Modulatory heat-transfer enhancement in grooved channels is investigated by direct numerical simulation of the Navier–Stokes and energy equations using the spectral element method. It is shown that oscillatory perturbation of the flow at the frequency of the least-stable mode of the linearized system results in subcritical resonant excitation and associated transport enhancement as the critical Reynolds number of the flow is approached. The Tollmien–Schlichting frequency theory that was presented in Part 1 of this paper is shown to accurately predict the optimal frequency for transport augmentation for small values of the modulatory amplitude, and the effect of the excited travelling-wave channel modes on the resulting temperature distribution is described. The importance of (non-trivial) geometry in the forced response of a flow is discussed, and grooved-channel flow is compared to (straight-channel) plane Poiseuille flow, for which no resonance excitation occurs owing to a zero projection of the forcing inhomogeneity on the dangerous modes of the system. For the particular grooved-channel geometry investigated, resonant oscillatory forcing at modulatory amplitudes as small as 20% of the mean flow results in a doubling of transport as measured by a time, space-averaged Nusselt number.

We experimentally determine the phase diagram for impacting ethanol droplets on a smooth, sapphire surface in the parameter space of Weber number $\mathit{We}$ versus surface temperature $T$. We observe two transitions, namely the one towards splashing (disintegration of the droplet) with increasing $\mathit{We}$, and the one towards the Leidenfrost state (no contact between the droplet and the plate due to a lasting vapour film) with increasing $T$. Consequently, there are four regimes: contact and no splashing (deposition regime), contact and splashing (contact–splash regime), neither contact nor splashing (bounce regime), and finally no contact, but splashing (film–splash regime). While the transition temperature $T_{L}$ to the Leidenfrost state depends weakly, at most, on $\mathit{We}$ in the parameter regime of the present study, the transition Weber number $\mathit{We}_{C}$ towards splashing shows a strong dependence on $T$ and a discontinuity at $T_{L}$. We quantitatively explain the splashing transition for $T<T_{L}$ by incorporating the temperature dependence of the physical properties in the theory by Riboux & Gordillo (Phys. Rev. Lett., vol. 113(2), 2014, 024507; J. Fluid Mech., vol. 772, 2015, pp. 630–648).

Measurements of the shear-stress variation along and the velocity profiles above a solid wavy wall bounding a turbulent flow are presented for waves with height-to-length ratios of 2a/λ = 0·0312 and 0·05. These are compared with previous measurements of the wall shear stress reported by Thorsness (1975) and by Morrisroe (1970) for 2a/λ = 0·012. The investigation covered a range of conditions from those for which a linear behaviour is observed to those for which a separated flow is just being initiated.

Pressure measurements indicate a linear response in that the spatial variation is described quite well by a single harmonic with a wavelength equal to that of the surface. However, the variation of τw for waves with 2a/λ = 0·0312 and 0·05 can be more rapid on the leeward side of the wave. The degree of departure from a sinusoidal variation increases with increasing wave height and fluid velocity and, from the results reported in this paper, it is suggested that nonlinear behaviour will become evident when au*/v [ges ] 27.

Many aspects of the flow for all three waves are described by a solution of the linear momentum equations previously presented by Thorsness (1975) and by Thorsness & Hanratty (1977). These include the phase and amplitude of the pressure profile and the first harmonic of the shear-stress profile and the velocity field outside the viscous wall region.

These results suggest that up to separation the flow is approximated quite well by linear theory. Nonlinearities affect the flow only in a region very close to the wave surface and are manifested by the appearance of higher harmonics in the variation of τw.

An analytical model is presented for the deformation of an ellipsoidal Newtonian droplet, suspended in another Newtonian fluid with different viscosity and zero interfacial tension. The theory is exact for any linear velocity field, and is not limited to small deformations. It encompasses some well-known special cases, such as Jeffery's equation for solid axisymmetric particles and Taylor's small-deformation theory for droplets. Example calculations exhibit droplet stretching, reorientation, and tumbling, and provide a reasonable match to available experimental data on transient and steady droplet shapes. The corresponding rheological theory for dilute dispersions is also derived, in a form that explicitly includes the effects of microstructure on dispersion rheology.

