On drag and lift coefficient computations by using hybrid meshing of physical domain rooted with regular obstacles Open Access

The characterization of the fluid is considerably more complex than is usually required, that is, each material has its own behavior when subjected to stress, deformation, or flow. On the basis of viscosity as a function of stress or deformation mode, the fluids are classified as Newtonian or non-Newtonian. The linear relation between the shear rate and stress was proposed by Sir Issac Newton, and fluids possessing such relations were termed Newtonian fluids. The said relation is also termed Newton’s law of viscosity. Newtonian fluids include water, organic solvents, and honey, to name just a few. The non-flow oriented isotropic small molecules are normally referred to as Newtonian fluids. One can accomplish the Newtonian fluids with an anisotropic massive molecule. The low meditation polymer solutions or protein could offer a constant viscosity, irrespective of the shear rate. Various examples may also exhibit Newtonian behavior at low shear levels. Regarding examination of Newtonian fluids, Newton again was the first one who constructed a differential equation to develop the relationship between the shear stress and strain. The Newtonian fluids are the simplest empirical fluid models to compensate for their viscosity. The physical property named viscosity characterizes basic fluid flow resistance. The development to examine the fluid flow includes the Navier–Stokes (NS) equations. NS equations are partial differential equations explaining the flow of viscous liquids. In mathematical terminology, the Navier–Stokes equations describe momentum conservation, mass conservation, and energy conservation. Typically, they are followed by a state equation relating to strain, temperature, and density. We derive by using the second law of Isaac Newton to fluid motion, by owing the summation of viscous diffusing terms as a fluid force and a pressure representing viscous flow. The NS equations are important as they explain the dynamics of many phenomena of interest to scientists and engineers. They can be used to model the atmosphere, ocean tides, the movement of water in a pipe, and the flow of air through a roof. The Navier–Stokes equations, in their complete and condensed forms, aid with aircraft and vehicle design, blood pressure test, power plant design, emissions analysis, and much more. This can be used in combination with Maxwell’s equations to model and test magnetohydrodynamics. Owing to the importance of NS equations, various researchers used such mathematical directory to investigate the viscous fluid flows subject to different configurations: Erdoḡan¹ considered the disk non-coaxial rotations and a stream at infinity, and the time-dependent Navier–Stokes equations were solved exactly for the flow. It was deduced that by using suitable initial conditions, a two-dimensional flow can be obtained, but the flow was three-dimensional when fluid far away from disk was impulsively initiated from the rest. The Laplace transform approach obtains an empirical solution that defines the flow at large and small periods after initialization. Ohmori² provided a computational approach for measuring the flow of incompressible and immiscible two viscous fluids with a moving interface. They used one-fluid model equations, NS equations, along with Boussinesq approximation. The used method was based on the Eulerian solution, using the mixed finite element method (FEM) with the pressure-implicit and velocity-explicit scheme. To solve the Navier–Stokes tridimensional evolutionary equation, a mixed Chebyshev pseudospectral-FEM was developed by Hou and Guo.³ The schemes’ convergence and stability were purely proven. The empirical results submitted demonstrate the advantages of this mixed process. Koh⁴ discussed numerical solution for the incompressible NS equations. Through the study of the governing equations and their discrete algebraic equations, it has been seen that boundary conditions are required to solve NS equations. The solution of the algebraic equations was obtained more conveniently in an explicit format by solving either the Poisson equation velocity or pressure. From the velocity endpoint conditions and momentum equations, the boundary conditions are required to report the solution of these Poisson equations. It also presented outcomes of numerical simulations for some designated unstable flows. The past and recent developments of the mathematical study of the fluid flow field having industrial standpoints can be assessed in Refs. 5–27.

There is consensus among researchers^28–30 that it is not easy to analytically solve the scheme of differential equations against the construction of obstacles to continuous fluid. Thus, the acceptable numerical solution is usually reported to examine the said flow field. The present attempt is also a numerical solution constructed subject to case-wise construction of circular, square, and triangular cylinders as a barrier in the divergent region of Convergent-Divergent (CD) channel just after the CD throat. Section I introduces the motivational analysis for the flow field in various configurations. Section II provides the mathematical formulation for convergent–divergent channel that has cylinders as a barrier. The simulation method is given in Sec. III. The fluid flow field analysis in convergent–divergent channel having mounted cylinders, namely, circle, square, and triangular-shaped, is given in Sec. IV, and the comparative analysis is carried out at low Reynolds number. The research is accompanied by representations of both contour and line diagrams. The whole work is summarized in Sec. V.

