Pressure-velocity Coupling Research Articles

Advancing Finite Difference Solutions for Two‐Dimensional Incompressible Navier–Stokes Equations Using Artificial Compressibility Method and Sparse Matrix Computation

In this article, a numerical scheme for solving two‐dimensional (2D) time‐dependent incompressible Navier–Stokes equations is presented. The artificial compressibility technique is used to incorporate a time derivative of pressure term to the continuity equation. It is employed for pressure–velocity coupling. The scheme consists of backward difference approximation for time derivatives and central difference approximation for spatial derivatives, implemented on a collocated grid. The discretization of the differential equations yields a system of algebraic equations with a block coefficient matrix. To solve this system efficiently, matrix inversion with sparse matrix computation is employed. The proposed numerical scheme is applied to solve three flow problems (lid‐driven cavity flow, rectangular channel flow, and Taylor–Green vortex problem) to validate the accuracy and applicability of the scheme. The results affirm the scheme’s capability to provide precise approximations for solutions to the Navier–Stokes equations. With slight modifications, the scheme can be applied to solve various flow problems with high accuracy, less memory usage, and reduced computational time.

Computational Fluid Dynamics Analysis of the Influence of Diameter at Inlet 2 on Heat Transfer and Fluid Flow in the Mixing Elbow

This study presents the CFD (Computational Fluid Dynamics) analysis of the mixing elbow by determining the influence of the variation in the diameter of the inlet 2 section, Diameter 2 with the use of the ANSYS FLUENT software. The flow of our problem being turbulent, was described by the Reynolds Averaged Navier-Stokes Equations (RANSE), the k-ε closure model and for the temperature field, the averaged energy conservation equation. To solve them numerically, the finite volume method and the SIMPLE scheme for Pressure-velocity coupling were chosen. The larger the Diameter 2, the more effective mixing areas, the more uniform the temperature and velocity at the outlet and the better the mixing. At the outlet, not only do the velocities and temperatures increase but their gradient also increases with an increase in Diameter 2 up to 0.025m, the presence of turbulence and the rate of heat transfer increase. For Diameter 2 equal to 0.030m, the curves are symmetrical, the temperature and velocity values ​​vary slightly over the entire outlet section, we have a better mixture but with a low heat transfer rate.

Estimation of heat transfer coefficient and friction factor with showering of aluminum nitride and alumina water based hybrid nanofluid in a tube with twisted tape insert

Twisted tape is one of the active thermal proficiency boosting technology which has been deeply examined because to consistent efficiency findings and easy implementations. Thermo-hydraulic effectiveness of tubes fitted with twisted tapes is becoming highly significant. Although twisted tapes can cause swirls and disturb boundary layers, this is the most widely used method for improving convection. In the present attempt, to enhance the heat transfer twisted tape is inserted in tube. In the current modern research, the effect of twisted tape, on the enhancement of thermal transport, Nusselt number and friction factor performance of AIN–Al2O3/water hybrid nanofluid is evaluating utilizing CFD and ANSYS-FLUENT software. the consequence of twisted pitch 44 mm, 66 mm, 88 mm, 100 mm and Reynolds numbers 800, 1200, 1600 and 2000 on Nusselt number, heat transfer coefficient and friction coefficient have been computed numerically with 0.01 to 0.04 volume friction of nanopowders. The commercial ANSYS-FLUENT code was used in this analysis utilizing the SIMPLE method for pressure–velocity coupling. The K-ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K - \\omega$$\\end{document} model and Navier Stokes equations are integrating utilizing finite volume method in ANSYS-FLUENT. It was observed that inserting the twisted tape in tube significantly improves the thermal conductivity as well as friction factor compared with the simple tube without turbulator.

