Two-dimensional Incompressible Fluid Flow Research Articles

Fractal structures in the chaotic advection of passive scalars in leaky planar hydrodynamical flows.

The advection of passive scalars in time-independent two-dimensional incompressible fluid flows is an integrable Hamiltonian system. It becomes non-integrable if the corresponding stream function depends explicitly on time, allowing the possibility of chaotic advection of particles. We consider for a specific model (double gyre flow), a given number of exits through which advected particles can leak, without disturbing the flow itself. We investigate fractal escape basins in this problem and characterize fractality by computing the uncertainty exponent and basin entropy. Furthermore, we observe the presence of basin boundaries with points exhibiting the Wada property, i.e., boundary points that separate three or more escape basins.

An MHD nanofluid flow with Marangoni laminar boundary layer over a porous medium with heat and mass transfer

ABSTRACT The current work considers two-dimensional incompressible fluid flow with heat transfer over a porous medium with mass transpiration and Marangoni boundary condition. The nanoparticles of graphene are mixed in water to improve thermal efficiency. It behaves as a heater upon increasing its solid volume fraction, Eckert number, and as a cooler upon increasing suction. This flow has remarkable interest due to its applications in glues, silicon wafers, heat exchangers, paints, crystal growth in space, etc. The ordinary differential equations (ODEs) are obtained from the considered partial differential equations (PDEs) using similarity transformation technique. Then the solutions are analytically recovered from the system of ODEs. The solutions of temperature and concentration fields are obtained in terms of Laguerre polynomial. The impacts of different parameters, such as Marangoni number, inverse Darcy number, Prandtl number, and Eckert number are analysed using graphical observations. Specially mentioned is Marangoni convection, which contains many industrial and engineering applications such as in nanotechnology, atomic reactors, soap films, semiconductor processing, and atomic reactors. This work adds the effect of graphene water nanofluid which enhance the thermal efficiency in a better way than the base fluid and explains the physical problem on the basis of chemically radiative Marangoni flow.

Modal analysis for incompressible fluid flow

This paper presents a numerical method for incompressible fluid flow. A difficulty in analyzing incompressible fluid flow is that the continuity equation has no time evolution term. In the marker and cell (MAC) method, Poisson’s equation is solved iteratively, which takes most of the computation time, and in the artificial compressibility method (ACM), pseudo-time iteration is necessary to solve for unsteady solutions. Here, modal analysis that uses the velocity eigenvectors corresponding to zero eigenvalues is proposed for analyzing two-dimensional incompressible fluid flow. The proposed method involves only about one third of the number of variables needed in the MAC method and the ACM, and it does not require iterative calculation of Poisson’s equation or pseudo-time iteration. Numerical results for a simple flow system and a cavity flow obtained using the proposed method are compared with those obtained using the ACM and the simplified MAC method. The results agree well, thereby validating the proposed modal analysis.

NUMERICAL STUDY OF CARREAU FLUID FLOW ALONG AN EXPONENTIAL CURVED STRETCHING SURFACE

A two-dimensional incompressible boundary layer Carneau fluid flow with heat-transfer analysis over a curved stretching surface is analyzed. The energy equation with the inclusion of thermal radiation and viscous dissipation effects is considered. The governing partial differential equations which govern such flow phenomena are transformed into suitable form of ordinary differential equations for integration by using stream function formulation. The developed non-linear problem has been solved by computational approach based on shooting technique using sixth-order Runge-Kutta method and Matlab built-in function bvp4c program. The effects of non-dimensional controlling parameters on temperature and velocity profile are analyzed with the aid of tables and figures. The surface drag force and Nusselt numbers are studied for the different values of the governing parameters. It is predicted that velocity of the fluid and boundary layer thickness is decreased when radius of curvature parameter δ is increased. Furthermore, the temperature profile dwindles for the growing values of δ. Other important information is that for shear-thinning fluid the velocity profile shows its increasing nature, whereas for shear-thickening fluid the opposite trend has been observed. For increasing values of curvature parameter δ from 2 to 1000, the temperature distribution and velocity profile is decreased. The radiative heat flux is included to enhance the temperature of the system, so, for the increasing values of radiation parameter <i>R<sub>d</sub></i> from 0.2-0.5 the temperature distribution is increased. Further, as the Biot number and Eckert number are increased from 0.2-2 and 0.1-1, respectively, the temperature distribution is increased.

