Unstable manifold computations for the two-dimensional plane Poiseuille flow

We follow the unstable manifold of periodic and quasi-periodic solutions in time for the Poiseuille problem, using two formulations: holding a constant flux or mean pressure gradient. By means of a numerical integrator of the Navier–Stokes equations, we let the fluid evolve from an initially perturbed unstable solution until the fluid reaches an attracting state. Thus, we detect several connections among different configurations of the flow such as laminar, periodic, quasi-periodic with two or three basic frequencies, and more complex sets that we have not been able to classify. These connections make possible the location of new families of solutions, usually hard to find by means of numerical continuation of curves, and show the richness of the dynamics of the Poiseuille flow. PACS 05.45.-a, 47.11.+j, 47.20.-k, 47.20.Ft     

Temperature distribution in a Newtonian fluid injected between two semi-infinite plates

The analysis of the temperature distribution in time and place of a hot heat-conducting Newtonian fluid injected between two cooled parallel plates is presented. The 2-dimensional flow has a free flow front moving with constant velocity. The kernel of the fluid remains almost at the inlet temperature, but at the walls boundary layers occur with steeply descending temperature. The inner solutions inside these boundary layers are determined. To this end, the total region is divided into three distinct regions: the region G_I far behind the flow front, the flow front region G_II, and the intermediate region G_III between G_I and G_II. The asymptotics owing to each region are presented. The fundamental small parameter here is the thickness-to-length ratio of the 2-dimensional flow region. In most of the cases, similarity solutions are found. In the flow front region, for the formulation of the inner solution a Wiener–Hopf technique is used. Via matching procedures, the separate boundary layers are linked to each other to form one global boundary layer for the whole front region. All calculations in this paper are performed by analytical means, and all results are in analytical form. Comparison of our results with numerical solutions shows perfect agreement.     

Double-diffusive convection for a heated cylinder submerged in a salt-stratified fluid layer

Double-diffusive convection due to a cylindrical source submerged in a salt-stratified solution is numerically investigated in this study. For proper simulation of the vortex generated around the cylinder, a computational domain with irregular shape is employed. Flow conditions depend strongly on the thermal Rayleigh number, Ra _T , and the buoyancy ratio, R _ρ. There are two types of onset of instability existing in the flow field. Both types are due to either the interaction of the upward temperature gradient and downward salinity gradient or the interaction of the lateral temperature gradient and downward salinity gradient. The onset of layer instability due to plume convection is due to the former, whereas, the onset of layer instability of layers around the cylinder is due to the latter. Both types can be found in the flow field. The transport mechanism of layers at the top of the basic plume belongs to former while that due to basic plume and layer around the cylinder are the latter. The increase in Ra _T reinforces the plume convection and reduces the layer numbers generated around the cylinder for the same buoyancy ratio. For the same Ra _T , the increase of R _ρ suppresses the plume convection but reinforces the layers generated around the cylinder. The profiles of local Nusselt number reflects the heat transfer characteristics of plume convection and layered structure. The profiles of averaged Nusselt number are between the pure conduction and natural convection modes and the variation is due to the evolution of layers. Received on 13 September 1996     

Bubble effect on Kelvin-Helmholtz instability

We derive boundary conditions at interfaces (contact discontinuities) for a class of Lagrangian models describing, in particular, bubbly flows. We use these conditions to study the Kelvin-Helmholtz instability that develops in the flow of two superposed layers of a pure incompressible fluid and a fluid containing gas bubbles, co-flowing with different velocities. We show that the presence of bubbles in one layer stabilizes the flow in some intervals of wavelengths.Received: 8 October 2002, Accepted: 5 May 2003PACS: 47.20.Ma, 47.55.Dz, 47.55.Kf Correspondence to: S.L. Gavrilyuk     

Complex dynamics in double-diffusive convection

The dynamics of a small Prandtl number binary mixture in a laterally heated cavity is studied numerically. By combining temporal integration, steady state solving and linear stability analysis of the full PDE equations, we have been able to locate and characterize a codimension-three degenerate Takens–Bogdanov point whose unfolding describes the dynamics of the system for a certain range of Rayleigh numbers and separation ratios near S=-1. PACS 44.25.+f, 47.20.Bp, 05.45.-a     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

