Nonlinear instability of developing streamwise vortices with applications to boundary layer heat transfer intensification through an extended Reynolds analogy

The intent of the present contribution is to explain theoretically the experimentally measured surface heat transfer rates on a slightly concave surface with a thin boundary layer in an otherwise laminar flow. As the flow develops downstream, the measured heat transfer rate deviates from the local laminar value and eventually exceeds the local turbulent value in a non-trivial manner even in the absence of turbulence. While the theory for steady strong nonlinear development of streamwise vortices can bridge the heat transfer from laminar to the local turbulent value, further intensification is attributable to the transport effects of instability of the basic steady streamwise vortex system. The problem of heat transport by steady and fluctuating nonlinear secondary instability is formulated. An extended Reynolds analogy for Prandtl number unity, Pr=1, is developed, showing the similarity between streamwise velocity and the temperature. The role played by the fluctuation-induced heat flux is similar to momentum flux by the Reynolds shear stress. Inferences from the momentum problem indicate that the intensified heat flux developing well beyond the local turbulent value is attributed to the transport effects of the nonlinear secondary instability, which leads to the formation of 'coherent structures' of the flow. The basic underlying pinions of the non-linear hydrodynamic stability problem are the analyses of J. T. Stuart, which uncovered physical mechanisms of nonlinearities that are crucial to the present developing boundary layers supporting streamwise vortices and their efficient scalar transporting mechanisms.     

Axial drive to nonlinear flow between rotating cylinders

Stability of pseudoplastic rotational flow between cylinders in presence of an independent axial component is investigated. The fluid is assumed to follow the Carreau model and mixed boundary conditions are imposed. The conservation of mass and momentum equations give rise to a four-dimensional low-order dynamical system, including additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. In absence of the axial flow, as the pseudoplasticity effects increases, the purely-azimuthal base flow loses its stability to the vortex structure at a lower critical Taylor number. Emergence of the vortices corresponds to the onset of a supercritical bifurcation also present in the flow of a linear fluid. However, unlike the Newtonian case, pseudoplastic Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Existence of an axial flow induced by a pressure gradient appears to further advance each critical point on the bifurcation diagram. In continuation, complete flow field together with viscosity maps is analyzed for different flow scenarios. Through evaluation of the Lyapunov exponent, flow stability and temporal behavior of the system for cases with and without axial flow are brought to attention.     

Aspects of linear and nonlinear instabilities leading to transition in pipe and channel flows

The failure of normal-mode linear stability analysis to predict a transition Reynolds number (Retr) in pipe flow and subcritical transition in plane Poiseuille flow (PPF) has led to the search of other scenarios to explain transition to turbulence in both flows. In this work, various results associated with linear and nonlinear mechanisms of both flows are presented. The results that combine analytical and experimental approaches indicate the strong link between the mechanisms governing the transition of both flows. It is demonstrated that the linear transient growth mechanism is based on the existence of a pair of least stable nearly parallel modes (having opposite phases and almost identical amplitude distributions). The analysis that has been applied previously to pipe flow is extended here to a fully developed channel flow predicting the shape of the optimized initial disturbance (a pair of counter-rotating vortices, CVP), time for maximum energy amplification and the dependence of the latter on Re. The results agree with previous predictions based on many modes. Furthermore, the shape of the optimized initial disturbance is similar in both flows and has been visualized experimentally. The analysis reveals that in pipe flow, the transient growth is a consequence of two opposite running modes decaying with an equal decay rate whereas in PPF it is due to two stationary modes decaying with different decay rates. In the first nonlinear scenario, the breakdown of the CVPs (produced by the linear transient growth mechanism) into hairpin vortices is followed experimentally. The associated scaling laws, relating the minimal disturbance amplitude required for the initiation of hairpins and the Re, are found experimentally for both PPF and pipe flow. The scaling law associated with PPF agrees well with the previous predictions of Chapman, whereas the scaling of the pipe flow is the same as the one previously obtained by Hof et al. indicating transition to a turbulent state. In the second nonlinear scenario, the base flow of pipe when it is mildly deviated from the Poiseuille profile by an axisymmetric distortion is examined. The nonlinear features reveal a Retr of approximately 2000 associated with the bifurcation between two deviation solutions.     

