Accuracy Verification of a 2D Adaptive Mesh Refinement Method by the Benchmarks of Lid-Driven Cavity Flows with an Arbitrary Number of Refinements

Abstract

The lid-driven cavity flow problem stands as a widely recognized benchmark in fluid dynamics, serving to validate CFD algorithms. Despite its geometric simplicity, the lid-driven cavity flow problem exhibits a complex flow regime primarily characterized by the formation of vortices at the centre and corners of the square domain. This study evaluates the accuracy of the 2D velocity-driven adaptive mesh refinement (2D VDAMR) method in estimating vortex centres in a steady incompressible flow within a 2D square cavity. The VDAMR algorithm allows for an arbitrary number of finite mesh refinements. Increasing the number of successive mesh refinements results in more accurate outcomes. In this paper, the initial coarse uniform grid mesh was refined ten times for Reynolds numbers $100\le Re\le 7500$. Results show that VDAMR accurately identifies vortex centres, with its findings closely aligning with benchmark data from six literature sources.

1. Introduction

Computational fluid dynamics (CFD) has evolved into a powerful tool for simulating fluid flow, heat transfer, and other related phenomena in diverse engineering applications. Discretizing the computing domain through a mesh is crucial to accurate and robust CFD simulations. Mesh refinement, a crucial aspect of this discretization process, involves adjusting the grid resolution to capture fine details and complex flow features and elevating computational accuracy without severely impacting computational cost.

High-quality mesh generation and mesh adaptivity remain major obstacles in the CFD workflow. Ensuring consistent and reliable flow solvers and residual values convergence presents challenges across various industrial contexts [1,2]. While many CFD software packages can achieve convergence in more straightforward scenarios, they often encounter difficulties with flows involving complex physics and geometries, such as those found in high-lift aircraft configurations. Existing solver techniques frequently need more robustness to ensure reliable convergence in such situations. Adaptive mesh refinement (AMR), another term for h-refinement, is a computational approach that enables users to optimize their simulations by dynamically adjusting the resolution of the computational mesh based on evolving flow characteristics and phenomena, thereby reducing computational costs. AMR techniques typically refine the computational mesh during the time-stepping process, selectively enhancing resolution in areas of interest or where refinement is deemed necessary [3]. Despite the potential of AMR strategies to enhance accuracy at a reduced computational cost, concerns regarding robustness, error estimation, management of complex geometries, and software complexity have limited their widespread adoption [1].

In addressing the challenges of accurately modelling fluid flow while preserving computational efficiency, the AMR technique has undergone significant evolution, driven by contributions from various researchers in the field of CFD. The origins of adaptive mesh refinement for CFD can be traced back to the 1980s and 1990s, encompassing seminal contributions from researchers like Berger and Oliger [4], Bell et al. [5], Friedel et al. [6], and Berger and Leveque [7]. Subsequently, the advancement of AMR has progressed alongside improvements in numerical techniques, computer hardware and software, and its application to various problems in fluid dynamics and computational science.

Li [8] introduced and presented a 2D AMR method derived from the qualitative theory of differential equations. This method refines a computational mesh based on numerically computed velocity fields. Henceforth, we refer to the 2D AMR approach introduced in [8] as the 2D velocity-driven adaptive mesh refinement (VDAMR) method. The efficacy and accuracy of the VDAMR method have been verified through the accurate locations of singular points, asymptotic lines, and closed streamlines [9], as well as through comparisons with established CFD benchmark experiments up to two refinements, including lid-driven cavity flow [10], 2D unsteady flow past a square cylinder [11], backward-facing step flow [12], and 2D flow over a wall-mounted plate [13]. In particular, the VDAMR method has been shown to be useful for capturing localized flow features, such as identifying accurate locations of the centre of vortices within refined cells [10,13]. The accuracy of the VDAMR method depends solely on the accuracy of the numerical methods used. The VDAMR method is a low-cost and robust method applicable to all incompressible fluid flows [10,11]. Li [14] analysed the computational cost of the 2D VDAMR method by applying it to a 2D lid-driven cavity flow. For a Reynolds number ($Re$) of 1000, the results in [14] indicated that a single application of mesh refinement reduces the computational cost to approximately 17.6% of that required for a uniform mesh to achieve similar accuracy in the velocity field. When the refinement method is applied twice, the cost is further reduced to about 3.5%. This efficiency is attributed to the refinement method’s focus on refining cells only around vortex centres, separation curves, and boundaries.

The lid-driven cavity flow problem stands as a widely recognized fluid dynamics benchmark, validating CFD algorithms [15,16,17,18,19,20,21,22,23,24,25,26]. Despite its geometric simplicity, the lid-driven cavity flow problem exhibits a complex flow regime primarily characterized by the formation of vortices at the centre and corners of the square domain. The combination of its simple geometry, the presence of various corner singularities, and the difficulty in approximating the exact solution made the lid-driven cavity one of the most common benchmarks in CFD.

