The mathematical formulation of the flowing liquid stream around and through confined multiply connected domains brings a complex differential system. Due to this, one cannot provide a complete description of flow field properties. The current pagination provides a numerical remedy by the use of artificial intelligence in this direction. To be more precise, fluid is considered in the rectangular channel. The circular, square, and triangular-shaped cylinders are positioned as an obstacle to the continuous fluid. The channel’s left wall is viewed as an inlet and two different velocity profiles are introduced at an inlet that are constant and parabolic profile. To discretize the computational domain, hybrid meshing is used. The variance in basic variables, namely, the velocity of the liquid and the distribution of the liquid pressure, is recorded via graphs. The nine different meshed grades are tested for the drag and lift coefficients around the right-angle triangle, square, and circular barrier. The neural networking model is constructed by using 99 datasets of sample values for drag coefficient when characteristic length, the density of fluid, the dynamic viscosity of the fluid, and mean flow velocity are taken as inputs. The training of the neural network takes up 69 slots (about 70%), while the testing and validation of the neural network each take up 15 slots (15%). The Levenberg–Marquardt backpropagation algorithm is used for training. We have observed that for the parabolic profile, the drag coefficient is higher in intensity for each obstacle compared to the constant profile, while the lift coefficient shows opposite patterns.

## I. INTRODUCTION

The pressure-induced flow between two reasonably moving plane walls was discussed by Liepmann and Bleviss^{1} for both porous and non-porous walls. It was discovered that for non-porous walls, even with a minor unfavorable pressure gradient given, the rise in magnetic field can create separation near the fixed wall after the electric field vanishes. It has been discovered that the magnetic field assists in postponing separation in the case of porous walls. Mazumder^{2} provided a precise solution to this problem. It investigated how the Coriolis force affected the resulting shear stresses for both steady and unsteady flow. The Couette velocity profile had a generic solution provided by Wendl.^{3} For configurations with large aspect ratios, it was demonstrated that Taylor’s traditional one-dimensional profile constituted a particular example of this solution. The results of the numerical evaluation provide information to determine when Taylor's profile should be substituted with the proposed solution and show that the difference between the two profiles was a logarithmic function of the aspect ratio. Jabbarzadeh *et al.*^{4} examined molecularly thin liquid films of alkanes under severe circumstances. The wall was represented by an atomic sinusoidal rough wall. Here, it was investigated how the roughness features affect the boundary condition. Here, the impact of the lubricating fluid's molecular length was also investigated. It also investigated how the rheological parameters of the lubricating film were affected by the characteristics of the wall roughness. Tan *et al.*^{5} incorporated the fractional calculus approach into the generalized second-grade fluid model. The generalized Mittag–Leffler function and the discrete inverse Laplace transform method were used to study the Couette flow for second-grade fluid. The problem was then solved precisely using an arbitrary fractional derivative. This gives researchers studying viscoelastic fluid mechanics a new analytical tool. Hayat *et al.*^{6} looked at the mobility of an electrically conducting, Oldroyd-B, and incompressible fluid between two infinitely long, non-conducting parallel plates in the presence of a constant transverse magnetic field. The upper plate was vibrating in its plane while the lower plate was at rest. This problem’s governing partial differential equation and its associated boundary conditions were analytically resolved. The graphs were drawn for various values of the problem’s dimensionless parameters, and an examination of the findings revealed that the magnetic field that was applied, the rotation, and the fluid’s material parameters all had a noticeable impact on the flow field. Dubrulle *et al.*^{7} presented a discussion and recommendations for Taylor–Coutte flow. This kind of geometry is typically found in astronomical or geophysical contexts. The recommendations they came up with were the outcome of a thorough review of the experimental data collected in various studies of the turbulent transport in Taylor–Couette flow and transition. In order for them to be applicable to all rotational shear flows, in general, and not just Taylor–Couette flow, they first establish a new set of control parameters that are based on dynamical considerations rather than geometrical ones. The transition thresholds were examined in order to determine how generally dependent on the control settings they were. They can get some broad suggestions on the turbulent to the laminar shear ratio by looking at the mean profiles. The extended study on Taylor–Couette flow can be assessed in Ref. 8. Saad and Ashmawy^{9} studied the erratic flow of an incompressible pair stress fluid between two parallel plates. While the other plate remained fixed, the upper plate was suddenly shifted with time-dependent velocity. Through the application of the Laplace transform technique, the issue was analytically resolved in the Laplace domain. Numerical methods were used to determine the fluid velocity's inverse transform. Plots of the velocity profiles for various timeframes and physical factors were made, and the numerical findings were then explained. Ellahi *et al.*^{10} studied the magnetized Couette–Poiseuille flow of pair stress fluid between parallel plates. Axial pressure gradient and homogeneous upper plate motion generated the flow. With homogeneous Hafnium and spherical particles, the effect of wall heating was considered. The Reynolds’ model, which describes viscosity as a function of temperature, was used. The flow equations were addressed using the Runge–Kutta approach with shooting. It was discovered that decreasing the values of the Hartman number caused the velocity to fall because heating the wall lessened the effects of viscous forces, which in turn caused the fluid’s velocity to decrease due to magnetic force resistance. One can assess some trustful developments on both Couette and Poiseuille flows in Refs. 11–14.