Newtonian time-independent flow is carried out in the CD channel. The coordinates of convergent–divergent channel are taken: (0, 0.6), (0.05, 0.6), (0.1, 0.55), (0.15, 0.55), (0.3, 0.5), (0.35, 0.5), (0.6, 0.55), (0.65, 0.55), (0.7, 0.6), (1.6, 0.6), (1.6, 0), (0.7, 0), (0.65, 0.05), (0.6, 0.05), (0.35, 0.1), (0.3, 0.1), (0.15, 0.05), (0.1, 0.05), (0.05, 0), and (0, 0). In the CD channel, the infinite circular-shaped obstacle is placed as an obstruction. The circular-shaped cylinder has a diameter of 0.1 m. This barrier is properly put after the CD throat in diverging region of the channel. The fluid flows from the CD channel inlet and hits the circular shield. It is well accepted among researchers that by constructing a mathematical model, one may study the viscous fluid flow. Navier–Stokes equations are mathematically strong to narrate the area of flow in this direction. The dimensional form of Navier–Stokes equations, subject to time-dependent field of incompressible flow, can be written as follows:

To forecast the flow field, we need to obtain the flow field parameters and use the following series of variables to this end:

By using Eq. (2) in Eq. (1), one can get

Here, the bar is lowered and the time-independent two-dimensional incompressible fluid flow in component form is read as

With (x₁, x₂) = (x, y)  and  (u₁, u₂) = (u, v), Eqs. (4) and (5) gives

The viscous fluid with the parabolic velocity profile is introduced from the inlet of the CD channel. At the CD channel walls, the no-slip condition is enforced. Obstacles’ exterior surface also presents no-slip status. The mathematical notation of the CD channel boundary conditions can be written as

The geometry of the problem is given in Fig. 1. In a divergent channel area, the interaction of the viscous fluid with the mounted infinite cylinders would encounter two forces, lift force and drag force. To measure these powers, we need to obtain the non-dimensional directories, namely, the coefficients of lift and drag. The drag coefficient is a statistical relationship against drag force, whereas the lift coefficient can help us research the difference in the lift values. In this context, the mathematical procedure is as follows:

Here, Eq. (9) is the lift coefficient relation, while Eq. (10) possesses the mathematical relation for the drag coefficient.

The present problem is planned to investigate the Newtonian fluid flow in a convergent–divergent channel that has fixed infinite cylinders with three regularly shaped barriers, namely, circular, square, and triangular. In the CD pipe, the mathematical model is considered to narrate the field of flow. Equations (6) and (7) are a set of partial differential equations and are here solved in accordance with Eq. (8), limit conditions by using the finite element method. We also performed line convergence along the outer surface of the mounting obstacles in terms of the drag coefficient and lift coefficient to demonstrate the disparity between the drag and lift forces. The description incentive can be obtained according to the finite element technique in Refs. 31 and 32. For simulation purposes, we adopted the fluid density (ρ_o = 1), fluid viscosity (μ = 0.001), and the reference velocity (U = 0.2), the perpendicular side is the characteristic length (L = 0.1) of the triangular-shaped cylinder, the side length of the square-shaped obstacle is viewed as the characteristic length (L = 0.1), while the diameter of the circular-shaped cylinder as an obstacle is used as the characteristic length (L = 0.1). The forces are estimated at a lower value of Reynolds number, that is, Re = 20.

We examined the difference in hydrodynamic forces observed by positioning three separate formed obstacles placed on the right-hand side of the CD throat in the convergent–divergent channel. The research is performed on a case by case basis.

