Two-dimensional Periodic Flow Research Articles

Stability of vortex flow in a modulated channel

Linear stability analysis of a fully developed two-dimensional periodic steady flow in a spatially modulated symmetric channel is investigated numerically. Both walls are sinusoidally modulated. The modulation amplitude is assumed to be small. The base steady flow is calculated using a regular perturbation expansion of the flow field coupled to a variable-step finite-difference scheme. The disturbance flow equations are derived within the framework of Floquet theory and solved using a two-point boundary value method. The implemented procedure is simple, direct, and very efficient. The accuracy of this method is established. The influence of geometric parameters on the threshold for the onset of instability is systematically investigated. It is established that the critical Reynolds number for the onset of hydrodynamic instability decreases as the wall modulation amplitude or wave number increases. Modulated channel flow is characterized by the formation of a recirculation region or vortices beneath the modulation crest as the Reynolds number exceeds a certain critical threshold. The relation between the thresholds for the onset of instability and vortex flow is examined over a wide range of geometric parameters. Vortex flow effects on the conditions for the onset of instability are systematically investigated. The parameter regime of stable vortex flow is established. It is found that the parameter regime of stable vortex flow increases significantly with the wall modulation wave number for small wave numbers. This increase become less pronounced as the wave number is further increased.

Equation-free/Galerkin-free POD-assisted computation of incompressible flows

We present a Galerkin-free, proper orthogonal decomposition (POD)-assisted computational methodology for numerical simulations of the long-term dynamics of the incompressible Navier–Stokes equations. The approach is based on the “equation-free” framework: we use short, appropriate initialized bursts of full direct numerical simulations (DNS) of the Navier–Stokes equations to observe, estimate, and accelerate, through “projective integration”, the evolution of the flow dynamics. The main assumption is that the long-term dynamics of the flow lie on a low-dimensional, attracting, and invariant manifold, which can be parametrized, not necessarily spanned, by a few POD basis functions. We start with a discussion of the consistency and accuracy of the approach, and then illustrate it through numerical examples: two-dimensional periodic and quasi-periodic flows past a circular cylinder. We demonstrate that the approach can successfully resolve complex flow dynamics at a reduced computational cost and that it can capture the long-term asymptotic state of the flow in cases where traditional Galerkin-POD models fail. The approach trades the overhead involved in developing POD-Galerkin and POD-nonlinear Galerkin codes, for the repeated (yet short, and on demand) use of an existing full DNS simulator. Moreover, since in this approach the POD modes are used to observe rather than span the true system dynamics, the computation is much less sensitive than POD-Galerkin to values of the system parameters (e.g., the Reynolds number) and the particular simulation data ensemble used to obtain the POD basis functions.

Long-wave convection in a two-layer system

Periodic regimes of large-scale convection in a two-layer system with a deformable interface are considered. In the limiting case of strong surface tension, an amplitude equation describing the interface deformation is derived. Two-dimensional periodic flow regimes are studied. It is shown that, depending on the governing parameters, both hard and soft excitation of convection is possible. In the presence of heating from above, the finite-amplitude excitation of convection was observed.

Nonlinear evolution of unstable fluid interface.

We study the coherent motion of bubbles and spikes in the Richtmyer-Meshkov instability for isotropic three-dimensional and two-dimensional periodic flows. For equations governing the local dynamics of the bubble, we find a family of regular asymptotic solutions parametrized by the principal curvature at the bubble top. The physically significant solution in this family corresponds to a bubble with a flattened surface, not to a bubble with a finite curvature. The evolution of the bubble front is described and the diagnostic parameters are suggested.

Transition in Convective Flows Heated Internally

The stability of internally heated convective flows in a vertical channel under the influence of a pressure gradient and in the limit of small Prandtl number is examined numerically. In each of the cases studied the basic flow, which can have two inflection points, loses stability at the critical point identified by the corresponding linear analysis to two-dimensional states in a Hopf bifurcation. These marginal points determine the linear stability curve that identifies the minimum Grashof number (based on the strength of the homogeneous heat source), at which the two-dimensional periodic flow can bifurcate. The range of stability of the finite amplitude secondary flow is determined by its (linear) stability against three-dimensional infinitesimal disturbances. By first examining the behavior of the eigenvalues as functions of the Floquet parameters in the streamwise and spanwise directions we show that the secondary flow loses stability also in a Hopf bifurcation as the Grashof number increases, indicating that the tertiary flow is quasi-periodic. Secondly the Eckhaus marginal stability curve, that bounds the domain of stable transverse vortices towards smaller and larger wavenumbers, but does not cause a transition as the Grashof number increases, is also given for the cases studied in this work.

