Finite Element Least Square Technique for Newtonian Fluid Flow through a Semicircular Cylinder of Recirculating Region via COMSOL Multiphysics

This article aims to study Newtonian fluid flow modeling and simulation through a rectangular channel embedded in a semicircular cylinder with the range of Reynolds number from 100 to 1500. The fluid is considered as laminar and Newtonian, and the problem is time independent. A numerical procedure of finite element’s least Square technique is implemented through COMSOL multiphysics 5.4. The problem is validated through asymptotic solution governed through the screen boundary condition. The vortex length of the recirculating region formed at the back of the cylinder and orientation of velocity field and pressure will be discussed by three horizontal and four vertical lines along the recirculating region in terms of Reynolds number. It was found that the two vortices of unequal size have appeared and the lengths of these vortices are increased with the increase Reynolds number. Also, the empirical equations through the linear regression procedure were determined for those vortices. The orientation of the velocity magnitude as well as pressure along the lines passing through the center of upper and lower vortices are the same.


Introduction
In the engineering sciences, the Navier-Stokes equations governed through Newton's law of motion possess the paramount importance and describe the physical phenomenon of flow in the form of nonlinear partial differential equations in the vector form. ey help to model so many physical problems such as blood flow in the human or animal body, manufacturing designs, manufacturing car, and aircraft designs, calculating drag force of air flow around wings, removal of pollutions, weather conditions, rivers, flow of oceans, etc. e literature studies reveal that when the fluid enters the region within a sufficient velocity magnitude and comes into contact with the bluff bodies of any shape, the formation of vortex shedding and recirculating flow is taken place at the back of the observed body. e size of this recirculating region or increasing with the increase in the velocity magnitude by which the fluid comes into the strike with these bluff bodies.
ese vortices are attached as well as detached periodically and creating a von Karman vortex region. e fluid which passes through the bodies created the vortices with the negative pressure, which means that, any object presented in the region might be attracted by these vortices. With the range of ratios 1.6 to 4 to the height of the rectangular channel with a diameter of the circle and ranging up to 5 the Stuart number, the vortex structure and its increment was studied by Rashid et al. [1]. It was derived that the formation of the recirculation region or vortex takes place in the back of circular obstacles when the fluid flows in the influence of the magnetic field. Moreover, to keep the drag force lower, a critical Stuart number was discussed. With the multigrid acceleration technique in the numerical finite volume method, the fluid flow through the lid-driven cavity which included a circular cylinder was observed by Faycal et al. [2]. e study was laminar and Newtonian and analyzed with the lower to high Reynolds number up to 1500. It was deduced that the formation of the recirculating region appeared at the back of the circular cylinder and length is directly proportional with the Reynolds number. An investigation of the heat transfer in the rectangular channel with circular obstacle and the fluid flow was discussed with the use of finite volume procedure using the Reynolds number from 50 to 180. Gholani and Dheman [3] found that the frequency of vortex production along the circular bluff, lift force, and drag force was showing the positive response with increasing nondimensional Reynolds number. A unified finite element technique of Galerkin's scheme of least square to deal with all types of compressible and incompressible fluid flows was suggested by Hauke and Haughes [4]. According to their findings if the Navier-Stokes equation can be represented as quasi-linear form with the pressure, velocity, and temperature as entropy variables, then a combined formulation of discretization can be derived to solve the fluid flow problems. An asymptotic solution was given by Elder [5] for the streamwise velocity at the outlet of the rectangular channel and placing a solid screen at the middle of the channel at an angle θ. e relationship was good enough to understand the behavior of flow before and after the screen. Memon et al. [6], with the help of comsol multiphysics 5.4, carried out graphical modeling for the fluid flow via the channel imposed with three screens at angles from-45°to 45°and at unit distance from each other. ey found that drag force is reduced by turning screens from 0 to 45°when moving in counterclockwise direction and also the process becomes faster by adding more screens. Memon et al. [7] presented a pattern of fluid flow behavior through the screens with the use of different resistance coefficient for the screens and described that the optimum velocity, as well as pressure at the exit of the channel will be optimized by increasing the resistance in the screens. e problem was solved through the FEM package COMSOL multiphysics 5.4. erefore, different researchers [8][9][10][11] put their efforts to describe the fluid flows through circular bluffs fixing at in the channel. By considering an unspecified Couette flow, the equation of motion of second grade fluid were worked out with the shift of variables followed by the finite integral transform with the slip and no-slip boundary states [12]. ey found that, for the slip flow, an alternative variation of the velocity very near the wall has been seen, which was quite a different behavior for the slip condition. An investigation [13] was done for the oscillation mode created by solving Burger's equation for a porous medium surrounded by the circular tube with the little boost of Fourier series which generated the time-dependent trapezoidal pressure gradient. It was revealed with the simulation that, for all kind of Burger's fluid, the drag forces at the wall are decreasing with the increase in porosity that also resulted with the increase in velocity. Later, it was Fomicheva et al. [14] who worked omas algorithm of implicit finite difference scheme over micropolar fluidic nature and the funnel flow followed by Navier-Stokes equations. e study contributed the velocity and pressure distribution for the fully coupled flow in the micropolar fluidic medium. e two articles [15,16] on the oscillatory behavior (Kamenev-type and Philos-type oscillation Criteria) and asymptotic behavior of the fluid were published and that based on working on the higher order differential equations with some additional terms. e criteria explained in the articles were also illustrated with the benchmark examples.
e article aims to observe the steady and Newtonian fluid flow through the rectangular channel fitted with a semicircular cylinder. e fluid behavior was predicted with ranging the Reynolds number from 100 to 1500. e simulation was obtained with COMSOL multiphysics 5.4. Firstly, we will try to validate our results with the asymptotic solution for the streamwise velocity at the outlet of the channel by fitting a solid screen at the middle of the channel including semicylinder and then without semicylinder. e vortex lengths formed at the back of the semicylinder is discussed with graphs in terms of the Reynolds number. e orientation of the velocity field and pressure along the three horizontal lines and four vertical lines on the recirculation region will be discussed.