COMBINED FORCED AND NATURAL CONVECTION FROM A SINGLE TRIANGULAR CYLINDER

Unsteady laminar confined and unconfined fluid flow and mixed (forced and free) convection heat transfer around equilateral triangular cylinders are investigated numerically. The computation model is a two-dimensional domain with blockage ratios of BR=0.5, 0.25, 0.2, 0.1, 0.05, and 0.0333, with the Reynolds numbers ranging from 100 to 200. The working fluid is water (Pr = 7). The effects of aiding and opposing thermal buoyancy are incorporated into the Navier-Stokes equations using the Boussinesq approximation. The Richardson number, which is a relative measure of free convection, is varied in the range -2 ≤ Ri ≤ 2. The governing equations are solved by using the Finite Volume Method with a second-order upwind scheme used for differencing of the convection terms, and the SIMPLE algorithm is used for the velocity-pressure coupling. A discussion of the effect of the blockage ratio on the mean drag, mean rms lift coefficients, the Strouhal number, and the mean Nusselt number is also presented. The iso-vorticity contours and dimensionless temperature field are generated to interpret and understand the underlying physical mechanisms. The results reveal that, in addition to the Richardson and Reynolds numbers, the blockage rate is effective in the vortex distribution in the channel. It has been determined that the vortices formed behind the cylinder spread to the channel with a decreasing blockage rate. Especially at high Reynolds numbers, both the drag coefficient and the mean Nusselt number are significantly affected by the blockage ratio. For Ri=0, the drag coefficients for BR=0.25 in comparison to the BR=0.05 case are about 9% and 29% larger for Re= 100 and 200, respectively. For BR

Effect of surface radiation on double diffusion convection in a low Mach number compressible gaseous mixture (H2–air)

This study aims to numerically investigate the effect of surface radiation on double diffusion convection in a low Mach number compressible gaseous mixture, specifically focusing on the H2–air system in a square cavity. The cavity is subjected to low horizontal temperature and concentration gradients and isolated from its flat walls. The mathematical model's equations were discretized using the finite volume method based on the semi-implicit method for pressure-linked equations revised algorithm for the pressure–velocity coupling. The radiosity method is employed to calculate the radiative heat exchange between the internal walls of the cavity. This work addresses a significant research gap by exploring the interplay between surface radiation, compressibility effects, and double diffusion convection. It makes a novel contribution to the field and has implications for combustion, astrophysics, and industrial heat exchange processes. Comparing the results with those obtained for a light gaseous mixture, the findings demonstrate that the presence of radiation considerably modifies the thermal, dynamic, and mass fields, as well as the thermophysical properties of the gaseous mixture in the case of heavy gaseous mixtures. These modifications can reach up to 20%.The outcomes of this study provide a foundation for further research and experimentation, with applications ranging from engineering to astrophysics.

Numerical simulation of thermal behavior of cerebral blood vessels using computational hemodynamic method

Nowadays, cardiovascular illnesses are among the leading causes of death in the world. Thus, many studies have been performed to diagnose and prevention of these diseases. Studies show that the computational hemodynamic method (CHD) is a very effective method to control and prevent the progression of this type of disease. In this computational paper, the impression of five non-Newtonian viscosity models (nNVMs) on cerebral blood vessels (CBV) is investigated by CHD. In this simulation, blood flow is supposed steady, laminar, incompressible, and non-Newtonian. The parameters of Nusselt number (Nu), dimensionless temperature (θ), pressure drop (Δp), and dimensionless average wall shear stress (DAWSS) are also investigated by considering the effects of heat generated by the body. Utilizing the FVM and SIMPLE scheme for pressure–velocity coupling is a good approach to investigating CBVs for five different viscosity models. In the results, it is shown that the θ and Δp+ increase with increasing Reynolds number (Re) in the CBVs. By enhancing the Re from 90 to 120 in the Cross viscosity model, the Δp+ changes about 1.391 times. The DAWSS grows by increasing the Re in all viscosity models. This increase in DAWSS leads to an increasing velocity gradient close to the cerebral vessel wall.

Physics‐based preconditioning of Jacobian‐free Newton–Krylov solver for Navier–Stokes equations using nodal integral method

