Professor Isaac Goldhirsch, the Raquel and Manuel Klachky Chair of Rheological Flows at the School of Mechanical Engineering of Tel-Aviv University, Israel, died unexpectedly on April 29 at age 60 while on sabbatical leave at the University of Erlangen–Nuremberg, Germany.

This paper presents a mathematical theory for premixed combustion under the influence of enthalpy fluctuations in the oncoming fresh mixture. On the basis of the assumptions of large activation energy and small Mach number, an analysis of the thermal, hydrodynamic and acoustic regions of a flame is performed to derive an interactive system that describes, on the first-principles basis, the intricate coupling between the flame and its spontaneously emitted acoustic waves. The system, in its general form, is strongly nonlinear and requires a numerical attack. In this paper, it is employed to analyse several fundamental physical processes in relatively simple cases in order to provide useful insights into the role of enthalpy fluctuations in combustion. First, the linear response of the flame to two- or three-dimensional small-amplitude enthalpy fluctuations is considered, and they are found to generate hydrodynamic motion. Secondly, enthalpy fluctuations are shown to radiate sound waves through their interaction with the flame. Thirdly, enthalpy fluctuations and the sound waves emitted by them modify the flame stability, and the analysis shows that a moderate level of enthalpy fluctuation may cause a strong subharmonic parametric instability. Finally, in the small-heat-release limit, an extended Michelson–Sivashinsky equation is derived to describe the nonlinear evolution of the flame under the influence of both the imposed enthalpy fluctuations and the induced acoustic waves. Numerical solutions suggest that the flame evolves into a time-periodic state and acquires a curved profile, which primarily vibrates in the longitudinal direction, while its overall shape remains almost unaltered.

Current macroscopic theories of two-phase flow in porous media are based on the extended Darcy's law and an algebraic relationship between capillary pressure and saturation. Both of these equations have been challenged in recent years, primarily based on theoretical works using a thermodynamic approach, which have led to new governing equations for two-phase flow in porous media. In these equations, new terms appear related to the fluid–fluid interfacial area and non-equilibrium capillarity effects. Although there has been a growing number of experimental works aimed at investigating the new equations, a full study of their significance has been difficult as some quantities are hard to measure and experiments are costly and time-consuming. In this regard, pore-scale computational tools can play a valuable role. In this paper, we develop a new dynamic pore-network simulator for two-phase flow in porous media, called DYPOSIT. Using this tool, we investigate macroscopic relationships among average capillary pressure, average phase pressures, saturation and specific interfacial area. We provide evidence that at macroscale, average capillary pressure–saturation–interfacial area points fall on a single surface regardless of flow conditions and fluid properties. We demonstrate that the traditional capillary pressure–saturation relationship is not valid under dynamic conditions, as predicted by the theory. Instead, one has to employ the non-equilibrium capillary theory, according to which the fluids pressure difference is a function of the time rate of saturation change. We study the behaviour of non-equilibrium capillarity coefficient, specific interfacial area, and its production rate versus saturation and viscosity ratio.

A major feature of our pore-network model is a new computational algorithm, which considers capillary diffusion. Pressure field is calculated for each fluid separately, and saturation is computed in a semi-implicit way. This provides more numerical stability, compared with previous models, especially for unfavourable viscosity ratios and small capillary number values.

When a droplet of liquid 1 falls through liquid 2 to eventually hit the liquid 2–liquid 1 interface, its initial impact on the interface can produce daughter droplets of liquid 1. In some cases, a partial coalescence cascade governed by self-similar capillary-inertial dynamics is observed, where the fall of the secondary droplets in turn continues to produce further daughter droplets. Results show that inertia and interfacial surface tension forces largely govern the process of partial coalescence. The partial coalescence is suppressed by the viscous force when Ohnesorge number is below a critical value and also by gravity force when Bond number exceeds a critical value. Generation of secondary drop is observed for systems of lower Ohnesorge number for liquid 1, lower and intermediate Ohnesorge number for liquid 2 and for low and intermediate values of Bond number. Whenever the horizontal momentum in the liquid column is more than the vertical momentum, secondary drop is formed. A transition regime from partial to complete coalescence is obtained when the neck radius oscillates twice. In this regime, the main body of the column can be fitted to power-law scaling model within a specific time range. We investigated the conditions and the outcome of these coalescence events based on numerical simulations using a coupled level set and volume of fluid method (CLSVOF).

