Single Relaxation Time Research Articles

Advanced thermal lattice Boltzmann method for the simulation of latent heat thermal energy in a porous storage unit

The current research expounds numerical investigation of key parameters effects, namely porosity ( ε= 0.4, 0.6, 0.7 and 0.8), Reynolds number (Re = 100, 200, 400 and 600) and Eckert number (Ec= 0, 1, 5 and 10) on the forced convective laminar flow and heat transfer through a horizontal porous channel filled with a metal foam structure impregnated with paraffin as a phase change material (PCM). The Darcy-Brinkman-Forchheimer model under the local thermal non-equilibrium (LTNE) condition is deemed at the representative elementary volume (REV) scale. The fully coupled equations of Navier-Stokes, Poisson’s equation, energy equations, and continuity equation were handled numerically via a thermal single relaxation time lattice Boltzmann method (TSRT-LBM). To facilitate implementation, all LB equations are based on the same speed discretization scheme (D2Q9). Three-population distribution functions were applied to simulate the fluid flow, and temperatures of the fluid and solid phases. Previously, the numerical model was validated by available cases. Then, a comprehensive investigation has been performed to investigate the influence of the aforementioned dimensionless numbers. All LBM results are found to be highly consistent with other numerical works. The outcomes reported that at lower porosities, the energy and exergy efficiencies increased with increasing Re and Ec. However, for large porosity values, the efficiencies were optimum for a critical Re ~ 400. To sum up, it can be stated that the implemented thermal lattice Boltzmann method has been demonstrated as a suitable approach study the thermal sensible energy storage.

Numerical analysis of forced convection heat transfer in a rectangular micro-channel totally filled with Ag/ water nano fluid in slip flow regime using the lattice Boltzmann method

Numerical simulation reported on heat transfer and fluid flow in a two-dimensional rectangular micro channel totally filled with Ag/water. The first –order slip/jump boundary conditions were uniformly imposed to the up and bottom walls. The governing conservation equations are translated in dimensionless form using the thermal Single Relaxation Time (T-SRT) modified Lattice Boltzmann Method (LBM) with double distribution functions (DDFs). The viscous dissipations effects are adopted into the energy equation. Effects of nanoparticle volume fraction φ, slip coefficient, B, on the flow of Nano fluid and heat transfer were studied. The results were interpreter in terms of slip velocity; temperature jump and Nusselt number. Based on the results found, it can be concluded that decreasing the values of slip coefficient enhances the convective heat transfer coefficient and consequently the Nusselt number (Nu) but increases the slip velocity at the wall and temperature jump values.

The structure factor in flowing wormlike micellar solutions

Abstract In this work we present theoretical and experimental studies on the structure factor in micellar systems. Cetyl trimethyl ammonium tosilate (CTAT) solutions in absence of salts exhibit a flow behavior similar to polymer solutions, while upon an increase in the ionic strength with NaCl, their behavior changes to that of a kinetic-controlled breakage-reformation process. The relaxation mechanism thus changes from that of a spectrum of relaxation times into an averaged dominant relaxation mode, representative of reversible kinetics. Micellar solutions exhibit mechanical instabilities under flow, which may be increased by concentration fluctuations. Elastic stresses increase concentration fluctuations, which are experimentally observed by rheo-optical techniques, such as Rheo-SALS, depicting characteristic “butterfly” patterns. Scattering patterns provide information on the structure factor, arising from concentration fluctuations of the flowing solution. Upon addition of salt to a CTAT solution, we observe agreement with the Cox-Merz rule, revealing that the solution is in the fast-breaking regime approaching a single relaxation time kinetics. This behavior cannot be predicted by polymer solution theories which do not incorporate the breakage-reformation kinetics. Here, we propose a kinetic model by which the structure factor can be calculated for these micellar solutions. A shift in the maximum of the scattering patterns as the shear rate is increased is predicted by the new theory, which can be explained by considering flow-induced changes in the structure of the solutions with the underlying kinetics of breakage-reformation. Agreement with experiments is exhibited.

The microscopic origins of stretched exponential relaxation in two model glass-forming liquids as probed by simulations in the isoconfigurational ensemble.