AbstractThe nodal integral methods (NIMs) have found widespread use in the nuclear industry for neutron transport problems due to their high accuracy. However, despite considerable development, these methods have limited acceptability among the fluid flow community. One major drawback of these methods is the lack of robust and efficient nonlinear solvers for the algebraic equations resulting from discretization. Since its inception, several modifications have been made to make NIMs more agile, efficient, and accurate. Modified nodal integral method (MNIM) and modified MNIM (M2NIM) are the two most recent and efficient versions of the NIM for fluid flow problems. M2NIM modifies the MNIM by replacing the current time convective velocity with the previous time convective velocity, leading to faster convergence albeit with reduced accuracy. This work proposes a preconditioned Jacobian‐free Newton–Krylov approach for solving the Navier–Stokes equation using MNIM. The Krylov solvers do not generally work well without an appropriate preconditioner. Therefore, M2NIM is used here as a preconditioner to accelerate the solution of MNIM. Due to pressure–velocity coupling in the Navier–Stokes equation, developing a quality preconditioner for these equations needs significant effort. The momentum equation is solved using the time‐splitting alternate direction implicit method. The velocities obtained from the solution are then used to solve the pressure Poisson equation. The computational results for the Navier–Stokes equation are presented to underscore the advantages of the developed algorithm.

Effect of a Sinusoidal Temperature Profile on Entropy Generation Due to Double-Diffusive Natural Convection in a Square Partly Porous Cavity

Abstract In this paper, entropy generation due to double-diffusive natural convection inside a partly porous enclosure under sinusoidal wall heating is numerically analyzed. The resulting dimensionless coupled partial differential equations are discretized using the finite volume method (FVM), while the pressure–velocity coupling is handled using the SIMPLE algorithm. To ensure the validity of the results obtained from the in-house FORTRAN code, a comparison is made with previous numerical and experimental works. The results of the study indicate that the dimensionless thickness of the porous layer (Δ), the thermal conductivity ratio (Rk), the angle of inclination of the cavity (α), and the buoyancy ratio (N) are critical factors in determining local distribution and global maximum value of entropy generation due to heat transfer and fluid friction. In contrast, their effect on entropy generation induced by concentration is insignificant, except for the buoyancy ratio parameter, where an enhancement of the global maximum of entropy generation is observed by increasing N. Moreover, it is found that Sψ (max) experiences a sharp decline as Δ varies from 0.2 to 0.8, resulting in a reduction of about 67% for the case with N=10 and 83% for the case with N=1. These results highlight the importance of carefully controlling system parameters to minimize energy losses and maximize system efficiency in heat transfer and fluid flow systems.

Magnetohydrodynamic (MHD) Natural Convection Flow of Titanium Dioxide Nanofluid Inside 3D Cavity Containing a Hot Block: Comparative with 2D Cavity

The natural convection of TiO2-Water-Nanofluid in a cubic cavity, containing a hot block under the influence of the magnetic field was studied numerically. The verticals walls are cold, the bottom wall is hot and the other walls (top, front and rear) are adiabatic. This work aims to visualize the importance of taking into account the three-dimensionality of the flow in the presence of magnetic field as well as the impact of the addition of nanoparticles on heat exchange rate evolution. The governing equations are solved using the finite volume method and the SIMPLER algorithm is used for pressure-velocity coupling. The problem was simulated at different Rayleigh numbers (103 ≤ Ra ≤ 106), Hartmann numbers (0 ≤ Ha ≤ 90) and inclination angles of the magnetic field (0 ≤ ω ≤ 135°) as well as nanoparticles volume fraction (φ = 0%, φ = 5%) with fixed Prandtl number (Pr = 7). The thermal conductivity and dynamic viscosity of the nanofluid are estimated by taking into account temperature-dependent properties, using Corcione’s correlations. Based on the cooling optimization of cold walls along with comparative analysis between 3D cavity and 2D cavity, the obtained results show that the buoyancy force enhances the heat exchange, while the magnetic field produces opposite effects. When the buoyancy force is dominated, the intensification of heat transfer becomes large, compared to the case where conduction is dominant. The qualitative difference between a 3D and 2D configuration is remarkable for higher Ra, and becomes smaller when the magnetic field is applied horizontally or vertically with relatively high intensity. But, quantitatively, the 3D flow is far from being considered as a 2D flow for all pertinent parameters control. Finally, adding nanoparticles enhances heat transfer for both configurations, the best transfer rate is obtained for ω = 0.