It is difficult to find the solution when obstacle is placed toward the path of incoming fluids. Here, the flow around hitches claims various real-world uses, such as the flow of water over marine nature, in mountain streams, flow via rivers and lakes, to mention just a few. The hitches placed being fixed in the path of fluid flows experienced two forces: drag and lift. In this context, one can find the motivation to investigate the flow toward submerged obstacles. Iftimie *et al.*^{15} investigated the asymptotic aspects of an ideal, time-dependent, incompressible two-dimensional flow. Finding the equation that the limit flow satisfied was the major goal. They noticed that the circulation around the obstruction affects the asymptotic behavior. In order to solve flow equations, they first constructed a series of approximate solutions in the full plane using the exact solutions. They then passed to the limit on the weak formulation of the equation. Lee^{16} conducted a numerical investigation into the features of flow around a single obstacle with a fixed height and variable length and width. The size of the recirculation zone behind the obstruction similarly grows as the obstacle’s length does, as does the flow distortion at its upwind side. Despite almost always having flow distortion close to the upwind side of the obstruction, the extent of the recirculation zone shrinks as the obstacle width grows. One portion of the flow across an obstruction went around it, and the other portion crossed over it. The distance between the obstruction and the location (reattachment point) where both flows converge was used to calculate the size of the recirculation zone. Flows were reattached at the obstacle surface and experienced recoveries when the obstacle width was reasonably large. The recirculation zone shrinks as a result of the flows’ pathways crossing and going around being compressed. This is supported by an analysis of the degree of flow distortion determined based on the change in wind direction. According to the findings, flow distortion was greatest close to the ground’s surface and got smaller as one went up. The frontal area percentage of flow distortion around an obstacle grows with its length. When the width is increased, the frontal area fraction near the obstacle’s upwind side does not vary much, but it changes more near the obstacle’s downwind side. Experimental research on an electronic Stokes flow around a disk submerged in a two-dimensional viscous liquid was conducted by Gusev *et al.*^{17} The circle obstruction causes a cumulative increase in resistivity. Gurzhi effect was no longer detectable if specular boundary conditions were present. The resistivity, on the other hand, falls with temperature in flow through a channel with a spherical obstruction. On the length scale of the disk size, they detected hydrodynamic regimes by adjusting the temperature. You^{18} gave two-dimensional viscous flow around a tiny obstruction some thought. The authors previously established that, if the size of the obstacle is smaller than a sufficient constant K times the kinematic viscosity, the solutions of the Navier–Stokes system around a tiny obstacle converge to solutions of the Euler system throughout the entire space. He demonstrates how this smallness restriction can be lifted if the Euler flow is antisymmetric. Hajimirzaie^{19} conducted an experimental study of the mean wake structures of circular and square cylinders with height-to-width aspect ratios at high Reynolds numbers. An isothermally heated circular cylinder was the subject of a numerical investigation by Al-Sumaily *et al.*^{20} The buoyant forces were acting in the opposite direction from the flow, which was moving vertically downward. To assess the impact of thermal buoyancy on the instantaneous flow behavior, vortex-shedding properties, thermal fields, and rate of heat transfer, a parametric analysis was carried out for different Reynolds and Richardson numbers. The simulations demonstrate that even at the greatest Reynolds number under investigation, the flow is still characterized by the typical periodic von Kármán vortex street as the Richardson number grows.

Recent research has shown that neural networking models are a valuable tool for fluid flow prediction. These models also known as deep learning models^{21} or deep neural networks, or DNNs have demonstrated promise in their ability to recognize intricate flow patterns^{22} and provide precise forecasts. Reconstructing flow fields, reduced-order modeling, turbulence modeling,^{23} optimization, control, and parameter estimation^{24} are a few examples of the applications of neural network modeling for fluid flow prediction. The field’s ongoing research^{25–27} keeps expanding the bounds of what neural networks can predict for fluid flow.

Schafer *et al.*^{28} offered the benchmark problem as a flow around a circular obstacle rooted in a rectangular channel. They considered a channel with a length of 2.2 m and a height of 0.41 m. A circular obstacle was installed at the center (0.2, 0.2) while fluid was initiated at the inlet with the parabolic profile. In such dimensions, at Reynolds number Re = 20, they offered numerical values of two well-used quantities, namely, drag coefficient = 5.579 535 and lift coefficient = 0.010 618. Motivated by the flow around obstacles and particular benchmark problem given by Schafer *et al.*,^{28} the novelty of this article includes:

Case-wise installation of triangular, squared, and circular obstacles within the same dimensions carried out in Ref. 28.

Use of two different velocity profiles at the inlet, namely, linear velocity profile and parabolic profile.

The simulation is done by using finite element-based Comsol Multiphysics software.

Optimized values of hydrodynamic forces in terms of drag coefficient and lift coefficient are given for each installation.