A cation-selective conducting particle is suspended in an electrolyte solution and is exposed to a uniformly applied electric field. The electrokinetic transport processes are described in a closed mathematical model, consisting of differential equations, representing the physical transport in the electrolyte, and boundary conditions, representing the physicochemical conditions on the particle boundary and at large distances away from it. Solving this mathematical problem would in principle provide the electrokinetic flow about the particle and its concomitant velocity relative to the otherwise quiescent fluid.

Using matched asymptotic expansions, this problem is analysed in the thin-Debye-layer limit. A macroscale description is extracted, whereby effective boundary conditions represent appropriate asymptotic matching with the Debye-scale fields. This description significantly differs from that corresponding to a chemically inert particle. Thus, ion selectivity on the particle surface results in a macroscale salt concentration polarization, whereby the electric potential is rendered non-harmonic. Moreover, the uniform Dirichlet condition governing this potential on the particle surface is transformed into a non-uniform Dirichlet condition on the macroscale particle boundary. The Dukhin–Derjaguin slip formula still holds, but with a non-uniform zeta potential that depends, through the cation-exchange kinetics, upon the salt concentration and electric field distributions. For weak fields, an approximate solution is obtained as a perturbation to a reference state. The linearized solution corresponds to a uniform zeta potential; it predicts a particle velocity which is proportional to the applied field. The associated electrokinetic flow is driven by two different agents, electric field and salinity gradients, which are of comparable magnitude. Accordingly, this flow differs significantly from that occurring in electrophoresis of chemically inert particles.

Turbulent plumes, which are seen in a wide number of industrial and natural flows, have been extensively studied; however, very little attention has been paid to plumes which have an internal mechanism for changing buoyancy. Such plumes arise in e.g. industrial chimneys, where species can react and change the density of the plume material. These plumes with chemical reaction are the focus of this study. An integral model describing the behaviour of a plume undergoing a second-order chemical reaction between a component in the plume (A) and a component in the surrounding fluid (B), which alters the buoyancy flux, is considered. The behaviour of a reactive plume is shown to depend on four dimensionless groups: the volume and momentum fluxes at the source, the parameter ϵ which indicates the additional buoyancy flux generated by the reaction and γ which is a dimensionless rate of depletion of species B. Additionally, approximate analytical solutions are sought for a reactive plume rising from a point source of buoyancy when species B is in great excess. These analytical results show excellent agreement with numerical simulations. It is also shown that the behaviour of a reactive plume in the far field is equivalent to an inert plume issuing from a virtual source downstream of the real source, and the dependence of the location of the virtual source on ϵ and γ is discussed. The effects of varying the volume flux at the source and the Morton source parameter Γ0 are further investigated by solving the full governing equations numerically. These solutions indicate that ϵ is important in determining the buoyancy generated by the reaction, and the length scale over which this reaction occurs depends on γ when γ > 1. It is also shown that when the dimensionless buoyancy ϵ < − 1, the reaction can cause the plume to collapse.

The heat transport and corresponding changes in the large-scale circulation (LSC) in turbulent Rayleigh–Bénard convection are studied by means of three-dimensional direct numerical simulations as a function of the aspect ratio Γ of a closed cylindrical cell and the Rayleigh number Ra. The Prandtl number is Pr = 0.7 throughout the study. The aspect ratio Γ is varied between 0.5 and 12 for a Rayleigh number range between 107 and 109. The Nusselt number Nu is the dimensionless measure of the global turbulent heat transfer. For small and moderate aspect ratios, the global heat transfer law Nu = A × Raβ shows a power law dependence of both fit coefficients A and β on the aspect ratio. A minimum of Nu(Γ) is found at Γ ≈ 2.5 and Γ ≈ 2.25 for Ra = 107 and Ra = 108, respectively. This is the point where the LSC undergoes a transition from a single-roll to a double-roll pattern. With increasing aspect ratio, we detect complex multi-roll LSC configurations in the convection cell. For larger aspect ratios Γ ≳ 8, our data indicate that the heat transfer becomes independent of the aspect ratio of the cylindrical cell. The aspect ratio dependence of the turbulent heat transfer for small and moderate Γ is in line with a varying amount of energy contained in the LSC, as quantified by the Karhunen–Loève or proper orthogonal decomposition (POD) analysis of the turbulent convection field. The POD analysis is conducted here by the snapshot method for at least 100 independent realizations of the turbulent fields. The primary POD mode, which replicates the time-averaged LSC patterns, transports about 50% of the global heat for Γ ≥ 1. The snapshot analysis enables a systematic disentanglement of the contributions of POD modes to the global turbulent heat transfer. Although the smallest scale – the Kolmogorov scale ηK – and the largest scale – the cell height H – are widely separated in a turbulent flow field, the LSC patterns in fully turbulent fields exhibit strikingly similar texture to those in the weakly nonlinear regime right above the onset of convection. Pentagonal or hexagonal circulation cells are observed preferentially if the aspect ratio is sufficiently large (Γ ≳ 8).

