Two-dimensional Time-dependent Flow Research Articles

Impact of thermal radiation on two-dimensional unsteady third-grade fluid flow over a permeable stretching Riga plate

In many fields, there are various applications of non-Newtonian fluids. Various complicated fluids (polymer melts, clay coatings and oil) belong to the category of non-Newtonian fluids. The third-grade fluid is one of the most important non-Newtonian fluid models. This paper has the primary object of heat transfer mechanism and boundary layer third-grade fluid flow under the effects of thermal radiation. The time-dependent two-dimensional flow is considered to flow above a permeable stretchable vertical Riga plate. For numerical solutions, the setup of ordinary differential equations (ODEs) is acquired by converting nonlinear governing equations through relevant similarity transformations. The nonlinear setup of ODEs is numerically solved with the aid of a suitable software such as MATLAB via its bvp4c technology. Graphs are sketched to discuss the various flow parameters’ significance for the expression of velocity and temperature fields. Tabulated values of surface drag force and heat transfer rate corresponding to the numerous pertinent parameters are described. The current analysis of the concerned flow mechanism concludes that the fluid parameters descend the temperature distribution but amplify the profile of the fluid velocity. The radiation parameter escalates the temperature field.

Heat and Mass Transfer in Unsteady Boundary Layer Flow of Williamson Nanofluids

In this paper, analytic approximation to the heat and mass transfer characteristics of a two-dimensional time-dependent flow of Williamson nanofluids over a permeable stretching sheet embedded in a porous medium has been presented by considering the effects of magnetic field, thermal radiation, and chemical reaction. The governing partial differential equations along with the boundary conditions were reduced to dimensionless forms by using suitable similarity transformation. The resulting system of ordinary differential equations with the corresponding boundary conditions was solved via the homotopy analysis method. The results of the study show that velocity, temperature, and concentration boundary layer thicknesses generally decrease as we move away from the surface of the stretching sheet and the Williamson parameter was found to retard the velocity but it enhances the temperature and concentration profiles near the surface. It was also found that increasing magnetic field strength, thermal radiation, or rate of chemical reaction speeds up the mass transfer but slows down the heat transfer rates in the boundary layer. The results of this study were compared with some previously published works under some restrictions, and they are found in excellent agreement.

Smart inertial particles

We performed a numerical study to train smart inertial particles to target specific flow regions with high vorticity through the use of reinforcement learning algorithms. The particles are able to actively change their size to modify their inertia and density. In short, using local measurements of the flow vorticity, the smart particle explores the interplay between its choices of size and its dynamical behaviour in the flow environment. This allows it to accumulate experience and learn approximately optimal strategies of how to modulate its size in order to reach the target high-vorticity regions. We consider flows with different complexities: a two-dimensional stationary Taylor-Green like configuration, a two-dimensional time-dependent flow, and finally a three-dimensional flow given by the stationary Arnold-Beltrami-Childress helical flow. We show that smart particles are able to learn how to reach extremely intense vortical structures in all the tackled cases.

Planar non-Newtonian confined laminar impinging jets: Hysteresis, linear stability, and periodic flow

This paper considers the linear stability of confined planar impinging jet flow of a non-Newtonian inelastic fluid. The rheology is shear rate dependent with asymptotic Newtonian behavior in the zero shear limit, and the analysis examines both shear thinning and shear thickening behavior. The planar configuration is such that the width of the inlet nozzle is smaller than the distance from the jet exit to the impinging surface, giving an aspect ratio e = 8 for which two-dimensional time dependent flow is readily manifest. For values of the power-law index n in the range 0.4≤n≤1.1, the bi-global linear stability of the laminar flow is analyzed for Newtonian Reynolds numbers Re ≲200. The calculations show that for certain values of n, including the Newtonian value n = 1, the steady flow exhibits multiplicity leading to hysteresis in the primary separation vortex reattachment point and a consequent jump in stability behavior. Even in the absence of hysteresis, relatively small changes in viscosity significantly affect stability characteristics. For Newtonian and mildly shear thinning or shear thickening fluids, an unstable flow shows a decaying perturbation growth rate as Re is increased, and for certain values of n, the flow may be restabilized at a larger Re before eventually becoming unstable again. This decay in the growth rate of the critical antisymmetric mode may be correlated as a function of the reattachment point RP of the primary separation vortex in the underlying steady flow. Representative results are analyzed in detail and discussed in the context of some experimental observations of time-dependent Newtonian impinging flow. The stability results are used to construct the neutral stability curve (n, Re) that displays multiplicity and contains several cusp points associated with flow restabilization and hysteresis. Integration of the full nonlinear equation reveals the structure of the time periodic flow field for both Newtonian and non-Newtonian fluids at Reynolds numbers well beyond the instability threshold.

