article distributed under the Creative Commons Attribution License, which permits unrestricted us... more article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Unsteady three-dimensional compressible flow characteristics inside a control valve are investigated using numerical simulations. The motion profiles of the armature are obtained for the given electromagnetic forces using a finite element analysis. The three-dimensional unsteady compressible flow characteristics inside the control valve are studied using computational fluid dynamics simulation. It was found that flows patterns inside the valve are as follows: 1) accelerating subsonic flow, 2) choked flow at the throat 3) supersonic expansion, 4) curved strong shock, 5) decelerating subsonic flow. With the inlet pressure stepping up, the free falling time is decreased and the operation frequency is increased. It is believed that present results can be used for the design of high-speed switching ...
Various types of internal defects occur during manufacturing and handling of composite materials.... more Various types of internal defects occur during manufacturing and handling of composite materials. It is practically impossible to manufacture composite structures without defects, making it crucial to understand the effect of defects on their failure behavior to maintain structural safety. In this work, the effect of pre-defects on the failure behavior of plain weave textile composites was studied. Unit cell configurations with symmetric, in-phase, and shifted fiber tow arrangements were considered. Inter-laced warp and fill tows and matrix pockets of plain weave unit cells were modeled in three-dimensional finite elements, and cohesive elements were inserted between all bulk elements to account for the fracture modes of the fiber and matrix direction failure of warp and fill tows, matrix pocket failure, and interface failure. Unit cell models containing pre-defects of voids, tow-matrix pocket separation, warp-fill tow separation, and cracks in the warp and fill tows were analyzed, ...
Journal of the Korean Society for Aeronautical & Space Sciences, 2005
The wake patterns and thrust characteristics of dual flapping airfoils in a biplane configuration... more The wake patterns and thrust characteristics of dual flapping airfoils in a biplane configuration are investigated using an unsteady panel method. To trace complicated wake shapes behind airfoils, a core addition scheme, a vortex core model, and the fourth order Runge-Kutta convection scheme are employed. Present results are verified by comparing them with flow visualization, exact solution and published computed results. The thickness and camber of thick airfoils has an effect of decreasing thrust. The airfoils produce maximum thrust when the phase angles between plunging and pitching motions are both 90 and 120 degrees. Thrust increases as the plunge velocity is increased, which is also found as the pitch amplitude is stepped up. Thrust decreases when the distance between the airfoils is less than 0.6c.
Unsteady three-dimensional compressible flow characteristics inside a control valve are investiga... more Unsteady three-dimensional compressible flow characteristics inside a control valve are investigated using numerical simulations. The motion profiles of the armature are obtained for the given electromagnetic forces using a finite element analysis. The three-dimensional unsteady compressible flow characteristics inside the control valve are studied using computational fluid dynamics simulation. It was found that flows patterns inside the valve are as follows: 1) accelerating subsonic flow, 2) choked flow at the throat 3) supersonic expansion, 4) curved strong shock, 5) decelerating subsonic flow. With the inlet pressure stepping up, the free falling time is decreased and the operation frequency is increased. It is believed that present results can be used for the design of high-speed switching electromagnetic control valves.
The wake shapes behind wings in formation flight are very important to the aerodynamics and perfo... more The wake shapes behind wings in formation flight are very important to the aerodynamics and performances of aircrafts. In the present study, a discrete vortex methood is extended to handle the wake rollups behind multiple wings. It was found that the relative distance between the wings and the rotational direction of the wingtip vortices have significant effect on the movement of the wingtip vortices. When the wings are close to each other, the wingtip vortices moved faster than the wings of large relative distances. The vortex pair of opposite signs generated from each wingtip has an effect of moving the wingtip vortices upward. The relative height between the wings has an effect of moving the wingtips along the centerline of each vortex. The wakeshape behind multiple wings is a function of the relative distances and thus is dependent on the configuration of the formation flight. In the futhre, a study on the vortex movement pattern will be studied.
