Numerical Investigation of Laminar Unsteady Wakes Behind Two Inline Square Cylinders Confined in a Channel

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1 Engineering Applications of Computational Fluid Mechanics ISSN: (Print) X (Online) Journal homepage: Numerical Investigation of Laminar Unsteady Wakes Behind Two Inline Square Cylinders Confined in a Channel Pratish P. Patil & Shaligram Tiwari To cite this article: Pratish P. Patil & Shaligram Tiwari (2009) Numerical Investigation of Laminar Unsteady Wakes Behind Two Inline Square Cylinders Confined in a Channel, Engineering Applications of Computational Fluid Mechanics, 3:3, , DOI: / To link to this article: Copyright 2009 Taylor and Francis Group LLC Published online: 19 Nov Submit your article to this journal Article views: 323 View related articles Citing articles: 3 View citing articles Full Terms & Conditions of access and use can be found at

2 Engineering Applications of Computational Fluid Mechanics Vol. 3, No. 3, pp (2009) NUMERICAL INVESTIGATION OF LAMINAR UNSTEADY WAKES BEHIND TWO INLINE SQUARE CYLINDERS CONFINED IN A CHANNEL Pratish P. Patil and Shaligram Tiwari* Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai , India * shaligt@iitm.ac.in (Corresponding Author) ABSTRACT: Two-dimensional numerical investigations have been carried out to study the behavior of unsteady wake for flow past an inline arrangement of square cylinders confined in a channel. The present work aims to study the influence of the relative size and arrangement of the two cylinders on vortex shedding characteristics in their wakes. A computational code has been developed using finite difference approach based on modified MAC algorithm for solution of governing flow equations. Effect of size ratio of the two cylinders, the cylinder spacing in streamwise direction and flow Reynolds number on the wake flow characteristics of the two cylinders have been presented. The temporal characteristics of the wake are analyzed using time series analysis. Results indicate that for a given Reynolds number, there exist a range of values of size ratio and cylinder spacing for which vortex shedding can occur in the wake of the upstream cylinder. Keywords: unsteady wakes, inline square cylinders, cylinder spacing, size ratio, Strouhal number, channel confinement 1. INTRODUCTION In the situations involving where solid fluid interactions, flow induced vibrations (FIV) and vortex induced vibrations (VIV) require special emphasis in their analysis, mainly because the instabilities appearing in flow fields induce instabilities in the neighboring solid structures. Vortex shedding and wake dynamics for flow past bluff bodies is an incomplete puzzle to fluid mechanists themselves. The classical problem of flow past obstacles is addressed in numerous research areas such as building aerodynamics, automotive design, sports, wind engineering, ocean structures, to name a few. Zdravkovich (1997) presented a thorough and comprehensive review of past research and highlighted probable challenges to be addressed in the study of flow past a circular cylinder. Franke, Rodi and Schönung (1990) carried out two-dimensional numerical computations to study laminar vortex shedding past circular (Re 5000) and square (Re 300) cylinders using finite volume algorithm and third order accurate QUICK scheme. They were successful in obtaining exact computational results for low Reynolds number but at higher values of Reynolds number their computations suffered from turbulent stochastic fluctuations. The works by Ramamurthy and Lee (1973) who studied triangular cylinders and Courchesne and Laneville (1979) who studied rectangular cylinders at higher Reynolds number of the order of 10 5 are worth mentioning. Zdravkovich (1985) reported that propagation of instabilities in the flow field and pattern of vortex shedding strongly depend on the relative arrangement of a pair or larger number of bluff bodies. The most commonly seen configurations are tube bundles in tandem as well as staggered arrangement as seen in the applications like heat exchangers or reactors and they form the basic field of interest for many researchers in fluid mechanics. For example, Schneider and Farge (2005), Farrant, Tan and Price (2000), Lam, Gong and So (2007), Beale and Spalding (1999), to name a few. Yen et al. (2008) conducted experimental study over two identical square cylinders having inline arrangement in a water tank using PIV and studied the effect of Reynolds number and intercylinder spacing on vortex shedding pattern behind both the cylinders. Xu and Zhou (2004) experimentally measured the vortex shedding frequencies with the help of hot wire anemometry for a pair of circular cylinders in tandem arrangement in the range of Reynolds number from 800 to 42,000. They observed that the vortex shedding behind both the cylinders remain highly correlated and the variation of Strouhal number with the inter-cylinder spacing depends on the flow regime. For smaller intercylinder spacing, the shear layer from the Received: 23 Jul. 2008; Revised: 24 Jan. 2009; Accepted: 10 Feb

