¼ 0.7 and 2.5, where g is the gap distance between the prisms and h is the height of prisms. They identified flip-flop and synchronized flow regimes for g* ¼ 0.7 and 2.5, respectively. In the flip-flop flow, the gap flow was biased, flip-flopping from a side to the other. On the other hand, in synchronized
* Corresponding author. Institute for Turbulence-Noise-Vibration Interaction and Control, Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China. E-mail addresses: [email protected], [email protected] (Md.M. Alam). https://doi.org/10.1016/j.oceaneng.2019.106797 Received 11 September 2019; Received in revised form 6 November 2019; Accepted 30 November 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Ramnarayan Mondal, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106797
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flow, the gap flow was symmetric, vortex shedding from the two prims occurring either an inphase or antiphase fashion. Alam and Zhou (2013) experimentally analyzed the features of wakes for g* ¼ 0–4.0 at Re ¼ 300. They distinguished four different flow regimes: (i) single bluff body regime (g*< 0.2), (ii) narrow and wide street regime (0.2 < g*< 1.1), (iii) transition regime (1.1 < g* < 1.4), and (iv) coupled-street regime (g* > 1.4). Ma et al. (2017) at Re ¼ 16–200 identified nine distinct flow patterns for g* ¼ 0–10. The nomenclatures of nine patterns are (i) steady-state wake, (ii) symmetric single vortex street, (iii) asymmetric single vortex street, (iv) irregular vortex shedding, (v) synchronized in-phase vortex shedding, (vi) synchronized anti-phase vortex shedding, (vii) synchronized inphase dominated vortex shedding with low-frequency modulation, (viii) synchronized antiphase dominated vortex shedding with low-frequency modulation, and (ix) asymmetric synchronized anti-phase vortex shedding mode. Recently, there is a surge of investigation on the flow across more than two square prisms. Sewatkar et al. (2009) numerically examined the flow over nine square prisms in side-by-side arrangements for Re ¼ 30–140 and 1 � g*� 4. At small g*, the secondary frequency caused by wake interaction accompanied the primary vortex shedding frequency. For large g* (¼ 3.0 and 4.0), the interaction among the wakes behind the cylinders was weak. The primary frequency thus dominated the flow. For five side-by-side square prisms at Re ¼ 150, Chatterjee et al. (2010) observed flip-flopping (g* ¼ 1.2), inphase and antiphase synchronized (g* ¼ 3), and non-synchronized (g* ¼ 4) flow structures. Recently, Zheng and Alam (2017) numerically investigated the flow physics for three side-by-side square prisms at Re ¼ 150 and g* ¼ 0.1–8.0. They identified five distinct flow structures: (i) base bleed flow, (ii) flip-flopping flow, (iii) symmetrically biased beat flow, (iv) non-biased beat flow, and (v) weak interaction flow. The detailed physical mechanism of the existence of the secondary frequency was clarified. All the above-mentioned studies on the multiple prisms are for the unconfined flow; that is, the side wall effect is negligible on the flow. Investigations on the fluid flow around confined rectangular prism(s) are very scarce. Davis et al. (1984) examined the effect of confining walls on the flow around a rectangular prism. The emphasis was given to understand the effect of g* (¼ 1.5 and 2.5), b/h (¼ 0.6, 1.0 and 1.7) and Re (¼ 100–1850) on the flow physics. The Strouhal number and average drag coefficient were increased when the side walls were brought close to the prisms. They observed that the average drag coefficient increases when Re is increased or b/h is decreased. Zhang et al. (2017) investi gated confined flow around two tandem rectangular prisms at Re ¼ 100 with a blockage ratio of 25%. They varied b/h from 1 to 4 and the gap spacing ratio between the prisms from 1 to 8. Regardless of b/h, two flow regimes were observed, namely steady flow regime for gap spacing ratio �3.0 and unsteady flow regime for gap spacing ratio �4.0. In many practical applications, rectangular structures appear in side-by-side under wall confinement, such as multiple prisms in a channel flow to enhance heat transfer, series of hard disks in CPUs, etc. Recently, in a CPU having nine hard disks in a x � y � z ¼ 1 � 3 � 3 configuration (x-direction representing the flow direction), it is identified that some hard disks became out of order when the cooling-fan-driven air flow reached a certain level. It is believed that flow-induced oscillations of the disks are the main issue behind the hard disks not working. Natu rally, investigations are direly needed on the flow-induced forces and flow around confined hard disks. The aim of this work to assimilate the dynamics of the flow around confined hard disks. Confined flow around three side-by-side (x � y � z ¼ 1 � 3 � 1) prisms having an aspect ratio of b/h ¼ 3.0 is investigated at Re ¼ 100. The focus is given on the effect of g* (¼ 0.8–3.4) on flow structures, Strouhal number (St), fluctuating lift coefficient (CʹL), mean lift coefficient (C‾L) and mean drag coefficient (C‾D). Generally, a prism with b/h < 2.8 is considered as a short prism where the shear layers from the leading edges roll up behind the prism, without reattaching on the side surfaces (Okajima et al., 1980; Takai and Sakamoto, 2006). On the other hand, a prism with b/h > 2.8 is considered as a long prism where the shear layers separating from the
leading edge reattach on the side surfaces of the prism. Presently b/h ¼ 3.0 is chosen as a representative of the elongated prisms. 2. Problem statement We considered two-dimensional incompressible viscous flow around three side-by-side identical rectangular prisms (P1, P2 and P3) placed between two parallel walls, as shown in Fig. 1. The aspect ratio (b/h) of each rectangular prism is the same (b/h ¼ 3), while the Reynolds number Re (¼ ρU∞h/μ) ¼ 100, where U∞, ρ and μ are the uniform inlet velocity, density, and dynamic viscosity of the fluid. The gap spacings (g) between two adjacent prisms and between the confining walls and outer prisms are equal. Numerical simulations are performed for g* (¼ g/h) ¼ 0.8, 1.0, 1.2, 1.5, 1.8, 2.0, 2.2, 2.5, 2.8, 3.0, 3.2 and 3.4, respectively. The two-dimensional Cartesian coordinate system (x, y) is considered such that the origin is at the centre of the middle prism (P2). The x-axis is positive along the streamwise direction, and the y-axis is perpendicular to the streamwise direction, directed upward. The inlet and outlet boundaries are assumed at a distance Xu ¼ 14.5h and Xd ¼ 30.5h from the front and rear faces of the prisms, respectively. Furthermore, the distance between the lateral boundary walls is denoted by Yl. 3. Computational details 3.1. Governing equations, boundary conditions and numerical method An unsteady, viscous, laminar, incompressible and Newtonian fluid flow is governed by the continuity and Navier-Stokes equations that in non-dimensional form can be written as
∂u* ∂v* þ ¼ 0; ∂x* ∂y*
(1) �
�
∂u* ∂u* ∂u* þ u* * þ v* * ¼ * ∂t ∂x ∂y
∂p* 1 ∂2 u* ∂2 u* þ þ ; ∂x* Re ∂x* 2 ∂y* 2
∂v* ∂v* ∂v* þ u* * þ v* * ¼ * ∂t ∂x ∂y
∂p* 1 ∂2 v* ∂2 v* þ þ : ∂y* Re ∂x* 2 ∂y* 2
�
(2a)
�
(2b)
The non-dimensional form of the Cartesian coordinate, velocity components, pressure and time are (x* ¼ x/h, y* ¼ y/h), (u* ¼ u/U∞, v* ¼ v/U∞), p* ¼ p/ρU2∞ and t* ¼ tU∞/h, respectively, where u and v are the velocity components in the x and y directions, respectively, p is the static pressure and t is the time. In the computational domain, the boundary conditions are pre scribed as follows. In engineering fields and in nature, there are many applications of confined flow where the approaching flow is either uniform or parabolic (Suzuki et al., 1993; Chakraborty et al., 2004; Sharma and Eswaran, 2005). At the inlet boundary (Xu ¼ 14.5h), a
Fig. 1. Schematic of three side-by-side rectangular prisms in the computa tional domain. 2
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uniform velocity profile is considered (u* ¼ 1, v* ¼ 0). No-slip boundary condition (u* ¼ 0, v* ¼ 0) is adopted on the prism surfaces and on the two side walls. At the outlet (Xd ¼ 30.5h), zero stress vectors are spec ified (∂u*/∂x* ¼ 0, ∂v*/∂x* ¼ ∂u*/∂y* ¼ 0). Utilizing the above boundary conditions, the governing equations (1) and (2) are solved numerically using finite volume method base com mercial solver Ansys-Fluent 15.0. To solve the equations, the compu tational domain is meshed as a combination of uniform and non-uniform quadrangular meshes using Ansys ICEM CFD. The coupling between the pressure and velocity is accessed using the scheme pressure implicit with splitting of operators (PISO). Convective terms are discretized using the second-order upwind scheme and second-order implicit discretization is adopted for time marching.