Singularity Formation in an Incompressible Boundary Layer on an Upstream Moving Wall under Given External Pressure

The two-dimensional laminar flow of a viscous incompressible fluid over a flat surface is considered at high Reynolds numbers. The influence exerted on the Blasius boundary layer by a body moving downstream with a low velocity relative to the plate is studied within the framework of asymptotic theory. The case in which a small external body modeled by a potential dipole moves downstream at a constant velocity is investigated. Formally, this classical problem is nonstationary, but, after passing to a coordinate system comoving with the dipole, it is described by stationary solutions of boundary layer equations on the wall moving upstream. The numerically found solutions of this problem involve closed and open separation zones in the flow field. Nonlinear regimes of the influence exerted by the dipole on the boundary layer with counterflows are calculated. It is found that, as the dipole intensity grows, the dipole-induced pressure acting on the boundary layer grows as well, which, after reaching a certain critical dipole intensity, gives rise to a singularity in the flow field. The asymptotics of the solution near the isolated singular point of the flow field is studied. It is found that, at this point, the vertical velocity grows to infinity, viscous stress vanishes, and no solution of the problem exists at higher dipole intensities.

Optimization of Magnetohydrodynamic Parameters in Two-Dimensional Incompressible Fluid Flow on a Porous Channel

Optimization of Magnetohydrodynamic Parameters in Two-Dimensional Incompressible Fluid Flow on a Porous Channel

Weakly viscous two-dimensional incompressible fluid flows using efficient isogeometric finite element method

This paper presents a straightforward and efficient numerical simulation method for solving the Navier–Stokes equations for weakly viscous incompressible fluids describing steady flow. Our approach utilizes isogeometric finite elements to handle higher-order partial differential operators associated with weakly viscous incompressible flow problems. Specifically, our numerical formulation employs a principle of virtual power (PVP)-based weak formulation that utilizes a stream-function field, which distinguishes it from the more commonly used bi-harmonic type formulations. The usage of a stream-function field ensures a pointwise divergence-free velocity condition, making the present method suitable for low to moderately high Reynolds number flow problems. In contrast to the bi-harmonic formulation, which is typically used for describing internal flow and requires special treatment of outflow boundary conditions, the PVP-based formulation is more general and does not require special treatment at the outflow boundary. It is also demonstrated that both bi-harmonic and PVP-based weak formulations yield identical results for internal flow problems. Our method employs non-uniform rational B-spline basis functions, and we present a simple stitching technique for imposing no-slip Dirichlet boundary conditions. Finally, we solve Poisson's equation to recover the pressure field. Furthermore, we use an appropriate Gaussian quadrature that is exact for splines to speed up the computation of various element matrices, especially for high polynomial degrees. The proposed formulation is evaluated by solving a set of numerical problems, including internal flow and channel flow problems.

Nonlinear Dynamics of Perturbations of a Plane Potential Fluid Flow: Nonlocal Generalization of the Hopf Equation

In this paper, we analytically study the two-dimensional unsteady irrotational flow of an ideal incompressible fluid in a half-plane whose boundary is assumed to be a linear sink. It is shown that the nonlinear evolution of perturbations of the initial uniform flow is described by a one-dimensional integro-differential equation, which can be considered as a nonlocal generalization of the Hopf equation. This equation can be reduced to a system of ordinary differential equations (ODEs) in the cases of spatially localized or spatially periodic perturbations of the velocity field. In the first case, ODEs describe the motion of a system of interacting virtual point vortex-sinks/sources outside the flow domain. In the second case, ODEs describe the evolution of a finite number of harmonics of the velocity field distribution; this is possible due to the revealed property of the new equation that the interaction of initial harmonics does not lead to generation of new ones. The revealed reductions made it possible to effectively study the nonlinear evolution of the system, in particular, to describe the effect of nonlinearity on the relaxation of velocity field perturbations. It is shown that nonlinearity can significantly reduce the relaxation rate by more than 1.5 times.

Physics-Informed Neural Network for Flow Prediction Based on Flow Visualization in Bridge Engineering

Wind loads can endanger the safety and stability of bridges, especially long-span cable-supported bridges. Therefore, it is important to evaluate the potential wind loads during the bridge design stage. Traditionally, wind load evaluation is performed by wind tunnel testing, which is relatively expensive. With the development of computational fluid dynamics and high-performance computing, numerical simulations are becoming more accessible for designers. However, the costs required for accurate numerical results are still high, especially for high-fidelity simulations. Under this condition, searching for a more efficient method to evaluate the wind loads in bridge wind engineering has become a new goal. It seems that flow visualization is a good entry point. Although flow visualization techniques have been developed in recent years, it remains difficult to extract velocity and pressure fields from images. To address this problem, physics-informed neural networks (PINNs) have been developed and validated. This study establishes a PINN to investigate the two-dimensional viscous incompressible fluid flow passing a generic bridge deck section. Two cases with different Reynolds numbers are tested. After careful training, it is found that the PINN can accurately extract the velocity and pressure fields from the concentration field and predict the drag and lift coefficients. The results demonstrate that PINNs are a promising method for extracting useful flow information from flow visualization data in engineering applications.

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.