3 upstream cylinder rolls up behind in the wake while for larger spacing vortex shedding is associated with both cylinders. Zhou, Cheng and Hung (2005) studied numerically the effect of a control plate on fluid forces appearing on a square cylinder for Re = 250. Their study shows that the extent of suppression of lift and drag forces on the downstream square cylinder due to the upstream control plate depends on the height and position of the control plate. The present work focuses on studying the effect of an upstream square cylinder on the unsteady wake characteristics of downstream inline square cylinder for varying size ratio and inter-cylinder spacing. Computations are performed employing a developed code for low values of flow Reynolds number ranging from Re = 100 to 250 for size ratio in the range of SR = 0.6 to 1.4 and intercylinder spacing in the range ls =2.0d to 5.0d. 2. PROBLEM STATEMENT The two-dimensional computational domain consists of two inline square cylinders confined in a channel of length L 1 =16d and width L 2 =6d as shown in Fig. 1. The downstream square cylinder is of a side length d, which is also chosen as the characteristic length scale. The upstream square cylinder is of a side length d 1. The ratio d 1 / d is defined as size ratio (SR). The spacing between the two cylinders is varied from ls=2d to 5d for every value of SR and Re chosen. Fig. 1 Computational domain. The ratio of the side of downstream cylinder to the width of the channel, d/l 2, is defined as blockage ratio (BR). Since, in contrast to flow past bluff bodies placed in infinite medium, the presence of laterally confining walls in a channel restricts the disturbed flow sidewise and imposes an additional source of disturbance in the form of transverse pressure gradient. In order to avoid the effect of larger blockage on transition of the boundary layer and the enveloping shear layer, the blockage ratio in the present computations is kept fixed at GOVERNING EQUATIONS AND BOUNDARY CONDITIONS 3.1 Governing equations Fluid motions are governed by basic conservation equations of mass and momentum. The mass conservation equation for an incompressible flow reduces to continuity equation given by Eq. (1). ui x i = 0 (1) Two-dimensional non-dimensionalized Navier- Stokes equations for unsteady laminar and incompressible flow in weak conservative form are given as 370

4 2 u ( uu i j) i p 1 ui + = + t xj xi Re xj x where u i j is the velocity component along (2) coordinate direction and Re is the flow Reynolds number. For a 2-D coordinate system, Eq. (2) collectively represents the x and y-components of the momentum equation for values of x i as x and y and u i as u and v respectively. The flow Reynolds number (Re) is defined as umdh Re = (3) υ where um is the mean velocity at inlet, Dh is the hydraulic diameter (equals side of downstream square cylinder, d) and υ is the kinematic viscosity of the fluid. The characteristic length and velocity are side of downstream square cylinder (d), and mean velocity at the inlet ( u m ) respectively. The physical time has been nondimensionalized as t = d/u m. 3.2 Boundary conditions The boundary conditions at inlet, exit, on confining channel walls and the cylinder surfaces are summarized in Table 1. x i ranging from 100 to 250. In order to reduce the complexities and computational time by incorporating possible simplifications such as order of discretization, discretization schemes etc, a simple computational algorithm employing low order modeling has been developed. Galletti et al. (2004) concluded that for simple flow simulations as in laminar flow regimes, low-order modeling offers an effective compromise between complexity of the physical phenomena and computational affordability. 4.1 Solution methodology The solution algorithm is based on the modified MAC method of Hirt and Cook (1972). A computational code has been developed employing the finite difference approach for discretization of governing equations. The convective terms are discretized using a weighted average of second order upwind and spacecentered scheme (hybrid scheme). The discretization of a simple convective term appearing in the momentum equation for staggered arrangement of variables on computational grid (Fig. 2) is given below. Table 1 Summary of boundary conditions for the computational domain. Boundary Top and bottom walls of channel Channel inlet Channel exit Cylinders surfaces Imposed boundary conditions No slip: u=v=0; p/ y=0 Parabolic velocity profile: u m =1.0, v=0 Second order Neumann type boundary condition: 2 2 u v = = x x No slip: u=v=0, p/ n=0 n: normal to cylinder surface 4. NUMERICAL TECHNIQUE AND GRID INDEPENDENCE Computations are carried out for combinations of five different size ratios ranging from 0.6 to 1.4, four different values of cylinder spacings, ls =2d to 5d, and four values of Reynolds number Fig. 2 ( ) A computational cell and its neighboring cells with pressure and velocity defined in staggered arrangement. uu uu = uu x Δx where, i, j u R u = R r L l + u i, j i+ 1, j 2 and u L (4) = ui 1, j + ui, j