The origin of stretched exponential relaxation in supercooled glass-forming liquids is one of the central questions regarding the anomalous dynamics of these fluids. The dominant explanation for this phenomenon has long been the proposition that spatial averaging over a heterogeneous distribution of locally exponential relaxation processes leads to stretching. Here, we perform simulations of model polymeric and small-molecule glass-formers in the isoconfigurational ensemble to show that stretching instead emerges from a combination of spatial averaging and locally nonexponential relaxation. The results indicate that localities in the fluid exhibiting faster-than-average relaxation tend to exhibit locally stretched relaxation, whereas slower-than-average relaxing domains exhibit more compressed relaxation. We show that local stretching is predicted by loose local caging, as measured by the Debye-Waller factor, and vice versa. This phenomenology in the local relaxation of in-equilibrium glasses parallels the dynamics of out of equilibrium under-dense and over-dense glasses, which likewise exhibit an asymmetry in their degree of stretching vs compression. On the basis of these results, we hypothesize that local stretching and compression in equilibrium glass-forming liquids results from evolution of particle mobilities over a single local relaxation time, with slower particles tending toward acceleration and vice versa. In addition to providing new insight into the origins of stretched relaxation, these results have implications for the interpretation of stretching exponents as measured via metrologies such as dielectric spectroscopy: measured stretching exponents cannot universally be interpreted as a direct measure of the breadth of an underlying distribution of relaxation times.

A hybrid scheme of single relaxation time lattice Boltzmann and finite volume methods coupled with discrete ordinates method for combined natural convection and volumetric radiation in an enclosure

This paper is focused on the application of hybrid Single relaxation time lattice Boltzmann and finite volume methods in conjunction with discrete ordinates method to simulate coupled natural convection and volumetric radiation in differentially heated enclosure, filled with an absorbing, emitting and non-scattering gray medium. In this work, the velocity and temperature fields are calculated using lattice Boltzmann and finite volume methods respectively, whereas the radiative term is computed by the discrete ordinates method. This study is carried out for Pr = 0.71, a Rayleigh number range of 103 ≤ Ra ≤ 106, an optical thickness with values 0 ≤ τ ≤ 100, a Planck number ranging in 0.001≤ Pl ≤ 100 and an aspect ratio varying between 0.5 ≤ Ar ≤ 2. Results are presented in terms of streamlines, isotherms, velocity profiles and average Nusselt number. Based on the obtained results, it can be concluded that the presence of volumetric radiation is noteworthy. Its effect, as a function of Rayleigh number and the radiative properties, yields significant changes on the behavior of streamlines and isotherms. In the taller enclosure, the increase of average total Nusselt number with increasing Rayleigh number is less significant than that in the case of the shallow enclosure.

Under-resolved and large eddy simulations of a decaying Taylor\u2013Green vortex with the cumulant lattice Boltzmann method

We present a comprehensive analysis of the cumulant lattice Boltzmann model with the three-dimensional Taylor–Green vortex benchmark at Reynolds number 1600. The cumulant model is investigated in several different variants, using regularization, fourth-order convergent diffusion and fourth-order convergent advection with and without limiters. In addition, a cumulant model combined with a WALE sub-grid scale model is being evaluated. The turbulence model is found to filter out the high wave number contributions from the energy spectrum and the enstrophy, while the non-filtered cumulant methods show good correspondence to spectral simulations even for the high wave numbers. The application of the WALE turbulence model appears to be counter productive for the Taylor–Green vortex at a Reynolds number of 1600. At much higher Reynolds numbers ({hbox {Re}}=160{,}000) a deviation from the ideal Kolmogorov theory can be observed in the absence of an explicit turbulence model. Cumulant models with fourth-order convergent diffusion show much better results than single relaxation time methods.

Single relaxation time entropic lattice Boltzmann methods: A developer’s perspective for stable and accurate simulations

There have been contradicting reports in the literature regarding the application of the Entropic LBM to various flow problems. In this paper, we aim to evaluate the various formulations of the Entropic LBM and elaborate on the construction and numerical implementation required for successfully running such simulations. Tests conducted on four different test cases over a large parameter range, along with comparisons with Lattice-BGK results, allows us to comment on the behaviour of four Entropic LBM formulations namely: the iterative Entropic LBM, the 1st and 3rd order Essentially Entropic LBM, and a simplified analytical Entropic LBM approach. The benchmark problems considered are: the athermal shocktube problem, the decay of a Taylor-Green vortex, the square lid driven cavity and, the unsteady flow past a square cylinder. Our simulations reveal the superiority of the iterative and Essentially Entropic LBM over the Lattice-BGK model in terms of numerical stability and operating parameter range at fixed grid sizes. In terms of computational cost, the Lattice-BGK and 1st order Essentially Entropic LBM are comparable, followed by the 3rd order Essentially Entropic LBM and the iterative Entropic LBM. Our results suggest that although the iterative and Essentially Entropic LBM schemes can be used as-is with continuum problems, care is needed for situations where shocks occur. Furthermore, the iterative Entropic LBM model suggests a parametric nature not displayed by the Essentially Entropic LBM schemes. With the simplified Entropic LBM, we find that additional numerical safeguards are needed for typical flow problems.

