Potential Flow Interactions among Multiple Immersed Bodies Using Body-Conforming Grid Generation and a Multi-Point Constraint Framework for Stream Function Determination

Potential Flow Interactions among Multiple Immersed Bodies Using Body-Conforming Grid Generation and a Multi-Point Constraint Framework for Stream Function Determination | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Short Report Potential Flow Interactions among Multiple Immersed Bodies Using Body-Conforming Grid Generation and a Multi-Point Constraint Framework for Stream Function Determination Anil Lal S, Mannu Yadav This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8425258/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract An efficient and robust methodology for the rapid generation of high-quality, body-conforming grids in two-dimensional domains containing multiple immersed solid bodies is employed to generate the mesh and to investigate potential flow interactions in a \(\:{90}^{\circ\:}\) flow turning configuration. The proposed approach significantly reduces the setup time for CFD analysis while maintaining grid quality in the vicinity of complex geometries. A systematic parametric study is carried out by varying the number of bodies, their sizes, and their locations upstream of the exit section in order to assess the resulting uniformity of the exit flow. In addition, the constant stream function values associated with the immersed bodies are computed using a multi-point constraint formulation implemented within a conjugate gradient method for symmetric systems, thereby clarifying the influence of potential interaction effects on the inter-body flow distribution. Mechanical Engineering Body Conforming Grids Coefficient of Variation Potential Flow Interactions Multi-point Constraints Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Introduction Flows involving multiple immersed bodies in confined channels arise in a wide range of engineering applications, including compact heat exchangers, diffuser and duct systems, flow distributors, turbine inlet passages, microfluidic devices, and experimental wind tunnel or water tunnel test sections. In such configurations, the interaction between neighboring bodies and the confining walls fundamentally alters the flow field compared to isolated body scenarios (Zdravkovich 1977 ). Even under the assumptions of incompressible and irrotational flow, the combined effects of geometric confinement, blockage, and mutual interference give rise to complex streamline patterns, non-uniform velocity distributions, and strong redistribution of mass flux within the channel (Zdravkovich 1977 ; Fox, McDonald, and Pritchard 1998 ; White 2011 ). Classical potential flow theory has long provided a useful analytical framework for understanding idealized body flow interactions. Early works by Lamb and Milne-Thomson established the mathematical foundations for inviscid, incompressible flows around solid boundaries using complex-variable techniques (Lamb 1932 ; Milne-Thomson 1968 ). Batchelor further clarified the physical interpretation of potential flow solutions and their relevance as limiting cases of high Reynolds number viscous flows (Batchelor 1967 ). These studies demonstrated that, although viscosity is neglected, potential flow models capture essential global features of the flow field, such as streamline topology, stagnation points, and pressure distribution trends. When multiple bodies are present, the flow domain becomes multiply connected, introducing additional mathematical and physical complexity (Lamb 1932 ; Milne-Thomson 1968 ). In such domains, the velocity potential and stream function are not uniquely determined unless appropriate global constraints are imposed. In particular, each immersed body is associated with an unknown constant value of the stream function, which governs how the incoming flow is partitioned among the inter-body gaps. The determination of these constants is crucial for obtaining physically meaningful solutions, as arbitrary or inconsistent values can lead to unrealistic circulation patterns and distorted inter-body flow distributions. The presence of channel walls further intensifies inter-body interactions. Confinement effects amplify velocity gradients, alter effective blockage ratios, and introduce strong wall-induced image influences that modify the flow field throughout the domain (Lamb 1932 ; Milne-Thomson 1968 ; Batchelor 1967 ; Katz and Plotkin 2001 ). Experimental and numerical studies have shown that, in confined channels, relatively small changes in body spacing or placement can produce disproportionately large changes in gap velocities and wake characteristics. This sensitivity has been documented for tandem, side-by-side, and staggered arrangements of cylinders and other bluff bodies (Zdravkovich 1977 ; Milne-Thomson 1968 ). Beyond fundamental fluid mechanics interest, the uniformity of the exit flow from a channel or duct is a key performance criterion in many practical systems. Non-uniform velocity profiles can degrade downstream performance, increase pressure losses, promote flow-induced vibrations, and lead to uneven mechanical loading, while in aerodynamic and experimental facilities, they compromise measurement accuracy and repeatability (Batchelor 1967 ; Fox, McDonald, and Pritchard 1998 ; White 2011 ). In channels containing internal bodies, such exit flow non-uniformity often arises as a direct consequence of inter-body interaction effects, including preferential flow through wider gaps, acceleration in narrow passages, and asymmetric streamline deflection induced by body placement. Even in the absence of viscous effects, potential flow solutions reveal pronounced non-uniformity when the flow partitioning among inter-body regions is not properly balanced, underscoring the relevance of potential flow analysis as a diagnostic and design tool for understanding and mitigating non-uniformity at an early stage (Milne-Thomson 1968 ; Katz and Plotkin 2001 ). From a computational standpoint, the simulation of multi-body flows in confined domains poses significant challenges related to grid generation and the consistent enforcement of global constraints. Traditional body-fitted grid generation for complex multi-body configurations is often time-consuming, particularly for repeated parametric studies, as observed in earlier three-dimensional mixed convection simulations involving heated bodies (Ajith and Lal 2021). Although unstructured meshes offer geometric flexibility, they may lead to numerical diffusion or poor element quality near closely spaced bodies, motivating the use of fast and robust body conforming grid generation strategies. For potential flow analysis, finite element and boundary element formulations based on Laplace equations provide computational efficiency and numerical robustness, specially when implemented with matrix-free or iterative solvers. In multiply connected domains, the determination of constant stream function values for immersed bodies is critical. This can be achieved in a mathematically consistent manner using multi-point constraint techniques that preserve system symmetry and enable efficient conjugate gradient solutions. As the present study does not explicitly detail the grid generation methodology or the implementation of the multi-point constraint formulation, a detailed exposition of these aspects may be found in our recent arXiv preprint (S and Yadav 2025 ). In this work, an integrated computational framework developed by the authors in (S and Yadav 2025 ) is used to investigate potential-flow interactions among multiple immersed cylinders with the objective of improving exit-flow uniformity in a channel downstream of a \(\:{90}^{\circ\:}\) bend. The framework combines a rapid body-conforming grid-generation strategy with a matrix-free finite element formulation for solving the governing Laplace equations, thereby enabling efficient and scalable simulations for complex multi-body configurations. A mathematically consistent multi-point constraint approach, embedded within a conjugate gradient solver for symmetric systems, is employed to determine the constant stream function values associated with each immersed body, allowing accurate representation of inter-body interaction effects. The methodology is applied to a \(\:{90}^{\circ\:}\) channel flow containing multiple internal cylinders, and the influence of body number, size, and placement on flow redistribution and