Prediction application of artificial intelligence is used for the prediction of drag coefficient.

The layout of the draft is as follows: Sec. I is devoted to a motivational literature review, while the description of the problem is given in Sec. II. The mathematical formulation is given in Sec. III and the adopted numerical scheme is outlined in Sec. IV. The description of the neural networking model is disclosed in Sec. V. The obtained outcomes are debated in Sec. VI. The results comparison with the existing literature is added in Sec. VII. The key outcomes are summarized in Sec. VIII. We are confident that the present neural networking analysis for fluid flow over obstacles will be a helping source for flow problems subject to installed obstruction having engineering applications.

## II. PROBLEM DESCRIPTION

In a smooth rectangular flow, the most technically involved fluid called Newtonian fluid is considered. The rectangular channel size is assumed to be 2.2 m while the channel height is taken to be 0.41 m. A channel inlet initiates the two profiles. The first type is the profile of constant velocity while the second type is the profile of parabolic. The velocity of the channel’s top and bottom walls is assumed to be zero. The rectangular channel’s right wall is defined with Neumann’s notable condition. The Newtonian fluid flows through a stream and interacts with three separate obstacles in form. The shape of an obstacle includes the right-angle triangle, square, and circle. The fluid is supposed to travel from the left wall toward the right wall and will strike with each obstacle case-wise.

## III. MATHEMATICAL MODELING

*ρ*is the fluid density,

*p*represents the pressure, $V\u20d7$ is the velocity field,

*μ*is viscosity, and $B\u20d7$ is the body force. To achieve dimensionless form of Eq. (2) and to avoid physical sizes dependency, the following setup is carried:

*L*

_{f}and the drag coefficient symbolic notation is

*D*

_{f}. The dimensionless representation for both quantities are

## IV. COMPUTATIONAL DIRECTORY

Many numerical methods are proposed to report the solutions to real-life problems involved in industrial and engineering areas. The Euler’s method, improved Euler method, midpoint method, Leapfrog method, trapezoidal method, modified trapezoidal method, Runge–Kutta methods, aka Heun’s method, predictor–corrector method, backward Euler method, and implicit midpoint method are the precious schemes to approximate the solution of ODEs. The method of lines, mesh-free methods, boundary element method, domain decomposition methods, multigrid methods, gradient discretization method, spectral method, finite volume method, and FEM are the schemes to approximate the solution of PDEs. Out of these, we carried the FEM^{29,30} for solution purposes. One can seek other techniques^{31–34} to solve flow-narrating differential equations. The simulation of concerned physical phenomena by means of the FEM is known by finite element analysis (FEA). The growth of biological cells, wave propagation, thermal transport, and fluid flow fields are a few examples of physical phenomena. The FEA is one of the pertinent numerical techniques to inspect such types of physical happening. For the present problem, the mathematical model is constructed for the fluid flow field in a channel, and the ultimate flow narrating differential equations are provided through Eqs. (6)–(8). Specifically, Eq. (6) is the continuity equation. Equation (7) and the equivalent Eq. (8) are the *x*-component of the momentum equation and the *y*-component. The fluid flow starts with two different speed profile categories, namely, the constant speed profile and the parabolic speed profile. Equation (9) refers to the constant velocity profile's mathematical definition while Eq. (10) admits the relation for the parabolic profile. The continuing incompressible viscous fluid strikes three different shaped obstacles (right angle triangle, square, and circular) and gives birth to both the powers of drag and lift. Equations (11) and (12) provide the dimensional mathematical inspiration for lifting force and drag force, respectively. The FEM is used commercially for the solution of the present problem. The parametric values adopted for a constant speed profile are *ρ* = 0.1, *μ* = 0.001, *D*_{m} = 0.1, *U*_{m} = *U*_{c} = 0.2, and Re = 20. For the parabolic velocity profile, the parametric values are chosen as follows: *ρ* = 0.1, *μ* = 0.001, *D*_{m} = 0.1, *U*_{max} = 0.3, *U*_{m} = 2/3*U*_{max} = 0.2, and Re = 20. The primitive variables are checked for each barrier, namely, velocity and pressure, and the measurement in this direction is expressed with graphs. The dependency of hydrodynamic forces (lift and drag) is inspected and offered by means of tables.

## V. NEURAL NETWORKING MODEL

Neural model . | Input . | Output . | |||
---|---|---|---|---|---|

Characteristic length (D_{m}) | Dynamic viscosity of fluid (μ) | Density of fluid (ρ) | Mean flow velocity (U_{m}) | Drag coefficient (DC) |

Neural model . | Input . | Output . | |||
---|---|---|---|---|---|

Characteristic length (D_{m}) | Dynamic viscosity of fluid (μ) | Density of fluid (ρ) | Mean flow velocity (U_{m}) | Drag coefficient (DC) |

*R*) and mean square error (

*MSE*) are as follows:

It is important to note that Schafer *et al.*^{28} offer the value of drag coefficient at a fixed value Re = 20. Our interest is to construct the neural model by engaging flow-affecting parameters, namely, characteristic length, the dynamic viscosity of a fluid, density of fluid, and mean flow velocity. Therefore, the constructed neural model can be utilized to obtain the hydrodynamic force over a large range of Reynolds numbers, even in cases when the real solution method is unable to yield the desired values.