Results are presented from a linear-stability analysis of the flow at the head of two-dimensional gravity-current fronts. The analysis was undertaken in order to clarify the instability mechanism that leads to the formation of the complex lobe-and-cleft pattern which is commonly observed at the leading edge of gravity currents propagating along solid boundaries. The stability analysis concentrates on the foremost part of the front, and is based on direct numerical simulation data of two-dimensional lock-exchange flows which are described in the companion paper, Härtel et al. (2000). High-order compact finite differences are employed to discretize the stability equations which results in an algebraic eigenvalue problem for the amplification rate, that is solved in an iterative fashion. The analysis reveals the existence of a vigorous linear instability that acts in a localized way at the leading edge of the front and originates in an unstable stratification in the flow region between the nose and stagnation point. It is shown that the amplification rate of this instability as well as its spanwise length scale depend strongly on Reynolds number. For validation, three-dimensional direct numerical simulations of the early stages of the frontal instability are performed, and close agreement with the results from the linear-stability analysis is demonstrated.

In an effort to resolve some controversies regarding the turbulent mixing-layer structure, the near field of a large (18 cm diameter) air jet has been investigated for the jet exit speed of 30 m s−1. The smoke-laden axisymmetric mixing layer has been illuminated by a thin sheet of laser light in an azimuthal plane passing through the jet axis. High-speed visualization films of the mixing layer in the region of its self-preservation (of which a few picture sequences depicting space-time evolutions of the structure of the layer are presented) reveal that most of the time the mixing layer is in a state of disorganization, consisting of relatively smaller scale, random and diffuse turbulent motions; only occasionally are organized distinct large-scale coherent structures formed. The survival distances of the large-scale structures are found to be comparable to their average sizes. The survival time of these structures is about one ‘turnover’ time, each being roughly about five times the local characteristic time scale of the mixing layer. It is seen that tearing is as dominant a mode of large-scale interaction as pairing is; large-scale structures are continually sheared and typically fragmented due to a segment on the high-speed side being torn and swept away from the slower-moving outer portion. Evolution of the large structures occur not primarily through complete pairing as widely believed but quite frequently through ‘fractional pairing’ between segments which have been torn from different upstream large-scale coherent structures or through ‘partial pairing’ when one structure captures only a part of another. The movies show that along with entrainment of non-vortical ambient fluid, radially outward ejection of vortical fluid into the ambient is an important aspect of jet mixing. From aligned displays of ciné film frame sequences, space-time trajectories of identifiable vortical fluid elements have been traced. The convection velocity variation across the shear layer and even the overall structure convection velocity measured from these trajectories agree with those determined from the wave-number-celerity spectra, obtained from double-Fourier transformation of longitudinal velocity space-time correlation measurements with hot-wires.

The visualization films do not bear out the two-street vortex ring model recently propounded by Lau. Based on our observations, we propose that tearing, ‘slippage’ and fractional and partial pairings are responsible for the observed radial variation of structure passage frequency, and the causes of the different coherent structures educed by Bruun on the high- and low-speed sides of the mixing layer and for Yule's failure in educing a coherent structure on the low-speed side of the layer.

Results of dynamic simulations of bubbles rising through a liquid are presented. The Reynolds number of the flow based on the radius and the terminal speed of bubbles is large compared to unity, and the Weber number, which is the ratio of inertial to surface tension forces, is small. It is assumed that the bubbles do not coalesce when they approach each other but rather bounce instantaneously, conserving the momentum and the kinetic energy of the system. The flow of the liquid is assumed to be irrotational and is determined by solving the many-bubble interaction problem exactly. The viscous force on the bubbles is estimated from the rate of viscous energy dissipation. It is shown that the random state of bubbly liquids under these conditions is unstable and that the bubbles form aggregates in planes transverse to gravity. These aggregates form even when the size distribution of the bubbles is non-uniform. While the instability results primarily from the nature of inertial interaction among pairs of bubbles, which causes them to be attracted toward each other when they are aligned in the plane perpendicular to gravity, it is shown that the presence of viscous forces facilitates the process.

The turbulent boundary layer over a flat plate was observed visually using two dye injectors and a normal bubble wire. One dye injector was a standard wall slot; the other one was a small Pitot probe which could be placed anywhere in the flow. The purpose of these experiments was to improve understanding of the relationships between the bursting of low-speed streaks near the wall and the flow field farther away from the wall.

On the basis of these observations, it seems that (i) each lift-up is associated with a disturbance which originates in the logarithmic region and is characterized by a mean motion towards the wall, and that (ii) such disturbances are generated by the interaction of an earlier burst from further upstream with the fluid motion in the logarithmic region.

