To resolve possible fluctuations about the mean motion of the Desnyansky-Novikov model for Kolmogorov's inertial flow, we have investigated two dynamical systems of the cascade process which are formally derivable from Burgers’ equation. The first cascade model produced no fluctuations, for its trajectory was identical with the Desnyansky–Novikov model's. Disappointingly, the second cascade system, which is similar to the Kerr–Siggia model, has also proved unable to engender fluctuations. This is because the second model when truncated consistently maps an arbitrary initial point into the attainable phase space of the first cascade model. However, when truncated inconsistently the trajectory of second model can exhibit a quite erratic and somewhat sporadic motion, thereby reflecting the apparently random motion of inviscid equilibrium solutions. Therefore, the observation of temporally intermittent fluctuations by a stationary Kerr–Siggia model is due to the inconsistent truncation produced by restricting energy dissipation for all but the upper truncation mode in their model.

This work builds on the foundation laid by Benney & Timson (Stud. Appl. Maths, vol. 63, 1980, pp. 93–98), who examined the flow near a contact line and showed that, if the contact angle is $18{0}^{\circ } $, the usual contact-line singularity does not arise. Their local analysis, however, does not allow one to determine the velocity of the contact line and their expression for the shape of the free boundary involves undetermined constants. The present paper considers two-dimensional Couette flows with a free boundary, for which the local analysis of Benney & Timson can be complemented by an analysis of the global flow (provided that the slope of the free boundary is small, so the lubrication approximation can be used). We show that the undetermined constants in the solution of Benney & Timson can all be fixed by matching the local and global solutions. The latter also determines the contact line’s velocity, which we compute among other characteristics of the global flow. The asymptotic model derived is used to examine steady and evolving Couette flows with a free boundary. It is shown that the latter involve brief intermittent periods of rapid acceleration of contact lines.

The so-called Crocco integral establishes a relation between the velocity and temperature distributions in steady boundary layer flow. It corresponds to an exact solution of the flow equations in the case of unity Prandtl number and an adiabatic wall, where it reduces to the condition that the total enthalpy remains constant throughout the boundary layer, irrespective of pressure gradient and compressibility. The effect of Prandtl number is usually incorporated by assuming a constant recovery factor across the entire boundary layer. Strictly, however, this modification is in conflict with the conservation-of-energy principle. In search of a more complete expression for the Crocco integral the present study applies an asymptotic solution approach to the energy equation in constant-property flow. The analysis of self-similar boundary layer solutions results in a formulation of the Crocco integral which correctly incorporates the effect of Prandtl number to first order, and that is complete in the sense that it satisfies the energy conservation requirement. Furthermore, the result is found to be applicable not only to self-similar boundary layers, but also to provide a solution to the laminar flow equations in general as well. The effect of varying properties is considered with regard to the extension of the expression to more general flow conditions. In addition to the asymptotic expression for the Crocco integral, asymptotic solutions are also obtained for the recovery factor for various classes of flows.

Unsteady viscous flow over irregular and fractal walls is discussed, and the flow generated by the longitudinal and transverse vibrations of an infinite periodic two-dimensional wall with cylindrical grooves is considered in detail. The behaviour of the Stokes layer and the functional dependence between the drag force and the frequency are illustrated in a broad band of frequencies for walls with sinusoidal corrugations and a family of walls with triangular asperities leading to fractal shapes. It is shown that, in the case of longitudinal oscillations, the drag force on a fractal wall with self-similar structure exhibits a power-law dependence on the frequency with an exponent that is related to the fractal dimension of the microstructure expressing the gain in surface area with increasing spatial resolution. Numerical evidence suggests that, in the case of transverse oscillations, the dissipative component of the drag force may show a power-law dependence on the frequency, but the exponent is not directly related to the geometry of the microstructure. The significance of these results on the behaviour of the drag force on walls with three-dimensional irregularities is discussed.