Numerous benchmark results are documented in the literature, including the works of Ghia et al. [21], Barragy and Carey [22], Botella and Peyret [23], Erturk et al. [24], Shapeev and Lin [25], and Perumal and Dass [26], among others. Ghia et al. [21] used the vorticity–stream function formulation of the 2D incompressible Navier–Stokes (NS) equations to study the effectiveness of the coupled strongly implicit multigrid method. They have presented solutions for several Reynolds numbers as high as $Re=10,000$ with the uniform grid mesh of resolution $129\times 129$. Barragy and Carey [22] have used a p-type finite element scheme for the fully coupled stream function–vorticity formulation of the NS equations. With this, they have presented solutions for $Re\le 12,500$ on a $257\times 257$ strongly graded and refined element mesh. Botella and Peyret [23] reported the solutions for the lid-driven cavity flow for $Re=1000$, computed by a Chebyshev collocation method with a grid mesh of $N=160$ (polynomial degree). Erturk et al. [24] have used the NS equations in stream function and vorticity formulation, which are solved numerically using a fine uniform grid mesh of $601\times 601$. The solution for flows computed with several high $Re\le 21,000$ are reported. Shapeev and Lin [25] employed the stream function formulation of the NS equations for the solution of the lid-driven cavity flow with several Reynolds numbers ($Re$). In [25], the problem is approached using a high-order finite element technique featuring exponential mesh refinement in proximity to the corners, coupled with analytical asymptotics of the flow near the corners. For $Re\le 25,000$, results using 49,156 triangular elements are presented in [25]. Perumal and Dass [26] have simulated the cavity flow using the lattice Boltzmann method with a single-relaxation-time (LBM-SRT) and a multi-relaxation-time (LBM-MRT) model. They provided solutions for $Re\le 7500$, obtained with a $201\times 201$ lattice. The findings in [26] confirmed that the LBM-MRT model produces qualitatively accurate results and exhibits significantly fewer spatial oscillations near geometric singularities in higher Reynolds number flows.

In the early works, e.g., [10,13], we examined the accuracy of the 2D VDAMR method with two refinements and used the finite volume methods to solve the NS equations numerically. As the 2D VDAMR technique can be iteratively applied, greater accuracy in locating vortex centres can be achieved if numerical velocity fields are sufficiently accurate and more refinements are implemented. Hence, the present study further verifies the accuracy of a 2D VDAMR method in estimating the locations of the centre of vortices for a steady incompressible flow in a 2D lid-driven square cavity using more mesh refinements. A finite element method is used to solve the NS equations numerically and compute steady-driven cavity flows. The range of $Re$ considered in this study is $100\le Re\le 7500$, which, according to several studies [15,16,21,22,23,24,25], is well in the steady-state range. We consider an initial coarse uniform grid mesh with a resolution of $20\times 20$ for $Re=100$, 400 and 1000; $50\times 50$ for $Re=2500$ and 5000; and $80\times 80$ for $Re=7500$. We perform ten mesh refinements for each of the six cases using the 2D VDAMR method. The accuracy of the 2D VDAMR method in identifying the locations of the centre of vortices is validated against the corresponding benchmark results presented in [21,22,23,24,25,26].

2. Materials and Methods

This section describes the governing equations, the computational domain and the mesh structure, the flow solver, and briefly, the 2D AMR method.

2.1. The Governing Equations and Computational Domain

The present study considers the finite element discretizations of the 2D steady, incompressible NS equations, which are governed by the continuity equation and the momentum equations in two directions defined by

$$\begin{array}{cc}\hfill \nabla ·\mathbf{V}& =0,\hfill \\ \hfill -\nu {\nabla }^2\mathbf{V}+\mathbf{V}·\nabla \mathbf{V}+{\displaystyle \frac{1}{\rho }}\nabla P& =0,\hfill \end{array}$$

(1)

where $\mathbf{V}=(u,v)$ denotes the velocity field in 2D with u and v as the velocity components in the x− and y− directions, respectively, $\nu$ is the kinematic viscosity, $\rho$ is the fluid density, and P represents the scalar pressure.

The lid-driven cavity consists of a square cavity with the dimensions of L. At the top boundary, a tangential unit velocity ($u=1,v=0$) is applied to drive the fluid flow in the cavity. The remaining three rigid walls are imposed with no-slip boundary conditions ($u=v=0$). The origin of the cartesian coordinate is located at the left lower corner of the cavity. The geometry of the computational domain is shown in Figure 1. The Reynolds number is defined as $Re=UL/\nu$, where U is the velocity of the lid, L is the length of the cavity, and $\nu$ is the kinematic viscosity of the fluid.

