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Howell Peregrine, who died on 20th March after a brief illness, had an exceptionally long and distinguished period of service for this Journal, first as Assistant Editor (1979–1980) then Associate Editor (1981–2007). In this capacity, he had editorial responsibility for some 50 papers every year, more than 1200 in total. This was a massive contribution to the continuing success of the Journal, to which it is appropriate and fitting that we pay grateful tribute.
In this paper we examine the role of thermal mass in buffering the interior temperature of a naturally ventilated building from the diurnal fluctuations in the environment. First, we show that the effective thermal mass which is in good thermal contact with the air is limited by the diffusion distance into the thermal mass over one diurnal temperature cycle. We also show that this effective thermal mass may be modelled as an isothermal mass. Temperature fluctuations in the effective thermal mass are attenuated and phase-shifted from those of the interior air, and therefore heat is exchanged with the interior air. The evolution of the interior air temperature is then controlled by the relative magnitudes of (i) the time for the heat exchange between the effective thermal mass and the air; (ii) the time for the natural ventilation to replace the air in the space with air from the environment; and (iii) the period of the diurnal oscillations of the environment. Through analysis and numerical solution of the governing equations, we characterize a number of different limiting cases. If the ventilation rate is very small, then the thermal mass buffers the interior air temperature from fluctuations in the environment, creating a near-isothermal interior. If the ventilation rate increases, so that there are many air changes over the course of a day, but if there is little heat exchange between the thermal mass and interior air, then the interior air temperature locks on to the environment temperature. If there is rapid thermal equilibration of the thermal mass and interior air, and a high ventilation rate, then both the thermal mass and the interior air temperatures lock on to the environment temperature. However, in many buildings, the more usual case is that in which the time for thermal equilibration is comparable to the period of diurnal fluctuations, and in which ventilation rates are moderate. In this case, the fluctuations of the temperature of the thermal mass lag those of the interior air, which in turn lag those of the environment. We consider the implications of these results for the use of thermal mass in naturally ventilated buildings.
The study analyses the cellular reaction zone structure of unstable methane–oxygen detonations, which are characterized by large hydrodynamic fluctuations and unreacted pockets with a fine structure. Complementary series of experiments and numerical simulations are presented, which illustrate the important role of hydrodynamic instabilities and diffusive phenomena in dictating the global reaction rate in detonations. The quantitative comparison between experiment and numerics also permits identification of the current limitations of numerical simulations in capturing these effects. Simulations are also performed for parameters corresponding to weakly unstable cellular detonations, which are used for comparison and validation. The numerical and experimental results are used to guide the formulation of a stochastic steady one-dimensional representation for such detonation waves. The numerically obtained flow fields were Favre-averaged in time and space. The resulting one-dimensional profiles for the mean values and fluctuations reveal the two important length scales, the first being associated with the chemical exothermicity and the second (the ‘hydrodynamic thickness’) with the slower dissipation of the hydrodynamic fluctuations, which govern the location of the average sonic surface. This second length scale is found to be much longer than that predicted by one-dimensional reaction zone calculations.
Thermal convection in a rapidly rotating spherical shell is investigated experimentally and numerically. The experiments are performed in water (Prandtl number P=7) and in gallium (P=0.025), at Rayleigh numbers R up to 80 times the critical value in water (up to 6 times critical in gallium) and at Ekman numbers E∼10−6. The measurements of fluid velocities by ultrasonic Doppler velocimetry are quantitatively compared with quasi-geostrophic numerical simulations incorporating a varying β-effect and boundary friction (Ekman pumping). In water, unsteady multiple zonal jets, weaker in amplitude than the non-axisymmetric flow, are experimentally observed and numerically reproduced at moderate forcings (R/Rc <40). In this regime, zonal flows and vortices share the same length scale. Gallium experiments and strongly supercritical convection experiments in water correspond to another regime. In these turbulent flows, the zonal motion amplitude U dominates the non-axisymmetric motion amplitude Ũ. As a result of the reverse cascade of kinetic energy, the characteristic Rhines length scale of zonal jets emerges, and the boundary friction becomes the main brake on the growth of the zonal flow. A scaling law U ∼ Ũ4/3 is then derived and verified both numerically and experimentally.