## VI. ANALYSIS

The flow in a rectangular domain around the right-angle triangle, square, and circular is examined case-wise, namely, CASE-I, CASE-II, and CASE-III.

### A. Case-I

At the inlet, the fluid is introduced with parabolic and constant profiles of velocity. It is believed that the rectangular channel's lower and upper walls are both at rest. The channel outlet is defined with the condition of Neumann. The said physical situation is controlled mathematically using the partial differential equations given by Eqs. (7) and (8) with boundary constraints reported in Eqs. (9) and (10). The formulated partial differential equations are non-linear, and therefore, the exact solution seems impossible. To propose a solution, we consider a computational method named the finite element method. The right-angle triangle as an obstacle is taken with vertices A (0.15, 0.15), B (0.25, 0.15), and C (0.15, 0.15). The meshing of the whole computational domain is performed and for better approximation, we have added the nine different meshing levels. The statistical data for the present case in terms of domain elements DE(s) and boundary elements BE(s) are summarized in Table II. It is noted that, for level 1, the computational domain is discretized into 1023 DE(s) and 117 BE(s). In computational terminology, this level is termed extremely coarse meshing. The next level admits 1695 DE(s) and 171 BE(s) and this level is called extra coarse meshing. The extra coarse level is better meshing as compared to the extremely coarse one. The third level consists of 2674 DE(s) and 220 BE(s) and it is known by coarser meshing. The level fourth is termed coarse meshing and it consists of 4934 DE(s) and 324 BE(s). Level 5 is identified as the normal level, and the channel meshes with 7610 DE(s) and 416 BE(s) uniformly at this level. In level 6, the channel is discretized in 13 637 DE(s) and 521 BE(s) in this case. The finer meshing is considered level 7. At this point, 31 477 DE(s) and 1093 BE(s) manage the channel. Levels 8 and 9 are known as extra fine and extremely fine, respectively. The rectangular channel is discretized into 74 877 DE(s) and 2099 BE(s) subject to extra fine meshing. The most suitable meshing level is extremely fine meshing. Such level contains 132 295 DE(s) and 2099 BE(s). The geometric illustration of extremely coarse and normal meshing is shown in Figs. 2(a) and 2(b), respectively. The solution of flow narrating partial differential equations when the right-angle triangle is considered as an obstacle is proposed as shown in Figs. 3(a)–3(d). To be more specific, Figs. 3(a) and 3(b) are plotted for a constant profile of velocity. Figs. 3(c) and 3(d) are plotted for a parabolic profile.

Mesh level . | DE(s) . | BE(s) . |
---|---|---|

1 | 1023 | 117 |

2 | 1695 | 171 |

3 | 2674 | 220 |

4 | 4934 | 324 |

5 | 7610 | 416 |

6 | 13 637 | 521 |

7 | 31 477 | 1093 |

8 | 74 877 | 2099 |

9 | 132 295 | 2099 |

Mesh level . | DE(s) . | BE(s) . |
---|---|---|

1 | 1023 | 117 |

2 | 1695 | 171 |

3 | 2674 | 220 |

4 | 4934 | 324 |

5 | 7610 | 416 |

6 | 13 637 | 521 |

7 | 31 477 | 1093 |

8 | 74 877 | 2099 |

9 | 132 295 | 2099 |

Particularly, Fig. 3(a) depicts the velocity variation when fluid is taken with constant velocity *u* = *U*_{c} = 0.2. It can be observed from the figure that the velocity strikes with the left face of the right-angle triangle. Then, the fluid bifurcation occurs in between 0.1 ≤ *x* ≤ 0.2 and such bifurcation travels down the stream up to *x* ≈ 0.9. For *x* > 0.9, the fluid regains its stream. The corresponding pressure plot is provided in Fig. 3(b). Maximum pressure at corner points of rectangular channel, namely, (0, 0) and (0, 041), is seen. Such points are called pressure singularities. They appear due to the interaction of no-slip conditions at walls with constant velocity initiated at an inlet. The pressure becomes linear down the stream. Figure 3(c) is plotted to visualize the velocity variation when the parabolic profile is taken at an inlet of channel. It is noted that the fluid with a parabolic velocity profile gains bifurcation from *x* = 0.1 to *x* = 0.2. This bifurcation travels toward the outlet up to *x* ≈ 1.1. The pressure variation in this case is provided in Fig. 3(d). One can observe that the pressure is maximum at the left face of an obstacle of a shaped right-angle triangle. Table III gives the numerical values of both the drag coefficient and lift coefficient. Such observations are recorded when the constant velocity profile is considered at an inlet. For better values of both drag and lift coefficients, we have evaluated these values up to nine levels. At the last meshing level, the trustful value of the drag coefficient is 5.6614, while the lift coefficient is noted as 0.731 16. Table IV provides the drag and lift coefficient values for the parabolic velocity case with a right-angle triangle-shaped obstacle. The values are evaluated for nine various mesh schemes, and the most suitable values are windup at the last level. The drag coefficient value at an extremely fine level is 6.9138, while the lift coefficient value at this level is noted as −0.107 07.