Boiling on a straight pin fin with variable thermal conductivity

This work theoretically investigated the thermal performance and stability characteristics of a straight pin fin subject to boiling considering a temperature-dependent thermal conductivity of fin, k=k _sat(1+b(T−T _sat)). Steady-state temperature distribution and the associated fin base heat flow were for the first time analytically found, whose stability characteristics were evaluated by linear stability analysis. A positive temperature coefficient b will raise both the fin's temperature and base heat flow. The corresponding stability for stable fin boiling was enhanced. A negative b results in an opposite trend. The use of a mean thermal conductivity in fin boiling calculations is discussed. Received on 3 November 1997     

Wall Effect on the Convective–Absolute Boundary for the Compressible Shear Layer

The linear stability of inviscid compressible shear layers is studied. When the layer develops at the vicinity of a wall, the two parallel flows can have a velocity of the same sign or of opposite signs. This situation is examined in order to obtain first hints on the stability of separated flows in the compressible regime. The shear layer is described by a hyperbolic tangent profile for the velocity component and the Crocco relation for the temperature profile. Gravity effects and the superficial tension are neglected. By examining the temporal growth rate at the saddle point in the wave-number space, the flow is characterized as being either absolutely unstable or convectively unstable. This study principally shows the effect of the wall on the convective–absolute transition in compressible shear flow. Results are presented, showing the amount of the backflow necessary to have this type of transition for a range of primary flow Mach numbers M ₁ up to 3.0. The boundary of the convective–absolute transition is defined as a function of the velocity ratio, the temperature ratio and the Mach number. Unstable solutions are calculated for both streamwise and oblique disturbances in the shear layer. Received 9 May 2001 and accepted 21 August 2001     

From global to local bifurcations in a forced Taylor–Couette flow

The unfolding due to imperfections of a gluing bifurcation occurring in a periodically forced Taylor–Couette system is analyzed numerically. In the absence of imperfections, a temporal glide-reflection Z₂ symmetry exists, and two global bifurcations occur within a small region of parameter space: a heteroclinic bifurcation between two saddle two-tori and a gluing bifurcation of three-tori. As the imperfection parameter increase, these two global bifurcations collide, and all the global bifurcations become local (fold and Hopf bifurcations). This severely restricts the range of validity of the theoretical picture in the neighborhood of the gluing bifurcation considered, and has significant implications for the interpretation of experimental results. PACS 47.20.Ky, 47.20.Lz, 47.20.Ft     

Predicting the Collapse of Turbulence in Stably Stratified Boundary Layers

The collapse of turbulence in a plane channel flow is studied, as a simple analogy of stably stratified atmospheric flow. Turbulence is parameterized by first-order closure and the surface heat flux is prescribed, together with the wind speed and temperature at the model top. To study the collapse phenomenon both numerical simulations and linear stability analysis are used. The stability analysis is nonclassical in a sense that the stability of a parameterized set of equations of a turbulent flow is analyzed instead of a particular laminar flow solution. The analytical theory predicts a collapse of turbulence when a certain critical value of the stability parameter δ/L (typically O(0.5–1)) is exceeded, with δ the depth of the channel and L the Obukhov length. The exact critical value depends on channel roughness to depth ratio z ₀/δ. The analytical predictions are validated by the numerical simulations, and good agreement is found. As such, for the flow configuration considered, the present framework provides both a tool and a physical explanation for the collapse phenomenon.     

