Unified Gas Kinetic Simulations of Lid-Driven Cavity Flows: Effect of Compressibility and Rarefaction on Vortex Structures

Abstract

We investigate and characterize the effect of compressibility and rarefaction on vortex structures in the benchmark lid-driven cavity flow. Direct numerical simulations are performed, employing the unified gas kinetic scheme to examine the changes in vortex generation mechanisms and the resulting flow structures at different Mach and Knudsen numbers. At high degrees of rarefaction, where inter-molecular interactions are minimal, the molecules mainly collide with the walls. Consequently, the dominant flow structure is a single vortex in the shape of the cavity. It is shown that increasing compressibility or decreasing rarefaction lead to higher molecular density in the cavity corners, due to more frequent inter-molecular collisions. This results in lower flow velocities, creating conditions conducive to the development of secondary and corner vortices. The physical processes underlying vortex formations at different Knudsen numbers, Mach numbers, and cavity shapes are explicated. A parametric map that classifies different regimes of vortex structures as a function of compressibility, rarefaction, and cavity shape is developed.

1. Introduction

Vortices are important features of fluid flows, as they lend structure to the flow field [1], play a key role in mass and momentum transport [2], and influence flow stability. Compressibility and rarefaction have a transformative effect on vortex structure in many applications of interest [3]. In this study, we focus on cavity flows, due to their importance in many engineering flows, such as micro-electro-mechanical systems (MEMS), space re-entry-vehicle tile cavities [4], and cooling devices. Lid-driven cavity flow is a widely accepted benchmark flow for investigating coherent structures in a simple geometric configuration [5]. It is ideally suited to the current investigation on the effects of Knudsen numbers on coherent vortex structures in fluid flows.

The study of rarefied gas flows within cavity configurations has significantly enhanced our understanding of heat and mass transfer in these complex systems. For instance, He, Qing, et al. [6] explored the dynamics of heat and mass transfer in a double-sided oscillatory lid-driven cavity, demonstrating the effects of compressibility, rarefaction, and lid oscillation frequency. Similarly, Zhu, M., et al. [7] conducted a computational study on rarefied gas flow and heat transfer in lid-driven cylindrical cavities, highlighting the impact of cavity geometry on flow characteristics. Moreover, Nabapure, D., et al. [8] utilized the direct simulation Monte Carlo (DSMC) method, to investigate flow behavior over an open cavity for different Mach and Knudsen numbers.

Analytical investigation of rarefied flows is challenging, as the flow physics cannot be accurately described by the Navier–Stokes equation. Naris and Valougeorgis [9] performed preliminary studies of Knudsen number effects on flow structures in cavity flows, using the linearized Boltzmann–BGK [10] equations. Since then, there have been other studies examining transport in rarefied cavity flows [11,12,13]. The work by [14] focused on rarefied thermally driven flow in a square cavity and rectangular bend for various Knudsen numbers. The study by [15] investigated the contributions of ballistic and collisional flows to anti-Fourier heat transfer in rarefied cavity flows, utilizing the DSMC method. While important progress has been made, there is a need for a detailed study to characterize the flow physics and vortical structures in different Knudsen and Mach number regimes.

The goal of this work is to explain the kinetic flow mechanisms underlying the transformation of vortex structures in rarefied cavity flows and to bridge the gap from continuum to rarefied vortex dynamics. Toward this end, we perform numerical simulations of two-dimensional lid-driven cavity flow, using the unified gas kinetic simulation (UGKS) approach, which is applicable over a wide range of Knudsen numbers. The UGKS method has proven to be effective for simulating both continuum and rarefied flows, as demonstrated by [16] in their work on high-speed flows and by [17] in their efficient simulation of multidimensional continuum and non-continuum flows. Additionally, Dai, L., et al. [18] utilized the UGKS to simulate a microchannel gas flow and heat transfer confined between isothermal and non-isothermal parallel plates, further establishing its versatility in various flow regimes.

The simulations performed in this study encompass a wide range of parameters, from low-speed incompressible to supersonic compressible, and from highly rarefied to continuum flows. The simulations also cover a variety of cavity shapes, including wide, square, and deep cavities. The computed flow structures are examined, to understand how compressibility, degree of rarefaction, and cavity aspect ratio affect the vortical structures. The specific objectives are to (a) explicate the physical mechanism of the development of vortex structures in rarefied flows and (b) propose a scheme for classifying vortex configurations in the Knudsen–Mach–Reynolds number parameter space. As previously mentioned, the benchmark lid-driven cavity flow is employed in this study, due its relevance in a variety of applications, including re-entry flows [4,5].