We present a new spectral method for the direct numerical simulation of magnetohydrodynamic turbulence at low magnetic Reynolds number. The originality of our approach is that instead of using traditional bases of functions, it relies on the basis of eigenmodes of the dissipation operator, which represents viscous and Joule dissipation. We apply this idea to the simple case of a periodic domain in the three directions of space, with a homogeneous magnetic field in the ez direction. The basis is then still a subset of the Fourier space, but ordered by growing linear decay rate |λ| (i.e. according to the least dissipative modes). We show that because the lines of constant energy tend to follow those of constant |λ| in the Fourier space, the scaling for the smallest scales |λmax| in a forced flow can be expressed, using this single parameter, as a function of the Reynolds number as , where kf is the forcing wavelength, or as a function of the Grashof number Gf, which gives a non-dimensional measure of the forcing, as |λmax|1/2/(2πkf) ≃ 0.47Gf0.20. This scaling is also found to be consistent with heuristic scalings derived by Alemany et al. (J. Mec., vol. 18, 1979, pp. 277–313) and Pothérat & Alboussière (Phys. Fluids, vol. 15, 2003, pp. 3170–3180) for interaction parameter S ≳ 1, and which we are able to numerically quantify as k⊥max/kf ≃ 0.5Re1/2 and kzmax/kf ≃ 0.8kfRe/Ha. Finally, we show that the set of least dissipative modes gives a relevant prediction for the scale of the first three-dimensional structure to appear in a forced, initially two-dimensional turbulent flow. This completes our numerical demonstration that the least dissipative modes can be used to simulate both two- and three-dimensional low-Rm magnetohydrodynamic (MHD) flows.

The understanding of the vortex formation process is currently driving a novel attempt to evaluate the performance of fluid dynamics in biological systems. The concept of formation time, developed for axially symmetric orifices, is here studied in two-dimensional flows for the generation of vortex pairs. The early stage of the formation process is studied with the single vortex model in the inviscid limit. Within this framework, the equation can be written in a universal form in terms of the formation time. The single vortex model properly represents the initial circular spiralling vortex sheet and its acceleration for self-induced motion. Then, an analysis is performed by numerical simulation of the two-dimensional Navier–Stokes equations to cope with the spatially extended vortex structure. The results do not show the pinch-off phenomenon previously reported for vortex rings. The two-dimensional vortex pair tends to a stably growing structure such that, while it translates and extends longitudinally, it remains connected to the sharp edge by a shear layer whose velocity is always about twice that of the leading vortex. At larger values of the Reynolds number the instability of the shear layer develops small-scale vortices capable of destabilizing the coherent vortex growth. The absence of a critical formation number for two-dimensional vortex pairs suggests further considerations for the development of concepts of optimal vortex formation from orifices with variable curvature or of a tapered shape.