Nonlinear convection of a viscoelastic fluid with inclined temperature gradient

The energy method is used to analyze the viscoelastic fluid convection problem in a thin horizontal layer, subjected to an applied inclined temperature gradient. The boundaries are considered to be rigid and perfectly conducting. Both linear and nonlinear stability analyses are carried out. The eigenvalue problem is solved by the Chebyshev Tau-QZ method and comparisons are reported between the results of the linear theory and energy stability theory.Received: 12 March 2004, Accepted: 19 April 2004, Published online: 17 September 2004PACS: 47.20 Ky, 47-27 Te, 83.60 Wc Correspondence to: P.N. Kaloni     

Quasi-periodicity and chaos in a differentially heated cavity

Convective flows of a small Prandtl number fluid contained in a two-dimensional vertical cavity subject to a lateral thermal gradient are studied numerically. The chosen geometry and the values of the material parameters are relevant to semiconductor crystal growth experiments in the horizontal configuration of the Bridgman method. For increasing Rayleigh numbers we find a transition from a steady flow to periodic solutions through a supercritical Hopf bifurcation that maintains the centro-symmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf bifurcation, the periodic solution loses stability in a subcritical Neimark–Sacker bifurcation, which gives rise to a branch of quasiperiodic states. In this branch, several intervals of frequency locking have been identified. Inside the resonance horns the stable limit cycles lose and gain stability via some typical scenarios in the bifurcation of periodic solutions. After a complicated bifurcation diagram of the stable limit cycle of the 1:10 resonance horn, a soft transition to chaos is obtained. PACS 44.25.+f, 47.20.Ky, 47.52.+j     

Effect of Low Rotation Rate on Steady Convection During the Solidification of a Ternary Alloy

We consider the problem of steady convective flow during the directional solidification of a horizontal ternary alloy system rotating at a constant and low rate about a vertical axis. Under the limit of large far-field temperature, the flow region is modeled to be composed of two horizontal mushy layers, which are referred to here as a primary layer over a secondary layer. We first determine the basic state solution and then carry out linear stability analysis to calculate the neutral stability boundary and the critical conditions at the onset of motion. We find, in particular, that there are two flow solutions and each solution exhibits two neutral stability boundaries, and the flow can be multi-modal in the low rotating rate case with local minima on each neutral boundary. The critical Rayleigh number and the wave number as well as the vertical volume flux increase with the rotation rate, but the flow is found to be less stabilizing as compared to the binary alloy counterpart flow. The effects of low rotation rate increase the solid fraction and the liquid fraction at certain vertically oriented fluid lines, and the highest value of such increase is at a horizontal level close to the interface between the two mushy layers.     

Surfactant systems for drag reduction: Physico-chemical properties and rheological behaviour

The first part of the work presents an overview of the physical chemistry of surfactants which in aqueous solutions reduce the frictional loss in turbulent pipe flow. It is shown that these surfactants form rodlike micelles above a characteristic concentraionc _t. The experimental evidence for rodlike micelles are reviewed and the prerequisites that the surfactant system must fulfill in order to form rodlike micelles are given. It is demonstrated by electrical conductivity measurements that the critical concentration for the formation of spherical micelles shows little temperature dependence, whereasc _t increases very rapidly with temperature. The length of the rodlike micelles, as determined by electric birefringence, decreases with rising temperature and increases with rising surfactant concentration. The dynamic processes in these micellar systems at rest and the influence of additives such as electrolytes and short chain alcohols are discussed.In the second part, the rheological behaviour of these surfactant solutions under laminar and turbulent flow conditions are investigated. Viscosity measurements in laminar pipe and Couette flow show the build-up of a shear induced viscoelastic state, SIS, from normal Newtonian fluid flow. A complete alignment of the rodlike micelles in the flow direction in the SIS was verified by flow birefringence. In turbulent pipe flow, drag reduction occurs in these surfactant systems as soon as rodlike micelles are present in the solution. The extent and type of drag reduction, i.e. the shape of the friction factor versus Reynolds number curve, depends directly on the size, number and surface charge of the rodlike micelles. The friction factor curve of each surfactant investigated changes in the same characteristic way as a function of temperature. For each surfactant, independent of concentration, an upper absolute temperature limit,T _L, for drag reduction exists which is caused by the micellar dynamics.T _L is influenced by the hydrophobic chain length and the counter-ion of the surfactant system. A first attempt is made to explain the drag reduction of surfactants by combining the results of these rheological measurements with the physico-chemical properties of the micellar systems.     

