INFLUENCE OF REYNOLDS NUMBER IN SEABED PROXIMITY IN STUDY OF FLOW AROUND A CIRCULAR CYLINDER INFLUÊNCIA DO NÚMERO DE REYNOLDS NA PROXIMIDADE DO

Flexible risers, widely used in offshore engineering, are long multilayer tubes designed to carry fluid — such as oil and natural gas — from the seabed to the sea platforms. In this work, the fluid passage was evaluated in a uniform flow around the cross-section of the tube in a twodimensional analysis. The seasonality and variation of sea currents over time make it necessary to study different flow regimes. Therefore, a variation of Reynolds number and type of flow is applied, thus obtaining results for the laminar regime. In the flow around an obstacle is interesting the study of the vortex shedding, fluid oscillations, drag and lift force and Strouhal number. Therefore, these parameters for two Reynolds numbers in the laminar regime are obtained through modeling using computational fluid dynamics software, ANSYSR c © Fluent, and then compared with the results of other models in the literature. The influence of seabed proximity, that is, the vertical position of the riser was investigated. Thus, the coefficients are obtained as a function of the distance. It could be observed that as the cylinder approaches the seabed, the values of coefficients varied significantly for both Reynolds number values studied, with some difference between them.


INTRODUCTION
Flexible risers, widely used in offshore engineering, are long multilayer tubes designed to carry fluid -such as oil and natural gas -from the seabed to the sea platforms. In this scenario, the risers have to be able to withstand efforts ranging from their own weight, water column pressure, to dynamic loads resulting from sea currents.
This study aims to evaluate the influence of sea currents on flexible risers. As for this interaction, the fluid passage was evaluated in an uniform flow around the cross-section of the tube in a two-dimensional analysis, miming the cross-section of the pipeline. It is no longer today that the study of flow around a circular cylinder has been the object of study for many researchers around the world. Such as (TRITTON, 1959) and (DENNIS;CHANG, 1970) the middle of the last century with experimental and numerical work. Currently, most of the studies are done using computers, through computational techniques, such as (STRINGER; ZANG; HILLIS, 2014), (ABDULAZIZ, 2017), (ROSETTI; VAZ; FUJARRA, 2012) and many others.
One of the main concern of this study is to evaluate the influence of seabed proximity in aerodynamic coefficients of a circular cylinder. If the flow around a cylinder in unbounded condition has been studied extensively, in contrast, the flow around a cylinder in the presence of a fixed boundary has received less attention (SARKAR; SARKAR, 2010). Although this disparity, it is possible to cite the contributions from (TANEDA, 1965;BEARMAN;ZDRAVKOVICH, 1978;SARKAR;SARKAR, 2010;NAMAZI-SALEH et al., 2017;TAWEKAL, 2015).
The motivation for this often comes from the cylindrical geometries used as structures and fluid transport in the offshore industry, like risers that have a circular cross-section. This use is particularly relevant due to the exploitation of new renewable energies, wind and marine energy technologies, many of which include cylindrical features that need to be assessed for their structural loading caused by vortex shedding.

Objectives
The first objective is to validate the modeling for flow around a circular cylinder, at Reynolds number equal to 100 and 1000. Later, this study aims to evaluate the influence of seabed proximity in the aerodynamic behavior of a circular cylinder. In practice, this analyse can be interpreted as the proximity of the oil riser to the seabed.

FLOW MODEL
The flow is predicted by enforcing the conservation of mass and momentum, respectively in Eq. 1 and Eq. 2, for viscous and incompressible flow.
Where ρ is the fluid density, p is the static pressure, v the velocity field,τ = µ ∇ v + ∇ v T is the stress tensor, µ is the viscosity, ρ g and F are the gravitational body force and external body forces.

