Numerical Analysis of Drag Force Acting on 2D Cylinder Immersed in Accelerated Flow

water

Technical Note Numerical Analysis of Drag Force Acting on 2D Cylinder Immersed in Accelerated Flow

Hyun A. Son 1,2, Sungsu Lee 1,* and Jooyong Lee 3

1 School of Civil Engineering, Chungbuk National University, Cheongju 28644, Korea; [email protected] 2 Naval Systems R&D Department, Daewoo Shipbuilding & Marine Engineering Co., Ltd., Geoje 53302, Korea 3 Department of Disaster Prevention Engineering, Chungbuk National University, Cheongju 28644, Korea; [email protected] * Correspondence: [email protected]

Received: 14 May 2020; Accepted: 16 June 2020; Published: 23 June 2020

Abstract: In this study, the drag exerted by an accelerating fluid on a stationary 2D circular cylinder is numerically investigated using Fluent 19.2 based on the finite-volume method. The SST k–ω model is chosen as the turbulence model because of its superiority in treating the viscous near-wall region. The results are compared to literature, and the numerical methods are validated. The acceleration of the inflow is analyzed for the range of 0.0981–9.81 m/s2, and the drag for each acceleration is compared. Additionally, the effect of the initial velocity on the drag acting on the circular cylinder is investigated at two initial velocities. As a result, a supercritical region, typically found under steady state conditions, is observed. Furthermore, vortex shedding is observed at a high initial velocity. This flow characteristic is explained via comparison with respect to the recirculation length and separation angle.

Keywords: sudden start; bluﬀ body; 2D circular cylinder; accelerated ﬂow; CFD

1. Introduction Sudden bursts of flow or impulsive motions of a body within the flow can be observed during natural phenomena, for example, in the case of structures affected by extreme atmospheric phenomena such as tornadoes, tropical cyclones, or thunderstorms. Such events also occur in several industrial and engineering applications. However, although these events are highly transient, and loads are exerted on structures for only a short period, wind-resistant designs assume that wind and its resulting structural loads are temporally stationary or quasi-steady. This leads to a discrepancy between the design loads of the structures and the actual loads they experience during strong wind events. In practical scenarios, unsteady forces due to gusts are much greater than their corresponding steady-state levels, which is called overshoot. In severe cases, these unsteady forces can result in the destruction of the structure. However, load-evaluation methods to account for overshoot have not been established. Therefore, the variation in unsteady force with time must be precisely investigated. As part of these efforts, Nomura et al. [1], Matsumoto et al. [2], and Takeuchi & Maeda [3] conducted specialized wind tunnels for a bluff body under a sudden increase (impulsive starting flow) in wind velocity. They explained that drag overshoot is generated by an inertial force proportional to du/dt and the formation of vortices. However, in the drag calculation, the steady-state drag coefficient was used instead of the unsteady-state coefficient. Rho et al. [4], Lee et al. [5], and Lee et al. [6] also conducted wind tunnel tests to investigate the difference in aerodynamic characteristics between non-accelerated and accelerated flows. They proposed that the drag in accelerated flow decreases compared to that of non-accelerated flow, and as the Reynolds number increases, the drag of the accelerated flow increases. However, the Reynolds number range was limited from 4 104 to 1.64 105; × ×

Water 2020, 12, 1790; doi:10.3390/w12061790 www.mdpi.com/journal/water Water 2020, 12, 1790 2 of 24 thus, the overall drag change for other Reynolds number ranges is unknown. Mason and Yang [7] performed wind tunnel experiments to investigate forces on a two-dimensional square cylinder during a period of rapid acceleration. The results show, in a fragmentary way, that when flow accelerated from an initial quiescent state, both the amplitude and frequency of force coefficient fluctuations exceeded those recorded during steady flow tests. Previous studies compared accelerated and non-accelerated flows in small Reynolds number ranges. Therefore, the studies with wind tunnel experiments provide limited understanding of the aerodynamic characteristics in accelerated flow. The overshoot phenomenon can be described as a mechanism wherein inertia is added to a system when an object accelerates or decelerates; this inertia is referred to as added mass. This added mass was studied in early potential flow theory, which does not consider the effect of viscosity or compressibility. There are several works, such as those by Lamb [8], Birkhoff [9], and Yih [10], that theoretically characterized the added mass for basic geometries. This characterization has been extended to a variety of complicated geometries in several textbooks (Keulegan & Carpenter [11]; Patton [12]; Kennard [13]; and Brennen [14]). The added mass depends on the wake shape, volume, and rate of change. The formation of the wake is primarily affected by viscosity. Therefore, Sarpkaya [15] considered vortices in his potential flow solution to simulate the effect of viscosity, and the use of potential flow has been extended to include more complicated flow dynamics. Sarpkaya and Garrison [16] also analyzed the strength, growth, and movement of a vortex behind a circular cylinder with constant acceleration. In addition, equations for lift and drag were obtained from the potential flow theory in terms of the flow and vortex characteristics. One of the important results of this study was the correlation between the fluid force and the distance using the dimensionless technique. In addition, Fackrell [17] developed a numerical model to determine the force on a 2D circular cylinder in the case of constant relative acceleration and applied the dimensionless technique to separate drag and added mass coefficients. However, that work was limited to the range of a/g = 0.2–1.0 for a low initial Reynolds number, where a and g are acceleration of flow and gravitational acceleration, respectively. Therefore, the influence of the initial Reynolds number or the change in drag for other accelerations remains unknown, especially at very low accelerations close to steady flow. In this study, the drag change according to initial velocity was compared considering low and high initial Reynolds numbers, and the acceleration range was extended to 0.01–1.0g. Because the fluid force acting on the bluff body is closely related to the shape of the wake, their relationship is investigated to identify the drag increase due to acceleration and to provide a basic understanding thereof.

2. Computational Methods

2.1. Governing Equations and Numerical Method The flow field is governed by the continuity and Navier–Stokes equations for a Newtonian and incompressible fluid. ∂u i = 0 (1) ∂xj ! ! ∂ui ∂ui ∂p ∂ ∂ui ∂uj ρ + uj = + µ + uj = 0 (2) ∂t ∂xj − ∂t ∂xj ∂xj ∂xi where ui is the velocity in i-th direction, ρ is the density, p is the pressure, µ is the coefficient of kinematic viscosity. This study assumes 2D flow around a circular cylinder, and literatures have shown that 2D computation can over-estimate the forces acting on the body to a great extent (Benim et al. [18]; Liuliu et al. [19]). However, it is the intention of this study to investigate the effects of accelerated inflow on the drag on the body compared to that in the steady flow condition, and the present results compared later with the results of experiments conducted in 3D flow showed the validity of present results. Water 2020, 12, x FOR PEER REVIEW 3 of 27 Water 2020, 12, 1790 3 of 24