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 6.0545 | 0.887 37 |

2 | 5.9330 | 0.881 82 |

3 | 5.7853 | 0.750 50 |

4 | 5.7328 | 0.745 43 |

5 | 5.7095 | 0.733 05 |

6 | 5.6896 | 0.735 84 |

7 | 5.6709 | 0.732 05 |

8 | 5.6637 | 0.733 64 |

9 | 5.6614 | 0.731 16 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 6.0545 | 0.887 37 |

2 | 5.9330 | 0.881 82 |

3 | 5.7853 | 0.750 50 |

4 | 5.7328 | 0.745 43 |

5 | 5.7095 | 0.733 05 |

6 | 5.6896 | 0.735 84 |

7 | 5.6709 | 0.732 05 |

8 | 5.6637 | 0.733 64 |

9 | 5.6614 | 0.731 16 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 7.2208 | 0.233 25 |

2 | 7.1363 | 0.175 79 |

3 | 7.0274 | −0.067 395 |

4 | 6.9871 | −0.079 413 |

5 | 6.9584 | −0.100 66 |

6 | 6.9430 | −0.100 06 |

7 | 6.9223 | −0.103 90 |

8 | 6.9147 | −0.103 40 |

9 | 6.9138 | −0.107 07 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 7.2208 | 0.233 25 |

2 | 7.1363 | 0.175 79 |

3 | 7.0274 | −0.067 395 |

4 | 6.9871 | −0.079 413 |

5 | 6.9584 | −0.100 66 |

6 | 6.9430 | −0.100 06 |

7 | 6.9223 | −0.103 90 |

8 | 6.9147 | −0.103 40 |

9 | 6.9138 | −0.107 07 |

### B. Case-II

The square-shaped obstacle is taken by vertices A (0.15, 0.15), B (0.25, 0.15), C (0.25, 0.25), and D (0.15, 0.15) in a rectangular path. The rest of the statement remains the same as the lower and upper walls have zero velocity and the outlet is carried out under the Neumann condition. The solution is proposed for this case using FEM. The meshing of the rectangular channel is performed, and for better approximation, we have added the nine different meshing levels. Table V offers the statistical data for the case of square obstacle in terms of domain elements DE(s) and boundary elements BE(s). For level 1, the computational domain is discretized into 1034 DE(s) and 120 BE(s). This is the extremely coarse level of meshing in computational terminology. An extra coarse meshing is known for level 2 and this level admits 1688 DE(s) and 176 BE(s). One can observe that the extra coarse level is better meshing as compared to the extremely coarse level. Level 3 is known by coursing meshing, and it consists of 2720 DE(s) and 226 BE(s). Level 4 is termed coarse meshing, and it consists of 5038 DE(s) and 334 BE(s). Level 5 is identified by normal meshing and the channel is meshed uniformly with 7802 DE(s) and 428 BE(s) at this level. At level 6, the rectangular channel is discretized in 13 834 DE(s) and 536 BE(s) in this case. Level 8 is named finer meshing. At this level, we have 31 912 DE(s) and 1120 BE(s). Levels 8 and 9 are known as extra fine and extremely fine, respectively. The rectangular channel is discretized into 76 458 DE(s) and 2146 BE(s) subject to extra fine meshing. The most suitable meshing level is extremely fine meshing. Such level contains 133 446 DE(s) and 2146 BE(s). The meshing illustration of an extremely coarse and normal meshing is shown in Figs. 4(a) and 4(b), respectively. The primitive variables are evaluated and provided through Figs. 5(a) and 5(b), namely, the velocity and the pressure. Both figures are for constant profile. Figures 5(c) and 5(d) are plotted for a parabolic profile at an inlet of a rectangular channel. To be more specific, the fluid flow initiated with a constant velocity profile *u* = *U*_{c} = 0.2 strikes with square obstacle, and bifurcation occurs. Figure 5(a) is plotted in this direction. It can be observed that the stagnation point is developed at the right face of the square. Furthermore, the fluid strikes a square obstacle and speedy travels down the stream of channel. The observed bifurcation range is 0.1 ≤ *x* ≤ 0.8. The pressure plot in this case is offered in Fig. 5(b). One can note that the pressure at the corners of the channel and left face of a square obstacle is maximum and becomes linear down the stream. The pressure singularities appear at (0, 0) and (0, 0.41). The fluid flow with parabolic profile is assumed and both the velocity and pressure variations are inspected. The velocity plot in this direction is offered in Fig. 5(c). It is observed that when fluid flow with a parabolic profile strikes a square-shaped obstacle it bifurcates, and such bifurcation occurs in between 0.1 ≤ *x* ≤ 0.9. Later, fluid gains uniformity down the stream. The corresponding pressure is recorded and given with the help of Fig. 5(d). Since the assumption of the parabolic profile is compatible with the no-slip condition of upper and lower walls, the singularities at the corners of the rectangular channel are eliminated. Furthermore, the stagnation point occurs on the left face of the square and hence the pressure value is maximum at this point. Table VI gives the numerical values of both coefficients. For better numerical values for both drag and lift coefficients, we have evaluated these values up to nine levels. At an extremely fine meshing, the trustful value of the drag coefficient is observed as 6.0834 while the lift coefficient is recorded as 0.110 31. When the fluid with a parabolic profile is initiated at an inlet and strikes the square shaped obstacles, both left and drag coefficients are observed and offered in Table VII. The value 6.9482 represents the drag coefficient and 0.086 056 is the lift coefficient. Both values are recorded at extremely fine meshing.

