Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 401 https://internationalpubls.com Comprehensive Numerical Analysis of Fluid Flow Characteristics in a Lid-Driven Cavity with a Central Obstacle: Flow Physics Dalmalka Praveen Kumar1*, Yadagiri Rameshwar2 E-mail: dalmalkapraveen@osmania.ac.in1 , rameshwar@osmania.ac.in2 Affiliation 1,2 Department of Mathematics, University College of Science, Osmania University, Hyderabad - 500007, India Article History: Received: 21-10-2024 Revised: 05-12-2024 Accepted: 12-12-2024 Abstract: Boundary layer flows and vortex dynamics are critical in understanding fluid motion in confined geometries. The lid-driven cavity (LDC) flow is a classical benchmark for analysing fluid behaviour under varying conditions, especially when obstacles are introduced. While past studies have explored primary and secondary vortex structures, the modulation of enstrophy and palinstrophy caused by obstacles has been underexplored. This study aims to bridge that gap by systematically investigating the effect of Reynolds numbers and obstacle aspect ratios on flow properties in a 2D LDC. To analyse the influence of Reynolds number (Re) and obstacle Aspect ratio (AR) on enstrophy (Z) and palinstrophy (P). To evaluate the enhancement in vortex dynamics and flow topology caused by introducing square obstacles at various aspect ratios within the cavity. To identify optimal configurations for improving fluid transport and mixing efficiency. An unsteady, laminar, incompressible flow of Newtonian fluid was simulated by the Finite volume-based Fractional Step Method. Square obstacles with aspect ratios AR=0L,0.4L, 0.5L, and 0.6L were placed at the centre of the cavity with numerical study for a range of Re=10,100,400, and 1000. The interaction between AR and Re dictates vortex dynamics in LDC flows. Higher AR disrupts flow symmetry through sharp-edge interactions, while increased Re elevates inertial effects, causing vortex splitting, secondary vortex formation, and improved fluid mixing. Obstacles boost Z and P by up to 53.08% and 50.85%, with the most significant increases for AR = 0.6L at Re = 1000 compared to a cavity without an obstacle. This suggests stronger vorticity gradients and enhanced mixing due to AR = 0.6L, as the obstacle occupies more of the cavity, restricting flow and creating stronger shear zones near the walls and the obstacle. At high Re, inertial forces dominate, leading to stronger vortex shedding, and more pronounced vorticity changes near sharp edges. Increasing Re from 10 to 1000 for AR = 0.6L raises Z and P by approximately 36.34% and 43.55%, illustrating a shift from viscous-dominated laminar to inertia-dominated flow, enhancing vortex strength, sharpening vorticity gradients, and improving mixing efficiency. The interplay between AR and Re dictates vortex dynamics, with higher AR disrupting symmetry and higher Re amplifying inertial effects, leading to vortex splitting, enhanced secondary vortices, and improved mixing. The increasing obstacle height amplifies Z and P, leading to improved fluid mixing and vortex dynamics, serving as effective tools for flow control and mixing enhancement in industrial and engineering applications. Keywords: Lid-driven cavity, Rectangle obstacle, Vortices development, Enstrophy, Palinstrophy, Mixing enhancement. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 402 https://internationalpubls.com 1. Introduction LDC problem with an obstacle has received considerable attention because of its vast applications in fluid dynamics; it has been investigated extensively and serves as a benchmark to test numerical methods and manifold code development [1,2]. Physics involved in LDC flow is relevant to vivid applications such as lake dynamics, heat exchangers, roll-coasting, galvanization, material processing, etc.[3,4].Burgraff [5] was the first to study the physical model of a 2D Lid-Driven Cavity (LDC) with block (obstacle), analysing the interactions between fluid flow and structure. He computed 2D LDCs to analyse velocity profiles, streamlines, and the analytical and numerical aspects of steady-separated flows. Pan and Acrivos [6] have investigated cavities of square and rectangular shapes with Creeping- flow streamline patterns, and cavities with finite and infinite depth. The authorized and accurate database is given by Ghia et al. [7] they reported comprehensive findings on the velocity fields along with the streamlines and vorticity contours for the different Re values ranging from 100 to 10000. Dhiman et al. [8] examined the way Reynolds number (1 ≤ Re ≤ 45) and power-law index (0.5 ≤ n ≤ 2.0) shape the steady cross-flow around an unconfined square cylinder and results indicate more extensive lateral disturbances for shear-thickening fluids, whereas shear-thinning fluids require finer grids to capture wake dynamics effectively. Hammami et al. [9] have studied the LDC in 3-D, and examined the unsteady nature of the flow in the cavity with and without an obstacle, they found that at critical value Re, the flow remains in a steady state for Re Recri. The results show that the essential value of Re without obstacle is greater than the critical value of Re with obstacle and the presence of an obstacle, the one-side LDC and two-sided LDC. Recently, two different papers authors, Huang et al. [10] and Rajan et al. [11] investigated 2-D conventional LDCs filled with various obstacles varying in sizes and shapes to clarify the effect of the size and position of the obstacle for different Re by using Lattice Boltzmann Method (LBM). Huang et al. (2020) [10] by LBM examined the internal 2-D circular obstacles with varying diameters below Re from 100 to 5000. When Renolds number Re increased, the secondary vortices