Choon-Bum Choi - Academia.edu (original) (raw)
Papers by Choon-Bum Choi
Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D)... more Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D) instability in flow past a rectangular cylinder ranging from a normal flat plate to a square cylinder with various aspect ratios (AR). The conventional modes A, B, and QP vanish as AR decreases towards a thin normal flat plate, whereas new modes A2 and QP2 emerge. For “thin” rectangular cylinders, mode QP2 becomes unstable first with increasing Reynolds number (Re). In a certain range of AR, mode A becomes unstable and then stable again as Re increases, resulting in a closed curve on the neutral stability diagram. For validation, 3D simulations were also performed.
Floquet stability analysis has been carried out to detect the onset of the secondary instability ... more Floquet stability analysis has been carried out to detect the onset of the secondary instability in the laminar flow past two identical square cylinders in tandem. Presence of a neighbouring cylinder alters flow topology, leading to change of stability characteristics. A parametric study has been performed with the gap between the two square cylinders as the key parameter. Three distinct patterns of base flow are found depending on the gap, and distinctive modes of the secondary instability are identified for each pattern of the base flow. The modes exhibit either odd reflection-translation symmetry or even RT symmetry. The critical Reynolds number and the corresponding dominant spanwise wavelength are presented for a wide range of the gap. Temporal and spatial characteristics of the dominant Floquet modes are described in detail. A hysteresis is also noticed in a certain range of the gap. Neutral stability curves are presented for some selected values of the gap. The averaged in-pl...
INTRODUCTION Vortex shedding from a cylindrical object immersed in cross-stream has been one of t... more INTRODUCTION Vortex shedding from a cylindrical object immersed in cross-stream has been one of the popular subjects in fluid mechanics, because it causes periodic force loading on the object, which could have negative effects on the object, such as vibration and noise-generation. Thus large effort has been made by researchers to control the vortex shedding for engineering purposes. Most of work has been focused on flow past a circular cylinder rather than a rectangular one because of its geometrical simplicity (Williamson, 1996; Perry et al., 1982). The main difference between the two is that separation points on a circular cylinder are dynamically determined by the flow field, while they should be fixed on some of the edges of a rectangular cylinder. A lot of work has been done on flow past a rectangular cylinder aligned with the main flow (Davis and Moore, 1982; Lyn et al., 1995; Obasaju et al., 1990), while relatively less attention has been paid to flow past an inclined rectang...
The large-scale vortices shed from a cylindrical object as a vortex generator can be used to enha... more The large-scale vortices shed from a cylindrical object as a vortex generator can be used to enhance heat transfer in a heat exchanger [1]. Furthermore, the large-scale vortices induce secondary vortices on the walls of a heat exchanger, which also play a positive role in heat transfer enhancement.© 2008 ASME
Physics of Fluids, Jun 1, 2014
Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D)... more Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D) instability in flow past a rectangular cylinder ranging from a normal flat plate to a square cylinder with various aspect ratios (AR). The conventional modes A, B, and QP vanish as AR decreases towards a thin normal flat plate, whereas new modes A2 and QP2 emerge. For “thin” rectangular cylinders, mode QP2 becomes unstable first with increasing Reynolds number (Re). In a certain range of AR, mode A becomes unstable and then stable again as Re increases, resulting in a closed curve on the neutral stability diagram. For validation, 3D simulations were also performed.
Transactions of the Korean Society of Mechanical Engineers B, 2008
Crossflow past an inclined square cylinder is numerically studied in the laminar range of Reynold... more Crossflow past an inclined square cylinder is numerically studied in the laminar range of Reynolds number (Re). We consider both the internal flow (Case A) where the cylinder is located inside a channel and the external flow (Case B) where the cylinder is positioned in an open domain. Flow characteristics change depending on Re and the inclination angle (IA) with respect to the main flow direction. For Case A, we show how the vortex shedding from the inclined square cylinder can enhance heat transfer from the channel walls, and present the optimum inclination angle for each Re to obtain the most effective heat transfer. For Case B, we present the contour diagrams for force coefficients on the cylinder as well as for Strouhal number of the vortex shedding on an Re-IA plane. The diagrams are useful in estimating flow-induced forces and the frequency of vortex shedding not only for a fixed inclined square cylinder but also for a slowly-rotating square cylinder.
