Turbulence modeling approaches on unsteady flow structures around a semi-circular cylinder

Ocean Engineering 200 (2020) 107051

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Turbulence modeling approaches on unsteady flow structures around a semi-circular cylinder Sercan Yagmur *, Sercan Dogan, Muharrem Hilmi Aksoy, Ilker Goktepeli Department of Mechanical Engineering, Faculty of Engineering and Natural Sciences, Konya Technical University, 42250, Konya, Turkey

A R T I C L E I N F O

A B S T R A C T

Keywords: Bluff body CFD Fluid-structure interaction LES RANS Semi-circular cylinder

This paper presents the flow field around a Semi-Circular Cylinder (SCC), a basic geometry for fluid mechanics, obtained numerically by Computational Fluid Dynamics (CFD) at Reynolds number of 3.2 � 104. Time-averaged flow characteristics such as vorticity, velocity fields with streamline patterns, Reynolds stress correlations, Turbulence Kinetic Energy (TKE), pressure and drag coefficients have been examined by using Reynoldsaveraged Navier-Stokes (RANS), Detached Eddy Simulation (DES) and Large Eddy Simulations (LES) turbu­ lence models. Several RANS approaches with turbulence modeling including SST k-ω, k-ε and their derivatives were used, while in the LES simulations, the dynamic Smagorinsky subgrid-scale model was employed. The obtained results of the present study have been compared with an experimental study from the open literature. The results of the investigation showed that when numerical findings are compared to experimental ones, the results of LES and DES turbulence models were found to be more consistent than those obtained by using k-ε and SST k-ω models. Time-averaged flow patterns have been found to be considerably symmetric with respect to the axis passing through the end of the curved surface of the SCC. The deviation of LES and DES results, and experimental results was less than 7% in terms of the drag coefficient.

1. Introduction Flow over a bluff-body is an important issue in a wide variety of engineering and industrial applications. In recent years, designs in many engineering applications have been done by Computational Fluid Dy­ namics (CFD). CFD is a method with high flexibility, accuracy and wide breadth of applications. In addition, it is possible to determine the flow structures with different turbulence models in a cost-effective and quick way (Kocaaslan et al., 2016; Babayigit et al., 2017). However, the im­ broglio in CFD is when it comes to choose the appropriate turbulence model to determine physically occurred complicated flow structures with laminar separations, turbulence transition regions, saddle points as well as the entropy movements around a bluff body. Some bluff bodies with complex geometric shapes have been considered in the studies concerning with the selection of the appro­ priate turbulence models. In the literature, there are some numerical studies using Spalart-Allmaras, Re-Normalization Group (RNG) k-ε, Shear Stress Transport (SST) and k-ω and Reynolds Stress Models (RSM) turbulence models on an airfoil (Villalpando et al., 2011); Reynolds-averaged Navier-Stokes (RANS), Detached Eddy Simulation

(DES) and Large Eddy Simulations (LES) turbulence models on a slender �s-P�erez et al., 2017); standard k-ε, standard k-ω and LES body (Nicola models on a vortex flowmeter (Ozgoren et al., 2015); RNG k-ε, SST k-ω and LES turbulence models employed on a surface-mounted cube (Dogan et al., 2017); RANS and DES models on Ahmed body (Guilmi­ neau et al., 2017); Embedded LES, Scale-Adaptive Simulation (SAS), unsteady-RANS (URANS) and RANS turbulence models on a container wagon (Maleki et al., 2017); URANS, SAS and Improved Delayed De­ tached Eddy Simulation (IDDES) turbulence models on a high speed train (Wang et al., 2017) and comparison between URANS and LES turbulence models on the unsteady effects in the stern region of ships (Kornev et al., 2011) have been encountered. What is more, a building via steady-RANS and unsteady-RANS models (Tominaga (2015); another building by Delayed Detached Eddy Simulation (DDES) and LES turbulence models (Liu and Niu, 2016); idealized morphologies via RNG k-ε, SST k-ω and SST γ Reθ methods (Kardan et al., 2018); different bluff bodies using OpenFOAM (Robertson et al., 2015); rectangular enclosures via standard k-ε, RNG k-ε, Realizable k-ε, RSM, standard k-ω and SST k-ω models (Altac and Ugurlubilek, 2016); different roof types of buildings by utilizing various turbulence models (Ntinas et al., 2018)

* Corresponding author. E-mail address: [email protected] (S. Yagmur). https://doi.org/10.1016/j.oceaneng.2020.107051 Received 27 September 2019; Received in revised form 17 December 2019; Accepted 30 January 2020 Available online 14 February 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.

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Ocean Engineering 200 (2020) 107051

