Two-dimensional Incompressible Flow Research Articles

The VM2D Open Source Code for Two-Dimensional Incompressible Flow Simulation by Using Fully Lagrangian Vortex Particle Methods

This article describes the open-source C++ code VM2D for the simulation of two-dimensional viscous incompressible flows and solving fluid-structure interaction problems. The code is based on the Viscous Vortex Domains (VVD) method developed by Prof. G. Ya. Dynnikova, where the viscosity influence is taken into account by introducing the diffusive velocity. The original VVD method was supplemented by the author’s algorithms for boundary condition satisfaction, which made it possible to increase the accuracy of flow simulation near the airfoil’s surface line and reduce oscillations when calculating hydrodynamic loads. This paper is aimed primarily at assessing the efficiency of the parallelization of the algorithm. OpenMP, MPI, and Nvidia CUDA parallel programming technologies are used in VM2D, which allow performing simulations on computer systems of various architectures, including those equipped with graphics accelerators. Since the VVD method belongs to the particle methods, the efficiency of parallelization with the usage of graphics accelerators turns out to be quite high. It is shown that in a real simulation, one graphics card can replace about 80 nodes, each of which is equipped with 28 CPU cores. The source code of VM2D is available on GitHub under GNU GPL license.

Effect of the lid-driven on mixed convection in an open flexible wall cavity with a partially heated bottom wall

The FSI-mixed convection heat transfer in a lid–driven vented cavity is studied in the present work. The heat source is set partially on the base of the cavity; the left flexible wall is cold, while the remaining walls are adiabatic. The top wall of the cavity moves in different cases depending on its movement; stationary wall (case 0), moving to the left (case I) and moving to the right (case II). The arbitrary Lagrangian-Eulerian formulation and finite element method are utilized to solve the governing equations. A two-dimensional incompressible laminar flow with an unsteady-state condition is used in the present model. The effects of relevant factors like the Richardson number 0.01≤ Ri ≤ 100 and the ratio of the velocities of moving top wall to that of the air entering the cavity, which is governed by Reynolds numbers ratio 1 ≤ Rer ≤ 4 have been studied. The outcomes demonstrate the feasibility of using a flexible wall and a special movement of the top wall. The percentage increase of the Nusselt number due to the flexible wall alone is about 4.5% while when the top wall moving to the left (case I), the augmentation rises to 28%.

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

A High Order Accurate Bound-Preserving Compact Finite Difference Scheme for Two-Dimensional Incompressible Flow

Computational study of consequence of effect of velocity slip on nanofluids with suspended CNTs

In this article, we have discussed two-dimensional steady, incompressible, and laminar hydromagnetic flow along with the transfer of heat above a plane plate that contains water and ethylene glycol-based nanofluids with multi- and single-wall CNTs. The velocity slip boundary condition is included in this research article to evaluate the nature of flow. The plate surface is subjected to variable wall temperature. The transfer of heat has been contemplated in the company of suspended CNTs above a flat plate. Navier–Stokes and the concept of the Prandtl boundary layer have been used to determine the governing equations of heat transfer and flow. Since the attained governing equations representing energy and flow are nonlinear PDEs, they are altered into nonlinear ODEs accompanied by analogous boundary limitations by applying some appropriate similarity transformations. Hence the "RK45" function is used to solve them numerically. The influence of governing parameters on the dimensionless flow and heat transfer profiles are studied and discussed thorough graphical plots .

Effective role of mineral oil and biological nanomaterial on thermal energy influenced by magnetic dipole and nanoparticle shape

This study of synovial fluid was conducted by considering two different nanofluid models over a two-dimensional stretched surface using nanoparticles of different shapes. We obtained remarkable results regarding the impact of nanoparticles on thermal performance. Through this study, we assessed heat and mass transfer and the involvement of magnetic dipole of chemically reactive species in two-dimensional steady incompressible flow. Heat generation was incorporated in the energy equation and a first-order chemical reaction was involved in the mass transport phenomenon. The concept of boundary layer was adopted to derive the physical problem in Cartesian coordinates, with results in the form of coupled partial differential equations (PDEs). The derived PDEs were highly non-linear, and exact solutions were not possible. Therefore, the PDEs were converted into non-linear ordinary differential equations (ODEs) using appropriate similarity transformation and then solved numerically via the finite element method. The impact of numerous emerging parameters on the solutions are displayed graphically, and the physical significance is discussed. An increment in Sc,Kc, and γ decelerated the solute field, while the concentration gradient increased with enhancement in Sc. Maximum acceleration in velocity for model-I was produced compared to acceleration in the velocity field for model-II.