2.2. Computational Mesh and the Flow Solver

For the case with $Re=100$, 400, and 1000, the initial mesh (${\mathrm{Mesh}}_0$) is a uniform grid mesh with a resolution of $20\times 20$, i.e., ${\mathrm{Mesh}}_0$ has 441 computational nodes. This corresponds to 1323 degrees of freedom (DOF). The DOF is given by the number of nodes multiplied by the number of dependent variables. For $Re=2500$ and 5000, ${\mathrm{Mesh}}_0$ is a uniform grid mesh with a resolution of $50\times 50$, corresponding to 2601 computational nodes and 7803 DOF. ${\mathrm{Mesh}}_0$ for $Re=7500$ is a finer initial mesh with a resolution of $80\times 80$, corresponding to 6561 computational nodes and 19,683 DOF.

The Finite Element Analysis simulation Toolbox (FEATool, version 1.16.3, https://www.featool.com/, accessed on 3 March 2024) in MATLAB (Version 23.2.0, R2023b) was used to numerically solve the steady 2D Navier–Stokes equations. We assumed that the convergence is achieved when the numerical solutions (velocity fields) are computed with residuals smaller than ${10}^{-12}$.

2.3. The 2D Adaptive Mesh Refinement Method

In the present study, we apply the same AMR method from [13]. We summarise the 2D AMR method and further refer the reader to Section 3 of [13] for a detailed description of the method.

Let ${\mathbf{V}}_l=\mathbf{AX}+{\mathbf{b}}^{\prime }$ be defined as a vector field on a triangle obtained through linear interpolation of vectors at the three vertices of the triangle in the domain of the velocity field, where

$$\mathbf{A}=\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right),{\mathbf{b}}^{\prime }=\left(\begin{array}{c}{b}_1^{\prime }\\ b_2^{\prime }\end{array}\right),\mathrm{and}\mathbf{X}=\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)$$

are a matrix of constants, a vector of constants, and a vector of spatial variables, respectively.

For the interpolated vector field ${\mathbf{V}}_l$ to satisfy the continuity equation for incompressible or steady-state flow, the equation $\nabla ·{\mathbf{V}}_l=\mathrm{trace}\left(\mathbf{A}\right)=0$ must be satisfied. However, the interpolated numerical velocity vector field generally does not satisfy the continuity equation. Hence, it is assumed that $f{\mathbf{V}}_l$ satisfies the continuity equation, where f is an unknown scalar function of the spatial variables $x_1$ and $x_2$. A differential equation is obtained when $f{\mathbf{V}}_l$ is substituted into the vector field $\mathbf{V}$ of the continuity equation $\nabla ·\mathbf{V}=0$, resulting in $\nabla ·f{\mathbf{V}}_l=0$. Solving the resulting differential equation for the four different Jacobian forms of the constant matrix $\mathbf{A}$ results in four distinct expressions of the function f.

Table 1. Jacobian forms of the constant matrix $\mathbf{A}$ and the corresponding distinct expressions of the function f ($C\ne 0$) for all possible cases of a non-mass conservative linear field. The value of C is set to 1 when the conditions (MC) are implemented in MATLAB (Version 23.2.0, R2023b).

Case	Jacobian	f
1	$\left(\begin{array}{cc}{r}_1& 0\\ 0& r_2\end{array}\right)$$(0\ne r_1\ne r_2\ne 0)$	${\displaystyle \frac{C}{\left(y_1+\frac{b_1}{r_1}\right)\left(y_2+\frac{b_2}{r_2}\right)}}$
2	$\left(\begin{array}{cc}{r}_1& 0\\ 0& 0\end{array}\right)$$(r_1\ne 0)$	${\displaystyle \frac{C}{y_1+\frac{b_1}{r_1}}}$
3	$\left(\begin{array}{cc}{r}_1& 0\\ 0& r_1\end{array}\right)$$(r_1\ne 0)$	${\displaystyle \frac{C}{{\left(y_1+\frac{b_1}{r_1}\right)}^2}}$
4	$\left(\begin{array}{cc}\mu & \lambda \\ -\lambda & \mu \end{array}\right)$$(\mu \ne 0,\lambda \ne 0)$	${\displaystyle \frac{C}{{\left(y_1+\frac{\mu b_1-\lambda b_2}{{\mu }^2+{\lambda }^2}\right)}^2+{\left(y_2+\frac{\lambda b_1+\mu b_2}{{\mu }^2+{\lambda }^2}\right)}^2}}$

shows the summary of the Jacobian forms of the constant matrix $\mathbf{A}$ with their corresponding distinct expressions of the function f for the four distinct cases in which the linear interpolations of the vector fields over triangular domains do not satisfy the law of mass conservation.