The two-dimensional problem of advection–dispersion associated with a non-aqueous-phase liquid (NAPL) pool is addressed using the boundary element method. The problem is appropriately posed with an inhomogeneous boundary condition taking into consideration the presence of the pool and the impermeable layer. We derive a Fredholm integral equation of the first kind for the concentration gradient along the pool location and compute the average mass transfer coefficient numerically using the boundary-element method. Numerical results are in agreement with asymptotic analytical solutions obtained for the cases of small and large Péclet number (Pex). The asymptotic solution for small Pex, which is obtained by applying a novel perturbation technique to the integral equation, is used to de-singularize the integral equation. Results predicted by this analysis are in good agreement with experimentally determined overall mass transfer coefficients.

This paper considers the manner in which turbulent premixed flames respond to a superposition of turbulent and narrowband disturbances. This is an important fundamental problem that arises in most combustion applications, as turbulent flames exist in hydrodynamically unstable flow fields and/or in confined systems with narrowband acoustic waves. This paper presents the first measurements of the sensitivity of the turbulent displacement speed to harmonically oscillating flame wrinkles. The flame is attached to a transversely oscillating, heated wire, resulting in the introduction of coherent, convecting wrinkles on the flame. The approach flow turbulence is varied systematically using a variable turbulence generator, enabling quantification of the effect of turbulent flow disturbances on the harmonic wrinkles. Mie scattering measurements are used to quantify the flame edge dynamics, while high speed particle image velocimetry is used to measure the flow field characteristics. By ensemble averaging the results, the ensemble-averaged flame edge and flow characteristics are recovered. For low turbulence intensities, sharp cusps are present in the negative curvature regions of the ensemble-averaged flame position, similar to laminar flames. These cusps are smoothed out at high turbulence intensities. The coherent, ensemble-averaged flame wrinkle amplitude decays with increasing turbulence intensity and with downstream distance. In addition, the ensemble-averaged turbulent flame speed is modulated in space and time. The most significant result of these measurements is the clear demonstration of the correlation between the ensemble-averaged turbulent flame speed and ensemble-averaged flame curvature, with the phase-dependent flame speed increasing in regions of negative curvature. These results have important implications on turbulent combustion physics and modelling, since quasi-coherent velocity disturbances are nearly ubiquitous in shear driven, high turbulent flows and/or confined systems with acoustic feedback. Specifically, these data clearly show that nonlinear interactions occur between the multi-scale turbulent disturbances and the more narrowband disturbances associated with coherent structures. In other words, conceptual models of the controlling physics in combustors with shear driven turbulence must account for the fundamentally different effects of spectrally distributed turbulent disturbances and more narrowband, quasi-coherent disturbances.

The dynamics of a plane interface separating two sheared, density and viscosity matched fluids in the vertical gap between parallel plate electrodes are studied computationally. A Couette profile is imposed onto the fluids by moving the rigid plates at equal speeds in opposite directions. In addition, a vertical electric field is applied to the shear flow by impressing a constant voltage difference on the electrodes. The stability of the initially flat interface is a very subtle balance between surface tension, inertia, viscosity and electric field effects. Under unstable conditions, the potential difference in the fluid results in an electrostatic pressure that amplifies disturbance waves on the two-fluid interface at characteristic wave lengths. Various mechanisms determining the growth rate of the most unstable mode are addressed in a systematic parameter study. The applied methodology involves a combination of numerical simulation and analytical work. Linear stability theory is employed to identify unstable parametric conditions of the perturbed Couette flow. Particular attention is given to the effect of the applied electric field on the instability of the perturbed two-fluid interface. The normal mode analyses are followed up by numerical simulations. The applied method relies on solving the governing equations for the fluid mechanics and the electrostatics in a one-fluid approximation by using a finite-volume technique combined with explicit tracking of the evolving interface. The numerical results confirm those of linear theory and, furthermore, reveal a rich array of dynamical behaviour. The elementary fluid instabilities are finger-like structures of interpenetrating fluids. For weakly unstable situations a single fingering instability emerges on the interface. Increasing the growth rates causes the finger to form a drop-like tip region connected by a long thinning fluids neck. Even more striking fluid motion occurs at higher values of the electric field parameter for which multiple fluid branches develop on the interface. For a pair of perfect dielectrics the vertical electric field was found to enhance interfacial motion irrespective of the permittivity ratio, while in leaky dielectrics the electric field can either stabilize or destabilize the interface, depending on the conductivity and permittivity ratio between the fluids.