Channel Description and Meshing.
Here, we choose the geometry to check the fluid flow around the vortex formed around the semicircle. Figure 1 is showing the structure of the channel. e channel is composed of two parallel plates of length 4 m each, and a semicircle of radius 0.2 m is placed near but not so much near the inlet. To analyze the laminar and Newtonian fluid with the property of fixed volume or incompressibility through the channel it is traditional to use the Reynolds number. e airflow is allowed to enter from the left entrance of the channel with U in depending on the Reynolds number. e channel is open from right-hand side and the pressure is taken zero there with the condition that no back fluid flow will be allowed. e upper and lower boundaries of the channel formed by two parallel plates are recognized as walls with slip conditions applying on them in order to observe zero viscous near the boundary.
Robust finite element technique of least Square Galerkin's scheme has been employed to achieve numerical approximation for the velocity field, as well as pressure. It is the beauty of all schemes depending on approximation to divide the whole domain into subdomains called elements. e chosen domain is divided by irregular triangular elements about 2886 with quality 0.64 as minimum and 0.945 with average quality, as shown in Figure 2. e quality we mean is the percentage of approximation from a single element.

Governing Equations and Boundaries.
e governing second-order PDEs derived with the conservative law of 2 Journal of Mathematics momentum (1) and mass (2) with steady-state (3) are considered to create a simulation of the fluid behavior while air is striking with the semicircular cylinder fixed between the parallel plates: where V � <u, v> are the velocity of the fluid with horizontal and vertical components u and v, respectively, and μ and ρ are the viscosity and density at the temperature T � 20°. As the fluid is allowed to enter with not any source, therefore F � 0. e behavior of the flow is testing at the region where the vortex is formed, and we call it the recirculating region in the article and described it through the nondimensional and obviously a traditional number, Reynolds number, which is described by equation (4): where L is the length of the channel and U in is the average velocity which becomes the source for the fluid to enter in the channel. e study has been carried out to apply the slip boundary conditions (5)- (7) which means the velocity along the upper and lower walls are not zero. Let n be the vector normal to the velocity; then, slip boundary conditions are defined as equations where We are getting help using the screen boundary condition by considering refraction coefficient η of 0.78 and resistance coefficient κ of 2.2. First, we calculate the streamwise velocity at the outlet by using only screen attached at the middle with inclination of 45°and with both the semicircular cylinder and screen. e results show the good agreement with the asymptotic solution. Let the + sign show the velocity orientation before upstream and − sign show the velocity downstream; then, we can determine the screen boundary condition and asymptotic solution by (8)-(10) and (11), respectively: e system of PDEs on behalf of these boundary conditions will be solved through COMSOL multiphysics 5.4 with Galerkin's Scheme of Least Square [4]:

Validation and Comparison.
e results obtained through the COMSOL multiphysics 5.4 are compared with the analytic solution. For the purpose, we solved the problem first by placing a semicylinder and a solid wire gauze in front of it to obtain the streamwise velocity given by equation (11). And then, we solved the same problem without the cylinder and with only solid wire gauze.

Streamline Presentation of the Velocity Field and Vortex
Lengths. In Figure 4, the velocity field through the streamlines influences the Reynolds number. We see at Re � 100, in Figure 4(a), the two vortices appeared at the back of a circular cylinder. In the paper, we call upper vortex as primary vortex and the lower vortex is the secondary vortex. Initially, the length of the primary vortex appears a little bit less than the secondary vortex. As the Reynolds number is growing, the lengths of the secondary as well as the primary are increasing and comparable with each other. We might look at the vortex forming at Re � 300 and 600 in Figure 4(b) and 4(c), where the primary vortex appeared greater than secondary vortex. At Re � 900 in Figure 4(d), we can also see that the secondary vortex is appearing bigger in length.
e lengths of these two vortices are influenced by the increasing Reynolds number. In Figure 4, we have calculated manually the length of these vortices and it is also clear that  Journal of Mathematics the lengths of these vortices are not growing equally, for example, the length of the secondary vortex is growing faster after Re � 750 when to compare with the primary vortex. From Figure 4, we are seeing that lengths of these vortices are approximately linearly related with the Reynolds number. erefore, we apply the linear regression process to find the empirical equations given by where PV L and SV L are the primary and secondary vortices, respectively.