In Table 1, ${\left(y_1,y_2\right)}^T={\mathbf{V}}^{-\mathbf{1}}\mathbf{X}$ and ${\left(b_1,b_2\right)}^T={\mathbf{V}}^{-\mathbf{1}}\mathbf{b}$ where $\mathbf{V}$ satisfies $\mathbf{AV}=\mathbf{VJ}$, and $\mathbf{J}$ is one of the Jacobian matrices in the table. For $f\notin \left\{0,\infty \right\}$, the vectors ${\mathbf{V}}_l$ and $f{\mathbf{V}}_l$ produce the same streamlines (for more details we refer the readers to Section 2.2 of [9]). Consequently, mass conservation (MC) conditions are that the computed functions f in Table 1 are not equal to zero or infinity at any point on the triangular domains.

A cell-by-cell AMR approach is followed on the quadrilateral grid meshes. A quadrilateral cell on which one of the conditions MC is not satisfied is subdivided. A cell is refined by connecting the mid-points of opposite sides into four equally smaller quadrilateral cells. Applying the AMR method to the initial mesh, ${\mathrm{Mesh}}_0$, produces the first refined mesh, ${\mathrm{Mesh}}_1$, and by repeating the procedure nine more times, we obtain the tenth refined mesh, ${\mathrm{Mesh}}_{10}$. The coordinates of the centre of the isolated refined cells in the area of interest are then taken as an estimate of the centre of vortices.

Figure 2 summarises the process by which an initial mesh (${\mathrm{Mesh}}_0$) is transformed into ${\mathrm{Mesh}}_2$ through the application of the flow solver and the mesh refinement algorithm VDMAR. For illustrative purposes, in Figure 2, ${\mathrm{Mesh}}_0$ is a uniform grid mesh with a resolution of $5\times 5$. This iterative procedure is repeated, with each successive mesh being refined, until the final mesh (${\mathrm{Mesh}}_{10}$) is obtained.

3. Accuracy Verification and Numerical Results

This section presents the tenth refined mesh, ${\mathrm{Mesh}}_{10}$, and the location of vortices estimated in a driven cavity flow for an $Re$ of 100, 400, 1000, 2500, 5000, and 7500. The computed velocity fields are compared with established benchmark data from the literature to verify local accuracy. The abbreviations BR, BL, and TL denote the bottom right, bottom left, and top left vortices, respectively. The numbers that follow these abbreviations indicate the vortices’ order of size, from largest to smallest. The primary vortex is abbreviated as PV.

3.1. Accuracy Verification

To validate the local accuracy of the numerically computed velocity fields for the cases with $100\le Re\le 7500$, through Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, we present the u-velocity profile along a vertical line and the v-velocity profile along a horizontal line passing through the geometric centre of the cavity. It can be seen from Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 that the velocity profiles for all cases agree very well with those obtained by Erturk et al. [24], with a grid mesh of $601\times 601$ using the stream function–vorticity method. The agreement between the present velocity profiles and the results of Erturk et al. [24] verifies the accuracy of the flow solver used in this study.

3.2. The Refined Meshes, ${\mathrm{Mesh}}_{10}$

Figure 9 shows the tenth refined mesh, ${\mathrm{Mesh}}_{10}$, for the flow with $Re=100$. As seen in ${\mathrm{Mesh}}_{10}$, the mesh is refined mostly near the boundaries and at the bottom in the vicinity of the vortices BR1 and BL1. The refinements also identify the location of the PV. For the flow with $Re=400$, a similar mesh refinement pattern as for the flow with $Re=100$ was observed. ${\mathrm{Mesh}}_{10}$ for $Re=400$ identified the location of two additional secondary vortices, BR2 and BL2, which appeared near the bottom right and left corners, respectively.

Figure 10 presents ${\mathrm{Mesh}}_{10}$ for the flow with $Re=1000$. Additionally, for this case, Figure 11a–e provides zoomed-in views highlighting the isolated refined cells containing the centres of the vortices PV, BR, and BL. In these zoomed-in sections, red circles are used for illustrative purposes to mark the locations of the vortex centres. Specifically, they indicate the positions of PV, BR1, BL1, BR2, and BL2 for $Re=1000$.

For the flow with $Re=2500$, a mesh refinement pattern similar to that of the flow with $Re=1000$ was observed. However, the refined mesh ${\mathrm{Mesh}}_{10}$ for $Re=2500$ identified the location of an additional secondary vortex, TL1, which appears near the top left corner. ${\mathrm{Mesh}}_{10}$ for the flow with $Re=5000$ is presented in Figure 12. Compared to the case with $Re=2500$, the refined mesh in Figure 12 identifies the location of additional secondary vortices, BR3 and BL3, which appear near the bottom right and left corners, respectively. For the flow with $Re=7500$, a mesh refinement pattern similar to that of the flow with $Re=5000$ was observed. ${\mathrm{Mesh}}_{10}$ for the flow with $Re=7500$ is shown in Figure 13.