Linear stability analysis of fluid–structure interaction problems with an immersed boundary method

In this work, we present a novel approach to perform the linear stability analysis of fluid–structure interaction problems. The underlying idea is the combination of a validated immersed boundary solver for the nonlinear coupled dynamics with Krylov-based techniques to obtain a robust and accurate global stability solver for elastic structures interacting with incompressible viscous flows. The computation of the leading eigenvalues of the linearized system is carried out in a matrix-free framework by adopting a classical Krylov subspace method. The proposed algorithm avoids the complex analytical linearization of the equations while retaining all the relevant aspects of the fully-coupled fluid–structure system.The methodology has been tested for several cases involving two-dimensional incompressible flows around elastically mounted circular cylinders. The obtained results show a good quantitative agreement with those available in the literature. Finally, the method was applied to investigate the linear stability of the laminar flow past two elastically mounted cylinders in tandem configuration at Re=100, revealing the existence of two complex dominant modes. For low values of the reduced velocity U∗, only one mode is found to be unstable and related to the stationary wake mode. The loss of stability of the second mode at U∗=4 marks the beginning of the lock-in region. We also show that for U∗=5 the modes interact, giving rise to the beating phenomenon observable in the nonlinear time evolution of the system. For larger values of the reduced velocity, the linear dynamics is governed by one dominant mode characterized by wider oscillations of the rear cylinder, matching the results of the nonlinear simulations.

Numerical investigation of heat transfer and fluid flow characteristics of ternary nanofluids through convergent and divergent channels.

The characteristics of nanomaterials have garnered significant attention in recent research on natural and forced convection. This study focuses on the forced convection characteristics of ternary nanofluids within convergent and divergent channels. The ternary nanofluid comprises titanium oxide (TiO2), zinc oxide (ZnO), and silver suspended in water, which serves as the base fluid. Using COMSOL Multiphysics 6.0, a reliable software for finite element analysis, numerical simulations were conducted for steady and incompressible two-dimensional flow. Reynolds numbers varying from 100 to 800 were employed to investigate forced convection. Additionally, we explored aspect ratios (channel height divided by the height of the convergent or divergent section) of -0.4, -0.2, 0, 0.2, and 0.4. Our findings revealed that only at aspect ratio a = 0.4 did the average outlet temperature increase as the Reynolds number rose, while other aspect ratios exhibited decreasing average temperatures with declining Reynolds numbers. Moreover, as the Reynolds number increased from 100 to 800 and the total volume fraction of the ternary nanofluids ranged from 0.003 to 0.15, there was a significant 100% enhancement in the average Nusselt number. For clarity, this article briefly presents essential information, such as the study's numerical nature, fluid properties (constant-property fluid), and the methodology (COMSOL Multiphysics 6.0, finite element analysis). Key conclusions are highlighted to enable readers to grasp the main outcomes at a glance. These details are also adequately covered in the manuscript to facilitate a comprehensive understanding of the research. The utilization of this emerging phenomenon holds immense potential in various applications, ranging from the development of highly efficient heat exchangers to the optimization of thermal energy systems. This phenomenon can be harnessed in scenarios in which effective cost management in thermal production is a critical consideration.

MHD Nanofluid boundary layer flow over a stretching sheet with viscous, ohmic dissipation

The objective of this research is to examine the steady incompressible two-dimensional hydromagnetic boundary layer flow of nanofluid passing through a stretched sheet in the influence of viscous and ohmic dissipations. The present problem is obtained with the help of an analytical technique called DTM-Pade Approximation. The mathematical modeling of the flow is considered in the form of the partial differential equation and is transformed into a differential equation through suitable similarity transformation. The force of fixed parameters like thermophoresis number Nt, Brownian motion number Nb, Prandtl number Pr, Lewis number Le, Magnetic field M, suction/injection S and Eckart number Ec are displayed with the aid of Figures. Our outcomes showed a greater trend in the velocity profile for the parameters of magnetics M, suction S, and nonlinear stretching parameter n. While the reverse trend is found against the temperature profile when the Prandtl number increases. Lewis number and other parameters have shown increasing behavior in the concentration profile.