different meshes M1, M2 and M3 are tested, consisting of the number of meshes 45,714, 71,610 and 130,863, respectively, to check the ade quacy of the grid resolution. The integral parameters (Strouhal number St, fluctuating lift coefficient CʹL, time-mean drag coefficient C‾D, and fluctuating drag coefficient CˊD) for three different mesh distributions are tabulated in Table 1. Here, the St and lift (CL) and drag (CD) co efficients are computed as St ¼
fh 2Fl 2Fd ; CL ¼ ; and CD ¼ U∞ ρhU 2∞ ρhU 2∞
(3)
where f, Fl and Fd are vortex shedding frequency, lift force and drag force, respectively. The results in Table 1 exhibits that the values of integral parameters have a maximum deviation of 3.77% for CˊD be tween meshes M1 and M2. On the other hand, meshes M2 and M3 render a negligible difference (<0.33%) in the results. Mesh M2 is thus assumed enough for the unconfined flow. The results obtained from the mesh distribution M2 are further compared with those in the literature in Table 2. The present results have a good agreement with those in the literature. Applying mesh distribution M2 to three rectangular prisms between two parallel walls, the mesh and time independence tests are done again as shown in Table 3. As the lateral boundaries, acting as confined walls, are now not free slip, a fine mesh is considered near the side walls such that the first grid point from a side wall is again 0.005h away. Again the grid space is increased with a geometric progression ratio of 1.03. The mesh distributions near the rectangular prism and side walls are shown in Fig. 2 (b, c). Space- and time-independent tests are done for g* ¼ 1.0, 1.5 and 2.8
3.2. Study of convergence and validation 3.2.1. Mesh independent test Before proceeding to extensive simulations on confined fluid flow around the three side-by-side rectangular prisms, the grid adequacy test is conducted for the unconfined flow around a square prism at Re ¼ 100 with symmetry boundary conditions (i.e. v* ¼ 0, ∂u*/∂y* ¼ 0) for lateral boundaries (blockage ratio ¼ 5%) with Xu ¼ 14.5h and Xd ¼ 30.5h. The computational domain consists of an O-grid system surrounding the prism and a quadrangular grid system away from the prism (Fig. 2). Grid points on the surface of the square prism are uniformly spaced, with the first grid point lying 0.005h away from the prism surface. The grid spacing is increased with an expansion rate of 1.03. At the far down stream field (x* � 14.5h), a uniform grid spacing is considered. Three
Fig. 2. Structured quadrangular grid distributions (a) in the first quadrant, (b) near a prism wall and (c) near a confined wall. g* ¼ 1.2. 3
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Table 1 Grid and time-step independent test for a single square prism at Re ¼ 100.
M1(Δt* ¼ 0.0779) M2(Δt* ¼ 0.0519) M3(Δt* ¼ 0.0259)
Element
Integral parameters
Deviation %
St
C ʹL
C‾D
CʹD
C ʹL
C‾D
C ʹD
45714 71610 130863
0.1463 0.1463 0.1463
0.1869 0.1841 0.1835
1.4985 1.4942 1.4898
0.0055 0.0053 0.0053
1.52 – 0.33
0.29 – 0.29
3.77 – 0
parameters for outer prisms (P1 and P3) are equal, the results of P1 only are presented. Considering all three values of g*, it is observed that the deviations in St, CʹL, C‾D and CʹD are less than 0.77%, 2.13%, 0.34% and 1.85%, respectively. The mesh density and time step of case 1 is thus adopted.
Table 2 Comparison of the integral parameters for a single square prism at Re ¼ 100. Present Sohankar et al. (1998) Sharma and Eswaran (2004) Sahu et al. (2009) Singh et al. (2009) Sen et al. (2011)
St
C ʹL
C‾D
C ʹD
0.1463 0.1460 0.1488 0.1486 0.1452 0.1470
0.1841 0.1560 0.1922 0.1880 0.1928 0.1600
1.4942 1.4770 1.4936 1.4878 1.5287 1.5100
0.0053 – 0.0054 – – 0.0055
4. Wake structures and forces Identification of flow regimes is of interest among the researchers and engineers in many applications. Depending on the flow structures and shedding frequency, three different regimes are identified, namely regime A: single vortex street (0.8 � g* <1.1), regime B: twin vortex streets (1.1 < g* < 2.1), and regime C: triple vortex streets (2.1 < g* � 3.4). The separation point between the two regimes is chosen as the midpoint of two consecutive simulations that show two different flow patterns. Each of the flow regimes is discussed in details in the
as presented in Table 3 where St, CʹL, C‾D and CʹD are compared for different grids and time-steps. For each value of g*, two different mesh distributions are considered where the mesh density in case 2 is double of that in case 1, where case 1 has the similar mesh density of M2 as validated for single isolated prism before. Since the values of the Table 3 Effect of mesh and time-step on the flow around three prisms at Re ¼ 100. g* ¼ 1.0
Case 1 Case 2
g* ¼ 1.5
Case 1 Case 2
g* ¼ 2.8
Case 1 Case 2
Element ¼ 198753, Δt* ¼ 0.0519 Element ¼ 403500, Δt* ¼ 0.0259 Element ¼ 226947, Δt* ¼ 0.0519 Element ¼ 465674, Δt* ¼ 0.0259 Element ¼ 281529, Δt* ¼ 0.0519 Element ¼ 606312, Δt* ¼ 0.0259
Prism 1 Prism 2 Prism 1 Prism 2 Prism 1 Prism 2 Prism 1 Prism 2 Prism 1 Prism 2 Prism 1 Prism 2
St
C ʹL
C‾D
C ʹD
0.3496 0.3496 0.3523 0.3523 0.2792 0.2792 0.2809 0.2809 0.2010 0.2010 0.2002 0.2002
0.5447 0.7064 0.5331 0.6942 0.3642 0.4254 0.3594 0.4182 0.1462 0.1291 0.1433 0.1271
7.1047 7.9142 7.1022 7.9159 4.3028 4.7477 4.2880 4.7329 2.4035 2.5605 2.3978 2.5552
0.0966 0.0707 0.0949 0.0695 0.0089 0.0162 0.0482 0.0159 0.0054 0.0017 0.0053 0.0017
Fig. 3. Sequential instantaneous snapshots of vorticity contours at g* ¼ 0.8 (regime A, single vortex street flow). (a) tˊ/T ¼ 0, (b) tˊ/T ¼ 0.4, (c) tˊ/T ¼ 0.65, (d) tˊ/T ¼ 0.91, (e) tˊ/T ¼ 1.16 and (f) tˊ/T ¼ 1.56, showing the process of for mation of a single vortex street of two rows of vortices. The tˊ is the time measured from the minimum lift of the middle prism and T is the vortex shedding period, identical for three cyl inders. Red and blue colors represent positive and negative vorticities, respectively. The flow structures for (a), (c) and (f) correspond to in stants (i), (ii) and (iii) are marked in the next figure. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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subsequent subsections.
process of the vortices results in the formation of a reverse Karman vortex street downstream. Fig. 4 shows the time histories of lift and drag coefficients of three prisms along with the power spectra of lift coefficients. The lift and drag coefficients of the middle prism are periodic and sinusoidal whereas those of outer prisms are periodic but not sinusoidal in nature. The lift signals (Fig. 4a) imply that the sheddings of the positive (from the lower side) vortices from prisms 1 and 2 are inphase (Figs. 3a and 4a), as are those of the negative (from the upper side) vortices from prisms 2 and 3 (Figs. 3f and 4a). The lift amplitude measured from minimum to maximum and time-mean drag coefficient are the same for the outer prisms while those are larger for the middle prism than for the outer prisms. The value of the time-mean lift coefficient is 0.41, 0 and 0.41 for prisms 1, 2 and 3, respectively, indicating attractive lift force for the outer prisms. The power spectra (Fig. 4c) show that the shedding frequency is the same for the three prisms. The second and third harmonics of vortex shedding frequency are observed. The harmonics emerge owing to the interaction among the vortices immediately behind the prisms. The second harmonic peak is quite strong for the outer prisms but absent for the middle prism. This is because the middle prism lying on the sym metric line experiences symmetric atmosphere on either side, an outer prism has an asymmetric atmosphere, one side surface facing the middle prism and the other side facing the wall.