3.3. Vortex Centre Locations

Table 2. The estimated location of the centre of vortices for the flow with $100\le Re\le 7500$ using the 2D VDAMR method and comparison of the present results with corresponding benchmark reference data.

$\mathbf{Re}$	Vortex	Present (${\mathbf{Mesh}}_{10}$) DOF: 557,922 ($\mathbf{Re}=100$) DOF: 575,820 ($\mathbf{Re}=400$) DOF: 615,669 ($\mathbf{Re}=1000$) DOF: 1,660,092 ($\mathbf{Re}=2500$) DOF: 1,751,523 ($\mathbf{Re}=5000$) DOF: 2,745,150 ($\mathbf{Re}=7500$)	Ghia et al. [21] grid size: $129\times 129$ DOF: 33,282	Barragy and Carey [22] grid size: $257\times 257$ DOF: 132,098	Botella and Peyret [23] grid size: $160\times 160$ DOF: 76,164	Erturk et al. [24] grid size: $601\times 601$ DOF: 722,402	Shapeev and Lin [25] DOF: 226,574	Perumal and Dass [26] lattice size: $201\times 201$ DOF: 121,203
100	PV	(0.615723, 0.737402)	(0.6172, 0.7344)					(0.6156, 0.7366)
	BR1	(0.942969, 0.0617188)	(0.9453, 0.0625)					(0.9405, 0.0681)
	BL1	(0.0335938, 0.0351562)	(0.0313, 0.0391)					(0.032, 0.0371)
400	PV	(0.554004, 0.605762)	(0.5547, 0.6055)					(0.5534, 0.6039)
	BR1	(0.884912, 0.12251)	(0.8906, 0.1250)					(0.8896, 0.1247)
	BL1	(0.0507812, 0.0476562)	(0.0508, 0.0469)					(0.0501, 0.05)
	BR2	(0.992578, 0.00742187)	(0.9922, 0.0078)					–
	BL2	(0.00302734, 0.00302734)	(0.0039, 0.0039)					–
1000	PV	(0.530762, 0.566113)	(0.5313, 0.5625)		(0.5308, 0.5652)	(0.5300, 0.5650)	(0.530790112, 0.565240557)	(0.5302, 0.5635)
	BR1	(0.862109, 0.112109)	(0.8594, 0.1094)		(0.8640, 0.1118)	(0.8633, 0.1117)	(0.86404006, 0.11180617)	(0.8612, 0.1112)
	BL1	(0.0833008, 0.0780273)	(0.0859, 0.0781)		(0.0833, 0.0781)	(0.0833, 0.0783)	(0.08327318, 0.078095725)	(0.0826, 0.0776)
	BR2	(0.992578, 0.00742187)	(0.9922, 0.0078)		(0.99232, 0.00765)	(0.9917, 0.0067)	(0.992324852, 0.007650979)	–
	BL2	(0.00488281, 0.00488281)	–		(0.00490, 0.00482)	(0.0050, 0.0050)	(0.0048426963, 0.0048452406)	–
2500	PV	(0.519629, 0.544043)		(0.5188822, 0.5434181)		(0.5200, 0.5433)	(0.5197769, 0.5439244)
	BR1	(0.833438, 0.090625)		(0.834232, 0.0907512)		(0.8333, 0.0900)	(0.8344014, 0.09075692)
	BL1	(0.0841992, 0.110918)		(0.08439557, 0.1109646)		(0.0833, 0.1117)	(0.08424181, 0.1110061)
	BR2	(0.9904594, 0.009384439)		(0.990370, 0.00932132)		(0.9900, 0.0083)	(0.9904594, 0.009384439)
	BL2	(0.00613281, 0.00621094)		(0.00602392, 0.00621139)		(0.0067, 0.0067)	(0.006129716, 0.006158831)
	TL1	(0.0434375, 0.889062)		(0.0432916, 0.8890354)		(0.0433, 0.8900)	(0.04300225, 0.8893601)
5000	PV	(0.514648, 0.535664)	(0.5117, 0.5352)	(0.5151064, 0.5358696)		(0.5150, 0.5350)	(0.5150937, 0.5352620)	(0.5147, 0.5331)
	BR1	(0.801758, 0.0730859)	(0.8086, 0.0742)	(0.8041016, 0.07248652)		(0.8050, 0.0733)	(0.8046254, 0.07274733)	(0.8051, 0.0733)
	BL1	(0.0727539, 0.136621)	(0.0703, 0.1367)	(0.07248652, 0.1370297)		(0.0733, 0.1367)	(0.07285071, 0.1370629)	(0.0733, 0.1367)
	BR2	(0.978359, 0.0188281)	(0.9805, 0.0195)	(0.9786017, 0.01881959)		(0.9783, 0.0183)	(0.9783735, 0.01877724)	(0.9801, 0.0194)
	BL2	(0.0078125, 0.00796875)	(0.0117, 0.0078)	(0.0078985, 0.007898571)		(0.0083, 0.0083)	(0.007849997, 0.007996012)	(0.0075, 0.0069)
	BR3	(0.998828, 0.001172)	–	(0.9987615, 0.001172679)		(0.9983, 0.0017)	(0.9987908, 0.001208945)	–
	BL3	(0.000468, 0.000468)	–	(0.0005185267, 0.0004534772)		–		–
	TL1	(0.0636719, 0.909141)	(0.0625, 0.9102)	(0.063488808, 0.9092488)		(0.0633, 0.9100)	(0.06335428, 0.9092566)	(0.0615, 0.9071)
7500	PV	(0.512842, 0.53208)	(0.5117, 0.5322)	(0.5132184, 0.532095)		(0.5133, 0.5317)	(0.5130967, 0.5318922)	(0.5107, 0.5308)
	BR1	(0.788037, 0.0648926)	(0.7813, 0.0625)	(0.790025, 0.06483485)		(0.79, 0.065)	(0.7903051, 0.06516917)	(0.7898, 0.0646)
	BL1	(0.064209, 0.15249)	(0.0645, 0.1504)	(0.06416178, 0.1525889)		(0.065, 0.1517)	(0.06425712, 0.1529439)	(0.0638, 0.1542)
	BR2	(0.950586, 0.0431641)	(0.9492, 0.043)	(0.9517405, 0.04218909)		(0.9517, 0.0417)	(0.9515559, 0.04215257)	(0.9474, 0.0475)
	BL2	(0.0109863, 0.0116699)	(0.0117, 0.0117)	(0.01117058, 0.01178644)		(0.0117, 0.0117)	(0.01106468, 0.01177737)	(0.0129, 0.0134)
	BR3	(0.997217, 0.0027832)	(0.9961, 0.0039)	(0.997342, 0.0026578)		(0.9967, 0.0033)	(0.9972834, 0.002715411)	–
	BL3	(0.000634766, 0.000634766)	–	(0.0006488255, 0.0007140747)		–	(0.000688091, 0.0006881077)	–
	TL1	(0.0636719, 0.911328)	(0.0664, 0.9141)	(0.06685477, 0.9116328)		(0.0667, 0.9133)	(0.06665739, 0.9114901)	(0.0652, 0.9071)