NUMERICAL METHOD
Discretization in finite volume is essential so that all equation terms can be solved. In this work the method of discretization used for gradient was least squares cell based and for the upwind scheme was second order. The solver that is chosen for laminar transient flow simulation in ANSYS R Fluent is explicit formulation.
The result is that a low Courant number is required to maintain numerical stability of which is given in Eq. (3), where ∆t is the time step and ∆x is the minimum cell width. It was used Courant equal the 0.2 in all cases in this study.
All the results presented in Section 4 were assessed in terms of the aerodynamic coefficients in the cylinder, such as the mean value of drag coefficient (Cd), fluctuation of lift coefficient (Cl) and Strouhal number (St).
According to (BLAZEK, 2015), the force coefficients, such as drag (Cd) and lift (Cl) coefficients on the cylinder, could be extracted from 4 and 5, where L is the length of the pipe that the 3rd dimension (z), F D and F L are drag and lift force respectively.
The mean drag coefficient, Cd can be obtained from 6 And the lift coefficient amplitude or rms of Cl,Cl can be obtained from 7, where n is the total number of points.
The Strouhal number could be extracted from 8, where f v is the vortex shedding frequency in Hz.

RESULTS AND DISCUSSION
This section presents the results obtained in the study. Firstly, an isolated cylinder was analyzed and results were compared with the literature in order to validate the computational model. The seabed influence was analyzed in the second part. The Reynolds number adopted was 100 and 1000. Revista Mundi, Engenharia e Gestão, Paranaguá, PR, v. 5, n. 2, p. 227-01, 227-14, 2020 DOI: 10.21575/25254782rmetg2020vol5n21167

Isolated Cylinder
The domain width measures 20m and the cylinder is centralized, with a diameter equal to 0.5m. The upstream distance since inlet until the center of cylinder is 8m and from there until the flow output has a distance of 20m. Also note that the 3rd dimension (z) was set to 1m. In order to ensure a two-dimensional analysis, there is no flow in this direction.
The outlet is just on downstream and a pressure boundary in ANSYS R Fluent the relative pressure is set to zero. When a fluid flows through a solid surface, it will come to a complete stop at the surface and the velocity relative to the surface (normal and tangential) is zero.
This condition, where a fluid is in direct contact with a solid and "sticks" onto the surface, is commonly known as the no-slip condition. That is, the cylinder will be set to 'no-slip condition', where pressure is set to zero gradient and velocities are set to zero, U x = U y = 0. So, the upper and lower wall assigned as 'slip' boundaries, shown in Fig. 1, allow the fluid velocity component parallel to the wall to be computed, while velocity normal to the wall is set to zero, U y = 0. Subsequently, in 4.2 will be presented the modification made for seabed proximity study.

Figure 1: Schematic illustration for boundaries in laminar flow
A uniform flow is specified at the inlet, whose Reynolds number is given as Eq. (9), for flow velocity U, where ρ is density of the fluid, D is diameter and µ is dynamic viscosity.
The mesh is developed so that the aspect ratio will be higher near the external domain boundaries and smaller near the cylinder. This arrangement is made since the flow is predicted to be a developed flow on that region. The mesh around the cylinder should be finer so that it can generate a more accurate results of simulation. The template consists of a body fitted hexahedral region surrounding the cylinder with unstructured wedges filling the remaining far field domain.
The mesh/grid near the cylinder is refined by modifying the grading scale and element size in the ANSYS R mesh generator. The mesh can be seen in the Fig. 2. This mesh has 4220 nodes and 6295 elements.  Table 1 shows the force coefficients obtained in this study and the results from others in the literature. Figures 3 depict the coefficients time histories. The comparison of the parameters with reference works was satisfactory. It was also generated the horizontal velocity profile for this simulation as follows in Fig. 4.