Numerical modeling of liquid spills from the damaged container and collision of two rising bubbles in partially filled enclosure using modified Volume-Of-Fluid (VOF) method

In the present work, two crucial shortcomings associated with Volume-Of-Fluid (VOF) model namely: (1) spurious interface smearing arising from false-diffusion errors, and (2) non-physical velocity fluctuation across the physical discontinuities, are systematically addressed, aiming to establish a unique methodological foundation and guidelines for the enhancement of interface-capturing techniques in handling multi-fluid flows. To accomplish this objective, first, a novel third-order bounded convection scheme is derived based on the Normalized Variable Diagram and Total Variation Diminishing concepts (NVD-TVD) and is then applied for the discretization of the volume-fraction equation. To cope with the instability issue induced by pressure fluctuation, the standard version of the implicit non-iterative PISO algorithm is first modified by incorporating the third pressure-correction step into the algorithm and is then utilized for the treatment of the pressure-velocity coupling. A feasibility and applicability of the proposed modifications in dealing with violent free-surface and multi-fluid flows are demonstrated against the five different challenging benchmark cases including two-dimensional dam-break flow over the dry bed, oil spill from the damaged container, single bubble rising, merging of two rising bubbles and two-fluid Rayleigh-Taylor Instability problems. The comparison of the obtained results with previously published literatures vividly corroborates the robustness and versatility of the modified VOF model in handling multi-fluid flows involving interface coalescence and breakup events. In the last staged, three new benchmark solutions namely (1) coalescence of two consecutive bubbles inside the partially filled enclosure, (2) two-dimensional three-fluid Rayleigh-Taylor Instability, and (3) oil/water spilling from the damaged tank are analyzed using the verified VOF method, aiming to provide a high-quality validation data for CFD simulations.

Numerical Investigation by Cut-Cell Approach for Turbulent Flow through an Expanded Wall Channel

The expanded wall channel backward-facing step (BFS) and axisymmetric diffuser plays an important role in the society of fluid dynamics. Using a cut-cell technique is an established new method to treat the inclined wall of an axisymmetric diffuser. Cut-cell handle to reach the shape of the inclined wall, an axisymmetric diffuser and complex geometry. It helps treat the boundary condition at the wall in an accurate physical way. The turbulent flow through the geometries is solved by using Reynolds averaged Navier-Stokes equations (RANS) with the standard k-ε model. A self-built FOTRAN code based on the finite volume method with the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm for pressure velocity coupling is established and examined with published experimental data for two different geometries backward-facing step (BFS) and axisymmetric diffuser. The results of the new technique reflect good agreement between the numerical results and the experimental data. A parametric study of the impact of area ratios (2, 2.5, 3, 3.5) in a backward-facing step on pressure, velocity, and turbulent kinetic energy. The angles (7°, 10°, 14°) and area ratios (2, 2.5, 3, 3.5) effect of an axisymmetric diffuser on the streamlines, local skin friction, pressure, velocity, turbulent kinetic energy, and separation zone.

Passive heating process enhancement of a new structure room using porous‐covering heat‐emitting channels

AbstractThis manuscript presents a numerical study of a passive heating process of a new structure room using porous‐covering heat‐emitting channels as small discrete radiators. The governing equations are discretized using the finite volume method and the SIMPLER algorithm is employed to handle the pressure–velocity coupling. The effect of porous‐cover to air thermal conductivity ratio (Kr), Darcy number (Da), Rayleigh number (Ra), thickness of the porous‐cover (Ep), and Biot number (Bi) on the heating process of the room is examined. The principal results of this investigation indicate that in the case with Kr > 1, numerical simulations have enabled us to find an optimal value of Ep (Epcr = 0.125) for which the average room temperature reaches a maximum value. In addition, an enhancement in the heating process in the room can reach 77.77% by increasing Ep, Ra, Da, and Kr. Also, a maximum increase in the average room temperature up to 97.77% is achieved at better thermal insulation (Bi = 10).

A solution to the pressure velocity coupling problem in computational fluid dynamics

A new method is shown to provide a solution to the long standing pressure-velocity coupling problem encountered in pressure-based Computational Fluid Dynamics. This problem occurs when the dependent variables are colocated on the computational mesh. A solution was found by requiring that the interpolation equations, used to relate the velocity and density at the control volume faces to the nodal values, be constrained to conserve mass. This is referred to as Mass Constrained Interpolation. It also leads to a strategy for deriving and testing boundary conditions. The method is demonstrated by comparing one-dimensional computational solutions to exact solutions for a wide range of incompressible and compressible flows.

