Two-dimensional Incompressible Flow Research Articles

STRUCTURAL BIFURCATION OF DIVERGENCE-FREE VECTOR FIELDS NEAR NON-SIMPLE DEGENERATE POINTS WITH SYMMETRY

In this study, topological features of an incompressible two-dimensional flow far from any boundaries is considered. A rigorous theory has been developed for degenerate streamline patterns and their bifurcation. The homotopy invariance of the index is used to simplify the differential equations of fluid flows which are parameter families of divergence-free vector fields. When the degenerate flow pattern is perturbed slightly, a structural bifurcation for flows with symmetry is obtained. We give possible flow structures near a bifurcation point. A flow pattern is found where a degenerate cusp point appears on the x-axis. Moreover, we also show that bifurcation of the flow structure near a non-simple degenerate critical point with double symmetry is generic away from boundaries. Finally, we give an application of the degenerate flow patterns emerging when index 0 and -2 in a double lid driven cavity and in two dimensional peristaltic flow.

2차원 비압축성 점성 열유동 해석을 위한 WMLS 기반 무격자 해법의 개발

In this study, a newly-developed Eulerian gridless solver for simulating two-dimensional unsteady incompressible viscous thermal flows in low Reynolds number is introduced. The Boussinesq approximation is applied to deal with heat transfer and spatial derivatives of governing equations are solved by WMLS(Weighted Moving Least Square) interpolations on a cloud of points. A fractional time step method is adopted and the Poission equation for pressure are solved by successively in a WMLS sense. Some heat transfer problems are solved by the developed solver for the validation and the results are compared with analytic solution and those from other CFD simulations.

Rotating hollow patterns in fluids

Stable rotating polygonal hollow patterns were predicted theoretically and measured for Leidenfrost drops confined by circular boundary. Liquid free surface modes occur with different number of edges. This two-dimensional incompressible and inviscid flow is controlled by surface tension, evaporation reaction and gravitation. The model is based on shallow water theory and predicts the existence of sharp rotational polygonal waves as solitary waves solutions. These patterns may represent a signature of universality in shallow water rotating models since similar rotating hollow polygons are noticed in tropical cyclones, fast rotated liquids, Saturn's hexagon, and some plasma systems.

Numerical solution of the Giesekus model for incompressible free surface flows without solvent viscosity

We present a numerical method for solving the Giesekus model without solvent viscosity. This paper is concerned with incompressible two-dimensional free surface flows and employs the finite difference method to solve the governing equations. The methodology involves solving the momentum equation using the implicit Euler scheme and an implicit technique for computing the pressure condition on the free surface. The nonlinear Giesekus constitutive equation is computed by a second order Runge–Kutta method. A novel analytic solution for channel flow is developed and is used to verify the numerical technique presented herein. Mesh refinement studies establish the convergence of the method for complex free surface flows. To demonstrate that the technique can deal with complicated free surface flows, the time-dependent flow produced by a fluid jet flowing onto a rigid surface is simulated and the influence of the parameter α on the jet buckling phenomenon is investigated. In addition, the simulation of the extrudate swell of a Giesekus fluid was carried out and the effect of the parameter α on the flow was similarly examined.

A study of the regularized lid-driven cavity’s progression to chaos

Computational simulations of a two-dimensional incompressible regularized lid-driven cavity flow were performed and analyzed to identify the dynamic behavior of the flow through multiple bifurcations which ultimately result in Eulerian chaotic flow. Semi-implicit, pseudo-spectral numerical simulations were performed at Reynolds numbers from 1000 to 25,000. Poincaré maps were used to identify transitions as the Reynolds number increased from stable-laminar flow to periodic flow, periodic flow to quasi-periodic flow, quasi-periodic flow to chaotic flow, and a sudden, brief return from chaotic flow to periodic flow. The first critical Reynolds number, near 10,250, is found in agreement with existing literature. An additional bifurcation is observed near a Reynolds number of 15,500. A power spectrum analysis, in which the novel concepts of frequency shredding and power capacity are introduced, was performed with the conclusion that no further bifurcations occurred at Reynolds numbers above 15,500 even though Eulerian chaos was not formally observed until Reynolds numbers above 18,000. While qualitative changes in the fluid flow’s attractor were apparent from trajectories in the phase space, mechanisms by which such changes can occur were apparent from power spectra of flow time histories. The novel power spectrum analysis may also serve as a new approach for characterizing multi-scale nonlinear dynamical systems.