presents the estimated coordinates of vortex centres for the cases with $100\le Re\le 7500$. The table compares the VDAMR estimated coordinates with the six reference centre locations reported by Ghia et al. [21], Barragy and Carey [22], Botella and Peyret [23], Erturk et al. [24], Shapeev and Lin [25], and Perumal and Dass [26]. The vortex centres reported in Table 2 by Perumal and Dass [26] were obtained using the LBM-MRT model. Additionally, the parameters, such as the grid resolution in terms of DOF for ${\mathrm{Mesh}}_{10}$, along with the grid mesh size and DOF from the corresponding references, are presented in Table 2. For each case, the errors relative to the reference vortex centres were computed as $\parallel (x_1,y_1)-(x_{\mathrm{ref}},y_{\mathrm{ref}})\parallel$/$\parallel (x_{\mathrm{ref}},y_{\mathrm{ref}})\parallel$, where $(x_1,y_1)$ are the VDAMR estimated coordinates and $(x_{\mathrm{ref}},y_{\mathrm{ref}})$ are the reference vortex centres. The summarised relative errors are presented in

Table 3. Summary of errors relative to the benchmark reference vortex centres from six reference studies in the literature. The errors are computed as $\parallel (x_1,y_1)-(x_{\mathrm{ref}},y_{\mathrm{ref}})\parallel$/$\parallel (x_{\mathrm{ref}},y_{\mathrm{ref}})\parallel$, where $(x_1,y_1)$ are the VDAMR estimated coordinates and $(x_{\mathrm{ref}},y_{\mathrm{ref}})$ are the reference vortex centres.