downstream uniformity is systematically analyzed using grid characteristics, body stream-function values, and quantitative exit flow uniformity metrics. METHODOLOGY The methodology involves the numerical solution of the Laplace equations governing the stream function \(\:\psi\:\) and the velocity potential \(\:\varphi\:\) using a mixed (triangular and quadrilateral) finite element discretization (Donea and Huerta 2003 ; Zienkiewicz and Taylor 2000 ; Lal and Jabir 2010). The grid construction strategy is specifically designed for flow configurations involving immersed solids. A simple Cartesian background grid is first employed to enclose the computational domain containing the immersed objects. The points at which the background grid intersects the solid boundaries are interpreted as boundary displacements of neighboring grid points located inside the solids. These displacement components are smoothly propagated to the surrounding grid using two Laplacian smoothing operators, one for each displacement component. This procedure results in a displacement superimposed Cartesian grid that accurately conforms to curved solid boundaries. A representative grid generated for a square domain containing three immersed cylindrical solids is shown in Fig. 1 . The mesh predominantly consists of regular rectangular or square cells, with a limited number of triangular elements appearing along the cylinder surfaces. These triangular elements arise locally where the Cartesian background grid intersects the curved body boundaries. FLOW CONFIGURATIONS The computational domain considered in this study is a unit square \(\:ABCD\) . As shown in Fig. 1 , the edge \(\:CE\) , of length \(\:0.25\hspace{0.25em}m\) , serves as the flow inlet, while the edge \(\:AD\) acts as the outlet. The remaining edges, namely \(\:AB\) , \(\:BC\) , and \(\:DE\) , are treated as impermeable solid walls. Figure 1 also illustrates one of the representative cases examined in this study, showing three circular immersed solid bodies together with the computational grid generated from a background \(\:100\times\:100\) uniform mesh. Computations are carried out for a base case without any immersed cylinders, as well as for several configurations involving three and six cylinders placed at different locations along the \(\:y\) -axis. The configuration without any cylinders is referred to as the base case. The boundary conditions are prescribed as follows. The stream function is set to \(\:\psi\:=0\) on the edge \(\:DE\) , and \(\:\psi\:=1.0\hspace{0.25em}{m}^{2}{s}^{-1}\) on the edges \(\:AB\) and \(\:BC\) . This specification ensures a total volumetric flow rate per unit width of \(\:1\hspace{0.25em}{m}^{2}{s}^{-1}\) within the enclosure. The same flow rate is maintained for all configurations considered; consequently, the expected average velocity at the outlet \(\:AD\) is \(\:1\hspace{0.25em}m\hspace{0.17em}{s}^{-1}\) . On the surfaces of the immersed bodies, the stream function is constrained to assume a constant value, which is determined as part of the solution procedure. For the velocity potential, the potential \(\:\varphi\:\) is prescribed as \(\:1\) and \(\:0\) on the inlet ( \(\:CE\) ) and outlet ( \(\:AD\) ) sections, respectively. On all solid boundaries, including the immersed body surfaces, the natural boundary condition is enforced, which effectively results in a zero normal gradient of \(\:\varphi\:\) . RESULTS AND DISCUSSION The cases investigated aim to interpret the influence of immersed solid bodies on a \(\:{90}^{\circ\:}\) turning flow, with particular emphasis on their role in redistributing the flow relative to the base configuration without any immersed cylinders. This section presents and discusses the results for the base configuration, the three-cylinder configurations, and the six-cylinder configurations. Base configuration The base configuration case corresponds to flow through a \(\:{90}^{\circ\:}\) turning channel bounded by three flat solid walls, namely \(\:AB\) , \(\:BC\) , and \(\:DE\) , with a geometric corner and an associated stagnation point at \(\:E\) . Figure 2 presents the computed flow net for this configuration. The flow entering through the inlet edge \(\:CE\) undergoes a turning motion within the domain due to the presence of the bounding walls. A significant portion of the flow is deflected downward and exits through the outlet edge \(\:DE\) , while the remaining flow initially moves toward the wall \(\:AB\) before turning and proceeding downstream. The differing flow path lengths associated with these deflections result in a non-uniform velocity distribution at the outlet. The coefficient of variation, defined as the ratio of the standard deviation to the average value for the exit velocity, is computed as \(\:14.42\%\) . The objective of the present work is to identify arrangements of immersed cylinders that promote a nearly uniform flow distribution at the exit, as quantified by a reduced coefficient of variation. In this context, two sets of configurations, consisting of three- and six-cylinders, respectively, are investigated. 3-Cylinder configurations Computations are performed to study the interaction of three cylindrical blocks placed in the base flow configuration at two different vertical locations, namely at heights of \(\:{h}_{1}=0.25\hspace{0.25em}m\) and \(\:{h}_{2}=0.5\hspace{0.25em}m\) measured from the bottom wall. The diameter of each cylinder is \(\:0.2\hspace{0.25em}m\) , and the gap between the cylinders is \(\:0.15\hspace{0.25em}m\) . Figures 3 and 4 show the computed flow nets corresponding to the heights of cylinder rows, \(\:{h}_{1}\) and \(\:{h}_{2}\) , respectively. It is observed that the presence of the cylinders displaces the streamlines toward the left. For the configuration with cylinders placed at the lower level, the streamlines exhibit non-uniform spacing that persists up to the exit. In contrast, when the cylinders are positioned at the mid-height, the streamlines become more uniformly spaced, indicating a smoothing of the flow over the relatively longer downstream distance available before the exit. Figure 5 presents the variation of the velocity component along the vertical direction at the outlet for the two three-cylinder arrangements, together with the base case. The exit velocity profile corresponding to the lower-level cylinder configuration exhibits noticeable fluctuations, whereas the configuration with cylinders placed at the higher- level results in a marginal smoothing of the velocity profile. This improvement is reflected in a small reduction in the coefficient of velocity variation at the exit. The coefficient of velocity variation for the configuration with cylinders placed at the lower level is \(\:13.64\%\) , whereas that for the configuration with cylinders placed at the higher level is \(\:13.41\%\) . Table 1 lists the constant stream-function values associated with the cylinder surfaces for the two arrangements. These values indicate a redistribution of the flow, with a reduced portion of the total flow passing through the right side of the channel for the configuration in which the cylinders are placed at the lower level. Table 1 Constant stream-function values of the three immersed cylinders arrangement for two vertical placements. Cylinder Lower level \(\:{h}_{1}=0.25\hspace{0.25em}m\) Higher level \(\:{h}_{2}=0.50\hspace{0.25em}m\) Left 0.1702 0.2171 Middle 0.5647 0.6246 Right 0.8862 0.9048 6-Cylinder configurations Computations are performed to study the interaction of six cylindrical blocks placed in the base flow configuration at three different vertical locations, at heights \(\:{h}_{1}=0.25\hspace{0.25em}m\) , \(\:{h}_{3}=0.375\hspace{0.25em}m\) , and \(\:{h}_{2}=0.5\hspace{0.25em}m\) measured from the bottom wall. The diameter of each cylinder is \(\:0.1\hspace{0.25em}m\) , and with the gap between the cylinders \(\:\delta\:=0.075\hspace{0.25em}m\) . Figures 6 , 7 , and 8 show the computed flow nets corresponding to the above cylinder row heights. Owing to the relatively smaller of half-size of the cylinder compared to the three-cylinder configuration, the disturbances of the streamlines are less, causing an effective compression of the streamlines towards the low velocity region in the left. It can be seen that the stream lines progressively become uniformly spaced with the lowering of the distance from the bottom wall. The constant body stream function values obtained for the six-cylinder arrangement are given in Table 