4.1. Regime A: Single vortex street (0.8 � g* < 1.1) For small g* (<1.1), a single body like flow appears. Fig. 3 shows instantaneous vorticity contours for g* ¼ 0.8. The snapshots are given at different time instants in about 1.5 vortex shedding periods, showing the evolution of vortices in the wake. Note that there are a total of four gaps, two between the prisms and two between the walls and prisms. Two shear layers form in each gap, spawning two counter-rotating vortices, respectively. A total of eight vortices are thus shed from the shear layers in one period of nominal vortex shedding. The growth and shedding of the vortices from the prisms occurs in an inphase fashion. An amal gamation and amputation process of the shed vortices leads to the evolution of the eight vortices in two vortices. A single wake consisting of two rows of vortices thus forms for x* > 6. As marked in the snapshot in Fig. 3a, eight vortices (U1, U2, M1, M2, L1, L2, W1, and W2) are linked to the eight shear layers, respectively. With the time elapsed, vortices U1, M1 and L1 are first pinched off and amalgamated, forming U1 þ M1 þ L1 vortex (Fig. 3d). Vortex W1 lifting up approaches the combined vortex U1 þ M1 þ L1 (Fig. 3d–f). Eventually, the four same-sign positive vortices merge together, forming U1 þ M1 þ L1 þ W1 vortex. Similarly, four negative-sign vortices form one united vortex U2 þ M2 þ L2 þ W0 2 downstream (Fig. 3f). Interestingly, three same-sign vortices from the prisms come close to each other and then the same-sign vortex from the corresponding wall joins them. Eventually, a single vortex street con sisting of positive and negative vortices forms in a similar fashion to a Karman vortex street. However, while a Karman vortex street is gener ally characterized by positive vortices on the lower side (y* < 0) and negative vortices on the upper side (y* > 0), here the vortex arrange ment is opposite, i.e. negative and positive vortices appearing on the lower and upper sides of the symmetry line (y* ¼ 0), respectively. See the vortex arrangement at x* > 9.0. That is, the merging and reversion
4.2. Regime B: Twin vortex streets (1.1 < g* < 2.1) Twin vortex streets are observed for 1.1 < g* < 2.1. Fig. 5 shows the instantaneous flow structures. The vortices from the prisms’ gap stretch in the transverse direction; each getting pinched off at the middle forms two small vortices. For example, M2 vortex in Fig. 5a splits into M0 2 and M00 2 vortices (Fig. 5b and c). The same happens for the positive vortex from the upper prism, U2 vortex splitting into U0 2 and U00 2 (Fig. 5a–c).
Fig. 4. Time histories of (a) lift and (b) drag coefficients of three prisms and (c) power spectra of lift coefficients of three prisms. g* ¼ 0.8. The flow structures (a) (c) and (f) in Fig. 3 correspond with the positions marked by (i), (ii) and (iii), respectively. 5
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Fig. 5. Sequential instantaneous snapshots of vorticity contours for g* ¼ 1.5 (regime B, twinvortex-street flow). T is the vortex shedding period and tˊ is the time delay measured from instant (i) corresponding to the minimum lift of the middle prism. Red and blue colors represent positive and negative vorticities, respectively. The flow structures for (a), (c), and (e) corre spond to instants (i), (iii) and (v) are marked in the next figure. (For interpretation of the refer ences to color in this figure legend, the reader is referred to the Web version of this article.)
While split vortices M0 2 and M00 2 from the middle prism are almost equal to each other, split vortex U0 2 is much weaker than U00 2 that goes to the wall side and dies out (Fig. 5c). The split of the negative vortex (from the upper side) from the upper prism does not take place, neither does the positive vortex from the lower prism. Now the same-signed vortices U00 2, M00 2, and L2 come close to each other and form a combined vortex U00 2 þ
M00 2 þ L2 lying on the lower side of y* ¼ 0. The vortex M0 2 stays on the upper side and snatches vorticity from the wall, keeping alive (Fig. 5d). The evolution and interactions of the vortices reveal that a positive vortex on the lower vortex street (x* > 8) is composed of two split positive vortices (U00 2 þ M00 2) from the middle and upper prisms and one intact positive vortex (L2) from the lower prism (Fig. 5f). On the other
Fig. 6. Time histories of (a) lift and (b) drag coefficients of three prisms and (c) power spectra of lift coefficients of three prisms. g* ¼ 1.5. 6
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hand, a positive vortex on the upper vortex street consists of a split positive vortex from the middle prism along with vorticity from the upper wall. In a similar fashion, two rows of negative vortices come into being in the wake. Two vortex streets thus feature the wake. The appearance of vortices in each street is again in a fashion of reverse Karman vortex street. Time histories of lift and drag coefficients and their power spectra are presented in Fig. 6. Similar to regime A, the three prisms shed vortices at the same frequency, St ¼ 0.2792 (Fig. 6c). Unlike regime A, the second or third harmonic frequency in the power spectra is absent. This is because the complex interaction between the vortices largely occurs in x* < 6 for regime A and in x* ¼ 6–9 for regime B. The lift signals of the three prisms indicates that the phase lag between the vortex sheddings from the outer prisms is zero and that between the middle and outer prisms is 15.5� . Time-mean lift coefficient is positive (0.35) for prism 1 and negative ( 0.35) for prism 3, while zero for prism 2. This implies that the outer prisms undergo repulsive lift forces. Fluctuation in drag is smaller for the middle prism than for the outer prisms. The middle prism undergoes a larger time-mean drag than the outer prisms.
St ¼ 0:1258 þ ¼ St0 þ
0:2189 g*
(4)
0:2189 g*
where St0 ¼ 0.1258 is the Strouhal number of a single rectangular (b/h ¼ 3) prism in an unconfined domain, which was obtained from the present simulations. The difference between estimated data and Eq. (4) plotted in Fig. 9a is less than 3.2%. The C‾L ¼ 0 for the middle prism (Fig. 9b). For a particular g*, the outer prisms’ C‾L are equal in magnitude but opposite in signs. Inter estingly, the variations in C‾L of the two outer prisms explicitly char acterize the flow classifications made above. In regime A, the C‾L is negative and positive for the upper and lower prisms, respectively, indicating attractive lift forces. On the other hand, an opposite scenario prevails for regime B, positive and negative C‾L complementing the upper and lower prisms, respectively. The lift forces on the two outer prisms are now repulsive. In regime B, with increasing g*, the C‾L grows for the upper prism and declines for the lower prism, reaching maximum and minimum at the border between regimes B and C. A gentle decrease in C‾L magnitude with g* occurs for the outer prisms in regime C. The CʹL and C‾D variations depicts that both outer prisms exhibit the same value (Fig. 9c and d). In regimes A and B, the CʹL is greater for the middle prism than for the outer prisms. On the contrary, an outer prism experiences a higher CʹL than the middle prism in regime C. The CʹL can be represented by a polynomial function of g*. The variations in CʹL1 (prism 1) and CʹL2 (prism 2) with g* can respectively be expressed as
4.3. Regime C: Triple vortex streets (2.1 < g* � 3.4) Now the gap between the prisms and between walls and prism is large enough, allowing each prism to shed vortices like an isolated prism (Fig. 7). Triple vortex streets thus feature the wake. Like regimes A and B, the vortex shedding frequencies of three prisms are the same which can also be observed from the power spectra of lift signal of three prisms (Fig. 8c). Still, repulsive lift forces appear for the outer prisms while amplitudes of the lift forces of the three prisms are approximately the same (Fig. 8a). The outer prisms have the same time-mean drag coeffi cient and the same fluctuating drag. The value of time-mean drag is smaller for outer prisms than for the middle prism while the fluctuation in drag is greater for outer prisms than the middle prism (Fig. 8b).
C’L1 ¼ 0:05 ¼
C’L0
0:34α1 þ 0:88α21
C’L2 ¼ 0:05 ¼
C’L0
0:34α1 þ 0:88α21
(5a)
and
0:62α1 þ 1:33α21
(5b)
0:62α1 þ 1:33α21 ;
where α1 ¼ (g*) 0.53 and CʹL0 ¼ 0.05 is the fluctuating lift coefficient of a single rectangular prism (b/h ¼ 3.0) in an unconfined domain. Fig. 9c shows that Eq. (5) has a good fit with the computed data, with CʹL1 and CʹL2 both approaching the single prism value (0.05) as g* → ∞. The C‾D of the middle prism is always greater than that of the outer prisms. The difference in C‾D between the middle prism and outer prisms is large in regimes A and B while small in regime C. The C‾D1 (prism 1) and C‾D2 (prism 2) can be represented as a function of g* which are of the forms
5. Effect of g* on forces and Strouhal number Fig. 9 illustrates the dependence of St, C‾D, CʹL, and C‾L, on g*. At the top of the figure, the span of each flow regime is marked. The St of each prism was estimated from spectral analyses of the fluctuating lift force acting on the prism. As shown before, all the three prisms have identical St; the only one data set is therefore plotted in the figure. With increasing g*, the St declines from 0.39 at g* ¼ 0.8 to 0.18 at g* ¼ 3.4 (Fig. 9a). A curve fit equation of St as a function of g*, using the least square method, is obtained as
CD1 ¼ 1:28 þ 0:67β1 þ 5:17β21 ;
and
CD2 ¼ 1:28 þ 0:87β1 þ 5:77β21 ;
where β1 ¼ ðg* Þ
(6a) 0:88
;
(6b)
Fig. 7. Sequential instantaneous snapshots of vorticity contours for g* ¼ 2.8 (regime C, triple-vortex-street flow). T is the vortex shedding period and tˊ is the time delay measured from instant (i) corresponding to the maximum lift of the middle prism. Red and blue colors represent positive and negative vorticities, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 7
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Fig. 8. Time histories of (a) lift and (b) drag coefficients of three prisms and (c) power spectra of lift coefficients of three prisms. g* ¼ 2.8. The flow structures (a) and (b) in Fig. 7 correspond with the positions marked by (i) and (ii), respectively in lift and drag coefficients.
Fig. 9. Effect of g*on (a) Strouhal number, (b) time-mean lift coefficient, (c) fluctuating lift coefficient, and (d) time-mean drag coefficient.
