Only recently, three-dimensional direct numerical simulations (DNS) with detailed chemistry have become possible. These types of simulations will have an important impact on combustion science because of the data they provide. Obtaining similar data sets through experimental characterization is very challenging and requires joint measurements of a multitude of chemical species concentrations in addition to temperature and velocity fields. Gruber et al. (J. Fluid Mech., vol. 658, 2010, pp. 5–32) have performed a detailed chemistry DNS to understand the physics of flame/wall interactions. They found that flames interact with streamwise vortical structures close to the wall, and that these structures can push the flame towards or away from the wall. The flames get extinguish when they are too close to the wall. Interestingly, the extinction process actually causes an order-of-magnitude increase in heat release, and therefore, strongly amplifies wall heat transfer.

A turbulent flame–wall interaction (FWI) configuration is studied using three-dimensional direct numerical simulation (DNS) and detailed chemical kinetics. The simulations are used to investigate the effects of the wall turbulent boundary layer (i) on the structure of a hydrogen–air premixed flame, (ii) on its near-wall propagation characteristics and (iii) on the spatial and temporal patterns of the convective wall heat flux. Results show that the local flame thickness and propagation speed vary between the core flow and the boundary layer, resulting in a regime change from flamelet near the channel centreline to a thickened flame at the wall. This finding has strong implications for the modelling of turbulent combustion using Reynolds-averaged Navier–Stokes or large-eddy simulation techniques. Moreover, the DNS results suggest that the near-wall coherent turbulent structures play an important role on the convective wall heat transfer by pushing the hot reactive zone towards the cold solid surface. At the wall, exothermic radical recombination reactions become important, and are responsible for approximately 70% of the overall heat release rate at the wall. Spectral analysis of the convective wall heat flux provides an unambiguous picture of its spatial and temporal patterns, previously unobserved, that is directly related to the spatial and temporal characteristic scalings of the coherent near-wall turbulent structures.

Direct numerical simulation is performed for the interaction between a deformable free surface and the homogeneous isotropic turbulent flow underneath. The Navier–Stokes equations subject to fully nonlinear free-surface boundary conditions are simulated by using a pseudospectral method in the horizontal directions and a finite-difference method in the vertical direction. Statistically, steady turbulence is generated by using a linear forcing method in the bulk flow below. Through investigation of cases of different Froude and Weber numbers, the present study focuses on the effect of surface deformation of finite amplitude. It is found that the motion of the free surface is characterized by propagating waves and turbulence-generated surface roughness. Statistics of the turbulence field near the free surface are analysed in detail in terms of fluctuations of velocity, fluctuations of velocity gradients and strain rates and the energy budget for horizontal and vertical turbulent motions. Our results illustrate the effects of surface blockage and vanishing shear stress on the anisotropy of the flow field. Using conditional averaging analysis, it is shown that splats and antisplats play an essential role in energy inter-component exchange and vertical transport.

Spontaneous film rupture from van der Waals instability is investigated in two dimensions. The focus is on pure liquids with clean interfaces. This case is applicable to metallic foams for which surfactants are not available. There are important implications in aqueous foams as well, but the main differences are noted. A thin liquid film between adjacent bubbles in a foam has finite length, curved boundaries (Plateau borders) and a drainage flow from capillary suction that causes it to thin. A full linear stability analysis of this thinning film shows that rupture occurs once the film has thinned to ‘tens’ of nanometres, whereas for a quiescent film with a constant and uniform thickness, rupture occurs when the thickness is ‘hundreds’ of nanometres. Plateau borders and flow are both found to contribute to the stabilization. The drainage flow leads to several distinct qualitative features as well. In particular, unstable disturbances are advected by the flow to the edges of the thin film. As a result, the edges of the film close to the Plateau borders appear more susceptible to rupture than the centre of the film.