An intake is a device for supplying external air into the internal engine of an aircraft. In orde... more An intake is a device for supplying external air into the internal engine of an aircraft. In order to reduce the radar reflection on the engine face for reducing the observability, intake has an S-shaped duct. When applied to s-duct of the fuselage, the uniformity of the flow at the engine face varies depending on the state of flow at the inlet of the duct. In this study, the effect of active flow control at the inlet of S-duct was analyzed using a commercial computational fluid dynamics tool. The purpose of this study is to investigate the effect of the active flow control factors for an RAE M 2129 S-duct. The study was conducted by changing the position, length, and mass flow of the active flow control, and the mass flow was within 1% of the s-duct internal mass flow. The OLHD method was applied to 15 models for three factors and the k-ω SST turbulence model was used to predict the flow separation and secondary flow by adverse pressure gradient. In conclusion, the performance of the S-duct was found to have the greatest influence on the position of the flow control, and it was confirmed that the performance was the worst when the position was 0 in the duct standard.
International Journal of Aeronautical and Space Sciences, 2002
Aerodynamic analysis of NACA wings moving with a constant speed over guideways are performed usin... more Aerodynamic analysis of NACA wings moving with a constant speed over guideways are performed using an indirect boundary element method (potential-based panel method). An integral equation is obtained by applying Green's theorem on all surfaces of the fluid domain. The surfaces over the wing and the guideways are discretized as rectangular panel elements. Constant strength singularities are distributed over the panel elements. The viscous shear layer behind the wing is represented by constant strength dipoles. The unknown strengths of potentials are determined by inverting the aerodynamic influence coefficient matrices constructed by using the no penetration conditions on the surfaces and the Kutta condition at the trailing edge of the wing. The aerodynamic characteristics for the wings flying over nonplanar ground surfaces are investigated for several ground heights.
Journal of the Korean Society for Aeronautical & Space Sciences, 2020
Behaviors of wake vortices generated by an aircraft affect the performance and flight stability o... more Behaviors of wake vortices generated by an aircraft affect the performance and flight stability of flying aircraft in formation flight. In the present study, the trajectories of the wake vortices behind airplanes in close formation flight were computed using a Fourier spectral method. The behavior of wake vortices showed complex patterns depending on the initial circulation and the relative positions between the vortices. In the initial stage, the wake vortex movement was affected by the nascent vortex. When the vortex becomes closer to the other vortex, then a new trajectory is formed. When the viscous effect becomes dominant, the core radius increases. Thus, a new vortex moving near the existing vortex can have strong interaction with each other, resulting in the complicated behavior of wake vortices. In the future, the ground effect on the behavior of the wake vortices during takeoff and landing will be studied.
Transactions of the Korean Society of Mechanical Engineers B, 2006
A numerical study on the propulsive characteristics of a biomimetic foil is done by developing an... more A numerical study on the propulsive characteristics of a biomimetic foil is done by developing an unsteady linearly-varying strength vortex method. A biomimetic foil is represented as a deforming foil with a tail fin. Present method is verified by comparing the simulated results with results using finite element and finite volume methods. A new boundary condition is imposed by considering the relative rotational velocity, which has not been included in the previous published literature. It is found that the undulation amplitude increases the thrust while maximum thickness is stepping down the thrust. It is also shown that there exists an optimal frequency for maximum thrust generation. It is believed that present results can be used in the investigation of the propulsive characteristics of the biomimetic deforming foil.
International Journal of Aerospace System Engineering, 2016
Experimental study on the unsteady aerodynamics analysis and power consumption of a folding wing ... more Experimental study on the unsteady aerodynamics analysis and power consumption of a folding wing is accomplished using a wind tunnel testing. A folding wing model is fabricated and actuated using servo motors. The flapping wing consists of an inboard main wing and an outboard folding wing. The aerodynamic forces and consumed powers of the flapping wing are measured by changing the flapping and folding wings inside a low-speed wind tunnel. In order to calculate the aerodynamic forces, the measured forces are modified using static test data. It was found that the effect of the folding wing on the flapping wing's total lift is small but the effect of the folding wing on the total thrust is larger than the main wing. The folding motion requires the extra use of the servo motor. Thus, the amount of the energy consumption increases when both the wings are actuated together. As the flight speed increases, the power consumption of the folding wing decreases which results in energy saving.