5 The upwinded velocities are given as follows: if u > 0, u = u, else u = u ; and if, else. R r i, j r i 1, j u > 0, u = u + L l i 1, j u = u l i, j Therefore, Rearranging the terms in above equation, On further simplification one gets ( uu) 1 u + u u u i, j i+ 1, j + i 1, j i, j = u x i, j Δx 2 2 ( uu) 1 ( )( = 4 i, j ( )( u i, j i 1, j ui, j+ ui+ 1, j ui, j+ ui+ 1, j+ ui, j ui+ 1, j x Δx ui 1, j+ ui, j ui 1, j+ ui, j+ ui 1, j ui, j ) ) ( uu) ( u + u )( u + u i, j i+ 1, j i, j i+ 1, j) + α ( u + u i, j i 1, j)( u u + i, j i+ 1, j) ( )( ) α ( u + u i 1, j i, j)( u u i 1, j i, j) 1 = x 4Δx u + u u + u i, j i 1, j i, j i 1, j i, j (5) where, α is the hybrid factor. The value α =1 gives a second order upwind scheme and α =0 gives a space-centered scheme. For the present computations, the hybrid factor has been chosen as α = 0.2. The diffusive terms and pressure gradient terms are discretized using central difference scheme. In the present algorithm point successive under-relaxation (SUR) technique is employed to solve the pressure Poisson equation using the following point iterative approach: p ( DIV ) ' o i, j = i, j 2δ t ω δx δ y 2 2 (6) where, δt, δx and δy are the temporal and spatial increments in X and Y-directions respectively, ' P i, j is the pressure correction, ( DIV ) i, j is the unbalanced divergence for the (i,j) th location and is the under-relaxation factor. In the present ω o computations, the under-relaxation factor of 0.2 is chosen. Even though in the original MAC algorithm, the pressure field is to be computed by solution of Poisson equation, Chorin (1967) showed that the pressure field and correct velocity field can be obtained by a simultaneous iteration procedure to obtain a divergence free flow field. Viecelli (1971) showed that the simultaneous iteration of pressure and velocity fields as developed by Chorin (1967) is equivalent to that of solving a pressure Poisson equation. 4.2 Grid independence study A simple two-dimensional Cartesian grid is generated with variables defined in staggered arrangement. The grid independence test is carried out using three different grid sizes, viz., , and , for flow past a single square cylinder (upstream cylinder being assumed to be absent, i.e. SR = 0.0) using the same computational code. Fig. 3 shows the variation of span-averaged non-dimensionalized pressure along length of the channel for the three grid sizes considered. The maximum deviation of span-averaged pressure at any location is within ± 1%. Also the Strouhal number for the three grid sizes is summarized in Table 2 and the maximum deviation is well within ± 0.6% with the mean value of

6 5. RESULTS AND DISCUSSION Fig. 3 Variation of span-averaged pressure along length of the channel compared for three different grid sizes, viz , and Table 2 Strouhal number for three different grid sizes. 5.1 Flow field behavior and validation of computation In order to understand the flow behavior for situation of flow past closely spaced bluff bodies (square cylinders in the present case), computations have been carried out for a domain size, L 1 =12d and L 2 =5d. Figures 4(a) and (b) present the contours of vorticity magnitude corresponding to instantaneous and time-averaged flow fields. The vorticity magnitude has been scaled as Ω Ω= Ω (7) max Grid size Δx Δy St St Cd Cd Fig. 4 Contour plots for non-dimensionalized vorticity magnitude: (a) instantaneous field (t=150) and (b) time-averaged field (Re=150, SR=0.6, ls=2.0d, L 1 =12, L 2 =5). 373