Thermal magnetoconvection in a rapidly rotating spherical shell is investigated numerically and experimentally in electrically conductive liquid gallium (Prandtl number P = 0.025), at Rayleigh numbers R up to around 6 times critical and at Ekman numbers E ∼ 10−6. This work follows up the non-magnetic study of convection presented in a companion paper (Gillet et al. 2007). We study here the addition of a z-invariant toroidal magnetic field to the fluid flow. The experimental measurements of fluid velocities by ultrasonic Doppler velocimetry, together with the quasi-geostrophic numerical simulations incorporating a three-dimensional modelling of the magnetic induction processes, demonstrate a stabilizing effect of the magnetic field in the weak-field case, characterized by an Elsasser number Λ < (E/P)1/3. We find that this is explained by the changes of the critical parameters at the onset of convection as Λ increases. As in the non-magnetic study, strong zonal jets of characteristic length scales ℓβ (Rhines length scale) dominates the fluid dynamics. A new characteristic of the magnetoconvective flow is the elongation of the convective cells in the direction of the imposed magnetic field, introducing a new length scale ℓφ. Combining experimental and numerical results, we derive a scaling law where U is the axisymmetric motion amplitude, Ũ s and Ũ φ are the non-axisymmetric radial and azimuthal motion amplitudes, respectively.
Lumped-parameter models (zero-dimensional) and wave-propagation models (one-dimensional) for pressure and flow in large vessels, as well as fully three-dimensional fluid–structure interaction models for pressure and velocity, can contribute valuably to answering physiological and patho-physiological questions that arise in the diagnostics and treatment of cardiovascular diseases. Lumped-parameter models are of importance mainly for the modelling of the complete cardiovascular system but provide little detail on local pressure and flow wave phenomena. Fully three-dimensional fluid–structure interaction models consume a large amount of computer time and must be provided with suitable boundary conditions that are often not known. One-dimensional wave-propagation models in the frequency and time domain are well suited to obtaining clinically relevant information on local pressure and flow waves travelling through the arterial system. They can also be used to provide boundary conditions for fully three-dimensional models, provided that they are defined in, or transferred to, the time domain.
Most of the one-dimensional wave propagation models in the time domain described in the literature assume velocity profiles and therefore frictional forces to be in phase with the flow, whereas from exact solutions in the frequency domain a phase difference between the flow and the wall shear stress is known to exist. In this study an approximate velocity profile function more suitable for one-dimensional wave propagation is introduced and evaluated. It will be shown that this profile function provides first-order approximations for the wall shear stress and the nonlinear term in the momentum equation, as a function of local flow and pressure gradient in the time domain. The convective term as well as the approximate friction term are compared to their counterparts obtained from Womersley profiles and show good agreement in the complete range of the Womersley parameter α. In the limiting cases, for Womersley parameters α → 0 and α → ∞, they completely coincide. It is shown that in one-dimensional wave propagation, the friction term based on the newly introduced approximate profile function is important when considering pressure and flow wave propagation in intermediate-sized vessels.
The entrainment across a stably stratified interface forced by convective motions is discussed in the light of the mixing efficiency of the entrainment process. The context is the convectively driven atmospheric boundary layer and we focus on the regime of equilibrium entrainment, i.e. when the boundary-layer evolution is in a quasi-steady state. The entrainment law is classically based on the ratio R of the negative of the heat flux at the interface to the heat flux at the ground surface. We propose a parameterization for R that involves the mixing efficiency and the thickness of the interface, which matches well the direct computation of R from a high-resolution large-eddy simulation. This result enables us to derive modified expressions for the classical entrainment laws (the so-called zero- and first-order models) as a function of the mixing efficiency. We show that, when the thickness of the interface is ignored (zero-order model), the scaling factor A in the entrainment law is the flux Richardson number. This parameterization of A is further improved when the thickness of the interface is considered (first-order model), as shown by the direct computation of A from the large-eddy simulation.
A numerical investigation of transonic and low-supersonic flows of dense gases of the Bethe–Zel'dovich–Thompson (BZT) type is presented. BZT gases exhibit, in a range of thermodynamic conditions close to the liquid/vapour coexistence curve, negative values of the fundamental derivative of gasdynamics. This can lead, in the transonic and supersonic regime, to non-classical gasdynamic behaviours, such as rarefaction shock waves, mixed shock/fan waves and shock splitting. In the present work, inviscid and viscous flows of a BZT fluid past an airfoil are investigated using accurate thermo-physical models for gases close to saturation conditions and a third-order centred numerical method. The influence of the upstream kinematic and thermodynamic conditions on the flow patterns and the airfoil aerodynamic performance is analysed, and possible advantages deriving from the use of a non-conventional working fluid are pointed out.
