Simultaneous planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) yield measurements of two-dimensional jet fluid concentration and velocity fields in turbulent crossflowing jets. The jet-to-crossflow velocity ratio is $r = 5.7$ and the jet exit Reynolds number is approximately 5000. The measurements are focused on the developing region of the flow. Two flow configurations are studied, one in which the jet nozzle is flush with the tunnel wall and the other where the nozzle protrudes into the uniform region of the tunnel flow. The jet nozzle in both cases is a simple pipe. The averaged scalar and velocity fields show a strong similarity in growth rates and centreline decay rates between the two nozzle configurations when using the centreline downstream coordinate $s$. This suggests that the flow geometry is less important than, for example, jet exit conditions in determining the initial flow development. The velocity fields show indications of jet-like scaling in the near field, and a possible approach to wake-like scaling in the far field. These canonical scalings are less evident in the scalar fields. Results are presented for the averaged scalar variance, $\langle C^{'2}\rangle$, the scalar flux components, $\langle u_{i}'C'\rangle$, the turbulent normal stresses, $\langle u^{\prime 2}\rangle$ and $\langle v^{\prime 2}\rangle$, and the turbulent shear stress, $\langle u'v'\rangle$. We also discuss the resolution issues bearing on the determination of small-scale fluctuations.

A peculiar multi-dimensional wave pattern in shock tube experiments on retrograde fluids is the ring formation phenomenon. When the incident shock is reflected by the endwall of a shock tube filled with retrograde fluid vapour, under certain conditions, rings of liquid/vapour mixture form behind the reflected shock. In this paper, we find the reason for such symmetry breaking and ring formation phenomena by studying a prototype model for flows with liquid/vapour phase transitions. The isothermal case of this model exhibits all major one-dimensional wave patterns observed in shock tube experiments on retrograde fluids. The symmetry is broken by the liquid or liquid/vapour drops initiated behind the reflected shock where the pressure is higher than the equilibrium pressure. Liquid drops become liquid/vapour mixture drops later. Using a liquid/vapour mixture drop at the reflecting wall as the initial condition for the prototype model, it is shown that this drop will evolve into a ring, yielding the explanation for ring formation. The outer fronts of rings are collapsing waves. Such waves are supersonic, as confirmed by some photographs of actual experiments. It is also shown that a liquid or liquid/vapour mixture drop cannot evolve into a ring if no reflecting wall is present, agreeing with experimental observation. A mechanism is also provided by the model for the emergence of the ‘button’ at the centre of, and the small-scale asperities at, the liquefaction shock front observed at the exit of the open-end shock tube.

The spreading of Newtonian fluids on smooth solid substrates is well understood; the speed of the contact line is given by a competition between capillary driving forces and viscous dissipation, yielding Tanner's law $R \propto t^{1/10}$. Here we study the spreading of non-Newtonian liquids, focusing on the two most common non-Newtonian flow properties, a shear-rate dependence of the viscosity and the existence of normal stresses. For the former, the spreading behaviour is found not to deviate strongly from Tanner's law. This is quite surprising given that, within the lubrication approximation, it can be shown that the contact line singularity disappears due to the shear-dependent viscosity. The experiments are compared with the predictions of the lubrication theory of power-law fluids. If normal stresses are present, again only small deviations from Tanner's law are found in the experiment. This can be understood by comparing viscous and normal stress contributions to the spreading; it turns out that only logarithmic corrections to Tanner's law survive, which are nonetheless visible in the experiment.

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.

The regions associated with high levels of vorticity and energy dissipation are studied in numerically simulated isotropic turbulence at $Re_\lambda = 168$. Their geometry and spatial distribution are characterized by means of box-counting methods. No clear scaling is observed for the box counts of intense strain rate and vorticity sets, presumably due to the limited inertial range, but it is shown that, even in that case, the box-counting method can be refined to characterize the shape of the intense structures themselves, as well as their spatial distribution. The fractal dimension of the individual vorticity structures, $D_\omega \rightarrow 1.1 \pm 0.1$, suggests that they tend to form filamentary vortices in the limit of high vorticity threshold. On the other hand, the intense dissipation structures have dimensions $D_{s} \simeq 1.7 \pm 0.1$, with no noticeable dependence on the threshold, suggesting structures in the form of sheets or ribbons. Statistics of the associated aspect ratios for different thresholds support these observations. Finally box counting is used to characterize the spatial distribution of the baricentres of the structures. It is found that the intense structures are not randomly distributed in space, but rather form clusters of inertial-range extent, implying a large-scale organization of the small-scale intermittent structures.