near the obstacle became smaller, resulting in strong primary vortices. Based on the AR of an obstacle, when the obstacle increases, it gives rise to a bigger size of the secondary vortices near the top of it. To analyse the vortex, the shift in the cavity, the stream function and the positions of the primary and secondary vortices were evaluated against existing data. Rajan et al. [11], by the LBM, investigated LDCs with circular, elliptic and square shapes of obstacles placed in it, which led to exhibiting exceptional circulation regions and structures in comparison with classical LDC for Re=100, 400, and 1000 and obstacles position and size contribute to dramatic change in flow structure of fluid such as generated vortices, Splitting vortices and recirculation near the obstacles. The Streamlines, vorticity, and centerline velocity profiles are illustrated by them. Bisht et al. [12] used the LBM to explore the non-Newtonian fluids flow around obstacles in LDC. They found that variables like Reynolds number, obstacle size, and shape significantly impact the flow patterns and vortex formation. The study highlighted notable flow structure changes depending on the obstacles' size and configuration (Bisht et al. 2021). The lid-driven is a crucial benchmark for studying incompressible flows in confined volumes, focusing on flow evolution as the Re increases. This review covers topics from flow instabilities and numerical treatments to the transition to chaos in two-dimensional and three- dimensional flows and various extensions of the lid-driven cavity system Kuhlmann et al. [13] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 403 https://internationalpubls.com (2018). Zhang et al. [14] (2022) conducted numerical work on LDC flow with embedded circular obstacles using spectral/hp element methods, focusing on different Re and obstacle diameters. The study visualized primary and induced vortices, capturing new vortex formations and demonstrating the efficiency and precision of the spectral/hp element methods. Requiring less preprocessing compared to the moving immersed boundary method and LBM. Bruneau & Saad [15] (2006), LDC problem discussed with global parameters such as enstrophy(Z), palinstrophy(P), and the qualitative essence of turbulent behaviour at high Reynolds numbers. We found the research gap focusing on less explored areas and findings to develop more comprehensive understanding of in vortex dynamics and flow topology for practical applications, Table 4 provides a clear, quantitative and followed by qualitative view of means by which increasing the Reynolds number along with aspect ratio of central squared obstacle enhances both enstrophy and palinstrophy as well flow dynamics (in Fig 4 and its description), with the changes reflecting the shift in flow behaviour from laminar to transitional and eventually more inertial- dominated flow which holds crucial insight into physics-involved flow dynamics in LDC with obstacles. We investigated the fundamental flow characteristics by analysing streamline (ψ) patterns and vorticity (ω) contour distributions, with particular attention to sharp-edge interactions near walls of obstacles in observing variations in flow separation, vortex structures, and global characteristics, Renolds number (Re) together AR scaling effects, flow mixing mechanisms, and flow control parameters. Hence, we critically analyse the effects, as stated above, of the LDC flow problem with or without obstacles by changing the AR of rectangular obstacles as well as Re numbers 10,100,400 and 1000. we adopted the Fractional step method [16] on a staggered grid. The present work is organized into 4 sections. In continuation, Section 2 describes the problem and the numerical method used, fundamental equations with the boundary conditions of the LDC problem. by validating code and also in Table 1-3 In Section 3, the results and discussion, flow characteristics such as streamline and vorticity contour in Figures 3-4 and its assessment by subsections 3a,3b and 3c with the support of Tables 4-6. Finally, Section 4, outlines the principles and conclusions drawn. 2. Methods Figure 1. Schematic of LDC flow with internal square obstacle at the centre of a square cavity. The governing equations in non-dimensional form are, the continuity equation (1) and momentum conservation equations (2) (as per Einstein’s summation convection.) as follows: by eqn (3) to (7). 𝝏𝒖𝒊 𝝏𝒙𝒊 = 0. (1) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 404 https://internationalpubls.com 𝝏𝒖𝒊 𝝏𝒕 + 𝝏(𝒖𝒋 𝒖𝒊) 𝝏𝒙𝒋 = −𝝏𝒑 𝝏𝒙𝒊 + ( 𝟏 𝑹𝒆 ) 𝝏𝟐𝒖𝒊 𝝏𝒙𝒋𝝏𝒙𝒋 . (2) Where 𝑢𝑖 is velocity component, t is time, p is pressure, Re is Reynolds number and 𝑥𝑗 are spatial coordinates for i, j which are spatial directions and summation is implied over repeated indices. Initial conditions: t=0, 0≤ 𝑥 ≤1, 0≤ 𝑦 ≤ 1: u=0; v=0 (3) Boundary conditions (BCs): t>0 Left wall 𝑥 =0, 0≤ 𝑦 ≤1, u=0, v=0, (4) Right wall: 𝑥 =L, 0≤ y ≤1, u=0, v=0, (5) Bottom wall: 𝑦 =0, 0≤x≤1, u =0, 𝑣 = 0, (6) Top wall: 𝑦 =1, 0≤x≤1, u=1, v=0, (7) The equations (1) and (2), are continuity and momentum equations resp., solved by Fractional step method. Figure 2. Vertical velocity along the x-axis. Horizontal velocity along the y-axis. Table 1: Comparison of locations of vortex centers with [7] (at AR= 0.0, Re = 100, 400, and 1000). Re, Validation, % error Upper Right Vortex Lower Left Vortex Lower Right Vortex x y x y x y Present (Re=100) 0.61718750 0.74218750 0.031250 0.03906250 0.94531250 0.06250 Ghai et al. [7] 0.6172 0.7344 0.0313 0.0391 0.9453 0.0625 Deviation (%) 0 1.063 0 0 0 0 Present (Re=400) 0.55468750 0.60546875 0.05468750 0.0468750 0.8906250 0.1250 Ghai et al. [7] 0.5547 0.6055 0.0508 0.0469 0.8906 0.125 Deviation (%) 0 0 7.662 0 0 