Effect of high frequency vibration on convection initiation

In [1] equations are derived which describe convection in a gravity force field and a criterion is introduced which determines the onset of convection.In the present study we consider the case when, in addition to the gravity forces, there are vibrational forces acting on a liquid enclosed in a vessel. These forces arise as a result of vibration of the vessel along the vertical axis. In order to study the effect of vibrations we use the method of averaging with respect to small vibrations [2,3]. The unknowns are sought in the form of the sum of two terms, one which varies slowly with time and one of small amplitude which varies rapidly.Averaged convsction equations are formulated (§1).We consider the case of a plane horizontal strip and, under the assumption of satisfaction of the stability exchange principle, we introduce two parameters which define the onset of convection. One of these parameters is already known [1]—the product of the Grashof and Prandtl numbers. The second, arising as a result of the action of the vibrational forces, is apparently introduced here for the first time. The conclusions concerning the effect of vibrations on the initiation of convection (§3) are made for a model problem, on the assumption of spatially periodic disturbances (without accounting for the actual boundary conditions). In this case the vibrations stabilize the relative state of rest, and we can choose the velocity so that stability will exist for all temperature gradients A [see (3.7)].It is found that in the presence of vibration (even small) the relative state of rest is stable with high temperature gradients. Moreover, if it is known that for given values of the vibration velocitya=a ₀and the temperature gradient A=A₀ the relative state of rest is stable, then we can, starting from the valuesa=0, A=0, by simultaneous variation ofa and A reach the indicated values (a ₀, A₀) while remaining in the region of stability. (We note that, generally speaking, the relative state of rest cannot be stabilized by simply increasing the gradient, since in the course of the increase the instability zone may be entered.) These conclusions are clearly valid only for vibrations with sufficiently high frequency which permit use of the averaging method. Justification of the method is not presented here.The study of the model problem gives an idea of the qualitative picture of the phenomenon. We would expect that the qualitative conclusions drawn are also valid under actual boundary conditions. The authors hope to carry out these calculations in the near future and present a justification for the proposed method.The authors wish to thank V. I. Yudovich for helpful advice and continued interest in this study.     

Velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium

In this paper the velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium are obtained for the governing equations (Kaviany [7]) following the technique adopted by Chandrashekara [2] which are concerned with the interesting situations of the existence of transverse, velocity and thermal boundary layers. Here the pressure gradient is just balanced by the first and second order solid matrix resistances for small permeability and observed that by increasing of the flow resistance the asymptotic value for the heat transfer rate increases. Further we concluded that the transverse boundary layers are thicker than that of axial boundary layers. Hence we evaluated the expressions for the boundary layer thickness, the shear stress at the semi-infinite plate and _T (the ratio of the thicknesses of the thermal boundary layer and momentum boundary layer). The variations of these quantities for different values of the porous parameterB and the flow resistanceF have been discussed in detail with the help of tables. The curves for velocity and temperature distributions have been plotted for different values ofB andF.Lastly we have evaluated the heat fluxq(x) and found that it depends entirely upon the Reynolds numberRe, Prandtl numberPr,B andF.     

Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents

The linear stability of a steady state solution of 2D Euler equations of an ideal fluid is being studied. We give an explicit geometric construction of approximate eigenfunctions for the linearized Euler operator L in vorticity form acting on Sobolev spaces on two dimensional torus. We show that each nonzero Lyapunov–Oseledets exponent for the flow induced by the steady state contributes a vertical line to the essential spectrum of L. Also, we compute the spectral and growth bounds for the group generated by L via the maximal Lyapunov–Oseledets exponent. When the flow has arbitrarily long orbits, we show that the essential spectrum of L on L₂ is the imaginary axis.     