Existence of local strong solutions to fluid–beam and fluid–rod interaction systems

We study an unsteady nonlinear fluid–structure interaction problem. We consider a Newtonian incompressible two-dimensional flow described by the Navier–Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear wave equation or a linear beam equation. The fluid and the structure systems are coupled via interface conditions prescribing the continuity of the velocities at the fluid–structure interface and the action-reaction principle. Considering three different structure models, we prove existence of a unique local-in-time strong solution, for which there is no gap between the regularity of the initial data and the regularity of the solution enabling to obtain a blow up alternative. In the case of a damped beam this is an alternative proof (and a generalization to non zero initial displacement) of the result that can be found in [20]. In the case of the wave equation or a beam equation with inertia of rotation, this is, to our knowledge the first result of existence of strong solutions for which no viscosity is added. The key points consist in studying the coupled system without decoupling the fluid from the structure and to use the fluid dissipation to control, in appropriate function spaces, the structure velocity.

CFD SIMULATION OF THERMO-AERAULIC FIELDS IN A CHANNEL WITH MULTIPLE BAFFLE PLATES

In this paper, a two-dimensional incompressible flow of a Newtonian fluid through a horizontal duct of rectangular section, where four flat rectangular baffle plates were inserted and fixed to the top and bottom walls in a periodically staggered manner, is examined and analyzed numerically using the finite volume method by means of commercial CFD software FLUENT 6.3. Researchers consider this situation as a significant issue in the field of heat exchangers, for which the fluid flow characterization, heat transfer and skin friction loss distribution, along with the existence and the extension of possible re-circulations must be determined. The aspect ratio of channel width-to-height, channel length-to-hydraulic diameter, baffle spacing-to-channel height ratio, and blockage ratio of baffle height-to-channel height are fixed at W/H = 1.321, L/Dh = 5.137, Pi/H = 0.972, and h/H = 0.547, respectively. The Reynolds-Averaged Navier-Stokes Equations are the governing flow equations for the problem investigated, with the energy equation. In particular, flow and temperature fields, dimensionless axial velocity profiles, skin friction coefficients, local and average Nusselt numbers, and thermal enhancement factor were presented at constant wall temperature condition along the upper and lower channel walls. The presence of the baffle plates in the whole domain analyzed causes a much high skin-friction loss, f/f0 = 10.829-25.412 but also provides a considerable heat transfer increase in the duct, Nu/Nu0 = 3.623-5.008, depending on the Re values. The enhancement thermal factor for fluid flowing in the baffled channel with larger flow rate is found to be higher than that with smaller flow rate. The enhancement thermal factor augments with the rise of Reynolds number and thus, the highest Reynolds number value, Re = 32,000, provides maximum thermal performance factor, TEF = 1.783. This indicates that the introducing the flat rectangular baffle plates into the flow in a staggered arrangement can improve the heat transfer efficiency inside the channel.

Multiobjective Optimal Control Methods for the Navier-Stokes Equations Using Reduced Order Modeling

In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly increases and makes its direct treatment infeasible. Reduced order modeling is a very popular method to reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation or optimization. In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. We consider a global, derivative free optimization method as well as a local, gradient based approach for which the optimality system is derived in two different ways. The methods are compared with regard to the solution quality as well as the computational effort and they are illustrated using the example of the two-dimensional incompressible flow around a cylinder.