and a tip-leakage flow with and without cavitation using LES turbulence model (Cheng et al., 2019) have been taken into account for the eval­ uation of the performance of different turbulence models. A number of studies using different turbulence models validated by the experimental results of flow around cylinders can be found from the open literature. However, to the best of our knowledge, there is a lack of an exact answer about which turbulence model is more suitable for the accurate or the reliable results respect to the experimental data on flow around the bluff bodies. In the previous studies involving the assessment of the turbulence models, there are several geometrical shapes for the basic models such as cylinders with different cross-sections. For instance; circular cylinders used in the numerical studies with different RANS turbulence models by Unal et al. (2010); SST-SAS and SST-URANS methods by Liu et al. (2017); RANS and SGS based turbulence models by Zhang (2017); a finite volume of pressure based method using SIMPLE algorithm by Chehreh and Javadi (2018), square cylinders considered by LES and RANS models by Rodi (1997); k-ε turbulence models with LES method by Bosch and Rodi (1998); LES and URANS models by Younis and Abrishamchi (2014); One- Eq.-SAS and the SST–SAS k-ω models by Elkhoury (2016) and triangular cylinders analyzed via LES turbulence model (Yagmur et al., 2015, 2017) are some popular geometrical structures encountered in the literature. Furthermore, some specific cases as square block and the surface-mounted cube have also been studied by RANS (Rusdin, 2017) and LES models (Lim et al., 2009), in terms of performance examination of the turbulence models. The studies are generally focused on the investigation of the turbulence models related with flow over these types of geometries. Especially for the bluff bodies, there are available empirical and numerical results such as drag coefficients in the literature for the validation. When the previous works in the literature are extensively viewed, there are restricted studies involving the assessment of the turbulence models in case of flow past the semi-circular cylinder (SCC). Relevant studies on circular cylinder (CC) are already available. The flow struc­ ture over the SCC has some application areas as side mirror of the ve­ hicles, skyscraper structures, abutments, vortex generator on vortex flowmeters, devices such as heat exchangers with different designs, thermal process in food industry, fibrous processing, paper industry, underwater vessels having flat base etc. (Barigou et al., 1998; Bhinder et al., 2012). For instance, a side mirror of a vehicle can be considered as SCC model. Owing to its position on the vehicle, it has a wake region and it produces drag and noise. Approximately 10% of the total drag acting on the vehicle is due to the side mirrors having 0.5% of the total pro­ jection surface area (Mimeau et al., 2014). In another study, Weaver and Veljkovic (2005) have used an open SCC and a parabolic cylinder in a wind tunnel. They have given the results of force coefficients as a function of incident angle for the flow characteristics, Strouhal number and amplitude response. Nada et al. (2007) have studied flow charac­ teristics over a SCC with various orientation angles experimentally and numerically. A wind tunnel was used in the experiments while the standard k-ε turbulence model has been employed for the 2-D numerical analyses. Based on the experimental and numerical results, the relative differences between these results have been found in the range of �20%. Farhadi et al. (2010) numerically investigated the laminar flow past a 2-D SCC under influence of a splitter plate placed in the wake region at different Reynolds numbers ranging from 100 to 500 for three different gaps (g ¼ 0.0D to 4.5D) and two different splitter lengths. Numerical study has been performed by finite volume discretization technique of Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme with Semi-Implicit Method for Pressure Linked Equations-Consistent (SIMPLEC) method. They reported that the drag coefficients were reduced by 6%–7% at 0 � g/D � 3 for 100 � Re � 500 by mounting the same splitter plate at the downstream of the model. Gode et al. (2011) numerically studied on flow past the SCC placed as parallel with respect to the ongoing flow in the case of Re � 120. Findings of their research revealed that Nusselt numbers were influ­ enced by both Reynolds and Prandtl numbers according to their

correlation. Han and Krajnovic (2013) validated a new method which is Very Large Eddy Simulation (VLES) with the experimental results at Re ¼ 3.6 � 104 for the application of passive flow control on a D-shaped cylinder. They obtained well-predicted results as the drag reduction of 18% via CFD results, while it was 17.5% on the experimental study. Kumar and Dhiman (2015) considered the flow characteristics of the SCC placed inside a horizontal channel at 1 � Re � 40 for different blockage ratios. The size of recirculation zone for the SCC was found to be increased with the Reynolds number for the same blockage ratio. Parthasarathy et al. (2017) prepared a study about flow and heat transfer characteristics of a row of SCC for various gap ratios at Re ¼ 100. Small increments for the values of average drag coefficient, Strouhal number and mean Nusselt number have been observed in terms of the gap ratio from 10 to 4. Sisodia et al. (2017) numerically examined the flow over an angled SCC with the downstream square cylinder at 10 � Re � 45. The time-averaged CD value was found minimum at 90� . Although there are some studies on the flow characteristics of the SCC in the literature, comparison of different models and their valida­ tion with the experimental data with respect to the flow structure, drag and pressure coefficient values have not been encountered. SCC has not been commonly considered comparing to circular cylinder (Hamed et al., 2017). In the present paper, the main aim is to investigate the flow characteristics over the SCC by different turbulence models. In addition, the numerical results have been compared to that of the experimental data. However, blunt trailing edge on flow characteristics are more transparently observed by considering the SCC in terms of unsteady flow structure than that of CC, especially for turbulence modeling, where, capturing the unsteady flow structure is a challenging issue. According to this issue, comparison of different turbulence models over the present problem in broad perspective seems to be of great interest for CFD analyses. 2. Methodology 2.1. Turbulence model Turbulent flows are under the influence of fluctuating and irregular velocity fields. These fluctuations mix transported quantities such as momentum and energy. Fluctuations can be in different scales and fre­ quencies. Considering the computer capacity and solution time, turbu­ lent flows are harder to simulate directly in the practical engineering applications. Instead of solving these types of fluid problems directly, the modified equations can be solved by adapting them to the problems. But in the modified equations including additional variables, it is required to define these variables in terms of known quantities. The flow is described by the conservation laws for continuity and momentum (Navier-Stokes) equations as follows:

∂ρ ∂ ðρui Þ ¼ 0 þ ∂t ∂xi � ∂ ∂ ρui uj ¼ ðρu Þ þ ∂t i ∂xi �

ρx’i x’j

(1) � �

∂ρ ∂ ∂ui ∂uj þ μ þ ∂xi ∂xj ∂xj ∂xi

2 ∂ul δij 3 ∂xl

�� þ

∂ � ∂xj (2)

This solution method provides available solution for the mean flow variables and evidently decreases the computational effort. For instance, the governing equations do not require time derivatives and the result is obtained in less solution time and economically for the steady-state flow condition. Therefore, the Reynolds-averaged approach is mostly usable for practical engineering applications, and the eddy viscosity turbulence models such as k-ε and k-ω and their derivatives are based on RANS. The other alternative approach for the transform Navier-Stokes equations is the filtering. It is basically a manipulation of the exact Navier-Stokes equations to extract only the eddies that are smaller than the filter size, which is usually taken as the grid size. The filtering 2

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Ocean Engineering 200 (2020) 107051

approximation also generates extra unknown terms which should be modeled (Smagorinsky, 1963; Lilly, 1966). The mean flow characteris­ tics are collected during the time-dependent analyses. In this case, LES provides an alternative approach in which the large eddies are calcu­ lated in a time-dependent analysis.