Optimal and robust control for the three-dimensional algebraically growing instability of a Blasius boundary layer is studied in the nonlinear regime. First, adjoint-based optimization is used to determine an optimal control in the form of a spanwise-uniform wall suction that attenuates the transient growth of a given initial disturbance, chosen to be the optimal perturbation of the uncontrolled flow. Secondly, a robust control is sought and computed simultaneously with the most disrupting initial perturbation for the controlled flow itself. Results for both optimal and robust control show that the optimal suction velocity peaks near the leading edge. In the robust-control case, however, the peak value is smaller, located farther downstream from the leading edge, and the suction profile is much less dependent on the control energy than in the optimal-control case.

A class of equilibria of the Euler equations is derived describing a vortex dipole of non-zero circulation interacting with the interface between a uniform shear layer and an irrotational region. The flow field and the shape of the deformed shear layer profile are given in terms of explicit mathematical formulae. Properties of the solutions are discussed. They share many qualitative features with a new translating quasi-geostrophic V-state solution recently found by McDonald (2004).

Recently a new system of field equations for the accurate description of flows in rarefied gases, called regularized 13-moment equations, was obtained by means of a hybrid gas kinetic approach. The first part of this paper discusses the relationship of the new system to classical high-order theories like the Burnett and super-Burnett equations as well as to modified models like the augmented and regularized Burnett equations. In the second part, shock structure calculations with the new theory are presented and compared to direct-simulation Monte Carlo (DSMC) solutions and to solutions of the Burnett models. Owing to additional higher-order dissipation in the system, the profiles are smooth for any Mach number, in contrast to the results of Grad's 13-moment case. The results show reliable quantitative agreement with DSMC simulations for Mach numbers up to $M_{0}\approx 3.0$. The agreement is better for Maxwell molecules than for hard spheres. The results of the augmented Burnett equations are comparable, but these equations are shown to be spatially unstable. Additionally, a validiation procedure for the new equations is presented by investigating the positivity of Grad's distribution function.

We consider the case of homogeneous turbulence in a conducting fluid that is exposed to a uniform external magnetic field at low to moderate magnetic Reynolds numbers (by moderate we mean here values as high as 20). When the magnetic Reynolds number is vanishingly small ($R_m \ll 1$), it is customary to simplify the governing magnetohydrodynamic (MHD) equations using what is known as the quasi-static (QS) approximation. As the magnetic Reynolds number is increased, a progressive transition between the physics described by the QS approximation and the MHD equations occurs. We show here that this intermediate regime can be described by another approximation which we call the quasi-linear (QL) approximation. For the numerical simulations performed, the predictions of the QL approximation are in good agreement with those of MHD for magnetic Reynolds number up to $R_m \sim 20$.

This article addresses the scaling and spectral properties of the advection–diffusion equation in closed two-dimensional steady flows. We show that homogenization dynamics in simple model flows is equivalent to a Schrödinger eigenvalue problem in the presence of an imaginary potential. Several properties follow from this formulation: spectral invariance, eigenfunction localization, and a universal scaling of the dominant eigenvalue with respect to the Péclet number $\mbox{\it Pe}$. The latter property means that, in the high-$\mbox{\it Pe}$ range (in practice $\mbox{\it Pe},\geq 10^2 \hbox{--} 10^3$), the scaling exponent controlling the behaviour of the dominant eigenvalue with the Péclet number depends on the local behaviour of the potential near the critical points (local maxima/minima). A kinematic interpretation of this result is also addressed.

A relation between the electric current and the flow rate of an electrospray operating in the cone–jet mode is proposed for the case of a very small meniscus, a long stationary jet, and a flow with important inertial effects. The finite size of the meniscus then affects the bulk-to-surface current transfer process, though the meniscus is still much larger than the drops of the spray. The result differs from the square-root law which is well-established for large menisci or very viscous flows.