Numerical investigation of porosity effect on a PCM's thermal performance in a porous rectangular channel via thermal lattice Boltzmann method

This paper performs a numerical assessment of the porosity effect on heat transfer under forced convection in an open-ended horizontal channel filled with a porous structure and a phase change material (PCM). To take into account forced convection between the solid matrix and the PCM, the dimensionless Darcy-Brinkman-Forchheimer equations and the local thermal non-equilibrium condition are considered. Simulations are achieved via the single relaxation time thermal lattice Boltzmann method using three distribution functions to handle the fluid velocity, and fluid and solid temperatures while translating the computational domain boundary conditions in terms of microscopic distribution functions. The numerical outcomes obtained were used to investigate the porosity effect (0.5 ≤ ε ≤ 0.9) on dynamic and thermal fields, entropy generation rate (Ns), and Bejan number (Be) of the considered system under influence of the Reynolds number (Re) during the charging and discharging processes. Based on the findings presented, it can be deduced that, among the various characteristics of metal foam such as conductivity and porosity, the decrease of the latter and the increase in Re speeds up the melting under the unsteady, forced and laminar convection. In addition, the overall irreversibility distribution in the proposed system is dominated by the heat transfer irreversibility.

Effects of pore geometry on the natural convective heat transmission in a porous cavity: a lattice Boltzmann simulation

In this study, a two-dimensional investigation on natural convection inside a porous cavity at pore-scale is performed using the lattice Boltzmann method with the focus on the effects of pores geometry. The model is independent of the shape, density, porosity, and arrangements of the pores, because the image of the porous cavity is binarized and given to the code as input. The influences of pores geometry, Rayleigh number, porosity, and pore/liquid thermal diffusivity ratio are studied. A D2Q9 multi-relaxation time and a D2Q9 single-relaxation time lattice Boltzmann model are used for the flow and energy equations, respectively, with two separate distribution functions approach. For this purpose, pores with square, circular, and star shapes are considered. The Rayleigh number is varied in the range of 103 to 106 for three different porosity values (0.43, 0.71, 0.85) and pore/liquid thermal diffusivity ratios of 6, 70, and 788, with water and air as working fluids. Local variations of hot wall Nusselt number revealed that the star, circular, and square pores have the highest peak values of the local Nusselt number, respectively. By contrast, the lowest local values occur in the porous cavity with circular pores. In the examined porous media, the maximum increase in the hot wall averaged Nusselt number is equal to 24.3% occurring at $$\varepsilon = 0.71$$ , when pores are varied from circular to star shapes at $${\text{Ra}} = 10^{4}$$ . Moreover, using the star pores with a porosity value of 0.43, the averaged Nusselt number on the hot wall is increased by 184.46% compared to the empty cavity at $${\text{Ra}} = 10^{3}$$ . Besides, it is shown that increasing the thermal diffusivity ratio higher than 70 has little effect on the hot wall averaged Nusselt number.

Thermal microscale gas flow simulation using wall function and bounce-back scheme: Modified lattice Boltzmann method

Rarefied gas flow with heat transfer was simulated utilizing a modified lattice Boltzmann method with two relaxation times (TRT) for the gas flow and single relaxation time (SRT) for heat transfer in the microchannel. The analytical function for capturing the effect of the Knudsen layer on the slip velocity was used. Anti-symmetric relaxation time was obtained (ASRT) by applying the boundary condition at the wall surfaces (bounce-back distribution functions) to improve the accuracy of slip velocity. The fluid temperature jump at the wall is considered by utilizing a second-order implicit scheme for capturing the temperature boundary condition at the wall. The Zou-He boundary conditions were also used for the flow at both the inlet and outlet of the microchannel. The bounce-back distribution functions were applied at the walls. In addition, a non-equilibrium temperature boundary condition is implemented at the geometry inlet. At the microchannel outlet, the zero-gradient temperature condition was also imposed. The simulation results were compared with those obtained from the linear Boltzmann method and direct simulation Monte Carlo (DSMC), and good agreement was found.