The change in settling velocity of inertial particles in cellular flow

The gravitational settling of small, spherical inertial particles in a simple, two-dimensional periodic cellular flow is investigated. The effects of adding motions of many length scales and unsteadiness are considered. It is found that in a single length scale steady cellular flow, with square drag law for the equation of motion, when the still fluid settling velocity of the particle V T <0.5 the average settling velocity of particles v 2 will be larger than V T by 120%. This occurs because the particles tend to concentrate in downward flow regions. As V T increases the particles are more uniformly sampling the flow field and hence this uneven distribution of particles is reduced. This uneven distribution of particles is also reduced by adding motion of many length scales and unsteadiness to the flow. In addition, it is found that particles can move in a seemingly chaotic manner without settling into a regular motion even in a steady cellular flow.

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude ε is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger ε, two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.

Penetrative convection in the isothermally incompressible fluid approximation

The onset of penetrative convection in an infinite horizontal fluid layer bounded by isothermal rigid or free nondeformable surfaces is numerically examined. It is assumed that the specific volume of the fluid depends quadratically on temperature and reaches a minimum inside the layer. The isothermally incompressible fluid convection model in which, as distinct from the Oberbeck-Boussinesq approximation, the thermal expansion is not assumed to be small is considered. Both the neutral stability curves of the conductive regime and the amplitudes of two-dimensional periodic and three-dimensional doubly-periodic convective flow are calculated. The results are compared with those previously obtained for the equations of penetrative convection in the Boussinesq approximation.

Numerical simulations of mixing in a two-dimensional periodic flow with buoyancy

Numerical simulations of unsteady mixed convection involving interface tracking between two immiscible fluids have been carried out, using spectral methods. A periodic oscillating flow is created by combining forced and natural convection with spatially periodic heating. The fluid motion that results is found to contain unsteady vortices in the flow which cause the stretching and folding that is required for efficient mixing processes. The interface between the two fluids is found to grow exponentially when the flow is unsteady, and the individual particles move with chaotically varying velocities. The interface growth rate is found to increase with increasing Prandtl number.

On the instability of wave-catalysed longitudinal vortices in strong shear

The inviscid instability of O(ε) two-dimensional periodic flows to spanwise-periodic longitudinal vortex modes in parallel O(1) shear flows is considered. In such cases, not only is the effect of fluctuations upon the mean state important but also the influence of the developing mean flow on the fluctuating part of the motion. The former is described by a generalized Lagrangian-mean formulation; the latter by a modified Rayleigh equation. Of specific interest is whether the spanwise distortion of the wave field feeds back to enhance or inhibit instability to longitudinal vortex form. Two cases are considered in detail: uniform shear between wavy walls and non-uniform shear beneath free-surface waves. In both cases wave distortion acts to inhibit, and in some circumstances curtail, instability for all but the shortest waves.

Some aspects of the freely propagating premixed flame in a spatially periodic flow field

The premixed flame situated in a spatially periodic flow field is examined using the passive propagation model with the local flame speed affected by stretch and nonequidiffusion. Numerical solution shows that the average flame speed increases with either increasing fluctuation amplitude or increasing wavelength of the imposed flow field, and that the flame surface can locally extinguish for sufficiently large fluctuation amplitude of the imposed flow. Perturbation solutions in the weakly wrinkled flame and the thin flame limits are presented. The formation of corners on the flame surface in the thin flame limit is illustrated, and the structure of the corner is further found to resemble that of the Bunsen flame. The premixed flame situated in a two-dimensional periodic flow field is also analyzed in the Huygens limit, leading to the observation that flame surface discontinuities exist in the form of cones.

Separatrix-map analysis of chaotic transport in planar periodic vortical flows.

We have studied chaotic transport in a two-dimensional periodic vortical flow under a time-dependent perturbation with period T, where the global diffusion occurs along the stochastic web. By using the Melnikov method we construct the separatrix map describing the approximate dynamics near the saddle separatrices. Focusing on small T, the width of the stochastic layer is calculated analytically using the residue criterion and the diffusion constant is computed using the random phase assumption and the theory of correlated random walks. The analytical results are in good agreement with the results of two different types of numerical simulations: integrations of Hamilton's equation of motion and iterations of the separatrix map, which establishes the validity of the use of the separatrix map.

Instability of spatially periodic flow to three-dimensional disturbances

The instability of two-dimensional periodic flows to three-dimensional disturbances is investigated. The problem is formulated for general periodic flows, and the eigenvalue problem is solved numerically for the rhombic cell flow with aspect ratio k = 0.5 and 1.0. The result of numerical calculation shows that the three-dimensional disturbance does not cause a change in the values of the critical Reynolds number Rc. Some decrease in damping rate is found for the three-dimensional disturbances of some set of wavenumbers, but it is not sufficient to change the value of Rc.

Stochastic diffusion in two-dimensional periodic flows

The global effective diffusion coefficient D* for a two-dimensional system of convective rolls with a time-dependent perturbation added is calculated. The perturbation produces a background diffusion coefficient D, which is calculated analytically using the Menlikov–Arnold integral. This intrinsic diffusion coefficient is then enhanced by the unperturbed flow to produce the global effective diffusion coefficient D*, which can be calculated theoretically for a certain range of parameters. The theoretical value agrees well with numerical simulations.