We initiate, in this case, a closed rounded obstacle called a triangle. The triangular object as an obstruction is positioned with vertices (0.4, 0.36), (0.4, 0.26), and (0.5, 0.26) to the right-hand side of the CD throat. The fluid started from the left wall hits the obstruction. We used FEM to study the flow regime. For the present case, seven separate meshing schemes, namely, A-1, A-2, A-3, A-4, A-5, A-6, and A-7, are brought. The CD channel is discretized in the first stage, that is, A-1, with 88 boundary elements (BEs) and 644 domain elements (DEs). With the addition of level A-2, the meshing is strengthened. At this stage, the CD channel with 112 BEs and 920 DEs is discretized. Information subject to meshing enhancements, such as A-3, A-4, A-5, A-6, and A-7, is given in Table I for triangular obstruction. The table also includes the description of the degrees of freedom and CPU time for each simulation level. The schematic representation for hybrid meshing levels, namely, A-1, A-2, A-3, A-4, A-5, A-6, and A-7, is given in Figs. 2(a)–2(g), respectively. Stage A-7 is viewed as a secure meshing to examine the flow in the CD channel around the triangular obstruction. At this point, 608 boundary elements and 15 460 domain elements discretize the geometry. The velocity and pressure are obtained at level A-7. Specifically, Fig. 3 gives the pressure description in the CD channel with an infinite triangular barrier. The viscous fluid trip with parabolic velocity profile from the CD channel inlet and hits the triangular obstruction. From Fig. 3, we can see that the friction on the left side of the triangular barrier is optimum. Later, the pressure in the divergent channel was normalized and differs linearly down the stream. Figure 4 offers the velocity profile, and in the CD path, the velocity distribution has an infinite triangular cylinder as an obstruction. As the viscous fluid meets with the triangular barrier, it bifurcates and moves the fluid in a symmetrical way. Upon connecting with the obstruction, the fluid velocity increased. The fluid speed was averaged down toward the outlet. For direct analysis, the line graph research is conducted to analyze the viscous flow at various CD channel locations. Figures 5(a)–5(d) are plotted in this regard. Specifically, Fig. 5(a) is the u-velocity line graph at x = 0. It validates the observation of the CD channel parabolic velocity profile from an inlet. The velocity of the fluid is tested at channel length x = 0.6. Figure 5(b) is a plot to this effect. It can be shown that the fluid begins bifurcation at this location due to a triangular obstruction built in a region of the divergent path. Clear bifurcation at the CD channel, x = 0.7, can be observed [see Fig. 5(c)]. The velocity of the fluid is evaluated at the channel location, x = 1.5, and provided through Fig. 5(d). One can see that the bifurcation effect is close to vanishing.

The effect of the obstacle mounted is intensely diminishing. The line integration is carried out along the triangular obstacles’ outer surface to achieve both the lift and drag values. Those values are in terms of the coefficient of drag and lift. The drag coefficient is noted, D_d = 9.5711, at level A-1, while the lift coefficient is reported, L_f = −0.645 79. Both the drag coefficient and the lift coefficient are rectified by conducting line integration along the triangular obstacles’ outer surface at various meshing proportions, namely, A-2, A-3, A-4, A-5, A-6, and A-7. In Table II, the values are listed. Owing to the grid independence, the trustworthy value of the drag and lift forces encountered by the triangular-shaped obstacle being mounted in the divergent region in terms of the coefficients is reported, D_d = 8.5343 and L_f = −0.993 88, respectively.

The square obstruction is placed in the CD channel in a divergent field. The square-shaped barrier’s lateral length is 0.1 m. In CD, the viscous flow regime is started with a parabolic velocity profile along with a square-shaped obstacle mounted before the CD throat. The flow system is tested at seven separate mesh rates, namely, A-1, A-2, A-3, A-4, A-5, A-6, and A-7. We discretized the CD channel in level A-1, with 90 BEs and 668 DEs. Level A-2 is an improved variant of CD channel meshing. We own 989 domain elements and 117 boundary elements at this level. In Table III, meshing statistics are given for A-3, A-4, A-5, A-6, and A-7. The description of the computation time and degrees of freedom for each meshing stage is also recorded in Table III. The geometric representation of the meshing levels, namely, A-1, A-2, A-3, A-4, A-5, A-6, and A-7, is given in Figs. 6(a)–6(g), respectively.

To inspect the flow regime around square obstruction being mounted in the divergent region of the CD channel, we choose level A-7 for simulation purposes. There is an analysis of the pressure distribution and the velocity distribution. Figures 7 and 8 are given in this regard. Indeed, Fig. 7 allows square pressure values in the CD channel. The friction on the left side of the square appears to be full. Furthermore, the pressure increases as the fluid reaches the convergent zone. We can see that downstream, the pressure decreases dramatically. Figure 8 records CD channel velocity distribution with square obstruction on the right-hand side of the CD throat. It is seen that where the pressure is maximum, the fluid strikes the square obstacle, and the region of the stagnation point appears. The bifurcation occurs, and the fluid is greatly accelerated. The fluid flow is stated in terms of line graph analysis. Figures 9(a)–9(d) are plotted in this regard. In-depth analysis of the u-velocity line graph is conducted to analyze the viscous fluid flow as a barrier (mounted square) in the CD channel. Figure 9(a) presents a line graph of x = 0 inlet u-velocity and supports the parabolic fluid speed at which the viscous fluid is emitted from the CD channel inlet. Figure 9(b) is u-velocity line graph at the channel length x = 0.6. One can see that because of the mounted square obstruction the fluid achieves bifurcation. You will note the major bifurcation at x = 0.7 in Fig. 9(c). Figure 9(d) offers the line graph study of the fluid at x = 1.5. One can see that the effect of the square cylinder mounted on the fluid velocity greatly reduces. The square cylinder mounted in the right vicinity of the CD throat was experiencing forces, including the drag force and lift force. To measure the lift coefficient and the drag coefficient, we have conducted line integration across the outer surface of the square obstacle. Such integration is carried out with the use of seven separate meshing rates, namely, A-1, A-2, A-3, A-4, A-5, A-6, and A-7. Table IV is constructed in this regard. At level A-1, the drag force is found in terms of the drag coefficient, D_d = 9.4938, while the lift force is measured in terms of the lift coefficient, L_f = −0.313 53. At level A-7, we obtain the conclusive and trustful force of drag and lift. The drag coefficient is measured, D_d = 8.4037, and the lifting coefficient is estimated, L_f = −0.087 879.