6. Flow in the gap
respectively. Eqs. (6 a, b) are also presented in Fig. 9d, showing less than 0.67% (prisms 1, 3) and 1.59% (prism 2) errors between fit curves and computed values. Variation in C‾D exhibits the same nature as that in St. At a lower g*, the blockage ratio is large, enhancing C‾D and St. An in crease in g* yields a smaller blockage ratio, C‾D, CʹL and St, all thus decrease with g*. A summary of the flow features of different flow re gimes are summarized in Table 4.
As explained before, there are two gaps between the prisms and two
between the prisms and wall. The time-mean streamwise velocity (u* ) is computed at the center (x* ¼ 0, y* ¼ � (g*þ1)/2, �3(g*þ1)/2) of each gap. The values of u* were equal in the two gaps between the prisms, as were in the two gaps between the prisms and wall. Therefore, only two sets of data, one (u* w 1 ) in the gap between the wall and prism 1 and the 8
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Table 4 Summary of features in different flow regimes. The subscripts ‘1’ and ‘2’ refer to prisms 1 and 2, respectively. The symbols ‘↗ ‘and ‘↘’ denote ‘increase’ and ‘decrease’, respectively, with increasing g*. Regime
Vortex street
Wake
Forces
A; 0.8 � g* <1.1 B; 1.1 < g* < 2.1 C; 2.1 < g* � 3.4
Single vortex street
Reverse Karman wake
Twin vortex streets
Reverse Karman wake
Triple vortex streets
Karman wake
CʹL1 < CʹL2; C‾D1 < C‾D2 C‾L1 ¼ -ve, C‾L3 ¼ þve; | C‾L1| ¼ |C‾L3| ↘ CʹL1 < CʹL2; C‾D1 < C‾D2 C‾L1 ¼ þve, C‾L3 ¼ -ve; | C‾L1| ¼ |C‾L3| ↗ CʹL1 > CʹL2; C‾D1 < C‾D2 C‾L1 ¼ þve, C‾L3 ¼ -ve; | C‾L1| ¼ |C‾L3| ↘
Fig. 10. Variations in time-mean streamwise velocity at gap between wall and outer prism (circle symbol) and between two prisms (square symbol). The lines represent the curve fits.
other (u* 1 2 ) between the prisms 1 and 2, are presented in Fig. 10. Obviously,
u*
1 2>
u*
w 1,
the difference between them being small with
increasing g*. The reason behind u* 1 2 > u* w boundary layer next to it while u* 1
2
1
is that the wall develops a
gets high because of shear layer
developments by the prisms. Again, u* w 1 and u* 1 with g* and obey the following relations u* w
1
u* 1
2
¼ 1:0 þ 1:41ðg* Þ ¼ 1:0 þ 1:47ðg* Þ
0:94
0:94
:
and
2
< g*< 2.1) is characterized by two vortex streets. The appearance of vortices in each street is again in a fashion of reverse Karman vortex street. In regime C (2.1 < g* � 3.4), each prism forms a vortex street that resembles a single isolated prism Karman street. Three streets thus feature the wake. The dependence on g* of C‾L and CʹL explicitly distinguishes the flow regimes. In regime A, the C‾L of the upper and lower prisms are negative and positive, respectively, C‾L magnitudes of both prisms waning with g*. The two outer prisms undergo an attractive lift force. The signs of the C‾L get reversed in regimes B and C, positive for the upper prism and negative for the lower prism, leading to a repulsive lift force between the prisms. The C‾L magnitude grows in regime B but declines in regime C. The CʹL in regimes A and B is greater for the middle prism than the outer prisms while the scenario is opposite in the regime C. The C‾D and St both decreases with increasing g*. While all three prisms have identical St at a given g*, the outer prisms have smaller C‾D than the middle prism. The smaller C‾D is attributed to the smaller ve locity in the gaps between the outer prisms and wall than in the gaps between the prisms. This is because the former gaps are characterized by a boundary layer on the wall side and a shear layer on the prism side while the latter gaps are characterized by shear layers on both sides. The gap flow velocity rapidly declines in either gaps as g* increases, which explains why C‾D and St diminish with g*. Variations in St, C‾D, CʹL and
both rapidly decay (7a) (7b)
Eqs. (7a, b) presented as the lines fit well with our computed results (Fig. 10). Comparing Figs. 9(d) and 10, it is observed that both C‾D and
u* decrease in a similar fashion with the increase of g*. So, it is of interest to find out a relation between the C‾D and u* . It is expected that C‾D of a
prism will be dictated by u* on the two sides (gaps) of the prism. That is,
u* w 1 and u* 1 2 both are associated with prism 1, and u* 1 2 being the same on the two sides is associated with prism 2. An average velocity ζ1 ¼ (u* w 1 þu* 1 2 )/2 is thus considered for prism 1 and ζ2 ¼ (u* 1 2 þu* 1 2 )/2 for prism 2. The correlation between ζ1 and C‾D1 (prism 1) is found as C‾D1 ¼ 0.86 þ 0.44 ζ31, while that for prism 2 is C‾D2 ¼ 0.86 þ 0.46 ζ32. The above stated relations have a good fit with computed results with a maximum deviation of less than 4.32%. One expects ζ1 and ζ2 → 1.0 when g* → ∞; the corresponding C‾D1 ¼ 1.3 and C‾D2 ¼ 1.32 which have deviations of 1.56% and 3.12%, respectively, from C‾D1 ¼ C‾D2 ¼ 1.28 that is the value of C‾D of a prism (b/h ¼ 3.0) in an un
u* with g* are expressed in empirical relations.
bounded domain. Furthermore, Figs. 9(a) and 10 depict that St and u*
decline with g*. Therefore, a relation between St and u* is expected.
Author contribution statement
gaps, i.e. ζ ¼ (2 u* w 1 þ 2u* 1 2 )/4. The St and ζ satisfy the relation St ¼ 0.0445 þ 0.1612 ζ. It is observed that the curve fit has a maximum deviation of 2.55% with the computed St. Expecting ζ → 1.0 for g* → ∞, we obtain St ¼ 0.1167 which has a deviation 7.2% with St0 (¼ 0.1258).
Ramnarayan Mondal: computing, data analysis, figure preparation, and original draft preparation. Md Mahbub Alam: formulation or evolution of overarching research aims, supervision, funding, reviewing, and editing. Rajesh Bhatt: computing, and validation.
7. Conclusions
Declaration of competing interest
Numerical simulations are conducted to investigate the flow around three side-by-side rectangular prisms in the presence of two confining walls at a Re ¼ 100. The dependence on gap spacing ratio g* (¼ 0.8–3.4) of C‾D, C‾L, CʹL, St and flow structures are examined. Based on the vortex structures, fluid forces and shedding frequency, the flow structures are classified into three distinct regimes A (single vortex street), B (twin vortex streets) and C (triple vortex streets). In regime A (0.8 � g*<1.1), each prism spawns vortices alternately, as do the two side walls. In one period of vortex shedding, six vortices from the prisms and two vortices from the two side walls are shed. They undergo splitting, amputation and amalgamation processes, and even tually, a single vortex street (two rows of vortices) forms downstream in the fashion of reverse Karman vortex street. The wake in regime B (1.1
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Since all the prisms have the same St, we get the average of u* at four
Acknowledgment The authors wish to acknowledge the support given by the National Natural Science Foundation of China through Grants 11672096 and 91752112 and by Research Grant Council of Shenzhen Government through grant JCYJ20180306171921088.
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Effect of side walls on flow around prisms Ramnarayan Mondal a, Md. Mahbub Alam a, b, *, Rajesh Bhatt a, c a
Institute for Turbulence-Noise-Vibration Interaction and Control, Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China Digital Engineering Laboratory of Offshore Equipment, Shenzhen, China c Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, India b
A R T I C L E I N F O
A B S T R A C T
Keywords: Wall effect Flow structure Forces Strouhal number
Two-dimensional laminar flow around three side-by-side rectangular prisms placed between two parallel walls is numerically investigated at a Reynolds number Re ¼ 100. The cross-sectional aspect ratio of each prism is b/h ¼ 3, where b and h are the prism width and height, respectively. The gap spacing (g) between the two adjacent prisms and between the wall and the outer prism is varied from g* (¼ g/h) ¼ 0.8 to 3.4. The focus is given on the influence of g* on flow structures, Strouhal number (St), and fluid forces. Three distinct flow regimes are identified, namely single vortex street (0.8 � g* < 1.1), twin vortex streets (1.1 < g* < 2.1), and triple vortex streets (2.1 < g* � 3.4). Lift forces on the outer prisms are attractive for single vortex street and repulsive for twin and triple vortex streets. The St being identical for the three prisms exponentially declines from 0.39 to 0.18 with g* increasing from 0.8 to 3.4. Fluctuating lift and time-mean drag also shrink with increasing g*, time-mean drag becoming higher for the middle prism than for the outer prisms. Empirical relations of St and forces are obtained as a function of g*.