Observations and Orientation of the Velocity Field.
e motive of this article, to explain the recirculating structure and the dynamics of the problem when the laminar and Newtonian fluid flow is entering the channel, is constructed by two parallel plates and a semicircle. In the session, we have to explain the orientation of the velocity field; for this purpose, we have drawn three lines L1, L2, and L3. ese three lines are passing through the vortices.
L1 and L2 are the lines that are passing through the center of primary and secondary vortices, respectively, and L3 is the line passing between the vortices. e orientation of the air velocity field at the lines L1 and L3 are described in Figure 5. From Figure 6 at Re � 100, the flow rate starts at 0.0004 m/sec, then comes down to zero, and then increases rapidly above. Also, at Re � 600, it can be seen that the fluid's velocity starts normally 0.0002 m/sec, then moves a parabolic path, reaches the zero flow rate of the fluid, and then increases rapidly. e same pattern can be seen for Re � 300, 900, 1200, and 1500 in Figure 6. e location where the speed of the fluid will become zero is the location where the center of the vortices is lying. It also can be shown from Figure 6 that the velocity of the fluid in the lines L1 and L2 shows the same pattern because these are the lines that are passing through the center of the vortices. It was also discussed that, after Re � 600, the secondary vortex is a little bit greater than the primary vortex; hence, we can see in Figure 6 for Re 1200 and 1500 that the fluid velocity has more parabolic fluctuation than in line L1.
In Figure 7, the velocity field orientation is presented at the line L3 which is passing between the vortices. At Re � 300-1500, we see that the velocity field is moving in a parabolic way up to the full length of the vortex and reaches at zero flow rate and then the fluid will become faster. We can get another advantage from Figure 7 that we can estimate the vortex length from the tracing of the parabolic path.
To trace the orientation of the velocity field through the y-axis, it is expressed by drawing the four vertical lines in the region of the vortices, as shown in Figure 8. In Figure 9 at Re � 100, we see that, as we go forward to the center the vortex, the flow rate is decreasing for all Reynolds number. Also, with the enhancement of the Reynolds number, these differences in the flow rates are declined as we see in Figure 9 for Re � 900, 1200, and 1500. We call this an M-shaped movement along with the vortex.

Orientation of the Pressure along Horizontal and Vertical
Lines. It seems from Figure 10 that the pressure exerted in the recirculating region is the same horizontal for the line 1 and line 2 for all Reynolds numbers. At Re � 100, we can see that the minimum pressure is − 18e − 18, and it is increasing   Moreover, when to expect the pressure along the line 3, which is between line 1 and line 2, is shown in Figure 11 and has the same behavior as in line 1 and 2, and also it is increasing in the same manner. e pressure along the recirculating region is enhanced with the enhancing of Re from 100 to 1500 and is negative which means that most of the fluid is attracted by the region. To observe the pressure along the line 3 which is between the two vortices, we see that it is not much different than when at the pressure along the line 1 and 2 except at Re � 1200.

Conclusion
Applying the finite element method, the Newtonian, laminar, steady-state, and incompressible fluid flow has been observed through the semicircular cylinder. e governing partial differential equation of Navier-Stokes and continuity equations were discretized and simulation was carried by emerging technology COMSOL multiphysics 5.4. An extra fine meshing was made to achieve convergence and ensure the validity by comparing the streamwise velocity obtained under screen boundary condition without semicircular cylinder and that of with semicircular cylinder under screen boundary condition. e results show a good agreement with the asymptotic solution. Furthermore, the velocity field orientation and the pressure along the recirculating region are described. e vortex lengths of the primary as well as secondary vortex formed along the boundary of semicircular cylinder were discussed in terms of the Reynolds number. We found that the vortex lengths of both vortices are increasing with enhancing the Reynolds number. e sizes of these two were different for all particular Reynolds number. Velocity magnitude of the flow through the center-line of the vortices follows the parabolic path up to the center of the vortex, becomes zero, and then increases up to the outlet of the channel for each Reynolds number. Also, for each Reynolds number, the orientation of the flow rate between the two vortices follows a parabolic path up to the full length of the vortex and then increases up to the outlet of the channel. By observing the fluid velocity on four vertical lines, it is observed that the fluid's velocity is maximum at the corner of the vortices for all Reynolds numbers. e pressure exerted by the fluid on the paths oriented through the center of the vortex is increasing up to the outlet. Also, through the path in between vortices, the pressure remains the same and increasing in the same pattern as in lines 1 and 2 through the vortices except for the result at Re � 1200. In the vortex region, the pressure along the four vertical lines was also declared and the pressure along these lines is fixed at decreasing at a comparable level as we move towards the outlet of the rectangular region. is difference will be ignored as we increase the Reynolds number.

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.