Influence of Shape Factor and Non-Linear Stretching of the Bullet-Shaped Object on the Mixed Convection Boundary Layer Flow and Heat Transfer with Viscous Dissipation and Internal Heat Generation

The two-dimensional incompressible axisymmetric mixed convection magnetohydrodynamic fluid flow and energy transfer over a bullet-shaped object with a non-linear stretching surface have been investigated. The main goal of this problem is to discuss the effect of the shape and size of the bullet-shaped object on the fluid velocity and temperature distributions. The present analysis has been performed in about two cases ε=0.0 and 2.0. Therefore, fluid velocity and temperature distributions have been investigated in two types of flow geometries such as the thicker surface (s ≥ 2) and the thinner surface (0 < s < 2) of the bullet-shaped object. The equations for momentum and heat transfer have been converted into ODEs by using suitable local similarity transformations. These equations have been performed with a recently developed spectral quasi-linearization method (SQLM). This method helps to identify the accuracy, validity, and convergence of the present solution. The novelty of the present work has been applying the recently developed numerical method to solve these highly nonlinear differential equations. The investigation shows that in the case of a thicker bullet-shaped object (s ≥ 2) the velocity and temperature profiles do not converse the far-field boundary condition asymptotically but cross the axis with an upright angle and the boundary layer structure has no definite shape whereas in the case of a thinner bullet-shaped object (0 < s < 2) the velocity profile converge the ambient condition asymptotically and the boundary layer structure has a definite shape. The innovation of this current work lies in the unification of relevant physical parameters into the governing equations and trying to explain how the flow properties are affected by these parameters.

Application of 2D adaptive mesh refinement method to estimation of the center of vortices for flow over a wall-mounted plate

This paper describes the application of an adaptive mesh refinement (AMR) method to estimate centers of vortices of two-dimensional (2D) incompressible fluid flow over a wall-mounted plate. Following the accuracy verification of the AMR method using the benchmarks of 2D lid-driven cavity flows and backward-facing step flows, this study considers the application of the AMR method to the flow over a wall-mounted plate. The AMR method refines a mesh using numerical solutions of the Navier–Stokes equations computed using an open-source flow solver, Navier2D. The AMR is applied to seven test cases considering flows with different Reynolds numbers ([Formula: see text]) and the estimated centers of vortices after the plate are reported. The results show that AMR can capture the location of the center of vortices within the once refined cells. Furthermore, improved estimation of vortex centers is obtained using twice refined meshes. The AMR aims to get a refined mesh which captures the characteristics of flow with required accuracy at a lower computational cost.

Measuring the effectiveness of extrapolation techniques associated with the multigrid method applied to the Navier-Stokes equations

In this work, we applied different extrapolation techniques in association with the multigrid method to discover which one is the most effective in reducing the iteration error and the processing time (CPU time), as well as in improving the convergence factors. The mathematical model studied refers to the two-dimensional laminar flow of an isothermal time-dependent incompressible fluid modeled by the Navier-Stokes equations, with , solved iteratively with the projection method and the Finite Volume Method. The extrapolation methods used were: Aitken, Empiric, Mitin, scalar Epsilon, scalar Rho, topological Epsilon, and topological Rho. A two-step application was performed: first, extrapolators methods were applied individually after the use of the multigrid method. Then, the best-performing extrapolation techniques were used in the second step, where they were applied between the cycles of the multigrid method. The methods that presented the best convergence properties in the first stage were topological and scalar Epsilon. In the second stage, both methods maintained their performance, however, the topological Epsilon method presented more significant convergence rates than the scalar Epsilon. The other parameters analyzed were: the storage memory peak, the dimensionless norm of the residual based on the initial estimate, and the error norms of iteration. Thus, it was possible to state which extrapolation technique performed best and to compare it with the multigrid method with no extrapolation, which in this study was the topological Epsilon method.

Energy balance for forced two-dimensional incompressible ideal fluid flow.

Cheskidov et al. (2016 Commun. Math. Phys. 348, 129-143. (doi:10.1007/s00220-016-2730-8)) proved that physically realizable weak solutions of the incompressible two-dimensional Euler equations on a torus conserve kinetic energy. Physically realizable weak solutions are those that can be obtained as limits of vanishing viscosity. The key hypothesis was boundedness of the initial vorticity in [Formula: see text], [Formula: see text]. In this work, we extend their result, by adding forcing to the flow. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Stagnation point flow of a viscous incompressible fluid

A steady two-dimensional incompressible viscous fluid flow in the vicinity of a stagnation point has been discussed. The reduced third order nonlinear ordinary differential equation of velocity has been solved using fourth order Runge-Kutta method with shooting technique. The qualitative (graphical) as well as quantitative (tabular) analysis of the velocity distribution has been made. Some of the important findings are (i) The viscous effect are confined in the boundary layer for fluids with very small viscosity. (ii) Outside the boundary layer, the viscous effect is negligible and the fluid is considered as inviscid. (iii) The difference in velocity between adjacent layers is decreasing for less viscous fluids.