TheWater shear 2020 stress, 12, x FOR transport PEER REVIEW (SST) k–ω turbulence model was chosen for its superiority in treating3 of 27 the viscous near-wall region, as well as to account for the effects of streamwise pressure gradients. The shear stress transport (SST) k–ω turbulence model was chosen for its superiority in treating These features are important for accurately modeling the boundary layer-separation process, which theThe viscous shear near-wallstress transport region, (SST) as well k–ω as turbulence to account model for the was eff ectschosen of streamwisefor its superiority pressure in treating gradients. is critical to obtain good results for the flow over bluff bodies. The pressure-implicit with splitting of theThese viscous features near-wall are important region, as for well accurately as to account modeling for the boundaryeffects of streamwise layer-separation pressure process, gradients. which is operator algorithm was adopted for the pressure–velocity coupling scheme that uses a relationship Thesecritical features to obtain are important good results for foraccurately the flow modeling over blu ffthebodies. boundary The pressure-implicitlayer-separation process, with splitting which of between the velocity and pressure corrections to enforce mass conservation and obtain the pressure is criticaloperator to algorithmobtain good was results adopted for the for flow the pressure–velocity over bluff bodies. couplingThe pressure-implicit scheme that useswith asplitting relationship of field. Solvers with second-order accuracy were chosen for pressure and momentum, turbulent kinetic operatorbetween algorithm the velocity was andadopted pressure for the corrections pressure–velocity to enforce coupling mass conservation scheme that and uses obtain a relationship the pressure energy, and specific dissipation-rate equations. The least-squares cell-based method was also used as betweenfield. Solvers the velocity with second-orderand pressure accuracycorrections were to chosenenforce formass pressure conservation and momentum, and obtain turbulent the pressure kinetic the gradient interpolation. field.energy, Solvers and with specific second-order dissipation-rate accuracy equations. were chosen The least-squaresfor pressure and cell-based momentum, method turbulent was also kinetic used as energy,the gradient and specific interpolation. dissipation-rate equations. The least-squares cell-based method was also used as 2.2the Computational gradient interpolation. Domain and Boundary Conditions 2.2.The Computationalschematic views Domain of a and2D circular Boundary cylinder Conditions and relevant boundary conditions are shown in 2.2 Computational Domain and Boundary Conditions Figure 1. The schematiccomputational views domain of a 2D was circular determined cylinder considering and relevant the boundarystudy of Fackrell conditions [17]. areFigure shown 1 in illustratesFigureThe the 1schematic. The computational computational views of domain a domain2D circular of size was ofcylinder determined 20𝑑 × and80𝑑 considering relevant with a cylinder boundary the study of diameter,conditions of Fackrell d are= [0.0127 17shown]. Figure m. in 1 TheFigure illustratesdistance 1. The from thecomputational computationalthe center of domain cylinder domain was to ofdetermined inlet size and of 20outlet consideringd 80wered with set the ato cylinderstudy 20𝑑 andof Fackrell of 60𝑑 diameter,, respectively, [17]. dFigure= 0.0127 1 × whileillustratesm. the The distance distancethe computational of from12𝑑 from thecenter domainthe center of of cylinder ofsize the of circular to20𝑑 inlet × cylinder80𝑑 and outlet with to a werethecylinder side set walls toof 20diameter, dwasand set.60 d d ,= respectively,0.0127 m. Thewhile distance the distance from the of center 12d from of cylinder the center to of inlet the circularand outlet cylinder were toset the to side20𝑑 wallsand 6 was0𝑑, respectively, set. while the distance of 12𝑑 from the center of the circular cylinder to the side walls was set.

Figure 1. Schematic views of the computational domain

The boundary conditionsFigureFigure of the1. 1.Schematic numericalSchematic views model views of ofthe are the computational shown computational in Figure domain domain. 2. A uniform velocity was imposed on the inlet of the domain with constant acceleration; a 0-Pa condition was defined at the outlet. TheThe boundaryno-slip boundary condition conditions conditions was of imposed ofthe the numerical numerical on the model su modelrface are of are shownthe shown body, in inFigure and Figure the 2. 2slipA. Auniform uniformwall was velocity velocity applied was was onimposed theimposed side onwalls, on the the inletwhere inlet of 𝑢 ofthe theis domain the domain initial with with velocity, constant constant t isacceleration; the acceleration; time, 𝜏 a 0-Pais a 0-Pathe condition wall condition shear was wasstress, defined defined y is atthe atthe the distanceoutlet.outlet. fromThe The no-slip the no-slip side condition wall. condition As wasthe was inflow imposed imposed veloci on on tythe the increases su surfacerface ofwith of the the time, body, body, the and inflow the slip becomes wallwall waswas turbulent appliedapplied on the side walls, where u0 is the initial velocity, t is the time, τw is the wall shear stress, y is the distance fromon thelaminar side walls,due to where flow instability;𝑢 is the initial however, velocity, the turbulencet is the time, was 𝜏 notis the modeled wall shear in the stress, inflow y is by the consideringdistancefrom the from the side uniformitythe wall. side wall. As of the Asinflow. inflow the inflow velocity veloci increasesty increases with with time, time, the inflowthe inflow becomes becomes turbulent turbulent from fromlaminarAs laminardescribed due due to in flow toFigure flow instability; 3, instability; the fluid however, was however, initiall the turbulence ythe allowed turbulence was to notachieve was modeled not a steady modeled in the flow inflow in and the by was inflow considering then by accelerated.consideringthe uniformity The the inflow uniformity of inflow. boundary of inflow. condition was defined by a user-defined function. As described in Figure 3, the fluid was initially allowed to achieve a steady flow and was then accelerated. The inflow boundary condition was Slipdefined wall, by a user-defined=0 function.

Slip wall, =0

Inlet, Outlet, No sl ip wall Inlet, Outlet, No sl ip wall No slip wall on cylinder surface,

FigureFigure 2.2. BoundaryBoundary conditions conditions. No slip wall on cylinder surface,

As described in Figure3, the fluid was initially allowed to achieve a steady flow and was then accelerated. The inflow boundary conditionFigure 2. was Boundary defined conditions by a user-defined function.

Water 2020, 12, x FOR PEER REVIEW 4 of 27

Water 2020, 12, x FOR PEER REVIEW 4 of 27 Water 2020, 12, 1790 4 of 24 𝑢

𝑢 𝑎 𝑎 𝑢

𝑢 𝑡 𝑡 Figure 3. Inflow boundary condition Figure 3. Inflow boundary condition. Figure 3. Inflow boundary condition 2.3 Validation of the Present Method 2.3. Validation of the Present Method 2.3 ValidationIn order of tothe validate Present Methodthe method employed in this study, the results were compared with the In order to validate the method employed in this study, the results were compared with the experimental results of Sarpkaya and Garrison [16], in which a vertical water tunnel with a quick- experimentalIn order to resultsvalidate of the Sarpkaya method and employed Garrison [in16 ],th inis whichstudy, a the vertical results water were tunnel compared with a quick-releasewith the release bottom gate was constructed to obtain a unidirectional, constant flow of fluid over a fixed experimentalbottom gate results was constructedof Sarpkaya toand obtain Garrison a unidirectional, [16], in which constant a vertical flow water of fluid tunnel over with a fixed a quick- circular circular cylinder. The experiments were conducted for the acceleration range of 0.155–0.994g. In this releasecylinder. bottom The gate experiments was constructed were conducted to obtain fora unid theirectional, acceleration constant range offlow 0.155–0.994 of fluid overg. In a this fixed study, study, the fluid properties were set to a density of 998.2 kg/m and a kinematic viscosity of 0.001003 circularthe fluid cylinder. properties The experiments were set to awere density conduc of 998.2ted for kg /them3 accelerationand a kinematic range viscosity of 0.155–0.994 of 0.001003g. In kgthis/m-s. kg/m-s. study, the fluid properties were set to a density of 998.2 kg/m and a kinematic viscosity of 0.001003 kg/m-2.3.1.s. Grid Convergence 2.3.1 Grid Convergence In the flow analysis around the bluff body, high gradients are formed along the surface of the 2.3.1cylinder. GridIn the Convergence flow Therefore, analysis it around is important the bluff to body, set the high grid gradients around theare cylinderformed along adequately. the surface Inaddition, of the cylinder. Therefore, it is important to set the grid around the cylinder adequately. In addition, consideringIn the flow the analysis time required around forthe computationalbluff body, high analysis, gradients an appropriate are formed total along number the surface of grids of shouldthe considering the time required for computational analysis, an appropriate total number of grids cylinder.be used, Therefore, and it is commonit is important to set the to gridsset the slightly grid fararound away the from cylinder the cylinder. adequately. Therefore, In addition, as shown in should be used, and it is common to set the grids slightly far away from the cylinder. Therefore, as consideringFigure4, thethe gridtime around required the cylinderfor computational was finely modeledanalysis, usingan appropriate hexahedra, total and thenumber other of regions grids are shown in Figure 4, the grid around the cylinder was finely modeled using hexahedra, and the other shouldcoarsely be used, modeled and it using is common tetrahedra. to set The the computational grids slightly far grid away was constructedfrom the cylinder. to calculate Therefore, the laminar as regions are coarsely modeled using tetrahedra.+ The computational grid was constructed to calculate shownregion in Figure near the 4, wall,the grid and around thus the they cylindervalue was was close finely to modeled one. using hexahedra, and the other the laminar region near the wall, and thus the 𝑦 value was close to one. regions are coarsely modeled using tetrahedra. The computational grid was constructed to calculate the laminar region near the wall, and thus the 𝑦 value was close to one.