1995). The turbulent kinetic energy k is similar to standard and RNG k-ε models; however, the equation of the dissipation rate, ε, is fairly different from them as follows: � � ∂ ∂ ∂ ∂k ðρkui Þ ¼ μ (9) ðρkÞ þ þ Gk ρε þ Sk ∂t ∂xi ∂xi ∂xj

2.1.1. The standard, RNG and realizable k-ε models Three derivatives of k-ε, which are commonly used in the literature and in the practical engineering applications, have similar forms, with transport equations for k and ε. The standard k-ε model is a semi-empirical form of transport equa­ tions for the turbulence kinetic energy (k) and its dissipation rate (ε). Standard k–ε model has some disadvantages on the cases including shear flow, the estimation of the size of the recirculation region in the downstream or flow separation point. In the calculation, the flow is accepted as fully turbulent and the effects of molecular viscosity are negligible (Launder and Spalding, 1983). The turbulence kinetic energy, k, and its dissipation rate of, ε, are calculated from the following transport equations: � � ∂ ∂ ∂ ∂k þ Gk ρε þ Sk ðρkui Þ ¼ μ (3) ðρkÞ þ ∂t ∂xi ∂xi ∂xj �

�

∂ ∂ ∂ ∂ε ε ðρεui Þ ¼ ðρεÞ þ þ C1ε G k μ ∂t ∂xi ∂xi ∂xj k

C2 ε ρ

ε2 k

�

h C1 ¼ max 0:43;

ρu’i u’j

∂uj ∂xi

(5)

�

�

C2ε ρ

ε2 k

Rε þ Sε

Cμ ρη3 ð1 η=η0 Þ ε2 k 1 þ βη3

pffiffiffiffiffi þ Sε vε

(10) (11)

ηþ5

�

∂ ∂ ∂ ∂ω ðρωui Þ ¼ Γω þ Gω þ Dω ðρωÞ þ ∂t ∂xi ∂xi ∂xj

Yω þ S ω

(13)

The terms of turbulence kinetic energy generation Gk is calculated as in k-ε turbulence model. Gω is the generation of ω. The terms of Γk and Γω are the main difference of the k-ω turbulence model from the k-ε tur­ bulence model which are indicated as the effective diffusity of k and ω. Yk and Yω describe the dissipation of k and ω, respectively. Derivation of S terms is also described as the user-defined source terms. 2.1.3. Large eddy simulation (LES) The Large Eddy Simulation is the turbulence model between the Direct Numerical Simulation (DNS) and RANS, which calculates large eddies directly. On the other hand, LES models small eddies. Just like all turbulence models, mesh independence study is required for all CFD analyses. Furthermore, LES requires substantially finer meshes than those used for RANS (Nicol� as-P�erez et al., 2017). Due to the fact that, finer meshes are required for appropriate resolution of wall layers and estimation of transition to turbulence in the boundary layers (Celik, 2003). Therefore, in this study, mesh independence study was per­ formed with respect to the LES model. In addition, the analysis needs to be transient with acceptable time steps, because the temporary resolu­ tion requirements are obtained by the dissipating scales, rather than the time-averaged flow or the energy-containing eddies. Moreover, the governing equations utilized for LES are provided by filtering the time-dependent Navier-Stokes equations (Rodi, 1997; Li et al., 2018). The incompressible Navier-Stokes equations are filtered as follows:

(7)

The basic difference between the RNG and standard k-ε models is due to the additional term in the ε equation given as: Rε ¼

kþ

η i

�

The values of constant terms are C1ε ¼ 1.44 and C2ε ¼ 1.92 due to the lots of experiments for main turbulent shear flows including homogeneous shear flows and disrupting isotropic grid turbulence. Sk and Sε are userdefined source terms. The Re-Normalization Group (RNG) k-ε model is determined from the instantaneous Navier-Stokes equations. The feature that distin­ guishes RNG k-ε model from standard k-ε model is the analytical deri­ vation results in a model with constants, different additional terms and functions in the transport equations (Yakhot et al., 1992). � � ∂ ∂ ∂ ∂k þ Gk ρε þ Sk ðρkui Þ ¼ μeff αk (6) ðρkÞ þ ∂t ∂xi ∂xi ∂xj

∂ ∂ ∂ ∂ε ε ðρεui Þ ¼ μ α þ C1ε Gk ðρεÞ þ ∂t ∂xi ∂xi eff k ∂xj k

ε2

2.1.2. Shear-stress transport (SST) k-ω In this turbulence model, the turbulent viscosity is replaced to explain the transport of the main turbulent shear stress. The SST k-ω model pieces together the virtues of the k–ε and k–ω models in terms of aerodynamic flows. Especially for boundary layers affected by reverse pressure gradients, the validation of this model has been encountered for fluid-structure interaction problems, free shear layers, zero or reverse pressure gradients in the boundary layers (Argyropoulos and Markatos, 2015). Other modifications are to add a cross-diffusion term in the ω equation and mixing function to provide that the model behaves more favorably in both near-wall and far-field regions (Menter, 1994). The transport equations for the SST k-ω are given as follows: � � ∂ ∂ ∂ ∂k ðρkui Þ ¼ (12) ðρkÞ þ Γk þ Gk þ Yk þ Sk ∂t ∂xi ∂xi ∂xj

where, the term of the Gk is the turbulence kinetic energy generation owing to the mean velocity gradients and determined as: Gk ¼

ρC2

The default values of constant terms are nearly similar to that of the RNG k-ε model and these are C1 ¼ 1.44 and C2 ¼ 1.9 for Realizable k-ε model.