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 405 https://internationalpubls.com Present (Re=1000) 0.531250 0.57031250 0.08593750 0.0781250 0.86718750 0.11718750 Ghai et al. [7] 0.5313 0.5625 0.0859 0.0781 0.8594 0.1094 Deviation (%) 0 1.39 0 0 0.91 7.12 Note: Deviation (%) for coordinates x, y of some vortex centres are taken to be zero since Ghai et al. [7] gave only four decimal round-offs and the present work gave full decimal precision so for most of the present work’s four decimal round-offs is exactly same. Table 2: Comparison of positions maxima/minima x, y corresponding to centrelines u, v velocity components and its values corresponding values of ψ and ω with Re = 100, 400, and 1000 [7] Note: Values compared as Ghai et al. [7] with 128x128 grid for Re =100,1000, and 256 x256 grid for Re =400. Deviation (%) for minimum/maximum points of coordinates x, and y of the centreline velocity profile is taken to be zero since Ghai et al. [7] gave only four decimal round-offs and the present work gave full decimal precision to the present work four decimal round-offs are the same. Re, Validation, % error y(min) u(min) x (max) v(max) x(min) v(min) ψ (min) ω Re=100 Present work 0.4531 25 - 0.210920 0.234375 0.175341 0.804688 - 0.245431 - 0.103389 - 3.192007 Ghai et al. [7] 0.4531 -0.21090 0.2344 0.17527 0.8047 -0.24533 - 0.103423 3.16646 Deviation (%) 0 0.009 0 0.041 0 0.041 0.033 0.81 Re=400 Present work 0.2812 50 - 0.327146 9 0.226562 5 0.301941 0 0.859375 0 - 0.449776 7 - 0.113852 - 2.292971 Ghai et al. [7] 0.2813 -0.32726 0.2266 0.30203 0.8594 -0.44993 - 0.113909 2.29469 Deviation (%) 0 0.035 0 0.030 0 0.034 0.050 0.075 Re=1000 Present work 0.1718 750 - 0.381450 1 0.156250 0 0.369357 8 0.906250 0 - 0.513362 0 - 0.117422 - 2.045723 Ghai et al. [7] 0.1719 -0.38289 0.1563 0.37095 0.9063 -0.51550 - 0.117929 2.04968 Deviation (%) 0 0.376 0 0.429 0 0.415 0.430 0.193 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 406 https://internationalpubls.com Table 3: Comparison of energy, enstrophy(Z) and palinstrophy(P) with grid 128x128 for Re =1000 with [15] Result Energy (KE) Enstrophy(Z) Palinstrophy(P) Present work 0.045635 18.405 0.15291 x 10+6 Bruneau and Saad [15] 0.043721 17.685 0.15280 x 10+6 Deviation (%) 3.55 4.07 0.072 3. RESULTS AND DISCUSSION The code validation of the numerical method is tested without an obstacle for our simulation, and we have excellent agreement between our work and literature evident in Tables 1-3 and Table 5 shows the cores of primary, secondary vortices and even minimum/maximum points of the centreline velocity profile are very accurately agreed with Ghai et al. [7]. Both the graphs present the same trends: the non-dimensional centre line velocity, and streamline plot are validated for Re=100,400 and 1000. All the iterations are carried out up to many iterations where convergence criteria of 10-8 were met. The top lid was moving with a constant velocity of 1(m/sec). Re used in this computation are 10,100,400 and 1000. The non-denationalised velocity values with vertical, and horizontal lines via the cavity centre are shown in Figure 2 followed by streamline and vorticity contour plots in Figures 3 and 4, with their global flow characteristics. All three deviations of global parameters are well under 4.08%, which suggests relatively small differences between the present and reference values. 𝑍 = 1 2 ∬ ||𝜔||2𝑑A A , 𝑃 = 1 2 ∬ ||∇𝜔||2𝑑A A , where 𝜔 = 𝜕𝑣 𝜕𝑥 − 𝜕𝑢 𝜕𝑦 is the vorticity denoting 𝑽ᵢⱼ = (𝑢ᵢⱼ, 𝑣ᵢⱼ) represents the velocity at the centre of a cell, for example kinetic energy 𝐸 = 1 2 ∬ ||𝑽||2 A 𝑑A, over domain A then E over the fluid region is 𝐸𝑓𝑙𝑢𝑖𝑑 = 1 2 ⋅ (fluid cell area) ⋅∑ ∑ (𝑢𝑖,𝑗 2 + 𝑣𝑖,𝑗 2 ),𝑖,𝑗 where each “fluid cell area” designates the area associated with a single fluid grid cell, and the summation spans all such cells within the fluid domain. For these Z and P are used in half of the work, validated against numerical results in Table 3 of Bruneau and Saad [15]. Streamline formations of the flow in enclosures with obstacles installed at the centre of the cavity are illustrated in Figure 3(d-l). In such cases, flow formation is not identical to the regular cavity without obstacles and with obstacles (compare Figure 3(a-d) and Figure 3(e-p)). These figures suggest that by increasing the Re, (column-wise images), a tiny vortex would develop on the bottom left and right corner of the bottom expand, as anticipated. Figure 3(a-d). Interestingly, the primary induced vortex grows as we increase Re and secondary vortices at both left- and right-hand sides reduce completely as we increase A.R. See Figure 3(e-p). The movement of the lid, which is uniform and is in the direction of the positive x-axis, generates a primary eddy (vortex) which encases the obstacle and hence holds a significant area in a cavity, leading to the induced eddy formed near the obstacle and the sharp corner obstacle improves the control and mixing flow characteristics. The top-right corner of the LDC and the leading edges of the rectangular obstruction are regions where local flow acceleration occurs due to flow compression, as per the principle of continuity. The recirculation area is positioned behind the obstacle, moving along the bottom of the cavity where the flow decelerates and reverses its direction. Thus, Figure 3 demonstrates the streamline pattern for Re=10,100,400,1000 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 407 https://internationalpubls.com employing the Fractional step method. In our work, we consider lower Re (Re=10,100,400) and a transitional Re (Re = 1000), along with the required AR, to effectively enhance mixing characteristics. In our earlier work (Kumar DP et al., [17]), we examined global parameters not only within the fluid domain but also across the entire domain including the obstacle region where velocity components are zero. This comprehensive investigation provided deeper physics insights —both quantitative and qualitative—into vortex development and fluid flow dynamics by varying the obstacle height H (while keeping its width W constant) in a LDCs with bottom-corner obstacles. a) b) c) d) e) f) g) h) i) j) k) l) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 408 https://internationalpubls.com m) n) o) p) Re=10 Re=100 Re=400 Re=1000 Figure 3. Variation of the streamlines with the aspect ratio in the first row is AR=0L, the Second is AR=0.4L, the third is AR=0.5L, and the fourth row is AR=0.6L with Re= 10, 100, 400 and 1000. 