Fine-scale scalar mixing in gas-phase planar turbulent jets is studied using measurements of three-component scalar gradient and scalar energy dissipation rate fields. Simultaneous planar Rayleigh scattering and planar laser-induced fluorescence, applied in parallel planes, yield the three-dimensional scalar field measurements. The spatial resolution is sufficient to permit differentiation in all three spatial directions. The data span a range of outer-scale Reynolds numbers from 3290 to 8330. Direct measurement of the thicknesses of scalar dissipation structures (layers) shows that the thicknesses scale with outer-scale Reynolds number as $\hbox{\it Re}_\delta^{-3/4}$, consistent with Kolmogorov/Batchelor scaling. Average layer thicknesses are described by the relation $\lambda_D=14.5\,\delta\, \hbox{\it Re}_\delta^{-3/4}\hbox{\it Sc}^{-1/2}$. There is no evidence here that Taylor scaling ($\lambda_D\propto\delta\, \hbox{\it Re}_\delta^{-1/2}$) plays a significant role in the scalar dissipation process. The present data resolve a range of length scales from the dissipation scales up to nearly the jet full width, and thus can be used in a priori testing of subgrid models for scalar mixing in large-eddy simulations (LES). Comparison of two models for subgrid scalar variance, a scale-similarity model and a gradient-based model, indicates that the scale-similarity model is more accurate at larger LES filter sizes.

This paper is the first part of a two-part study on the mechanisms of the receptivity to disturbances of a Mach 4.5 flow over a flat plate by using both direct numerical simulations (DNS) and linear stability theory (LST). The main objective of the current paper is to study the linear stability characteristics of the boundary-layer wave modes and their mutual resonant interactions. The numerical solutions of both steady base flow and unsteady flow induced by forcing disturbances are obtained by using a fifth-order shock-fitting method. Meanwhile, the LST results are used to study the supersonic boundary-layer stability characteristics relevant to the receptivity study. It is found that, in addition to the conventional first and second modes, there exist a family of stable wave modes in the supersonic boundary layer. These modes play a very important role in the receptivity process of excitation of the unstable Mack modes, especially the second mode. These stable modes are termed mode I, mode II, etc., in this paper. Though mode I and mode II waves are linearly stable, they can have resonant (synchronization) interactions with both acoustic waves and the Mack-mode waves. Therefore, the stable wave modes such as mode I and mode II are critical in transferring wave energy between the acoustic waves and the unstable second mode. The effects of frequencies and wall boundary conditions for the temperature perturbations on the boundary-layer stability and receptivity are also studied.

In this paper, we continue to study the mechanisms of the receptivity of the supersonic boundary layer to free-stream disturbances by using both direct numerical simulation and linear stability theory. Specifically, the receptivity of a Mach 4.5 flow over a flat plate to free-stream fast acoustic waves is studied. The receptivity to free-stream slow acoustic waves, entropy waves and vorticity waves will be studied in the future. The oblique shock wave induced by the boundary-layer displacement plays an important role in the receptivity because the free-stream disturbance waves first pass through the shock before entering the boundary layer. A high-order shock-fitting scheme is used in the numerical simulations in order to account for the effects of interactions between free-stream disturbance waves and the oblique shock wave. The results show that the receptivity of the flat-plate boundary layer to free-stream fast acoustic waves leads to the excitation of both Mack modes and a family of stable modes, i.e. mode I, mode II, etc. It is found that the forcing fast acoustic waves do not interact directly with the unstable Mack modes. Instead, the stable mode I waves play an important role in the receptivity process because they interact with both the forcing acoustic waves and the unstable Mack-mode waves. Through the interactions, the stable mode I waves transfer wave energy from the forcing fast acoustic waves to the second Mack-mode waves. The effects of incident wave angles, forcing wave frequencies, and wall temperature perturbation conditions on the receptivity are studied. The results show that the receptivity mechanisms of the second mode are very different from those of modes I and II, which leads to very different receptivity properties of these discrete wave modes to free-stream fast acoustic waves with different incident wave angles, frequencies, and different wall boundary conditions. The maximum receptivities of the second mode, mode I and mode II to planar free-stream fast acoustic waves are obtained when incident wave angles approximately equal 26°, 45°, and 18°, respectively. The results of receptivity to a beam of free-stream fast acoustic waves show that the leading edge is one of the most efficient regions for receptivity.