(4)

þ Sε

�

∂ ∂ ∂ ∂k þ ρC1 Sε ðρεui Þ ¼ μ ðρεÞ þ ∂t ∂xi ∂xi ∂xj

(8)

In this term, the constant Cμ ¼ 0.0845, η�Sk/ε, η0 ¼ 4.38, β ¼ 0.012. Because of that, the RNG is a refinement of the Standard k–ε model which is more sensitive to the prediction of the effects of rapid strain and streamlines curvature compared to standard k-ε model. The values of constants terms are also different from the standard k-ε model and these are C1ε ¼ 1.44 and C2ε ¼ 1.92. In addition to the standard and RNG derivatives of k-ε model, ANSYS-Fluent ensures the Realizable module. As having the modifica­ tions of the other derivatives of k-ε model, it is used for the problems with rotating shear flows, boundary-free shear flows such as mixing layers, planar and round jets, channel flows, flat plate boundary layers with and without a pressure gradient and separated flows. Within the framework of turbulent flow physics, the Realizable module includes specific mathematical limitation on the normal stresses (Shih et al.,

∂ρ ∂ ðρui Þ ¼ 0 þ ∂t ∂xi

(14) �

� ∂ ∂ ∂ ∂u ðρu Þ þ ρui uj ¼ μ i ∂t i ∂xj ∂xj ∂xj

�

∂p ∂xi

∂τij ∂xj

(15)

where, τij represents the subgrid-scale stress, and it is described as:

3

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τij � ρui uj

Ocean Engineering 200 (2020) 107051

2.2. Computational domain and boundary conditions

(16)

ρui uj

The Navier-Stokes equation form for LES (Eqs. (14) and (15)) demon­ strates similarity to the Reynolds-averaged Navier-Stokes equation (Eqs. (1) and (2)). The time dependent variables reveal the significant dif­ ference, because these are filtered quantities in LES equations, which are mean quantities in RANS equations. Due to the filtering, the subgridscale stresses are unknown and modeled as:

τij

1 τkk δij ¼ 3

2μt Sij

In the present study, the computational 3-D flow domain was created by taking into account of the experimental study of Hamed et al. (2017). A smooth SCC having dc ¼ 50 mm diameter is placed vertically in the air flow domain within the dimensions -6.0dc � x � 20.0dc in the stream­ wise direction, -6.0dc � y � 6.0dc in the vertical direction and -5.0dc � z � 5.0dc in the transverse direction as shown in Fig. 1. No-slip boundary conditions are described on the physical surfaces of the SCC model. At the inlet boundary condition of the flow domain, the uniform velocity inlet is defined as U∞ ¼ 9.5 m/s to obtain a Reynolds number of Re¼U∞dc ∕υ � 3.2 � 104, where υ is the kinematic viscosity of the air which is υ � 1.46x10 5 m2/s, is defined with the zero-pressure gradient. At the outlet boundary condition, pressure outlet value is specified as zero. Pressure outlet boundary condition has been defined as a static (gauge) pressure at the outlet condition. The boundary is open to the atmosphere which means the gauge pressure is to be set zero (Dogrul et al., 2016). The free-slip condition is applied at the top, bottom and side walls of the flow domain. The finite volume method is integrated to solve the transient and incompressible Navier-Stokes equations with the pressure and velocity coupling with the splitting of the coupled-type algorithm. The second order implicit scheme was used for the pressure and the momentum spatial discretizations with Green-Gauss solver to take the physical diffusion into account. It is significantly influenced by numer­ ical diffusion; as a result of the central differencing schemes, it should be appropriate approximation to conduct the spatial discretization. The second order implicit scheme is also used for time integration.

(17)

This is the eddy viscosity model which is mostly used as subgrid-scale model. In this equation, μt is the subgrid-scale turbulent viscosity, and Sij is the rate of strain tensor and determined as follows: � � 1 ∂ui ∂uj Sij � þ (18) 3 ∂xj ∂xi The eddy viscosity μt is modeled in accordance with the SmagorinskyLilly model:

μt ¼ ρL2s jSj

(19)

and Smagorinsky constant: qffiffiffiffiffiffiffiffiffiffiffiffi jSj � 2Sij Sij

(20)

The CS was taken as 0.1 from homogeneous isotropic turbulence in the inertial sub-range. The mixing length, Ls, for subgrid scale is computed: 0 1 1

LS ¼ min@κd; CS V3 A

(21)

2.3. Grid and time independence studies Numerical results show that the unsteady prediction results are more accurate than those of the steady results, due to the three-dimensional unsteady turbulent flow regime in the downstream of the SCC. There­ fore, the transient analyses were performed with different time step size to obtain the time independent results according to the Grid Structure III shown in Fig. 3. In ITTC Standard for periodic phenomenon like vortex shedding applications, it is advised to use at least 100 time steps per period (Anonymous, 2011). In this study, vortex shedding frequency was calculated and its value was 3.03 Hz at Reynolds number of 3.2 � 104. The period of the vortex shedding has been found as 0.33 s. Ac­ cording to ITTC, the time-step size is recommended to be around 0.0033 s. Therefore, in this study, the time step size has been defined between 0.0001 s and 0.05 s. Based on this result, the dimensionless time step sizes are set to Δt1¼0.95dc/U∞, Δt2¼0.095dc/U∞, Δt3¼0.0095dc/U∞ and Δt4¼0.0019dc/U∞; 4.0x102, 4.0x103, 4.0x104 and 2.0x105 number of time steps, respectively, whose multiplications are equal to totally 20 s in actual time for all cases. Each time step consists of 20 iterations. Furthermore, the convergence criteria of the numerical 3-D velocity components are detected at each time step by controlling the residuals of the all equations, which are fixed to be solved by setting their variations less than 10 7. During the analyses, the mean streamwise velocity is controlled for the study of time step independence as can be seen in