From the solution of the creeping-flow equations, the drag force on a sphere becomes infinite when the gap between the sphere and a smooth wall vanishes at constant velocity, so that if the sphere is displaced towards the wall with a constant applied force, contact theoretically may not occur. Physically, the drag is finite for various reasons, one being the particle and wall roughness. Then, for vanishing gap, even though some layers of fluid molecules may be left between the particle and wall roughness peaks, conventionally it may be said that contact occurs. In this paper, we consider the example of a smooth sphere moving towards a rough wall. The roughness considered here consists of parallel periodic wedges, the wavelength of which is small compared with the sphere radius. This problem is considered both experimentally and theoretically. The motion of a millimetre size bead settling towards a corrugated horizontal wall in a viscous oil is measured with laser interferometry giving an accuracy on the displacement of 0.1 $\mu$m. Several wedge-shaped walls were used, with various wavelengths and wedge angles.

From the results, it is observed that the velocity of the sphere is, except for small gaps, similar to that towards a smooth plane that is shifted down from the top of corrugations. Indeed, earlier theories for a shear flow along a corrugated wall found such an equivalent smooth plane. These theories are revisited here. The creeping flow is calculated as a series in the slope of the roughness grooves. The cases of a flow along and across the grooves are considered separately. The shift is larger in the former case. Slightly flattened tops of the wedges used in experiments are also considered in the calculations. It is then demonstrated that the effective shift for the sphere motion is the average of the shifts for shear flows in the two perpendicular directions. A good agreement is found between theory and experiment.

The stress and conformational relaxation of an initially straight flexible polymer is studied through Brownian dynamics simulations covering a broad range of time scales and polymer lengths. At short times $t\,{\ll}\, N^{-2}$, the strong stress component scales as $\sigma_{11}\,{\sim}\, N^3$ (‘1’ is the direction of the original alignment), while the weak component is $\sigma_{22} \,{\sim}\, N$ (where $N$ is the polymer length). At intermediate times $N^{-2}\,{\ll}\, t\,{\ll}\, N^{2}$, the stress decay is shown to be anisotropic: $\sigma_{11} \,{\sim}\, N^2 t^{-1/2}$ while $\sigma_{22} \,{\sim}\, N^{1/2} t^{-1/4}$. At long times $t\,{\gg}\, N^2$, both stress components show the same exponential decay associated with the reduction of the chain's length, while their magnitudes are still different; $\sigma_{11}\,{=}\, \mbox{\it O}(N)$ while $\sigma_{22} \,{=}\, \mbox{\it O}(1)$. The configuration relaxation is studied over the same extended time periods by employing scaling laws for the evolution of the eigenvalues of the gyration tensor. After the short-time free diffusion, the configuration relaxation is anisotropic at intermediate times: the chain's width grows as $R_{\perp} \,{\sim}\, N^{-1/4} t^{3/8}$ while its length is reduced as $R_{\parallel}(0)\,{-}\,R_{\parallel} \,{\sim}\, t^{1/2}$. During long times, the polymer length shows an exponential decay towards the equilibrium coil-like shape. The polymer chain remains aligned along the original direction until late in long times where the chain rotation is shown to be significant. The chain is shown to be far from equilibrium during the entire transient relaxation. By focusing on the chain's longitudinal relaxation, a quasi-steady equilibrium of the link tensions in shown to exist along the chain's length; the mechanism is identical to that found in Grassia & Hinch (1996) based on a sideways motion model. This mechanism also explains the ability of the FENE model to describe the longitudinal relaxation of the flexible bead-rod chain. A comparison with experimental findings from initially straight single tethered DNA molecules is also included.

Flame holes and flame disks in a laminar axisymmetric counterflow configuration are numerically investigated for unity Lewis number, with the strain rate as the control parameter. The temporal evolution of the topological structure of flame holes and flame disks is described in detail for different representative strain rates. It is found that corresponding to each given strain rate, there exists a critical hole (disk) radius $r_c$ that separates the shrinking and expanding hole (disk) regimes. The value of $r_c$ decreases monotonically with the increase (decrease) of strain rate and reaches a finite minimum at the extinction (ignition) limit of the strain rate, which indicates that one cannot ignite a mixing layer by an infinitesimal energy source, nor can one quench a diffusion flame by making an infinitesimal extinction hole on it. An examination of the phase diagrams of flame holes (disks) justifies the existence of a unique edge-flame velocity $v_f$ as a smooth continuous function of the hole (disk) radius $r_f$ in the entire range $0 < r_f < \infty$, with the strain rate (or equivalently, Damköhler number) as a parameter. For the flame hole case, it is found that in the final stage of collapse of a hole, the edge-flame velocity is essentially proportional to the inverse of the hole radius, except when the strain rate is very close to the extinction limit. Flame interactions induced by overlapping of pre-heat zones are mainly responsible for the acceleration of the edge-flame velocity when the hole radius approaches zero, and it is further enhanced by the focusing effects of hole curvature in the plane of the stoichiometric surface. For the flame disk, the increasing heat loss rate plays a major role on the acceleration of the shrinking speed when the disk radius approaches zero.