An approach to accelerate the convergence of SIMPLER algorithm for convection-diffusion problems of fluid flow with heat transfer and phase change

A new segregated pressure-based algorithm is developed to solve two-dimensional incompressible fluid flows with convective heat transfer and liquid to solid phase change by the finite volume method. The predictive-correction scheme is based on the SIMPLER algorithm with n added iterative cycles to increase the accuracy of pressure in satisfying the discretized continuity equation. The performance of SIMPLERnP is compared with SIMPLER and IDEAL in the solution of the laminar fluid flow in a lid driven cavity, natural heat convection inside a square cavity, convective heat transfer in the annular space between two concentric cylinders and conjugate natural heat convective solidification of a binary alloy that included the transient heat conduction in the thick mold walls. The results indicate that the proposed algorithm is highly stable, even when a value of 0.99 is used for the under-relaxation of the velocity components, and allows significant improvements in the reduction of the number of iterations and time of computation.

Determination of Reynolds shear stress from the turbulent mean velocity profile and spectral characteristics of Orr-Sommerfeld -Squire equations

Theoretically, based on a waveguide model, the expression of the tangential stress is formulated for steady, two-dimensional incompressible fluid flow over a flat plate in turbulent boundary layer. It is dependent on some factors, one of them, the behaviour of the last damping mode eigenvalues, and eigenfunctions, that are deduced from solution Orr-Sommerfeld equation by spectral Chebyshev collocation Method. Verification of the latter method is investigated by comparison the deduced formula of turbulent tangential stress with experimental data. In addition to, weight factors in this expression are connected to define the condition of dynamical system solution for multiple 3-wave resonance. This system is solved numerically, and the dynamic invariant is normalized to obtain the time average of the square modulus harmonic, and sub harmonics amplitudes by theorem Birkhoff-Khinchin. Comparison is made between the time-averaged and the phase average for the square modulus of harmonic, and sub harmonic amplitudes that defined on the unit sphere, in the state of multiple 3-wave resonance.

Viscous heating and cooling process in a mixed convection cavity with free-slip effect

Both no-slip and free-slip effects are enforced to the lid-driven cavity walls with Robin boundary condition. Viscous heating and cooling process happen when the shear force applied at the top lid is removed and the cavity is exposed to natural convection. This numerical study is performed for a two-dimensional laminar incompressible fluid flow with unsteadiness. The governing equations of time-dependent vorticity-stream function and thermal energy equation are solved using a finite difference method. Dissipation of viscous-heating of the cavity flow to the ambient is observed. The in-house Matlab® code is firstly verified with Ansys Fluent® finite element method at steady state. With no-slip boundary condition, temperature profile shows a reduction with acumination style at the beginning of the cooling process whereas a blunted profile is shown in free-slip boundary condition with temperature profile reduces dawdlingly. This is due to the movement of fluid is still able to create viscous heating at the outset of cooling process and this effect is even more pronounced at high Reynolds number when subject to free-slip boundary condition.

On the Use of Probability-Based Methods for Estimating the Aerodynamic Boundary-Layer Thickness

In this paper, known probabilistic methods for estimating the thickness of the boundary layer of a two-dimensional laminar flow of viscous incompressible fluid are extended to three-dimensional laminar flows of a viscous compressible medium. Their applicability to the problems of boundary-layer stability is studied with the LOTRAN3 software package, which allows us to compute the position of laminar-turbulent transition in three-dimensional aerodynamic configurations.

A deterministic model for bubble propagation through simple and cascaded loops of microchannels in power-law fluids

This paper investigates the path selection of bubbles suspended in different power-law carrier liquids in microfluidic channel networks. A finite volume-based numerical method is used to analyze the two-dimensional incompressible fluid flow in microchannels, while the volume of fluid method is used to capture the gas–liquid interface. To instill the influences of shear thinning, Newtonian, and shear-thickening fluids, the range of power-law indices (n) is varied from 0.3 to 1.5. We have validated our numerical model with the available literature data in good agreement. We have investigated the nonlinearity in the hydrodynamic resistance which arises due to single-phase non-Newtonian fluid flow. The path selection of a bubble in power-law fluids is examined from the perspective of velocity distribution and bubble deformation. We have found that the bubble indeed goes to the channel with a higher flow rate for all power-law fluids, but interestingly it did not always take the shorter route channel at a junction for n = 0.3. Our results suggest that long channels need not be more resistant for every fluid and that the longest arm becomes the least resistant resulting in the bubble leading into the long arm at a junction for shear-thinning fluid. We have proposed a deterministic model that enables predicting the second bubble path in a single bubble system for any location of the first bubble. We believe that the present study results will help design future generation microfluidic systems for efficient drug delivery and biomedical and biochemical applications.