Numerical investigation has been carried out for laminar flow past an inclined square cylinder in... more Numerical investigation has been carried out for laminar flow past an inclined square cylinder in cross freestream. In particular, inclination of a square cylinder with respect to the main flow direction can cause sudden shift of the separation points to other edges, resulting in drastic change of flow-induced forces on the cylinder such as Strouhal number (St) of vortex shedding, drag and lift forces on the cylinder, depending upon the inclination angle. Collecting all the numerical results obtained, we propose contour diagrams of drag/lift coefficients and Strouhal number on an Re-Angle plane. This study would be the first step towards understanding flow-induced forces on cylindrical structures under a strong gust of wind from the viewpoint of wind hazards.
A parametric study has been carried out to elucidate the characteristics of flow past a square cy... more A parametric study has been carried out to elucidate the characteristics of flow past a square cylinder inclined with respect to the main flow for Re150. Reynolds number and angle of incidence are the key parameters which determine the flow characteristics. This study would be the first step towards understanding flow pattern past a cylindrical structure under a strong gust of wind from the viewpoint of wind hazards. A complete classification of flow pattern has been obtained in the laminar region.
Transactions of the Korean Society of Mechanical Engineers B, 2006
Interference of the wakes behind two nearby bluff bodies is important in many engineering applica... more Interference of the wakes behind two nearby bluff bodies is important in many engineering applications. In this investigation, secondary instability (SI) in the flow past two square cylinders in side-by-side or tandem arrangements has been numerically studied via a Floquet analysis. An immersed boundary method was employed to implement the cylinders in the computational domain. The distance between the neighboring
Transactions of The Korean Society of Mechanical Engineers B, 2008
A numerical study has been carried out to investigate heat transfer enhancement in channel flow u... more A numerical study has been carried out to investigate heat transfer enhancement in channel flow using large-scale vortices. A square cylinder, inclined with respect to the main flow direction, is located at the center of the channel flow, generating a separation region and Karman vortices. Two cases are considered; one with a fixed blockage ratio and the other one with a fixed cylinder size. In both cases, the flow characteristics downstream of the cylinder significantly change depending on the inclination angle. As a result, heat transfer from channel wall is significantly enhanced due to increased vertical-velocity fluctuations induced by the large-scale vortices shed from the cylinder. Quantitative results as well as qualitative physical explanation are presented to justify the effectiveness of the inclined square cylinder as a vortex generator to enhance heat transfer from channel wall.
Transactions of The Korean Society of Mechanical Engineers B, 2009
Effect of computational domain size in simulating of periodic obstacle flow has been investigated... more Effect of computational domain size in simulating of periodic obstacle flow has been investigated for the flow past tube banks. Reynolds number, defined by freestream velocity ( ∞) and cylinder diameter (d), was fixed as 200, and center-to-center distance (P) as 1.5d. In-line square array and staggered square array were considered. Drag coefficient, lift coefficient and Strouhal number were calculated depending on domain size. Circular cylinders were implemented on a Cartesian grid system by using an immersed boundary method. Boundary condition is periodic in both streamwise and lateral directions. Previous studies in literature often use a square domain with a side length of P, which contains only one cylinder. However, this study reveals that the domain size is improper. Especially, RMS values of flow-induced forces are most sensitive to the domain size.
Physics of Fluids, 2013
The onset of the three-dimensional (3D) instability in the flow past two side-by-side square cyli... more The onset of the three-dimensional (3D) instability in the flow past two side-by-side square cylinders has been studied via Floquet stability analysis. The gap between the two cylinders is the key parameter that alters flow topology, leading to change of stability characteristics. Depending on the gap, four distinct flow patterns have been identified for base flow. For each pattern of base flow, distinctive Floquet modes of 3D instability have been found. The spatial and temporal characteristics of the Floquet modes are similar to those of mode A or mode B of the single square cylinder case, namely, odd or even Reflection-Translation symmetries. Six distinct Floquet modes have been identified in total, and the spatial and temporal characteristics of each mode are described in detail. The criticality of Floquet modes, including their critical Reynolds number and the associated dominant spanwise wave number, is presented for each gap considered in the current study. For a small gap, the flow past the two cy...