7 Fig. 5 Contour plots for non-dimensionalized pressure: (a) instantaneous field (t=150) and (b) time-averaged field (Re=150, SR=0.6, ls=2.0d, L 1 =12, L 2 =5). The prime observation from Fig. 4 is the entrapment of the vortices which are formed behind the upstream cylinder in the inter-cylinder spacing, which is encompassed between the shear layers separating from the upstream cylinder. The shear layers separating out from the upstream cylinder provide the medium for transport of vorticity from the wake of upstream cylinder to that of the downstream cylinder. The length and curvature of these shear layers are the governing parameters for both the stability of the shear layers and their capacity of vorticity transport. The rate of vorticity generation in the upstream wake and later its transport to the downstream wake decide the nature and periodicity of vortex shedding in the wake behind the downstream cylinder. The spacing between the two cylinders is responsible for the vorticity storage and also partially for controlling the voticity transport through the shear layers. Hence, the spacing between the cylinders (ls) can be identified as an important parameter that governs the flow field. The second parameter influencing the shear layer curvature and its strength is the size of the upstream square cylinder relative to that of the downstream cylinder. This parameter is identified as the size ratio (SR = d 1 /d). Free flow past an obstacle differs from flow past an obstacle in the presence of wall confinement. This is primarily due to the fact that the wall boundary behaves as an additional source of vorticity (Lighthill, 1963). As a matter of fact, the secondary vortical structures are observed near the channel walls in Figs. 4(a) and (b). Figures 5(a) and (b) present the contour plots of instantaneous and time-averaged pressure fields, which have been scaled as p p p = p p inlet inlet max (8) Such a scaling of pressure considers the pressure drop as a fraction of the local maximum of pressure drop that occurs in the channel. The flow approaches the forward stagnation point of the high pressure zone in front of the upstream cylinder and later accelerates in the narrow region between the side faces of the upstream cylinder and the channel walls causing a steep drop in the pressure as can be seen from Fig. 5(a). The 374

8 apparently symmetric pressure field in the wake behind the upstream cylinder in Fig. 5(a) is due to the absence of vortex shedding. However, the pressure field in the unsteady wake of the downstream cylinder shows asymmetry. Fig. 6 Dependence of Strouhal number upon Reynolds number compared for flow past a single square cylinder against results of Breuer et al. (2000), Galletti et al. (2004) and Mukhopadhyay, Biswas and Sundararajan (1992). Figure 6 presents the validation of computed results by comparing the Strouhal number in the wake of a single square cylinder. A comparison of computed Strouhal number for different values of Reynolds numbers against various results reported in literature by Breuer et al. (2000), Galletti et al. (2004) and Mukhopadhyay, Biswas and Sundararajan (1992) is made for a common blockage ratio of BR = The computational results of Breuer et al. (2000), Galletti et al. (2004) and Mukhopadhyay, Biswas and Sundararajan (1992) are distinctly different from each other. For low values of Reynolds number (Re = ), the Strouhal number in the present computations is slightly overestimated as compared to the results of Breuer et al. (2000) and Galletti et al. (2004). The difference diminishes for higher values of Re. This comparison shows that the computed results offer a close agreement with the benchmark results in literature. 5.2 Effect of size ratio (SR) Figures 7(a) (e) present the effect of SR on instantaneous flow field in the neighborhood of both cylinders for five different values of SR. All the streamline windows correspond to the flow field with fixed flow Reynolds number of Re = 150 and inter-cylinder spacing of ls =3.0d. It is observed that for smaller size ratios, i.e. SR 1.0, there exists periodic vortex shedding in the wake of the downstream cylinder. On the other hand, the wake of the upstream cylinder does not demonstrate any apparent vortex shedding. However, this does not mean that the twin vortices are stationary in temporal sense. In the present situation of flow past two inline square cylinders, the upstream cylinder is expected to generate vorticity in its wake alternately from its lateral surfaces. In due course of time, the vortex gets fed by the vorticity supplied from either of the lateral side and grows in size. The smaller inter-cylinder spacing restricts the detachment of the fully grown vortex from the upstream cylinder s rear face. Consequently, the vortex tends to store extra vorticity by growing in size lengthwise till it spreads over the entire available inter-cylinder spacing. This vortex is termed source vortex. The instant at which the source vortex is no longer capable to grow in size in order to accept the incessantly generated vorticity, it begins to feed the vortex adjoining it which is termed sink vortex. Further on, both the source and sink vortices demonstrate a balanced growth in size and shape and appear like Föppl s twin vortices (Föppl, 1913). Thereby, vorticity storage ensues in the inter-cylinder spacing. This explains the apparent flow symmetry in the wake behind the upstream cylinder. The vorticity transport from the sink vortex residing in the wake of the upstream cylinder to the wake of the downstream cylinder takes place through the neighboring shear layer separated from the upstream cylinder. The source and sink vortices continue to interchange among themselves. We infer that at any given instant only one shear layer in the neighborhood of the sink vortex acts as the vorticity transport medium between the upstream and downstream cylinder wakes and both shear layers do not function simultaneously. The vortex shedding behind the downstream cylinder vanishes for larger size ratios (SR > 1.0: Figs. 7d and 7e) mainly because the separated shear layers from the upstream cylinder entirely encompass the downstream cylinder in its wake and the downstream cylinder will no longer see any approaching flow. Secondly, these widely located shear layers are more stable and they eliminate the possibility of vortex shedding in the wake of the downstream cylinder. 375