2. Vortex Dynamics in Continuum Flow Regime

We first examine the vorticity evolution equation in the continuum regime, to establish the baseline flow features and physical mechanisms in a cavity flow. Then, we introduce the added complexities arising from the effects of rarefaction.

The evolution of vorticity ($\overrightarrow{\omega }$) in the continuum regime is governed by the following equation derived from the fundamental Navier–Stokes equations:

$${\displaystyle \frac{D\overrightarrow{\omega }}{Dt}}={\displaystyle \frac{\partial \overrightarrow{\omega }}{\partial t}}+\left(\overrightarrow{u}·\overrightarrow{\nabla }\right)\overrightarrow{\omega }=\underset{I}{\underbrace{\left(\overrightarrow{\omega }·\overrightarrow{\nabla }\right)\overrightarrow{u}}}-\underset{II}{\underbrace{\overrightarrow{\omega }\left(\overrightarrow{\nabla }·\overrightarrow{u}\right)}}+\underset{III}{\underbrace{{\displaystyle \frac{1}{{\rho }^2}}\overrightarrow{\nabla }\rho \times \overrightarrow{\nabla }p}}+\underset{IV}{\underbrace{\overrightarrow{\nabla }\times \left({\displaystyle \frac{\overrightarrow{\nabla }·\overrightarrow{\overrightarrow{\tau }}}{\rho }}\right)}}+\underset{V}{\underbrace{\overrightarrow{\nabla }\times \overrightarrow{B}}},$$

(1)

where $\overrightarrow{u}, \rho , p, \tau$, and $\overrightarrow{B}$ are the velocity, density, pressure, viscous stress tensor, and body force of the continuum fluid. Each of the terms on the right-hand side of Equation (1) signifies the different processes through which vorticity evolves in the continuum regime. Term I on the right-hand side gives rise to ‘vortex stretching’, a key process in the incompressible turbulence cascade. In compressible flows, vortex stretching due to ‘dilatation’ is also important and is given by term $II$. Term $III$ represents the generation of vorticity due to baroclinic mechanisms. Viscous and body force contributions to vorticity evolution are given in terms $IV$ and V, respectively.

Typical flow structures observed in 2D continuum cavity flows are illustrated in Figure 1. These continuum vortex structures are well-established in the literature [19], and they help validate the current results at the low Knudsen number limit. A high-Reynolds-number square cavity flow is typically composed of a primary vortex (PV) in the center and three eddies: upper upstream eddy (UUE), upstream secondary eddy (USE), and downstream secondary eddy (DSE) in the corners. For the sake of classification, the large steady recirculation regions are defined as vortices, while the smaller unsteady recirculation regions that frequently appear near the cavity corners are called eddies. These structures are then used as the baseline, to assess various changes brought about by compressibility and rarefaction. It will be seen later that these corner eddies play a critical role in the formation of bigger vortical structures under appropriate circumstances.

Vorticity in transitional and rarefied regimes: While the continuum vorticity Equation (1) is not strictly valid in transitional and rarefied regimes, the definition of a ‘vortex structure’ is still possible. The continuum terminologies $I-V$ in Equation (1) may break down in the rarefied regime, with the viscous dissipation ($IV$) being the earliest to deviate upon the onset of kinetic effects. Therefore, it is imperative to examine the vorticity dynamics in rarefied regimes using kinetic features such as molecular number density and collision frequency.

3. Computational Methodology and Simulation Parameters

The numerical simulations in this study employ the unified gas kinetic scheme (UGKS), owing to its validity over a wide range of Knudsen numbers. The UGKS has a wide range of applicability, in terms of Mach and Knudsen numbers, as it directly solves the Boltzmann equation. It is also well-suited for multi-scale flows that exhibit continuum and rarefied features in different parts of the flow. For such flows, the alternative approach, the DSMC method, can be very expensive [12,20]. Additionally, the UGKS is capable of handling complex boundary conditions, and it lends itself to efficient high-performance computing, due to its inherent structure [21,22]. The fundamentals of this approach can be found in the pioneering works of Xu and coworkers [20,21,22,23]. As the name implies, the UGKS solves the kinetic Boltzmann equation rather than the continuum Navier–Stokes equation. The molecular velocity phase space is first discretized, and the Boltzmann equation for the particle velocity probability distribution function (pdf) is solved for each velocity cell. The UGKS methodology initially proposed by Xu and Huang [20] discretizes the entire phase space; it has delivered promising preliminary results in both the rarefied and continuum regimes [21,22]. The discretization is based on either the Gauss–Hermite or the Newton–Cotes quadrature, depending upon the degree of rarefaction needed [12,24]. Further computational efficiency is achieved by using the simple Bhatnagar–Gross–Krook approximation for the collision operator within each ensemble:

$${\displaystyle \frac{\partial f}{\partial t}}+u_j{\displaystyle \frac{\partial f}{\partial x_j}}+{\displaystyle \frac{\partial F_{j}f}{\partial u_j}}={\displaystyle \frac{g-f}{\tau }}.$$

(2)

Here, f is the particle velocity pdf and g is the equilibrium Maxwellian pdf corresponding to each discretized ensemble. The macroscopic properties are computed by integrating over all the ensembles [12,25]. While significant progress has been made in the literature to couple continuum and discrete solvers into hybrid schemes [26,27], limitations persist, due to the constraint on the time step to the mean collision time and the grid size to the mean free path. The efficiency of the UGKS is further enhanced by the fact that the time step and cell size are governed by the Courant–Friedrichs–Lewy (CFL) condition, rather than being restricted by the mean collision time or mean free path [12]. This allows for larger time steps and grid sizes in simulations, which significantly improves computational efficiency without sacrificing accuracy. A comprehensive validation of grid size and CFL number was conducted in previous studies, where comparison was performed against the DSMC method. Based on these validations, we employ the computational parameters that were found adequate in previous studies, as detailed in Refs. [5,12,28,29].

The physics underlying rarefaction effects can be effectively examined in two-dimensional (2D) flows [5]. In the 2D lid-driven cavity flows considered for this study, the parameters affecting the vorticity dynamics are as follows: (i) the extent of compressibility, characterized via the lid Mach number:

$$M_{lid}\equiv {\displaystyle \frac{U_{lid}}{a}};$$

(3)

(ii) the degree of rarefaction quantified by the Knudsen number:

$$Kn\equiv {\displaystyle \frac{{\lambda }_{\infty }}{L}};$$

(4)

and, (iii) the cavity aspect ratio:

$$AR\equiv {\displaystyle \frac{\mathrm{width}\mathrm{of}\mathrm{the}\mathrm{cavity}}{\mathrm{depth}\mathrm{of}\mathrm{the}\mathrm{cavity}}}.$$

(5)

In the above equations, $U_{lid}, a, {\lambda }_{\infty }, L$ denote lid velocity, speed of sound, molecular free path in free-stream conditions corresponding to the lid boundary condition, and global length scale, respectively. The Reynolds number, which is also important, is not an independent parameter, as it is a function of Mach and Knudsen numbers [28]:

$$Re\sim {\displaystyle \frac{Ma}{Kn}}$$

(6)

In the simulations performed in the study, the characteristic lengthscale (cavity length − L) varies from 1000${\lambda }_{\infty }$ to ${\lambda }_{\infty }$, to simulate conditions ranging from a continuum to an extremely rarefied flow. Correspondingly, the $Kn$ ranges from 0.001 to 1.0. The lid Mach number is varied from 0.1 to 3.0 in steps of 0.1. Hence, the cavity flow in incompressible, compressible subsonic, and supersonic regimes is simulated. The aspect ratios considered in this study correspond to a square, deep, and wide cavity, as shown in

Table 1. Height and width for various cavity sizes.

AR	Height	Width
1.0	1.0 L	1.0 L
2.5	2.5 L	1.0 L
0.4	1.0 L	2.5 L

.

All the cavity walls, including the lid, are set to be isothermal, maintaining a temperature of $T_{wall}$, which is set to the reference temperature $T_{ref}=273$ K. The initial temperature within the cavity is also set to the reference temperature. The CFL number is 0.9 and the grid points in each direction are uniformly spaced. The grid sensitivity study results suggest that the spacing of $L/90$ is adequate. Thus, the computations are performed over grids of size $N_x\times N_y=90\times 90$ for $AR=1$; $90\times 225$ for $AR=2.5$; and $225\times 90$ for $AR=0.4$. Here, $N_x$ and $N_y$ are the numbers of cells along x and y, respectively. For cases of extreme non-equilibrium, characterized by high $Ma$ and $Kn$, the velocity space is discretized using 100 × 100 Newton–Cotes quadrature points, to ensure accurate resolution of the flow dynamics. For lower $Kn$ cases, a 28 × 28 Gauss–Hermite quadrature is employed [12,30]. This approach balances computational efficiency with the need for precision across different flow regimes. The results used in the analysis are obtained after a statistically steady state of the flow is reached.