In this paper, a wind-tunnel test is accomplished to investigate the roll characteristics of a va... more In this paper, a wind-tunnel test is accomplished to investigate the roll characteristics of a variable-span wing flying inside a channel. The factors that affect the roll characteristics of the wing were identified by analyzing the measured data; accordingly, when the wing is flying without both the ground and sidewall effects, the asymmetric wing extension causes the roll moment. Both the ground and the sidewall can increase the roll moment, but when the wing is affected by both the ground and the sidewall, the roll moment does not increase as much as the case where the wing is only affected by the ground. Also, the aerodynamic characteristics of the flying wing inside a channel are the nonlinear function of the wing height and the gap between the wingtip and the sidewall, both of which should be considered in a study of the stability and the flight control of the wing-in-ground effect of the vehicle flying inside a channel.
Journal of the Korean Society for Aeronautical & Space Sciences, 2007
Insect and birds in nature flap their wings to generate fluid dynamic forces that are required fo... more Insect and birds in nature flap their wings to generate fluid dynamic forces that are required for locomotion. To develop a feasible flapping MAV, it is of crucially important to study the fundamental relations between flapping motion and thrust generation. In this paper, the onset conditions of the thrust generation of a heaving flat plate is investigated using a Lattice-Boltzmann method. For a fixed heaving amplitude of h/C
In this paper, the fluid dynamic forces and performances of a moving airfoil in the low Reynolds ... more In this paper, the fluid dynamic forces and performances of a moving airfoil in the low Reynolds number flow is addressed. In order to simulate the necessary propulsive force for the moving airfoil in a low Reynolds number flow, a lattice-Boltzmann method is used. The critical Reynolds and Strouhal numbers for the thrust generation are investigated for the four propulsion types. It was found that the Normal P&D type produces the largest thrust with the highest efficiency among the investigated types. The leading edge of the airfoil has an effect of deciding the force production types, whereas the trailing edge of the airfoil plays an important role in augmenting or reducing the instability produced by the leading edge oscillation. It is believed that present results can be used to decide the optimal propulsion types for the given Reynolds number flow.
The prediction of the separation trajectories of external stores released from a military aircraf... more The prediction of the separation trajectories of external stores released from a military aircraft is an important task in the aircraft design area having the objective to define the operational and release envelopes. This paper presents the results obtained for store separation by employing commercial softwares, FLUENT and CFD-FASTRAN. FLUENT treats the rigid body motion by employing a remeshing scheme. CFD-FASTRAN uses Chimera(overset) grid and interpolations. It was found that, for the prediction of the trajectories and behavior of the stores separated from the wing, both codes show the good agreement with the experimental results.
The unsteady evolution of trailing vortex sheets behind a wing in ground effect is simulated usin... more The unsteady evolution of trailing vortex sheets behind a wing in ground effect is simulated using an unsteady discrete vortex panel method. The ground effect is included by image method. The present method is validated by comparing the simulated wake roll-up shapes to published numerical results. When a wing is flying in a very close proximity to the ground, the optimal wing loading is parabolic rather than elliptic. Thus, a theoretical model of wing load distributions is suggested, and unsteady vortex evolutions behind lifting lines with both elliptic and parabolic load distributions are simulated for several ground heights. For a lifting line with elliptic and parabolic loading, the ground has the effect of moving the wingtip vortices laterally outward and suppressing the development of the vortex. When the wing is in a very close proximity to the ground, the types of wing load distributions does not affect much on the overall wake shapes, but parabolic load distributions make the wingtip vortices move more laterally outward than the elliptic load distributions.
The development of empirical model for the hydraulic transport of sand-water mixtures is importan... more The development of empirical model for the hydraulic transport of sand-water mixtures is important for the design of economical solid-liquid transportation system in chemical and waste-disposal industries. The hydraulic transport characteristics of sand-water mixtures in circular pipelines are numerically investigated by using the FLUENT commercial software. Eulerian granular multiphase (EGM) model with the k-e turbulent model is used for the computation. Present method is validated by the computed values with the measured data. The effect of the concentration and pipe sizes on the relative solid effect is numerically investigated. It is found that the effect of the volumetric delivered concentration on both hydraulic gradient and solid effect increases as the Reynolds number decreases. When the Reynolds number is small, the increase in the volumetric delivered concentration has an effect of decreasing the hydraulic gradient whereas the solid effect increases with the volumetric delivered concentration stepping up. The effect of the pipe diameter is not the critical parameter for deciding the values of the relative solid effect in the sand-water mixture transportation.