3a) Effect of Reynolds number and obstacle Aspect Ratio on fluid flow and vortex dynamics: Obstacle’s aspect ratio AR=0L: In the case of Re=10,100,400 and 1000, AR=0, streamlines of LDC show us the primary vortex (larger circulation pattern in the middle) and secondary vortex (smaller circulation pattern in bottom corners) due to the geometrical shape of the square of a lid. In Figure 3(a-d), a common feature is flow, which is dominated by a large central vortex. For Re=10, Figure 3a, we observe that the induced primary vortex is positioned upwards, in the middle of the enclosure, and very small secondary eddies are induced at the bottom corners. Induced vortices formed due to the fluid flow are disrupted by a boundary or obstacle that causes the flow to separate or shear. As we increase Re to 100, we see the primary vortex shift to a bit right and the expansion of the secondary vortices. Further, with a rise in Re to 400, we see the primary vortex shifted to a bit down from the previous case position and much increased in size of the secondary vortices. Also, with a rise in Re to 1000, we see the primary vortex shifted a bit down, which is almost to the geometric centre of the cavity from the previous case position and much more enlarged in size of the secondary vortices due to the low viscosity related to Re=1000, which gives rise to increase in velocity and mobility of the fluid. Here, we observe that flow is characterised as laminar at low Re since viscous forces dominate over inertial forces. Moreover, vortices are small, stable and slowly evolving with no complex behaviour exhibition and an organised flow with smooth streamlines. This contrasts with higher Re and the prevalence of turbulent and more complicated patterns. In the view of a dynamical system, all cores are called centres, featuring clockwise rotation in a 2-D phase portrait by looking at steady-state flow solutions that look similar. Obstacle’s aspect ratio AR=0.4L: In the case of Re=10, AR=0.4 from Figure 3e, the generated vortex is bifurcated, producing two sub- vortices, which have almost a vertical centrosymmetric with a right-side larger sub-vertex. Further, it is also interesting to note that both subvortices of the top are unified into one vortex with two centres on the left- and right-hand sides, and almost near the centre of them, a saddle point (x, y) = (0.483,0.869) exist. From Figure 3f and Re=100, a primary vortex has expanded to form a single induced vortex with one core on the hand side by the uniting of the two sub-vortices, and its saddle point has disappeared along with the vertex centre shifted in the direction of the lid motion. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 409 https://internationalpubls.com Additionally, a tiny vortex can be observed in the left- and right-bottom enclosure’s corners. From Figure 3h, in Re=400, the induced vortex, which is rotating clockwise in scoop-like form, becomes bigger and approaches close to the top right vicinity of the obstacle, and for Re greater than or equal to 400 as it grows, a lower corner vortex is induced which become increasingly visible. Finally, for Re=1000, due to the sharp slope of the rectangular obstacle, the bigger induced vortex moves downwards on the right side and will break down in parts by flow driven by the direction of lid movement. Also, a significant enhancement in the area of the bottom left- and right-hand side vortices is observed when compared to Re=10,100,400. As we observed in Figure 3(a-h) when compared to without obstacle (Figure 3d with Figure 3h), we see the primary/secondary vortices size drastically reduced due to the installation of obstacles at the geometric centre of the cavity. Obstacle’s aspect ratio AR=0.5L: In the case of Re=10, AR=0.5 (Fig 3i), same as AR=0.5 but with reduced size of the generated vortex, which is bifurcated, producing two sub-vortices which have exactly a vertical centrosymmetric, both subvortices of the top are merged containing two cores on the left and right sides, next to the centre of them a saddle point (x, y) = (0.504,0.889) exist. In the case of Re=100, AR=0.5 (Figure 3j), we have a similar form, which becomes bigger but has a reduced vertex size due to pattern and flow field behaviour about saddle point disappearing as seen in Fig 3f the induced vortex, in scoop-like the bigger induced vortex moves downwards between the obstacle's right side and the cavity's right wall. Also, a new extra vortex (ternary vortex) is caused by the core at (x,y) =(0.678,0.231) at the bottom right side of the obstacle. We observed in Figures 3h and 3l that the secondary vortex size was reduced due to the increased aspect ratio of the obstacle. Obstacle’s aspect ratio AR=0.6L: In the case of Re=10 with a new saddle point (x, y) = (0.512,0.909), also for Re =100 for AR=0.6 from Figure 3(j-l), the same pattern globally and behaviour of flow field but with further reduced induced vortex. Interestingly, in the cavity with the most significant obstacle (i.e., AR=0.6L) at Re=400, we