Figure 4. Mesh in the computational domain. Figure 4. Mesh in the computational domain Figure 4. Mesh in the computational domain

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Three cases of grid configurations, listed in Table1, were used to determine the appropriate grid size and distribution. To consider the importance of the height of the first cells next to the circular cylinder wall, Mesh 2 was modeled such that y+ = 1, and Meshes 1 and 3 were modeled with y+ values less than and greater than 1, respectively. At this time, Mesh 3 was modeled such that the total number of grids and the number of grids around the cylinder were greater than those of Mesh 2 to determine the effect of the height of the first layer.

Table 1. Mesh information for grid convergence.

First Grid Height (m) Total Number of Grids Number of Grids around the Cylinder Mesh 1 1.9 10 6 454,012 940 × − Mesh 2 2.9 10 6 229,211 800 × − Mesh 3 5.9 10 6 376,680 1600 × −

Figure5 shows the variation in drag coe fficient with nondimensional time, compared to the experimental results of Sarpkaya and Garrison [16]. The acceleration given here is a = 9.81 m/s2 and the drag coefficient is defined as Fd C ∗ = (3) d aρA Water 2020, 12, x FOR PEER REVIEW 6 of 27 where Fd is the drag force per unit length, and A is the area of the circular cylinder.

Figure 5. Comparison of different grid configurations (experimental data from Sarpkaya and FigureGarrison 5. Comparison [16]). of different grid configurations (experimental data from Sarpkaya and Garrison [16]) Keulegan and Carpenter [11] reported that the drag coefficient in a flow with constant acceleration 2.3.2correlated Choice of with Turbulence the relative Model dimensionless displacement and was not function of time or Reynolds number. Based on this, Sarpkaya and Garrison [16] proposed an analysis using a relative displacement If a bluff body is immersed in a flow with constant acceleration, the surrounding flow enters the of s/d = 0.5at2/d where s is the displacement of the fluid flow. In this study, for a more intuitive turbulent region as the velocity increases. The method used to simulate the turbulence of the understanding, it was denoted as a nondimensional time tˆ, defined as unsteady state has a crucial effect on the analytical accuracy. The most common methods to analyze turbulent2 flow are is the direct numerical simulation at (u u0)t tˆ = = − (4) (DNS), the large eddy simulation (LES) and the2 dReynolds-averaged2d Navier–Stokes (RANS) equation approach. DNS numerically solves the Navier–Stokes equations without the use of a turbulenceAs can model. be seen Using from FigureDNS,5 it, Mesh is possible 3 case withto ob thetain largest results first that grid are height relatively deviates close the to most actual from the experimental results. When the first grid height is decreased by 2.9 10 6 m, the C value is closer phenomena with the cost of a large amount of computing resources. LES× also− enablesd simulation∗ of real phenomena with sufficient accuracy but requires a large amount of computational resources since it resolves directly large eddies while small eddies are modeled. On the other hand, RANS in general models turbulence of all the scales and offers relatively reasonable results, which saves the computational. In these perspective, this study employs RANS in order for higher computational efficiency with less requirement for resources, and we consider the k–ε model, standard k–ω model, and SST k–ω model among the two-equation models of RANS. In order to determine the two-equation model for this study, the drag coefficients computed by using different models were compared with experimental data, and the results are shown in Figure ∗ 6. In the case of the k–ε model, the initial value of 𝐶 is closer to 2.3 upon inspection. This deviates from the experimental results; thus, this model cannot be used. Furthermore, the k–ε model and standard k– ω model show that vortex shedding occurs in the early stage of the simulation, whereas the SST k–ω model does not. In addition, Sarpkaya’s experimental results do not indicate vortex shedding, and hence these models are unrealistic. The remaining model, the SST k–ω model, shows excellent agreement with only a slight difference at larger values of 𝑡̂; hence, this model is considered applicable and chosen as the turbulence model.

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to the experimental result than in the case of Mesh 3. However, decreasing the first grid height further by 1.9 10 6 m does not affect the results. Mesh 2 is then selected for the simulations to ensure that × − the high velocity gradients are resolved and to reduce the computational cost. In Mesh 2, y+ = 1 was satisfied for all cases studied, including the most critical case of a/g = 1 at tˆ = 10.

2.3.2. Choice of Turbulence Model If a bluff body is immersed in a flow with constant acceleration, the surrounding flow enters the turbulent region as the velocity increases. The method used to simulate the turbulence of the unsteady state has a crucial effect on the analytical accuracy. The most common methods to analyze turbulent flow are is the direct numerical simulation (DNS), the large eddy simulation (LES) and the Reynolds-averaged Navier–Stokes (RANS) equation approach. DNS numerically solves the Navier–Stokes equations without the use of a turbulence model. Using DNS, it is possible to obtain results that are relatively close to actual phenomena with the cost of a large amount of computing resources. LES also enables simulation of real phenomena with sufficient accuracy but requires a large amount of computational resources since it resolves directly large eddies while small eddies are modeled. On the other hand, RANS in general models turbulence of all the scales and offers relatively reasonable results, which saves the computational. In these perspective, this study employs RANS in order for higher computational efficiency with less requirement for resources, and we consider the k–ε model, standard k–ω model, and SST k–ω model among the two-equation models of RANS. In order to determine the two-equation model for this study, the drag coefficients computed by using different models were compared with experimental data, and the results are shown in Figure6. In the case of the k–ε model, the initial value of Cd∗ is closer to 2.3 upon inspection. This deviates from the experimental results; thus, this model cannot be used. Furthermore, the k–ε model and standard k– ω model show that vortex shedding occurs in the early stage of the simulation, whereas the SST k–ω model does not. In addition, Sarpkaya’s experimental results do not indicate vortex shedding, and hence these models are unrealistic. The remaining model, the SST k–ω model, shows excellent agreementWater 2020 with, 12, x only FOR aPEER slight REVIEW difference at larger values of tˆ; hence, this model is considered applicable7 of 27 and chosen as the turbulence model.

FigureFigure 6. 6.ComparisonComparison of different of diﬀ erentturbulence turbulence models models (experimental (experimental data from data Sarpkaya from Sarpkayaand Garrison and [16])Garrison [16]).

2.3.3 Time Step Convergence In order to determine the time step size, three different values of 0.001 s, 0.0001 s and 0.00001 s were considered. Figure 7 shows the comparison of the computed drag coefficients with experimental ∗ data. Both results with the time step of 0.0001 s and 0.00001 s show that 𝐶 is closer to the experimental data while the result with 0.001 s depicts larger discrepancy. And it was found that the fluid particles were not allowed to pass more than one cell with the time step size of 0.0001s, including the most critical case of a/g = 1 at 𝑡̂ =10. Therefore, the time step size of 0.0001s was selected for this study.

Figure 7. Comparison of different time steps (experimental data from Sarpkaya and Garrison [16])

2.4 Scope of study The fluid is initially allowed to achieve a fully developed flow at 𝑅𝑒 = 40 and 1000 and is then accelerated with the accelerations of 0.0981–9.81 m/s2 as listed in Table 2. The Reynolds number, 𝑅𝑒 is defined as

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Figure 6. Comparison of different turbulence models (experimental data from Sarpkaya and Garrison Water[16])2020 , 12, 1790 7 of 24

2.3.3 Time Step Convergence 2.3.3. Time Step Convergence In order to determine the time step size, three different values of 0.001 s, 0.0001 s and 0.00001 s were considered.In order to determineFigure 7 shows the time the comparison step size, three of the di ffcomputederent values drag of coefficients 0.001 s, 0.0001 with s experimental and 0.00001 s ∗ data.were considered.Both results Figure with 7the shows time the step comparison of 0.0001 of s theand computed 0.00001 drags show coe thatfficients 𝐶 with is closer experimental to the experimentaldata. Both results data withwhile the the time result step with of 0.0001 0.001 ss anddepicts 0.00001 larger s show discrepancy. that Cd∗ is And closer it was to the found experimental that the fluiddata particles while the were result not with allowed 0.001 to s depictspass more larger than discrepancy. one cell with And the ittime was step found size that of 0.0001s, the fluid including particles thewere most not critical allowed case to of pass a/g more= 1 at than𝑡̂ =10 one. Therefore, cell with the time step step size size of of 0.0001s 0.0001s, was including selected the for mostthis study.critical case of a/g = 1 at tˆ = 10. Therefore, the time step size of 0.0001s was selected for this study.