2 . The distribution provides direct insight into the redistribution of flow caused by inter-body interference at different vertical locations. The difference in stream function values between adjacent bodies represents the volumetric flow passing through the corresponding sub-channels; hence, variations in the absolute values and their gradients across the row of cylinders reflect changes in local flow distribution. Table 2 Constant stream-function values of the immersed cylinders for three vertical placements of the six-cylinder system. The value of \(\:h\) is in meters. Cyl-No \(\:{h}_{1}=0.25\) \(\:{h}_{3}=0.375\) \(\:{h}_{2}=0.50\) 1 0.0912 0.1003 0.1176 2 0.2262 0.2470 0.2825 3 0.3550 0.3825 0.4253 4 0.4737 0.5032 0.5458 5 0.5807 0.6089 0.6471 6 0.6737 0.6983 0.7302 At the lowest placement ( \(\:{h}_{1}=0.25\hspace{0.25em}m\) ), the body stream value associated with cylinder 1 is the smallest among all configurations. This indicates a reduction in the flow passing on the right side of the first cylinder, which can be attributed to the proximity of the cylinder row to the bounding wall. The wall-induced constraint strengthens the interaction between the wall and the upstream cylinders, thereby diverting a portion of the flow toward the interior of the channel. Simultaneously, the comparatively lower body stream value for cylinder 6 implies an increase in the flow on its left side, indicating enhanced flow recovery toward the downstream side of the row. The combined effect results in a more balanced redistribution of flow across the entire width of the channel. As the cylinder row is raised to the intermediate height ( \(\:{h}_{3}=0.375\hspace{0.25em}m\) ), the influence of the wall weakens, and the inter-cylinder interference becomes more symmetric. While the overall trend of increasing body stream values from cylinder 1 to cylinder 6 is preserved, the differences between successive values increase slightly. At the highest placement ( \(\:{h}_{2}=0.5\hspace{0.25em}m\) ), the body stream values for both cylinder 1 and cylinder 6 increase noticeably. This indicates that the flow tends to remain biased toward the outer portions of the channel, with reduced influence from wall-induced redirection. In this configuration, the potential flow interaction among the cylinders is dominated primarily by mutual interference rather than by confinement effects, resulting in comparatively larger variations in the flow passing between successive cylinders. Figure 9 presents the variation of the velocity component along the vertical direction at the outlet for the six-cylinder configuration with the cylinder row placed at three different vertical locations, \(\:{h}_{1}=0.25\hspace{0.25em}m\) , \(\:{h}_{3}=0.375\hspace{0.25em}m\) , and \(\:{h}_{2}=0.5\hspace{0.25em}m\) , together with the corresponding base (empty channel) case. In comparison to the empty channel, which exhibits a relatively large coefficient of velocity variation of \(\:14.42\%\) , the presence of the six-cylinder row leads to a substantial reduction in exit flow non-uniformity for all three placements. Among the three configurations, the lowest placement ( \(\:{h}_{1}=0.25\hspace{0.25em}m\) ) produces a noticeably smoother exit velocity profile than the empty channel, with the coefficient of velocity variation reduced to \(\:7.51\%\) . Although this configuration is characterized by a relatively higher flow on the left side of cylinder 6 and a reduced flow on the right side of cylinder 1, the exit velocity distribution remains significantly more uniform than the base case. When the cylinder set is placed at the intermediate height ( \(\:{h}_{3}=0.375\hspace{0.25em}m\) ), a further marginal smoothing of the exit velocity profile is observed, particularly in the middle region of the outlet. This reduction in velocity fluctuations leads to the minimum coefficient of velocity variation among the three cases, with a value of \(\:7.08\%\) . The dip in velocity variation in the central region compensates for the residual asymmetry near the side boundaries, thereby yielding the most uniform quantitative flow distribution at the exit. In contrast, raising the cylinder row to the highest level ( \(\:{h}_{2}=0.5\hspace{0.25em}m\) ) results in a partial loss of this uniformity, with the coefficient of velocity variation increasing to \(\:9.04\%\) . This trend indicates a weaker influence of the cylinders on redistributing the flow across the channel width as their distance from the bounding wall increases. Overall, the six-cylinder configuration demonstrates a marked improvement in exit-flow uniformity compared to the empty channel. While all three placements are effective in reducing velocity non-uniformity, the intermediate height \(\:{h}_{3}=0.375\hspace{0.25em}m\) provides the optimum balance between wall interaction and inter-cylinder interference, resulting in the minimum coefficient of velocity variation at the outlet. CONCLUSION An efficient computational framework for analyzing potential-flow interactions among multiple immersed bodies has been presented. The methodology combines a simple body-conforming grid-generation strategy based on a Cartesian background mesh with mixed triangular and quadrilateral finite elements for the Laplace equations governing the stream function and velocity potential. A multi-point constraint approach is employed to consistently determine the constant stream-function values associated with multiple immersed bodies. The framework has been applied to a \(\:{90}^{\circ\:}\) turning channel to investigate the effect of immersed cylindrical bodies on flow redistribution and exit flow uniformity. For the base configuration without immersed bodies, the exit flow exhibits significant non-uniformity, with a coefficient of velocity variation of \(\:14.42\%\) . The introduction of immersed cylinders leads to a systematic reduction in exit-flow non-uniformity by redistributing the streamlines within the channel. For the three-cylinder configuration, a modest improvement in exit-flow uniformity is observed when the cylinders are placed at a higher vertical location, consistent with the increased downstream development length available for flow smoothing. In contrast, the six-cylinder configuration of half diameter compared to the three-cylinder configuration produces a substantial improvement in exit flow uniformity for all placements considered. Among these, the intermediate placement at \(\:{h}_{3}=0.375\hspace{0.25em}m\) yields the minimum coefficient of velocity variation ( \(\:7.08\%\) ), resulting from a favorable balance between wall-induced effects and inter-cylinder potential-flow interference. Overall, the results demonstrate that both the number of immersed bodies and their vertical placement play a critical role in controlling flow redistribution in confined turning flows. The proposed methodology provides a robust and efficient tool for parametric studies aimed at identifying optimal body arrangements for enhancing flow uniformity in practical engineering applications. References Ajith KS, Anil S, Lal (2021) Effects of Prandtl Number on Three-Dimensional Coherent Structures in the Wake Behind a Heated Cylinder. J Appl Fluid Mech 14:515–526 Batchelor GK (1967) An Introduction to Fluid Dynamics. Cambridge University Press Donea J, Huerta A (2003) Finite Element Methods for Flow Problems. Wiley Fox RW, McDonald AT, Pritchard PJ (1998) Introduction to Fluid Mechanics, 7th edn. Wiley Katz J, Plotkin A (2001) Low-Speed Aerodynamics, 2nd edn. Cambridge University Press Lal S, Anil, Jabir E (2010) A Hybrid Finite Element–Finite Volume Method for Incompressible Flow Through Complex Geometries Using Mixed Grids. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 224 (1): 23–41 Lamb H (1932) Hydrodynamics, 6th edn. Cambridge University Press Milne-Thomson LM (1968) Theoretical Hydrodynamics, 5th edn. Macmillan Yadav M (2025) A Computational Approach for Multi-Body Potential-Flow Interaction Effects Using Matrix-Free FEM and Body-Conforming Grids. https://arxiv.org/abs/2512.12232 White FM (2011) Fluid Mechanics, 7th edn. McGraw-Hill Zdravkovich MM (1977) Review of Flow Interference Between Two Circular Cylinders in Various Arrangements. J Fluids Eng 99:618–633 Zienkiewicz OC, Taylor RL (2000) The Finite Element Method, Vol. 3: Fluid Dynamics . 5th ed. Butterworth–Heinemann Additional Declarations The authors declare no competing interests. 