where, the von Karman constant is described as κ, d is the distance to the closest wall, and V is the volume of the computational cell. 2.1.4. Detached Eddy Simulation (DES) DES approach, also known as the hybrid LES/RANS, was first pro­ posed by (Spalart, 1997). DES turbulence model uses both the RANS models near the wall and LES model in the wake region of flow where unsteady and chaotic structure of flow occurred (Spalart, 2000). In DES models, RANS equations are used for the outside of the wake region of the flow to reduce the calculation time compared to the use of LES in the whole flow volume. For instance, it includes RANS simulation in the boundary layers and LES run in the unsteady separated regions (Zhang et al., 2019). There are three options of subgrid turbulence modeling for DES turbulence model as Spalart-Allmaras, Realizable k-ε and SST k-ω. For the simulations carried out in this work, SST k–ω based DES models were employed. Actually, in the analyses, DES performs depending on a turbulent length scale, Lt, to decide which approach is used during the simulation. At the inlet boundary conditions, DES starts the computation with the SST model, the formulation is the same as that of standard SST model except the length scale used in the calculation of the dissipation rate. This is modified by local grid spacing, Δ. If the turbulent length scale is greater than the grid spacing, which is common in the regions with large eddies and chaotic flow nature, LES is activated in the DES formulation. The activation of LES or the switching to SST model in DES is controlled by a blending factor, F (Menter and Kuntz, 2003). Gener­ ally, this factor takes the form as: F¼

Lt CDES Δ

(22)

where CDES is a constant.

Fig. 1. Schematic view of the computational domain and boundary conditions. 4

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Ocean Engineering 200 (2020) 107051

CD ¼

2FD

(23)

ρU2∞ A

where, FD is the drag force in the streamwise direction, ρ is the fluid density and A is the projection area of the front view. The variations of mean CD distribution with different grid sizes are presented in Table 1. As seen, there is a quite change of CD among Grid Structures I-II-III but no significant difference between Grid Structures III and IV, which was under 1%. In addition to the mean value of CD, the variations of the normalized time-averaged streamwise velocity components along the xaxis and y-axis at the downstream of the bluff body were examined. As shown in Fig. 4a, the saddle point is occurred approximately at x/dc ¼ 1.52 for Grid Structures III and IV, while, it is observed nearly at x/dc ¼ 1.4 and x/dc ¼ 1.28 for Grid Structures I and II, respectively. Apparently, along the y-axis in Fig. 4b, the mean streamwise velocity distributions of Grid Structures I and II are different from Grid Structures III and IV at the location of the saddle point at x/dc ¼ 1.52. The saddle point was also determined nearly at x/dc ¼ 1.6 in the experimental study (Hamed et al., 2017). In addition, according to the solution time per time step, a comparison between the grid structures was conducted. The results revealed that the Grid Structure III was more consistent than the Grid Structure IV and the difference between their results was about 40%. According to the results given in Table 1 and Fig. 4, due to the fact that the computer capacity and solution time cost are of great importance in CFD analyses, further analyses were performed with the Grid Structure

Fig. 2. Variation of normalized time-averaged streamwise velocity component along the downstream of SCC at different time step sizes.

Fig. 2. The analysis has been performed by using the LES turbulence model and it showed that the differences between the results of the time step Δt3 and Δt4 are negligible. For instance, the saddle point in the downstream of the SCC is nearly occurred at x/dc ¼ 1.52 for Δt3 and Δt4, it is occurred at x/dc ¼ 8 for Δt2 and not observed within the flow domain limits for Δt1. Therefore, the analysis has been continued by using the time step size Δt3, considering the total solution time cost and ITTC Standard. Another important factor that influences the solution time and ac­ curacy of the results is the grid size. To obtain a good performance from the CFD analyses, enough element size is needed to solve the chaotic flow structure in the past of the bluff body. In CFD studies, the refine­ ment factor represents the ratio of two grid sets, which is advised to be greater than 1.3 (Celik et al., 2008; Tezdogan et al., 2015; Sezen et al., 2018; Dogrul, 2019). For a better prediction, the grid density should be increased in the near of the high gradient velocity distribution regions and grid structures with fine quality which should be built close to the fluid-solid interaction regions. In this study, the computational domain was divided into seven sub-domains for each grid arrangement as given in Table 1. Structured grids with quadrilateral elements were used in all the sub-domains as can be seen in Fig. 3. In this figure, the grid structure was compressed toward the rigid wall and four sets of grid arrangements were built for the grid independence study to ensure that the results are convincing. In this study, the refinement factor is between 1.68 and 2.28. Therefore, the results of the grid independence study in terms of the mean drag coefficient and the normalized mean streamwise velocity along the x-axis and y-axis have been presented in Table 1 and Fig. 4, respectively. The drag coefficient is the dimensionless form of the drag force which is acting on the surfaces of the bluff body in the streamwise direction given as follows:

Table 1 The grid arrangements and the mean CD distributions for grid independence study.

Grid size of subdomain a (105) Grid size of subdomain b (105) Grid size of subdomain c (105) Grid size of subdomain d (105) Grid size of subdomain e (105) Total Grid Size (106) Max skewness Mean CD CPU hour Solution time per time step

Grid Structure I

Grid Structure II

Grid Structure III

Grid Structure IV

0.5

1.14

2.0

3.375

1.0

2.28

4.0

6.75

2.5

5.70

10.0

16.875

2.0

4.56

8.0

13.50

1.0

2.28

4.0

6.75

1.0 0.013 1.247 ~ 100h ~9s

2.28 0.012 1.138 ~133h ~12s

4.0 0.010 1.107 ~200h ~18s

6.7 0.009 1.115 ~333h ~30s

Fig. 3. Computational non-uniform grid arrangement of the solution domain with seven subdomains (a-e) and detailed view around the SCC. 5

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Fig. 4. Variations of normalized time-averaged streamwise velocity component along the (a) x-axis and (b) y-axis of the downstream of the SCC for different grid arrangements.