1. Introduction The problem of fluid flow around slender structures is of interest among the researchers and scientists for their enormous applications in the fields of aerodynamics (e.g. wind flow around high rise buildings, chimney stacks, overhead power line bundles), offshore engineering (e. g. flow past bridge piers, offshore platform, drilling rigs and riser sys tem), and subsea engineering (e.g. flow around sub-sea cables and pipelines). These structures very often appear in a group, making the flow physics more complicated than a single structure. For the closelyspaced structures, the complicated flow arises due to the mutual inter action between shear layers and between wake vortices. Because of everything getting compact in the modern engineering, applications of flow around multiple structures in wall proximity get broader, such as the flow around prisms/cylinders in a channel to enhance flow mixing and heat transfer, the flow around drive bays in CPUs, etc. It is therefore essential to assimilate the complicated flow around closely spaced multiple structures with a view to their practical applications. Due to the geometric simplicity, the structures are mostly found in circular, square or rectangular shapes. A comprehensive review of the wakes behind different bluff body shapes (including square, circular, and triangular cylinders) is can be found in Derakhshandeh and Alam, 2019.
Flow around multiple cylindrical structures of circular or square shape have been extensively investigated in the literature through ex periments (e.g. Okajima, 1982; LeGal et al., 1990; Sumner et al., 1999; Alam and Sakamoto, 2005; Alam et al., 2003; Alam and Zhou, 2007; Alam, 2014; Bai and Alam, 2018) and numerical simulations (e.g. Sohankar, 2006, 2008, 2014; Ding et al., 2007; Carini et al., 2014; Maiti, 2012; Maiti and Bhatt, 2014; Alam, 2016; Alam et al., 2017; Chen et al., 2018; Zheng and Alam, 2019; Zhou et al., 2019). One of the key features of the flow around a circular cylinder is that the flow separation point oscillates following the vortex shedding and also depends on the Rey nolds number. On the other hand, for the flow around sharp edge bodies like square/rectangular prisms, the flow separation point is fixed at the corners and less sensitive to Reynolds number. The effect of gap spacing between the two side-by-side square prisms on the wake formation and vortex shedding was reported in the litera ture (e.g. Agrawal et al., 2006; Alam and Zhou, 2013; Ma et al., 2017). Agrawal et al. (2006) studied the vortex shedding and wake at a low Reynolds number Re ¼ 73 for the gap spacing g* (¼ g/h) ¼ 0.7 and 2.5, where g is the gap distance between the prisms and h is the height of prisms. They identified flip-flop and synchronized flow regimes for g* ¼ 0.7 and 2.5, respectively. In the flip-flop flow, the gap flow was biased, flip-flopping from a side to the other. On the other hand, in synchronized
* Corresponding author. Institute for Turbulence-Noise-Vibration Interaction and Control, Harbin Institute of Technology (Shenzhen), Shenzhen, 518055, China. E-mail addresses: [email protected], [email protected] (Md.M. Alam). https://doi.org/10.1016/j.oceaneng.2019.106797 Received 11 September 2019; Received in revised form 6 November 2019; Accepted 30 November 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Ramnarayan Mondal, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106797
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flow, the gap flow was symmetric, vortex shedding from the two prims occurring either an inphase or antiphase fashion. Alam and Zhou (2013) experimentally analyzed the features of wakes for g* ¼ 0–4.0 at Re ¼ 300. They distinguished four different flow regimes: (i) single bluff body regime (g*< 0.2), (ii) narrow and wide street regime (0.2 < g*< 1.1), (iii) transition regime (1.1 < g* < 1.4), and (iv) coupled-street regime (g* > 1.4). Ma et al. (2017) at Re ¼ 16–200 identified nine distinct flow patterns for g* ¼ 0–10. The nomenclatures of nine patterns are (i) steady-state wake, (ii) symmetric single vortex street, (iii) asymmetric single vortex street, (iv) irregular vortex shedding, (v) synchronized in-phase vortex shedding, (vi) synchronized anti-phase vortex shedding, (vii) synchronized inphase dominated vortex shedding with low-frequency modulation, (viii) synchronized antiphase dominated vortex shedding with low-frequency modulation, and (ix) asymmetric synchronized anti-phase vortex shedding mode. Recently, there is a surge of investigation on the flow across more than two square prisms. Sewatkar et al. (2009) numerically examined the flow over nine square prisms in side-by-side arrangements for Re ¼ 30–140 and 1 � g*� 4. At small g*, the secondary frequency caused by wake interaction accompanied the primary vortex shedding frequency. For large g* (¼ 3.0 and 4.0), the interaction among the wakes behind the cylinders was weak. The primary frequency thus dominated the flow. For five side-by-side square prisms at Re ¼ 150, Chatterjee et al. (2010) observed flip-flopping (g* ¼ 1.2), inphase and antiphase synchronized (g* ¼ 3), and non-synchronized (g* ¼ 4) flow structures. Recently, Zheng and Alam (2017) numerically investigated the flow physics for three side-by-side square prisms at Re ¼ 150 and g* ¼ 0.1–8.0. They identified five distinct flow structures: (i) base bleed flow, (ii) flip-flopping flow, (iii) symmetrically biased beat flow, (iv) non-biased beat flow, and (v) weak interaction flow. The detailed physical mechanism of the existence of the secondary frequency was clarified. All the above-mentioned studies on the multiple prisms are for the unconfined flow; that is, the side wall effect is negligible on the flow. Investigations on the fluid flow around confined rectangular prism(s) are very scarce. Davis et al. (1984) examined the effect of confining walls on the flow around a rectangular prism. The emphasis was given to understand the effect of g* (¼ 1.5 and 2.5), b/h (¼ 0.6, 1.0 and 1.7) and Re (¼ 100–1850) on the flow physics. The Strouhal number and average drag coefficient were increased when the side walls were brought close to the prisms. They observed that the average drag coefficient increases when Re is increased or b/h is decreased. Zhang et al. (2017) investi gated confined flow around two tandem rectangular prisms at Re ¼ 100 with a blockage ratio of 25%. They varied b/h from 1 to 4 and the gap spacing ratio between the prisms from 1 to 8. Regardless of b/h, two flow regimes were observed, namely steady flow regime for gap spacing ratio �3.0 and unsteady flow regime for gap spacing ratio �4.0. In many practical applications, rectangular structures appear in side-by-side under wall confinement, such as multiple prisms in a channel flow to enhance heat transfer, series of hard disks in CPUs, etc. Recently, in a CPU having nine hard disks in a x � y � z ¼ 1 � 3 � 3 configuration (x-direction representing the flow direction), it is identified that some hard disks became out of order when the cooling-fan-driven air flow reached a certain level. It is believed that flow-induced oscillations of the disks are the main issue behind the hard disks not working. Natu rally, investigations are direly needed on the flow-induced forces and flow around confined hard disks. The aim of this work to assimilate the dynamics of the flow around confined hard disks. Confined flow around three side-by-side (x � y � z ¼ 1 � 3 � 1) prisms having an aspect ratio of b/h ¼ 3.0 is investigated at Re ¼ 100. The focus is given on the effect of g* (¼ 0.8–3.4) on flow structures, Strouhal number (St), fluctuating lift coefficient (CʹL), mean lift coefficient (C‾L) and mean drag coefficient (C‾D). Generally, a prism with b/h < 2.8 is considered as a short prism where the shear layers from the leading edges roll up behind the prism, without reattaching on the side surfaces (Okajima et al., 1980; Takai and Sakamoto, 2006). On the other hand, a prism with b/h > 2.8 is considered as a long prism where the shear layers separating from the
leading edge reattach on the side surfaces of the prism. Presently b/h ¼ 3.0 is chosen as a representative of the elongated prisms. 2. Problem statement We considered two-dimensional incompressible viscous flow around three side-by-side identical rectangular prisms (P1, P2 and P3) placed between two parallel walls, as shown in Fig. 1. The aspect ratio (b/h) of each rectangular prism is the same (b/h ¼ 3), while the Reynolds number Re (¼ ρU∞h/μ) ¼ 100, where U∞, ρ and μ are the uniform inlet velocity, density, and dynamic viscosity of the fluid. The gap spacings (g) between two adjacent prisms and between the confining walls and outer prisms are equal. Numerical simulations are performed for g* (¼ g/h) ¼ 0.8, 1.0, 1.2, 1.5, 1.8, 2.0, 2.2, 2.5, 2.8, 3.0, 3.2 and 3.4, respectively. The two-dimensional Cartesian coordinate system (x, y) is considered such that the origin is at the centre of the middle prism (P2). The x-axis is positive along the streamwise direction, and the y-axis is perpendicular to the streamwise direction, directed upward. The inlet and outlet boundaries are assumed at a distance Xu ¼ 14.5h and Xd ¼ 30.5h from the front and rear faces of the prisms, respectively. Furthermore, the distance between the lateral boundary walls is denoted by Yl. 3. Computational details 3.1. Governing equations, boundary conditions and numerical method An unsteady, viscous, laminar, incompressible and Newtonian fluid flow is governed by the continuity and Navier-Stokes equations that in non-dimensional form can be written as
∂u* ∂v* þ ¼ 0; ∂x* ∂y*
(1) �
�
∂u* ∂u* ∂u* þ u* * þ v* * ¼ * ∂t ∂x ∂y
∂p* 1 ∂2 u* ∂2 u* þ þ ; ∂x* Re ∂x* 2 ∂y* 2
∂v* ∂v* ∂v* þ u* * þ v* * ¼ * ∂t ∂x ∂y
∂p* 1 ∂2 v* ∂2 v* þ þ : ∂y* Re ∂x* 2 ∂y* 2
�
(2a)
�
(2b)
The non-dimensional form of the Cartesian coordinate, velocity components, pressure and time are (x* ¼ x/h, y* ¼ y/h), (u* ¼ u/U∞, v* ¼ v/U∞), p* ¼ p/ρU2∞ and t* ¼ tU∞/h, respectively, where u and v are the velocity components in the x and y directions, respectively, p is the static pressure and t is the time. In the computational domain, the boundary conditions are pre scribed as follows. In engineering fields and in nature, there are many applications of confined flow where the approaching flow is either uniform or parabolic (Suzuki et al., 1993; Chakraborty et al., 2004; Sharma and Eswaran, 2005). At the inlet boundary (Xu ¼ 14.5h), a
Fig. 1. Schematic of three side-by-side rectangular prisms in the computa tional domain. 2
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uniform velocity profile is considered (u* ¼ 1, v* ¼ 0). No-slip boundary condition (u* ¼ 0, v* ¼ 0) is adopted on the prism surfaces and on the two side walls. At the outlet (Xd ¼ 30.5h), zero stress vectors are spec ified (∂u*/∂x* ¼ 0, ∂v*/∂x* ¼ ∂u*/∂y* ¼ 0). Utilizing the above boundary conditions, the governing equations (1) and (2) are solved numerically using finite volume method base com mercial solver Ansys-Fluent 15.0. To solve the equations, the compu tational domain is meshed as a combination of uniform and non-uniform quadrangular meshes using Ansys ICEM CFD. The coupling between the pressure and velocity is accessed using the scheme pressure implicit with splitting of operators (PISO). Convective terms are discretized using the second-order upwind scheme and second-order implicit discretization is adopted for time marching.