Lattice Boltzmann Simulation of Blood Flow Properties and Vessel Geometry in an Open and Closed Vessel: A Numerical Study

In the present article, Lattice Boltzmann method is utilized to simulate two-dimensional incompressible viscous flow in an open and closed microchannel (vessel). The main focus of the present research is to study physical parameters of blood flow in a vessel. To find the effect of oscillatory flow inside the vessel, we take account of the Reynolds number from 0.05 to 1.5 for numerical computation in the present manuscript in an open straight vessel. In addition, the accuracy of Poiseuille Law is investigated for blood flow in open vessel too. For this purpose, the effect of the vessel diameter and blood viscosity on the blood flow is studied numerically. As extra results, the effect of blood injection to a coronary bifurcation with two closed ends are studied. The blood pressure drop is high at the beginning of the vessel (pressure variation is high between the adjacent points along the vessel), but after the path along the vessel, the speed of dropping pressure decreases and the pressure difference between the adjoining points decreases along the vessel. Finally, the present results have been compared with the available experimental and numerical results that show good agreements.

Numerical simulation of coupled fluid flow and heat transfer with phase change using the Finite Pointset Method

The purpose of this work is to carry out a meshfree implementation for the numerical simulation of two-dimensional transient incompressible flows coupled with heat transfer where phase change is present. The Finite Pointset Method is applied in order to solve the involved partial differential equations where the corresponding classical or strong formulation is directly used instead of the corresponding weak form as needed for some other meshfree approaches. The incorporation of the boundary conditions is done in a direct and simple manner. The simplicity and efficiency of this numerical method are demonstrated on several two-dimensional benchmark problems where solidification and melting processes occurs.

Finite element analysis of magnetohydrodynamic flow over flat surface moving in parallel free stream with viscous dissipation and Joule heating

PurposeThe purpose of this paper is to investigate two-dimensional viscous incompressible magnetohydrodynamic boundary layer flow and heat transfer of an electrically conducting fluid over a continuous moving flat surface considering the viscous dissipation and Joule heating.Design/methodology/approachSuitable similarity variables are introduced to reduce the governing nonlinear boundary layer partial differential equations to ordinary differential equations. A numerical solution of the resulting two-point boundary value problem is carried out by using the finite element method with the help of Gauss elimination technique.FindingsA comparison of obtained results is made with the previous work under the limiting cases. Behavior of flow and thermal fields against various governing parameters like mass transfer parameter, moving flat surface parameter, magnetic parameter, Prandtl number and Eckert number are analyzed and demonstrated graphically. Moreover, shear stress and heat flux at the moving surface for various values of the physical parameters are presented numerically in tabular form and discussed in detail.Originality/valueThe work is relatively original, as very little work has been reported on magnetohydrodynamic flow and heat transfer over a continuous moving flat surface. Viscous dissipation and Joule heating are neglected in most of the previous studies. The numerical method applied to solve governing equations is finite element method which is new and efficient.

Immersed boundary-simplified lattice Boltzmann method for incompressible viscous flows

An immersed boundary-simplified lattice Boltzmann method is developed in this paper for simulations of two-dimensional incompressible viscous flows with immersed objects. Assisted by the fractional step technique, the problem is resolved in a predictor-corrector scheme. The predictor step solves the flow field without considering immersed objects, and the corrector step imposes the effect of immersed boundaries on the velocity field. Different from the previous immersed boundary-lattice Boltzmann method which adopts the standard lattice Boltzmann method (LBM) as the flow solver in the predictor step, a recently developed simplified lattice Boltzmann method (SLBM) is applied in the present method to evaluate intermediate flow variables. Compared to the standard LBM, SLBM requires lower virtual memories, facilitates the implementation of physical boundary conditions, and shows better numerical stability. The boundary condition-enforced immersed boundary method, which accurately ensures no-slip boundary conditions, is implemented as the boundary solver in the corrector step. Four typical numerical examples are presented to demonstrate the stability, the flexibility, and the accuracy of the present method.

Finite element computational procedure for convective flow of nanofluids in an annulus

In the present study, the detailed procedures of Galerkin weighted residual technique of finite element method (FEM) for solving two-dimensional incompressible natural convective flow of nanofluids using nonhomogeneous dynamic model are discussed for the first time. The physical domain is discretized by using unstructured triangular elements. The governing partial differential equations of nanofluids are made dimensionless using the suitable transformation of variables for weak formulations. The method of weighted residuals is used for obtaining the approximate solutions. This approach typically leads to a sparse and indefinite matrix that is difficult to solve efficiently. The formation of an indefinite matrix is avoided in the present work by introducing an artificial compressibility term in the continuity equation. Unequal order interpolation functions are used for pressure, velocity, temperature and concentration variables. The coefficient matrices are calculated using interpolation functions. Assembling of triangular elements in the discretized domain is discussed elaborately. The process of calculating boundary integrals is also discussed. The Newton-Raphson iteration technique along with Euler-backward scheme is used to solve the global matrix. The sample results are obtained for the convective flow of nanofluids in a concentric annulus. It shows that the annulus of having higher thickness is the best performer enhancing convective heat transfer rates.