Onset of Taylor-Görtler instability in accelerated circular Couette flow

Summary The onset of Taylor-G?rtler vortices in a developing Couette flow induced by the inner cylinder rotating with time-dependent manner is analyzed using linear theory. It is well known that there is a critical Taylor number Ta _c at which Taylor vortices set in between two concentric cylinders. For Ta > Ta _c Taylor-G?rtler vortices are detected experimentally at a certain elapsed time. In the present study the critical time t _c to represent the onset of a fastest growing instability, which then grows as toroidal vortices, is analyzed using the propagation theory. Available experimental data indicate that for large Ta secondary motion is detected starting from a certain time t _m ≈ 4t _c. This means that the growth period of initiated instabilities is needed for secondary motion to be detected experimentally. The new measures to represent the onset of a fastest growing instability in the primary time-dependent Couette flow are suggested.     

Behaviour of spiral vortices on a rotating cone in axial flow

Summary The purpose of the present paper is to investigate experimentally in detail the boundary layer transition process and the behaviour of spiral vortices appearing in the transition range of the boundary layer on a 30°-cone, rotating in axial flow. Counterrotating spiral vortices in the transition range are visualized with a white smoke method, and observed the time dependent behaviour of them using a drum camera and a light sheet illumination method with a stroboscope flash light. The light passes a slit in order to illuminate only a thin sheet in the flow. With this method, the time dependent growing up and breaking down process of these spiral vortices is greatly clarified. A hot wire anemometer is also used for measuring in the flow field quantitatively. The results show that the spiral vortices are generated in the thin region of the steep shear velocity gradients near the wall. As the vortices grow up in z-direction, they are strongly distorted by the mean velocity field there, and finally they are teared off.With 7 Figures     

On the development of Görtler vortices in wall jet flow

The development of Görtler vortices in wall jet flow over curved surfaces is considered in both the linear and nonlinear growth régimes. It is shown, using asymptotic methods based on the largeness of the wavenumber of the vortices, that this hydrodynamic instability is prone to occur more readily on concave rather than convex surfaces. It is found that after passing the position of neutral stability, the flow develops a surprising structure quite unlike that produced in the Blasius boundary-layer. Once the flow is into the unstable regime, the effect of increasing the Görtler number is to move the vortices away from the wall.     

Linear stability of non-Newtonian annular liquid sheets

Summary This paper reports a linear stability analysis of a non-Newtonian annular liquid sheet that is surrounded by nonviscous fluids in relative axial motion to it. It is shown that for a stress free basic flow the dispersion relation giving the absolute and convective instability mechanisms can be immediately obtained from the dispersion relation for a Newtonian sheet by introducing a wavenumber dependent viscosity. The stability behavior of the sheet is investigated numerically by a continuation algorithm, by which the solution branches of the dispersion relation, relevant for the stability information, can be traced. The results give a stability picture which covers the whole range of annular sheets from the cylindrical jet to the plane liquid curtain.     

Some indications from instability results about the effectiveness of wall heating as a control option for channel flow

This paper is a review of some of our recent work on the effect of wall heating on the stability of laminar flow in a channel. The summary of our results, some of them unexpected, is as follows. Viscosity stratification has very little effect on transient growth, whereas it results in linear mode stabilising or destabilising by an order of magnitude. It has hitherto been accepted that heat diffusivity does not affect stability. This is however true only for linear instability, transient growth is affected by an order of magnitude. Unusually, the growth is spanwise-independent and not in the form of streamwise vortices. It is also shown that flow is destabilised by secondary modes as the viscosity ratio increases. However, the viscosity ratio has no role in the selection of the pattern of Λ vortices.     