Re	Vortex	Ghia et al. [21]	Barragy and Carey [22]	Botella and Peyret [23]	Erturk et al. [24]	Shapeev and Lin [25]	Perumal and Dass [26]
100	PV	0.0035					0.0008
	BR1	0.0026					0.0073
	BL1	0.0911					0.0513
400	PV	0.0009					0.0024
	BR1	0.0069					0.0058
	BL1	0.0109					0.0345
	BR2	0.0005					–
	BL2	0.2238					–
1000	PV	0.0047		0.0012	0.0017	0.0011	0.0035
	BR1	0.0044		0.0022	0.0014	0.0022	0.0015
	BL1	0.0224		0.0006	0.0024	0.0006	0.0072
	BR2	0.0005		0.0003	0.0011	0.0003	–
	BL2	–		0.0095	0.0234	0.0080	–
2500	PV		0.0013		0.0011	0.0003
	BR1		0.0010		0.0008	0.0012
	BL1		0.0014		0.0086	0.0007
	BR2		0.0001		0.0012	0.0000
	BL2		0.0126		0.0790	0.0060
	TL1		0.0002		0.0011	0.0006
5000	PV	0.0040	0.0007		0.0010	0.0008	0.0035
	BR1	0.0085	0.0030		0.0040	0.0036	0.0041
	BL1	0.0160	0.0032		0.0036	0.0029	0.0036
	BR2	0.0023	0.0002		0.0005	0.0001	0.0019
	BL2	0.2767	0.0099		0.0502	0.0041	0.1093
	BR3	–	0.0001		0.0007	0.0001	–
	BL3	–	0.0763		–	0.0212	–
	TL1	0.0017	0.0002		0.0010	0.0004	0.0033
7500	PV	0.0016	0.0005		0.0008	0.0004	0.0034
	BR1	0.0091	0.0025		0.0025	0.0028	0.0024
	BL1	0.0129	0.0007		0.0068	0.0027	0.0105
	BR2	0.0015	0.0016		0.0019	0.0015	0.0057
	BL2	0.0432	0.0134		0.0432	0.0082	0.1387
	BR3	0.0016	0.0037		0.0007	0.0001	–
	BL3	–	0.0002		–	0.0775	–
	TL1	0.0042	0.0035		0.0039	0.0032	0.0049

.

Our results for $Re=100$ align closely with those reported by Ghia et al. [21] and Perumal and Dass [26]. The relative errors in the estimated coordinates of the vortex centres underscore the precision of our findings. Specifically, the relative error in the centre location of PV and BR1 is less than 0.0073 compared to results reported by Ghia et al. [21] and Perumal and Dass [26]. A minor discrepancy exists for the estimated centre location of BL1, where the relative error is slightly higher, approaching 0.09. However, our overall results remain highly consistent with those reported in [21,26], reinforcing the robustness and accuracy of our computational approach in capturing the flow dynamics at $Re=100$.

The results for the flow at $Re=400$ also show a good agreement with the findings of Ghia et al. [21] and Perumal and Dass [26]. The relative errors in the centre locations of PV, BR1, and BR2 are remarkably low, with discrepancies of less than 0.007 compared to the result reported in [21,26]. This indicates a high level of precision in the VDAMR method. Additionally, the relative error in the centre location of BL1 is similarly minimal, being under 0.04. However, a significant deviation is observed in the centre location of BL2, where our results exhibit a relative error of 0.2238. This significant difference suggests a divergence in the detection of BL2 between our study and Ghia et al. [21], highlighting a potential area for further investigation.

The results for $Re=1000$ demonstrate a high level of agreement with the findings of Botella and Peyret [23], Erturk et al. [24], Shapeev and Lin [25], and Perumal and Dass [26]. The relative errors in the centre locations of PV and secondary vortices (BR1, BR2) are impressively low, all being under 0.0035. For the secondary vortices BL1 and BL2, the relative error is slightly higher but remains under 0.025. Notably, our findings align most closely with those reported by Shapeev and Lin [25], where the relative errors for the centre locations of all vortices are consistently below 0.008. This agreement suggests that our methodology and computational approach are robust and in agreement with the results of Shapeev and Lin [25] for $Re=1000$.

For $Re=2500$, the VDAMR results show a remarkable agreement with those reported by Barragy and Carey [22], Erturk et al. [24], and Shapeev and Lin [25]. The comparison reveals that the relative errors in the centre locations of PV, BR1, BL1, BR2, and TL1 between our work and the studies mentioned above are consistently under 0.0086. Even though the location of BL2 exhibits a slightly higher relative error, it remains under 0.08. Among the reference studies, the closest alignment is observed with the results of Shapeev and Lin [25], where the relative errors for all VDAMR estimated centre locations are impressively less than 0.006.

The comparison of the present results for $Re=5000$ with those reported by Ghia et al. [21], Barragy and Carey [22], Erturk et al. [24], Shapeev and Lin [25], and Perumal and Dass [26] reveals a high level of agreement in the estimated locations of the vortex centres, with relative errors in the centre locations of PV, BR1, BR2, BR3, and TL1 being less than 0.009, and less than 0.016 for BL1. This indicates a consistent alignment across these studies, showcasing the robustness and reliability of the VDAMR method. Notably, a significant discrepancy is observed in the centre location of BL2 when compared to the result reported by Ghia et al. [21] and Perumal and Dass [26]. However, the VDAMR method’s estimated centre location of BL2 aligns closely with the findings of Barragy and Carey [22] and Shapeev and Lin [25], with relative errors of 0.0099 and 0.0041, respectively. This underscores the compatibility and accuracy of VDAMR in relation to these studies. Additionally, the centre location of BL3, which is only reported by Barragy and Carey [22] and Shapeev and Lin [25], shows good agreement with the VDAMR estimates, evidenced by relative errors of 0.0763 and 0.0212, respectively. Overall, the closest and hence the best agreement of the present results for $Re=5000$ is observed with the results of Shapeev and Lin [25], where the relative errors for the centre locations of PV, BR1, BL1, BR2, BR3, and TL1 are all under 0.004, and for BL3, the relative error is 0.0212. This comparison underscores the precision and efficacy of VDAMR in capturing vortex centre locations accurately, aligning well with the values reported in the literature.