Numerical analysis of heat transfer enhancement in grooved vertical Multi-Cylinders at various groove geometries

In this study, the natural convection enhancement in grooved vertical multi-cylinders at various groove geometries is investigated. The effects of several grooves ranging from 3 to 7, groove thickness ranging from 0.25 to 1 mm, and cylinder surface temperature ranging from 350 to 500 K at different Rayleigh numbers are examined. The current study was simulated using the finite volume method using CFD with a laminar steady-state condition. The SIMPLE scheme is used for the pressure–velocity coupling discretization and the second-order upwind method is utilized to discretize the momentum and energy equations. The results obtained from the present research show that the presence of grooves on the cylinders will increase the heat transfer surface, create and intensify the secondary flow and mixing, and ultimately increase the heat transfer. Moreover, by increasing the number of grooves and its thickness, the amount of heat transfer increases dramatically. It’s also found that the groove thickness parameter's effectiveness on heat transfer is more than the groove number parameter. Ultimately, it’s demonstrated that using grooved cylinders leads to a 14 % augmentation in Nusselt number in comparison with employing plain cylinders.

Numerical study of turbulent flow and heat transfer in a novel design of serpentine channel coupled with D-shaped jaggedness using hybrid nanofluid

This study aimed to examine numerically the effects of a dimpled surface over a mini-channel heat exchangeron the flow characteristics and heat transfer across a serpentine channel with a uniformrectangular cross-section. The dimples were arranged in parallel with a spanwise (y/d) distance of 3.125 and streamwise (x/d) distance of 11.25 along just one side of the serpentine channel's surface.Turbulent flow regime with Reynolds number ranging from 5 × 103 to 20 × 103 in the channel with the surface modificationwas studied using water and various volume concentrations(φ = 0.1%, 0.33%, 0.75%, 1%) ofAl2O3-Cu/waterhybrid nanofluid as the coolant to achieve a three-step passive heat transfer enhancement. Applying the Finite Volume Method (FVM), RNG k-e turbulence model, and a constant heat flux of 50 kW/m2, simulations were run assuming the mixture of Al2O3-Cunanoparticles homogenous using ANSYS 2020 R1. The second-order upwind approach is used forapproximation of solution and discretizationwith SIMPLE pressure–velocity coupling. Taking heat transfer increment and pressure drop penalty into consideration,the dimpled serpentine channel provides a 1.47-times improvement in thermal efficiency using water as the coolant, andthe dimpled channel with 1% vol.Al2O3-Cu/water nanofluid enhanced thermal efficiency by a remarkable maximumof 2.67-times at Re 5 × 103. The study also indicates that thermal efficiency increased with an increasing volume concentration of the nanofluid and increment in thermal efficiency gradually decreased as the Re increased.

An alternative full multigrid SIMPLEC approach for the incompressible Navier–Stokes equations

An alternative approach to solve the steady-state incompressible Navier–Stokes equations using the multigrid (MG) method is presented. The mathematical model is discretized using the finite volume method with second-order approximation schemes in a uniform collocated (nonstaggered) grid. MG is employed through a full approximation scheme-full MG algorithm based on V-cycles. Pressure-velocity coupling is ensured by means of a developed modified SIMPLEC algorithm which uses independent V-cycles for relaxing the pressure-correction and momentum equations. The coarser grids are used only internally in these cycles. All other original SIMPLEC steps can be performed only on the finest grid of the current full MG level. The model problem of the lid-driven flow in the unitary square cavity is used for the tests of the numerical model. Computational performance is measured through error and residual decays and execution times. Good performances were obtained for a wide range of Reynolds numbers, with speedups of orders as high as Linear relationships between execution times and grid sizes were observed for low and high Re values ( and 7,500). For intermediate Re values ( and 1,000), the linear trend was observed from more refined grids (512 onwards).

Numerical Study of Viscoplastic Flows Using a Multigrid Initialization Algorithm

In this paper, an innovative methodology to handle the numerical simulation of viscoplastic flows is proposed based on a multigrid initialization algorithm in conjunction with the SIMPLE procedure. The governing equations for incompressible flow, which consist of continuity and momentum equations, are solved on a collocated grid by combining the finite volume discretization and Rhie and chow interpolation for pressure–velocity coupling. Using the proposed solver in combination with the regularization scheme of Papanastasiou, we chose the square lid-driven cavity flow and pipe flow as test cases for validation and discussion. In doing so, we study the influence of the Bingham number and the Reynolds number on the development of rigid areas and the features of the vortices within the flow domain. Pipe flow results illustrate the flow’s response to the stress growth parameter values. We show that the representation of the yield surface and the plug zone is influenced by the chosen value. Regarding viscoplastic flows, our experiments demonstrate that our approach based on using the multigrid method as an initialization procedure makes a significant contribution by outperforming the classic single grid method. A computation speed-up ratio of 6.45 was achieved for the finest grid size (320 × 320).