Physics of Fluids, 2012
Floquet stability analysis has been carried out to detect the onset of the secondary instability ... more Floquet stability analysis has been carried out to detect the onset of the secondary instability in the laminar flow past two identical square cylinders in tandem. Presence of a neighbouring cylinder alters flow topology, leading to change of stability characteristics. A parametric study has been performed with the gap between the two square cylinders as the key parameter. Three distinct patterns of base flow are found depending on the gap, and distinctive modes of the secondary instability are identified for each pattern of the base flow. The modes exhibit either odd reflection-translation symmetry or even RT symmetry. The critical Reynolds number and the corresponding dominant spanwise wavelength are presented for a wide range of the gap. Temporal and spatial characteristics of the dominant Floquet modes are described in detail. A hysteresis is also noticed in a certain range of the gap. Neutral stability curves are presented for some selected values of the gap. The averaged in-plane vorticity of the do...
Journal of Wind Engineering and Industrial Aerodynamics, 2012
ABSTRACT Numerical investigation has been performed for flows past an inclined square cylinder in... more ABSTRACT Numerical investigation has been performed for flows past an inclined square cylinder in the range of Re≤300 to elucidate the characteristics of the laminar three-dimensional wake behind the cylinder depending upon Reynolds number and angle of incidence (θ). An immersed boundary method was used for implementation of the cylinder on a Cartesian grid system. Both Floquet stability analysis and full three-dimensional simulation were employed to detect the onset of the secondary instability leading to three-dimensional flow, and to provide quantitative flow data. It was revealed that mode A becomes more unstable for “symmetric” flow configuration, while mode C is dominant for asymmetric flows in the range of 10°≤θ≤25°. The most unstable three-dimensional modes predicted by the Floquet stability analysis were well confirmed by the full 3D simulations which were conducted at Re=150, 200, 250, and 300, with varying angle of incidence. The full 3D simulations also provided the key flow characteristics such as mean flow-induced force/moment coefficients and Strouhal number of vortex shedding. It was seen that they are sensitive to slight inclination of the square cylinder, and the Re effects are insignificant except for mean lift coefficient. Vortical structures in the wake, taken from the results of the Floquet stability analysis and the full 3D simulations, respectively, were visualized by Q-contours, revealing good agreement between the two.
Journal of Heat Transfer, 2009
Heat transfer enhancement in channel flow by using an inclined vortex generator has been numerica... more Heat transfer enhancement in channel flow by using an inclined vortex generator has been numerically investigated. A square cylinder is located on the centerline of laminar channel flow, which is subject to a constant heat flux on the lower channel wall. As the cylinder is inclined with some angle of attack with respect to the main flow direction, flow characteristics change downstream of the cylinder, and significantly affect heat transfer on the channel wall. A parametric study has been conducted to identify the cause, and to possibly find the optimal inclination angle. It turns out that the increased periodic fluctuation of the vertical velocity component in the vicinity of the channel walls is responsible for the heat transfer enhancement. The large fluctuation is believed to be induced by the large-scale vortices shed from the inclined square cylinder, as well as by the secondary vortices formed near the channel walls.
Physics of Fluids, 2010
A parametric study has been carried out to elucidate the characteristics of flow past a square cy... more A parametric study has been carried out to elucidate the characteristics of flow past a square cylinder inclined with respect to the main flow in the laminar flow regime. Reynolds number and angle of incidence are the key parameters which determine the flow characteristics. Location of separation point is greatly affected by angle of incidence, thus determining the flow field around the square cylinder. The critical Reynolds number for periodic vortex shedding at each angle of incidence considered is obtained by using Stuart-Landau equation. Attempt is made to classify the related flow patterns from a topological point of view, resulting in three distinct patterns in total. A comprehensive analysis of the effects of Reynolds number and angle of incidence on flow-induced forces on the square cylinder is presented. Collecting all the results obtained, contour diagrams of force and moment coefficients, Strouhal number, rms of lift-coefficient fluctuation, as well as a flow-pattern diagram are proposed for the ranges of the two parameters considered in the current investigation. Finally, a Floquet stability analysis is presented to detect the onset of the secondary instability leading to three-dimensional flow. The proposed diagrams and the Floquet stability analysis shed light on better physical understanding of the flow past a square cylinder, which should be useful in many engineering applications.
Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D)... more Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D) instability in flow past a rectangular cylinder ranging from a normal flat plate to a square cylinder with various aspect ratios (AR). The conventional modes A, B, and QP vanish as AR decreases towards a thin normal flat plate, whereas new modes A2 and QP2 emerge. For “thin” rectangular cylinders, mode QP2 becomes unstable first with increasing Reynolds number (Re). In a certain range of AR, mode A becomes unstable and then stable again as Re increases, resulting in a closed curve on the neutral stability diagram. For validation, 3D simulations were also performed.
Floquet stability analysis has been carried out to detect the onset of the secondary instability ... more Floquet stability analysis has been carried out to detect the onset of the secondary instability in the laminar flow past two identical square cylinders in tandem. Presence of a neighbouring cylinder alters flow topology, leading to change of stability characteristics. A parametric study has been performed with the gap between the two square cylinders as the key parameter. Three distinct patterns of base flow are found depending on the gap, and distinctive modes of the secondary instability are identified for each pattern of the base flow. The modes exhibit either odd reflection-translation symmetry or even RT symmetry. The critical Reynolds number and the corresponding dominant spanwise wavelength are presented for a wide range of the gap. Temporal and spatial characteristics of the dominant Floquet modes are described in detail. A hysteresis is also noticed in a certain range of the gap. Neutral stability curves are presented for some selected values of the gap. The averaged in-pl...
INTRODUCTION Vortex shedding from a cylindrical object immersed in cross-stream has been one of t... more INTRODUCTION Vortex shedding from a cylindrical object immersed in cross-stream has been one of the popular subjects in fluid mechanics, because it causes periodic force loading on the object, which could have negative effects on the object, such as vibration and noise-generation. Thus large effort has been made by researchers to control the vortex shedding for engineering purposes. Most of work has been focused on flow past a circular cylinder rather than a rectangular one because of its geometrical simplicity (Williamson, 1996; Perry et al., 1982). The main difference between the two is that separation points on a circular cylinder are dynamically determined by the flow field, while they should be fixed on some of the edges of a rectangular cylinder. A lot of work has been done on flow past a rectangular cylinder aligned with the main flow (Davis and Moore, 1982; Lyn et al., 1995; Obasaju et al., 1990), while relatively less attention has been paid to flow past an inclined rectang...
The large-scale vortices shed from a cylindrical object as a vortex generator can be used to enha... more The large-scale vortices shed from a cylindrical object as a vortex generator can be used to enhance heat transfer in a heat exchanger [1]. Furthermore, the large-scale vortices induce secondary vortices on the walls of a heat exchanger, which also play a positive role in heat transfer enhancement.© 2008 ASME
Physics of Fluids, Jun 1, 2014
Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D)... more Floquet stability analysis has been carried out to detect the onset of the three-dimensional (3D) instability in flow past a rectangular cylinder ranging from a normal flat plate to a square cylinder with various aspect ratios (AR). The conventional modes A, B, and QP vanish as AR decreases towards a thin normal flat plate, whereas new modes A2 and QP2 emerge. For “thin” rectangular cylinders, mode QP2 becomes unstable first with increasing Reynolds number (Re). In a certain range of AR, mode A becomes unstable and then stable again as Re increases, resulting in a closed curve on the neutral stability diagram. For validation, 3D simulations were also performed.
Transactions of the Korean Society of Mechanical Engineers B, 2008
Crossflow past an inclined square cylinder is numerically studied in the laminar range of Reynold... more Crossflow past an inclined square cylinder is numerically studied in the laminar range of Reynolds number (Re). We consider both the internal flow (Case A) where the cylinder is located inside a channel and the external flow (Case B) where the cylinder is positioned in an open domain. Flow characteristics change depending on Re and the inclination angle (IA) with respect to the main flow direction. For Case A, we show how the vortex shedding from the inclined square cylinder can enhance heat transfer from the channel walls, and present the optimum inclination angle for each Re to obtain the most effective heat transfer. For Case B, we present the contour diagrams for force coefficients on the cylinder as well as for Strouhal number of the vortex shedding on an Re-IA plane. The diagrams are useful in estimating flow-induced forces and the frequency of vortex shedding not only for a fixed inclined square cylinder but also for a slowly-rotating square cylinder.