Mixing in laminar flows by chaotic advection

It is pointed out that mixing of clouds of fluid particles can take place by pure advection in laminar flows. This is illustrated in a simple model of flow consisting of the superposition of steady flow and a free oscillation mode in a shallow layer of water in a rectangular basin. Here the generation of regions of chaotic advection of fluid particles occurs via homoclinic bifurcation of the stable and unstable manifolds of an equilibrium point. Similar behaviours are found in plane reversible maps that model the essential geometric properties of the Poincare map associated with the set of ordinary differential equations describing advection in two-dimensional periodic flows with reflection symmetry and discretized time periodicity.

The energy spectrum in the universal range of two-dimensional turbulence

Direct numerical simulation of two-dimensional Navier-Stokes equations at large Reynolds numbers is made by the spectral method with 13642 modes starting from a high-symmetric random initial velocity field. Two wavenumber ranges, governed by different similarity laws are observed after the enstrophy dissipation rate η(t) takes the maximum value. At small wavenumbers the energy spectrum is stationary in time, while at larger wavenumbers it decays according to the similarity law predicted by the enstrophy cascade theory, and the shape of the energy spectrum E(k,t) is expressed by E(k,t) = Aη(t)1/6v3/2(k/kd)-3exp[ –√2A(k/kd)], where k is the wavenumber, t the time, v the kinematic viscosity of fluid, kd = η(t)1/6v1/2 the dissipation wavenumber, and A ≈ 1.6. Concerning the enstrophy dissipation rate the following properties are observed: (i) As the Reynolds number R increases, the time of maximum enstrophy dissipation rate is delayed, probably in proportion to ln R. (ii) It approaches finite positive values in the inviscid limit if the above-mentioned time-lag is taken into account, (iii) It decays inversely proportionally to the cubic of time, so that the enstrophy is expressed as a sum of a constant term and a term which decays inversely proportionally to the square of time. This paper discusses why power laws of the energy spectrum observed in most of previously reported direct numerical simulations of two-dimensional periodic flows were steeper than k-3.

Onset and development of turbulence in two-dimensional periodic shear flows

We investigate numerically the time evolution of a two-dimensional flow submitted to a spatially periodic shear force. Initially, the flow is at equilibrium, the forcing balancing viscous stresses. At Reynolds numbers slightly above critical, a large-scale, linear instability drives the fluid towards a stable laminar state. At larger Reynolds number turbulence finally develops after several transient states. These transient states are described by measuring the divergence rate of linearized trajectories from the turbulent flow. This rate gives asymptotically a measure of the first Lyapunov exponent of the flow. We find that the first Lyapunov exponent scales as the characteristic frequency of the flow at large scale. We show here data on incompressible, isothermal and perfect gas (subsonic) two-dimensional flows with unit Prandtl number, and Reynolds number around 30.

Computation of the dimension of two-dimensional turbulence.

The Lyapunov dimension of an incompressible, homogeneous, two-dimensional periodic shear flow with moderate resolution (from ${16}^{\mathrm{*}}$16 up to ${128}^{\mathrm{*}}$128 points) is calculated. The results show that the dimension of the chaotic attractor is much lower than the total number of degrees of freedom represented in the numerical system, and depends essentially on the scale at which energy is injected in the system, and not (or very weakly) on the Reynolds number, suggesting that only a fraction of the modes at a scale larger than that of the forced mode is turbulent.

The instability of rhombic cell flows

Instability of two-dimensional periodic flows with rhombic cell structure represented by the stream function ψ=cos kx+cos y is investigated. Stability characteristics are obtained for the Reynolds number R=1, 2, 3 and 4 and the ratio of the diagonals of the cell k=1, ½ and ¼. Variation of the critical Reynolds number Rc with k is obtained, and the square cell flow (k=1) is found to be most stable (Rc=√2). It is found that Rc → 1 as k → 0, which leads to a finite gap between this limiting Rc and Rc=√2 for K=0 (ψ=cos y).

Wave-induced longitudinal-vortex instability in shear flows

The instability of two-dimensional periodic flows to spanwise-periodic ‘longitudinal- vortex’ modes is examined. The undisturbed state comprises a parallel shear flow and a two-dimensional O (e) wave field as encountered in, say, water-wave or hydrodynamic-stability theories. When the mean shear is weak, of order e 2 , the present theory reduces to that of Craik (1977) and Leibovich (1977 b , 1980). For stronger but still weak shear, of order e, it is established that the Craik-Leibovich instability mechanism is essentially unchanged, apart from scaling factors. For strong O (1) shear flows, the governing equations are derived by using, in part, a generalized Lagrangian-mean formulation. The resultant eigenvalue problem for the longitudinal-vortex instability is then more complex, but simplifies in the case of small spanwise spacing of the vortices, in the inviscid limit. An example is given of flows that exhibit instability in this limiting case. Such instability seems likely to occur for a wide class of periodic shear flows. Complementary physical interpretations of the instability mechanism are discussed.