Bubble rising and interaction in ternary fluid flow: a phase field study.

Bubble-droplet interaction is essential in the gas-flotation technique employed in wastewater treatment. However, due to the limitations of experimental methods, the details of the fluid flow involved have not been fully understood. Therefore, a phase field model for a three-phase flow was developed to study the rise of a single bubble and bubble-droplet interactions. The fluid-fluid interfaces are tracked by the Cahn-Hilliard equation, which is coupled with the Navier-Stokes equations with an equivalent volumetric force substituted for interfacial tensions. The model was discretized using an explicit finite difference method on a half staggered grid, and the pressure velocity coupling was tackled using the projection method. The in-house code was written in Fortran and run with the help of OpenMP, a shared memory parallelism. The model was validated against experiments with gratifying agreement achieved. Bubble-droplet interaction was simulated in two distinct situations: the first features a gas bubble crossing the interface between two other phases, and the second features a gas bubble chasing from behind an oil droplet in a surrounding fluid of the third phase.

HEAT TRANSFER ENHANCEMENT IN A CHANNEL WITH INCLINED BAFFLES UNDER PULSATING FLOW: A CFD STUDY

This study numerically investigated hydraulic and thermal performance in a channel with inclined baffles under pulsating flow conditions. The baffles were placed in a staggered arrangement. The governing equations were discretized with the finite volume method (FVM), and the pressure-velocity coupling was handled by the SIMPLE algorithm. The Strouhal number (St: 0.5,1, 2, 3, and 4), pulsation amplitude (A: 0.2, 0.5, and 0.8), and Reynolds number (200 ≤ Re ≤ 1000) were changed. The top and bottom surfaces of the channel were kept at <i>T</i><sub>ω</sub> = 350 K, and thermal improvement and friction factor were calculated for a pulsating cycle. The results were given in terms of thermal enhancement (η), relative friction factor (<i>f</i><sub>rel</sub>), and performance evaluation criteria (PEC). The flow and temperature contours were presented to determine the impacts of the pulsation frequency, the pulsation amplitude, and the Reynolds number. The results showed that the pulsation amplitude and the pulsation frequency contributed remarkably to thermal enhancement with increasing Reynolds numbers, while the heat transfer improved significantly depending on pulsation parameters together with a slight rise in friction factor. The highest thermal enhancement achieved was about 1.47 at <i>Re</i> = 1000, <i>A</i> = 0.8, and <i>St</i> = 4. The highest PEC obtained was approximately 1.12 at <i>Re</i> = 1000, <i>A</i> = 0.2, and <i>St</i> = 4.

Influence of the Inclination Angle on Mixed Convection and Heat Transfer in a “T” Shaped Double Enclosure

The effect of the tilt angle on mixed convection and related heat transfer in a “T” shaped double enclosure with four heated obstacles on the bottom surface is numerically investigated. The considered obstacles are constantly kept at a relatively high (fixed) temperature, while the cavity’s upper wall is cooled. The finite volume approach is used to solve the mass, momentum, and energy equations with the SIMPLEC algorithm being exploited to deal with the pressure-velocity coupling. Emphasis is put on the influence of the tilt angle on the solution symmetry, flow structure, and heat exchange through the walls. The following parameters and related ranges are considered: Rayleigh number 104 ≤ Ra ≤ 5.105, tilt angle 0° ≤ φ ≤ 90°, Reynolds number 100 ≤ Re ≤ 1000, Prandtl number Pr = 0.72, block height B = 0.5, opening width C = 0.15, and distance between blocks D = 0.5. The results reveal different branches of solutions on varying Re and φ . They also show that the symmetry of the solution regarding the P2 axis is retained for all cases with no tilt and for values of Re between 100 and 1000.