Self-sustained oscillations between two tandem cylinders at Reynolds number 1,000

This study focuses on the self-sustained oscillatory flow characteristics between two tandem circular cylinders of equal diameter placed in a uniform inflow. The Reynolds number (Re _D ), based on the cylinder diameter, was around 1,000 and all experiments were performed in a recirculating water channel. The streamwise distance between two tandem cylinders ranged within 1.5 ≤ X _c/D ≤ 7.0. Here X _c denotes the center-to-center distance between two tandem cylinders. For all experiments studied herein, quantitative velocity measurements were performed using hot-film anemometer and the LDV system. The laser sheet technique was employed for qualitative flow visualization. The wavelet transform was applied to elucidate the temporal variation and phase difference between two spectral components of the velocity signals detected in the flow field. The remarkable finding was that when two tandem circular cylinders were spaced at a distance within 4.5 ≤ X _c/D ≤ 5.5, two symmetrical unstable shear layers with a certain wavelength were observed to impinge onto the downstream cylinder. The responding frequency (f _u ), measured between these two cylinders, was much higher than the natural shedding frequency behind a single isolated cylinder at the same Re _D . This responding frequency decreased as the distance X _c/D increased. Not until X _c/D ≥ 6.0, did it recover to the natural shedding frequency behind a single isolated cylinder. Between two tandem cylinders, the Strouhal numbers (St _c = f _u X _c/U_c) maintained a nearly constant value of 3, indicating the self-sustained oscillating flow characteristics with a wavelength X _c/3. Here U _c is the convection speed of the unstable shear layers between two tandem cylinders. At Re _D = 1,000, the self-sustained oscillating characteristics between two tandem circular cylinders were proven to exhibit a sustained flow pattern, not just a sporadic phenomenon.     

Convective-Absolute Transition of Non-Isothermal Compressible Shear Layer

The linear stability of inviscid compressible shear layers is studied. When the layer develops at the vicinity of a wall, the two parallel flows can have velocity of the same sign or of opposite sign. This situation is examined in order to obtain first hints on the stability of separated flows in the compressible regime. The shear layer is described by an hyperbolic tangent profile for the velocity component and the Crocco relation for the temperature profile. The gravity effects and the superficial tension are neglected. By examining the temporal growth rate at the saddle point in the wave number space, the flow is characterized as being either absolutely unstable or convectively unstable. This study principally shows the non-isothermal effect on the absolute-convective transition in compressible shear flow. Results are presented, showing the amount of the backflow necessary to have this type of transition for a range of primary flow Mach number M ₁ up to 3.0. The boundary of the absolute-convective transition is defined as a function of the velocity ratio, the temperature ratio and the Machnumber.     

Miscible Thermo-Viscous Fingering Instability in Porous Media. Part 2: Numerical Simulations

The development of the thermo-viscous fingering instability of miscible displacements in homogeneous porous media is examined. In this first part of the study dealing with stability analysis, the basic equations and the parameters governing the problem in a rectilinear geometry are developed. An exponential dependence of viscosity on temperature and concentration is represented by two parameters, thermal mobility ratio β _T and a solutal mobility ratio β _C , respectively. Other parameters involved are the Lewis number Le and a thermal-lag coefficient λ. The governing equations are linearized and solved to obtain instability characteristics using either a quasi-steady-state approximation (QSSA) or initial value calculations (IVC). Exact analytical solutions are also obtained for very weakly diffusing systems. Using the QSSA approach, it was found that an increase in thermal mobility ratio β _T is seen to enhance the instability for fixed β _C , Le and λ. For fixed β _C and β _T , a decrease in the thermal-lag coefficient and/or an increase in the Lewis number always decrease the instability. Moreover, strong thermal diffusion at large Le as well as enhanced redistribution of heat between the solid and fluid phases at small λ is seen to alleviate the destabilizing effects of positive β _T . Consequently, the instability gets strictly dominated by the solutal front. The linear stability analysis using IVC approach leads to conclusions similar to the QSSA approach except for the case of large Le and unity λ flow where the instability is seen to get even less pronounced than in the case of a reference isothermal flow of the same β _C , but β _T  = 0. At practically, small value of λ, however, the instability ultimately approaches that due to β _C only.