Numerical investigation has been carried out for laminar flow past an inclined square cylinder in... more Numerical investigation has been carried out for laminar flow past an inclined square cylinder in cross freestream. In particular, inclination of a square cylinder with respect to the main flow direction can cause sudden shift of the separation points to other edges, resulting in drastic change of flow-induced forces on the cylinder such as Strouhal number (St) of vortex shedding, drag and lift forces on the cylinder, depending upon the inclination angle. Collecting all the numerical results obtained, we propose contour diagrams of drag/lift coefficients and Strouhal number on an Re-Angle plane. This study would be the first step towards understanding flow-induced forces on cylindrical structures under a strong gust of wind from the viewpoint of wind hazards.
A parametric study has been carried out to elucidate the characteristics of flow past a square cy... more A parametric study has been carried out to elucidate the characteristics of flow past a square cylinder inclined with respect to the main flow for Re150. Reynolds number and angle of incidence are the key parameters which determine the flow characteristics. This study would be the first step towards understanding flow pattern past a cylindrical structure under a strong gust of wind from the viewpoint of wind hazards. A complete classification of flow pattern has been obtained in the laminar region.
Transactions of the Korean Society of Mechanical Engineers B, 2006
Interference of the wakes behind two nearby bluff bodies is important in many engineering applica... more Interference of the wakes behind two nearby bluff bodies is important in many engineering applications. In this investigation, secondary instability (SI) in the flow past two square cylinders in side-by-side or tandem arrangements has been numerically studied via a Floquet analysis. An immersed boundary method was employed to implement the cylinders in the computational domain. The distance between the neighboring
Transactions of The Korean Society of Mechanical Engineers B, 2008
A numerical study has been carried out to investigate heat transfer enhancement in channel flow u... more A numerical study has been carried out to investigate heat transfer enhancement in channel flow using large-scale vortices. A square cylinder, inclined with respect to the main flow direction, is located at the center of the channel flow, generating a separation region and Karman vortices. Two cases are considered; one with a fixed blockage ratio and the other one with a fixed cylinder size. In both cases, the flow characteristics downstream of the cylinder significantly change depending on the inclination angle. As a result, heat transfer from channel wall is significantly enhanced due to increased vertical-velocity fluctuations induced by the large-scale vortices shed from the cylinder. Quantitative results as well as qualitative physical explanation are presented to justify the effectiveness of the inclined square cylinder as a vortex generator to enhance heat transfer from channel wall.
Transactions of The Korean Society of Mechanical Engineers B, 2009
Effect of computational domain size in simulating of periodic obstacle flow has been investigated... more Effect of computational domain size in simulating of periodic obstacle flow has been investigated for the flow past tube banks. Reynolds number, defined by freestream velocity ( ∞) and cylinder diameter (d), was fixed as 200, and center-to-center distance (P) as 1.5d. In-line square array and staggered square array were considered. Drag coefficient, lift coefficient and Strouhal number were calculated depending on domain size. Circular cylinders were implemented on a Cartesian grid system by using an immersed boundary method. Boundary condition is periodic in both streamwise and lateral directions. Previous studies in literature often use a square domain with a side length of P, which contains only one cylinder. However, this study reveals that the domain size is improper. Especially, RMS values of flow-induced forces are most sensitive to the domain size.
Physics of Fluids, 2013
The onset of the three-dimensional (3D) instability in the flow past two side-by-side square cyli... more The onset of the three-dimensional (3D) instability in the flow past two side-by-side square cylinders has been studied via Floquet stability analysis. The gap between the two cylinders is the key parameter that alters flow topology, leading to change of stability characteristics. Depending on the gap, four distinct flow patterns have been identified for base flow. For each pattern of base flow, distinctive Floquet modes of 3D instability have been found. The spatial and temporal characteristics of the Floquet modes are similar to those of mode A or mode B of the single square cylinder case, namely, odd or even Reflection-Translation symmetries. Six distinct Floquet modes have been identified in total, and the spatial and temporal characteristics of each mode are described in detail. The criticality of Floquet modes, including their critical Reynolds number and the associated dominant spanwise wave number, is presented for each gap considered in the current study. For a small gap, the flow past the two cy...