We describe a new approach to modelling three-dimensional rotating stratified flows under the Boussinesq approximation. This approach is based on the explicit conservation of potential vorticity, and exploits the underlying leading-order geostrophic and hydrostratic balances inherent in these equations in the limit of small Froude and Rossby numbers. These balances are not imposed, but instead are used to motivate the use of a pair of new variables expressing the departure from geostrophic and hydrostratic balance. These new variables are the ageostrophic horizontal vorticity components, i.e. the vorticity not directly associated with the displacement of isopycnal surfaces. The use of potential vorticity and ageostrophic horizontal vorticity, rather than the usual primitive variables of velocity and density, reveals a deep mathematical structure and appears to have advantages numerically. This change of variables results in a diagnostic equation, of Monge–Ampère type, for one component of a vector potential ${\varphib}$, and two Poisson equations for the other two components. The curl of ${\varphib}$ gives the velocity field while the divergence of ${\varphib}$ is proportional to the displacement of isopycnal surfaces. This diagnostic equation makes transparent the conditions for both static and inertial stability, and may change form from (spatially) elliptic to (spatially) hyperbolic even when the flow is statically and inertially stable. A numerical method based on these new variables is developed and used to examine the instability of a horizontal elliptical shear zone (modelling a jet streak). The basic-state flow is in exact geostrophic and hydrostratic balance. Given a small perturbation however, the shear zone destabilizes by rolling up into a street of vortices and radiating inertia–gravity waves.

Linear stability of Couette flow modified by transpiration applied at the lower wall is considered. It is shown that transpiration can induce flow instability resulting in the appearance of streamwise-vortex-like structures. It is argued that the instability is driven by centrifugal forces associated with streamline curvature. The conditions leading to the onset of the instability depend on the amplitude and wavelength of the transpiration and can be expressed in terms of the critical Reynolds number. The global critical conditions describing the minimum critical Reynolds number required for the onset of the instability for the specified amplitude of the transpiration regardless of its wavelength are also given. The threshold amplitude required for the onset varies approximately as ${\sim}\hbox{\it Re}^{-1.15}$ for large $Re$, where the Reynolds number used is based on the velocity difference between the walls and the channel half-width. The existence of a global threshold, below which the instability cannot occur regardless of the amplitude of the transpiration, has been demonstrated. This threshold corresponds approximately to $\hbox{\it Re}=84$.

A powerful, easy-to-use analytic technique for nonlinear problems, the homotopy analysis method, is employed to give analytic solutions of magnetohydrodynamic viscous flows of non-Newtonian fluids over a stretching sheet. For the so-called second-order and third-order power-law fluids, the explicit analytic solutions are given by recursive formulas with constant coefficients. Also, for real power-law index and magnetic field parameter in a quite large range, an analytic approach is proposed. All of our analytic results agree well with numerical ones. In particular, a simple analytic formula of the dimensionless velocity gradient at the wall is found, which is accurate for all real power-law indices and magnetic field parameters. This analytic formula can give sufficiently accurate results for the skin friction on the moving sheet that it would find wide application in industries. Physically, they indicate that the magnetic field tends to increase the skin friction, and that this effect is more pronounced for shear-thinning than for shear-thickening fluids.

The theory of viscous potential flow is applied to the problem of finding the rise velocity $U$ of a spherical cap bubble (see Davies & Taylor 1950; Batchelor 1967). The rise velocity is given by \frac{U}{\sqrt{gD}}=-\frac{8}{3}\frac{\nu(1+8s)}{\sqrt{gD^3}}+ \frac{\sqrt{2}}{3}\left[ 1-2s-\frac{16s\sigma}{\rho gD^2}+ \frac{32v^2}{gD^3}(1+8s)^2\right]^{1/2}, \nonumber where $R = D/2$ is the radius of the cap, $\rho$ and $\nu$ are the density and kinematic viscosity of the liquid, $\sigma$ is surface tension, $r(\theta) = R(1 + s\theta^2)$ and $s = r''(0)/D$ is the deviation of the free surface from perfect sphericity $r(\theta)=R$ near the stagnation point $\theta = 0$. The bubble nose is more pointed when $s < 0$ and blunted when $s > 0.$ A more pointed bubble increases the rise velocity; the blunter bubble rises slower. The Davies & Taylor (1950) result arises when $s$ and $\nu$ vanish; if $s$ alone is zero, \[\frac{U}{\sqrt{gD}}= -\frac{8}{3}\frac{\nu}{\sqrt{gD^3}}+\frac{\sqrt{2}}{3} \left[ 1+\frac{32\nu^2}{gD^3}\right]^{1/2},\] showing that viscosity slows the rise velocity. This equation gives rise to a hyperbolic drag law \[C_D =6+32/R_e,\] which agrees with data on the rise velocity of spherical cap bubbles given by Bhaga & Weber (1981).