4. Flow Physics and Vortical Structures at Different $\mathit{Ma}$ and $\mathit{Kn}$

The changes in vortex structure as a function of the Knudsen number cannot be easily explained in terms of macroscopic fluid mechanics processes. In rarefied flows, explanations of the flow physics must be developed in terms of molecular number density and collision frequency. The number density, which is defined as the number of molecules per unit volume, is equal to the density normalized with the molecular mass. In this section, we first present the contours of collision frequency at different Mach numbers, Knudsen numbers, and cavity aspect ratios. Then, we proceed to examine the vortical structures and explain the changes, in terms of molecular number density and collision frequency. Finally, we develop a flow-structure map that delineates the Mach–Knudsen number parameter space into different regions of distinct vortex configurations.

4.1. Square Cavity

Figure 2 shows the collision frequency (CF) contours for a square cavity at different Mach and Knudsen numbers. For better illustration purposes, the collision frequency contour values are normalized with the quantity $a_{ref}Kn/L$, where $a_{ref}$ is the speed of sound at the reference state and L is the characteristic length. The reference collision frequency is used, as it incorporates important flow geometry dimensions and molecular properties. The collision frequency contours in Figure 2a,d,e are qualitatively similar to one another. The contours in Figure 2b,c,f exhibit a different pattern. Figure 3 shows the streamlines inside a square cavity at various Knudsen and Mach numbers. The gray-scale coloring of these streamlines represents the velocity magnitude non-dimensionalized with the lid velocity, and the background color contours represent the density. It is worth noting that the velocity magnitude significantly decreases with the increasing depth of the cavity.

In the baseline low $Kn$-$Ma$ case (Figure 3a), the primary vortex occupies most of the flow domain. At this Reynolds number, UUE is absent and the USE and DSE are very small in size. It is also important to mention that viscous transport due to inter-molecular collisions dominates in this continuum low-speed flow regime. At $Kn=0.005$, as $Ma$ increases, the size of the USE and DSE grows larger, as seen in Figure 3b. This is because the effective Reynolds number is higher in Figure 3b than in Figure 3a, as can be inferred from Equation (6). However, if $Ma$ is held at $0.3$ and $Kn$ is increased to $0.05$ (Figure 3c) then the USE and DSE become significantly weaker than the baseline case. In the case of high $Kn$ and $Ma$ (Figure 3d), the USE and DSE are completely absent. With increasing Knudsen numbers, the Reynolds number goes down from the baseline case and the ballistic transport begins to dominate [5]. Consequently, the molecules collide mostly with the wall rather than amongst themselves. It was observed in [5] that this leads to thermal transport from the cold upstream flow to the hot downstream wall. A second outcome is that the aggregate molecular flow takes the shape of the cavity, leading to a single primary vortex. With regard to density, it is nearly uniform throughout the cavity at low $Kn$, due to frequent inter-molecular collisions. However, at high $Kn$, the density varies significantly within the cavity. The lowest density occurs at the center of the primary vortex. The highest density occurs along the downstream wall and the bottom corners of the cavity, which experience the most collisions. The collision frequency in different parts of the flow will be examined in detail later.

At the continuum high-speed limit, the collision frequency becomes dominant in the USE and DSE regions (see Figure 2a,d,e). The fact that high collision frequency, high density, and low-velocity magnitude occur simultaneously in the USE and DSE regions provides an important insight. The USE and DSE regions are made up of a near-stagnant cluster of entrapped molecules that collide with other molecules in this cluster as well as the cavity walls. This cluster of entrapped molecules forms a barrier that prevents external molecules from penetrating the USE/DSE region. Thus, the external streamlines are diverted away from these near-stagnant USE/DSE regions. These USE/DSE regions gain angular momentum from the outer primary vortex, eventually forming secondary eddies. It should be noted that low-Mach-number or high-Knudsen-number flow patterns do not yield favorable conditions for the secondary eddies to develop at the USE/DSE regions. Hence, only a primary vortex prevails in such conditions, where the flow is driven by the moving lid, with only the cavity walls guiding the streamlines.