notice that a single merge primary vortex in Re=100 as we increase Re=400, it gets split into two more induced vortices at the top near the centre and top right region of the rectangular obstacle. As we further increased Re to 1000, we observed in Figure 3p that the primary vortex ruptures and produced primary vortices on two sides of the obstacle. Thus leading to efficient control and improved mixing of fluid flow. Table 4: Variation of Enstrophy (Z), Palinstrophy (P) with Reynolds Number (Re) and AR in LDC flow. Re Aspect ratio (AR) (Height(H) ) Enstro -phy (Z) of fluid region % ↑ in Z for correspon d-ing Re from without obstacle % ↑ in Z (from correspon d- ing Re to Re=1000) Palinstro -phy (P) of fluid region % ↑ in P for correspondin g Re from without obstacle % ↑ in P (from correspon d ing Re to Re=1000) 10 0.0L 11.679 - 122.59% 4.1x10+04 - 119.51% Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 410 https://internationalpubls.com 0.40L 14.525 24.37% 57.78% 4.8x10+04 17.07% 85.42% 0.50L 17.135 46.71% 57.20% 5.3x10+04 29.27% 78.00% 0.60L 21.658 85.48% 36.34% 6.2x10+04 51.22% 43.55% 100 0.0L 11.679 - 59.20% 4.1x10+04 - 42.86% 0.40L 14.928 27.83% 23.59% 4.8x10+04 17.07% 43.55% 0.50L 17.459 49.53% 24.69% 5.4x10+04 29.27% 39.06% 0.60L 21.906 87.57% 21.53% 6.3x10+04 53.66% 18.67% 400 0.0L 14.840 - 26.40% 4.7x10+04 - 18.42% 0.40L 17.920 20.77% 9.90% 5.5x10+04 17.02% 23.61% 0.50L 20.188 36.06% 10.37% 6.1x10+04 29.79% 21.92% 0.60L 24.282 63.63% 8.61% 7.1x10+04 51.06% 7.23% 100 0 0.0L 19.286 - - 5.9x10+04 - - 0.40L 22.921 18.85% - 7.0x10+04 18.64% - 0.50L 25.465 32.01% - 7.7x10+04 30.51% - 0.60L 29.528 53.08% - 8.9x10+04 50.85% - Note: The table summarizes the enstrophy (Z) and palinstrophy (P) values for different Reynolds numbers (Re) and obstacle aspect ratio (AR), alongside percentage increases relative to reference cases (without obstacle and Re=1000). Symbols ‘-’ are not required as they are zero since they represent percentages on the same values. a) b) c) d) . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 411 https://internationalpubls.com e) f) g) h) i) j) k) l) m) n) o) p) Re=10 Re=100 Re=400 Re=1000 Figure 4. Vorticity contour plots show variations with different aspect ratios: the first row displays AR=0L (no obstacle), the second row AR=0.4L, the third row AR=0.5L, and the fourth row AR=0.6L. Each column presents results at Reynolds numbers Re of 10, 100, 400, and 1000. 3b) Effect of Reynolds number and obstacle Aspect Ratio on enstrophy and palinstrophy in LDC: The vorticity contour plots in the provided Figure 4 and Table 4 provide key insights into the dynamics of enstrophy (Z) and palinstrophy (P) in LDC flows are governed by the complex interplay of Reynolds number (Re) and obstacle height (aspect ratio, AR), which influence the flow structure, energy redistribution, and mixing intensity. This study demonstrates variations in AR and Re affect vortex formation, energy cascades, and mixing efficiency through detailed analysis of the flow fields, with a particular focus on Z (a measure of vorticity intensity) and P (a metric for vorticity gradient magnitudes). In the absence of obstacles (AR=0.0L), the cavity flow exhibits a dominant primary vortex whose structure is dictated primarily by the balance of viscous and inertial forces. At low Re =10, viscous effects dominate, leading to weak rotational structures and minimal Z with 11.679) and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 412 https://internationalpubls.com P with 4.1x10+04) values. As Re increases, inertial effects amplify vorticity production and induce secondary vortex formation near the cavity corners. This results in higher Z and P values, with sharp gradients appearing at the boundaries of the primary vortex, which reflects the redistribution of energy into smaller flow structures. For Re=1000, the flow becomes increasingly dominated by inertial forces, resulting in stronger vortex interactions, enhanced mixing, and significant increases in both Z (with 19.286) and P (with 5.9x10+04) values. The introduction of obstacles modifies the flow by disrupting the primary vortex, enhancing secondary vortex formation and even tertiary vortex if present. For intermediate obstacle aspect ratios (AR=0.4L,0.5L), the sharp edges of the obstacle act as the positions for vorticity production due to flow separation and recirculation zones which are evident as shown in Figure 4e-4l. At Re = 10, the obstacle induces minor increases in Z by 24.37% and 46.71%, and in P by 17.07% and 29.27% for AR = 0.4L and 0.5L, respectively, compared to the reference case without the obstacle, as the flow remains largely governed by viscous effects. However, at Re=100 and Re=400, inertial forces drive significant vortex stretching, splitting and interactions with the obstacle's sharp edges particularly near top right the sharp edge of the obstacle and stagnatin points on top left and right corners of cavity, which amplify the energy in the flow's rotational structures and enhance Z and P as shown in values of Figure 4. At Re=1000, the interaction between the primary vortex and the obstacle sharpens vorticity gradients, producing stronger secondary vortices associated with higher inertial forces in the flow and leading to the third-largest peak in both Z with 22.92 and P with 7.0x10+04 values. The highest obstacle height (AR=0.6L) demonstrates the most pronounced effects across all Re. The larger obstacle increases flow separation and promotes the formation of smaller, high-energy vortices, which are indicative of an enhanced cascade of energy to smaller scales. At low Re, the obstacle moderately increases Z (21.66%) and P (51.22%) compared to the reference case without the obstacle, reflecting stronger