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08:37:05","extension":"png","order_by":20,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":21207,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/ff137bd87e9592e69e5bca20.png"},{"id":98929455,"identity":"6e6f7ccd-0434-42cc-a3da-9908c547c1b1","added_by":"auto","created_at":"2025-12-24 08:37:06","extension":"xml","order_by":21,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":55947,"visible":true,"origin":"","legend":"","description":"","filename":"rs84252580structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/db05eab1f3dc432e8e824855.xml"},{"id":98929457,"identity":"824ff146-63f9-4047-b216-5b0f597d571b","added_by":"auto","created_at":"2025-12-24 08:37:06","extension":"html","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":65706,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/204538fbfcb9ff905ba25dc1.html"},{"id":99310858,"identity":"7fef16dc-0ba6-44c0-b1b5-23f9ce0eba53","added_by":"auto","created_at":"2025-12-31 16:13:28","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":278528,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic of the computational domain and corresponding grid for a 90° bend flow containing three immersed cylinders.\u003c/p\u003e","description":"","filename":"Picture1.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/933822bf571c3c11ac559f92.png"},{"id":99310257,"identity":"1ae70dea-d213-4978-bf38-80952850d9a8","added_by":"auto","created_at":"2025-12-31 16:12:22","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":305138,"visible":true,"origin":"","legend":"\u003cp\u003eFlow net for the base configuration.\u003c/p\u003e","description":"","filename":"Picture2.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/bf8ec3e5b9bbf1c4a2bbe675.png"},{"id":99310039,"identity":"4bb8a75e-c7a0-4e8e-a1a1-6c66d333c9a4","added_by":"auto","created_at":"2025-12-31 16:11:43","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":303168,"visible":true,"origin":"","legend":"\u003cp\u003e: Flow net for the three-cylinder configuration with cylinders placed at 0.2m from the bottom edge of the domain.\u003c/p\u003e","description":"","filename":"Picture3.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/f58c7f149b91fd616488dd43.png"},{"id":98929431,"identity":"701f0a4d-2650-495c-ba99-1bc1a195abb9","added_by":"auto","created_at":"2025-12-24 08:37:05","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":308186,"visible":true,"origin":"","legend":"\u003cp\u003eFlow net for the three-cylinder configuration with cylinders placed at 0.5m from the bottom edge of the domain.\u003c/p\u003e","description":"","filename":"Picture4.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/9d1a2a83a30f28ded5a5590d.png"},{"id":99310380,"identity":"bb1e72a8-a59f-4004-b2b9-4c42e9cc11c5","added_by":"auto","created_at":"2025-12-31 16:12:41","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":61095,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of the exit velocity for the base configuration and the two configurations of the three-cylinder arrangement.\u003c/p\u003e","description":"","filename":"Picture5.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/32a6c28170ae66fa261f87f8.png"},{"id":99310894,"identity":"c3ecdb90-8bc6-4ee9-a37d-3e4de6b66417","added_by":"auto","created_at":"2025-12-31 16:13:31","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":1784523,"visible":true,"origin":"","legend":"\u003cp\u003eFlow net for the six-cylinder configuration with cylinders placed at 0.25 m from the bottom edge of the domain.\u003c/p\u003e","description":"","filename":"Picture6.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/4f45ef771418a539a1cf5197.png"},{"id":98929448,"identity":"df4788db-c7a9-4a26-ba76-c7445e736f34","added_by":"auto","created_at":"2025-12-24 08:37:05","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":1715966,"visible":true,"origin":"","legend":"\u003cp\u003eFlow net for the six-cylinder configuration with cylinders placed at 0.375m from the bottom edge of the domain.\u003c/p\u003e","description":"","filename":"Picture7.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/eaf0ae715c2c4281b97e211d.png"},{"id":98929450,"identity":"c4e4288a-bee6-4fd2-bd6f-5cc10b1bee81","added_by":"auto","created_at":"2025-12-24 08:37:05","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":306053,"visible":true,"origin":"","legend":"\u003cp\u003eFlow net for the six-cylinder configuration with cylinders placed at 0.5m from the bottom edge of the domain.\u003c/p\u003e","description":"","filename":"Picture8.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/5319f100cb641700cb59d31d.png"},{"id":98929441,"identity":"d2c0955c-555e-415d-9846-2d6b5328b997","added_by":"auto","created_at":"2025-12-24 08:37:05","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":67777,"visible":true,"origin":"","legend":"\u003cp\u003eVariation of the exit velocity for the base configuration and the three configurations of the six-cylinder arrangement.\u003c/p\u003e","description":"","filename":"Picture9.png","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/4c734b150426db8bec2e879a.png"},{"id":99787918,"identity":"c2a9a627-146f-4a82-8083-d38823447396","added_by":"auto","created_at":"2026-01-08 12:41:27","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":5807767,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8425258/v1/84916809-7f6b-4c30-aad1-e4be153d8fb0.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003ePotential Flow Interactions among Multiple Immersed Bodies Using Body-Conforming Grid Generation and a Multi-Point Constraint Framework for Stream Function Determination\u003c/p\u003e","fulltext":[{"header":"Introduction","content":"\u003cp\u003eFlows involving multiple immersed bodies in confined channels arise in a wide range of engineering applications, including compact heat exchangers, diffuser and duct systems, flow distributors, turbine inlet passages, microfluidic devices, and experimental wind tunnel or water tunnel test sections. In such configurations, the interaction between neighboring bodies and the confining walls fundamentally alters the flow field compared to isolated body scenarios (Zdravkovich \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1977\u003c/span\u003e). Even under the assumptions of incompressible and irrotational flow, the combined effects of geometric confinement, blockage, and mutual interference give rise to complex streamline patterns, non-uniform velocity distributions, and strong redistribution of mass flux within the channel (Zdravkovich \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1977\u003c/span\u003e; Fox, McDonald, and Pritchard \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1998\u003c/span\u003e; White \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2011\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eClassical potential flow theory has long provided a useful analytical framework for understanding idealized body flow interactions. Early works by Lamb and Milne-Thomson established the mathematical foundations for inviscid, incompressible flows around solid boundaries using complex-variable techniques (Lamb \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e1932\u003c/span\u003e; Milne-Thomson \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1968\u003c/span\u003e). Batchelor further clarified the physical interpretation of potential flow solutions and their relevance as limiting cases of high Reynolds number viscous flows (Batchelor \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1967\u003c/span\u003e). These studies demonstrated that, although viscosity is neglected, potential flow models capture essential global features of the flow field, such as streamline topology, stagnation points, and pressure distribution trends.\u003c/p\u003e \u003cp\u003eWhen multiple bodies are present, the flow domain becomes multiply connected, introducing additional mathematical and physical complexity (Lamb \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e1932\u003c/span\u003e; Milne-Thomson \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1968\u003c/span\u003e). In such domains, the velocity potential and stream function are not uniquely determined unless appropriate global constraints are imposed. In particular, each immersed body is associated with an unknown constant value of the stream function, which governs how the incoming flow is partitioned among the inter-body gaps. The determination of these constants is crucial for obtaining physically meaningful solutions, as arbitrary or inconsistent values can lead to unrealistic circulation patterns and distorted inter-body flow distributions.