The vaporization of a droplet in rectilinear motion relative to a stagnant gaseous atmosphere is addressed for the limit of low Reynolds numbers and slow variation of the droplet velocity. Approximations are introduced that enable a formal asymptotic analysis to be performed with a minimum of complexity. It is shown that, under the conditions addressed, there is an inner region in the vicinity of the droplet within which the flow is nearly quasi-steady except during short periods of time when the acceleration changes abruptly, and there is a fully time-dependent outer region in which departures of velocities and temperatures from those of the ambient medium are small. Matched asymptotic expansions, followed by a Green's function analysis of the outer region enable expressions to be obtained for the velocity and temperature fields and for the droplet drag and vaporization rate. The results are applied to problems in which the droplet experiences constant acceleration, constant deceleration and oscillatory motion. The results, which identify dependences on the Prandtl number and the transfer number, are intended to be compared with experimental measurements on droplet behaviours in time-varying flows.
It is generally believed that a viscous, non-resistive plasma will eventually decay to a magnetostatic state, probably possessing contact discontinuities. We prove that even in the presence of a decaying forcing, the kinetic energy of the system tends to zero, which justifies the belief that the limit state is static. Regarding the magnetic field, the fact that the magnetic energy remains bounded proves the existence of weak sequential limits of the field as the time goes to infinity, but this does not imply that the magnetic field tends to a single state: we present an example where there is no limit, even in a weak sense. One additional condition upon the velocity, however, is enough to guarantee existence of a single limit magnetic configuration.
Motivated by the problem of microfluidic mixing, optimal control of advective mixing in Stokes fluid flows is considered. The velocity field is assumed to be induced by a finite set of spatially distributed force fields that can be modulated arbitrarily with time, and a passive material is advected by the flow. To quantify the degree of mixedness of a density field, we use a Sobolev space norm of negative index. We frame a finite-time optimal control problem for which we aim to find the modulation that achieves the best mixing for a fixed value of the action (the time integral of the kinetic energy of the fluid body) per unit mass. We derive the first-order necessary conditions for optimality that can be expressed as a two-point boundary value problem (TPBVP) and discuss some elementary properties that the optimal controls must satisfy. A conjugate gradient descent method is used to solve the optimal control problem and we present numerical results for two problems involving arrays of vortices. A comparison of the mixing performance shows that optimal aperiodic inputs give better results than sinusoidal inputs with the same energy.
We have investigated both experimentally and numerically the time evolution of clouds of particles settling under the action of gravity in an otherwise pure liquid at low Reynolds numbers. We have found that an initially spherical cloud containing enough particles is unstable. It slowly evolves into a torus which breaks up into secondary droplets which deform into tori themselves in a repeating cascade. Owing to the fluctuations in velocity of the interacting particles, some particles escape from the cloud toroidal circulation and form a vertical tail. This creates a particle deficit near the vertical axis of the cloud and helps in producing the torus which eventually expands. The rate at which particles leak from the cloud is influenced by this change of shape. The evolution toward the torus shape and the subsequent evolution is a robust feature. The nature of the breakup of the torus is found to be intrinsic to the flow created by the particles when the torus aspect ratio reaches a critical value. Movies are available with the online version of the paper.
Rotating convection is analysed numerically in a cylinder of aspect ratio one, for Prandtl number about 7. Traditionally, the problem has been studied within the Boussinesq approximation with density variation only incorporated in the gravitational buoyancy term and not in the centrifugal buoyancy term. In that limit, the governing equations admit a trivial conduction solution. However, the centrifugal buoyancy changes the problem in a fundamental manner, driving a large-scale circulation in which cool denser fluid is centrifuged radially outward and warm less-dense fluid is centrifuged radially inward, and so there is no trivial conduction state. For small Froude numbers, the transition to three-dimensional flow occurs for Rayleigh number R ≈ 7.5 × 103. For Froude numbers larger than 0.4, the centrifugal buoyancy stabilizes the axisymmetric large-scale circulation flow in the parameter range explored (up to R = 3.5 × 104). At intermediate Froude numbers, the transition to three-dimensional flow is via four different Hopf bifurcations, resulting in different coexisting branches of three-dimensional solutions. How the centrifugal and the gravitational buoyancies interact and compete, and the manner in which the flow becomes three-dimensional is different along each branch. The centrifugal buoyancy, even for relatively small Froude numbers, leads to quantitative and qualitative changes in the flow dynamics.