Vortex motion in the early stages of unsteady flow around a circular cylinder

In this paper, processes in the early stages of vortex motion and the development of flow structure behind an impulsively-started circular cylinder at high Reynolds number are investigated by combining the discrete vortex model with boundary layer theory, considering the separation of incoming flow boundary layer and rear shear layer in the recirculating flow region. The development of flow structure and vortex motion, particularly the formation and development of secondary vortex and a pair of secondary vortices and their effect on the flow field are calculated. The results clearly show that the flow structure and vortices motion went through a series of complicated processes before the symmetric main vortices change into asymmetric: development of main vortices induces secondary vortices; growth of the secondary vortices causes the main vortex sheets to break off and causes the symmetric main vortices to become “free” vortices, while a pair of secondary vortices is formed; then the vortex sheets, after breaking off, gradually extend downstream and the structure of a pair of secondary vortices becomes relaxed. These features of vortex motion look very much like the observed features in some available flow field visualizations. The action of the secondary vortices causes the main vortex sheets to break off and converts the main vortices into free vortices. This should be the immediate cause leading to the instability of the motion of the symmetric main vortices. The flow field structure such as the separation position of boundary layer and rear shear layer, the unsteady pressure distributions and the drag coefficient are calculated. Comparison with other results or experiments is also made. This work was presented at the First Asian Congress of Fluid Mechanics, Bangalore in December 1980.     

Phase Slips and Josephson Weak Links in Superfluid Helium

When superfluid ⁴He flows through a submicron aperture, the velocity is limited by a critical value which marks the onset of quantized vortex creation. The evolution of the vortices causes the quantum phase across the aperture to change by 2π, leading to a detectable drop in flow energy. Recent studies of these phase slip events have provided new insights into the nucleation mechanisms for quantum vortices. By contrast, superfluid ³He passing through a submicron aperture exhibits nonlinear hydrodynamics, characterized by a Josephson-like current phase relation. Recent experiments have revealed a multitude of effects analogous to phenomena observed in superconductors. The experiments also reveal unexpected effects such as bistability, π-states, and novel dissipation mechanisms. PACS numbers: 67.57.-z, 74.50.+r, 67.40.Hf, 67.40. Vs.     

Direct resonance analysis and modeling for a turbulent boundary layer over a corrugated surface

Summary Effects of surface corrugation on turbulent flow in a boundary layer are studied using a model based on the direct resonance theory. The induced mean flow due to weakly nonlinear waves, superimposed on the mean and fluctuating components of turbulence, is determined. The mean turbulent flow is affected by the surface corrugation throughout the boundary layer. The corrugated surface generates higher harmonics and affects the streamwise vortices generated by the waves superimposed on turbulence whose mean flow includes secondary induced mean flow components due to the corrugation.     

Flow visualization studies of vortices

An actual vortex in the Kármán vortex street downstream of a circular cylinder has a core of finite dimension which increases downstream. The circulation of the vortex is nearly constant. The ratiob/a which is 0.281 according to the theory of Kármán, grows from 0.2 to 0.4 in the near wake. In the flow about a circular cylinder rotating in a uniform flow, a Kármán vortex street, Görtler-type vortices and Taylor vortices are generated at the same time. In the flow about a circular cylinder impulsively started with a constant velocity, the primary twin vortices behind the cylinder induce secondary twin vortices near the separation point. At the beginning of the motion, the separation does not occur even though a reverse flow is observed in the boundary layer. Mutual slip-through of a pair of vortex rings was achieved by increasing the Reynolds number. A vortex ring rebounds from a plane surface due to the separation of the flow on the surface induced by the vortex ring, and the secondary vortex ring is formed from the separated shear layer.     

Numerical prediction of eddy structure in a shear-driven cavity

We present in this paper a detailed numerical study of the vortical flow structure in a confined lid-driven cavity which is defined by a depth-to-width aspect ratio of 1:1 and a span-to-width aspect ratio of 3:1. In this study we have carefully examined the computed data that the useful to gain an in-depth knowledge of the complex interactions among secondary eddies, primary eddies, and spiraling spanwise motions. Chief of conclusions drawn from this study is to explain how the secondary eddies are intimately coupled with the primary recirculating flow. We also enlighten in this paper why spiraling vortices inside the upstream secondary eddy tend to destabilize the incompressible flow system and aid development of laminar instabilities.     

Study on boundary layer transition of a rotating disk

Summary Behaviour of spiral vortices being generated in transition regime of a disk rotating in otherwise undisturbed fluid is experimentally studied in detail. Through visualizations of the transition regime by using close-up camera, new striped flow patterns originating along the axis of spital vortices are found to be ring-like vortices which occur on the surfaces of each spiral vortices. Mechanism of the spiral vortex is clarified by cutting the vortices by strobo slit light. It is also found out experimentally that the phase velocity of the vortices is zero.     