Fully turbulent Taylor–Couette flow between independently rotating cylinders is investigated experimentally in a wide-gap configuration ($\unicode[STIX]{x1D702}=0.5$) around the maximum transport of angular momentum. In that regime turbulent Taylor vortices are present inside the gap, leading to a pronounced axial dependence of the flow. To account for this dependence, we measure the radial and azimuthal velocity components in horizontal planes at different cylinder heights using particle image velocimetry. The ratio of angular velocities of the cylinder walls $\unicode[STIX]{x1D707}$, where the torque maximum appears, is located in the low counter-rotating regime ($\unicode[STIX]{x1D707}_{max}(\unicode[STIX]{x1D702}=0.5)=-0.2$). This point coincides with the smallest radial gradient of angular velocity in the bulk and the detachment of the neutral surface from the outer cylinder wall, where the azimuthal velocity component vanishes. The structure of the flow is further revealed by decomposing the flow field into its large-scale and turbulent contributions. Applying this decomposition to the kinetic energy, we can analyse the formation process of the turbulent Taylor vortices in more detail. Starting at pure inner cylinder rotation, the vortices are formed and strengthened until $\unicode[STIX]{x1D707}=-0.2$ quite continuously, while they break down rapidly for higher counter-rotation. The same picture is shown by the decomposed Nusselt number, and the range of rotation ratios, where turbulent Taylor vortices can exist, shrinks strongly in comparison to investigations at much lower shear Reynolds numbers. Moreover, we analyse the scaling of the Nusselt number and the wind Reynolds number with the shear Reynolds number, finding a communal transition at approximately $Re_{S}\approx 10^{5}$ from classical to ultimate turbulence with a transitional regime lasting at least up to $Re_{S}\geqslant 2\times 10^{5}$. Including the axial dispersion of the flow into the calculation of the wind amplitude, we can also investigate the wind Reynolds number as a function of the rotation ratio $\unicode[STIX]{x1D707}$, finding a maximum in the low counter-rotating regime slightly larger than $\unicode[STIX]{x1D707}_{max}$. Based on our study it becomes clear that the investigation of counter-rotating Taylor–Couette flows strongly requires an axial exploration of the flow.

Steady and unsteady velocity components over a backward-facing circular arc are measured by laser-Doppler velocimetry. A periodic disturbance is added to the mean flow and the response of unsteady separation is investigated. Special attention is given to the distribution and the flux of vorticity.

Continents exert a thermal blanket effect to the mantle underneath by locally accumulating heat and modifying the flow structures, which in turn affects continent motion. This dynamic feedback is studied numerically with a simplified model of an insulating plate over a thermally convecting fluid with infinite Prandtl number at Rayleigh numbers of the order of $10^{6}$. Several plate–fluid coupling modes are revealed as the plate size varies. In particular, small plates show long durations of stagnancy over cold downwellings. Between long stagnancies, bursts of velocity are observed when the plate rides on a single convection cell. As plate size increases, the coupled system transitions to another type of short-lived stagnancy, in which case hot plumes develop underneath. For an even larger plate, a unidirectional moving mode emerges as the plate modifies impeding flow structures it encounters while maintaining a single convection cell underneath. These identified modes are reminiscent of some real cases of continent–mantle coupling. Results show that the capability of a plate to overcome impeding flow structures increases with plate size, Rayleigh number and intensity of internal heating. Compared to cases with a fixed plate, cases with a freely drifting plate are associated with higher Nusselt number and more convection cells within the flow domain.