Energy stability theory has been formulated for two-dimensional buoyancy–thermocapillary convection in a layer with a free surface. The theory yields a critical Rayleigh number RE for which R < RE is a sufficient condition for stability of the layer. RE emerges from the variational formulation as an eigenvalue of a nonlinear system of Euler–Lagrange equations. For the case of small capillary number (large mean surface tension) explicit values are obtained for RE. The analogous linear-theory results for this case are obtained in terms of a critical Rayleigh number RL. These are compared. It is found that the existence of the deformable interface can lead to a stabilization relative to the case of a planar interface. This result is explained in physical terms. The energy theory is then generalized to include general flow problems having three-dimensional disturbances, non-Newtonian bulk fluids and general interfacial mechanics such as surface viscosity and elasticity.

Three-dimensional numerical simulations of laminar flow around a circular cylinder with sinusoidal variation of cross-section along the spanwise direction, named ‘wavy cylinder’, are performed. A series of wavy cylinders with different combinations of dimensionless wavelength (λ/Dm) and wave amplitude (a/Dm) are studied in detail at a Reynolds number of Re = U∞Dm/ν = 100, where U∞ is the free-stream velocity and Dm is the mean diameter of a wavy cylinder. The results of variation of mean drag coefficient and root mean square (r.m.s.) lift coefficient with dimensionless wavelength show that significant reduction of mean and fluctuating force coefficients occurs at optimal dimensionless wavelengths λ/Dm of around 2.5 and 6 respectively for the different amplitudes studied. Based on the variation of flow structures and force characteristics, the dimensionless wavelength from λ/Dm = 1 to λ/Dm = 10 is classified into three wavelength regimes corresponding to three types of wake structures. The wake structures at the near wake of different wavy cylinders are captured. For all wavy cylinders, the flow separation line varies along the spanwise direction. This leads to the development of a three-dimensional free shear layer with periodic repetition along the spanwise direction. The three-dimensional free shear layer of the wavy cylinder is larger and more stable than that of the circular cylinder, and in some cases the free shear layer even does not roll up into a mature vortex street behind the cylinder. As a result, the mean drag coefficients of some of the typical wavy cylinders are less than that of a corresponding circular cylinder with a maximum drag coefficient reduction up to 18%. The r.m.s. lift coefficients are greatly reduced to practically zero at optimal wavelengths. In the laminar flow regime (60 ≤ Re ≤ 150), the values of optimal wavelength are Reynolds number dependent.

A two-dimensional mathematical model is described for the calculation of the depth-averaged velocity and temperature or concentration distribution in open-channel flows, an essential feature of the model being its ability to handle recirculation zones. The model employs the depth-averaged continuity, momentum and temperature/concentration equations, which are solved by an efficient finite-difference procedure. The ‘rigid lid’ approximation is used to treat the free surface. The turbulent stresses and heat or concentration fluxes are determined from a depth-averaged version of the so-called k, ε turbulence model which characterizes the local state of turbulence by the turbulence kinetic energy k and the rate of its dissipation ε. Differential transport equations are solved for k and ε to determine these two quantities. The bottom shear stress and turbulence production are accounted for by source/sink terms in the relevant equations. The model is applied to the problem of a side discharge into open-channel flow, where a recirculation zone develops downstream of the discharge. Predicted size of the recirculation zone, jet trajectories, dilution, and isotherms are compared with experiments for a wide range of discharge to channel velocity ratios; the agreement is generally good. An assessment of the numerical accuracy shows that the predictions are not influenced significantly by numerical diffusion.

The partially cavitating two-dimensional hydrofoil problem is treated using nonlinear theory by employing a low-order potential-based boundary-element method. The cavity shape is determined in the framework of two independent boundary-value problems; in the first, the cavity length is specified and the cavitation number is unknown, and in the second the cavitation number is known and the cavity length is to be determined. In each case, the position of the cavity surface is determined in an iterative manner until both a prescribed pressure condition and a zero normal velocity condition are satisfied on the cavity. An initial approximation to the nonlinear cavity shape, which is determined by satisfying the boundary conditions on the hydrofoil surface rather than on the exact cavity surface, is found to differ only slightly from the converged nonlinear result.