4.2. Deep Cavity

The normalized collision frequency contours in the deep cavity are illustrated in Figure 4. At low Knudsen numbers, the collision frequency increases as one traverses deeper into the cavity. The effect is much more dominant in the high-speed case (Figure 4d,e). As discussed earlier for the case of a square cavity, the effect of an increasing lid Mach number is similar to the effect of a decreasing Knudsen number—the transition from Figure 4a,b is comparable to the transition from Figure 4b–e.

Figure 5 shows the streamlines at different $Kn$ and $Ma$. Vorticity and velocity distribution within the cavity are indicated by the background color contours and streamline shades, respectively. In general, the velocity magnitude is very small, deep inside the cavity, as indicated by the lighter shade of the streamlines. The flow structures reveal that the number of vortices increases from one (Figure 5a) to three (Figure 5c) as the Knudsen number decreases. The primary vortex (vortex closest to the lid) has the maximum vortex strength, and it significantly decreases for secondary and subsequent higher-order vortices. The continuum regime exhibits the highest number of vortices for a given cavity depth. This can be explained by the increase in the frequency of inter-molecular collisions with decrease in the Knudsen number. Conversely, increasing the Knudsen number reduces inter-molecular collisions, leading to the dominance of a single vortex. This flow structure is consistent with the change in thermal transport properties with increasing Knudsen numbers reported by [12]. Figure 6 illustrates the density contours for a deep cavity at different Mach and Knudsen numbers. From this figure, two important inferences can be drawn. First, the effect of decreasing the lid Mach number is equivalent to the effect of increasing the flow Knudsen number. At the same Knudsen number, a lower lid Mach number leads to a simpler flow structure, consistent with the findings of [9]. This is also equivalent to reducing the Reynolds number. The second set of inferences pertains to density. Near the continuum regime, density within the cavity does not vary much with increase in the Mach number. At higher Mach numbers, the density at the bottom of the cavity is significantly higher than at the top, except at the top upstream corner (Figure 6c,d). From a micro-scale perspective, this effect can be attributed to the increasing entrapment of the molecules near the bottom half of the cavity, as can be inferred from the collision frequency contours.

Except at high Knudsen numbers, the collision frequency, density, and velocity magnitude deep inside the cavity favor the formation of a near-stagnant region, as in the case of a square cavity. The appearances of higher-order vortices beneath the primary vortex in a deep cavity (given suitable conditions) can be understood as follows: with increasing depth of the cavity, the USE and DSE grow in size, as more molecules become entrapped in these regions. Eventually, the DSE and USE merge, to form a secondary vortex beneath the primary vortex. This has also been observed for very-low-speed cavity flows, in a previous study by Naris and Valougeorgis [9]. A subsequent increase in depth can lead to the formation of tertiary and higher-order vortices, with progressive reduction in vorticity strength.

4.3. Wide Cavity

Figure 7 shows the collision frequency contours in wide cavities at different Mach and Knudsen numbers. Flow structures within a wide cavity are illustrated in Figure 8 for different Knudsen numbers at $Ma=3.0$. The streamline shade represents the velocity magnitude and the background color indicates the vorticity magnitude. It is immediately evident that the primary vortex dominates at high Knudsen and Mach numbers—Figure 8a. The flow domain consists only of a primary vortex, as the collision frequency is not favorable near the cavity corners (Figure 7b,c,f). The streamlines are turned, primarily due to the molecular collisions with the cavity walls. With a decrease in the Knudsen number or an increase in the lid Mach number, collision frequencies and densities become more suitable for the generation of secondary eddies near the cavity corners (Figure 7a,e). The USE and DSE emerge with a slight decrease in the Knudsen number, as seen in Figure 8b. In these cases, the core of the primary vortex is on the right (downstream) side of the cavity.

As the lid Mach number increases further, the USE grows in size as a consequence of more molecules being entrapped (Figure 7d). This eventually stabilizes, to form a secondary vortex (SV). The counter-rotating secondary vortex is centered on the upstream side of the cavity and is characterized by higher density—Figure 8c. The SV pushes the PV downstream, but a distorted portion of PV remains between the lid and the SV on the upstream side. Tertiary and higher-order vortices form with an increase in the cavity width, given that a favorable condition prevails. It should also be noted that the vortex strength and the velocity magnitude are smaller for the secondary vortex and much smaller for the USE and DSE. Figure 9 shows the density contours at different Mach and Knudsen numbers for wide cavities. At high Mach numbers, density is higher at the bottom corners compared to the cavity center (Figure 9d). High-speed continuum flows in wide cavities give rise to a denser left half (upstream side) compared to the downstream side (Figure 9c). In low-speed flows, the downstream side is generally denser than the upstream side, as evidenced by comparing Figure 9b,d. In general, the effect of decreasing flow speed is similar to the effect of an increasing Knudsen number [12], as observed earlier.