Mesh level . | DE(s) . | BE(s) . |
---|---|---|

1 | 1034 | 120 |

2 | 1688 | 176 |

3 | 2720 | 226 |

4 | 5038 | 334 |

5 | 7802 | 428 |

6 | 13 834 | 536 |

7 | 31 912 | 1120 |

8 | 76 458 | 2146 |

9 | 133 446 | 2146 |

Mesh level . | DE(s) . | BE(s) . |
---|---|---|

1 | 1034 | 120 |

2 | 1688 | 176 |

3 | 2720 | 226 |

4 | 5038 | 334 |

5 | 7802 | 428 |

6 | 13 834 | 536 |

7 | 31 912 | 1120 |

8 | 76 458 | 2146 |

9 | 133 446 | 2146 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 6.5268 | 0.150 70 |

2 | 6.3356 | 0.128 51 |

3 | 6.2269 | 0.085 685 |

4 | 6.1480 | 0.112 62 |

5 | 6.1288 | 0.104 91 |

6 | 6.1085 | 0.108 34 |

7 | 6.0910 | 0.108 71 |

8 | 6.0834 | 0.110 31 |

9 | 6.0834 | 0.110 31 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 6.5268 | 0.150 70 |

2 | 6.3356 | 0.128 51 |

3 | 6.2269 | 0.085 685 |

4 | 6.1480 | 0.112 62 |

5 | 6.1288 | 0.104 91 |

6 | 6.1085 | 0.108 34 |

7 | 6.0910 | 0.108 71 |

8 | 6.0834 | 0.110 31 |

9 | 6.0834 | 0.110 31 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 7.4390 | 0.141 61 |

2 | 7.2048 | 0.143 63 |

3 | 7.1141 | 0.039 853 |

4 | 7.0234 | 0.087 319 |

5 | 6.9991 | 0.081 720 |

6 | 6.9795 | 0.084 804 |

7 | 6.9580 | 0.083 834 |

8 | 6.9490 | 0.086 138 |

9 | 6.9482 | 0.086 056 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 7.4390 | 0.141 61 |

2 | 7.2048 | 0.143 63 |

3 | 7.1141 | 0.039 853 |

4 | 7.0234 | 0.087 319 |

5 | 6.9991 | 0.081 720 |

6 | 6.9795 | 0.084 804 |

7 | 6.9580 | 0.083 834 |

8 | 6.9490 | 0.086 138 |

9 | 6.9482 | 0.086 056 |

### C. Case-III

The circle with a radius of 0.05 m having centered at (0.2, 0.2) m is considered to be ongoing fluid in a rectangular channel. The rest of the assumption includes the zero velocity of both the lower and upper walls. The right wall of the rectangular channel is carried with Neumann condition. The meshing description with the channel having a circular obstacle is illustrated in Table VIII. In information, 882 DE(s) and 108 BE(s) discrete the flow as an extremely coarse. An extra coarse meshing is known for the second level and this level admits 1512 DE(s) and 160 BE(s). The third stage is called course meshing and is made up of 2422 DE(s) and 206 BE(s). The coarse meshing is considered the fourth stage and is made up of 4522 DE(s) and 306 BE(s). At level five, the channel is meshed uniformly with 6722 DE(s) and 388 BE(s) at this level. At the sixth level, we have 12 162 DE(s) and 484 BE(s). At level seven, the 28 244 DE(s) and 1040 BE(s) manage the flow. The eight-one extra fine level is made up of 69 592 DE(s) and 2014 BE(s). Extremely fine meshing is called tier nine. It is made up of 125 658 DE(s) and BE(s) for 2014. Figures 6(a) and 6(b) show the meshing illustrations subject to an extremely coarse, normal, and extremely fine meshing. The velocity profile and pressure plot for the case of constant velocity are offered in Figs. 7(a) and 7(b), respectively. For the parabolic velocity profile assumption at an inlet of the channel, the velocity and pressure outcomes are depicted in Figs. 7(c) and 7(d), respectively. The numerical values of both coefficients for each case are calculated and provided in Tables IX and X. Since the values are evaluated at nine meshing levels. The trustful values of drag and lift coefficient at level nine named extremely fine are 4.7944 and 0.045 508, respectively, when the velocity profile is taken constant at an inlet. The drag lift is 5.5811 and the lift coefficient is 0.010 207 for the case of parabolic profile.