article distributed under the Creative Commons Attribution License, which permits unrestricted us... more article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Unsteady three-dimensional compressible flow characteristics inside a control valve are investigated using numerical simulations. The motion profiles of the armature are obtained for the given electromagnetic forces using a finite element analysis. The three-dimensional unsteady compressible flow characteristics inside the control valve are studied using computational fluid dynamics simulation. It was found that flows patterns inside the valve are as follows: 1) accelerating subsonic flow, 2) choked flow at the throat 3) supersonic expansion, 4) curved strong shock, 5) decelerating subsonic flow. With the inlet pressure stepping up, the free falling time is decreased and the operation frequency is increased. It is believed that present results can be used for the design of high-speed switching ...
Various types of internal defects occur during manufacturing and handling of composite materials.... more Various types of internal defects occur during manufacturing and handling of composite materials. It is practically impossible to manufacture composite structures without defects, making it crucial to understand the effect of defects on their failure behavior to maintain structural safety. In this work, the effect of pre-defects on the failure behavior of plain weave textile composites was studied. Unit cell configurations with symmetric, in-phase, and shifted fiber tow arrangements were considered. Inter-laced warp and fill tows and matrix pockets of plain weave unit cells were modeled in three-dimensional finite elements, and cohesive elements were inserted between all bulk elements to account for the fracture modes of the fiber and matrix direction failure of warp and fill tows, matrix pocket failure, and interface failure. Unit cell models containing pre-defects of voids, tow-matrix pocket separation, warp-fill tow separation, and cracks in the warp and fill tows were analyzed, ...
Journal of the Korean Society for Aeronautical & Space Sciences, 2005
The wake patterns and thrust characteristics of dual flapping airfoils in a biplane configuration... more The wake patterns and thrust characteristics of dual flapping airfoils in a biplane configuration are investigated using an unsteady panel method. To trace complicated wake shapes behind airfoils, a core addition scheme, a vortex core model, and the fourth order Runge-Kutta convection scheme are employed. Present results are verified by comparing them with flow visualization, exact solution and published computed results. The thickness and camber of thick airfoils has an effect of decreasing thrust. The airfoils produce maximum thrust when the phase angles between plunging and pitching motions are both 90 and 120 degrees. Thrust increases as the plunge velocity is increased, which is also found as the pitch amplitude is stepped up. Thrust decreases when the distance between the airfoils is less than 0.6c.
Unsteady three-dimensional compressible flow characteristics inside a control valve are investiga... more Unsteady three-dimensional compressible flow characteristics inside a control valve are investigated using numerical simulations. The motion profiles of the armature are obtained for the given electromagnetic forces using a finite element analysis. The three-dimensional unsteady compressible flow characteristics inside the control valve are studied using computational fluid dynamics simulation. It was found that flows patterns inside the valve are as follows: 1) accelerating subsonic flow, 2) choked flow at the throat 3) supersonic expansion, 4) curved strong shock, 5) decelerating subsonic flow. With the inlet pressure stepping up, the free falling time is decreased and the operation frequency is increased. It is believed that present results can be used for the design of high-speed switching electromagnetic control valves.
The wake shapes behind wings in formation flight are very important to the aerodynamics and perfo... more The wake shapes behind wings in formation flight are very important to the aerodynamics and performances of aircrafts. In the present study, a discrete vortex methood is extended to handle the wake rollups behind multiple wings. It was found that the relative distance between the wings and the rotational direction of the wingtip vortices have significant effect on the movement of the wingtip vortices. When the wings are close to each other, the wingtip vortices moved faster than the wings of large relative distances. The vortex pair of opposite signs generated from each wingtip has an effect of moving the wingtip vortices upward. The relative height between the wings has an effect of moving the wingtips along the centerline of each vortex. The wakeshape behind multiple wings is a function of the relative distances and thus is dependent on the configuration of the formation flight. In the futhre, a study on the vortex movement pattern will be studied.