An application of spectral linearisation method on the steady two-dimensional radial flow of Au-water and Ag-water nanofluids between two inclined plane walls

The present work purveys the steady two-dimensional viscous incompressible radial flow of Au-water and Ag-water nanofluids between plane walls which either converge or diverge in the presence of MHD effect. A uniform magnetic field is applied. The governing partial differential equations of the present physics and their appropriate boundary conditions are initially cast into the dimensionless forms to reduce into the ordinary differential equation. The resulting equation thus formed is then solved by adopting the spectral linearisation method (SLM) to get the accurate numerical solution. Solution errors and residual norms were analysed to elaborate the convergence and accuracy of the numerical solution. The present results are validated with favourable comparisons to previously published results due to the current investigations' unique cases. Parametric study of the governing parameters, namely the magnetic field strength parameter M, Reynolds number Re, angle of inclination α, and the volume fraction parameter ɸ on the non-dimensional velocity is conducted. The analysis discloses that the profiles of the flow are largely impacted by the physical parameters.

Optimal L2 error analysis of first-order Euler linearized finite element scheme for the 2D magnetohydrodynamics system with variable density

In this paper, a first-order Euler semi-implicit finite element scheme is proposed for solving the two-dimensional incompressible magnetohydrodynamics flows with variable density numerically. The proposed FEM algorithm is unconditionally stable at the full discrete level, which is a key issue in designing efficient algorithms for the multi-physical field problems. When the time step size τ and the mesh size h both are sufficiently small, optimal spatial error estimates in L 2 -norm are presented for the density, velocity and magnetic field by using energy techniques based on uniform regularities of solutions to the time-discrete scheme and the method of mathematical induction. Finally, numerical results are given to confirm the theoretical analysis.

Application of Successive Linearization Method on Steady Radial Flow of Nanofluids Between Inclined Plane Walls

The present work purveys the heat transfer enhancement in the steady two-dimensional viscous incompressible radial flow of Au-Water and Ag-Water nanofluids in the presence of MHD effect between the stationary convergent/divergent channel walls which are permitted to stretch or shrink. A uniform magnetic field is applied. The governing partial differential equations of the present physics and their appropriate boundary conditions are initially cast into dimensionless forms to reduce into the ordinary differential equations. The resulting equations thus formed are then solved by adopting the Successive Linearization Method (SLM) to get the accurate numerical solution. Solution errors and residual norms are analyzed to elaborate the convergence and accuracy of the numerical solution. The behavior of thermal conductivity of both types of nanofluids is examined for converging channel and diverging channel cases under the uniform magnetic field effect. The present results are validated with favorable comparisons with previously published results as the current investigations’ unique cases. A parametric study of the governing parameters, namely the magnetic field strength parameter, Reynolds number, angle of inclination, and the stretching parameter on the non-dimensional velocity and temperature, is conducted. Analysis discloses that the profiles of the flow are largely impacted by the physical parameters. It is noticed that the magnetic parameter deploys an enhancing influence on fluid velocity profile as well as heat transfer rate, and the effect of the magnetic field is less pronounced on Au-water nanofluid than that of the Ag-water nanofluid. The fluid velocity increases as the values of Re increase for both the nanofluids in the convergent channel and decreases in the case of the divergent channel. Fluid temperature increases as Re increases for the divergent channel. The velocity of both the nanofluids increases as the angle of inclination of the plates increases.

Significance of activation energy and entropy optimization in radiative stagnation point flow of nanofluid with cross-diffusion and viscous dissipation

The basic aim of this work is to examine the activation energy, thermal radiation, and the Joule heating influence in an incompressible two-dimensional dissipative stagnation point flow of a nano liquid past a curved stretching sheet. Furthermore, this work includes the investigation of entropy generation with cross-diffusion. The basic partial differential equations (PDEs) arising from the current study have been converted into ordinary differential equations (ODEs) by applying some dimensionless transformations. The solver bvp4c has been used for the numerical solutions of resulting ODEs. Various graphical results for the velocity, skin friction coefficient, temperature, Nusselt number, concentration, and Sherwood number have been plotted to show the impact of diverse values of the involved flow parameters. The skin friction coefficient is diminished by increasing the curvature and nanoparticles concentration whereas the heat transport rate reduces with the increase of Eckert and Hartmann number. Similarly, the escalating values of Soret number, chemical reaction rate parameter, and the temperature difference parameter enhance mass transportation. Furthermore, the Bejan number decreases with an increase in Brinkman and Hartmann number whereas the fluid’s entropy rises for the growing values of thermal radiation and temperature difference parameter.