This paper develops a general theory for linear propagation of acoustic waves in a gas enclosed in a two-dimensional channel and in a circular tube subject to temperature gradient axially and extending infinitely. A ‘narrow-tube approximation’ is employed by assuming that a typical axial length is much longer than a span length, but no restriction on a thickness of thermoviscous diffusion layer is made. For each case, basic equations in this approximation are reduced to a spatially one-dimensional equation in terms of an excess pressure by making use of a method of Fourier transform. This equation, called a thermoacoustic-wave equation, is given in the form of an integro-differential equation due to memory by thermoviscous effects. Approximations of the equations for a short-time and a long-time behaviour from an initial state are discussed based on the Deborah number and the Reynolds number. It is shown that the short-time behaviour is well approximated by the equation derived previously by the boundary-layer theory, while the long-time behaviour is described by new diffusion equations. It is revealed that if the diffusion layer is thicker than the span length, the thermoviscous effects give rise to not only diffusion but also wave propagation by combined action with temperature gradient, and that negative diffusion may occur if the gradient is steep.

Direct numerical simulations are employed to investigate the effects of discrete suction orifices at the wall on the disturbance evolution in laminar three-dimensional boundary-layer flows with favourable pressure gradient. Suction panels with many suction orifices can excite unstable crossflow (CF) modes even if the orifice spacing is smaller than the chordwise/spanwise wavelengths of unstable modes, caused by imperfections in the orifice order or suction strength. It has been found that the most unstable steady vortex mode leads to strong CF vortices that invoke turbulence by secondary instability even on the active suction panel. The deliberate excitation and support of stabilizing vortices that have less than two-thirds of the spanwise wavelength of the most amplified ones – known from the upstream flow deformation or micrometre-sized roughness elements technique – by the orifice order on the panel can secure the desired stabilizing effect of suction and lower the necessary suction amount significantly.

Subtropical ocean waters are susceptible to the occurrence of salt finger instability. The effect of salt fingers in modifying water mass properties may depend upon the ubiquitous presence of oceanic shear produced by internal wave motion. We present an experimental study of the buoyancy fluxes produced by sugar–salt fingers in the presence of a laminar shear flow. As is commonly done in the laboratory, sugar (the slower diffuser) was used as a proxy for salt, and salt (the faster diffuser compared to sugar) was used as a proxy for cold. Sugar–salt fingers, initially aligned vertically, were observed to tilt when a shear flow was imposed. A consistent decrease in the salt fluxes was measured as the Reynolds number (Re) was increased by increasing the shear velocity magnitude. Through regression analysis, the salt fluxes were found to depend upon the Reynolds number as Re−0.025, Re−0.1 and Re−0.34, for density ratio values (Rρ) equal to 1.2, 1.54 and 2.1 respectively. The salt fluxes produced by the sheared fingers were also found to decrease by one order of magnitude when Rρ increased from 1.2 to 2.1. A computation of the salt Nusselt number revealed that the finger fluxes approach molecular flux values when Rρ = 2.1 and Re ≃ 140.

A global shock solution for the nonlinear shallow water equations (NSWEs) is found by assigning proper seaward boundary data that preserve a constant incoming Riemann invariant during the shock wave evolution. The correct shock relations, entropy conditions and asymptotic behaviour near the shoreline are provided along with an in-depth analysis of the main quantities along and behind the bore. The theoretical analysis is then applied to the specific case in which the water at the front of the shock wave is still. A comparison with the Shen & Meyer (J. Fluid Mech., vol. 16, 1963, p. 113) solution reveals that such a solution can be regarded as a specific case of the more general solution proposed here. The results obtained can be regarded as a useful benchmark for numerical solvers based on the NSWEs.