FigureFigure 7. 7. ComparisonComparison of of different different time time steps steps (experim (experimentalental data data from from Sarpkaya Sarpkaya and and Garrison Garrison [16]) [16]). 2.4. Scope of Study 2.4 Scope of study The fluid is initially allowed to achieve a fully developed flow at Re = 40 and 1000 and is then The fluid is initially allowed to achieve a fully developed flow at 𝑅𝑒 = 40 and 1000 and is then accelerated with the accelerations of 0.0981–9.81 m/s2 as listed in Table2. The Reynolds number, Re is accelerated with the accelerations of 0.0981–9.81 m/s2 as listed in Table 2. The Reynolds number, 𝑅𝑒 defined as is defined as ρvd Re = (5) µ

For a low Reynolds number (Re = 40), the flow is steady and symmetric, and the wake is laminar, whereas for a high Reynolds number (Re = 1000), the flow is unsteady, and the vortex street is fully turbulent. However, in the case of high initial velocity (Re0 = 1000), only the three accelerations of a = 9.81, 0.981, and 0.4905 m/s2 are studied. Fackrell [17] extended the study on unidirectional constant acceleration of an initial stationary fluid to include a flow with a small initial velocity (Re0 < 40), and the case of low Reynolds number in this study was set with reference to this.

Table 2. Computational cases.

a/g a (m/s2) 1 9.81 0.5 4.905 0.1 0.981 0.05 0.4905 0.01 0.0981 Water 2020, 12, x FOR PEER REVIEW 8 of 27

𝜌𝑣𝑑 𝑅𝑒 = (5) 𝜇 For a low Reynolds number (𝑅𝑒 = 40), the flow is steady and symmetric, and the wake is laminar, whereas for a high Reynolds number (𝑅𝑒 = 1000), the flow is unsteady, and the vortex street is fully turbulent. However, in the case of high initial velocity (𝑅𝑒 = 1000), only the three accelerations of a = 9.81, 0.981, and 0.4905 m/s2 are studied. Fackrell [17] extended the study on unidirectional constant acceleration of an initial stationary fluid to include a flow with a small initial velocity (𝑅𝑒 < 40), and the case of low Reynolds number in this study was set with reference to this.

Table 2. Computational cases

a/g a (m/𝐬𝟐) 1 9.81 0.5 4.905 0.1 0.981

Water 2020, 12, 1790 0.05 0.4905 8 of 24 0.01 0.0981

3.3. Drag Drag in in Accelerated Accelerated Flow Flow at at Low Low Initial Initial Velocity Velocity 3.1. Overshoot of Drag Force 3.1 Overshoot of drag force Figure8 shows the drag coe fficient versus Reynolds number when the flow is accelerated with Figure 82 shows the drag coefficient versus Reynolds number when the flow is accelerated with a = 9.81 m/s2 from the initial constant velocity (Re0 = 40). When the flow accelerated, an overshoot a = 9.81 m/s from the initial constant velocity (𝑅𝑒 = 40). When the flow accelerated, an overshoot phenomenon of the fluid force is evident. The drag coefficient Cd is defined as phenomenon of the fluid force is evident. The drag coefficient 𝐶 is defined as

𝐹Fd 𝐶Cd == (6) 11 2 (3) 2𝜌𝑣ρv𝐴A 2 The value of C reached 23,320, which is much greater than that if the motion was steady. However, The value of 𝐶d reached 23,320, which is much greater than that if the motion was steady. the C values constantly decreased after overshoot. A comparison of C values of non-accelerated and However,d the 𝐶 values constantly decreased after overshoot. A comparisond of 𝐶 values of non- acceleratedaccelerated and ﬂow accelerated for various flow acceleration for various ranges acceleration is presented ranges in theis presented following in section. the following section.

FigureFigure 8. 8. OvershootOvershoot phenomen phenomenonon of of drag drag force force. 3.2. Diﬀerence between Non-Accelerated and Accelerated Flow

3.2 DifferenceIn Section between 2.3, the Non-Acce computationallerated and modeling Accelerated and Flow numerical methods were verified via comparison with the experimental results of Sarpkaya and Garrison [16]. Based on this, the fluid force was analyzed with varying ranges of acceleration from the low initial velocity (Re0 = 40). Figure9 shows a comparison between the drag Cd of a circular cylinder obtained experimentally by Bullivant [20] and that obtained numerically in this study. The experimental result was measured when the flow was fully developed with constant velocity at each Reynolds number. In contrast, the numerical results describe the instantaneous fluid force during acceleration. As shown in Figure9, the drag of accelerated flow is higher than that of non-accelerated flow. The difference between accelerated and non-accelerated flow decreases as the acceleration decreases. In the cases of a/g = 1 and 0.5, the drag steadily decreased, except in the ranges of 30,000 Re 50,000 ≤ ≤ and 15,000 Re 40,000, respectively, where a constant value was maintained for some time. In the ≤ ≤ cases of a/g = 0.1, 0.5, and 0.01, the supercritical region observed at a high Reynolds number in the case of steady flow was also found for the ranges of 6000 Re 10,000, 4000 Re 6000, and 1300 ≤ ≤ ≤ ≤ ≤ Re 1800, respectively. Moreover, except in this region, the drag constantly decreased regardless of ≤ whether the region was subcritical or critical. In Figure9, Re0 is the Reynolds number defined with the initial velocity. Water 2020, 12, x FOR PEER REVIEW 9 of 27

In Section 2.3, the computational modeling and numerical methods were verified via comparison with the experimental results of Sarpkaya and Garrison [16]. Based on this, the fluid force was analyzed with varying ranges of acceleration from the low initial velocity (𝑅𝑒 = 40). Figure 9 shows a comparison between the drag 𝐶 of a circular cylinder obtained experimentally by Bullivant [20] and that obtained numerically in this study. The experimental result was measured when the flow was fully developed with constant velocity at each Reynolds number. In contrast, the numerical results describe the instantaneous fluid force during acceleration. As shown in Figure 9, the drag of accelerated flow is higher than that of non-accelerated flow. The difference between accelerated and non-accelerated flow decreases as the acceleration decreases. In the cases of a/g = 1 and 0.5, the drag steadily decreased, except in the ranges of 30,000 𝑅𝑒 50,000 and 15,000 𝑅𝑒 40,000, respectively, where a constant value was maintained for some time. In the cases of a/g = 0.1, 0.5, and 0.01, the supercritical region observed at a high Reynolds number in the case of steady flow was also found for the ranges of 6000 𝑅𝑒 10,000, 4000 𝑅𝑒 6000, and 1300 𝑅𝑒 1800, respectively. Moreover, except in this region, the drag constantly decreasedWater 2020, 12regardless, 1790 of whether the region was subcritical or critical. In Figure 9, 𝑅𝑒 is9 the of 24 Reynolds number defined with the initial velocity.