For the case with $Re=7500$, the VDAMR estimated centre locations of PV, BR1, BR2, and TL show good agreement with the results of Ghia et al. [21], Barragy and Carey [22], Erturk et al. [24], Shapeev and Lin [25], and Perumal and Dass [26]. The relative errors in the estimated PV, BR1, BR2, and TL centre locations are all less than 0.009. The estimated centre location of BL1 aligns closely with that reported by Barragy and Carey [22], with a relative error of 0.0007. There is a significant discrepancy in the estimated centre location of BL2 compared to the results reported by Perumal and Dass [26], with a relative error of 0.1387. The present location of BL2 aligns most closely with those reported by Barragy and Carey [22] and Shapeev and Lin [25], with relative errors of 0.0134 and 0.0082, respectively. Moreover, the present estimated centre location of BL3 is also in good agreement with the literature, with results only reported by Barragy and Carey [22] and Shapeev and Lin [25] showing a relative error of less than 0.08. The closest alignment of the present results for the case with $Re=7500$ is observed with those presented in [25].

Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19 illustrate the streamlined contours and the flow characteristics near the vortex centres for flows with $Re=100$, $Re=400$, $Re=1000$, $Re=2500$, $Re=5000$, and $Re=7500$, respectively. In these figures, the red dots, added for illustrative purposes, mark the locations of the vortex centres as estimated using the VDAMR method.

4. Discussion

The agreement between the VDAMR method’s estimated vortex centre locations and the reference data from Ghia et al. [21], Barragy and Carey [22], Botella and Peyret [23], Erturk et al. [24], and Shapeev and Lin [25] is notably strong. Across a range of Reynolds numbers ($Re$), the estimated locations of the vortex centres align closely with the benchmark reference values, demonstrating the robustness and accuracy of the VDAMR method. The only significant deviations occur for the BL2 vortex centres at $Re=400$ and $Re=5000$, where there is a marked discrepancy with the results reported by Ghia et al. [21]. However, this disagreement is specific to the findings of Ghia et al. [21], as the VDAMR estimations for BL2 align well with the results from Barragy and Carey [22], Botella and Peyret [23], Erturk et al. [24], and Shapeev and Lin [25]. This consistency with multiple reference studies underscores the reliability of the VDAMR method across various Reynolds numbers. Moreover, the adaptive mesh refinement inherent in the VDAMR method allows for enhanced resolution in critical regions, ensuring precise capture of complex flow structures. The minor discrepancies can be attributed to differences in numerical schemes and boundary conditions in the six reference studies [21,22,23,24,25,26].

Building on the success of the 2D VDAMR method in accurately estimating vortex centres, future work will explore its application to flows with higher Reynolds numbers, where more complex and turbulent phenomena are present. In our previous works [27,28], we successfully verified the accuracy of the 3D VDAMR using analytical velocity fields. Future work will focus on evaluating the accuracy of the VDAMR using 3D computational velocity fields. This includes adapting the method to handle 3D flow problems, requiring additional refinement in the mesh adaptation process and further validation against 3D benchmark data. Additionally, investigating the method’s performance in real-world engineering applications, such as aerodynamic simulations or industrial fluid dynamics, will be a key focus. These extensions aim to enhance the method’s versatility and applicability to a broader range of fluid dynamics problems.

5. Conclusions

The accuracy of the 2D VDAMR method was rigorously validated through the analysis of six distinct cases of 2D lid-driven cavity flow, spanning a Reynolds number range of $100\le Re\le 7500$. A very coarse initial mesh was iteratively refined ten times. The resulting estimated vortex centre locations demonstrated a high level of agreement with established benchmarks across six references. This agreement was further supported by the numerically computed u− and v− velocity profiles, which fit very well with the benchmark data. The main conclusions that can be drawn are as follows:

• The present results, along with the errors relative to the benchmark reference vortex centres, affirm the robustness and precision of the 2D VDAMR method in estimating the centres of vortices.
• This study demonstrates that applying the VDAMR method improves estimation accuracy with more refinements. Since 2D VDAMR can be applied a finite number of times, further refinements can be made if the results are not accurate enough in certain cases.
• The VDAMR method’s performance confirms its potential in applications for simulating fluid dynamics problems, mainly where detailed vortex dynamics are crucial.