We study the fluctuating motion of a Brownian-sized probe particle as it is dragged by a constant external force through a colloidal dispersion. In this nonlinear-microrheology problem, collisions between the probe and the background bath particles, in addition to thermal fluctuations of the solvent, drive a long-time diffusive spread of the probe's trajectory. The influence of the former is determined by the spatial configuration of the bath particles and the force with which the probe perturbs it. With no external forcing the probe and bath particles form an equilibrium microstructure that fluctuates thermally with the solvent. Probe motion through the dispersion distorts the microstructure; the character of this deformation, and hence its influence on the probe's motion, depends on the strength with which the probe is forced, Fext, compared to thermal forces, kT/b, defining a Péclet number, Pe = Fext/(kT/b), where kT is the thermal energy and b the bath particle size. It is shown that the long-time mean-square fluctuational motion of the probe is diffusive and the effective diffusivity of the forced probe is determined for the full range of Péclet number. At small Pe Brownian motion dominates and the diffusive behaviour of the probe characteristic of passive microrheology is recovered, but with an incremental flow-induced ‘microdiffusivity’ that scales as Dmicro ~ DaPe2φb, where φb is the volume fraction of bath particles and Da is the self-diffusivity of an isolated probe. At the other extreme of high Péclet number the fluctuational motion is still diffusive, and the diffusivity becomes primarily force induced, scaling as (Fext/η)φb, where η is the viscosity of the solvent. The force-induced microdiffusivity is anisotropic, with diffusion longitudinal to the direction of forcing larger in both limits compared to transverse diffusion, but more strongly so in the high-Pe limit. The diffusivity is computed for all Pe for a probe of size a in a bath of colloidal particles, all of size b, for arbitrary size ratio a/b, neglecting hydrodynamic interactions. The results are compared with the force-induced diffusion measured by Brownian dynamics simulation. The theory is also compared to the analogous shear-induced diffusion of macrorheology, as well as to experimental results for macroscopic falling-ball rheometry. The results of this analysis may also be applied to the diffusive motion of self-propelled particles.

Oscillatory pipe flows of aqueous Carbopol solutions are investigated both experimentally and analytically. Using the PIV technique, the velocity profiles are measured and compared with the numerical solutions based on an elasto-viscoplastic rheological model, in which an elastic spring is serially connected to a regularized Bingham viscoplastic model. The rheological parameters, such as shear modulus of elasticity, yield stress and viscosity, are estimated from steady-shear measurements. Good agreement between the experiments and the model results is observed. It is apparent that the elasticity plays an important role in the unsteady flows of the soft yield-stress fluid studied herein.

Recent experiments in a turbulent boundary layer by Gerashchenko et al. (J. Fluid Mech., vol. 617, 2008, pp. 255–281) showed that the variance of inertial particle accelerations in the near-wall region increased with increasing particle inertia, contrary to the trend found in homogeneous and isotropic turbulence. This behaviour was attributed to the non-trivial interaction of the inertial particles with both the mean shear and gravity. To investigate this issue, we perform direct numerical simulations of channel flow with suspended inertial particles that are tracked in the Lagrangian frame of reference. Three simulations have been carried out considering (i) fluid particles, (ii) inertial particles with gravity and (iii) inertial particles without gravity. For each set of simulations, three particle response times were examined, corresponding to particle Stokes numbers (in wall units) of 0.9, 1.8 and 11.8. Mean and r.m.s. profiles of particle acceleration computed in the simulation are in qualitative (and in several cases quantitative) agreement with the experimental results, supporting the assumptions made in the simulations. Furthermore, by comparing results from simulations with and without gravity, we are able to isolate and quantify the significant effect of gravitational settling on the phenomenon.

Recent predictions from competing theoretical models have disagreed about the stability/instability of bi-periodic patterns of surface waves on deep water. We present laboratory experiments to address this controversy. Growth rates of modulational perturbations are compared to predictions from: (i) inviscid coupled nonlinear Schrödinger (NLS) equations, according to which the patterns are unstable and (ii) dissipative coupled NLS equations, according to which they are linearly stable. For bi-periodic wave patterns of small amplitude and nearly permanent form, we find that the dissipative model predicts the experimental observations more accurately. Hence, our experiments support the claim that these bi-periodic wave patterns are linearly stable in the presence of damping. For bi-periodic wave patterns of large enough amplitude or subject to large enough perturbations, both models fail to predict accurately the observed behaviour, which includes frequency downshifting.

In this paper, we solve numerically the entire set of equations associated with the electro-thermo-convective phenomena that take place in a planar layer of dielectric liquid heated from below and subjected to unipolar injection. For the first time the whole set of coupled equations is solved: Navier–Stokes equations, electrohydrodynamic (EHD) equations and the energy equation. We first validate the numerical simulation by comparing the electro-convection stability criteria with ones obtained with a stability approach. The numerical solution of the electro-thermo-convection problem is then presented entirely with a detailed analysis of stability parameters. In particular, the relation between fluid velocity, non-dimensional electrical parameter T, Rayleigh number Ra and Prandtl number Pr is given. An analytical model is presented in order to understand the flow behaviour at some critical conditions. The way that the onset of motion passes from purely electrical convection to purely thermal convection is, in particular, investigated and explained in detail. Finally, a result on the heat transfer enhancement due to electro-convection is exhibited and compared with data from experimental works available in this field.