Figure 9. Comparison of drag coefficient Cd between non-accelerated flow (Bullivant [20]) and 𝐶 Figureaccelerated 9. Comparison flow (Re0 = of40). drag coefficient between non-accelerated flow (Bullivant [20]) and accelerated flow (𝑅𝑒 =40) 3.3. Drag Changes over Dimensionless Time and Flow Time 3.3 Drag Changes over Dimensionless Time and Flow Time According to Figures 10–12, the Cd value of a circular cylinder immersed in accelerated flow ˆ increasesAccording while toCd ∗Figuresdecreases 10–12, with thet and 𝐶 Re value. This of is a because circularC cylinderd is dimensioned immersed by in a squareaccelerated term offlow the ∗ increasesvelocity andwhile the 𝐶 denominator decreases with of Cd ∗𝑡containŝ and 𝑅𝑒. an This acceleration is because term. 𝐶 is At dimensioned the same time, by a the square fluid term force ofF theincreases velocity over and time. the denominator Therefore, the of drag𝐶 ∗ contains coefficient anvaries acceleration greatly term. according At the tosame the time, dimensionless the fluid Water 2020, 12, x FOR PEER REVIEW 10 of 27 forcemethod; F increases in this paper, overC dtime.or Cd ∗Therefore,is used as necessary.the drag coefficient varies greatly according to the ∗ dimensionless method; in this paper, 𝐶 or 𝐶 is used as necessary. As mentioned earlier, Sarpkaya and Garrison [16] proved that there is a correlation among drag ∗ coefficients (𝐶 ) in terms of dimensionless distance (s/d), and Fackrell [17] also showed that correlation among the data was very good for a/g = 1, 0.5, and 0.2. In this study, we found that the correlation of the data is good for a/g = 1 and 0.5. However, as can be seen in Figure 10, for a/g = 0.1, ∗ 0.05, and 0.5, the gradient of 𝐶 increases as the acceleration decreases. Furthermore, Figure 11, which shows the variation in 𝐶 according to dimensionless time.

Figure 10. C ∗ Re = Figure 10. DimensionlessDimensionless time time variation variation of of drag drag coefficient coeﬃcient 𝐶d∗ ((𝑅𝑒0 =4040).)

Figure 11. Dimensionless time variation of drag coefficient 𝐶 (𝑅𝑒 =40)

According to Figures 10 and 11, a lower acceleration results in larger 𝐶 at the same dimensionless time. However, more flow time is required to achieve the same change in dimensionless time with lower accelerations. For example, the case of a/g = 10 takes 10 times longer ∗ to achieve the flow time of the case of a/g = 1. Figure 12 shows the variation in 𝐶 with flow time t. ∗ The results show that the smaller the acceleration, the smaller the value of 𝐶 . In addition, the inflection point at which the slope increases sharply is shown in the cases of a/g = 0.1, 0.05, and 0.01.

Water 2020, 12, x FOR PEER REVIEW 10 of 27

∗ Water 2020, 12, 1790Figure 10. Dimensionless time variation of drag coefficient 𝐶 (𝑅𝑒 =40) 10 of 24

Water 2020, 12, x FOR PEER REVIEW 11 of 27

Figure 11. Dimensionless time variation of drag coeﬃcient Cd (Re0 = 40). Figure 11. Dimensionless time variation of drag coefficient 𝐶 (𝑅𝑒 =40)

Figure 12. Temporal variation of drag coefficient Cd∗∗ (Re0 = 40). Figure 12. Temporal variation of drag coefficient 𝐶 (𝑅𝑒 =40) As mentioned earlier, Sarpkaya and Garrison [16] proved that there is a correlation among drag 3.4 Vortex Formation and Development coefficients (Cd∗) in terms of dimensionless distance (s/d), and Fackrell [17] also showed that correlation amongFigures the data 13–17 was show very the good streamlines for a/g = 1, for 0.5, various and 0.2. values In this of study, 𝑡̂ and we a/ foundg. If the that circular the correlation cylinder is of immersedthe data is in good unidirectional for a/g = 1 andconstant 0.5. However, flow with as a can high be Reynolds seen in Figure number, 10, forthea /symmetricalg = 0.1, 0.05, pattern and 0.5, becomesthe gradient unstable of Cd and∗ increases vortex asshedding the acceleration occurs behind decreases. the cylinder. Furthermore, In the Figurecase of 11accelerated, which shows flow, theas thevariation velocity in Cincreases,d according the to vortex dimensionless forms and time. grows, but vortex shedding does not appear. These resultsAccording are consistent to Figures with 10 Sarpkayaand 11, a lowerand accelerationGarrison's [16] results experiment, in larger C dwhichat the sameshowed dimensionless no vortex sheddingtime. However, in the morerange flow of 𝑡 timê 30. is required This indicates to achieve that the even same if the change Reynolds in dimensionless number of timethe fully with developedlower accelerations. steady flow For and example, that of the the case accelerate of a/g =d10 unsteady takes 10 flow times match longer instantaneously, to achieve the flow the time wake of situationthe case ofbehinda/g = the1. Figure object 12at thatshows moment the variation is different. in C dThe∗ with size flow of th timee vortext. The tends results to increase show that as the the dimensionlesssmaller the acceleration, time increases the smaller because the the value energy of Cd ∗generated. In addition, by theacceleration inflection contributes point at which to the the increasingslope increases size sharplyof the vortex. is shown In inaddition, the cases as of tha/eg =dimensionless0.1, 0.05, and time 0.01. increases, the length of the recirculation length gradually grows, except for the case of a/g = 0.01, where it starts to dwindle at 𝑡 =6. Furthermore, as per the correlation between the drag coefficients, shown in Figures 10 and 11, respectively, for the cases of a/g = 1 and 0.5, the wake shape behind the cylinder according to 𝑡̂ is very similar.

Water 2020, 12, 1790 11 of 24

3.4. Vortex Formation and Development Figures 13–17 show the streamlines for various values of tˆ and a/g. If the circular cylinder is immersed in unidirectional constant flow with a high Reynolds number, the symmetrical pattern becomes unstable and vortex shedding occurs behind the cylinder. In the case of accelerated flow, as the velocity increases, the vortex forms and grows, but vortex shedding does not appear. These results are consistent with Sarpkaya and Garrison’s [16] experiment, which showed no vortex shedding in the range of tˆ < 30. This indicates that even if the Reynolds number of the fully developed steady flow and that of the accelerated unsteady flow match instantaneously, the wake situation behind the object at that moment is different. The size of the vortex tends to increase as the dimensionless time increases because the energy generated by acceleration contributes to the increasing size of the vortex. In addition, as the dimensionless time increases, the length of the recirculation length gradually grows, except for the case of a/g = 0.01, where it starts to dwindle at tˆ = 6. Furthermore, as per the correlation Water 2020, 12, x FOR PEER REVIEW 12 of 27 between the drag coefficients, shown in Figures 10 and 11, respectively, for the cases of a/g = 1 and 0.5, the wake shape behind the cylinder according to tˆ is very similar.

(a) 𝑡̂ =2 (𝑅𝑒 = 8967) (d) 𝑡̂ =8 (𝑅𝑒 = 17882)

(b) 𝑡̂ =4 (𝑅𝑒 = 12662) (e) 𝑡̂ =10 (𝑅𝑒 = 20002)

a g Re = FigureFigure 13. 13.StreamlinesStreamlines around around circul circularar cylinder cylinder (a/g ( =/ 1,= 𝑅𝑒1, =400 40).)