III. All numerical computations were performed on the Dell workstation with 64 GB RAM, Intel Xeon CPU E5-2630, 2.6 GHz.

the field of view. Additionally, the saddle point could also provide some information on the other flow patterns. One of them is the time-averaged cross-stream velocity contours that are presented in the right column of Fig. 5. Due to the blockage effect, the cross-stream velocity is induced in the negative and positive directions. Then, after the flow passing the SCC, two different symmetrical pairs of clusters were formed in the downstream. The cluster regions with comparatively smaller size, were centered at nearly x/dc ¼ 0.6 in the experimental result (Hamed et al., 2017). In CFD results, the centers of small clusters are found at x/dc ¼ 0.52 and x/dc ¼ 0.31 by LES and DES turbulence models, respectively. However, for SST k-ω, Realizable and standard k-ε models, the small clusters are occurred approximately near the surface of the SCC. Besides, two small clusters are also obtained by RNG k-ε model, however, compared to the experimental results, these clusters did not physically occur. Furthermore, considerable time-averaged cross-stream velocity contours, large size clusters, occur between x/dc ¼ 1.0 and x/dc ¼ 3.0 in the experimental and LES results, where the centers of these clusters are aligned with the saddle points. However, for the rest of the turbulence models, the effect of large size clusters is observed nearly up to 2.0dc and also the centers of these clusters related with the saddle points. Gener­ ally, except for RNG k-ε model, the effect of the time averaged cross-stream velocity is diminished beyond x/dc ¼ 3.5, and can be neglected for this reason. In RNG k-ε model, due to the formation of the second reverse flow, the influence of cross-stream velocity continues to the further. Another flow characteristic is the instantaneous and the timeaveraged vorticities that were normalized with dc/U∞ and given in Fig. 6. The instantaneous vorticity ω*¼ω dc/U∞ results clarify the von Karman vortex street owing to the vortex shedding on the experimental and numerical studies. As in the experimental study (Hamed et al., 2017), the flow separation occurs from the top and the bottom corners of the SCC at all CFD results. The vortex shedding in the downstream of the SCC coming from the top side rotates in the clockwise direction, while the one from bottom side turns in the counter clockwise direction, and this situation creates the shear layer between the upcoming flow and the vortex shedding. When the flow separates from the body surface, small scale eddies roll up along the downstream. These small eddies appear more prominently in the LES and the DES turbulence models results, which are similar to the experimental study, whereas it is not captured by the rest of the turbulence models. Similarly, the instantaneous vorticity structures of the experimental, LES and DES results, the time-averaged vorticity patterns <ω*¼ω dc/U∞> as shown in the right column of Fig. 6 are also in good agreement. The small-scale vorticity suppresses each other and is not visible in the time averaged vorticity patterns, while the large-scale symmetrical positive and negative

3. Results and discussion In this study, a numerical investigation of a flow around the SCC was considered by using different turbulence models. To do so, LES, DES, SST k-ω and Realizable, RNG, standard k-ε turbulence models at Re � 3.2 � 104 were used for simulations and the obtained results were reported in Figs. 5–11. The dimensions of all images were normalized with the diameter, as characteristic length, of the SCC as x/dc and y/dc. The distributions of the values were given in the legend bar for the flow patterns. The maximum layers of the flow patterns are displayed as a red background color, while the minimum layers are indicated with a blue one. 3.1. Instantaneous and time-averaged flow patterns The distribution of non-dimensional time averaged streamwise and cross-stream velocity components have been given in Fig. 5. The time-averaged streamwise and cross-stream velocities were normalized with the free stream velocity (U∞). As emphasized in the left column of Fig. 5, the flow structures in the wake region are almost equally symmetrical with respect to the centerline of the body. The flow accelerates after passing the top and bottom corners of the SCC and reaches its maximum value near that region. The location of the saddle point in the experimental study (Hamed et al., 2017) was approximately determined at x/dc ¼ 1.6. Referring to the streamline topology, the nearest result to the experimental study was found to be x/dc ¼ 1.52 by LES turbulence model with 5% of relative error. For the rest of turbulence models in the present study, the saddle point was nearly obtained at x/dc ¼ 1.17 for DES and Realizable k-ε turbulence models, and for standard k-ε model and SST k-ω model, it was found at x/dc ¼ 1.04 and x/dc ¼ 0.92, respectively. In addition, the error magnitude values were found as 23.8%, 39.48%, 25.0% and 32.5% for of DES, SST k-ω, Realizable k-ε and standard k-ε models respectively. For RNG module of k-ε turbulence model, the flow structure in the down­ stream of the SCC was significantly different compared to those of the rest of applied CFD models in this study as well as to the experimental one. In the downstream of the SCC, two symmetrical foci points with similar sizes were observed. These foci cycling towards the opposite directions are consisted in the experimental and numerical results except for RNG k-ε model, however in this model, two symmetrical large-scale foci points are observed after the first saddle point. Needless to say that the reverse flows have continued to extend beyond x/dc ¼ 5 in 6

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Fig. 5. Comparison of the normalized time-averaged streamwise velocity () and superimposed streamline topology (left column) and cross-stream velocity () (right column) around the SCC a-experimental (Hamed et al., 2017) and with various turbulence models; -b- LES; -c- DES; -d- SST k-ω; -e- Realizable k-ε; -fstandard k-ε; -g- RNG k-ε models.

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Fig. 6. Comparison of the normalized instantaneous vorticity (ω*) (left column) and the time-averaged vorticity (<ω*>) (right column) around the SCC by -aexperimental study (Hamed et al., 2017) and with various turbulence models -b- LES; -c- DES; -d- SST k-ω; -e- Realizable k-ε; -f- standard k-ε; -g- RNG k-ε.