Phase-field lattice Boltzmann method with two-relaxation-time model for dendrite growth of a binary alloy with melt convection

To perform accurate, stable, and efficient phase-field lattice Boltzmann (PF-LB) simulations of the solidification of alloys with melt convection, a two-relaxation-time (TRT) model for calculating the collision term in the LB equation and an interpolated bounce-back (IBB) model for imposing a non-slip condition at a solid-liquid interface—(TRT-IBB model)—were introduced. The simulations of the plane Poiseuille flow and the dendrite growth of a binary alloy with forced convection were performed to confirm the computational accuracy and acceleration of PF-LB simulations upon introducing the TRT-IBB model. For dendrite growth with forced convection, the simulations of the TRT-IBB model were in good agreement with those of the conventional single-relaxation-time (SRT) model and the dissipative drag force (DDF) model for imposing the non-slip condition at the diffuse interface (SRT-DDF model). In addition, this study confirmed that the TRT-IBB model enabled stable simulations with ~500-times larger time increment than the SRT-DDF model. Thus, the developed TRT-IBB model was found to be extremely effective in the acceleration of PF-LB simulations for the solidification of alloys with melt convection.

Thermally nonlinear response of an exponentially graded disk using the Lord-Shulman model

The response of a disk made from a heterogeneous material within the framework of Lord-Shulman theory is investigated in this article. The theory admits a single relaxation time to avoid the infinite speed of temperature wave. It is assumed that all of the thermomechanical properties except for Poisson ratio and thermal relaxation time vary exponentially through the radial direction of the disk. Two coupled equations, namely, the axisymmetric equation of motion and the energy equation are obtained for the disk. The energy equation is kept in the nonlinear form and the linearization of the previous investigations is not performed in this work. These two equations are provided in a dimensionless presentation. After that, using the generalized differential quadrature method, a system of nonlinear algebraic equations is established. Following the Newmark time marching method, the temporal evolution of radial displacement and temperature at the nodal points of the disk are obtained. Comparison study is given to assure the validity of the proposed solution method. After that, novel numerical results are given to discuss the effects of involved parameter. It is shown that, under the nonlinear analysis, the magnitudes of radial and hoop stresses and radial displacement are underestimated.

Heat transport in semiconductor crystals: Beyond the local-linear approximation

We extend the application of the nonlocal theory of Mahan and Claro [Phys. Rev. B 38, 1963 (1988)] to solve the steady-state Boltzmann–Peierls transport equation within the framework of the single mode relaxation time approximation using the modified Debye–Callaway model. We consider the case of a semi-infinite semiconductor (SC) crystal with a boundary condition at its top surface that can be considered reasonably representative of time domain thermoreflectance (TDTR) and frequency domain thermoreflectance (FDTR) techniques. The approach allows us to obtain three different contributions to the heat flux density current that shed further light on the fundamental role of nonlocality and nonlinearity in heat transport by phonons in SC crystals. Through their intrinsic and implicit shuffling effect of the crystal momentum, phonon–phonon Normal scattering processes play a key role in the onset of thermal conduction as they introduce the temperature Laplacian as a second driving potential force for the heat flux density current in addition to the conventional Fourier's temperature gradient. The developed model suits quite fairly to interpret the frequency behavior of the reduced effective thermal conductivity of SC crystals that is observed in TDTR and FDTR experiments. We obtain an expression of the effective thermal conductivity of the SC crystal that is characterized with a universal spectral suppression function that captures and describes the role, the weight, and the contribution of quasi-ballistic and non-diffusive phonons. The spectral suppression function only depends on the ratio between the phonon mean free path and the thermal penetration depth as defined based on the diffusive Fourier's law.