9 Fig. 7 Streamlines for instantaneous field (t=250) showing wake flow behavior for Re=150, ls=3.0d and different size ratios: (a) SR=0.6, (b) SR=0.8, (c) SR=1.0, (d) SR=1.2 and (e) SR=1.4. Figures 8(a) (d) present the transverse velocity signal and the respective Fourier spectra for a fixed value of flow Reynolds number (Re = 200) and four different values of size ratio. The presence of a single dominant peak in the Fourier spectra of velocity signals for SR values from 0.6 to 1.0 confirms the occurrence of planar vortex shedding in the wake of the downstream cylinder without any transitions being detected. An increase in the size ratio results in a decrease in the amplitude of the velocity signal. Spectral power (S ω ) of the single dominant peak in power spectra for all the velocity signals continuously decreases with an increase in size ratio. The Strouhal frequency of vortex shedding corresponding to the peak location in the Fourier spectra shows a gradual decrease with increase in size ratio (SR = 0.6 to 1.0). However, for SR = 1.2, the absence of velocity signal is due to the absence of vortex shedding. Similar situation can be extrapolated for SR = 1.4, for which signal and spectra are not presented in Fig

10 Fig. 8 Time signal of transverse velocity (v) in the wake of second cylinder and its Fourier spectra for Re=150 and ls=3.0d; (a) SR=0.6, (b) SR=0.8, (c) SR=1.0 and (d) SR=1.2. Fig. 9 Variation of lift coefficient (C l ) behind the downstream cylinder for Re=150 and ls=3.0d: (a) SR=0.6, (b) SR=0.8, (c) SR=1.0 and (d) SR=

11 The signals for lift coefficients (C l ) corresponding to the four cases of Fig. 8 have been shown in Figs. 9(a) (d). The lift coefficient for either cylinder may be written in terms of dimensional as well as nondimensional pressure as C l = lower suface p dl dz 1/ 2ρ u upper surface 2 m p dl dz = lower suface p dl for unit thickness in the normal direction (z-direction). Here ρ is density of the fluid and u m is the mean velocity at inlet to the channel. The nature of variation of lift coefficients is qualitatively similar to the nature of velocity signals presented above. Figure 10 shows the variation of drag coefficients (C d1 and C d2 ) with SR on the upstream and downstream cylinders respectively for Re = 150 and ls =3.0d. The drag coefficient on either of the two cylinders in terms of pressure becomes C d front surface p dl dz rear surface 2 um p dl dz = = 1/2 ρ front surface p dl upper surface rear surface (9) pdl p dl (10) drag force on the downstream cylinder becomes negative and acts as a thrust force against the free stream. The downstream cylinder has a negative C d value which becomes more negative with an increasing size ratio because of increased pressure drop in the wake of the larger upstream cylinder. Fig. 10 Variation of drag coefficient of both cylinders with SR for Re=150 and ls=3.0d. The skin-friction drag is found negligible in magnitude compared to pressure drag for all the computational cases considered in the present study. An increasing value of SR results in an increase of drag on the upstream cylinder and a decrease on the downstream cylinder. Since the upstream cylinder experiences positive pressure difference across it, C d for the upstream cylinder is positive. With an increasing SR, the differential pressure across the upstream cylinder increases, resulting in an increase of positive C d for the upstream cylinder. Zdravkovich (1987) identified that the flow past multiple bluff bodies in cross flow is associated with mystifying paradoxes. He observed that for the inline arrangement of two cylindrical bodies which are closely spaced, the 5.3 Effect of cylinder spacing (ls) Figures 11(a) (d) present the effect of ls on instantaneous flow field behavior and streamline structure in the neighborhood of both inline square cylinders for four different values of ls. All the streamline windows correspond to the flow field with a fixed flow Reynolds number of Re = 150 and fixed size ratio, i.e. SR = 1.0. The flow behavior in the vicinity of both square cylinders demonstrates very distinct nature for different values of inter-cylinder spacing (ls). Fig. 11(a) shows that for very small value of inter-cylinder spacing, ls =2.0d, vortex shedding is completely absent in the wake of both cylinders, as they nearly imitate the behavior of a single but less bluff body having much higher critical Reynolds number corresponding to onset of vortex shedding. For higher values of ls =3.0d and 4.0d, vortex shedding is observed in the wake of the downstream cylinder as can be seen from Figs. 11(b) and (c). On the other hand, even for larger value of ls =5.0d, wakes behind both a cylinders exhibit periodic vortex shedding. It can also be noticed that the half-a-wavelength of the vortex street in the wake of the upstream cylinder 378