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vorticity patterns are occurred in the wake region of the SCC. As in the experimental results, the time-averaged vorticity patterns in the region beyond the saddle point lose their effect for all CFD results. The closest results to the experimental base study (Hamed et al., 2017) were ob­ tained by LES turbulence model between the values <ω*> ¼ �5 x 10 3, while DES model is somewhat structurally similar to these results. It is worth mentioning that near the back side of the SCC, since the direction of the vortex shedding changes continuously, the time-averaged vorticity patterns are nearly equal to zero for the LES and DES results which are in good agreement with the experimental results. However, in the SST k-ω turbulence model, the clusters of negative and positive vorticity patterns approach to the back surface of the SCC. For the de­ rivatives of k-ε turbulence model with the Realizable and standard modules, the results are approximately the same, but in the RNG mod­ ule, a different vorticity structure is formed as in the velocity contours. In fluid dynamics, Turbulence Kinetic Energy (TKE is the averaged kinetic energy per unit mass associated with eddies in turbulent flow, emphasized in the normalized form as follows, TKE ¼ < u’2 þ v’2 > =U2∞ , where u0 is the velocity fluctuations and defined as u0 ¼u-U, u is the instantaneous velocity and U is the local mean velocity. In the Fig. 7a, the base study (Hamed et al., 2017) showed a single peak of the TKE observed in the wake region of the body around x/dc ¼ 1.8 at the symmetry axis. Similar to the experimental results, the effects of the velocity fluctuations begin within the first diameter downstream of the SCC in the LES turbulence model. The maximum value of the normalized TKE ¼ 0.0133 is determined around x/dc ¼ 1.76 by the LES turbulence model. For the rest of the turbulence models, the position and the strength of the TKE changed according to the levels of the velocity fluctuations and the vortex shedding frequency of the flow. In the DES and SST k-ω turbulence models, the maximum value is nearly the same with that of the LES, while it is located around x/dc ¼ 1.2 and x/dc ¼ 1.0, respectively. For the Realizable and standard k-ε models, the maximum (normalized) TKE values were found as TKE ¼ 0.008 and TKE ¼ 0.01, respectively. However, the RNG module showed different structure due to the observed vortex shedding structure. The Reynolds stress correlations < u’ v’ > =U2∞ in Fig. 8 are decided on the momentum transfer in the downstream of the SCC. Due to the time-dependent behavior of the vortex shedding and the flow entrain­ ment into the wake region of the SCC, the momentum transfer occurred. As pointed out in the experimental study of Hamed et al. (2017), the small and large scales of asymmetric clusters have occurred with respect to the symmetry axis in the results of LES and DES, appropriate with the lateral mean shear. As in the normalized TKE, the center of maximum 2 Reynolds stress correlations < u’ v’ > =U∞ ¼ �0.1 was also nearly found at x/dc ¼ 1.76 and x/dc ¼ 1.2 by LES and DES turbulence models, respectively. The Reynolds stresses are approximated by an eddy vis­ cosity times the mean rate-of-strain tensor. The strain rate tensor is obtained by the Boussinesq approximation for the k-ω and k-ε turbulence models. (Schmitt, 2007). Therefore, all the components of the Reynolds stresses are not available in the k-ω and k-ε turbulence models and their derivatives since these turbulence models are two-equations based models. Hence, the flow characteristics of Reynolds stress results are only given by LES and DES turbulence models. 3.2. One-dimensional profiles of flow characteristics Additional insights over the effect of the SCC on its wake region have been presented by one-dimensional flow profiles of , TKE and u’v’/U2∞ along the x-axis of symmetry plane of the SCC, i.e., y∕dc ¼ 0, and at x∕dc ¼ 1.2 locations in the range of y∕dc ¼ �1.35, which were selected for comparison purpose since these locations have also been used in the experimental study of Hamed et al. (2017). In Fig. 9, the superposition of the results of all turbulence models and the experi­ mental base study for the comparison was shown. The u∕U∞ profiles along the axis of symmetry (Fig. 9a) indicate that velocity recovery was

Fig. 7. Comparison of the normalized TKE around the SCC -a- experimental (Hamed et al., 2017) study and with various turbulence models -b- LES; -c- DES; -d- SST k-ω; -e- Realizable k-ε; -f- standard k-ε; -g- RNG k-ε.

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2 Fig. 8. Comparison of the normalized Reynolds stress correlations ( < u’ v’ > =U∞ ) around the SCC, -a- experimental (Hamed et al., 2017) with various turbulence models -b- LES; -c- DES models.

rapid. As can also be seen in Fig. 5, recirculation bubbles with different dimensions have been observed for each case. As it was pointed out above, the saddle point of experimental study of Hamed et al. (2017) occurred at x∕dc ¼ 1.6 in the present study, and the closest result is obtained by LES model at x∕dc ¼ 1.52. Needless to say that, x∕dc ¼ 5.0 is not enough for the full recovery for all of the treated cases. However, the recovery rate is getting slower after the point of approximately x∕dc ¼ 1.75 for the models of DES, SST k-ω, Realizable and standard k-ε. The recovery rate for decreasing trend is observed after nearly x∕dc ¼ 2.35 for both LES result and experimental result of Hamed et al. (2017). These profiles also show the location where the velocity has the global mini­ mum. The global minimum was found at x∕dc ¼ 0.25 for RNG k-ε, x∕dc ¼ 0.35 for SST k-ω, x∕dc ¼ 0.50 for standard k-ε, x∕dc ¼ 0.60 for DES, x∕dc ¼ 0.70 for Realizable k-ε, x∕dc ¼ 0.92 for LES and x∕dc ¼ 1.00 for experimental study of Hamed et al. (2017). In Fig. 9b indicates that, along the x axis of symmetry, the turbulent kinetic energy had the same decreasing trend after nearly x∕dc ¼ 3.00 for all the cases. The global

maximum was observed at x∕dc ¼ 0.65 for RNG k-ε, x∕dc ¼ 0.80 for SST k-ω, x∕dc ¼ 1.30 for standard k-ε, x∕dc ¼ 1.1 for DES, x∕dc ¼ 1.35 for. Realizable k-ε, x∕dc ¼ 1.60 for LES while the global maximum for the experimental study of Hamed et al. (2017) was reported to be at x∕dc ¼ 1.75.The symmetrical flow patterns are reported in Fig. 9 c, d and e where the normalized streamwise velocity components, TKE values and Reynolds stress correlations were given for x∕dc ¼ 1.2. As demonstrated in Fig. 9c, the normalized streamwise velocity components reach the uniform value approximately at y∕dc ¼ �0.95 for RNG k-ε, y∕dc ¼ �0.83 for SST k-ω, y∕dc ¼ �0.70 for standard k-ε, y∕dc ¼ �0.63 for DES, y∕dc ¼ �0.64 for Realizable k-ε, y∕dc ¼ �0.55 for LES and y∕dc ¼ �0.62 for experimental study of Hamed et al. (2017). After the point of approxi­ mately y∕dc ¼ �0.75, except the RNG k-ε model, the TKE values are getting closer to zero as can be seen in Fig. 9d. This situation is clearly supported by the Reynolds stress correlations of Fig. 9e. Due to the effect of fluctuations on both TKE values and Reynolds stress correlations; the results of LES and DES turbulence models demonstrated a tendency 10