Modified lattice Boltzmann solution for non-isothermal rarefied gas flow through microchannel utilizing BSR and second-order implicit schemes

Thermal microscale gas flow was simulated into a coplanar microchannel was simulated at a broad range of Knudsen numbers. Attempts were made to improve the accuracy of slip velocity on walls using a modified model with two relaxation times based upon the mesoscopic method. The temperature jump of fluid flow at the wall was captured by a model with a single relaxation time using a second-order implicit method. The Zou–He boundary conditions were employed at both inlet and outlet boundaries, and bounce-back/specular reflection distribution functions were applied to the impermeable walls. The non-equilibrium distribution functions were also used as the inlet temperature boundary condition. A fully developed temperature profile was considered at the microchannel outlet. A pressure ratio of 2 was considered in the simulations, and various parameters such as dimensionless pressure, pressure deviation from the linear pressure, dimensionless velocity at various Knudsen numbers, centerline velocity and slip velocity of the fluid, centerline temperature and fluid temperature on the wall, Nusselt number with changing Knudsen and Prandtl numbers, parameter k along the microchannel length and Cf·Re values were evaluated in the slip and transition flow regimes. The results of the direct simulation Monte Carlo were used to evaluate the correctness of the numerical model. The consistency of the two methods indicated the accuracy of the proposed method.

Numerical investigation of combined force convective–radiative heat transfer in a horizontal channel with lattice Boltzmann method

To simulate heat transfer processes, there are numerous numerical methods available; in the meantime, the lattice Boltzmann method is a novel method for analyzing a type of engineering problem that has been very much considered. In the present study, to analyze the heat transfer problem of fluid flow between two horizontal parallel planes and solve the governing equations, such as momentum equations, energy and radiation transfer, the lattice Boltzmann method was used, whereas energy and momentum equations are solved by two relaxation time models to increase stability in the lattice Boltzmann method. For radiation intensity, the radiative transfer equation is solved by the single relaxation time model, and then radiation flux divergence is added into the energy equation. The geometry governing the problem is a horizontal and two-dimensional channel with constant temperature walls. After explaining the lattice Boltzmann method and solving governing equations, the results were validated, and the effect of radiation heat transfer parameters (Planck number–Optical thickness–Scattering albedo) on the heat transfer problem for fluid flow between two horizontal plates was determined. By examining the impact of the related parameters, it was found that increasing the optical thickness, decreasing the Planck number and reducing the scattering albedo tend to enhance the effect of radiative heat transfer. More importantly, the optical thickness parameter has the most influence on the radiative heat transfer.

A high‐order phase‐field based lattice Boltzmann model for simulating complex multiphase flows with large density ratios

SummaryBased on phase‐field theory, we develop a simple and robust single relaxation time (SRT) lattice Boltzmann (LB) model for simulating complex multiphase flows with large density ratios (up to 2000). The approach utilizes two LB equations (LBE), one is used to describe the interface behavior and the other is used to calculate the hydrodynamic properties. To improve the accuracy and stability in capturing interface, the high‐order LB model derived through the fourth‐order Chapman‐Enskog expansion analysis is applied to Cahn‐Hilliard equation. For solution of the flow field, a modified particle distribution function in the pressure‐velocity formulation is constructed to correctly consider the effect of local density variation and continuous pressure field. With such improvement, the proposed multiphase LB model is able to maintain numerical stability for the problem with very large density ratio, lower relaxation parameter and complex interface. For validation, a series of benchmark cases are carried out. Specially, the full potential of the proposed model is validated by bubble bursting at a free surface, bubble rising and droplet splashing with a density ratio of 2000. In all test cases, the obtained numerical results agree well with the reference data or the analytical solutions.

Comparison between synthetic and rosemary-based antioxidants for the deep frying of French fries in refined soybean oils evaluated by chemical and non-destructive rapid methods

Using natural antioxidants instead of synthetics ones has been the tendency for retarding the oil deterioration during repeated deep frying process. Concerning this, the comparison between synthetic tertiarybutyl hydroquinone (TBHQ) and rosemary-based antioxidants in frying French fries was hereby evaluated. The quality and stability of frying oils with rosemary-based antioxidants showed higher efficiency than TBHQ regarding oxidation parameters (i.e., chemical indices, sensory, etc.), where rosmarinic acid (RA) was the most effective, followed by rosemary extracts (RE) and carnosic acid (CA). LF-NMR results were highly correlated (R2 = 0.909–0.998) to the change in physicochemical properties tested, where RA could effectively regulate the relaxation spectrum (T2) change and decrease single component relaxation time (T2W). The PCA graph of NIRS also revealed the dynamic change of antioxidant effectiveness in accordance with that obtained by chemical methods. Hence, both LF-NMR and NIRS can be expected as rapid and efficient methods for future monitoring the frying process.