WaterWater2020 2020, 12, ,12 1790, x FOR PEER REVIEW 13 of12 27 of 24

(a) 𝑡̂ =2 (𝑅𝑒 = 6351) (d) 𝑡̂ =8 (𝑅𝑒 = 12656)

(b) 𝑡̂ =4 (𝑅𝑒 = 8961) (e) 𝑡̂ =10 (𝑅𝑒 = 14150)

Figure 14. Streamlines around circular cylinder (a/g = 0.5, Re0 = 40). Figure 14. Streamlines around circular cylinder (a/g = 0.5, 𝑅𝑒 =40)

WaterWater2020 2020, 12,, 12 1790, x FOR PEER REVIEW 14 of13 27 of 24

(a) 𝑡̂ =2 (𝑅𝑒 = 2985) (d) 𝑡̂ =8 (𝑅𝑒 = 5682)

(b) 𝑡̂ =4 (𝑅𝑒 = 4030) (e) 𝑡̂ =10 (𝑅𝑒 = 6348)

Figure 15. Streamlines around circular cylinder (a/g = 0.1, Re0 = 40). Figure 15. Streamlines around circular cylinder (a/g = 0.1, 𝑅𝑒 =40)

WaterWater2020 2020, 12, 179012, x FOR PEER REVIEW 15 14of 27 of 24

(a) 𝑡̂ =2 (𝑅𝑒 = 2035) (d) 𝑡̂ =8 (𝑅𝑒 = 4031)

(b) 𝑡̂ =4 (𝑅𝑒 = 2861) (e) 𝑡̂ =10 (𝑅𝑒 = 4501)

Figure 16. Streamlines around circular cylinder (a/g = 0.05, Re0 = 40). Figure 16. Streamlines around circular cylinder (a/g = 0.05, 𝑅𝑒 =40)

WaterWater2020 2020, 12, 12 1790, x FOR PEER REVIEW 16 of15 27 of 24

(a) 𝑡̂ =2 (𝑅𝑒 = 933) (d) 𝑡̂ =8 (𝑅𝑒 = 1825)

(b) 𝑡̂ =4 (𝑅𝑒 = 1305) (e) 𝑡̂ =10 (𝑅𝑒 = 2036)

Figure 17. Streamlines around circular cylinder (a/g = 0.01, Re0 = 40). Water 2020, 12, x FORFigure PEER 17. REVIEW Streamlines around circular cylinder (a/g = 0.01, 𝑅𝑒 =40) 17 of 27 The recirculation length and separation angle are closely related to the drag acting on the circular The recirculation length and separation angle are closely related to the drag acting on the circular cylinder and are a part of the criteria for quantifying the size of the vortices. The recirculation length cylinder and are a part of the criteria for quantifying the size of the vortices. The recirculation length and separation angle are deﬁned as Lr and θ, as shown in Figure 18. and separation angle are defined as 𝐿 and 𝜃, as shown in Figure 18.

Figure 18. Deﬁnition of recirculation length and separation angle. Figure 18. Definition of recirculation length and separation angle

The recirculation length at low Reynolds numbers, in the range of 20 𝑅𝑒 80, has been studied by Acrivos et al. [21]. In this study, vortex shedding did not appear, even at high Reynolds numbers. Therefore, to quantitatively evaluate the size of the vortex, the variation in the recirculation length according to the dimensionless time in each case is shown in Figure 19. The results show that at higher acceleration and dimensionless time, the recirculation length and its rate of increase are greater. In particular, in the cases of a/g = 1 and 0.5, the recirculation length increases linearly with the relative time (𝑡̂), whereas in the cases of a/g = 0.1 and 0.05, its rate of increase was found to decrease. In addition, as shown in Figure 20, the separation angle also increases the size of the vortex by causing it to move in the windward direction along the surface of the cylinder as a/g and 𝑡̂ increase. These trends can be seen in Figures 13–17. This change in the vortex behind the cylinder is directly related to the variation in 𝐶 shown in Figure 11. Ultimately, when the size of vortex is larger, 𝐶 is decreased. In the case of a/g = 0.01, the recirculation length decreases after 𝑡̂ = 6, and this is considered to influence the recovery of the 𝐶 value.

Figure 19. Recirculation length versus 𝑡̂ for various accelerations

Water 2020, 12, x FOR PEER REVIEW 17 of 27

The recirculation length and separation angle are closely related to the drag acting on the circular cylinder and are a part of the criteria for quantifying the size of the vortices. The recirculation length and separation angle are defined as 𝐿 and 𝜃, as shown in Figure 18.

Water 2020, 12, 1790 16 of 24 Figure 18. Definition of recirculation length and separation angle

The recirculation length at low Reynolds numbers, in the range of 20 Re 80, has been studied The recirculation length at low Reynolds numbers, in the range of ≤20 ≤ 𝑅𝑒 80, has been studiedby Acrivos by Acrivos et al. [21 et]. al. In [21]. this study,In this vortex study, shedding vortex shedding did not appear,did not evenappear, at higheven Reynolds at high Reynolds numbers. numbers.Therefore, Therefore, to quantitatively to quantitatively evaluate evaluate the size the of the size vortex, of the vortex, the variation the vari ination the recirculationin the recirculation length lengthaccording according to the to dimensionless the dimensionless time time in each in each case case is shown is shown in Figure in Figure 19. 19. The The results results show show that that at athigher higher acceleration acceleration and and dimensionless dimensionless time, time, the recirculationthe recirculation length length and itsand rate its of rate increase of increase are greater. are greater.In particular, In particular, in the cases in the of acases/g = 1 of and a/g 0.5, = 1 theand recirculation 0.5, the recirculation length increases length linearlyincreases with linearly the relative with thetime relative (tˆ), whereas time in(𝑡̂), the whereas cases of ina/ gthe= 0.1 cases and of 0.05, a/g its = rate0.1 ofand increase 0.05, its was rate found of increase to decrease. was In found addition, to decrease.as shown In in addition, Figure 20 as, the shown separation in Figure angle 20, alsothe sepa increasesration the angle size also of the increases vortex the by causingsize of the it to vortex move byin thecausing windward it to move direction in the along windward the surface direction of the cylinderalong the as asurface/g and tˆofincrease. the cylinder These as trends a/g and can be𝑡̂ increase.seen in Figures These trends13–17. can This be change seen in in Figures the vortex 13–17. behind This the change cylinder in the is directlyvortex behind related the to the cylinder variation is in C shown in Figure 11. Ultimately, when the size of vortex is larger, C is decreased. In the case of directlyd related to the variation in 𝐶 shown in Figure 11. Ultimately, dwhen the size of vortex is a/g = 0.01, the recirculation length decreases after tˆ = 6, and this is considered to inﬂuence the recovery larger, 𝐶 is decreased. In the case of a/g = 0.01, the recirculation length decreases after 𝑡̂ = 6, and of the C value. this is consideredd to influence the recovery of the 𝐶 value.

Water 2020, 12, x FOR PEER REVIEW 18 of 27 Figure 19. Recirculation length versus tˆ for various accelerations. Figure 19. Recirculation length versus 𝑡̂ for various accelerations

Figure 20. Separation angle versus tˆ for various accelerations. Figure 20. Separation angle versus 𝑡̂ for various accelerations

4. Drag in Accelerated Flow at High Initial Velocity

4.1 Difference between Non-Accelerated and Accelerated Flow In Section 3, the drag characteristics of circular cylinders immersed in flow that is accelerated from a low initial velocity (𝑅𝑒 = 40) were analyzed. In this section, flow accelerated from a high initial velocity (𝑅𝑒 = 1000) is considered. First, various acceleration ranges are examined. Then, the effect of initial velocity on drag change is analyzed in Section 5.

Figure 21 shows the logarithmic distribution of 𝐶 obtained experimentally in the case of non- accelerated flow by Bullivant [20] and that obtained numerically in the case of accelerated flow in this study. As evident from Figure 21, the 𝐶 values of accelerated flow were higher for all ranges than those of non-accelerated flow. Furthermore, the lower the acceleration, the smaller the difference between the drag values of the accelerated and non-accelerated flows. It is noticeable that 𝐶 values of accelerated flow decrease with time and oscillate when approaching the value of the non- accelerated flow in all acceleration ranges. In addition, the supercritical regions did not appear, unlike in the case of low initial velocity.

Water 2020, 12, 1790 17 of 24

4. Drag in Accelerated Flow at High Initial Velocity

4.1. Difference between Non-Accelerated and Accelerated Flow In Section3, the drag characteristics of circular cylinders immersed in flow that is accelerated from a low initial velocity (Re0 = 40) were analyzed. In this section, flow accelerated from a high initial velocity (Re0 = 1000) is considered. First, various acceleration ranges are examined. Then, the effect of initial velocity on drag change is analyzed in Section5. Figure 21 shows the logarithmic distribution of Cd obtained experimentally in the case of non-accelerated flow by Bullivant [20] and that obtained numerically in the case of accelerated flow in this study. As evident from Figure 21, the Cd values of accelerated flow were higher for all ranges than those of non-accelerated flow. Furthermore, the lower the acceleration, the smaller the difference between the drag values of the accelerated and non-accelerated flows. It is noticeable that Cd values of accelerated flow decrease with time and oscillate when approaching the value of the non-accelerated flow in all acceleration ranges. In addition, the supercritical regions did not appear, unlike in the case Water 2020, 12, x FOR PEER REVIEW 19 of 27 of low initial velocity.