We extend our previous analysis of closed-form equations for finite Knudsen number flow and scalar transport that result from the Boltzmann–Bhatnagar–Gross–Krook (BGK) kinetic theory with constant relaxation time. Without approximation, we obtain closed-form equations for arbitrary spatial dimension and flow directionality which are local differential equations in space and integral equations in time. These equations are further simplified for incompressible flow and scalars. The particular case of no-flow scalar transport admits analytical solutions that exhibit ballistic behaviour at short times while behaving diffusively at long times. It is noteworthy that, even with constant relaxation time BGK microphysics, quite complex macroscopic descriptions result that would be difficult to obtain using classical constitutive models or continuum averaging.

This paper presents analyses of particle image velocimetry measurements from a boundary layer on a flat plate subject to grid-generated free-stream turbulence. The pre-transition region and early stages of breakdown to turbulent spots are explored by means of quadrant analysis and quadrant hole analysis. By isolating the contributors to the Reynolds shear stresses, it is possible to identify coherent structures within the flow that are responsible for the production of TKE. It is found that so called ‘ejection’ events are the most significant form of disturbance, exhibiting the largest amplitude behaviour with increased negative spanwise vorticity. Sweep events become increasingly large close to the wall with increased Reynolds number and intermittency.

A model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behaviour of the very large scale motions. The model is derived by treating the nonlinearity in the perturbation equation (involving the Reynolds stress) as an unknown forcing, yielding a linear relationship between the velocity field response and this nonlinearity. We do not assume small perturbations. We examine propagating helical velocity response modes that are harmonic in the wall-parallel directions and in time, permitting comparison of our results to experimental data. The steady component of the velocity field that varies only in the wall-normal direction is identified as the turbulent mean profile. A singular value decomposition of the resolvent identifies the forcing shape that will lead to the largest velocity response at a given wavenumber–frequency combination. The hypothesis that these forcing shapes lead to response modes that will be dominant in turbulent pipe flow is tested by using physical arguments to constrain the range of wavenumbers and frequencies to those actually observed in experiments. An investigation of the most amplified velocity response at a given wavenumber–frequency combination reveals critical-layer-like behaviour reminiscent of the neutrally stable solutions of the Orr–Sommerfeld equation in linearly unstable flow. Two distinct regions in the flow where the influence of viscosity becomes important can be identified, namely wall layers that scale with R+1/2 and critical layers where the propagation velocity is equal to the local mean velocity, one of which scales with R+2/3 in pipe flow. This framework appears to be consistent with several scaling results in wall turbulence and reveals a mechanism by which the effects of viscosity can extend well beyond the immediate vicinity of the wall. The model reproduces inner scaling of the small scales near the wall and an approach to outer scaling in the flow interior. We use our analysis to make a first prediction that the appropriate scaling velocity for the very large scale motions is the centreline velocity, and show that this is in agreement with experimental results. Lastly, we interpret the wall modes as the motion required to meet the wall boundary condition, identifying the interaction between the critical and wall modes as a potential origin for an interaction between the large and small scales that has been observed in recent literature as an amplitude modulation of the near-wall turbulence by the very large scales.

The presence of the free surface adds an element of difficulty to the development of numerical and theoretical methods for the performance prediction of surface-piercing hydrofoils. Existing methods of analysis for two-dimensional surface-piercing hydrofoils or blade sections of a surface-piercing propeller solve either a linear problem, assuming a thin section and ventilated surface along with linear free-surface boundary conditions, or a nonlinear problem in a self-similar setting. Both these approaches cannot be used when the effects of gravity are important, which is the case when a craft is operating at low speeds. A two-dimensional boundary-element-method-based numerical scheme is presented here that overcomes these drawbacks by solving the fully ventilated flow past a surface-piercing hydrofoil of finite dimensions and includes the whole gamut of nonlinear free-surface interactions. The unique aspect of the numerical scheme is that fully nonlinear boundary conditions are applied on the free surface which allows for the accurate modelling of the jet generated on the wetted boundary and the ventilated surface formed on the suction side as a result of the passage of the hydrofoil through the free surface. Moreover, the effects of gravity can be considered to take into account the influence of the Froude number. Ventilated-surface shapes predicted by the present scheme are compared with existing experimental results and are shown to be in good agreement.