\u003c/p\u003e \u003cp\u003eThe presence of channel walls further intensifies inter-body interactions. Confinement effects amplify velocity gradients, alter effective blockage ratios, and introduce strong wall-induced image influences that modify the flow field throughout the domain (Lamb \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e1932\u003c/span\u003e; Milne-Thomson \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1968\u003c/span\u003e; Batchelor \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1967\u003c/span\u003e; Katz and Plotkin \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). Experimental and numerical studies have shown that, in confined channels, relatively small changes in body spacing or placement can produce disproportionately large changes in gap velocities and wake characteristics. This sensitivity has been documented for tandem, side-by-side, and staggered arrangements of cylinders and other bluff bodies (Zdravkovich \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e1977\u003c/span\u003e; Milne-Thomson \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1968\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eBeyond fundamental fluid mechanics interest, the uniformity of the exit flow from a channel or duct is a key performance criterion in many practical systems. Non-uniform velocity profiles can degrade downstream performance, increase pressure losses, promote flow-induced vibrations, and lead to uneven mechanical loading, while in aerodynamic and experimental facilities, they compromise measurement accuracy and repeatability (Batchelor \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e1967\u003c/span\u003e; Fox, McDonald, and Pritchard \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1998\u003c/span\u003e; White \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). In channels containing internal bodies, such exit flow non-uniformity often arises as a direct consequence of inter-body interaction effects, including preferential flow through wider gaps, acceleration in narrow passages, and asymmetric streamline deflection induced by body placement. Even in the absence of viscous effects, potential flow solutions reveal pronounced non-uniformity when the flow partitioning among inter-body regions is not properly balanced, underscoring the relevance of potential flow analysis as a diagnostic and design tool for understanding and mitigating non-uniformity at an early stage (Milne-Thomson \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1968\u003c/span\u003e; Katz and Plotkin \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2001\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eFrom a computational standpoint, the simulation of multi-body flows in confined domains poses significant challenges related to grid generation and the consistent enforcement of global constraints. Traditional body-fitted grid generation for complex multi-body configurations is often time-consuming, particularly for repeated parametric studies, as observed in earlier three-dimensional mixed convection simulations involving heated bodies (Ajith and Lal 2021). Although unstructured meshes offer geometric flexibility, they may lead to numerical diffusion or poor element quality near closely spaced bodies, motivating the use of fast and robust body conforming grid generation strategies. For potential flow analysis, finite element and boundary element formulations based on Laplace equations provide computational efficiency and numerical robustness, specially when implemented with matrix-free or iterative solvers. In multiply connected domains, the determination of constant stream function values for immersed bodies is critical. This can be achieved in a mathematically consistent manner using multi-point constraint techniques that preserve system symmetry and enable efficient conjugate gradient solutions. As the present study does not explicitly detail the grid generation methodology or the implementation of the multi-point constraint formulation, a detailed exposition of these aspects may be found in our recent arXiv preprint (S and Yadav \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2025\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eIn this work, an integrated computational framework developed by the authors in (S and Yadav \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2025\u003c/span\u003e) is used to investigate potential-flow interactions among multiple immersed cylinders with the objective of improving exit-flow uniformity in a channel downstream of a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{90}^{\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e bend. The framework combines a rapid body-conforming grid-generation strategy with a matrix-free finite element formulation for solving the governing Laplace equations, thereby enabling efficient and scalable simulations for complex multi-body configurations. A mathematically consistent multi-point constraint approach, embedded within a conjugate gradient solver for symmetric systems, is employed to determine the constant stream function values associated with each immersed body, allowing accurate representation of inter-body interaction effects. The methodology is applied to a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{90}^{\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e channel flow containing multiple internal cylinders, and the influence of body number, size, and placement on flow redistribution and downstream uniformity is systematically analyzed using grid characteristics, body stream-function values, and quantitative exit flow uniformity metrics.\u003c/p\u003e"},{"header":"METHODOLOGY","content":"\u003cp\u003eThe methodology involves the numerical solution of the Laplace equations governing the stream function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\psi\\:\\)\u003c/span\u003e\u003c/span\u003e and the velocity potential \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varphi\\:\\)\u003c/span\u003e\u003c/span\u003e using a mixed (triangular and quadrilateral) finite element discretization (Donea and Huerta \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2003\u003c/span\u003e; Zienkiewicz and Taylor \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Lal and Jabir 2010). The grid construction strategy is specifically designed for flow configurations involving immersed solids. A simple Cartesian background grid is first employed to enclose the computational domain containing the immersed objects. The points at which the background grid intersects the solid boundaries are interpreted as boundary displacements of neighboring grid points located inside the solids. These displacement components are smoothly propagated to the surrounding grid using two Laplacian smoothing operators, one for each displacement component. This procedure results in a displacement superimposed Cartesian grid that accurately conforms to curved solid boundaries.\u003c/p\u003e \u003cp\u003eA representative grid generated for a square domain containing three immersed cylindrical solids is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The mesh predominantly consists of regular rectangular or square cells, with a limited number of triangular elements appearing along the cylinder surfaces. These triangular elements arise locally where the Cartesian background grid intersects the curved body boundaries.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eFLOW CONFIGURATIONS\u003c/h2\u003e \u003cp\u003eThe computational domain considered in this study is a unit square \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:ABCD\\)\u003c/span\u003e\u003c/span\u003e. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the edge \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:CE\\)\u003c/span\u003e\u003c/span\u003e, of length \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.25\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e, serves as the flow inlet, while the edge \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:AD\\)\u003c/span\u003e\u003c/span\u003e acts as the outlet. The remaining edges, namely \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:AB\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:BC\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:DE\\)\u003c/span\u003e\u003c/span\u003e, are treated as impermeable solid walls. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e also illustrates one of the representative cases examined in this study, showing three circular immersed solid bodies together with the computational grid generated from a background \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:100\\times\\:100\\)\u003c/span\u003e\u003c/span\u003e uniform mesh.\u003c/p\u003e \u003cp\u003eComputations are carried out for a base case without any immersed cylinders, as well as for several configurations involving three and six cylinders placed at different locations along the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:y\\)\u003c/span\u003e\u003c/span\u003e-axis. The configuration without any cylinders is referred to as the base case.\u003c/p\u003e \u003cp\u003eThe boundary conditions are prescribed as follows. The stream function is set to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\psi\\:=0\\)\u003c/span\u003e\u003c/span\u003e on the edge \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:DE\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\psi\\:=1.0\\hspace{0.25em}{m}^{2}{s}^{-1}\\)\u003c/span\u003e\u003c/span\u003e on the edges \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:AB\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:BC\\)\u003c/span\u003e\u003c/span\u003e. This specification ensures a total volumetric flow rate per unit width of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:1\\hspace{0.25em}{m}^{2}{s}^{-1}\\)\u003c/span\u003e\u003c/span\u003e within the enclosure. The same flow rate is maintained for all configurations considered; consequently, the expected average velocity at the outlet \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:AD\\)\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:1\\hspace{0.25em}m\\hspace{0.17em}{s}^{-1}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eOn the surfaces of the immersed bodies, the stream function is constrained to assume a constant value, which is determined as part of the solution procedure. For the velocity potential, the potential \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varphi\\:\\)\u003c/span\u003e\u003c/span\u003e is prescribed as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:1\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0\\)\u003c/span\u003e\u003c/span\u003e on the inlet (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:CE\\)\u003c/span\u003e\u003c/span\u003e) and outlet (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:AD\\)\u003c/span\u003e\u003c/span\u003e) sections, respectively. On all solid boundaries, including the immersed body surfaces, the natural boundary condition is enforced, which effectively results in a zero normal gradient of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varphi\\:\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e"},{"header":"RESULTS AND DISCUSSION","content":"\u003cp\u003eThe cases investigated aim to interpret the influence of immersed solid bodies on a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{90}^{\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e turning flow, with particular emphasis on their role in redistributing the flow relative to the base configuration without any immersed cylinders. This section presents and discusses the results for the base configuration, the three-cylinder configurations, and the six-cylinder configurations.\u003c/p\u003e\n\u003ch3\u003eBase configuration\u003c/h3\u003e\n\u003cp\u003eThe base configuration case corresponds to flow through a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{90}^{\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e turning channel bounded by three flat solid walls, namely \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:AB\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:BC\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:DE\\)\u003c/span\u003e\u003c/span\u003e, with a geometric corner and an associated stagnation point at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:E\\)\u003c/span\u003e\u003c/span\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the computed flow net for this configuration. The flow entering through the inlet edge \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:CE\\)\u003c/span\u003e\u003c/span\u003e undergoes a turning motion within the domain due to the presence of the bounding walls. A significant portion of the flow is deflected downward and exits through the outlet edge \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:DE\\)\u003c/span\u003e\u003c/span\u003e, while the remaining flow initially moves toward the wall \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:AB\\)\u003c/span\u003e\u003c/span\u003e before turning and proceeding downstream. The differing flow path lengths associated with these deflections result in a non-uniform velocity distribution at the outlet. The coefficient of variation, defined as the ratio of the standard deviation to the average value for the exit velocity, is computed as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:14.42\\%\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe objective of the present work is to identify arrangements of immersed cylinders that promote a nearly uniform flow distribution at the exit, as quantified by a reduced coefficient of variation. In this context, two sets of configurations, consisting of three- and six-cylinders, respectively, are investigated.\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003e3-Cylinder configurations\u003c/b\u003e \u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eComputations are performed to study the interaction of three cylindrical blocks placed in the base flow configuration at two different vertical locations, namely at heights of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{1}=0.25\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{2}=0.5\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e measured from the bottom wall. The diameter of each cylinder is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.2\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e, and the gap between the cylinders is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.15\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e. Figures\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e show the computed flow nets corresponding to the heights of cylinder rows, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{1}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{2}\\)\u003c/span\u003e\u003c/span\u003e, respectively.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIt is observed that the presence of the cylinders displaces the streamlines toward the left. For the configuration with cylinders placed at the lower level, the streamlines exhibit non-uniform spacing that persists up to the exit. In contrast, when the cylinders are positioned at the mid-height, the streamlines become more uniformly spaced, indicating a smoothing of the flow over the relatively longer downstream distance available before the exit.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e presents the variation of the velocity component along the vertical direction at the outlet for the two three-cylinder arrangements, together with the base case. The exit velocity profile corresponding to the lower-level cylinder configuration exhibits noticeable fluctuations, whereas the configuration with cylinders placed at the higher- level results in a marginal smoothing of the velocity profile. This improvement is reflected in a small reduction in the coefficient of velocity variation at the exit. The coefficient of velocity variation for the configuration with cylinders placed at the lower level is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:13.64\\%\\)\u003c/span\u003e\u003c/span\u003e, whereas that for the configuration with cylinders placed at the higher level is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:13.41\\%\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e lists the constant stream-function values associated with the cylinder surfaces for the two arrangements. These values indicate a redistribution of the flow, with a reduced portion of the total flow passing through the right side of the channel for the configuration in which the cylinders are placed at the lower level.