vortex structures even in laminar regimes. At Re=100 and Re=400, the obstacle generates higher vorticity gradients due to enhanced interactions between the primary and secondary vortices especially near obstacle sharp edges where the flow undergoes abrupt directional changes. This will lead even more to higher Re which is evidenced by sharper velocity profiles, sharp changes in vorticity across the fluid domain and increased Z and P values. Thus, for Re=1000, the obstacle produces intense vortex breakdown and turbulent mixing, maximizing Z (53.08% increase) and P (50.85% increase), due to the concentrated shear layers near the obstacle and the enhanced energy transfer to smaller scales. Larger central obstacles in LDC flows fundamentally alter the flow dynamics by intensifying vortex generation and interactions, leading to multiple vortical structures. Thus, in essence, the increased obstacle height enhances enstrophy levels through stronger velocity gradients and vorticity production. This results in a more complex flow topology with modified shear layer behaviour and distinct high- vorticity regions, as evidenced in both streamline and vorticity contour plots. Even higher Reynolds numbers drive the dominance of inertial forces over viscous effects, promoting vortex stretching and rotation, especially when obstacles induce flow separation and shear-layer instability that generate steep vorticity gradients. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 413 https://internationalpubls.com Table 5. Position of generated primary vortex and secondary vortex centres for LDC installed rectangular obstacles. Note: x and y (not bold letter) coordinates of the centre of the primary vortex and x and y (x, y bold letters) coordinates of the centre of the secondary vortex's left and right vertices in Table 5, whereas in Table 6, x and y (bold letters) coordinates of stagnation point on obstacles. Dash indicates (-) in Table 5, no primary or secondary vortices cores, whereas Table 6 indicates no stagnation points observed. (Re) Vortex produced by the obstacle Left secondary vortex Right secondary vortex Aspect ratio Left vortex above the obstacle Right vortex above the obstacle x y x y x y x y 10 AR=0 Primary Vortex (without obstacle) 0.0391 0.0391 0.9609 0.0391 x 0.5156 y 0.7656 AR=0.4 x y x y 0.0234 0.0156 0.9844 0.0234 0.3906 0.8672 0.6563 0.8672 AR=0.5 0.3203 0.8906 0.7109 0.8906 0.0156 0.0156 0.9844 0.0156 AR=0.6 0.2578 0.9141 0.7656 0.9141 0.0156 0.0156 0.9922 0.0156 100 AR=0 Primary Vortex (no obstacle) 0.0391 0.0313 0.9453 0.0625 x 0.6172 y 0.7422 AR=0.4 x y x y 0.0156 0.0234 0.9844 0.0234 - - 0.7421 0.8594 AR=0.5 - - 0.7813 0.8828 0.0156 0.0156 0.9844 0.0156 AR=0.6 - - 0.8125 0.9063 0.0156 0.0156 0.9922 0.0156 400 AR=0 Primary Vortex (no obstacle) 0.0547 0.0469 0.8906 0.125 x 0.55469 y 0.60936 AR=0.4 x y x y 0.0313 0.0313 0.9688 0.0313 - - 0.7969 0.8125 AR=0.5 - - 0.8281 0.8516 0.0234 0.0234 0.9844 0.0234 AR=0.6 - - 0.8516 0.8281 0.0234 0.0234 0.9844 0.0234 1000 AR=0 Primary Vortex (no obstacle) 0.0859 0.0781 0.8672 0.1172 x 0.53125 y 0.57031 AR=0.4 x y x y 0.0703 0.0547 0.9063 0.1094 - - 0.8047 0.6875, and 0.7109 AR=0.5 - - 0.8516 0.7969 0.0469 0.0391 0.9531 0.0625 AR=0.6 0.6519 0.9063 0.8750 0.8516 0.0313 0.0234 0.9766 0.0313 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 414 https://internationalpubls.com Table 6. Position of Stagnation points on the right side of the obstacle and max. and min. of p, u, v values. Reynolds number (Re) Aspect Ratio (AR) Stagnation points Pressure distribution Velocity distribution x y Max.(p) Min.(p) Max.(u) and Max.(v) Min.(u) and Min.(v) u v u v 10 AR=0 - - 19.384 -18.606 1 0.3694 -0.2081 -0.3986 AR=0.4 0.703 0.703 20.20 -19.64 1 0.3668 -0.1874 -0.3852 AR=0.5 0.75 0.745 20.84 -20.39 1 0.3655 -0.1875 -0.3802 AR=0.6 0.797 0.793 22.08 -21.82 1 0.3622 -0.1886 -0.3752 100 AR=0 - - 2.4351 -1.6040 1 0.3132 -0.2429 -0.5330 AR=0.4 0.703 0.6838 2.4101 -1.7272 1 0.3139 -0.2080 -0.4748 AR=0.5 0.75 0.7374 2.4398 -1.8090 1 0.3143 -0.2059 -0.4590 AR=0.6 0.797 0.7855 2.5256 -1.9641 1 0.3150 -0.2033 -0.4383 400 AR=0 - - 1.0696 -0.2916 1 0.3199 -0.3365 -0.6364 AR=0.4 0.703 0.581 1.0012 -0.3361 1 0.2403 -0.2317 -0.6154 AR=0.5 0.75 0.667 0.9809 -0.3557 1 0.2398 -0.2128 -0.5990 AR=0.6 0.797 0.745 0.9658 -0.3833 1 0.2395 -0.2192 -0.5753 1000 AR=0 - - 0.7460 -0.0727 1 0.3772 -0.3841 -0.6591 AR=0.4 0.703 0.394 0.6903 -0.0949 1 0.2811 -0.2916 -0.6613 AR=0.5 0.75 0.491 0.6689 -0.1067 1 0.2523 -0.2620 -0.6586 AR=0.6 0.797 0.643 0.6470 -0.1234 1 0.2430 -0.2418 -0.6491 3c) Velocity distribution: The analysis of velocity distribution in a lid-driven cavity (LDC) flow with embedded obstacles highlights critical flow dynamics shaped by varying aspect ratios (AR) and Reynolds numbers (Re) see (Table 6). The maximum horizontal velocity (max(u)) remains consistently at 1 across all configurations, indicating a strong influence on the top wall's motion. At Re = 100, min(u) increases by approximately 16.3% from AR = 0 to AR = 0.6, indicating a reduction in reverse flow intensity. Similarly, at Re = 400, min(u) increases by about 34.8% from AR = 0 to AR = 0.6, showing greater flow stabilization with increasing AR. At Re = 400, min(v) decreases by approximately 9.6% from AR = 0 to AR = 0.6, underscoring substantial fluctuations in downward flow. However, the minimum horizontal velocity (min(u)) varies with a mean of approximately -0.234 and a standard deviation of 0.031, reflecting moderate variability as AR and Re change. This variability suggests that the reverse flow's intensity is significantly influenced by the cavity's geometric constraints and flow parameters. And enhancing the induced vortex to manage the system’s process parameters and mixing In contrast, the vertical velocity component (v) exhibits greater sensitivity. The mean maximum vertical velocity (max(v)) is about 0.320, with a standard deviation of 0.045, indicating a relatively stable upward flow across different conditions. However, the minimum vertical velocity (min(v)) presents a broader Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 415 https://internationalpubls.com spread, with a mean of -0.541 and a higher standard deviation of 0.085, indicating significant fluctuations in the downward flow. These statistical insights suggest that while the horizontal flow is more consistent, the vertical flow is more susceptible to changes in AR and Re, leading to substantial variations in the overall flow dynamics, which are important for optimizing fluid behaviour. 