The dynamics of radiative, near-limit, stretched premixed flames was investigated analytically and numerically, with emphasis on pulsating stability for sub-unity Lewis numbers. The analysis includes both flame stretch and order-unity heat loss, and yields a dispersion equation for the stability of radiative stretched flames subjected to symmetrical and asymmetrical perturbations. The dispersion equation reduces to that of the classical thermo-diffusional stability analysis of Sivashinsky in the limit of small heat loss, small stretch rate, and infinite flame separation distance. Results show that sub-limit flames are stable near both radiation and stretch extinction limits, and that oscillation occurs only at moderate flame stretch rates. The unstable regime and the stability diagram were obtained. Numerical simulation with detailed chemical kinetics and transport models yielded results that are in good agreement with theory. The present work also provides a satisfactory explanation of the experimental results obtained in microgravity.

The paper is the second in a series concerned with the development of the quasi-mode concept in the context of boundary layers. The evolution of localized two-dimensional perturbations in boundary layers without inflection points is considered within the framework of linear inviscid theory. Making use of the results of Part 1 (Shrira & Sazonov 2001) for monochromatic perturbations it is shown that arbitrary broadband initial perturbations tend to a universal asymptotic regime at large times $t$. We refer to the phenomenon of the emergence of the asymptotic regime as ‘adjustment’. The regime itself corresponds to a slow dynamics of long-wave triple-deck-type perturbations and is described well as the evolution of a single quasi-mode, which allows dramatic simplification of its description. At the asymptotic stage the spatio-temporal structure of the perturbation is explicitly described in terms of Fresnel's functions with coefficients specified by certain integrals of the initial distribution. Asymptotically the perturbation represents a sharply localized group of decaying oscillations which propagate with celerity approaching the mean flow velocity at the surface. For generic perturbations, the decay, in terms of streamwise velocity, is $t^{-1/2}$. The envelope of the group is formed by the Landau damping intrinsic to the quasi-modes, and the length of the group and the number of oscillations in the group grow with time as $t^{2/3}$ and $t^{1/6}$, respectively. The evolution of the non-quasi-modal part is also investigated. The vorticity perturbation is found to form a vortex patch shaped like a comet tail and advected by the mean flow. The picture of evolution established for generic perturbations is found to hold for several classes of non-generic but physically relevant initial distributions; the corresponding solutions are presented and discussed. The analytical results have been confirmed by the direct numerical simulation of the linearized primitive equations.

Microbubble-laden homogeneous and isotropic turbulent flow is investigated by using direct numerical simulation of the three-dimensional Navier–Stokes equations and computing the bubble trajectories with Lagrangian tracking. The bubble motion is calculated by taking into account the effect of fluid acceleration plus added mass, drag, gravity, and in particular the lift force, which had been neglected in many previous simulations. By comparing the results from simulations with and without lift, we find the effect of the lift force to be crucial: for passive bubbles, i.e. bubbles without backreaction on the flow (one-way coupling), the lift enhances the accumulation of bubbles on the downward flow side of vortices, resulting in a considerably reduced rise velocity of bubbles in turbulent flow, compared to still water. This also has consequences for the active bubble case, i.e. for bubbles with backreaction on the flow (two-way coupling): the energy spectrum of the turbulence is modified {non-uniformly}. Because of the combined effect of preferential bubble clustering in downflow zones and the local buoyant transfer, which reduces the vertical fluid velocity fluctuations, large-scale motions (small wavenumbers $k$) are suppressed. In contrast, small-scale motions (large wavenumbers $k$) are enhanced due to the local bubble forcing. The net effect turns out to be a reduction of the energy dissipation rate.