different meshes M1, M2 and M3 are tested, consisting of the number of meshes 45,714, 71,610 and 130,863, respectively, to check the ade quacy of the grid resolution. The integral parameters (Strouhal number St, fluctuating lift coefficient CʹL, time-mean drag coefficient C‾D, and fluctuating drag coefficient CˊD) for three different mesh distributions are tabulated in Table 1. Here, the St and lift (CL) and drag (CD) co efficients are computed as St ¼
fh 2Fl 2Fd ; CL ¼ ; and CD ¼ U∞ ρhU 2∞ ρhU 2∞
(3)
where f, Fl and Fd are vortex shedding frequency, lift force and drag force, respectively. The results in Table 1 exhibits that the values of integral parameters have a maximum deviation of 3.77% for CˊD be tween meshes M1 and M2. On the other hand, meshes M2 and M3 render a negligible difference (<0.33%) in the results. Mesh M2 is thus assumed enough for the unconfined flow. The results obtained from the mesh distribution M2 are further compared with those in the literature in Table 2. The present results have a good agreement with those in the literature. Applying mesh distribution M2 to three rectangular prisms between two parallel walls, the mesh and time independence tests are done again as shown in Table 3. As the lateral boundaries, acting as confined walls, are now not free slip, a fine mesh is considered near the side walls such that the first grid point from a side wall is again 0.005h away. Again the grid space is increased with a geometric progression ratio of 1.03. The mesh distributions near the rectangular prism and side walls are shown in Fig. 2 (b, c). Space- and time-independent tests are done for g* ¼ 1.0, 1.5 and 2.8
3.2. Study of convergence and validation 3.2.1. Mesh independent test Before proceeding to extensive simulations on confined fluid flow around the three side-by-side rectangular prisms, the grid adequacy test is conducted for the unconfined flow around a square prism at Re ¼ 100 with symmetry boundary conditions (i.e. v* ¼ 0, ∂u*/∂y* ¼ 0) for lateral boundaries (blockage ratio ¼ 5%) with Xu ¼ 14.5h and Xd ¼ 30.5h. The computational domain consists of an O-grid system surrounding the prism and a quadrangular grid system away from the prism (Fig. 2). Grid points on the surface of the square prism are uniformly spaced, with the first grid point lying 0.005h away from the prism surface. The grid spacing is increased with an expansion rate of 1.03. At the far down stream field (x* � 14.5h), a uniform grid spacing is considered. Three
Fig. 2. Structured quadrangular grid distributions (a) in the first quadrant, (b) near a prism wall and (c) near a confined wall. g* ¼ 1.2. 3
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Table 1 Grid and time-step independent test for a single square prism at Re ¼ 100.
M1(Δt* ¼ 0.0779) M2(Δt* ¼ 0.0519) M3(Δt* ¼ 0.0259)
Element
Integral parameters
Deviation %
St
C ʹL
C‾D
CʹD
C ʹL
C‾D
C ʹD
45714 71610 130863
0.1463 0.1463 0.1463
0.1869 0.1841 0.1835
1.4985 1.4942 1.4898
0.0055 0.0053 0.0053
1.52 – 0.33
0.29 – 0.29
3.77 – 0
parameters for outer prisms (P1 and P3) are equal, the results of P1 only are presented. Considering all three values of g*, it is observed that the deviations in St, CʹL, C‾D and CʹD are less than 0.77%, 2.13%, 0.34% and 1.85%, respectively. The mesh density and time step of case 1 is thus adopted.
Table 2 Comparison of the integral parameters for a single square prism at Re ¼ 100. Present Sohankar et al. (1998) Sharma and Eswaran (2004) Sahu et al. (2009) Singh et al. (2009) Sen et al. (2011)
St
C ʹL
C‾D
C ʹD
0.1463 0.1460 0.1488 0.1486 0.1452 0.1470
0.1841 0.1560 0.1922 0.1880 0.1928 0.1600
1.4942 1.4770 1.4936 1.4878 1.5287 1.5100
0.0053 – 0.0054 – – 0.0055
4. Wake structures and forces Identification of flow regimes is of interest among the researchers and engineers in many applications. Depending on the flow structures and shedding frequency, three different regimes are identified, namely regime A: single vortex street (0.8 � g* <1.1), regime B: twin vortex streets (1.1 < g* < 2.1), and regime C: triple vortex streets (2.1 < g* � 3.4). The separation point between the two regimes is chosen as the midpoint of two consecutive simulations that show two different flow patterns. Each of the flow regimes is discussed in details in the
as presented in Table 3 where St, CʹL, C‾D and CʹD are compared for different grids and time-steps. For each value of g*, two different mesh distributions are considered where the mesh density in case 2 is double of that in case 1, where case 1 has the similar mesh density of M2 as validated for single isolated prism before. Since the values of the Table 3 Effect of mesh and time-step on the flow around three prisms at Re ¼ 100. g* ¼ 1.0
Case 1 Case 2
g* ¼ 1.5
Case 1 Case 2
g* ¼ 2.8
Case 1 Case 2
Element ¼ 198753, Δt* ¼ 0.0519 Element ¼ 403500, Δt* ¼ 0.0259 Element ¼ 226947, Δt* ¼ 0.0519 Element ¼ 465674, Δt* ¼ 0.0259 Element ¼ 281529, Δt* ¼ 0.0519 Element ¼ 606312, Δt* ¼ 0.0259
Prism 1 Prism 2 Prism 1 Prism 2 Prism 1 Prism 2 Prism 1 Prism 2 Prism 1 Prism 2 Prism 1 Prism 2
St
C ʹL
C‾D
C ʹD
0.3496 0.3496 0.3523 0.3523 0.2792 0.2792 0.2809 0.2809 0.2010 0.2010 0.2002 0.2002
0.5447 0.7064 0.5331 0.6942 0.3642 0.4254 0.3594 0.4182 0.1462 0.1291 0.1433 0.1271
7.1047 7.9142 7.1022 7.9159 4.3028 4.7477 4.2880 4.7329 2.4035 2.5605 2.3978 2.5552
0.0966 0.0707 0.0949 0.0695 0.0089 0.0162 0.0482 0.0159 0.0054 0.0017 0.0053 0.0017
Fig. 3. Sequential instantaneous snapshots of vorticity contours at g* ¼ 0.8 (regime A, single vortex street flow). (a) tˊ/T ¼ 0, (b) tˊ/T ¼ 0.4, (c) tˊ/T ¼ 0.65, (d) tˊ/T ¼ 0.91, (e) tˊ/T ¼ 1.16 and (f) tˊ/T ¼ 1.56, showing the process of for mation of a single vortex street of two rows of vortices. The tˊ is the time measured from the minimum lift of the middle prism and T is the vortex shedding period, identical for three cyl inders. Red and blue colors represent positive and negative vorticities, respectively. The flow structures for (a), (c) and (f) correspond to in stants (i), (ii) and (iii) are marked in the next figure. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)
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subsequent subsections.
process of the vortices results in the formation of a reverse Karman vortex street downstream. Fig. 4 shows the time histories of lift and drag coefficients of three prisms along with the power spectra of lift coefficients. The lift and drag coefficients of the middle prism are periodic and sinusoidal whereas those of outer prisms are periodic but not sinusoidal in nature. The lift signals (Fig. 4a) imply that the sheddings of the positive (from the lower side) vortices from prisms 1 and 2 are inphase (Figs. 3a and 4a), as are those of the negative (from the upper side) vortices from prisms 2 and 3 (Figs. 3f and 4a). The lift amplitude measured from minimum to maximum and time-mean drag coefficient are the same for the outer prisms while those are larger for the middle prism than for the outer prisms. The value of the time-mean lift coefficient is 0.41, 0 and 0.41 for prisms 1, 2 and 3, respectively, indicating attractive lift force for the outer prisms. The power spectra (Fig. 4c) show that the shedding frequency is the same for the three prisms. The second and third harmonics of vortex shedding frequency are observed. The harmonics emerge owing to the interaction among the vortices immediately behind the prisms. The second harmonic peak is quite strong for the outer prisms but absent for the middle prism. This is because the middle prism lying on the sym metric line experiences symmetric atmosphere on either side, an outer prism has an asymmetric atmosphere, one side surface facing the middle prism and the other side facing the wall.