In a variety of applications, most notably microfluidics design, slip-based boundary conditions have been sought to characterize fluid flow over patterned surfaces. We focus on laminar shear flows over surfaces with periodic height fluctuations and/or fluctuating Navier scalar slip properties. We derive a general formula for the ‘effective slip’, which describes equivalent fluid motion at the mean surface as depicted by the linear velocity profile that arises far from it. We show that the slip and the applied stress are related linearly through a tensorial mobility matrix, and the method of domain perturbation is then used to derive an approximate formula for the mobility law directly in terms of surface properties. The specific accuracy of the approximation is detailed, and the mobility relation is then utilized to address several questions, such as the determination of optimal surface shapes and the effect of random surface fluctuations on fluid slip.

The steady axisymmetric flow internal and external to a deformable viscous liquid drop falling through a quiescent gas under the action of gravity is computed by solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted quadrilateral mesh. Considering typical values of the density and viscosity for common liquids and gases, numerical solutions are first computed for the liquid-to-gas density ratio ρ = 1000 and viscosity ratio μ from 50 to 1000. Visually noticeable drop deformation is shown to occur when the Weber number We ~ 5. For μ ≥ 100, drops of Reynolds number Re < 200 tend to have a rounded front and flattened or even dimpled rear, whereas those at Re > 200 a flattened front and somewhat rounded rear, with that at Re = 200 exhibiting an almost fore–aft symmetric shape. As an indicator of drop deformation, the axis ratio (defined as drop width versus height) increases with increasing We and μ, but decreases with increasing Re. By tracking the solution branches around turning points using an arclength continuation algorithm, critical values of We for the ‘shape instability’ are determined typically within the range of 10 to 20, depending on the value of Re (for Re ≥ 100). The drop shape can change drastically from prolate- to oblate-like when μ < 80 (for 100 ≤ Re ≤ 500). For example, for μ = 50 a drop at Re ≥ 200 exhibits a prolate shape when We < 10 and an upside-down button mushroom shape when We > 10. The various solutions computed at ρ = 1000 with the associated values of drag coefficient and drop shapes are found to be almost invariant at other values of ρ (e.g. from 500 to 1500) as long as the value of ρ/μ2 is fixed, despite the fact that the internal circulation intensity changes according to the value of μ. The computed values of drag coefficient are shown to agree quite well with an empirical formula for rigid spheres with the radius of the sphere replaced by the radius of the cross-sectional area.

The dynamics of fuel-lean (equivalence ratio φ = 0.5) premixed hydrogen/air atmospheric pressure flames are investigated in open cylindrical tubes with diameters of d = 1.0 and 1.5 mm using three-dimensional numerical simulations with detailed chemistry and transport. In both cases, the inflow velocity is varied over the range where the flames can be stabilized inside the computational domain. Three axisymmetric combustion modes are observed in the narrow tube: steady mild combustion, oscillatory ignition/extinction and steady flames as the inflow velocity is varied in the range 0.5 ≤ UIN ≤ 500 cm s−1. In the wider tube, richer flame dynamics are observed in the form of steady mild combustion, oscillatory ignition/extinction, steady closed and open axisymmetric flames, steady non-axisymmetric flames and azimuthally spinning flames (0.5 ≤ UIN ≤ 600 cm s−1). Coexistence of the spinning and the axisymmetric modes is obtained over relatively wide ranges of UIN. Axisymmetric simulations are also performed in order to better understand the nature of the observed transitions in the wider tube. Fourier analysis during the transitions from the steady axisymmetric to the three-dimensional spinning mode and to the steady non-axisymmetric modes reveals that the m = 1 azimuthal mode plays a dominant role in the transitions.