The purpose of this article is to show that the Navier–Stokes equations can be rewritten as a set of linearized inhomogeneous Euler equations (in convective form) with source terms that are exactly the same as those that would result from externally imposed shear-stress and energy-flux perturbations. These results are used to develop a mathematical basis for some existing and potential new jet noise models by appropriately choosing the ‘base flow’ about which the linearization is carried out.

This paper examines the low-Reynolds-number migration of an insulating and rigid particle that is freely suspended in a viscous liquid metal and subject to uniform ambient electric and magnetic fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B}.$ Under the same physical assumptions as Part 1, a whole boundary formulation of the problem is established. It allows the determination of the particle rigid-body motion without calculating the modified electric field and the flow induced by the Lorentz body force in the fluid domain. The advocated boundary approach, well-adapted for future numerical implementation, makes it possible to obtain an analytical expression for the translational velocity of any ellipsoidal particle (the simplest case of non-spherical orthotropic particles). The behaviour of a spheroid is carefully investigated and discussed both without and with gravity. The migration of this simple non-spherical particle is found to depend on both its nature (prolate or oblate) and the ambient uniform fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B}.$ The spheroid translates without rotation, and not necessarily parallel to $\hbox{\bf\itshape E}\wedge \hbox{\bf\itshape B}.$ For adequately selected fields $\hbox{\bf\itshape E}$ and $\hbox{\bf\itshape B},$ the spheroid may either migrate parallel or anti-parallel to a sphere and even be motionless.

We implement a linear stability analysis for the viscous annular jet surrounded by inviscid motionless gas subject to three-dimensional disturbances, and an analytical dispersion relation is presented. With this relation, we are able to obtain evidence that the axisymmetric mode is the most unstable mode of the system. The evidence for general cases is based on the numerical results obtained by solving the transcendental dispersion relation for a wide range of physical parameters. The results for two special cases, a very thin annular jet and an annular jet with disturbances of very short axial wavelength, are obtained from two rigorous analytical approaches.

A model is developed for the adjustment of the spatially averaged time-mean flow of a deep turbulent boundary layer over small roughness elements to a canopy of larger three-dimensional roughness elements. Scaling arguments identify three stages of the adjustment. First, the drag and the finite volumes of the canopy elements decelerate air parcels; the associated pressure gradient decelerates the flow within an impact region upwind of the canopy. Secondly, within an adjustment region of length of order $L_c$ downwind of the leading edge of the canopy, the flow within the canopy decelerates substantially until it comes into a local balance between downward transport of momentum by turbulent stresses and removal of momentum by the drag of the canopy elements. The adjustment length, $L_c$, is proportional to (i) the reciprocal of the roughness density (defined to be the frontal area of canopy elements per unit floor area) and (ii) the drag coefficient of individual canopy elements. Further downstream, within a roughness-change region, the canopy is shown to affect the flow above as if it were a change in roughness length, leading to the development of an internal boundary layer. A quantitative model for the adjustment of the flow is developed by calculating analytically small perturbations to a logarithmic turbulent velocity profile induced by the drag due to a sparse canopy with $L/L_c \ll 1$, where $L$ is the length of the canopy. These linearized solutions are then evaluated numerically with a nonlinear correction to account for the drag varying with the velocity. A further correction is derived to account for the finite volume of the canopy elements. The calculations are shown to agree with experimental measurements in a fine-scale vegetation canopy, when the drag is more important than the finite volume effects, and a canopy of coarse-scale cuboids, when the finite volume effects are of comparable importance to the drag in the impact region. An expression is derived showing how the effective roughness length of the canopy, $\z0eff$, is related to the drag in the canopy. The value of $\z0eff$ varies smoothly with fetch through the adjustment region from the roughness length of the upstream surface to the equilibrium roughness length of the canopy. Hence, the analysis shows how to resolve the unphysical flow singularities obtained with previous models of flow over sudden changes in surface roughness.

Using an observation by Lamb, namely that continuum sound wave propagation in sufficiently narrow channels is quasi-steady and isothermal, we obtain analytical predictions for the propagation of sound waves at small scales under non-continuum transport. We also extend Lamb's approach to include the effects of inertia and heat conduction for wave propagation at larger characterisitc scales descibed by continuum transport (no-slip and slip-flow regimes). Our theoretical predictions are compared to molecular-based direct Monte Carlo solutions of the Boltzmann equation. Very good agreement is found between theory and numerical solutions.