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Fig. 9. The variations of flow characteristics along x-axis at symmetry plane of the SCC: -a- normalized streamwise velocity , -b- TKE; spanwise profiles at x∕dc ¼ 1.2 locations with the range of y∕dc ¼ �1.35: -c- normalized streamwise velocity , -d- TKE and -e- Reynolds stress correlations u’v’/U2∞.

similar to those of the experimental data of the base study.

and bottom sides of the SCC and its minimum value was CP ¼ -1.36. Generally, the negative pressure region disappeared beyond the saddle point in the wake region of the bluff body. Considering the similarity of the experimental and LES results in terms of instantaneous and timeaveraged flow patterns in Figs. 5 to 9, the pressure coefficient ob­ tained by the LES turbulence model can be regarded as real-like condition. As mentioned in the grid independence study, the mean CD history is another important issue on flow over bluff bodies. The drag force is composed of the pressure force acting on bluff body and the friction force from its surface. The friction force is affected by the fluid because of the shear stress patterns and therefore, the friction force is signifi­ cantly important in terms of laminar flow. Conversely, the pressure force is important for turbulent flow, because it depends on the pressure dif­ ference between the upstream and the downstream of the bluff body. The time-dependent CD behavior obtained by LES turbulence model is given in Fig. 11. As seen in the time dependent history of CD, the fluc­ tuations are occurred owing to the vortex shedding in the shear layer. In addition, the time-averaged mean CD was found to be 1.107 with the LES

3.3. Pressure and drag coefficients The pressure coefficient is a dimensionless number which describes the relative pressure values around a body in a flow field as defined in Eq. (24). P P∞ CP ¼ � 1 2ρU2∞

(24)

where: P is the local static pressure, P∞ is the static pressure in the free stream. In Fig. 10, stagnation point where the pressure coefficient dis­ tribution reaches its maximum value of CP ¼ 1.0 at the first contact point of flow to the SCC. That is at this point the flow stagnates, and the total pressure is equal to the static pressure. With the flow acceleration, a negative pressure zone is formed in the vicinity of the top and bottom corners of SCC and continues at the downstream of the SCC. The nega­ tive pressure coefficient structure was found to be symmetric at the top 11

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Fig. 11. The time history of CD by the LES turbulence model.

turbulence model. For the rest of the turbulence models, the mean values of CD were calculated and compared with the experimental results of Hamed et al. (2017), then reported in Table 2. In this table, ΔCD represents the difference between CD in the ex­ periments and CFD. It worth mentioning that in the experimental study, the momentum method approximation was used by Hamed et al. (2017) to determine the mean CD. In all turbulence models except the RNG k-ε and SST k-ω, the maximum difference value between the experimental and CFD results is found to be below 7%. 4. Conclusions The flow characteristics over the SCC at Re ¼ 3.2 � 104 have been obtained numerically by six different turbulence models which are the most commonly used in the literature and engineering applications, namely: LES, DES, SST k-ω and derivatives of k-ε. The CFD results were compared with those of the experimental study found in the literature. The present paper has supported quantitative results about the turbu­ lence model at the upstream and downstream of the SCC. At the beginning of numerical analyses, the grid and time indepen­ dence studies were done by LES turbulence model. As a result of these studies, the non-dimensional time step was found to be Δt ¼ 0.0095dc/ U∞ and the flow volume was created with about four million grid size. The preferred lower time step and higher grid size increase the total solution time, and it requires a powerful computing capacity. Taken together, when the experimental and CFD results are compared, except the RNG k-ε model, flow characteristics practically looked similar to one another. In this context, some important flow characteristics such as streamline topology superimposed streamwise and cross-stream velocity contours, instantaneous and time-averaged vorticity patterns, TKE, Reynolds stress correlations and pressure and drag coefficients were compared in detail. When all the time-averaged flow patterns are examined, it is determined that the closest result to the experimental result is obtained by the LES turbulence model. The DES turbulence model follows the LES turbulence model in terms of flow structure similarity, especially in the instantaneous vorticity results, findings revealed that the LES and DES turbulence models have been very successful in capturing small scale vorticity components, and the Table 2 Variations of the mean CD for each turbulence models and experimental result.

Fig. 10. Comparison of the mean pressure coefficient distribution around the SCC with various turbulence models. -a- LES; -b- DES; -c- SST k-ω; -d- Realizable k-ε; -e- standard k-ε; -f- RNG k-ε.

12

Method

Mean CD

(|ΔCD|/CD,

Experimental (Hamed et al., 2017) LES DES SST k-ω Realizable k-ε standard k-ε RNG k-ε

1.16 1.107 1.226 1.584 1.136 1.235 2.749

– 4.57 5.68 36.5 2.07 6.47 113.7

experimental)

x100 (%)

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results were in good agreement with the experimental study. However, the outcomes demonstrated that SST k-ω, Realizable and standard de­ rivatives of k-ε turbulence models showed relatively similar results in terms of flow characteristics when compared to one another. The evidence from this study suggests for this type of flow problem, the flow around the bluff body, the LES turbulence model leads to the closest result to the real-like conditions. However, this turbulence model is only used for 3-D flow analysis. This is a disadvantage for practical engineering applications, because it increases the number of the grid and total solution time, and requires high computer capacity. According to the outcomes of this study, if a 2-D analysis is required, needless to say, the DES turbulence model may be recommended over other applied turbulence models.

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