Thermal Radiation Effects on Squeezing Flow Casson Fluid between Parallel Disks

In this paper, we investigate the thermal radiation effects in a time-dependent two-dimensional flow of a Casson fluid between two parallel disks when upper disk is taken to be impermeable and lower one is porous. Suitable similarity transforms are employed to convert governing partial differential equations into system of ordinary differential equations. Well known Homotopy Analysis Method (HAM) is employed to obtain the expressions for velocity and temperature profiles. Effects of different physical parameters such as squeeze number $S$, Prandtl number $Pr$, Eckert number $Ec$ and the dimensionless length on the flow are also discussed with the help of graphs for velocity and temperature coupled with a comprehensive discussions. The skin friction coefficient and local Nusselt number along with convergence of the series solutions obtained by HAM are presented in tabulated form, while numerical solution is obtained by $RK-4$ method and comparison shows an excellent agreement between both the solutions.

Nonlinear radiation effects on squeezing flow of a Casson fluid between parallel disks

Abstract In this paper, we investigate the nonlinear radiation effects in a time-dependent two-dimensional flow of a Casson fluid squeezed between two parallel disks when the upper disk is taken to be impermeable and the lower one is porous. Suitable similarity transforms are employed to convert governing partial differential equations into the system of ordinary differential equations. A well known Homotopy Analysis Method (HAM) is employed to obtain the expressions for velocity and temperature profiles. Effects of different physical parameters such as squeeze number S , Eckert number Ec and the dimensionless length on the flow when keeping Pr = 7 are also discussed with the help of graphs for velocity and temperature coupled with comprehensive discussions. Mathematica Package BVPh2.0 is utilized to formulate the total error of the system for both the suction and injection cases. The skin friction coefficient and local Nusselt number are presented with graphical aids for emerging parameters.

Refining finite-time Lyapunov exponent ridges and the challenges of classifying them.

While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model.

Topological chaos, braiding and bifurcation of almost-cyclic sets

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler et al., “Topological chaos and periodic braiding of almost-cyclic sets,” Phys. Rev. Lett. 106, 114101 (2011)]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or “ghost rods” around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes’ flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence, we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.

Simulation of micellar-polymeric water-flooding in a system of wells

A mathematical model of the time-dependent two-dimensional flow of a two-phase multicomponent incompressible fluid through a porous medium is proposed for the micellar-polymeric flooding of oil reservoirs. The oil displacement process is investigated numerically using an implicit first-order-accurate upwind scheme with integration over the nonlinearity on a uniform grid under the assumption of plane-radial motion in the neighborhood of the wells. The influence of the nonuniform permeability of the porous medium on the efficiency of the proposed method of improving oil recovery is analyzed using a five-point slug injection scheme.

Topology tracking for the visualization of time-dependent two-dimensional flows

The paper presents a topology-based visualization method for time-dependent two-dimensional vector elds. A time interpolation enables the accurate tracking of critical points and closed orbits as well as the detection and identication of structural changes. This completely characterizes the topology of the unsteady ow. Bifurcation theory provides the theoretical framework. The results are conveyed by surfaces that separate subvolumes of uniform ow behavior in a three-dimensional space-time domain.

Particle dynamics and mixing in the frequency driven "Kelvin cat eyes" flow.