4. Discussion Numerical results show that the Fractional step method which is more efficient can accurately simulate the LDC problem. This study presented in the paper reveals that the phenomenon of fluid flow is linked to varying the Renolds number (Re), and obstacle Aspect ratio (AR) where increasing Re and AR significantly enhances both enstrophy (Z) and Palinstrophy (P), with larger obstacles and high inertial effects contributing to pronounced vortex dynamics, energy redistribution, and mixing efficiency. Understanding the flow behaviours change with varying AR and Re is critical for optimizing designs in engineering applications, including turbulent mixing, energy-efficient fluid systems, and flow control technologies. This work addresses a significant gap by quantifying the effects of obstacles on both large-scale vorticity intensity (Z) and fine-scale flow gradient sensitivity (P), offering insights applicable across industrial and scientific domains. The following are key insights of the work: 1) At higher Re, inertial forces dominate, vortices interact more strongly and mixing is enhanced as reflected in sharp increases in enstrophy (Z) which is a measure of vorticity intensity with increasing Re (as in Table 4) from 11.679 at Re=10 to 19.286 at Re=1000 for without obstacle (AR=0) marking a 122.59% increase, showing a strong correlation between Re and vorticity intensity. The slower growth at higher Re (26.40% for Re=400 to 1000) reflects saturation of vorticity production which is true in all AR as shown in Table 4. 2) In the impact of obstacle aspect ratio (AR) view, obstacle presence amplifies Z by enhancing vortex generation, with moderate ARs (0.4L and 0.5L) obstacles displaying nearly identical trends. Their increases (57.78% and 57.20% for Re=10 to 1000) suggest similar effects on flow separation and recirculation zones, despite slight divergence at mid-Re values (0.5L exhibits a higher 24.69% increase for Re=100 to 1000). For AR = 0.6L, despite having the smallest overall growth (36.34%), larger obstacles significantly disrupt flow symmetry and induce more pronounced vortex breakdown, causing Z to increase in unique ways. The smaller percentage increases at higher Re (8.61% for Re=400 to 1000) reflect energy redistribution into finer-scale turbulence. In the vortex dynamics view, the complexity of the flow, reflected in the enstrophy levels, intensifies with larger obstacles and higher Re owing to pronounced shear layers and enhanced rotational energy. 3) In the gradient magnitude sensitivity aspect Palinstrophy (P), representing vorticity gradient magnitudes, increases sharply with Re, reflecting higher energy concentration in flow structures evident from AR=0.0L, without obstacles, P increases sharply (119.51%) from Re=10 to Re=1000, showing uninhibited energy transfer into concentrated vortices. The sharp decline at higher Re (18.42% for Re=400 to 1000) mirrors diminishing shear-layer gradients. The role of obstacles in intermediate obstacles ARs (0.4L and 0.5L), at AR = 0.4L exhibits the largest growth overall (85.42%) and remains more sensitive at intermediate Re (23.61% for Re=400 to 1000). This suggests that moderate obstacles optimize the formation of high-gradient flow zones. AR=0.5L follows similar trends to AR=0.4L with slightly reduced values, achieving 78.00% growth for Re=10 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 416 https://internationalpubls.com to1000 and 21.92% for Re=400 to1000. The slightly lower P increase may reflect subtler flow separation effects compared to 0.4L. 4) The largest obstacles AR=0.6L, exhibit the lowest overall growth (43.55%), consistent with diminishing gradient intensities in flows disrupted by significant vortex breakdown. The low growth at higher Re (7.23% for Re=400 to 1000) indicates energy is increasingly distributed across broader scales rather than concentrated in small vortices. Enhanced energy cascades, shear-layer instabilities, and increased velocity gradients near obstacles all contribute to higher P (8.9 x 10+04) value, particularly at higher Re. 5) We investigate the influence of aspect ratio (AR) and Reynolds number (Re) on a centred obstacle, with AR varying from 0L to 0.6L. As Re increases (from Re=10 to Re=1000), the size of the generated primary vortex grows significantly, accompanied by increases in velocity magnitude and vorticity above the obstacle. Notably, for non-zero AR configurations by increasing AR, the secondary vortices in the bottom-right and left corner are significantly diminished or completely vanish compared to the case without an obstacle (AR=0), as illustrated in Figure 3(a–c). Additionally, stagnation points were observed both on the obstacle's surface and along the domain boundaries, reflecting the intricate flow dynamics induced by the obstacle's presence evident in Figure 3 and corresponding Table 6 for it. 6) The aspect ratio (AR) and Reynolds number (Re) have a core impact on pressure (p) distribution. At lower Re (for Re=10), increasing AR significantly enhances maximum pressure (max(p)), whereas, at higher Re (for Re=1000), the influence of AR on max(p) diminishes, leading to a more complex interplay between fluid mixing and pressure gradients as in Table 6. 