An experimental study of energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model to simulate the dissipation due to wave breaking are reported in this paper. Measured wave surface elevations are used to examine the characteristic time and length scales associated with wave groups and local breaking waves, and to estimate and parameterize the energy dissipation and dissipation rate due to wave breaking. Numerical tests using the eddy viscosity model are performed and we find that the numerical results well capture the measured energy loss. In our experiments, three sets of characteristic time and length scales are defined and obtained: global scales associated with the wave groups, local scales immediately prior to breaking onset and post-breaking scales. Correlations among these time and length scales are demonstrated. In addition, for our wave groups, wave breaking onset predictions using the global and local wave steepnesses are found based on experimental results. Breaking time and breaking horizontal length scales are determined with high-speed imaging, and are found to depend approximately linearly on the local wave steepness. The two scales are then used to determine the energy dissipation rate, which is the ratio of the energy loss to the breaking time scale. Our experimental results show that the local wave steepness is highly correlated with the measured dissipation rate, indicating that the local wave steepness may serve as a good wave-breaking-strength indicator. To simulate the energy dissipation due to wave breaking, a simple eddy viscosity model is proposed and validated with our experimental measurements. Under the small viscosity assumption, the leading-order viscous effect is incorporated into the free-surface boundary conditions. Then, the kinematic viscosity is replaced with an eddy viscosity to account for energy loss. The breaking time and length scales, which depend weakly on wave breaking strength, are applied to evaluate the magnitude of the eddy viscosity using dimensional analysis. The estimated eddy viscosity is of the order of 10−3 m2s−1 and demonstrates a strong dependence on wave breaking strength. Numerical simulations with the eddy viscosity estimation are performed to compare to the experimental results. Good agreement as regards energy dissipation due to wave breaking and surface profiles after wave breaking is achieved, which illustrates that the simple eddy viscosity model functions effectively.

Experimental investigation of the generation and decay of coherent structures, namely, streaks (accompanied by a counter-rotating vortex pair) and hairpin vortices in pipe flow, is carried out by artificial injection of continuous disturbances. Flow visualization and velocity measurements show that for small amplitudes of disturbances (v0) streaks are produced, and increasing v0 produces instability waves on the streaks, which further break down into an array of hairpin vortices. However, the streaks and hairpins decay along the downstream direction (X). In fact, the critical value of v0 required for the initiation of hairpins at a given Re (Reynolds number) varies with the streamwise distance (in contrast to the previously found scaling of v0 ~ Re−1, valid only close to the location of injection, i.e. smaller X). This is a consequence of the decay of the coherent structures in the pipe. Moreover, the hairpins have been found to decay more slowly with increasing Re. Measurements of energy in the cross-sectional plane of the pipe, and maps of disturbance velocity at various X-locations show the transient growth and decay of energy for relatively low v0. For higher v0 and Re the energy has been seen to increase continuously along the length of the pipe under observation. Owing to the increase in the cross-sectional area occupied by the disturbance along the X-direction, it is observed that energy can transiently increase even when the total disturbance magnitude is decreasing. Observing the similarity of the present work and other investigations wherein decay of turbulence in pipe flow is found, a schematic illustration of the transition surface for pipe flow on a v0−Re−X, three-dimensional coordinate system is presented.

The topology of the composite flow fields reconstructed by linear superposition of a two-dimensional boundary layer flow with an embedded laminar separation bubble and its leading three-dimensional global eigenmodes has been studied. According to critical point theory, the basic flow is structurally unstable; it is shown that in the presence of three-dimensional disturbances the degenerate basic flow topology is replaced by a fully three-dimensional pattern, regardless of the amplitude of the superposed linear perturbations. Attention has been focused on the leading stationary eigenmode of the laminar separation bubble discovered by Theofilis et al. (Phil. Trans. R. Soc. Lond. A, vol. 358, 2000, pp. 3229–3324); the composite flow fields have been fully characterized with respect to the generation and evolution of their critical points. The stationary global mode is shown to give rise to a three-dimensional flow field which is equivalent to the classical U-shaped separation, defined by Hornung & Perry (Z. Flugwiss. Weltraumforsch., vol. 8, 1984, pp. 77–87), and induces topologies on the surface streamlines that are resemblant to the characteristic stall cells observed experimentally.