Numerical Study on Heat Transfer to an Arc Absorber Designed for a Waste Heat Recovery System around a Cement Kiln

A numerical study on combined free convection, forced convection, and radiation heat transfers from an industrial isothermal rotating cylinder (cement kiln) is carried out in this work. The investigation is done by the study of two-dimensional (2D) incompressible turbulent flow around the kiln under steady- and unsteady-state solutions. The results of this study show that the average Reynolds and Rayleigh numbers around the cylindrical kiln are 647,812.1 and 1.75986 × 1011, respectively. A heat absorber is specifically designed around the kiln, according to the available space around the kiln, in a sample cement factory. The study investigates the effect of an added absorber on the heat transfer features, for both constant heat flux and constant temperature, on the kiln. The temperature distribution along the absorber circumference is obtained for designing an efficient thermoelectric waste heat recovery system as a future study. It is observed that the contribution of the radiative heat transfer is significant in the total heat transferred from the kiln to the absorber.

A mixed characteristic boundary condition for simulating viscous incompressible fluid flows around a hydrofoil

Smoothed particle hydrodynamics (SPH) is a meshfree method and has been widely used in the field of fluid dynamics. However, there is still a lack of robust approach to implement the far-field boundary condition in the simulation of free stream flow past the rigid body. To solve this problem, on the basis of Lastiwka et al.’s work, an improved non-reflecting boundary condition, called mixed characteristic boundary condition is proposed in this paper. The mixed characteristic boundary condition is tested by two-dimensional incompressible flows with a perturbation at the initial moment. The numerical results show that the mixed characteristic boundary condition can effectively suppress the reflection of waves on far-field boundaries. Finally, flows past a hydrofoil are simulated within the framework of the kernel gradient free SPH in two kinds of non-reflecting boundary condition. The results indicate that the present boundary condition is more robust and suitable for SPH than the one adopted by Lastiwka et al.

Meshfree numerical simulation of free surface thermal flows in mould filling processes using the Finite Pointset Method

The purpose of this work is to carry out a meshfree implementation for the numerical simulation of two-dimensional transient incompressible free surface flows coupled with heat transfer. The Finite Pointset Method is applied in order to solve the involved partial differential equations where the corresponding classical or strong formulation is directly used instead of the corresponding weak form as needed for some other meshfree approaches. The incorporation of the boundary conditions is done in a direct and simple manner. The simplicity and efficiency of this numerical method are demonstrated on two complex two-dimensional mould filling processes.

Development of an efficient gas kinetic scheme for simulation of two-dimensional incompressible thermal flows.

In this work, an efficient gas kinetic scheme is presented for simulation of two-dimensional incompressible thermal flows. In the scheme, the macroscopic governing equations for mass, momentum, and energy conservation are discretized by the finite volume method and the numerical fluxes at the cell interface are reconstructed by the local solution of the Boltzmann equation. To compute these fluxes, two distribution functions are involved. One is the circular function, which is used to calculate the numerical fluxes of mass and momentum equations. Due to the incompressible limit, the circle at the cell interface can be approximately considered to be symmetric so that the expressions for the conservative variables and numerical fluxes at the cell interface can be given explicitly and concisely. Another one is the D2Q4 model, which is utilized to compute the numerical flux of the energy equation. By following the process for derivation of numerical fluxes of mass and momentum equations, the numerical flux of the energy equation can also be given explicitly. The accuracy, efficiency, and stability of the present scheme are validated by simulating several thermal flow problems. Numerical results showed that the present scheme can provide accurate numerical results for incompressible thermal flows at a wide range of Rayleigh numbers with less computational cost than that needed by the thermal lattice Boltzmann flux solver (TLBFS), which has been proven to be more efficient than the thermal lattice Boltzmann method (TLBM).