The Reynolds number dependence of the structure and statistics of wall-layer turbulence remains an open topic of research. This issue is considered in the present work using two-component planar particle image velocimetry (PIV) measurements acquired at the Surface Layer Turbulence and Environmental Science Test (SLTEST) facility in western Utah. The Reynolds number (δu τ/ν) was of the order 106. The surface was flat with an equivalent sand grain roughness k + = 18. The domain of the measurements was 500 < yu τ/ν < 3000 in viscous units, 0.00081 < y/δ < 0.005 in outer units, with a streamwise extent of 6000ν/u τ. The mean velocity was fitted by a logarithmic equation with a von Kármán constant of 0.41. The profile of u′v′ indicated that the entire measurement domain was within a region of essentially constant stress, from which the wall shear velocity was estimated. The stochastic measurements discussed include mean and RMS profiles as well as two-point velocity correlations. Examination of the instantaneous vector maps indicated that approximately 60% of the realizations could be characterized as having a nearly uniform velocity. The remaining 40% of the images indicated two regions of nearly uniform momentum separated by a thin region of high shear. This shear layer was typically found to be inclined to the mean flow, with an average positive angle of 14.9°.
The phenomenon of bursting, in which streaks in turbulent boundary layers oscillate and then eject low-speed fluid away from the wall, has been studied experimentally, theoretically and computationally for more than 50 years because of its importance to the three-dimensional structure of turbulent boundary layers. Five new three-dimensional solutions of turbulent plane Couette flow are produced, one of which is periodic while the other four are relative periodic. Each of these five solutions demonstrates the breakup and re-formation of near-wall coherent structures. Four of our solutions are periodic, but with drifts in the streamwise direction. More surprisingly, two of our solutions are periodic, but with drifts in the spanwise direction, a possibility that does not seem to have been considered in the literature. It is argued that a considerable part of the streakiness observed experimentally in the near-wall region could be due to spanwise drifts that accompany the breakup and re-formation of coherent structures. A new periodic solution of plane Couette flow is also computed that could be related to transition to turbulence.
The violent nature of the bursting phenomenon implies the need for good resolution in the computation of periodic and relative periodic solutions within turbulent shear flows. This computationally demanding requirement is addressed with a new algorithm for computing relative periodic solutions one of whose features is a combination of two well-known ideas – namely the Newton–Krylov iteration and the locally constrained optimal hook step. Each of the six solutions is accompanied by an error estimate.
Dynamical principles are discussed that suggest that the bursting phenomenon, and more generally fluid turbulence, can be understood in terms of periodic and relative periodic solutions of the Navier–Stokes equation.
We study steady vertical propagation of a crack filled with buoyant viscous fluid through an elastic solid with large effective fracture toughness. For a crack fed by a constant flux Q, a non-dimensional fracture toughness K=Kc /(3μQm 3/2)1/4 describes the relative magnitudes of resistance to fracture and resistance to viscous flow, where Kc is the dimensional fracture toughness, μ the fluid viscosity and m the elastic modulus. Even in the limit K ≫ 1, the rate of propagation is determined by viscous effects. In this limit the large fracture toughness requires the fluid behind the crack tip to form a large teardrop-shaped head of length O(K 2/3) and width O(K 4/3), which is fed by a much narrower tail. In the head, buoyancy is balanced by a hydrostatic pressure gradient with the viscous pressure gradient negligible except at the tip; in the tail, buoyancy is balanced by viscosity with elasticity also playing a role in a region within O(K 2/3) of the head. A narrow matching region of length O(K −2/5) and width O(K −4/15), termed the neck, connects the head and the tail. Scalings and asymptotic solutions for the three regions are derived and compared with full numerical solutions for K ≤ 3600 by analysing the integro-differential equation that couples lubrication flow in the crack to the elastic pressure gradient. Time-dependent numerical solutions for buoyancy-driven propagation of a constant-volume crack show a quasi-steady head and neck structure with a propagation rate that decreases like t −2/3 due to the dynamics of viscous flow in the draining tail.