12 is ~ 2d 1, which is similar to the wavelength of the vortex street in the wake of single square cylinder. This is because the bubble size of the shed vortex is always anticipated to be of the size of the characteristic dimension of the bluff obstacle causing it. The wavelength is also dependent on the Reynolds number and thereby the Strouhal number (St). The fluid physics behind the sudden change in flow structure for ls =4.0d to 5.0d can be explained as follows. For the present computational domain, for the formation of the vortex street with at least two distinct and oppositely oriented vortices, the available space in the wake should be a little more than one wavelength of a regular vortex street (i.e. > 4d 1 ). Hence in Fig. 11(c), for ls =4.0d, not enough space is available for the formation of a one wavelength of the vortex street and so vortex shedding is absent in the wake of the upstream cylinder. Extending the same reasoning, it can also be deduced that the mode of vortex shedding in the wakes of both cylinders will also depend strongly on the inter-cylinder spacing. In other words, the spacing between the two cylinders influences the nature of synchronous or asynchronous vortex shedding in the wakes of both inline cylinders. Consequently, ls becomes one of the most crucial parameter governing the flow behavior in the vicinity of bluff bodies having inline arrangement. Fig. 11 Streamlines for instantaneous field (t=250) showing wake flow behavior for Re=150, SR=1.0 and different cylinder spacings; (a) ls=2.0d, (b) ls=3.0d, (c) ls=4.0d and (d) ls=5.0d. 379

13 Fig. 12 Time signal of transverse velocity (v) in the wake of second cylinder and its Fourier spectra for Re=150 and SR=1.0; (a) ls=2.0d, (b) ls=3.0d, (c) ls=4.0d and (d) ls=5.0d. Fig. 13 Variation of drag coefficient of both cylinders with ls for Re=150 and SR=1.0. Figures 12(a) (d) present the transverse velocity signal of a wake element and their respective Fourier spectra for a fixed value of flow Reynolds number (Re = 150) and four different values of cylinder spacings (ls). An increase in the cylinder spacing results in an increase in the amplitude of the velocity signal. The onset of vortex shedding takes place in the wake for ls =3.0d or greater. For ls =4.0d, the Strouhal frequency is slightly greater than for ls =3.0d. With further increase in spacing between the cylinders, the Strouhal frequency starts decreasing as seen from Figs. 12(c) and 12(d). This is primarily due to the fact that at larger separation, no more vorticity storage takes place within the inter-cylinder spacing, but rather the rear surfaces of both cylinders participate in shedding the vortices. Since the width of the wake bubble increases as apparent from the amplitude of the velocity signal in Fig. 12(d), the frequency of shedding in the wake of the downstream cylinder decreases. The dependence of drag coefficients (C d1 and C d2 ) on the cylinder spacing (ls) for upstream and downstream cylinders is shown in Fig. 13 for Re = 150 and SR = 1.0. The results indicate that the upstream cylinder drag coefficient remains nearly constant with increase of separation between the upstream and downstream cylinders. This is expected because the increase in intercylinder spacing (ls) should not have significant upstream effects on the flow field structure in the vicinity of the upstream cylinder. On the other hand, for the downstream cylinder, the drag coefficient shows a noticeable increase with increase in cylinder spacing. However, the downstream C d does not become positive till ls 4d. Moreover, for the downstream cylinder, the negative value of C d decreases with increasing ls, i.e. the forward thrust force on the downstream cylinder decreases with increase in ls. This is 380