4.1. Regime A: Single vortex street (0.8 � g* < 1.1) For small g* (<1.1), a single body like flow appears. Fig. 3 shows instantaneous vorticity contours for g* ¼ 0.8. The snapshots are given at different time instants in about 1.5 vortex shedding periods, showing the evolution of vortices in the wake. Note that there are a total of four gaps, two between the prisms and two between the walls and prisms. Two shear layers form in each gap, spawning two counter-rotating vortices, respectively. A total of eight vortices are thus shed from the shear layers in one period of nominal vortex shedding. The growth and shedding of the vortices from the prisms occurs in an inphase fashion. An amal gamation and amputation process of the shed vortices leads to the evolution of the eight vortices in two vortices. A single wake consisting of two rows of vortices thus forms for x* > 6. As marked in the snapshot in Fig. 3a, eight vortices (U1, U2, M1, M2, L1, L2, W1, and W2) are linked to the eight shear layers, respectively. With the time elapsed, vortices U1, M1 and L1 are first pinched off and amalgamated, forming U1 þ M1 þ L1 vortex (Fig. 3d). Vortex W1 lifting up approaches the combined vortex U1 þ M1 þ L1 (Fig. 3d–f). Eventually, the four same-sign positive vortices merge together, forming U1 þ M1 þ L1 þ W1 vortex. Similarly, four negative-sign vortices form one united vortex U2 þ M2 þ L2 þ W0 2 downstream (Fig. 3f). Interestingly, three same-sign vortices from the prisms come close to each other and then the same-sign vortex from the corresponding wall joins them. Eventually, a single vortex street con sisting of positive and negative vortices forms in a similar fashion to a Karman vortex street. However, while a Karman vortex street is gener ally characterized by positive vortices on the lower side (y* < 0) and negative vortices on the upper side (y* > 0), here the vortex arrange ment is opposite, i.e. negative and positive vortices appearing on the lower and upper sides of the symmetry line (y* ¼ 0), respectively. See the vortex arrangement at x* > 9.0. That is, the merging and reversion
4.2. Regime B: Twin vortex streets (1.1 < g* < 2.1) Twin vortex streets are observed for 1.1 < g* < 2.1. Fig. 5 shows the instantaneous flow structures. The vortices from the prisms’ gap stretch in the transverse direction; each getting pinched off at the middle forms two small vortices. For example, M2 vortex in Fig. 5a splits into M0 2 and M00 2 vortices (Fig. 5b and c). The same happens for the positive vortex from the upper prism, U2 vortex splitting into U0 2 and U00 2 (Fig. 5a–c).
Fig. 4. Time histories of (a) lift and (b) drag coefficients of three prisms and (c) power spectra of lift coefficients of three prisms. g* ¼ 0.8. The flow structures (a) (c) and (f) in Fig. 3 correspond with the positions marked by (i), (ii) and (iii), respectively. 5
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Fig. 5. Sequential instantaneous snapshots of vorticity contours for g* ¼ 1.5 (regime B, twinvortex-street flow). T is the vortex shedding period and tˊ is the time delay measured from instant (i) corresponding to the minimum lift of the middle prism. Red and blue colors represent positive and negative vorticities, respectively. The flow structures for (a), (c), and (e) corre spond to instants (i), (iii) and (v) are marked in the next figure. (For interpretation of the refer ences to color in this figure legend, the reader is referred to the Web version of this article.)
While split vortices M0 2 and M00 2 from the middle prism are almost equal to each other, split vortex U0 2 is much weaker than U00 2 that goes to the wall side and dies out (Fig. 5c). The split of the negative vortex (from the upper side) from the upper prism does not take place, neither does the positive vortex from the lower prism. Now the same-signed vortices U00 2, M00 2, and L2 come close to each other and form a combined vortex U00 2 þ
M00 2 þ L2 lying on the lower side of y* ¼ 0. The vortex M0 2 stays on the upper side and snatches vorticity from the wall, keeping alive (Fig. 5d). The evolution and interactions of the vortices reveal that a positive vortex on the lower vortex street (x* > 8) is composed of two split positive vortices (U00 2 þ M00 2) from the middle and upper prisms and one intact positive vortex (L2) from the lower prism (Fig. 5f). On the other
Fig. 6. Time histories of (a) lift and (b) drag coefficients of three prisms and (c) power spectra of lift coefficients of three prisms. g* ¼ 1.5. 6
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hand, a positive vortex on the upper vortex street consists of a split positive vortex from the middle prism along with vorticity from the upper wall. In a similar fashion, two rows of negative vortices come into being in the wake. Two vortex streets thus feature the wake. The appearance of vortices in each street is again in a fashion of reverse Karman vortex street. Time histories of lift and drag coefficients and their power spectra are presented in Fig. 6. Similar to regime A, the three prisms shed vortices at the same frequency, St ¼ 0.2792 (Fig. 6c). Unlike regime A, the second or third harmonic frequency in the power spectra is absent. This is because the complex interaction between the vortices largely occurs in x* < 6 for regime A and in x* ¼ 6–9 for regime B. The lift signals of the three prisms indicates that the phase lag between the vortex sheddings from the outer prisms is zero and that between the middle and outer prisms is 15.5� . Time-mean lift coefficient is positive (0.35) for prism 1 and negative ( 0.35) for prism 3, while zero for prism 2. This implies that the outer prisms undergo repulsive lift forces. Fluctuation in drag is smaller for the middle prism than for the outer prisms. The middle prism undergoes a larger time-mean drag than the outer prisms.
St ¼ 0:1258 þ ¼ St0 þ
0:2189 g*
(4)
0:2189 g*
where St0 ¼ 0.1258 is the Strouhal number of a single rectangular (b/h ¼ 3) prism in an unconfined domain, which was obtained from the present simulations. The difference between estimated data and Eq. (4) plotted in Fig. 9a is less than 3.2%. The C‾L ¼ 0 for the middle prism (Fig. 9b). For a particular g*, the outer prisms’ C‾L are equal in magnitude but opposite in signs. Inter estingly, the variations in C‾L of the two outer prisms explicitly char acterize the flow classifications made above. In regime A, the C‾L is negative and positive for the upper and lower prisms, respectively, indicating attractive lift forces. On the other hand, an opposite scenario prevails for regime B, positive and negative C‾L complementing the upper and lower prisms, respectively. The lift forces on the two outer prisms are now repulsive. In regime B, with increasing g*, the C‾L grows for the upper prism and declines for the lower prism, reaching maximum and minimum at the border between regimes B and C. A gentle decrease in C‾L magnitude with g* occurs for the outer prisms in regime C. The CʹL and C‾D variations depicts that both outer prisms exhibit the same value (Fig. 9c and d). In regimes A and B, the CʹL is greater for the middle prism than for the outer prisms. On the contrary, an outer prism experiences a higher CʹL than the middle prism in regime C. The CʹL can be represented by a polynomial function of g*. The variations in CʹL1 (prism 1) and CʹL2 (prism 2) with g* can respectively be expressed as
4.3. Regime C: Triple vortex streets (2.1 < g* � 3.4) Now the gap between the prisms and between walls and prism is large enough, allowing each prism to shed vortices like an isolated prism (Fig. 7). Triple vortex streets thus feature the wake. Like regimes A and B, the vortex shedding frequencies of three prisms are the same which can also be observed from the power spectra of lift signal of three prisms (Fig. 8c). Still, repulsive lift forces appear for the outer prisms while amplitudes of the lift forces of the three prisms are approximately the same (Fig. 8a). The outer prisms have the same time-mean drag coeffi cient and the same fluctuating drag. The value of time-mean drag is smaller for outer prisms than for the middle prism while the fluctuation in drag is greater for outer prisms than the middle prism (Fig. 8b).