Density in vortical structures: Figure 10 plots the average density within three regions of a square cavity as a function of the Knudsen number and the lid Mach number. The three regions considered are (a) the primary vortex (PV) region, (b) the upstream secondary eddy (USE) region, and (c) the downstream secondary eddy (DSE) region. At $Kn=1.0$, ballistic transport dominates and the average density in the three regions does not change significantly with the Mach number. However, at low Knudsen numbers, the average density of the PV region decreases with the increase in the Mach number. Interestingly, in regions of USE and DSE, the average density increases with the Mach number. The rate of increase is larger with increasing Mach numbers. This is consistent with the collision frequency results, which indicate a reduced number of collisions at the center of the cavity with the increase in the Mach numbers. In contrast, the corner regions experience increased collisions. As compressibility increases at high Knudsen numbers, there is a marked increase in molecular density in the corners of the cavity. This is due to increased inter-molecular collisions, which enhance momentum transfer and trap molecules in the corners.

Vortex structure regimes: To summarize, Figure 11 delineates the $Kn-Ma$ parameter space into different regimes of vortex structure configuration. The vortex structure maps are prepared from a set of 360 simulations of square, deep, and wide cavities. The delineation boundaries are computed based on the Gaussian naive Bayes model [31]. It should be noted from Figure 12 that only the primary vortex exists in the flow domain at high degrees of rarefaction and low lid Mach numbers, which also corresponds to low Reynolds numbers. The eddies and higher-order vortices appear at higher Reynolds number continuum regimes. The classification remains almost unchanged for deep and wide cavities, with the difference being that the wide cavities are more sensitive to changes in the degree of rarefaction.

5. Conclusions

The effect of Mach and Knudsen numbers on vortex structures in rarefied cavity flows is examined in this study. The key observation is that the reduction of inter-molecular collisions results in a significant simplification of flow structure and a reduction in the intensity of vortical motion. A highly rarefied lid-driven cavity comprises only a primary vortex driven by the moving lid and shaped by the bounding cavity walls. The streamlines form a closed loop as a consequence of the molecules colliding principally against the cavity walls. There is insufficient accumulation of molecules at the corners to initiate the formation of corner eddies. Favorable conditions for corner eddy formation are (a) high density of molecule accumulation, (b) high collision frequency, and (c) low-velocity magnitude. As the degree of rarefaction decreases, more molecules accumulate in the DSE/USE regions. Small velocity magnitude and high collision frequency at these regions give rise to the formation of a near-stagnant cluster of molecules. These regions then act as an independent collection of fluid particles creating a barrier that deflects oncoming external molecules. The near-stagnant particles start to gain angular momentum from the colliding external molecules, to form secondary eddies. As the cavity depth is increased, more molecules become entrapped in the DSE/USE regions, allowing the eddies to grow in size. With a further increase in the cavity depth, the DSE and USE merge to form a secondary vortex. The process replicates itself with an increase in the cavity depth, given favorable flow conditions. As the width of the cavity increases, the USE region grows in size, leading to the formation of a secondary vortex. The shape of the primary vortex distorts, with a narrow region between the secondary vortex and the ‘no-slip’ moving lid. Vortex structure classification maps are generated in $Kn-Ma$ parameter space for square, deep, and wide cavities. In general, it is observed that the evolution of flow structures in wide cavities is more sensitive to the degree of rarefaction than that for deep cavities.

The parametric map characterizes the behavior of vortex dynamics in different flow regimes. In applications like micro-mechanical devices and space re-entry, the flow conditions can vary widely, leading to different combinations of these parameters. This map provides a clear overview of how heat- and mass-transfer processes—which are closely tied to the behavior of vortical structures within the cavity—respond to changes in these parameters. By classifying the flow regimes based on $Ma$ and $Kn$, the parametric map enables us to characterize and optimize flow behavior in various engineering contexts, ensuring better performance and reliability in systems operating under different conditions. Overall, this study provides important insights into the effect of rarefaction on vortex structure for different applications.