14 because an increase in ls results in a decrease in the pressure drop in the wake of the upstream cylinder, i.e. the pressure in the wake of the upstream cylinder increases. This reduces the negative pressure differential across the downstream cylinder and thereby the upstream thrust acting on it. The reduced pressure drop in the wake of the upstream cylinder with increase in ls is due to the spread of vortex more than what is required to hold a natural amount of vorticity. Hence, the vorticity stored in the space between the two cylinders decreases with increase in ls. It is important to note that for ls 4d, the value of C d does not become positive in magnitude. For ls =5.0d, the C d of the downstream cylinder shows a steep rise to a positive value which becomes higher than C d of the upstream cylinder. This is because for ls =5.0d, there is no more vorticity storage in the region of inter-cylinder spacing and the upstream cylinder wake demonstrates the regular periodic vortex shedding. Consequently, the downstream cylinder experiences the flow stagnation on its front surface, which results in an increased pressure on the front side of the downstream cylinder. On the other hand, there is less pressure region faced by the rear surface of the downstream cylinder due to the usual wake. This causes a large drag force in the streamwise direction. Hence, the C d value of the downstream cylinder for the situation of ls =5.0d becomes highly positive. This indicates that for given cylinder sizes and ls >5.0d, both the cylinders will get exposed to an approach flow and would bear a positive value of drag coefficient. Fig. 14 Time signal of transverse velocity (v) in the wake of second cylinder and its Fourier spectra for SR=1.0 and ls=3.0d; (a) Re=100, (b) Re=150, (c) Re=200 and (d) Re= Effect of flow Reynolds number (Re) Figures 14(a) (d) present the transverse velocity signals and respective Fourier spectra of a representative fluid particle located in the wake of the downstream cylinder for ls =3.0d, SR =1.0 and four different values of Reynolds numbers, viz. Re = 100, 150, 200 and 250. It is observed that with an increase in Reynolds number, the amplitude as well as frequency of the velocity signal increases. The Fourier spectra clearly indicate the presence of a single dominant frequency which increases with an increase of Reynolds number. It is known from literature (Zdravkovich, 1997) that an increase in Reynolds 381

15 number causes an increase in the Strouhal frequency of a single bluff body and also for a row of bluff bodies. However, the relationship depends upon the geometrical configuration of the bluff bodies in the cross flow. Figure 15 shows the variation of drag coefficients (C d1 and C d2 ) of the upstream and downstream cylinders with respect to Reynolds number (Re) for SR = 1.0 and ls =3.0d. With an increase in Re, the drag force in streamwise direction increases due to an increase in pressure drop in the wake of the upstream cylinder. On the other hand, with increase in Re, the value of C d for the downstream cylinder decreases and becomes more negative. This is again due to the above-mentioned increase in pressure drop in the wake of the upstream cylinder, resulting in an increase of negative pressure differential across the downstream cylinder. Fig. 15 Variation of drag coefficient of both cylinders with Re for ls=3.0d and SR=1.0. Fig. 16 Temporal evolution of wake over one vortex shedding period behind second cylinder for SR=0.6, Re=150 and ls=2.0d; (a) t=150, (b) t= , (c) t=151.31, (d) t= , (e) t= and (f) t= (time period of vortex shedding, T=3.275). 5.5 Temporal characteristics of the unsteady wake Figures 16(a) (f) present the temporal evolution of the coupled unsteady wake of both cylinders for ls =2.0d, Re = 200 and SR = 0.6 over one time period of vortex shedding behind the downstream cylinder. Fig. 16(f) corresponds to an elapsed time of one vortex shedding period after reappearance of the same vortical flow structure shown in Fig. 16(a). The main objective of the figure is to demonstrate the behavior of the unsteady confined vortical loop between the two cylinders and also to show the periodic nature of the wake behind the downstream cylinder. Figure 17(a) presents the variation of spanaveraged pressure along the length of the channel for fixed values of Reynolds number (Re = 150) and cylinder spacing (ls =3.0d) and five different values of size ratios (SR = 0.6, 0.8, 1.0, 1.2 and 1.4). Even though the location of the forward stagnation point of the upstream cylinder changes due to the change in size ratio, the maximum peak 382