CFD-based simulation of heat transfer in a rectangular channel

Heat transfer is an important phenomenon in the industrial sector. Thus, the simulation is made to compute the distribution of heat in a rectangular channel in this paper. A heated rod is inserted at the center of the rectangular channel. The fluid flowing in the rectangular channel is considered to be a viscous fluid. Navier–Stokes equations of motion for laminar flow are used. The medium for the fluid motion is considered to be a porous medium. Heat transfer is computed for nonlinear two-dimensional incompressible and unsteady flows. The Fourier’s law of heat conduction is used for the transmission of heat in the rectangular channel. The Finite Element Method (FEM) is applied to the solution of the problem. For different values of the permeability parameter, Prandtl number and Rayleigh number, the graphic solution for the velocity and temperature fields is shown.

A semi-analytical passive strategy to examine a magnetized heterogeneous mixture having sodium alginate liquid with alumina and copper nanomaterials near a convectively heated surface of a stretching curved geometry

In this article, the authors have analyzed a steady and incompressible two-dimensional flow of sodium alginate (SA)-based viscous ternary nanofluid including copper and alumina nanoparticles near a stretchable curved surface. A uniform magnetic field is exerted on the nanofluid flow along the radial direction. Furthermore, an appropriate convective heating process and zero wall mass flux are adopted in this study as boundary conditions. The chemical reaction, variable internal heating, thermophoresis, Brownian motion, and Arrhenius activation energy impacts are assumed. A robust semi-analytical method abbreviated as HAM (i.e., homotopy analysis method) is chosen in this analysis to get accurate solutions for the modeled problem. Additionally, HAM results are compared extensively with an outstanding tabular/graphical agreement and consistency. The impressions of the embedded factors on the velocity profile, the temperature variation, the concentration distribution, the skin friction factor, and Nusselt's number of sodium alginate-based copper nanofluid, alumina nanofluid, and copper-alumina hybrid nanofluid are deliberated comprehensively. The results showed that the velocity distribution of SA-based nanofluid and hybrid nanofluid decreases as the nanoparticles' volume fraction and magnetic factor increase, whereas the temperature distribution of SA-based nanofluid and hybrid nanofluid improves as the nanoparticles' volume fraction escalates. The rising solid volume fraction and magnetic factor increase the strength of the surface drag force of SA-based copper nanofluid and alumina nanofluid. The increasing nanoparticles' volume fraction and internal heat source parameter upsurge the heat transfer rate of SA-based copper nanofluid and alumina nanofluid. It is also revealed that the hybrid nanofluid is highly affected by the involved physical factors as compared to the case of monotype nanofluids.

Discrete representations of orbit structures of flows for topological data analysis

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called the flow of finite type, are in one-to-one correspondence with discrete structures such as trees/graphs and sequences of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of two-dimensional (2D) incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse–Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows when applied to data brought by measurements, experiments, and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem.

Unsteady aerodynamic theory for membrane wings

We study analytically the dynamic response of membrane aerofoils subject to arbitrary, small-amplitude chord motions and transverse gusts in a two-dimensional inviscid incompressible flow. The theoretical model assumes linear deformations of an extensible membrane under constant tension, which are coupled aeroelastically to external aerodynamic loads using unsteady thin aerofoil theory. The structural and aerodynamic membrane responses are investigated for harmonic heave oscillations, an instantaneous change in angle of attack, sinusoidal transverse gusts and a sharp-edged gust. The unsteady lift responses for these scenarios produce aeroelastic extensions to the Theodorsen, Wagner, Sears and Küssner functions, respectively, for a membrane aerofoil. These extensions incorporate for the first time membrane fluid–structure interaction into the expressions for the unsteady lift response of a flexible aerofoil. The indicial responses to step changes in the angle of attack or gust profile are characterised by a slower lift response in short times relative to the classical rigid-plate response, while achieving a significantly higher asymptotic lift at long times due to aeroelastic camber. The unsteady lift for harmonic gusts or heaving motions follows closely the rigid plate lift responses at low reduced frequencies but with a reduced lift amplitude and greater phase lag. However, as the reduced frequency approaches the resonance of the fluid-loaded membrane, the lift response amplitude increases abruptly and is followed by a sharp decrease. This behaviour reveals a frequency region, controlled by the membrane tension coefficient, for which membrane aerofoils could possess substantial aerodynamic benefits over rigid aerofoils in unsteady flow conditions.