Numerical simulations of fully turbulent weak fountain flow are used to provide direct evidence for the scaling behaviour of fountain flow over the Froude number range Fr = 0.1–2.1 and Reynolds number range Re = 20–3494. For very weak flow at Fr < 0.4, the flow mean penetration height, Zm, scales with Zm/R0 = A1Fr2/3 + A2Fr2/3 where R0 is the source radius. A1 and A2 are constants which quantify the separate effects of the radial acceleration of fountain fluid from the source (A1) and the backpressure from the surrounding intrusion, if present, on the upflow (A2). The evidence presented in this work suggests that the mechanisms for the two parts in the scaling of Zm scale with Fr2/3. The intrusion behaviour varies with the Reynolds number (Re) but there is no Re affect on the fountain penetration height. For Re < 250 the radial intrusion flow is subcritical and has different behaviour. For Fr between 0.4 and 2.1 the effect of source momentum flux increases and the flow structure changes to one where there is a coherent upflow and a cap region where the flow stagnates and then reverses. The two regions have separate scaling behaviour such that the overall height, through this transition range of Froude numbers, can be described by Zm/R=C1Fr2/3 + C2Fr2, where C1 and C2 are constants. Over this transition range the effect of source velocity profile is more significant than the Reynolds number effects and the effect of inlet turbulence is minor.

The steady plane boundary-layer flows of velocity field {u(x, y), v(x, y)} induced by continuous moving surfaces are revisited in this paper. It is shown that the governing balance equations, as well as the asymptotic condition u(x, ∞) = 0 at the outer edge of the boundary layer are invariant under arbitrary translations y → y + y0(x) of the transverse coordinate y. The wall conditions, i.e. the prescribed stretching velocity u(x, 0) ≡ Uw(x) and the transpiration velocity v(x, 0) ≡ Vw(x) distributions, however, undergo in general substantial changes. The consequences of this basic symmetry property on the structure of the solution space are investigated. It is found that starting with a primary solution which describes the boundary-layer flow induced by an impermeable surface, infinitely many translated solutions can be generated which form a continuous group, the translation group of the given primary solution. The elements of this group describe boundary-layer flows induced by permeable surfaces stretching under transformed wall conditions, Uw(x) → Ũw(x) = u[x, y0(x)] and Vw(x) → Ṽw(x) = v[x, y0(x)] − y′0(x)u[x, y0(x)], respectively. In this way, starting with a known solution {u(x, y), v(x, y)} so that the inverse y0(x) = u−1(x, Ũw) of u[x, y0(x)] exists, a new solution {ũ(x, y), ṽ(x, y)} corresponding to any desired stretching velocity distribution Ũw(x) can be prepared. It also turns out that several exact solutions discovered during the latter decades are not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.

Instability waves in a laminar planar jet are extracted using hybrid unsteady-flow simulation combining particle tracking velocimetry (PTV) and direct numerical simulation (DNS). Unsteady velocity fields on a laser sheet in a water tunnel are measured with time-resolved PTV; subsequently, PTV velocity fields are rectified in a least squares sense so that the equation of continuity is satisfied, and they are transplanted to a two-dimensional incompressible Navier–Stokes solver by setting a multiple of the computational time step equal to the frame rate of the PTV system. As a result, the unsteady hybrid velocity field approaches that of the measured one over time, and we can simultaneously acquire the unsteady pressure field. The resultant set of flow quantities satisfies the governing equations, and their resolution is comparable to that of numerical simulation with the noise level much lower than the original PTV data. From hybrid unsteady velocity fields, we extract eigenfunctions using bi-orthogonal decomposition as a spatial problem for viscous instability. We also investigate stability/convergence characteristics of the hybrid simulation referring to linear stability analysis.

Particle image velocimetry experiments were performed to study the impact of realistic roughness on the spatial structure of wall turbulence at moderate Reynolds number. This roughness was replicated from an actual turbine blade damaged by deposition of foreign materials and its features are quite distinct from most roughness characterizations previously considered as it is highly irregular and embodies a broad range of topographical scales. The spatial structure of flow over this rough surface near the outer edge of the roughness sublayer is contrasted with that of smooth-wall flow to identify any structural modifications due to roughness. Hairpin vortex packets are observed in the outer layer of the rough-wall flow and are found to contribute heavily to the Reynolds shear stress, consistent with smooth-wall flow. While similar qualitative consistency is observed in comparisons of smooth- and rough-wall two-point correlations, some quantitative differences are also apparent. In particular, a reduction in the streamwise extent of two-point correlations of streamwise velocity is noted which could be indicative of a roughness-induced modification of outer-layer vortex organization. Proper orthogonal decomposition analysis reveals the streamwise coherence of the larger scales to be most sensitive to roughness while the spatial characteristics of the smaller scales appear relatively insensitive to such effects.