The "Kelvin cat eyes" stream function is used as a simple fluid flow model to study particle dynamics, mixing and transport in a two-dimensional time-dependent flow field. Lagrangian formulation is used to describe the motion of small spherical particles present in the flow. Individual particle trajectories, under the influence of various flow parameters are studied. The equation describing the motion of these particles constitutes a set of first-order nonlinear differential equations describing a dynamical system. The time-dependent Eulerian flow field is studied as a nonintegrable Hamiltonian system in order to get insight into the underlying nonlinear properties of the system, which directly influence its complicated transport and mixing behavior. Chaotic advection (Lagrangian turbulence) was observed for heavy particles (high Stokes numbers) while no stochastic behavior was observed for light particles. The introduction of perturbation had only a limited effect on individual particle trajectories. However, the introduction of perturbation caused a shrinking of the phase space where bounded stochastic or quasi-periodic motion occurs. This phenomenon can lead to a better understanding of the link between the behavior of the underlying flow in the Hamiltonian formulation and the dynamics of the passive scalars in the Lagrangian description. The Eulerian flow field itself was found to behave chaotically under the influence of a periodic perturbation, because the stable and unstable manifolds associated with neighboring hyperbolic points intersected. This coincides with the better mixing of the fluid. Stochasticity was also discovered close to the periodic points of the system using Poincare maps. Mixing and transport properties are analyzed as a function of the perturbation frequency. (c) 2001 American Institute of Physics.

Exact Solutions for Two-Dimensional Time-Dependent Flow and Deformation Within a Poroelastic Medium

Exact analytic solutions are derived for the time-dependent deformation of a poroelastic medium within a two-dimensional finite domain. Solutions are given with a specific set of boundary conditions for the case of a source of fluid at an arbitrary point and for an applied pressure on the boundary. These solutions are ideal for testing numerical schemes for poroelastic flow and deformations due to their relative simplicity.

Instationary highly turbulent hydrogen tube combustion processes in obstacle areas

The interaction between combustion and turbulence in hydrogen-air mixtures in the early flame development phase is reviewed from the safety point of view. Detailed knowledge about the acceleration of hydrogen-air flames in real physical environments is needed to avoid deflagration-to-detonation transition (DDT). Depending on the initial and boundary conditions, the burning velocity of the same hydrogen-air mixture can vary by an order of magnitude. A time-dependent two-dimensional flow algorithm, using a direct numerical simulation method (DNS) and including large-scale eddy simulation in complex channel areas, is presented to calculate flame folding in the early burning phase after ignition. Complex mapping functions to realize different obstacle areas in tubes are presented. The effects of combustion on the structure of turbulence in the flow field, in front of the flame surface and effects of flow field instability on the flame development in tubes containing line or expanded obstacles are discussed.

Analysis of Aeroacoustic Source Distribution by Means of Numerical Simulation.

This study deals numerically with the aerodynamic sound radiated from the flow over a circular cylinder at a Reynolds number of l03. The standard acoustic analogy method is applied. The incompressible time-dependent two-dimensional flow around the cylinder is calculated by means of the finite difference method, then the variables are substituted into the approximate analytic solution of a nonhomogeneous wave equation. The result of far-field sound calculation using a solution which Howe expresses as a volume integral form is compared with that using another one which Curle expresses as a surface integral form. Good agreement is obtained between them. Howe's equation is used to investigate the sound source distribution in the flow field. The sign of the pressure fluctuation radiated from a small sound source region is opposite to those radiated from the neighboring ones. Most of the pressure fluctuations interfere and vanish as a consequence of volume integration. The remaider is observed as far-field sound. It is shown that there are some regions which contribute more significantly to the sound generation than the other regions on average in time.

A fast kinematic dynamo in two-dimensional time-dependent flows

A time-continuous, constant-resistivity version of the fast dynamo model introduced by Bayly & Childress (1988) is studied numerically. The expected dynamo mechanism in this context is described and is shown to be operative in the simulations. Exponential growth of the fastest growing mode is observed, with the growth rate for the smallest resistivity attempted (1/Rm = 10-4) agreeing well with the Bayly–Childress model. It is argued, based on the long- and short-wavelength behaviour of the mode for different resistivities, that the growth rates obtained for the Rm = 104 case should persist as Rm → ∞.

The Linear Shallow Water Theory: A Mathematical Justification

The authors provide in the space of square integrable measurable functions a mathematical justification for the “shallow water” theory for time-dependent two-dimensional flows of an inviscid, irrotational, incompressible fluid moving under the influence of gravity. A by-product of this investigation is a new derivation of the “shallow water” equations.