Unstructured, Pressure-Based Algorithm Using Implicit Runge–Kutta for Moving Boundary Problems

An unstructured, pressure-based algorithm, IRK-SIMPLER, is developed for simulating two-dimensional incompressible flows over moving bodies. The Navier–Stokes equations in the arbitrary Lagrangian Eulerian form are discretized using collocated, median-dual control volumes with a flux correction method. The semidiscrete form is integrated in time using a diagonally implicit Runge–Kutta method. The unstructured discretization of the spatial computational domain is composed of a circular body-fitted grid, a rectangular patch grid, and a background grid. The motion of the bodies is decomposed into a series of rotations and translations. Rotation is achieved by rotating the body-fitted grid along with the body and flipping the edges of the appropriate cells. Translation is achieved by morphing the patch grid as the body moves. For large translational displacements, the rectangular patch grid is moved with the body and rematched with the background grid to obtain a new conforming grid. The algorithm is validated for multiple incompressible flow cases.

Energy conservation in the limit of filtered solutions for the 2D Euler equations

We consider energy conservation in a two-dimensional incompressible and inviscid flow through weak solutions of the filtered-Euler equations, which describe a regularized Euler flow based on a spatial filtering. We show that the energy dissipation rate for the filtered weak solution with vorticity in L p , p > 3/2 converges to zero in the limit of the filter parameter. Although the energy defined in the whole space is not finite in general, we formally extract a time-dependent part, which is well-defined for filtered solutions, from the energy and define the energy dissipation rate as its time-derivative. Moreover, the limit of the filtered weak solution is a weak solution of the Euler equations and it satisfies a local energy balance in the sense of distributions. For the case of p = 3/2, we find the same result as p > 3/2 by assuming Onsager’s critical condition for the family of the filtered solutions.

Investigation of Inner Elliptical Pin-Fins Configuration on Magnetoconvective Heat Transfer

This paper deals with Lattice Boltzmann computation of two-dimensional incompressible flow past two heat elliptical sinks vertically attached to the horizontal walls of a cavity in the presence of magnetic field. A two dimensional nine-velocity square lattice Boltzmann (D2Q9) model is used. The obtained results show that the applied Lattice Boltzmann method (LBM) can effectively capture vortex shedding structures and other features inside such complex flow. The present study has been carried out for Hartmann numbers with panoply of heat sinks location. Results show that the heat sinks positions greatly influence the heat transfer rate depending on the Hartmann number.

Nonlinear response of passively flapping foils

This study numerically investigates two-dimensional incompressible flows over elastically mounted foils, undergoing semi-passive and fully passive motion. In our strongly coupled numerical models, we employ linear and cubic stiffness and damping terms in order to examine their highly nonlinear response. The undamped model of the fully passive system exhibits various responses from periodic to chaotic and then to flip-over for the reduced velocity, ranging from 1 to 10. However, introducing the cubic damping terms causes a significant decrease in the magnitude of plunging and pitching amplitudes without affecting the onset point of bifurcation. Also, plunging and pitching amplitudes are altered significantly after the point of onset. Furthermore, the performance metrics of each passive system are computed for power generation applications to demonstrate that semi-passive system attain efficiency up to 20% for a pitching amplitude of 50° with the excitation frequency in the narrow range of 0.15 to 0.20. On the other hand for a fully passive system, the efficiency of around 34% is obtained near the onset point of a bifurcation with a low mass ratio and linear damping terms. However, introducing cubic damping terms causes degradation in efficiency to bring it down to 14−20% for a wide range of reduced velocity.