Asymptotic equivalence of forcing terms in the lattice Boltzmann method within second-order accuracy.

We show the asymptotic equivalence of two forcing schemes in the lattice Boltzmann method (LBM) within second-order accuracy through the asymptotic analysis instead of the Chapman-Enskog analysis. We consider the single relaxation time LBM with the following two forcing schemes: the simplest scheme by He etal. [J. Stat. Phys. 87, 115 (1997)10.1007/BF02181482] (referred to as He forcing); the most popular scheme by Guo etal. [Phys. Rev. E 65, 046308 (2002)10.1103/PhysRevE.65.046308] (referred to as Guo forcing). It has been shown by using the Chapman-Enskog analysis that the He forcing leads the unphysical terms in the macroscopic equations due to the spatial and time derivatives of the body force, whereas the Guo forcing does not lead such terms. However, we find by using the asymptotic analysis that the order of the unphysical terms is comparable to or less than (Δx)^{3} for the continuity equation and (Δx)^{4} for the Navier-Stokes equations (where Δx is the lattice spacing). Therefore, not only the Guo forcing but also the He forcing give the macroscopic flow velocity and pressure for incompressible viscous fluid with relative errors of O[(Δx)^{2}]. To verify the result of the asymptotic analysis, we simulate two benchmark problems in which the body force is changed in space and time: a generalized Taylor-Green problem and a natural convection problem. As a result, we find that the calculated results of macroscopic variables by the He forcing converge to those by the Guo forcing at the second-order convergence rate. Therefore, we can conclude that the He forcing and the Guo forcing are equivalent within the second-order accuracy even for the space- and time-dependent body force.

Nonlinear generalized thermoelasticity of FGM finite domain based on Lord–Shulman theory

Current investigation deals with the nonlinear generalized thermoelasticity of a functionally graded material (FGM) layer. It is assumed that the elasticity modulus, thermal conductivity, thermal expansion coefficient, mass density and specific heat capacity follow the exponential type of property distribution while the thermal relaxation time is assumed to be constant for the FGM layer. The governing motion and energy equations for the layer are obtained under the assumptions of Lord and Shulman with a single relaxation time. The energy equation is kept in its nonlinear form and the assumption made in the previous investigations to linearize the energy equation is not performed in the current research. Two coupled equations are presented in a dimensionless presentation and solved by means of the generalized differential quadrature and Newmark time marching method. Numerical results of this study are validated for the case of homogeneous layer. After that, novel numerical results are provided to investigate the response of a layer subjected to thermal shock. As expected, the exponential index of the FGM layer, boundary conditions, coupling parameter, relaxation time and nonlinearity are all major factors on the response of the thermally affected layer. It is verified that under thermally nonlinear analysis the magnitudes of stress and displacement are underestimated. Also, temperature and displacement enhance as the exponential index of the FGM layer increases.

Simulation of Lid-Driven Cavity Flow with Internal Circular Obstacles

The Lattice Boltzmann method (LBM) has been applied for the simulation of lid-driven flows inside cavities with internal two-dimensional circular obstacles of various diameters under Reynolds numbers ranging from 100 to 5000. With the LBM, a simplified square cross-sectional cavity was used and a single relaxation time model was employed to simulate complex fluid flow around the obstacles inside the cavity. In order to made better convergence, well-posed boundary conditions should be defined in the domain, such as no-slip conditions on the side and bottom solid-wall surfaces as well as the surface of obstacles and uniform horizontal velocity at the top of the cavity. This study focused on the flow inside a square cavity with internal obstacles with the objective of observing the effect of the Reynolds number and size of the internal obstacles on the flow characteristics and primary/secondary vortex formation. The current LBM has been successfully used to precisely simulate and visualize the primary and secondary vortices inside the cavity. In order to validate the results of this study, the results were compared with existing data. In the case of a cavity without any obstacles, as the Reynolds number increases, the primary vortices move toward the center of the cavity, and the secondary vortices at the bottom corners increase in size. In the case of the cavity with internal obstacles, as the Reynolds number increases, the secondary vortices close to the internal obstacle become smaller owing to the strong primary vortices. In contrast, depending on the sizes of the obstacles ( R / L = 1/16, 1/6, 1/4, and 2/5), secondary vortices are induced at each corner of the cavity and remain stationary, but the secondary vortices close to the top of the obstacle become larger as the size of the obstacle increases.