In this paper, we perform direct numerical simulation (DNS) of turbulent boundary layers at Mach 5 with the ratio of wall-to-edge temperature Tw/Tδ from 1.0 to 5.4 (Cases M5T1 to M5T5). The influence of wall cooling on Morkovin's scaling, Walz's equation, the standard and modified strong Reynolds analogies, turbulent kinetic energy budgets, compressibility effects and near-wall coherent structures is assessed. We find that many of the scaling relations used to express adiabatic compressible boundary-layer statistics in terms of incompressible boundary layers also hold for non-adiabatic cases. Compressibility effects are enhanced by wall cooling but remain insignificant, and the turbulence dissipation remains primarily solenoidal. Moreover, the variation of near-wall streaks, iso-surface of the swirl strength and hairpin packets with wall temperature demonstrates that cooling the wall increases the coherency of turbulent structures. We present the mechanism by which wall cooling enhances the coherence of turbulence structures, and we provide an explanation of why this mechanism does not represent an exception to the weakly compressible hypothesis.

A compressible air-flow model is introduced for the thin film dynamics of a highly rotating squeeze-film thrust bearing. The lubrication approximation to the Navier–Stokes equations for compressible flow leads to a modified Reynolds equation incorporating additional rotation effects. To investigate the dynamics of the system, the axial position of the bearing stator is prescribed by a finite-amplitude periodic forcing. The dynamics of the squeeze-film are modelled in the uncoupled configuration where the axial position of the rotor is fixed. The coupled squeeze-film bearing dynamics are investigated when the axial position of the rotor is modelled as a spring-mass-damper system that responds to the film dynamics. Initially the uncoupled squeeze-film dynamics are considered at low operating speeds with the classical Reynolds equation for compressible flow. The limited value of the linearized small-amplitude results is identified. Analytical results indicate that finite-amplitude forcing needs to be considered to gain a complete understanding of the dynamics. Using a Fourier spectral collocation numerical scheme, the periodic bearing force is investigated as a nonlinear function of the frequency and amplitude of the stator forcing. High-speed bearing operation is modelled using the modified Reynolds equation. A steady-state analysis is used to identify the effect of rotation and the rotor support properties in the coupled air-flow–structure model. The unsteady coupled dynamics are computed numerically to determine how the rotor support structures and the periodic stator forcing influence the system dynamics. The potential for resonant rotor behaviour is identified through asymptotic and Fourier analysis of the rotor motion for small-amplitude, low-frequency oscillations in the stator position for key values of the rotor stiffness. Through the use of arclength continuation, the existence of resonant behaviour is identified numerically for a range of operating speeds and forcing frequencies. Changes in the minimum rotor–stator clearance are presented as a function of the rotor stiffness to demonstrate the appearance of resonance.

Many growing plant cells undergo rapid axial elongation with negligible radial expansion. Growth is driven by high internal turgor pressure causing viscous stretching of the cell wall, with embedded cellulose microfibrils providing the wall with strongly anisotropic properties. We present a theoretical model of a growing cell, representing the primary cell wall as a thin axisymmetric fibre-reinforced viscous sheet supported between rigid end plates. Asymptotic reduction of the governing equations, under simple sets of assumptions about the fibre and wall properties, yields variants of the traditional Lockhart equation, which relates the axial cell growth rate to the internal pressure. The model provides insights into the geometric and biomechanical parameters underlying bulk quantities such as wall extensibility, and shows how either dynamical changes in wall material properties or passive fibre reorientation may suppress cell elongation.

An optimal transient growth analysis is compared with experimental observation for the steady flow through an abrupt, axisymmetric stenosis of varying stenosis degree. Across the stenosis range, a localized sinuous convective shear-layer instability type is predicted to dominate. A comparison of the shape and development of the optimal modes is made with experimental dye visualizations. The presence of the same sinuous-type disturbance immediately upstream of the highly chaotic region observed in the experimental flow is consistent with the optimal growth predictions. This, together with the fact that the flow is unstable globally only at much higher Reynolds numbers, suggests bypass transition.