Turbulence measurements for rough-wall boundary layers are presented and compared to those for a smooth wall. The rough-wall experiments were made on a three-dimensional rough surface geometrically similar to the honed pipe roughness used by Shockling, Allen & Smits (J. Fluid Mech. vol. 564, 2006, p. 267). The present work covers a wide Reynolds-number range (Re θ = 2180–27 100), spanning the hydraulically smooth to the fully rough flow regimes for a single surface, while maintaining a roughness height that is a small fraction of the boundary-layer thickness. In this investigation, the root-mean-square roughness height was at least three orders of magnitude smaller than the boundary-layer thickness, and the Kármán number (δ+), typifying the ratio of the largest to the smallest turbulent scales in the flow, was as high as 10100. The mean velocity profiles for the rough and smooth walls show remarkable similarity in the outer layer using velocity-defect scaling. The Reynolds stresses and higher-order turbulence statistics also show excellent agreement in the outer layer. The results lend strong support to the concept of outer layer similarity for rough walls in which there is a large separation between the roughness length scale and the largest turbulence scales in the flow.
Brenner (Physica A, vol. 349, 2005a, b, pp. 11, 60) has recently proposed modifications to the Navier–Stokes equations that are based on theoretical arguments but supported only by experiments having a fairly limited range. These modifications relate to a diffusion of fluid volume that would be significant for flows with high density gradients. So the viscous structure of shock waves in gases should provide an excellent test case for this new model. In this paper we detail the shock structure problem and propose exponents for the gas viscosity–temperature relation based on empirical viscosity data that is independent of shock experiments. We then simulate monatomic gas shocks in the range Mach 1.0–12.0 using the Navier–Stokes equations, both with and without Brenner's modifications. Initial simulations showed that Brenner's modifications display unphysical behaviour when the coefficient of volume diffusion exceeds the kinematic viscosity. Our subsequent analyses attribute this behaviour to both an instability to temporal disturbances and a spurious phase velocity–frequency relationship. On equating the volume diffusivity to the kinematic viscosity, however, we find the results with Brenner's modifications are significantly better than those of the standard Navier–Stokes equations, and broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. Brenner's modifications add only two terms to the Navier–Stokes equations, and the numerical implementation is much simpler than conventional extended hydrodynamic models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.
We consider freely decaying two-dimensional isotropic turbulence. It is usually assumed that, in such turbulence, the energy spectrum at small wavenumber, k, takes the form E(k → 0) ∼ Ik 3, where I is the two-dimensional version of Loitsyansky's integral. However, a second possibility is E(k → 0) ∼ Lk, where the pre-factor, L, is the two-dimensional analogue of Saffman's integral. We show that, as in three dimensions, L is an invariant and that E ∼ Lk spectra arise whenever the eddies possess a significant amount of linear impulse. The conservation of L is shown to be a direct consequence of the principle of conservation of linear momentum. We also show that isotropic turbulence dominated by a cloud of randomly located monopole vortices has a singular energy spectrum of the form E(k → 0) ∼ Jk −1, where J, like L, is an invariant. However, while E ∼ Jk −1 necessarily implies the existence of a sea of monopoles, the converse need not be true: a sea of monopoles whose spatial locations are not entirely random, but constrained in some way, need not give a E ∼ Jk −1 spectra. The constraint imposed by the conservation of energy is particularly important, ruling out E ∼ Jk −1 spectra for certain classes of initial conditions. Finally, we provide simple explicit examples of random vorticity fields with E ∼ Ik 3, E ∼ Lk and E ∼ Jk −1 spectra.
In this paper we consider the dynamics of droplets attached to rough or chemically inhomogeneous solid substrates with a circular contact line as they are deformed in subcritical and supercritical simple shear flows. Our main interest is concentrated on identifying the portions of the contact line where the contact angle hysteresis condition is first violated, i.e. the portions of the contact line which slide first. To address this physical problem, we employ our fully implicit time integration algorithm for interfacial dynamics in Stokes flow. Our study reveals that droplets with small and moderate initial angles show an early period where both upstream and downstream sliding are equally favourable as well as a late downstream-favoured period. By contrast, droplets with large initial angles, after a rather small early equally favourable period, show a large period where downstream sliding is more favourable than the upstream sliding. Owing to the surface tension force, droplets with intermediate initial angles are shown to be more stable. Droplets with different viscosity ratio show similar behaviour with respect to the onset of interfacial sliding; however, the viscosity ratio strongly affects the rate of the interfacial deformation and the equilibrium conditions. An asymptotic behaviour for very small or large viscosity ratios is shown to exist.