Figure 21. Comparison of drag coefficient Cd between non-accelerated flow (Bullivant [20]) and Figure 21. Comparison of drag coefficient 𝐶 between non-accelerated flow (Bullivant [20]) and accelerated flow (Re0 = 1000). accelerated flow (𝑅𝑒 = 1000) 4.2. Drag Changes over Dimensionless Time and Flow Time 4.2 Drag Changes over Dimensionless Time and Flow Time Figures 22 and 23 show the variations in Cd∗ and Cd, respectively, with tˆ. As mentioned earlier, ∗ it wasFigures confirmed 22 and that 23 show the results the variations for several in 𝐶 di ff erentand 𝐶 accelerations, respectively, showed with 𝑡̂. good As mentioned correlation earlier, in the itcase was of confirmed low initial that velocity. the results In the for high several initial different velocity accelerations cases, it can be showed seen that good the correlationCd∗ and Cd invalues the ∗ casecorrelated of low initial well with velocity. respect In tothe dimensionless high initial velocity time. cases, it can be seen that the 𝐶 and 𝐶 values correlatedFigure well 24 with illustrates respect the to dimensionless variation in C time.d∗ with time. The results show that the smaller the ∗ acceleration,Figure 24 the illustrates smaller thethe valuevariation of C din∗. The𝐶 inflectionwith time. point The atresults which show the slope that increasesthe smaller sharply the ∗ acceleration,appears in the the case smaller of a/ gthe= 1value only. of 𝐶 . The inflection point at which the slope increases sharply appears in the case of a/g = 1 only.

∗ Figure 22. Dimensionless time variation of drag coefficient 𝐶 (𝑅𝑒 = 1000)

Water 2020, 12, x FOR PEER REVIEW 19 of 27

Figure 21. Comparison of drag coefficient 𝐶 between non-accelerated flow (Bullivant [20]) and

accelerated flow (𝑅𝑒 = 1000)

4.2 Drag Changes over Dimensionless Time and Flow Time ∗ Figures 22 and 23 show the variations in 𝐶 and 𝐶, respectively, with 𝑡̂. As mentioned earlier, it was confirmed that the results for several different accelerations showed good correlation in the ∗ case of low initial velocity. In the high initial velocity cases, it can be seen that the 𝐶 and 𝐶 values correlated well with respect to dimensionless time. ∗ Figure 24 illustrates the variation in 𝐶 with time. The results show that the smaller the Water 2020, 12, 1790 ∗ 18 of 24 acceleration, the smaller the value of 𝐶 . The inflection point at which the slope increases sharply appears in the case of a/g = 1 only.

Water 2020, 12, x FOR PEER REVIEW 20 of 27

Water 2020, 12, x FOR PEER REVIEW 20 of 27 Figure 22. Dimensionless time variation of drag coeﬃcient∗ Cd∗ (Re0 = 1000). Figure 22. Dimensionless time variation of drag coefficient 𝐶 (𝑅𝑒 = 1000)

Figure 23. Dimensionless time variation of drag coefficient 𝐶 (𝑅𝑒 = 1000) Figure 23. Dimensionless time variation of drag coeﬃcient Cd (Re0 = 1000). Figure 23. Dimensionless time variation of drag coefficient 𝐶 (𝑅𝑒 = 1000)

Figure 24. Temporal variation of drag coeﬃcient ∗C (Re = 1000). Figure 24. Temporal variation of drag coefficient 𝐶 d(𝑅𝑒∗ =0 1000) ∗ Figure 24. Temporal variation of drag coefficient 𝐶 (𝑅𝑒 = 1000) 4.3 Vortex Shedding Formation Development 4.3 VortexFigures Shedding 25–27 showFormation the stream Developmentlines for various values of 𝑡̂ and a/g. As shown previously, in the case Figuresof low initial 25–27 velocity,show the thestream sizelines of symmetrical for various valuesvortices of tends𝑡̂ and to a /gincrease. As shown as the previously, dimensionless in the timecase ofis lowincreased. initial velocity,In contrast, the insize the of casesymmetrical of high initialvortices velocity, tends to asymmetrical increase as the pair dimensionless vortices are generatedtime is increased. at the beginning, In contrast, and inthe the vortex case is ofthen high created initial and velocity, destroyed asymmetrical alternately. pairVortex vortices shedding are alsogenerated gradually at the develops beginning, as timeand the passes. vortex Because is then of created the growth and destroyed of this wake, alternately. the supercritical Vortex shedding region isalso not gradually shown, but develops 𝐶 rapidly as time approaches passes. Because the drag of theof the growth non- acceleratedof this wake, flow the withsupercritical oscillations, region as

shownis not shown, in Figure but 21. 𝐶 rapidly approaches the drag of the non-accelerated flow with oscillations, as shown in Figure 21.

Water 2020, 12, 1790 19 of 24

4.3. Vortex Shedding Formation Development Figures 25–27 show the streamlines for various values of tˆ and a/g. As shown previously, in the case of low initial velocity, the size of symmetrical vortices tends to increase as the dimensionless time is increased. In contrast, in the case of high initial velocity, asymmetrical pair vortices are generated at the beginning, and the vortex is then created and destroyed alternately. Vortex shedding also gradually develops as time passes. Because of the growth of this wake, the supercritical region is not shown, butWaterCd rapidly 2020, 12, approachesx FOR PEER REVIEW the drag of the non-accelerated ﬂow with oscillations, as shown in Figure21 of 27 21.

(a) 𝑡̂ =2 (𝑅𝑒 = 9927) (d) 𝑡̂ =8 (𝑅𝑒 = 18842)

(b) 𝑡̂ =4 (𝑅𝑒 = 13622) (e) 𝑡̂ =10 (𝑅𝑒 = 20950)

Figure 25. Streamlines around circular cylinder (a/g = 1, Re0 = 1000). Figure 25. Streamlines around circular cylinder (a/g = 1, 𝑅𝑒 = 1000)

Water 2020, 12, x FOR PEER REVIEW 22 of 27

Water 2020, 12, 1790 20 of 24

(a) 𝑡̂ =2 (𝑅𝑒 = 3822) (d) 𝑡̂ =8 (𝑅𝑒 = 6639)

(b) 𝑡̂ =4 (𝑅𝑒 = 4990) (e) 𝑡̂ =10 (𝑅𝑒 = 7309)

Figure 26. Streamlines around circular cylinder (a/g = 0.1, Re0 = 1000). Figure 26. Streamlines around circular cylinder (a/g = 0.1, 𝑅𝑒 =1000)

Water 2020, 12, 1790 21 of 24 Water 2020, 12, x FOR PEER REVIEW 23 of 27

(a) 𝑡̂ =2 (𝑅𝑒 = 2995) (d) 𝑡̂ =8 (𝑅𝑒 = 4990)

(b) 𝑡̂ =4 (𝑅𝑒 = 3821) (e) 𝑡̂ =10 (𝑅𝑒 = 5461)

Figure 27. Streamlines around circular cylinder (a/g = 0.05, Re0 = 1000). Figure 27. Streamlines around circular cylinder (a/g = 0.05, 𝑅𝑒 = 1000) 5. Comparison of Fluid Force by Initial Velocity

Thus far, the fluid forces for two initial velocities have been analyzed with respect to several accelerations. Figure 28 presents the effect of the initial state of flow past a circular cylinder on the drag change. When the flow is accelerated at Re0 = 40, the supercritical region is prominent due to the variation in the vortex size. In contrast, in the case of Re0 = 1000, the oscillation of Cd occurred due to the generation and development of vortex shedding. As a result, the influence of the initial state is transmitted even when the acceleration continues, which eventually leads to a difference in the drag between the two initial velocities. The formation of vortex shedding is evident from the variation in lift coefficient with Reynolds number, as shown in Figure 29. The lift coefficient Cl was defined as follows:

Fl Cl = (7) 1 2 2 ρv A where Fl is the lift force per unit length. In the case of Re0 = 40, Cl converged to 0. In the case of Re0 = 1000, the value of Cl varied due to vortex shedding, and the amplitude also increased with time.