Bifurcations of numerically simulated thermocapillary flows in axially symmetric float zones

Two-dimensional time-dependent flow in an axially symmetric liquid bridge of Si at zero g with flat ends at the melting temperature θM of Si driven by motions of the free, lateral surface is studied numerically. The system is energized by a ring heater H at temperature θH>θM that surrounds the midcircle m of the bridge. The radiation from H creates a ∇σS(δS is surface temperature), which in turn creates a ∇σ (σ is surface tension) that drives oscillations of its free surface and thermocapillary flow in the bridge. For aspect ratios A of (1)/(2) , (2)/(3) , and 1 it is found that as the relevant control parameter Δθ≡θH−θM is increased, the flow in the bridge bifurcates differently depending on A. For A= (1)/(2) and (2)/(3) the flow bifurcates from steady state to small-amplitude periodic oscillations, breaking symmetry, to large-amplitude, period-tripled ones, to their subharmonics, then becomes aperiodic, and finally chaotic. Toroidal cells in the bridge oscillate to-and-fro along its axis as a compound pendulum, and they change in shape, number, and strength of rotation over the period of an oscillation. For A=1, the bifurcation is from steady state to asymmetric oscillations steadily growing in amplitude as Δθ increases and then directly to semiperiodic ones and thence to chaos. Agreement between the present results and recent experimental and numerical ones, are also described.

Potential flows in general relativity: Nonlinear and time-dependent solutions.

We present several new calculations of subsonic relativistic fluid flows in fixed background spacetimes. For irrotational, isentropic, perfect-fluid flows, the relativistic Euler and continuity equations can be formulated as a potential problem which is very accessible to numerical and analytic techniques. Schwarzschild and Kerr background spacetimes are considered with the inner boundary condition determining whether the gravitating source is a black hole or a hard sphere. For stationary flows with a polytropic equation of state $P=K{n}^{\ensuremath{\gamma}}$, the potential equation is nonlinear and elliptic. We compute several examples of stationary, axisymmetric flows about a hard sphere moving through an asymptotically homogeneous medium. For a $P=\ensuremath{\rho}$ (i.e., $\ensuremath{\gamma}=2$) equation of state the potential equation is linear and can be solved analytically for steady-state flows. We solve the hyperbolic system to calculate various time-dependent two-dimensional flows past a hard sphere. For hard spheres larger than a critical size $\frac{{R}_{\mathrm{lim}}}{M}$, there is an asymptotic velocity (of the sphere through the medium) above which the steady state cannot be achieved. We use the time-dependent code to study the behavior of the stationary and nonstationary cases. The code is also employed to examine the transition to steady-state accretion flow onto a black hole moving through a homogeneous medium. We also present a new, analytic, three-dimensional solution for flow past a rotating hard sphere. These flows represent excellent test problems for numerical relativity. Multidimensional fluid solutions may be used to benchmark codes as well as provide realistic numerical data for developing three-dimensional visualization methods. Because of the numerical tractability of the potential formulation, great precision can be achieved with modest computational resources.

On the numerical simulation of two-dimensional time-dependent flows of oldroyd fluids part 1: basic method and preliminary results

Abstract An investigation is carried out into numerical techniques to be used to solve time-dependent flow of Oldroyd liquids. A finite-difference method is derived which is stable for low elasticity but which must be modified by stress and vorticity smoothing in order to obtain solutions at higher values of elasticity. The Two-dimensional flow past a circular cylinder is used as a test problem. It is found that, at low elasticity, although the early development of the flow is quite different from the equivalent Newtonian situation, these differences diminish as time goes on and one is left with streamlines which differ only very slightly from the Newtonian flow field. As elasticity is increased, however, the initial departure from Newtonian streamlines persists and one sees develop areas of recirculating fluid near to the cylinder surface. The effective radius of the cylinder is thus increased and a much larger area of fluid is disturbed by the presence of the cylinder. These predictions are compared with available experimental evidence.

A numerical study of vortex shedding from rectangles

The purpose of this paper is to present numerical solutions for two-dimensional time-dependent flow about rectangles in infinite domains. The numerical method utilizes third-order upwind differencing for convection and a Leith type of temporal differencing. An attempted use of a lower-order scheme and its inadequacies are also described. The Reynolds-number regime investigated is from 100 to 2800. Other parameters that are varied are upstream velocity profile, angle of attack, and rectangle dimensions. The initiation and subsequent development of the vortex-shedding phenomenon is investigated. Passive marker particles provide an exceptional visualization of the evolution of the vortices both during and after they are shed. The properties of these vortices are found to be strongly dependent on Reynolds number, as are lift, drag, and Strouhal number. Computed Strouhal numbers compare well with those obtained from a wind-tunnel test for Reynolds numbers below 1000.