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eConstant stream-function values of the three immersed cylinders arrangement for two vertical placements.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCylinder\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLower level\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{1}=0.25\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eHigher level\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{2}=0.50\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLeft\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.1702\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2171\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMiddle\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.5647\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.6246\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eRight\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8862\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.9048\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003e6-Cylinder configurations\u003c/b\u003e \u003c/p\u003e \u003cp\u003eComputations are performed to study the interaction of six cylindrical blocks placed in the base flow configuration at three different vertical locations, at heights \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{1}=0.25\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{3}=0.375\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{2}=0.5\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e measured from the bottom wall. The diameter of each cylinder is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.1\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e, and with the gap between the cylinders \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\delta\\:=0.075\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e. Figures\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e, \u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e show the computed flow nets corresponding to the above cylinder row heights. Owing to the relatively smaller of half-size of the cylinder compared to the three-cylinder configuration, the disturbances of the streamlines are less, causing an effective compression of the streamlines towards the low velocity region in the left. It can be seen that the stream lines progressively become uniformly spaced with the lowering of the distance from the bottom wall.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe constant body stream function values obtained for the six-cylinder arrangement are given in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The distribution provides direct insight into the redistribution of flow caused by inter-body interference at different vertical locations. The difference in stream function values between adjacent bodies represents the volumetric flow passing through the corresponding sub-channels; hence, variations in the absolute values and their gradients across the row of cylinders reflect changes in local flow distribution.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eConstant stream-function values of the immersed cylinders for three vertical placements of the six-cylinder system. The value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:h\\)\u003c/span\u003e\u003c/span\u003e is in meters.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCyl-No\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{1}=0.25\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{3}=0.375\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{2}=0.50\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.0912\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.1003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.1176\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.2262\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.2470\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.2825\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.3550\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.3825\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.4253\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.4737\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5032\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.5458\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.5807\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.6089\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.6471\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.6737\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.6983\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.7302\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eAt the lowest placement (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{1}=0.25\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e), the body stream value associated with cylinder 1 is the smallest among all configurations. This indicates a reduction in the flow passing on the right side of the first cylinder, which can be attributed to the proximity of the cylinder row to the bounding wall. The wall-induced constraint strengthens the interaction between the wall and the upstream cylinders, thereby diverting a portion of the flow toward the interior of the channel. Simultaneously, the comparatively lower body stream value for cylinder 6 implies an increase in the flow on its left side, indicating enhanced flow recovery toward the downstream side of the row. The combined effect results in a more balanced redistribution of flow across the entire width of the channel.\u003c/p\u003e \u003cp\u003eAs the cylinder row is raised to the intermediate height (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{3}=0.375\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e), the influence of the wall weakens, and the inter-cylinder interference becomes more symmetric. While the overall trend of increasing body stream values from cylinder 1 to cylinder 6 is preserved, the differences between successive values increase slightly.\u003c/p\u003e \u003cp\u003eAt the highest placement (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{2}=0.5\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e), the body stream values for both cylinder 1 and cylinder 6 increase noticeably. This indicates that the flow tends to remain biased toward the outer portions of the channel, with reduced influence from wall-induced redirection. In this configuration, the potential flow interaction among the cylinders is dominated primarily by mutual interference rather than by confinement effects, resulting in comparatively larger variations in the flow passing between successive cylinders.\u003c/p\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e presents the variation of the velocity component along the vertical direction at the outlet for the six-cylinder configuration with the cylinder row placed at three different vertical locations, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{1}=0.25\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{3}=0.375\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{2}=0.5\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e, together with the corresponding base (empty channel) case. In comparison to the empty channel, which exhibits a relatively large coefficient of velocity variation of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:14.42\\%\\)\u003c/span\u003e\u003c/span\u003e, the presence of the six-cylinder row leads to a substantial reduction in exit flow non-uniformity for all three placements.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAmong the three configurations, the lowest placement (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{1}=0.25\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e) produces a noticeably smoother exit velocity profile than the empty channel, with the coefficient of velocity variation reduced to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:7.51\\%\\)\u003c/span\u003e\u003c/span\u003e. Although this configuration is characterized by a relatively higher flow on the left side of cylinder 6 and a reduced flow on the right side of cylinder 1, the exit velocity distribution remains significantly more uniform than the base case.