In this case, in a convergent–divergent channel, we found the circular-shaped cylinder as an obstacle. The circular cylinder diameter is 0.1 m. The velocity in a CD channel is introduced with parabolic velocity and is analyzed numerically. To disclose the solution, the seven separate meshing levels, namely, A-1, A-2, A-3, A-4, A-5, A-6, and A-7, are used. In level A-1, we discretized the CD channel with 82 BEs and 546 DEs, including a circular barrier. The meshing for the CD channel is strengthened by providing level A-2 with 797 domain elements and 103 boundary elements. The CD channel meshing discretization for levels A-3, A-4, A-5, and A-6 is shown in Table V. The findings are measured with the CD channel being mounted on the right side of the CD throat at level A-7 with a circular obstruction. We hold 12 938 domain elements in this level and 558 boundary elements in it. Table V also cites CPU time and degrees of freedom for each level. For the mounted circular cylinder, the schematic view of meshing levels, namely, A-1, A-2, A-3, A-4, A-5, A-6, and A-7, is given in Figs. 10(a)–10(g), respectively.

To track fluid flow in a CD channel, we use level A-7 for simulation purposes. The pressure and velocity profiles were given in a CD channel that was subject to fluid flow. In this direction Figs. 11 and 12 are given. Specifically, Fig. 11 provides pressure transfer with circular obstruction in the CD domain. We can note that the pressure on the left side of the channel tends to be full. The pressure intensity is linearly variable down the channel. Figure 12 owns the movement of fluid flow through a circular cylinder mounted in the immediate vicinity of the CD throat. The area of the stagnation point is located on the left side of the circular cylinder, and due to the mounted circular cylinder, the fluid begins bifurcation. The fluid flow is accelerated due to collision. The effect of mounted barriers on fluid flow is found to decrease significantly.

In terms of line graph analysis, the variance of viscous fluid flow is checked at various channel locations, namely, x = 0, 0.6, 0.7, and 1.5. In this direction, we have plotted Figs. 13(a)–13(d). Specifically, Fig. 13(a) includes an inlet line graph of u-velocity x = 0. This validates the concept of parabolic velocity profile from which viscous material flows from the CD channel inlet. The fluid hits in this direction with circular cylinder, and bifurcation of u-velocity at x = 0.6 is seen in Fig. 13(b). In Fig. 13(c), the effect of the mounted circular obstruction on the CD channel fluid flow at length x = 0.7. Figure 13(d) includes a line graph of u-velocity at channel length x = 1.5. It is found that the fluid velocity downstream is diminished by the impact of the mounted barrier. The limitless fixed circular cylinder encountered two types of quantities, namely, the drag force felt as a drag coefficient by the circular cylinder is found, D_d = 7.7741, and the lift coefficient is registered, L_f = −0.067 218, at level A-1. The grid independence is checked, and we give lift and drag coefficients at various meshing levels and provided in Table VI. The positive values are stated at level A-7. The lift coefficient L_f = −0.056 515, whereas the drag coefficient D_d = 6.9934.

To evaluate the hydrodynamic forces, including drag force and lifting force faced by installed circular, square, and triangular cylinders in the right vicinity of the CD throat, a numerical analysis is performed. In terms of differential equations, the mathematical structure is built and solved by using the finite element technique. It is found that the addition of the convergent and divergent nozzle increases the velocity of the fluid, and as a consequence, the pressure results in its highest value on the left side of each barrier. The divergent nozzle tends to minimize speed up to a degree. Even the presence of thhe CD nozzle states an inciting drag force values encountered by the construction of circular, square, and triangular barriers to the continuous stream. The coefficient of drag encountered by the square-shaped cylinder is greater than that of the circular-shaped cylinder. In contrast with both square and circular cylinders, the triangular-shaped cylinder felt the greater drag force on the right side of the CD throat.

The authors would like to thank Prince Sultan University Saudi Arabia for their technical support through the TAS research lab. K. U. Rehman owns a post-doctoral fellowship under the supervision of Professor Wasfi Shatanawi at TAS lab.

The authors have no conflicts to disclose.

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