Mesh level . | DE(s) . | BE(s) . |
---|---|---|

1 | 882 | 108 |

2 | 1512 | 160 |

3 | 2422 | 206 |

4 | 4522 | 306 |

5 | 6722 | 388 |

6 | 12 162 | 484 |

7 | 28 244 | 1040 |

8 | 69 592 | 2014 |

9 | 125 658 | 2014 |

Mesh level . | DE(s) . | BE(s) . |
---|---|---|

1 | 882 | 108 |

2 | 1512 | 160 |

3 | 2422 | 206 |

4 | 4522 | 306 |

5 | 6722 | 388 |

6 | 12 162 | 484 |

7 | 28 244 | 1040 |

8 | 69 592 | 2014 |

9 | 125 658 | 2014 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 5.1671 | −0.055 807 |

2 | 4.8794 | −0.038 878 |

3 | 4.8317 | −0.057 457 |

4 | 4.8197 | −0.036 253 |

5 | 4.8014 | 0.008 605 2 |

6 | 4.8040 | 0.012 904 |

7 | 4.8005 | 0.046 219 |

8 | 4.7965 | 0.045 115 |

9 | 4.7944 | 0.045 508 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 5.1671 | −0.055 807 |

2 | 4.8794 | −0.038 878 |

3 | 4.8317 | −0.057 457 |

4 | 4.8197 | −0.036 253 |

5 | 4.8014 | 0.008 605 2 |

6 | 4.8040 | 0.012 904 |

7 | 4.8005 | 0.046 219 |

8 | 4.7965 | 0.045 115 |

9 | 4.7944 | 0.045 508 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 6.0207 | −0.099 437 |

2 | 5.6119 | −0.102 890 |

3 | 5.5788 | −0.133 150 |

4 | 5.5796 | −0.091 933 |

5 | 5.5892 | 0.050 458 |

6 | 5.5930 | 0.038 479 |

7 | 5.5863 | 0.010 429 |

8 | 5.5812 | 0.010 423 |

9 | 5.5811 | 0.010 207 |

Mesh level . | Drag coefficient (D_{f})
. | Lift coefficient (L_{f})
. |
---|---|---|

1 | 6.0207 | −0.099 437 |

2 | 5.6119 | −0.102 890 |

3 | 5.5788 | −0.133 150 |

4 | 5.5796 | −0.091 933 |

5 | 5.5892 | 0.050 458 |

6 | 5.5930 | 0.038 479 |

7 | 5.5863 | 0.010 429 |

8 | 5.5812 | 0.010 423 |

9 | 5.5811 | 0.010 207 |

The neural networking model is constructed to predict the value of the drag coefficient experienced by a circular obstacle being installed in a rectangular channel. In detail, we consider characteristic length, the dynamic viscosity of the fluid, the density of the fluid, and the mean flow velocity upon which the Reynolds number depends.

The total 99 values are collected for drag coefficient by performing line integration around circular obstacles. The training of the neural model is done by considering 69 random sample values of the drag coefficient. Figure 8(a) shows the ultimate outcome subject to the performance of the neural model. As shown in Fig. 8(a), the mean square error is considerable in the early stages but decreases in subsequent epochs. At epoch 292, the validation performance is 2.9701 × 10^{−6}. The corresponding error plot is shown in Fig. 8(b). From both graphs, we can conclude that the training of the neuronal working model is completed successfully with very low values of error. The performance of the neural networking model is tested by owning values of the coefficient of determination. Figures 9(a)–9(d) are plotted in this regard. Particularly, the regression plot of the training for the ANN model to predict the drag coefficient is shown in Fig. 9(a). The plots of regression for validation and testing against 15 sample values of drag coefficient are shown in Figs. 9(b) and 9(c), respectively.

The collective plot for training, testing, and validation is offered in Fig. 9(d). From the figures, we observed that the value of the coefficient of regression approaches R = 1, which suggests a strong correlation between the predicted values of DC by ANN and targeted values. For each dataset, the percentage error plot of the forecast values of DC with the desired values of DC is shown in Fig. 10(a). Detailing Fig. 10(a), it can be seen that the percentage error values from the mean line are very small. Most of the data holds the percentage error between −0.1 and 0.1, which is happily acceptable and hence the constructed neural model is best for prediction. Using an artificial neural networking model, Fig. 10(b) compares drag coefficient values with predicted values for drag coefficient. We have observed that the predicted values of DC and actual values are an excellent match. The numerical values of *MSE* and *R* for training, testing, and validation dataset are summarized in Table XI.

Level . | MSE
. | R
. |
---|---|---|

Training | 4.3499 × 10^{−6} | 9.999999 × 10^{−1} |

Validation | 2.9701 × 10^{−6} | 9.999999 × 10^{−1} |

Testing | 2.9355 × 10^{−6} | 9.999999 × 10^{−1} |

Level . | MSE
. | R
. |
---|---|---|

Training | 4.3499 × 10^{−6} | 9.999999 × 10^{−1} |

Validation | 2.9701 × 10^{−6} | 9.999999 × 10^{−1} |

Testing | 2.9355 × 10^{−6} | 9.999999 × 10^{−1} |

For constructed ANN, from graphical and tabular outcomes we can conclude that the created neural network has the ability to forecast DC values very accurately for the circular obstacle toward a wide range of flow parameters (that actually control Reynolds number), namely, characteristic length, density of the fluid, the dynamic viscosity of the fluid, and mean flow velocity.