\u003c/p\u003e \u003cp\u003eWhen the cylinder set is placed at the intermediate height (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{3}=0.375\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e), a further marginal smoothing of the exit velocity profile is observed, particularly in the middle region of the outlet. This reduction in velocity fluctuations leads to the minimum coefficient of velocity variation among the three cases, with a value of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:7.08\\%\\)\u003c/span\u003e\u003c/span\u003e. The dip in velocity variation in the central region compensates for the residual asymmetry near the side boundaries, thereby yielding the most uniform quantitative flow distribution at the exit.\u003c/p\u003e \u003cp\u003eIn contrast, raising the cylinder row to the highest level (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{2}=0.5\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e) results in a partial loss of this uniformity, with the coefficient of velocity variation increasing to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:9.04\\%\\)\u003c/span\u003e\u003c/span\u003e. This trend indicates a weaker influence of the cylinders on redistributing the flow across the channel width as their distance from the bounding wall increases.\u003c/p\u003e \u003cp\u003eOverall, the six-cylinder configuration demonstrates a marked improvement in exit-flow uniformity compared to the empty channel. While all three placements are effective in reducing velocity non-uniformity, the intermediate height \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{3}=0.375\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e provides the optimum balance between wall interaction and inter-cylinder interference, resulting in the minimum coefficient of velocity variation at the outlet.\u003c/p\u003e"},{"header":"CONCLUSION","content":"\u003cp\u003eAn efficient computational framework for analyzing potential-flow interactions among multiple immersed bodies has been presented. The methodology combines a simple body-conforming grid-generation strategy based on a Cartesian background mesh with mixed triangular and quadrilateral finite elements for the Laplace equations governing the stream function and velocity potential. A multi-point constraint approach is employed to consistently determine the constant stream-function values associated with multiple immersed bodies.\u003c/p\u003e \u003cp\u003eThe framework has been applied to a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{90}^{\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e turning channel to investigate the effect of immersed cylindrical bodies on flow redistribution and exit flow uniformity. For the base configuration without immersed bodies, the exit flow exhibits significant non-uniformity, with a coefficient of velocity variation of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:14.42\\%\\)\u003c/span\u003e\u003c/span\u003e. The introduction of immersed cylinders leads to a systematic reduction in exit-flow non-uniformity by redistributing the streamlines within the channel.\u003c/p\u003e \u003cp\u003eFor the three-cylinder configuration, a modest improvement in exit-flow uniformity is observed when the cylinders are placed at a higher vertical location, consistent with the increased downstream development length available for flow smoothing. In contrast, the six-cylinder configuration of half diameter compared to the three-cylinder configuration produces a substantial improvement in exit flow uniformity for all placements considered. Among these, the intermediate placement at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{h}_{3}=0.375\\hspace{0.25em}m\\)\u003c/span\u003e\u003c/span\u003e yields the minimum coefficient of velocity variation (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:7.08\\%\\)\u003c/span\u003e\u003c/span\u003e), resulting from a favorable balance between wall-induced effects and inter-cylinder potential-flow interference.\u003c/p\u003e \u003cp\u003eOverall, the results demonstrate that both the number of immersed bodies and their vertical placement play a critical role in controlling flow redistribution in confined turning flows. The proposed methodology provides a robust and efficient tool for parametric studies aimed at identifying optimal body arrangements for enhancing flow uniformity in practical engineering applications.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAjith KS, Anil S, Lal (2021) Effects of Prandtl Number on Three-Dimensional Coherent Structures in the Wake Behind a Heated Cylinder. J Appl Fluid Mech 14:515\u0026ndash;526\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBatchelor GK (1967) An Introduction to Fluid Dynamics. Cambridge University Press\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDonea J, Huerta A (2003) Finite Element Methods for Flow Problems. Wiley\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eFox RW, McDonald AT, Pritchard PJ (1998) Introduction to Fluid Mechanics, 7th edn. Wiley\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKatz J, Plotkin A (2001) Low-Speed Aerodynamics, 2nd edn. Cambridge University Press\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLal S, Anil, Jabir E (2010) A Hybrid Finite Element\u0026ndash;Finite Volume Method for Incompressible Flow Through Complex Geometries Using Mixed Grids. \u003cem\u003eProceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering\u003c/em\u003e 224 (1): 23\u0026ndash;41\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLamb H (1932) Hydrodynamics, 6th edn. Cambridge University Press\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMilne-Thomson LM (1968) Theoretical Hydrodynamics, 5th edn. Macmillan\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eYadav M (2025) A Computational Approach for Multi-Body Potential-Flow Interaction Effects Using Matrix-Free FEM and Body-Conforming Grids. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://arxiv.org/abs/2512.12232\u003c/span\u003e\u003cspan address=\"https://arxiv.org/abs/2512.12232\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWhite FM (2011) Fluid Mechanics, 7th edn. McGraw-Hill\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZdravkovich MM (1977) Review of Flow Interference Between Two Circular Cylinders in Various Arrangements. J Fluids Eng 99:618\u0026ndash;633\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZienkiewicz OC, Taylor RL (2000) \u003cem\u003eThe Finite Element Method, Vol. 3: Fluid Dynamics\u003c/em\u003e. 5th ed. Butterworth\u0026ndash;Heinemann\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Body Conforming Grids, Coefficient of Variation, Potential Flow Interactions, Multi-point Constraints","lastPublishedDoi":"10.21203/rs.3.rs-8425258/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8425258/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAn efficient and robust methodology for the rapid generation of high-quality, body-conforming grids in two-dimensional domains containing multiple immersed solid bodies is employed to generate the mesh and to investigate potential flow interactions in a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{90}^{\\circ\\:}\\)\u003c/span\u003e\u003c/span\u003e flow turning configuration. The proposed approach significantly reduces the setup time for CFD analysis while maintaining grid quality in the vicinity of complex geometries. A systematic parametric study is carried out by varying the number of bodies, their sizes, and their locations upstream of the exit section in order to assess the resulting uniformity of the exit flow. In addition, the constant stream function values associated with the immersed bodies are computed using a multi-point constraint formulation implemented within a conjugate gradient method for symmetric systems, thereby clarifying the influence of potential interaction effects on the inter-body flow distribution.\u003c/p\u003e","manuscriptTitle":"Potential Flow Interactions among Multiple Immersed Bodies Using Body-Conforming Grid Generation and a Multi-Point Constraint Framework for Stream Function Determination","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-24 08:37:00","doi":"10.21203/rs.3.rs-8425258/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"d83050f6-880a-4095-a751-f02473e141f3","owner":[],"postedDate":"December 24th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":60069717,"name":"Mechanical Engineering"}],"tags":[],"updatedAt":"2025-12-24T08:37:00+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-24 08:37:00","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8425258","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8425258","identity":"rs-8425258","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}