16 pressure attained is same for all the cases. The drop in pressure continues till the rear face of the downstream cylinder for smaller value of size ratio, i.e. SR = 0.6. For aspect ratio of SR =0.8, pressure almost remains constant in the narrow region between the channel walls and the downstream cylinder and then begins to recover. For larger values of SR, i.e. SR = 1.0, 1.2 and 1.4, no drop in pressure is observed in the blockage region of the downstream cylinder; instead it starts recovering from the core location of the vortex bubble in the wake of the upstream cylinder. The span-averaged pressure variation along the length of the channel for different values of the cylinder spacing (Re = 150 and SR = 1.0) has been shown in Fig. 17(b). Apparently, with increase in ls, up to ls =4d, the recovery of pressure begins from a more downstream location. However, for ls =5d, the nature of pressure variation is seen to be much different. The pressure recovery starts from the wake of the upstream cylinder and then further drop in pressure takes place when the flow encounters the downstream cylinder. This is in perfect agreement with arguments presented before because for ls >4d, the separated shear layers from the lateral surfaces of the upstream cylinder wrap in its wake and they separate again after the downstream cylinder. Fig. 18 Variation of Strouhal numbers (St 1 and St 2 ): (a) with size ratio (SR) for Re=150 and ls=2.0d, (b) with cylinder spacing (ls) for SR=1.0 and Re=150 and (c) with Reynolds number (Re) for SR=1.0 and ls=3.0d. Fig. 17 Variation of span-averaged pressure along length of the channel with (a) different size ratios (SR) for Re=150 and ls=3.0d and (b) different values of cylinder spacing (ls) for Re=150 and SR=1.0. Fig. 18(a) presents the variation of Strouhal number with size ratio for different values of Reynolds number. For ls =3.0d and a given Reynolds number (Re = 150), no vortex shedding occurs in the wake of the upstream cylinder while the Strouhal number in the wake of the downstream cylinder decreases with increase in SR. An increase in value of SR results in the attachment of boundary layer over longer distance on the lateral surface of the downstream cylinder causing increase in the length of vortex bubble in its wake. The strained shear layer imparts more stability to the unsteady wake of the downstream cylinder, resulting in the decrease of vortex shedding frequency. Fig. 18(b) presents the dependence of wake Strouhal frequency on 383

17 cylinder spacing (ls) for SR =1.0 and Re = 150. Since the onset of vortex shedding behind the upstream cylinder occurs only when ls 5d, no value of St 1 exists for ls <5d. The dependence of wake frequency upon Reynolds number is commonly known in the form of St-Re relationships. In the wake of a single square cylinder (SR = 0.0), the Strouhal number initially increases and then decreases with increase in Reynolds number (Davis, Moore and Purtell, 1984). However, the Strouhal number in the wake of the downstream cylinder in the inline arrangement of two square cylinders shown in Fig. 18(c) increases continuously with increase in Reynolds number over the considered range of Reynolds number. This is mainly due to the large value of SR considered (SR = 1.0). Such an increase is apparent from the velocity signals and their spectra shown in Fig CONCLUSIONS Two-dimensional numerical computations have been carried out using a developed computational code to study the flow field characteristics of the coupled unsteady wake for flow past an inline arrangement of square cylinders confined in a channel. Effects of relative cylinder sizes, cylinder spacing and flow Reynolds number have been considered for the flow structure and temporal characteristics of the combined wake behind both cylinders. It is observed that the vortex shedding behind the downstream cylinder vanishes for larger size ratios (SR > 1.0). The drag coefficient for the upstream cylinder remains positive and increases with increase in SR. On the other hand, the drag coefficient for the downstream cylinder becomes more negative with an increasing value of SR. The effect of increase in the cylinder spacing manifests itself first in an increase and then a decrease of wake Strouhal frequency in the wake of the downstream cylinder. For cylinder spacing less than 2d (ls <2d), no vortex shedding is observed behind both cylinders. With an increase in ls from 3d to 4d, the influence of coupled shear layers results in an increase Strouhal number. For ls =5d, independent vortex shedding takes place in both wakes and the Strouhal number for the downstream cylinder wake again decreases. The drag coefficient for the upstream cylinder remains almost constant with increase in cylinder spacing. However, the drag coefficient for the downstream cylinder increases slowly for 2d < ls <4d and then drastically for ls =5d. The detailed mechanism for the above observations has been presented in the text. NOMENCLATURE BR blockage ratio (= d/l 2 ) C d drag coefficient C l lift coefficient D h hydraulic diameter DIV unbalanced divergence in a cell d side of downstream square cylinder (characteristic dimension for length scale) d 1 side of upstream square cylinder f frequency of vortex shedding L 1 length of computational domain L 2 width of computational domain ls inter-cylinder spacing p pressure correction p non-dimensionalized pressure p pressure p inlet inlet pressure p max maximum pressure Re Reynolds number SR size ratio (ratio of sides of upstream and downstream square cylinders) St Strouhal number S ω spectral power density t non-dimensional time U av span-averaged streamwise velocity u velocity component in x-direction or flow direction v velocity component in y-direction or cross-stream direction u velocity component in i th coordinate i direction u mean velocity at the inflow section X Y m non-dimensional streamwise co-ordinate non-dimensional cross-stream co-ordinate Greek symbols α hybrid factor ω spectral frequency of a time-series ω o under-relaxation factor Ω non-dimensionalized vorticity magnitude Ω max maximum vorticity magnitude φ transport property υ kinematic viscosity Δ x grid size in streamwise direction Δ y grid size in transverse direction 384

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