## VII. RESULTS VALIDATION

It is important to note that Eqs. (6)–(8) along with boundary conditions Eqs. (9) and (10) are non-linear in character and an exact solution in this regard seems difficult. Therefore, a numerical scheme, namely, the finite element method by using Comsol Multiphysics software is adopted to find the best possible solution to evaluate the hydrodynamic forces subject to three various shaped obstacles, namely, triangular, square, and circular. The grid independence is given in Table XII. In detail, for all three cases with linear velocity profiles, we considered an ideal number of grid sizes and observed that the corresponding values of drag and lift coefficient at level 8 and level 9 are an excellent match. Therefore, we adopted the grid sizes of 132 295, 133 446, and 125 658 as an optimum grid size for triangular, square, and circular obstacles, respectively. Furthermore, the obtained results are validated by constructing a comparison with Schafer *et al.*;^{28} see Table XIII. For comparison, the length, height of channel, circular obstacle, center of obstacle, parabolic profile at inlet, and Re = 20 are taken same as taken in Ref. 28. One can see from Table XIII that we have an excellent match subject to both drag and lift coefficient values, which leads to surety of present results.

CASE-I—triangular-shaped obstacle | ||||

Meshing level | DE(s) | BE(s) | D_{f} | L_{f} |

8 | 74 877 | 2099 | 5.6637 | 0.733 64 |

9 | 132 295 | 2099 | 5.6614 | 0.731 16 |

CASE-I—triangular-shaped obstacle | ||||

Meshing level | DE(s) | BE(s) | D_{f} | L_{f} |

8 | 74 877 | 2099 | 5.6637 | 0.733 64 |

9 | 132 295 | 2099 | 5.6614 | 0.731 16 |

CASE–II–square-shaped obstacle | ||||

Meshing level | DE(s) | BE(s) | D_{f} | L_{f} |

8 | 76 458 | 2146 | 6.0834 | 0.110 31 |

9 | 133 446 | 2146 | 6.0834 | 0.110 31 |

CASE–II–square-shaped obstacle | ||||

Meshing level | DE(s) | BE(s) | D_{f} | L_{f} |

8 | 76 458 | 2146 | 6.0834 | 0.110 31 |

9 | 133 446 | 2146 | 6.0834 | 0.110 31 |

CASE–III–circular-shaped obstacle | ||||

Meshing level | DE(s) | BE(s) | D_{f} | L_{f} |

8 | 69 592 | 2014 | 4.7965 | 0.045 115 |

9 | 125 658 | 2014 | 4.7944 | 0.045 508 |

CASE–III–circular-shaped obstacle | ||||

Meshing level | DE(s) | BE(s) | D_{f} | L_{f} |

8 | 69 592 | 2014 | 4.7965 | 0.045 115 |

9 | 125 658 | 2014 | 4.7944 | 0.045 508 |

## VIII. CONCLUSION

The incompressible fluid flow in the finite length rectangular channel is examined by using artificial neural networking. Two separate velocity profiles hold the channel inlet. The ongoing fluid interacts with three different types of obstacles. The nature of such obstacles comprises the right-angle triangle, square, and circular cylinders. The flow-controlling mathematical equations are solved numerically. The characteristic length, the dynamic viscosity of the fluid, the density of the fluid, and the mean flow velocity are considered as inputs, and DC is considered to output for the construction of the ANN model. The Levenberg–Marquardt backpropagation algorithm is used to train the ANN model, and prediction is done for the drag coefficient.

The conclusive outcomes of this article are as follows:

In the case of a constant velocity assumption at an inlet, the singularities of pressure are detected at the corner of the rectangular channel.

Considering parabolic velocity at an inlet, results removal of the singularities of pressure.

The variation in pressure down the stream is noticed linearly.

The parabolic profile initiated at an inlet is the compatible choice with the no-slip assumption at both the upper and lower walls of the rectangular channel.

The constructed neural networking model is the best ANN model to predict the drag coefficient for circular obstacles up to a wide range of parameters.

The drag coefficient is a significant subject to each obstacle in the case of parabolic profile with respect to the constant velocity profile while opposite trends are noticed for the lift coefficient.

The pressure strength is higher at the left face of each obstacle in the case of the parabolic velocity profile as compared to a constant velocity profile.

## ACKNOWLEDGMENTS

The authors would like to thank Prince Sultan University, Saudi Arabia, for the technical support through the TAS Research Laboratory.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Khalil Ur Rehman**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal). **Wasfi Shatanawi**: Investigation (equal); Methodology (equal); Software (equal); Supervision (equal). **Zead Mustafa**: Data curation (equal); Formal analysis (equal); Methodology (equal); Visualization (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request.