Physics of Fluids, 2012
Floquet stability analysis has been carried out to detect the onset of the secondary instability ... more Floquet stability analysis has been carried out to detect the onset of the secondary instability in the laminar flow past two identical square cylinders in tandem. Presence of a neighbouring cylinder alters flow topology, leading to change of stability characteristics. A parametric study has been performed with the gap between the two square cylinders as the key parameter. Three distinct patterns of base flow are found depending on the gap, and distinctive modes of the secondary instability are identified for each pattern of the base flow. The modes exhibit either odd reflection-translation symmetry or even RT symmetry. The critical Reynolds number and the corresponding dominant spanwise wavelength are presented for a wide range of the gap. Temporal and spatial characteristics of the dominant Floquet modes are described in detail. A hysteresis is also noticed in a certain range of the gap. Neutral stability curves are presented for some selected values of the gap. The averaged in-plane vorticity of the do...
Journal of Wind Engineering and Industrial Aerodynamics, 2012
ABSTRACT Numerical investigation has been performed for flows past an inclined square cylinder in... more ABSTRACT Numerical investigation has been performed for flows past an inclined square cylinder in the range of Re≤300 to elucidate the characteristics of the laminar three-dimensional wake behind the cylinder depending upon Reynolds number and angle of incidence (θ). An immersed boundary method was used for implementation of the cylinder on a Cartesian grid system. Both Floquet stability analysis and full three-dimensional simulation were employed to detect the onset of the secondary instability leading to three-dimensional flow, and to provide quantitative flow data. It was revealed that mode A becomes more unstable for “symmetric” flow configuration, while mode C is dominant for asymmetric flows in the range of 10°≤θ≤25°. The most unstable three-dimensional modes predicted by the Floquet stability analysis were well confirmed by the full 3D simulations which were conducted at Re=150, 200, 250, and 300, with varying angle of incidence. The full 3D simulations also provided the key flow characteristics such as mean flow-induced force/moment coefficients and Strouhal number of vortex shedding. It was seen that they are sensitive to slight inclination of the square cylinder, and the Re effects are insignificant except for mean lift coefficient. Vortical structures in the wake, taken from the results of the Floquet stability analysis and the full 3D simulations, respectively, were visualized by Q-contours, revealing good agreement between the two.
Journal of Heat Transfer, 2009
Heat transfer enhancement in channel flow by using an inclined vortex generator has been numerica... more Heat transfer enhancement in channel flow by using an inclined vortex generator has been numerically investigated. A square cylinder is located on the centerline of laminar channel flow, which is subject to a constant heat flux on the lower channel wall. As the cylinder is inclined with some angle of attack with respect to the main flow direction, flow characteristics change downstream of the cylinder, and significantly affect heat transfer on the channel wall. A parametric study has been conducted to identify the cause, and to possibly find the optimal inclination angle. It turns out that the increased periodic fluctuation of the vertical velocity component in the vicinity of the channel walls is responsible for the heat transfer enhancement. The large fluctuation is believed to be induced by the large-scale vortices shed from the inclined square cylinder, as well as by the secondary vortices formed near the channel walls.
Physics of Fluids, 2010
A parametric study has been carried out to elucidate the characteristics of flow past a square cy... more A parametric study has been carried out to elucidate the characteristics of flow past a square cylinder inclined with respect to the main flow in the laminar flow regime. Reynolds number and angle of incidence are the key parameters which determine the flow characteristics. Location of separation point is greatly affected by angle of incidence, thus determining the flow field around the square cylinder. The critical Reynolds number for periodic vortex shedding at each angle of incidence considered is obtained by using Stuart-Landau equation. Attempt is made to classify the related flow patterns from a topological point of view, resulting in three distinct patterns in total. A comprehensive analysis of the effects of Reynolds number and angle of incidence on flow-induced forces on the square cylinder is presented. Collecting all the results obtained, contour diagrams of force and moment coefficients, Strouhal number, rms of lift-coefficient fluctuation, as well as a flow-pattern diagram are proposed for the ranges of the two parameters considered in the current investigation. Finally, a Floquet stability analysis is presented to detect the onset of the secondary instability leading to three-dimensional flow. The proposed diagrams and the Floquet stability analysis shed light on better physical understanding of the flow past a square cylinder, which should be useful in many engineering applications.