C’L1 ¼ 0:05 ¼
C’L0
0:34α1 þ 0:88α21
C’L2 ¼ 0:05 ¼
C’L0
0:34α1 þ 0:88α21
(5a)
and
0:62α1 þ 1:33α21
(5b)
0:62α1 þ 1:33α21 ;
where α1 ¼ (g*) 0.53 and CʹL0 ¼ 0.05 is the fluctuating lift coefficient of a single rectangular prism (b/h ¼ 3.0) in an unconfined domain. Fig. 9c shows that Eq. (5) has a good fit with the computed data, with CʹL1 and CʹL2 both approaching the single prism value (0.05) as g* → ∞. The C‾D of the middle prism is always greater than that of the outer prisms. The difference in C‾D between the middle prism and outer prisms is large in regimes A and B while small in regime C. The C‾D1 (prism 1) and C‾D2 (prism 2) can be represented as a function of g* which are of the forms
5. Effect of g* on forces and Strouhal number Fig. 9 illustrates the dependence of St, C‾D, CʹL, and C‾L, on g*. At the top of the figure, the span of each flow regime is marked. The St of each prism was estimated from spectral analyses of the fluctuating lift force acting on the prism. As shown before, all the three prisms have identical St; the only one data set is therefore plotted in the figure. With increasing g*, the St declines from 0.39 at g* ¼ 0.8 to 0.18 at g* ¼ 3.4 (Fig. 9a). A curve fit equation of St as a function of g*, using the least square method, is obtained as
CD1 ¼ 1:28 þ 0:67β1 þ 5:17β21 ;
and
CD2 ¼ 1:28 þ 0:87β1 þ 5:77β21 ;
where β1 ¼ ðg* Þ
(6a) 0:88
;
(6b)
Fig. 7. Sequential instantaneous snapshots of vorticity contours for g* ¼ 2.8 (regime C, triple-vortex-street flow). T is the vortex shedding period and tˊ is the time delay measured from instant (i) corresponding to the maximum lift of the middle prism. Red and blue colors represent positive and negative vorticities, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 7
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Fig. 8. Time histories of (a) lift and (b) drag coefficients of three prisms and (c) power spectra of lift coefficients of three prisms. g* ¼ 2.8. The flow structures (a) and (b) in Fig. 7 correspond with the positions marked by (i) and (ii), respectively in lift and drag coefficients.
Fig. 9. Effect of g*on (a) Strouhal number, (b) time-mean lift coefficient, (c) fluctuating lift coefficient, and (d) time-mean drag coefficient.
6. Flow in the gap
respectively. Eqs. (6 a, b) are also presented in Fig. 9d, showing less than 0.67% (prisms 1, 3) and 1.59% (prism 2) errors between fit curves and computed values. Variation in C‾D exhibits the same nature as that in St. At a lower g*, the blockage ratio is large, enhancing C‾D and St. An in crease in g* yields a smaller blockage ratio, C‾D, CʹL and St, all thus decrease with g*. A summary of the flow features of different flow re gimes are summarized in Table 4.
As explained before, there are two gaps between the prisms and two
between the prisms and wall. The time-mean streamwise velocity (u* ) is computed at the center (x* ¼ 0, y* ¼ � (g*þ1)/2, �3(g*þ1)/2) of each gap. The values of u* were equal in the two gaps between the prisms, as were in the two gaps between the prisms and wall. Therefore, only two sets of data, one (u* w 1 ) in the gap between the wall and prism 1 and the 8
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Table 4 Summary of features in different flow regimes. The subscripts ‘1’ and ‘2’ refer to prisms 1 and 2, respectively. The symbols ‘↗ ‘and ‘↘’ denote ‘increase’ and ‘decrease’, respectively, with increasing g*. Regime
Vortex street
Wake
Forces
A; 0.8 � g* <1.1 B; 1.1 < g* < 2.1 C; 2.1 < g* � 3.4
Single vortex street
Reverse Karman wake
Twin vortex streets
Reverse Karman wake
Triple vortex streets
Karman wake
CʹL1 < CʹL2; C‾D1 < C‾D2 C‾L1 ¼ -ve, C‾L3 ¼ þve; | C‾L1| ¼ |C‾L3| ↘ CʹL1 < CʹL2; C‾D1 < C‾D2 C‾L1 ¼ þve, C‾L3 ¼ -ve; | C‾L1| ¼ |C‾L3| ↗ CʹL1 > CʹL2; C‾D1 < C‾D2 C‾L1 ¼ þve, C‾L3 ¼ -ve; | C‾L1| ¼ |C‾L3| ↘
Fig. 10. Variations in time-mean streamwise velocity at gap between wall and outer prism (circle symbol) and between two prisms (square symbol). The lines represent the curve fits.
other (u* 1 2 ) between the prisms 1 and 2, are presented in Fig. 10. Obviously,
u*
1 2>
u*
w 1,
the difference between them being small with
increasing g*. The reason behind u* 1 2 > u* w boundary layer next to it while u* 1
2
1
is that the wall develops a
gets high because of shear layer
developments by the prisms. Again, u* w 1 and u* 1 with g* and obey the following relations u* w
1
u* 1
2
¼ 1:0 þ 1:41ðg* Þ ¼ 1:0 þ 1:47ðg* Þ
0:94
0:94
:
and
2
< g*< 2.1) is characterized by two vortex streets. The appearance of vortices in each street is again in a fashion of reverse Karman vortex street. In regime C (2.1 < g* � 3.4), each prism forms a vortex street that resembles a single isolated prism Karman street. Three streets thus feature the wake. The dependence on g* of C‾L and CʹL explicitly distinguishes the flow regimes. In regime A, the C‾L of the upper and lower prisms are negative and positive, respectively, C‾L magnitudes of both prisms waning with g*. The two outer prisms undergo an attractive lift force. The signs of the C‾L get reversed in regimes B and C, positive for the upper prism and negative for the lower prism, leading to a repulsive lift force between the prisms. The C‾L magnitude grows in regime B but declines in regime C. The CʹL in regimes A and B is greater for the middle prism than the outer prisms while the scenario is opposite in the regime C. The C‾D and St both decreases with increasing g*. While all three prisms have identical St at a given g*, the outer prisms have smaller C‾D than the middle prism. The smaller C‾D is attributed to the smaller ve locity in the gaps between the outer prisms and wall than in the gaps between the prisms. This is because the former gaps are characterized by a boundary layer on the wall side and a shear layer on the prism side while the latter gaps are characterized by shear layers on both sides. The gap flow velocity rapidly declines in either gaps as g* increases, which explains why C‾D and St diminish with g*. Variations in St, C‾D, CʹL and
both rapidly decay (7a) (7b)
Eqs. (7a, b) presented as the lines fit well with our computed results (Fig. 10). Comparing Figs. 9(d) and 10, it is observed that both C‾D and
u* decrease in a similar fashion with the increase of g*. So, it is of interest to find out a relation between the C‾D and u* . It is expected that C‾D of a
prism will be dictated by u* on the two sides (gaps) of the prism. That is,
u* w 1 and u* 1 2 both are associated with prism 1, and u* 1 2 being the same on the two sides is associated with prism 2. An average velocity ζ1 ¼ (u* w 1 þu* 1 2 )/2 is thus considered for prism 1 and ζ2 ¼ (u* 1 2 þu* 1 2 )/2 for prism 2. The correlation between ζ1 and C‾D1 (prism 1) is found as C‾D1 ¼ 0.86 þ 0.44 ζ31, while that for prism 2 is C‾D2 ¼ 0.86 þ 0.46 ζ32. The above stated relations have a good fit with computed results with a maximum deviation of less than 4.32%. One expects ζ1 and ζ2 → 1.0 when g* → ∞; the corresponding C‾D1 ¼ 1.3 and C‾D2 ¼ 1.32 which have deviations of 1.56% and 3.12%, respectively, from C‾D1 ¼ C‾D2 ¼ 1.28 that is the value of C‾D of a prism (b/h ¼ 3.0) in an un
u* with g* are expressed in empirical relations.
bounded domain. Furthermore, Figs. 9(a) and 10 depict that St and u*
decline with g*. Therefore, a relation between St and u* is expected.
Author contribution statement
gaps, i.e. ζ ¼ (2 u* w 1 þ 2u* 1 2 )/4. The St and ζ satisfy the relation St ¼ 0.0445 þ 0.1612 ζ. It is observed that the curve fit has a maximum deviation of 2.55% with the computed St. Expecting ζ → 1.0 for g* → ∞, we obtain St ¼ 0.1167 which has a deviation 7.2% with St0 (¼ 0.1258).
Ramnarayan Mondal: computing, data analysis, figure preparation, and original draft preparation. Md Mahbub Alam: formulation or evolution of overarching research aims, supervision, funding, reviewing, and editing. Rajesh Bhatt: computing, and validation.
7. Conclusions
Declaration of competing interest
Numerical simulations are conducted to investigate the flow around three side-by-side rectangular prisms in the presence of two confining walls at a Re ¼ 100. The dependence on gap spacing ratio g* (¼ 0.8–3.4) of C‾D, C‾L, CʹL, St and flow structures are examined. Based on the vortex structures, fluid forces and shedding frequency, the flow structures are classified into three distinct regimes A (single vortex street), B (twin vortex streets) and C (triple vortex streets). In regime A (0.8 � g*<1.1), each prism spawns vortices alternately, as do the two side walls. In one period of vortex shedding, six vortices from the prisms and two vortices from the two side walls are shed. They undergo splitting, amputation and amalgamation processes, and even tually, a single vortex street (two rows of vortices) forms downstream in the fashion of reverse Karman vortex street. The wake in regime B (1.1
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Since all the prisms have the same St, we get the average of u* at four
Acknowledgment The authors wish to acknowledge the support given by the National Natural Science Foundation of China through Grants 11672096 and 91752112 and by Research Grant Council of Shenzhen Government through grant JCYJ20180306171921088.
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