Seabed Proximity
In this case, the influence of the vertical position of the riser (depth) to a bottom wall was investigated. Thus, the force coefficients were analyzed through the ratio of the asymmetry of the domain, measured by distance j in Fig. 5.
The change to the base case is the approximation of the cylinder to the seabed. For this, the condition of the lower wall was also changed to 'No-slip condition'. Fig. 5 shows the new configuration, where j represents the distance from the cylinder to the seabed.  Table 2. In Figure 6 it is possible to observe the variations of the aerodynamic coefficients with gap ratio, for Re = 100 and Re = 1000. The same information are represented in Fig. 7, where the variations of each coefficient with j/D are represented separately.
From these results, it is possible to draw some comments: • There is a significant change of flow parameters such as Strouhal number, lift and drag coefficient.
• When the cylinder touched the wall there is no regular shedding of vortices, as evidenced by in Figure 9(b). This inhibition is stronger for a higher value of Reynolds number, equal to 1000 in our study. In this case, the wall inhibited the formation of a vortex street and there is a lack of regular vortex shedding. Some authors, such as (TANEDA, 1965;BEARMAN;ZDRAVKOVICH, 1978;SARKAR;SARKAR, 2010;TAWEKAL, 2015) reported the same behavior in Strouhal number. Also, as evidenced by (TANEDA, 1965), which conducted an experimental campaign with Re = 170 in a similar situation, for j/D = 0.1 there is an only single row of vortices, while for j/D = 0.6 a strong regular street of vortices was observed. In our simulations, this is remarkable by comparing the results from j/D = 0 to j/D = 1. In the last situation, the CL responses become more regular, mainly for Re = 1000.
• The experimental campaign from (BEARMAN; ZDRAVKOVICH, 1978) shown that the pressure distribution around the cylinder for small gaps was characterized by a displacement of the front stagnation point towards the gaps and by boundary layer thickness interaction from both cylinder and wall. The movement of the frontal stagnation point towards the wall is associated with the generation of an upward lift, while the movements of separation points with an increase of the base pressure result in a reduced drag coefficient (SARKAR; SARKAR, 2010). The lower shear layer of the cylinder is suppressed because of the wall boundary layer, which also affects the base pressure (SARKAR; SARKAR, 2010). This behavior could explain the lower values ofCd for j/D = 0 in our experiments.
The highest value ofCd andCl for j/D = 1 in the range of j/D between 0 and 1.5 and Re = 1000 can be associated to the highly correlated of boundary layer thickness between the seabed and the cylinder. This influence is weak for Re = 100, maybe due to the thinner boundary layer in both cylinder and wall. Revista Mundi, Engenharia e Gestão, Paranaguá, PR, v. 5, n. 2, p. 227-01, 227-14, 2020 DOI: 10.21575/25254782rmetg2020vol5n21167 • The mean flow around the near-wall cylinder is not symmetric, therefore a non-zero mean lift must exist. This remark is contrary to the case of a free cylinder where the mean lift coefficient is always zero.
• In general, it can also be observed that when the cylinder was in the seabed, the values of Cd andCl varied significantly, which indicates that there was no asymmetry in the flow • As the ratio j/D increases, that is, there is the detachment of the cylinder from the lower wall, it begin to perceive the tendency of return of the values of the studied parameters to near the values of the base case, indeed it happens too.

CONCLUSIONS
The objective of this work was satisfactorily achieved. First, the values of Cd andCl for the base case converse perfectly well with experimental cases in the literature for both Reynolds number studied, as well as with other numerical cases in past studies.
In this way the modeling done on ANSYS R fluent could be considered as good when compared to other applied numerical methods in previous research and other softwares. And thus having a reliability in the model for the analyze made in this study (seabed proximity).
It can be concluded that the influence of seabed proximity in drag and lift coefficients increases proportionally to the Reynolds number of simulation, as well as the frequencies. As far as Strouhal number, what was observed was a minor influence of the Reynolds number on this parameter when the distance between the cylinder and the seabed varied.
And with this it will be possible undertake new analyzes as a study of neighborhood effect for example, and also it will be arrive at a modeling closer to real cases of the offshore industry, like risers.