An intake is a device for supplying external air into the internal engine of an aircraft. In orde... more An intake is a device for supplying external air into the internal engine of an aircraft. In order to reduce the radar reflection on the engine face for reducing the observability, intake has an S-shaped duct. When applied to s-duct of the fuselage, the uniformity of the flow at the engine face varies depending on the state of flow at the inlet of the duct. In this study, the effect of active flow control at the inlet of S-duct was analyzed using a commercial computational fluid dynamics tool. The purpose of this study is to investigate the effect of the active flow control factors for an RAE M 2129 S-duct. The study was conducted by changing the position, length, and mass flow of the active flow control, and the mass flow was within 1% of the s-duct internal mass flow. The OLHD method was applied to 15 models for three factors and the k-ω SST turbulence model was used to predict the flow separation and secondary flow by adverse pressure gradient. In conclusion, the performance of the S-duct was found to have the greatest influence on the position of the flow control, and it was confirmed that the performance was the worst when the position was 0 in the duct standard.
International Journal of Aeronautical and Space Sciences, 2002
Aerodynamic analysis of NACA wings moving with a constant speed over guideways are performed usin... more Aerodynamic analysis of NACA wings moving with a constant speed over guideways are performed using an indirect boundary element method (potential-based panel method). An integral equation is obtained by applying Green's theorem on all surfaces of the fluid domain. The surfaces over the wing and the guideways are discretized as rectangular panel elements. Constant strength singularities are distributed over the panel elements. The viscous shear layer behind the wing is represented by constant strength dipoles. The unknown strengths of potentials are determined by inverting the aerodynamic influence coefficient matrices constructed by using the no penetration conditions on the surfaces and the Kutta condition at the trailing edge of the wing. The aerodynamic characteristics for the wings flying over nonplanar ground surfaces are investigated for several ground heights.
Journal of the Korean Society for Aeronautical & Space Sciences, 2020
Behaviors of wake vortices generated by an aircraft affect the performance and flight stability o... more Behaviors of wake vortices generated by an aircraft affect the performance and flight stability of flying aircraft in formation flight. In the present study, the trajectories of the wake vortices behind airplanes in close formation flight were computed using a Fourier spectral method. The behavior of wake vortices showed complex patterns depending on the initial circulation and the relative positions between the vortices. In the initial stage, the wake vortex movement was affected by the nascent vortex. When the vortex becomes closer to the other vortex, then a new trajectory is formed. When the viscous effect becomes dominant, the core radius increases. Thus, a new vortex moving near the existing vortex can have strong interaction with each other, resulting in the complicated behavior of wake vortices. In the future, the ground effect on the behavior of the wake vortices during takeoff and landing will be studied.
Transactions of the Korean Society of Mechanical Engineers B, 2006
A numerical study on the propulsive characteristics of a biomimetic foil is done by developing an... more A numerical study on the propulsive characteristics of a biomimetic foil is done by developing an unsteady linearly-varying strength vortex method. A biomimetic foil is represented as a deforming foil with a tail fin. Present method is verified by comparing the simulated results with results using finite element and finite volume methods. A new boundary condition is imposed by considering the relative rotational velocity, which has not been included in the previous published literature. It is found that the undulation amplitude increases the thrust while maximum thickness is stepping down the thrust. It is also shown that there exists an optimal frequency for maximum thrust generation. It is believed that present results can be used in the investigation of the propulsive characteristics of the biomimetic deforming foil.
International Journal of Aerospace System Engineering, 2016
Experimental study on the unsteady aerodynamics analysis and power consumption of a folding wing ... more Experimental study on the unsteady aerodynamics analysis and power consumption of a folding wing is accomplished using a wind tunnel testing. A folding wing model is fabricated and actuated using servo motors. The flapping wing consists of an inboard main wing and an outboard folding wing. The aerodynamic forces and consumed powers of the flapping wing are measured by changing the flapping and folding wings inside a low-speed wind tunnel. In order to calculate the aerodynamic forces, the measured forces are modified using static test data. It was found that the effect of the folding wing on the flapping wing's total lift is small but the effect of the folding wing on the total thrust is larger than the main wing. The folding motion requires the extra use of the servo motor. Thus, the amount of the energy consumption increases when both the wings are actuated together. As the flight speed increases, the power consumption of the folding wing decreases which results in energy saving.