Two-Dimensional Simulation of the Navier-Stokes Equations for Laminar and Turbulent Flow around a Heated Square Cylinder with Forced Convection

Few studies jointly investigate thermal and turbulent effects. In general, these subjects are treated separately. The purpose of this paper is to use the Immersed Boundary Method (IBM) coupled with the Virtual Physical Model (VPM) to investigate incompressible two-dimensional Newtonian flow around a heated square cylinder at constant temperature on its surface with forced convection and turbulence. The VPM model dynamically evaluates the force that the fluid exerts on the immersed surface and the thermal exchange between both in the Reynolds numbers (Re) window 40 ≤ Re ≤ 5×103 . For simulations of turbulence the Smagorinsky and Spalart-Allmaras models are used. The first model uses the Large Eddy Simulation (LES) methodology and is based on the local equilibrium hypothesis for small scales associated with the Boussinesq hypothesis, such that the energy injected into the spectrum of the turbulence balances the energy dissipated by convective effects. The second model uses the concept Unsteady Reynolds Averaged Navier-Stokes Equations (URANS), with only one transport equation for turbulent viscosity, being calibrated in pressure gradient layers. The goal of this work is to analyse the combination of the heat-transfer phenomena with the turbulence for the thermo-fluid-structure interaction in a square cylinder. For this, it was developed a C/C++ code that requires low computational costs in regards to memory and computer facilities. It is observed that, with the increase of the Reynolds number, an increase of the drag coefficient occurs, as well as reinforces the influence of the pressure distribution downstream of the cylinder, which is strongly influenced by the formation and detachment of vortices on the upper and lower sides of the square cylinder.

On some coupled PDE-ODE systems in fluid dynamics

In this note we will present some existence and uniqueness issues for three coupled PDE-ODE systems. The common frame is that they arise as the asymptotical dynamics of a regular, incompressible two-dimensional flow interacting with: • points at which the vorticity is highly concentrated (point vortices); • an obstacle shrinking to a steady point; • rigid bodies contracting to moving massive particles. We will mainly focus on the last situation, corresponding to the article [11], which is a joint work with Christophe Lacave.

Effect of turbulence on zonal jet flows in equivalent-barotropic quasi-geostrophic model

The linear stability of zonal flows superposed upon a sinusoidal, slightly supercritical background flow in an equivalent-barotropic quasi-geostrophic model on a \(\beta\) plane is investigated using asymptotic analysis. Consistent with results of full numerical simulations, all the steady isolated zonal jet solutions were found to be linearly unstable, and specifically, the mode independent of longitudinal direction was found to be most unstable, even when the amplitude equation of the zonal flows has longitudinal dependence. References K. Obuse, S. Takehiro and M. Yamada, Long-time asymptotic states of forced two-dimensional barotropic incompressible flows on a rotating sphere . Phys. Fluids 22 :156601, 2010. doi:10.1063/1.3407652 R. K. Scott and L. M. Polvani, Forced-dissipative shallow-water turbulence on the sphere and the atmospheric circulation of the giant planets . J. Atmos. Sci. 64 :3158–3176, 2007. doi:10.1175/JAS4003.1 A. J. Manfroi and W. R. Young, Slow evolution of zonal jets on the beta plane . J. Atmos. Sci. 56 :784–800, 1999. 2.0.CO;2>doi:10.1175/1520-0469(1999)056 2.0.CO;2 K. Obuse, S. Takehiro and M. Yamada, Linear stability of steady zonal jet flows induced by a small-scale forcing on a \(\beta \) plane . Physica D 240 :1825–1834, 2011. doi:10.1016/j.physd.2011.08.009 S. Takehiro, M. Odaka, K. Ishioka, M. Ishiwatari, Y.-Y. Hayashi and SPMODEL Development Group, A series of hierarchical spectral models for geophysical fluid dynamics . Nagare Multimedia, 2006. http://www.nagare.or.jp/mm/2006/spmodel/