In this paper, a high-order Boussinesq model is used to conduct a systematic numerical study of crescent (or horseshoe) water wave patterns in a tank, arising from the instability of steep deep-water waves to three-dimensional disturbances. The most unstable phase-locked (L2) crescent patterns are investigated, and comparisons with experimental measurements confirm the quantitative accuracy of the model. The unstable growth rate is also investigated, as are the effects of variable nonlinearity. The dominant physical mechanism is clearly demonstrated (through time and space series analysis) to be the established quintet resonant interaction, involving the primary wave with a pair of symmetric satellites. A numerical investigation into oscillating crescent patterns is also included, and a detailed account of the complicated oscillation cycle is presented. These patterns are shown to arise from quintet resonant interactions involving the primary wave with two unsymmetric satellite pairs. Pre-existing methods for analysing the stability of steep deep-water plane waves subject to three-dimensional perturbations are extended to provide accurate quantitative estimates for the oscillation period. A possible explanation for their selection in experiments is also provided. Finally, we use the model to conduct a series of experiments involving competition between various unstable modes. The results generally show that multiple instabilities can grow simultaneously, provided that they are of roughly equivalent strength. Results using random perturbations also match observations in physical experiments both in the form (i.e. two- or three-dimensional) and the location of the initial instability. The computational results are the first examples of highly nonlinear (to the breaking point) deep-water wave modeling in two horizontal dimensions with a Boussinesq model. The efficiency of the model has allowed for a quantitative study of these phenomena at significantly larger spatial and temporal scales than have been demonstrated previously, providing new insight into the complicated physical processes involved.

We consider the displacement of a ‘formation fluid’ by an ‘injection fluid’ along a temperature gradient in a reactive porous rock. We assume that the formation fluid is in chemical equilibrium initially and we allow the injection fluid to differ from the formation fluid both in temperature and in chemical composition. Two types of reaction front can form as the injection fluid propagates into the rock: (i) ‘depletion fronts’ caused by any undersaturation of the injection fluid and (ii) ‘thermal reaction fronts’ caused by any difference in temperature between the injection fluid and the trailing edge of the formation fluid. As flow proceeds the temperature gradient in the formation fluid is advected at the thermal propagation speed, but the fluid itself moves more quickly at the interstitial fluid speed. This difference in speed causes formation fluid to be carried out of chemical equilibrium and a ‘gradient reaction’ results. The subsequent dissolution or precipitation alters the amount of reactant present on the rock and this alters the propagation speed of any subsequent depletion front. We derive the six dimensionless parameters controlling the evolution of the system and calculate approximate solution profiles for the limit of fast kinetics and negligible diffusion. These solutions give insight into the physical processes governing the solution and are compared with numerical solutions of the full equations. The behaviour of the system is strongly influenced by three dimensionless parameters in particular. The parameters $J_{C}$ and $J_{T}$ quantify the chemical and thermal difference between the fluids at the point of injection and they control the relative positions of the depletion and thermal fronts at short times. On the other hand, the parameter $\alpha$ quantifies the background thermal gradient and controls the relative positions of the fronts at long times. In the cases where the depletion front moves from one side of the thermal front to the other, we derive approximate expressions for the time at which this ‘crossover’ occurs.

The initial value problem of the nonlinear evolution, shoreline motion and flow velocities of long waves climbing sloping beaches is solved analytically for different initial waveforms. A major difficulty in earlier work utilizing hodograph-type transformation when solving either boundary value or initial value problems has been the specification of equivalent boundary or initial condition in the transformed space. Here, in solving the initial value problem, the transformation is linearized in space at $t{=}0$, then the full nonlinear transformation is used to solve the initial value problem of the nonlinear shallow-water wave equations. A solution method is presented to describe the most physically realistic initial waveforms and simplified equations for the runup–rundown motions and shoreline velocities. This linearization of the initial condition does not appear to affect the subsequent nonlinear evolution, as shown through comparisons with earlier studies. Comparisons with runup results from solutions of the boundary value problem suggest the same variation with the runup laws. The methodology presented here appears simpler than earlier work as it does not involve the numerical calculation of singular elliptic integrals.