In this paper, a wind-tunnel test is accomplished to investigate the roll characteristics of a va... more In this paper, a wind-tunnel test is accomplished to investigate the roll characteristics of a variable-span wing flying inside a channel. The factors that affect the roll characteristics of the wing were identified by analyzing the measured data; accordingly, when the wing is flying without both the ground and sidewall effects, the asymmetric wing extension causes the roll moment. Both the ground and the sidewall can increase the roll moment, but when the wing is affected by both the ground and the sidewall, the roll moment does not increase as much as the case where the wing is only affected by the ground. Also, the aerodynamic characteristics of the flying wing inside a channel are the nonlinear function of the wing height and the gap between the wingtip and the sidewall, both of which should be considered in a study of the stability and the flight control of the wing-in-ground effect of the vehicle flying inside a channel.
Journal of the Korean Society for Aeronautical & Space Sciences, 2007
Insect and birds in nature flap their wings to generate fluid dynamic forces that are required fo... more Insect and birds in nature flap their wings to generate fluid dynamic forces that are required for locomotion. To develop a feasible flapping MAV, it is of crucially important to study the fundamental relations between flapping motion and thrust generation. In this paper, the onset conditions of the thrust generation of a heaving flat plate is investigated using a Lattice-Boltzmann method. For a fixed heaving amplitude of h/C
In this paper, the fluid dynamic forces and performances of a moving airfoil in the low Reynolds ... more In this paper, the fluid dynamic forces and performances of a moving airfoil in the low Reynolds number flow is addressed. In order to simulate the necessary propulsive force for the moving airfoil in a low Reynolds number flow, a lattice-Boltzmann method is used. The critical Reynolds and Strouhal numbers for the thrust generation are investigated for the four propulsion types. It was found that the Normal P&D type produces the largest thrust with the highest efficiency among the investigated types. The leading edge of the airfoil has an effect of deciding the force production types, whereas the trailing edge of the airfoil plays an important role in augmenting or reducing the instability produced by the leading edge oscillation. It is believed that present results can be used to decide the optimal propulsion types for the given Reynolds number flow.
The prediction of the separation trajectories of external stores released from a military aircraf... more The prediction of the separation trajectories of external stores released from a military aircraft is an important task in the aircraft design area having the objective to define the operational and release envelopes. This paper presents the results obtained for store separation by employing commercial softwares, FLUENT and CFD-FASTRAN. FLUENT treats the rigid body motion by employing a remeshing scheme. CFD-FASTRAN uses Chimera(overset) grid and interpolations. It was found that, for the prediction of the trajectories and behavior of the stores separated from the wing, both codes show the good agreement with the experimental results.
The unsteady evolution of trailing vortex sheets behind a wing in ground effect is simulated usin... more The unsteady evolution of trailing vortex sheets behind a wing in ground effect is simulated using an unsteady discrete vortex panel method. The ground effect is included by image method. The present method is validated by comparing the simulated wake roll-up shapes to published numerical results. When a wing is flying in a very close proximity to the ground, the optimal wing loading is parabolic rather than elliptic. Thus, a theoretical model of wing load distributions is suggested, and unsteady vortex evolutions behind lifting lines with both elliptic and parabolic load distributions are simulated for several ground heights. For a lifting line with elliptic and parabolic loading, the ground has the effect of moving the wingtip vortices laterally outward and suppressing the development of the vortex. When the wing is in a very close proximity to the ground, the types of wing load distributions does not affect much on the overall wake shapes, but parabolic load distributions make the wingtip vortices move more laterally outward than the elliptic load distributions.
The development of empirical model for the hydraulic transport of sand-water mixtures is importan... more The development of empirical model for the hydraulic transport of sand-water mixtures is important for the design of economical solid-liquid transportation system in chemical and waste-disposal industries. The hydraulic transport characteristics of sand-water mixtures in circular pipelines are numerically investigated by using the FLUENT commercial software. Eulerian granular multiphase (EGM) model with the k-e turbulent model is used for the computation. Present method is validated by the computed values with the measured data. The effect of the concentration and pipe sizes on the relative solid effect is numerically investigated. It is found that the effect of the volumetric delivered concentration on both hydraulic gradient and solid effect increases as the Reynolds number decreases. When the Reynolds number is small, the increase in the volumetric delivered concentration has an effect of decreasing the hydraulic gradient whereas the solid effect increases with the volumetric delivered concentration stepping up. The effect of the pipe diameter is not the critical parameter for deciding the values of the relative solid effect in the sand-water mixture transportation.
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