Water 2020, 12, x FOR PEER REVIEW 24 of 27

5. Comparison of Fluid Force by Initial Velocity Thus far, the fluid forces for two initial velocities have been analyzed with respect to several accelerations. Figure 28 presents the effect of the initial state of flow past a circular cylinder on the drag change. When the flow is accelerated at 𝑅𝑒 =40, the supercritical region is prominent due to the variation in the vortex size. In contrast, in the case of 𝑅𝑒 = 1000, the oscillation of 𝐶 occurred due to the generation and development of vortex shedding. As a result, the influence of the initial stateWater is 2020transmitted, 12, 1790 even when the acceleration continues, which eventually leads to a difference22 in of 24 the drag between the two initial velocities.

Water 2020, 12, x FOR PEER REVIEW 25 of 27

Figure 28. Cd versus Re for ﬂow past circular cylinder at two initial velocities Figure 28. 𝐶 versus 𝑅𝑒 for flow past circular cylinder at two initial velocities

The formation of vortex shedding is evident from the variation in lift coefficient with Reynolds number, as shown in Figure 29. The lift coefficient 𝐶 was defined as follows: 𝐹 𝐶 = 1 (4) 𝜌𝑣𝐴 2 where 𝐹 is the lift force per unit length. In the case of 𝑅𝑒 = 40, 𝐶 converged to 0. In the case of 𝑅𝑒 = 1000, the value of 𝐶 varied due to vortex shedding, and the amplitude also increased with time.

Figure 29. Cl versus Re for flow past circular cylinder of two different initial velocity. Figure 29. 𝐶 versus 𝑅𝑒 for flow past circular cylinder of two different initial velocity 6. Conclusions 6. Conclusion In this study, the effects of accelerated flow on a 2D bluff body were analyzed numerically. ToIn investigate this study, the the drag effects increase of accelerated and its relationship flow on witha 2D thebluff form body and were development analyzed ofnumerically. the wake behind To investigatethe cylinder, the thisdrag study increase employed and its arelationship finite-volume with based the form computational and development fluid dynamics of the wake method behind using theFluent cylinder, 19.2 this with study the SST employed k–ω model a finite-volume for turbulence based eff ects.computational fluid dynamics method using Fluent Fackrell19.2 with [17 the] showed SST k–ω that model good for correlation turbulence among effects. results was achieved when using dimensionless parametersFackrell in[17] the showed limited rangethat ofgood acceleration correlation at a /gamong= 0.2, 0.5results and 1.0was with achieved the low initialwhen Reynolds using dimensionlessnumber used. parameters This study in extendedthe limited the range scope of ofacceleration the investigation at a/g = by0.2, considering 0.5 and 1.0 diwithfferent the initiallow initialReynolds Reynolds numbers number (Re0 used.= 40and This 1000) study and exten theded acceleration the scope range of the was investigation extended to 0.01–1.0by consideringg. differentIn initial addition, Reynolds a dimensionless numbers (𝑅𝑒 time =tˆ 40was and introduced 1000) and to the aid accele in intuitiveration range understanding, was extended and to the 0.01–1.0changeg. in drag according to tˆ was analyzed. Because the drag acting on the body and the wake are closelyIn addition, related, a the dimensionless separation angle time and𝑡̂ was the introduced recirculation to lengthaid in intuitive were investigated. understanding, The numericaland the changeresults in obtained drag according are summarized to 𝑡̂ was asanalyzed. follows: Because the drag acting on the body and the wake are closely related, the separation angle and the recirculation length were investigated. The numerical results obtained are summarized as follows: (1) The results show that the initial drag increases due to overshoot, and that the drag decreases as dimensionless time increases. In addition, the calculated drag in this study is in good agreement with that from previous studies, clearly showing an increase in comparison with the drag in the flow with constant velocity. ∗ (2) The value of 𝐶 decreases as 𝑅𝑒 and 𝑡̂ increase, whereas the value of 𝐶 increases as 𝑅𝑒 ∗ and 𝑡̂ increase. When 𝐶 is expressed over time t, the inflection point at which the slope increases sharply is observed. The results also show that the smaller the acceleration, the smaller the value. (3) As in the previous study, it was confirmed that the correlation of data is good for a/g = 1 and ∗ 0.5, but in the case of a/g = 0.1, 0.05, and 0.01, where the flow is relatively steady, the value of 𝐶 increases. (4) In the case of the low initial Reynolds number, as the acceleration and 𝑡̂ increase, the recirculation length gradually increases, and the recirculation length decreases after 𝑡̂ = 6 only when a/g = 0.01. Similarly, the separation location moves toward the windward direction along the surface of cylinder, which results in the increase of the size of the vortex as the acceleration and 𝑡̂ increase.

This variation in the size of the vortex behind the cylinder is directly related to the 𝐶; when the size of vortex is larger, 𝐶 is decreased. (5) In the case of the high initial Reynolds number, an asymmetrical pair of vortices is generated at the beginning. Then, the vortex is created and destroyed alternately. Vortex shedding also

Water 2020, 12, 1790 23 of 24

(1) The results show that the initial drag increases due to overshoot, and that the drag decreases as dimensionless time increases. In addition, the calculated drag in this study is in good agreement with that from previous studies, clearly showing an increase in comparison with the drag in the flow with constant velocity. (2) The value of Cd decreases as Re and tˆ increase, whereas the value of Cd ∗ increases as Re and tˆ increase. When Cd∗ is expressed over time t, the inflection point at which the slope increases sharply is observed. The results also show that the smaller the acceleration, the smaller the value. (3) As in the previous study, it was confirmed that the correlation of data is good for a/g = 1 and 0.5, but in the case of a/g = 0.1, 0.05, and 0.01, where the flow is relatively steady, the value of Cd ∗ increases. (4) In the case of the low initial Reynolds number, as the acceleration and tîncrease, the recirculation length gradually increases, and the recirculation length decreases after tˆ = 6 only when a/g = 0.01. Similarly, the separation location moves toward the windward direction along the surface of cylinder, which results in the increase of the size of the vortex as the acceleration and tˆ increase. This variation in the size of the vortex behind the cylinder is directly related to the Cd; when the size of vortex is larger, Cd is decreased. (5) In the case of the high initial Reynolds number, an asymmetrical pair of vortices is generated at the beginning. Then, the vortex is created and destroyed alternately. Vortex shedding also gradually develops with time. Due to the growth of this wake, the supercritical region is not shown, but Cd rapidly approaches the value of the non-accelerated flow with oscillations. (6) In the case of a low initial Reynolds number, vortex shedding does not occur, but the supercritical region is prominent due to the variation in the vortex size, whereas in the case of a high initial Reynolds number, the drag force fluctuates due to the generation and development of vortex shedding. This shows that the influence of the initial Reynolds number propagates in the accelerated flow, resulting in a change in the fluid force. As a final remark, the drag force in an accelerated flow field over a stationary circular cylinder is considerably different from that in a constant velocity flow field at the same Reynolds number. Therefore, the load evaluation of the bluff body in an accelerated flow must be considered based on the unsteady state conditions.

Author Contributions: Conceptualization, H.A.S. and S.L.; methodology, H.A.S. and S.L.; software, H.A.S. and S.L.; validation, H.A.S.; formal analysis, H.A.S.; investigation, H.A.S., S.L., and J.L.; resources, H.A.S., S.L., and J.L.; data curation, H.A.S. and S.L.; writing—original draft preparation, H.A.S. and S.L.; writing—review and editing, H.A.S. and S.L.; visualization, H.A.S. and S.L.; supervision, S.L.; project administration, S.L.; funding acquisition, S.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by a grant (2018-MOIS31-009) from Fundamental Technology Development Program for Extreme Disaster Response funded by Korean Ministry of Interior and Safety(MOIS). Conﬂicts of Interest: The authors declare no conﬂict of interest.

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