7) Low and intermediate Reynolds numbers (Re) are selected to enhance mixing due to the large vortex generated as the aspect ratio (AR) of the obstacle increases. This results in a more compact flow region and increased velocity to maintain continuity. The primary vortex moves clockwise, while the secondary and tertiary vortices move counterclockwise, even with an obstruction. Localized velocity fluctuations occur near the leading edges of the obstruction, but overall flow characteristics remain consistent with the turbulent regime. At intermediate Reynolds numbers (Re = 400 to 1000), flow separation behind the obstacle becomes more pronounced, and evident of vortex shedding or the formation of wake regions. 8) Pressure changes are most pronounced near the obstruction, particularly at the leading edges, where flow separation and localized acceleration cause pressure dips. While Bernoulli's principle can partially explain these dips along specific streamlines, the overall pressure field is influenced by complex vortex dynamics and viscous effects. The rest of the cavity generally exhibits a more uniform pressure distribution, resembling an unobstructed scenario. However, the obstruction induces flow restrictions and creates low-pressure zones that alter the flow structure and pressure gradients throughout the cavity, especially at higher Reynolds numbers. 5. Acknowledgments: The authors express their profound gratitude to the Indian Institute of Technology Bhubaneswar (IIT Bhubaneswar) for their invaluable support and for providing access to the high-performance Computational Lab for an extended one-month duration, which was instrumental in conducting this research. Special thanks are extended to Dr. Yogesh G. Bhumkar, Associate Professor, School of Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 417 https://internationalpubls.com Mechanical Sciences, for his unwavering encouragement, insightful guidance, and commitment to fostering excellence in research. His mentorship played a pivotal role in achieving meaningful outcomes. 6. Conflict of interest: Authors have no conflict of interest References: [1] Kuhlmann, Hendrik. (2018). The Lid-Driven Cavity. Cham: Springer Nature Switzerland. doi:10.1007/978-3-319- 91494-7_8. [2] Shankar, P. N., & Deshpande, M. D. (2000). Fluid mechanics in the driven cavity. Annual Review of Fluid Mechanics, 32, 93–136. doi: 10.1146/annurev.fluid.32.1.93. [3] Chhabra, R. P., & Richardson, J. F. (1999). Non-Newtonian flow and rheology in turbulent processes. Oxford: Butterworth-Heinemann. [4] Nakamura, M., & Sawada, T. (1988). Numerical study on the flow of a non-Newtonian fluid through an axisymmetric stenosis. ASME Journal of Biomechanical Engineering, 110(2), 137–143. doi:10.1115/1.3168588. [5] Burggraf, O. R. (1966). Analytical and numerical studies of the structure of steady separated flows. Journal of Fluid Mechanics, 24(1), 113–151. doi:10.1017/S002211206600046X. [6] Pan, F., & Acrivos, A. (1967). Steady flows in rectangular cavities. Journal of Fluid Mechanics, 28(4), 643–665. doi:10.1017/S0022112067001663. [7] Ghia, U., Ghia, K. N., & Shin, C. T. (1982). High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics, 48(3), 387–411. doi:10.1016/0021- 9991(82)90058-4. [8] Dhiman, A. K., Chhabra, R. P., & Eswaran, V. (2006). Steady Flow of Power-law Fluids Across a Square Cylinder. Chemical Engineering Research and Design, 84(4), 300–310. doi:10.1205/cherd05017 [9] Hammami, F., Ghorbel, A., & Cheikh, N. B. (2017). Computational analysis of fluid flow due to a two-sided lid driven cavity with a circular cylinder. International Journal of Computational Fluid Dynamics, 31(7), 317–328. doi:10.1080/10618562.2017.1330795. [10] Huang, T., Yang, B., & Chen, F. (2020). Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles. Applied Sciences, 10(19), 6675. doi:10.3390/app10196675. [11] Rajan, I., & Perumal, D. (2021). Flow dynamics of lid-driven cavities with obstacles of various shapes and configurations using the Lattice Boltzmann method. Journal of Thermal Engineering, 83-102. https://doi.org/10.18186/THERMAL.869135. [12] Bisht, M., Kumar, P., & Patil, D. (2021). Non-Newtonian power-law fluid flow over obstacles embedded inside a cavity. Physics of Fluids, 33, 043111. https://doi.org/10.1063/5.0046655. [13] Kuhlmann, H., & Romanò, F. (2018). The Lid-Driven Cavity. Computational Methods in Applied Sciences. https://doi.org/10.1007/978-3-319-91494-7_8. [14] Zhang, J., Xiao, B., & Yang, W. (2022). Numerical Study of Lid-Driven Square Cavity Flow with Embedded Circular Obstacles Using Spectral/hp Element Methods. Applied Sciences. https://doi.org/10.3390/app122211711. [15] Bruneau, C., & Saad, M. (2006). The 2D lid-driven cavity problem revisited. Computers & Fluids, 35, 326-348. https://doi.org/10.1016/J.COMPFLUID.2004.12.004. [16] Kim, J., & Moin, P. (1985). Application of a fractional step method to incompressible Navier Stokes equations. Journal of Computational Physics, 59(2), 308–323. doi:10.1016/0021-9991(85)90148-2. [17] Kumar DP, Rameshwar Y, Ali R, Nazari S, Kallel M, Galal AM. Computational analysis of Newtonian fluid in a 2- D square cavity with kinetic energy, enstrophy, and palinstrophy in the presence of bottom corner obstacle . Results Phys. 2024;108086. https://doi.org/10.1016/j.rinp.2024.108086 https://doi.org/10.1063/5.0046655 https://doi.org/10.1007/978-3-319-91494-7_8 https://doi.org/10.3390/app122211711 https://doi.org/10.1016/J.COMPFLUID.2004.12.004