The boundary element method is then extended to treat the partially cavitating three-dimensional hydrofoil problem. The three-dimensional kinematic and dynamic boundary conditions are applied on the hydrofoil surface underneath the cavity. The cavity planform at a given cavitation number is determined via an iterative process until the thickness at the end of the cavity at all spanwise locations becomes equal to a prescribed value (in our case, zero). Cavity shapes predicted by the present method for some three-dimensional hydrofoil geometries are shown to satisfy the dynamic boundary condition to within acceptable accuracy. The method is also shown to predict the expected effect of foil thickness on the cavity size. Finally, cavity planforms predicted from the present method are shown to be in good agreement to those measured in a cavitating three-dimensional hydrofoil experiment, performed in MIT's cavitation tunnel.

Buoyancy-driven flows resulting from the introduction of fluid of one density into a crack embedded in an elastic solid of different density are analysed. Scaling arguments are used to determine the regimes in which different combinations of the buoyancy force, elastic stress, viscous pressure drop and material toughness provide the dominant pressure balance in the flow. The nonlinear equations governing the shape and rate of spread of the propagating crack are formulated for the cases of vertical propagation of buoyant fluid released into a solid of greater density and of lateral propagation of fluid released at an interface between an upper layer of lesser density and a lower layer of greater density. Similarity solutions of these equations are derived under the assumption that the volume of fluid is given by Qtα, where Q and α are constants. Both laminar and turbulent flows are considered.

Fluid fracture is an important mechanism for the transport of molten rock from the region of production in the Earth's mantle to surface eruptions or near-surface emplacement. The theoretical solutions provide simple models which describe the relation between the elastic and fluid-mechanical phenomena involved in the vertical transport of melt through the Earth's lithosphere and in the lateral intrusion of melt at a neutral-buoyancy level close to the Earth's surface.

It is common to relate the dynamic contact angle $\theta_d$ to the relative speed between the substrate and the contact line; theory suggests $\theta_d^3 \propto U$. In fact, available physical models show that the dynamic angle involves speed logarithmically and in a model-dependent manner. Experimental data consistent with this interpretation are cited.

The current understanding of fundamental processes in atmospheric clouds, such as nucleation, droplet growth, and the onset of precipitation (collision–coalescence), is based on the assumption that droplets in undiluted clouds are distributed in space in a perfectly random manner, i.e. droplet positions are independently distributed with uniform probability. We have analysed data from a homogeneous cloud core to test this assumption and gain an understanding of the nature of droplet transport. This is done by examining one-dimensional cuts through clouds, using a theory originally developed for x-ray scattering by liquids, and obtaining statistics of droplet spacing. The data reveal droplet clustering even in cumulus cloud cores free of entrained ambient air. By relating the variance of droplet counts to the integral of the pair correlation function, we detect a systematic, scale-dependent clustering signature. The extracted signal evolves from sub- to super-Poissonian as the length scale increases. The sub-Poisson tail observed below mm-scales is a result of finite droplet size and instrument resolution. Drawing upon an analogy with the hard-sphere potential from the theory of liquids, this sub-Poisson part of the signal can be effectively removed. The remaining part displays unambiguous clustering at mm- and cm-scales. Failure to detect this phenomenon until now is a result of the previously unappreciated cumulative nature, or ‘memory,’ of the common measures of droplet clustering.

In the first part of this paper presents a gas-kinetic scheme based on the Bhatnagar–Gross–Krook (BGK) model for the microflow simulations in the near continuum flow regime. The current method improves the previous gas-kinetic BGK Navier–Stokes (BGK–NS) solver by (i) implementing a general non-equilibrium state based on the Chapman–Enskog expansion of the BGK model up to the Knudsen number squared $(\hbox{\it Kn})^2$ in the gas distribution function, (ii) using the compatibility condition to evaluate all high-order time derivative terms in the Chapman–Enskog expansion, and (iii) implementing the kinetic boundary condition for the gas distribution function to obtain ‘slip’ boundary automatically. As a result, the gas-kinetic BGK–Burnett scheme improves the validity of the BGK–NS solver for the microchannel flow simulations even in the slip flow regime, where the Navier–Stokes equations with the slip boundary conditions are considered to be legitimately valid. Owing to the correction to the heat transport in the energy flux, the Prandtl number in the gas-kinetic BGK–Burnett method can take any value to capture both viscous and heat conduction effects. Since the current method is based on the direct evaluation of the gas distribution function and captures its time evolution, it is different from those methods that are based on the macroscopic Burnett or extended hydrodynamic equations. The second part of this paper is about the application of the newly developed gas-kinetic BGK–Burnett method in the microchannel flows. First, we verify the method in the pressure- and external-force-driven Poiseuille flows, where the reliable direct simulation Monte Carlo (DSMC) results are available. In the study of Poiseuille flow with $\hbox{\it Kn}=0.1$, the qualitative differences in the pressure distribution in the cross-stream direction between the Navier–Stokes and the DSMC results are resolved by the gas-kinetic BGK–Burnett scheme. It demonstrates that the BGK–Burnett method could give a more realistic description of flow motion than the Navier–Stokes method even in the slip flow regime. After that, the current method is used to simulate the microchannel flows, where the experimental data are available. In this study, the similarity in the pressure distribution along the straight microchannel is verified first. Then, the mass flow rates for different gases, such as argon, helium and nitrogen, in the long microchannel of submicron height are computed and compared with the experimental measurements.