Generalized K entropy and Gibbs entropy of a system of point vortices

A generalization of the Kolmogorov-Sinay entropy (K entropy) is proposed and applied to a polygonal system of point vortices. It is demonstrated that the generalized K entropy (GK entropy) not only characterizes the local instability and mixing in a dynamical system, but also describes its structural rearrangements, in particular, a transition of the vortex system from one stationary state to another according to some scenario of vortex combination. The Gibbs entropy is constructed, its relation to the GK entropy is established, and it is demonstrated that both describe the transition of a dynamical system to equilibrium.     

Shear-flow Instability in Two-component Bose-Einstein Condensates

We theoretically study Kelvin-Helmholtz instability in phase-separated two-component Bose-Einstein condensates with shear flow at T=0. The stability of the shear flow state is investigated with the Gross-Pitaevskii and the Bogoliubov-de Gennes models, compared with a hydrodynamic model. The dynamics of the instability is revealed by numerically solving the Gross-Pitaevskii equation. In the nonlinear development, singly-quantized vortices are released from the interface between the two condensates.     

Magnetic field contribution to the Lorentz model

The classical Lorentz model of dielectric dispersion is based on the microscopic Lorentz force relation and Newton's second law of motion for an ensemble of harmonically bound electrons. The magnetic field contribution in the Lorentz force relation is neglected because it is typically small in comparison with the electric field contribution. Inclusion of this term leads to a microscopic polarization density that contains both perpendicular and parallel components relative to the plane wave propagation vector. The modified parallel and perpendicular polarizabilities are both nonlinear in the local electric field strength.     

Experimental study of the development of a helium jet during acoustic action

A helium jet subjected to acoustic action is experimentally studied by the particle image velocimetry method. The mechanisms of flow stability loss are discussed. An asymmetric instability mode is shown to develop in the jet subjected to the action of sound, which leads to jet division. Jet vibrations are found to be accompanied by a secondary instability, which manifests itself in the formation of vortices inside the jet.     

Effect of fibrils on curvature- and rotation-induced hydrodynamic stability

Flow of a suspension of water and nano-fibrillated cellulose (NFC) in a curved and rotating channel is studied experimentally and theoretically. The aim is to investigate how NFC affects the stability of the flow. This flow is subject to a centrifugal instability creating counter-rotating vortices in the flow direction. These rolls can be both stabilised and destabilised by system rotation, depending on direction and velocity of the rotation. Flow visualisation images with pure water and an NFC/water suspension are categorised, and stability maps are constructed. A linear stability analysis is performed, and the effect of fibrils is taken into account assuming straight fibrils and constant orientation distributions, i.e., without time-dependent flow-orientation coupling. The results show that NFC has a less stabilising effect on the primary flow instability than indicated from the increase in viscosity measured by a rotary viscometer, but more than predicted from the linear stability analysis. Several unknown parameters (the most prominent being fibril aspect ratio and the interaction parameter in the rotary diffusion) appear in the analysis.     

CMC对石墨-H_2O分散液稳定性的影响

选用羧甲基纤维素钠(CMC)作为分散剂,制备了稳定性较好的石墨-H2O分散液。通过测定石墨-H2O分散液的吸光度和Zeta电位,探讨了不同CMC添加量、pH值、球磨时间对石墨-H2O分散液稳定性的影响,并分析了作用机理。结果表明:以石墨质量为基准,CMC用量大于4.5%时,分散液稳定性较好;该体系中,主要通过静电斥力使石墨粒子保持稳定,Zeta电位绝对值与吸光度有良好的对应关系,Zeta电位绝对值越高,吸光度越大,分散液稳定性越好;pH值对分散液的稳定性影响很大,pH=9～10时,分散液稳定性最好;随着球磨时间的增大,石墨粒径越来越小,球磨180min时,得到平均粒径为2.284μm的石墨-H2O分散液;石墨-H2O分散液的粒径分布及粘度曲线表明,加入CMC使石墨粒子在水中达到良好的分散。