We present results from our experiments with the irregular reflection of shock waves in argon. We compare the data with the results we obtained numerically; the assumptions for the computational code were that we had unsteady, two-dimensional, compressible, inviscid, flow of a perfect gas. When precautions were taken to reduce the effects of the gas viscosity on the experimental data, we obtained very good agreement between the numerical and the experimental results for the ramp Mach number and the trajectory path triple-point angle, but there were discrepancies with the wave-angle data. The discrepancies were ascribed to the sensitivity of the data to both viscosity and to a singularity. We show that there are actually two weak irregular wave reflections, namely a classic Mach reflection (MR) and a new type, that we call a von Neumann reflection (NR). The structure of the NR is discussed in some detail, and so are the transition criteria for the various wave systems.

The stability of plane channel flow between compliant walls is investigated for disturbances which have the same symmetry, with respect to the channel centreline, as the Tollmien–Schlichting mode of instability. The interconnected behaviour of flow-induced surface waves and Tollmien–Schlichting waves is examined both by direct numerical solution of the Orr–Sommerfeld equation and by means of an analytic shear layer theory. We show that when the compliant wall properties are selected so as to give a significant stability effect on Tollmien–Schlichting waves, the onset of divergence instability can be severely disrupted. In addition, travelling wave flutter can interact with the Tollmien–Schlichting mode to generate a powerful instability which replaces the flutter instability identified in studies based on a potential mean-flow model. The behaviour found when the mean-flow shear layer is fully accounted for may be traced to singularities in the wave dispersion relation. These singularities can be attributed to solutions which represent Tollmien–Schlichting waves in rigid-walled channels. Such singularities will also be found in the dispersion relation for the case of Blasius flow. Thus, similar behaviour can be anticipated for Blasius flow, including the disruption of the onset of divergence instability. As a consequence, it seems likely that previous investigations for Blasius flow will have yielded very conservative estimates for the optimal stabilization that can be achieved for Tollmien–Schlichting waves for the purposes of laminar-flow control.

An electron beam technique was used to measure density profiles of strong shock waves in argon with high accuracy. The experimental results are compared with the results of theoretical models. Of the models that are available in enough detail for comparison, the best agreement with experiment is shown by the direct simulation Monte-Carlo method (Bird 1968), assuming an inverse 12th power molecular interaction force law. It is shown that the density maximum slope thickness is not sufficient for a detailed description of the shock wave structure.

The constant initial speed of propagation (V) of heavy gravity currents, of density ρC, released from behind a lock and along the bottom boundary of a tank containing a linearly stratified fluid has been measured experimentally and calculated numerically. The density difference, bottom to top, of the stratification is (ρb−ρ0) and its intrinsic frequency is N. For a given ratio of the depth of released fluid (h) to total depth (H) it has been found that the dimensionless internal Froude number, Fr = V/NH, is independent of the length of the lock and is a logarithmic function of a parameter R = (ρC−ρ0)/(ρb−ρ0), except at small values of h/H and R close to unity. This parameter, R, is one possible measure of the relative strength of the current (ρC−ρ0) and stratification (ρb−ρ0). The distance propagated by the current before this constant velocity regime ended (Xtr), scaled by h, has been found to be a unique function of Fr for all states tested. After this phase of the motion, for subcritical values of Fr, i.e. less than 1/π, internal wave interactions with the current resulted in an oscillation of the velocity of its leading edge. For supercritical values, velocity decay was monotonic for the geometries tested. A two-dimensional numerical model incorporating a no-slip bottom boundary condition has been found to agree with the experimental velocity magnitudes to within ±1:5%.