An initial value problem with relevance to jet noise is investigated. A plane parallel jet flow is subjected to a spatially localized initial disturbance and is then left to evolve according to the two-dimensional compressible Navier–Stokes equations. The hydrodynamic response is in the form of a convecting vortex packet. The Ffowcs Williams–Hawkings approach is formulated in the time domain and used to extrapolate from the simulated near field to the acoustic far field. The predominant downstream sound radiation comes from an early stage of nonlinear development of the vortex packet. Two simplified models to account for the radiation are introduced, based on nonlinear mode interactions on a prescribed base flow. The first uses two sets of linearized Euler equations, coupled via the inviscid Lilley–Goldstein acoustic analogy. This formulation separates the linear sound field from the sound field driven by nonlinear interactions; qualitative agreement of the latter with the Navier–Stokes computations demonstrates the importance of nonlinear interactions. The second model uses combinations of linear inviscid eigenmodes to drive the sound field, which allows extraction of the dominant mode interactions responsible for the observed radiation pattern. The results indicate that a difference-wavenumber nonlinear interaction mechanism dominates sound radiation from subsonic instability modes in shear flows.

We consider the direct numerical simulation (DNS) of a homogeneously turbulent flow in combination with a premixed flame. The combustion takes place in the flamelet regime which means that combustion occurs in a very thin layer, called the flame front. The position of the flame front is modelled by means of the $G$-equation, in which the flame front is represented by an isosurface $G_0$ of a scalar field $G({\bm x},t)$. The flow is described by the Navier–Stokes equations in the low-Mach-number limit, which allows for the inclusion of gas expansion due to the temperature increase by the combustion. The advantage of the low-Mach-number approximation is that efficient numerical methods, used for incompressible flows, can be applied to solve the discretized equations.

The calculations are carried out in a box with homogeneous isotropic turbulence. In addition, a uniform mean velocity is imposed with a inflow boundary condition at $x\,{=}\,0$. The inflow velocity is adjusted such that the mean position of the flame is stabilized at a fixed position. This allows us to use time averaging to obtain accurate statistics, which are very difficult to obtain when the flame is allowed to propagate. In the $y$- and $z$-direction, periodic boundary conditions are applied.

The numerical code has been checked with a well-known theoretical result, the so-called Darrieus–Landau instability of a thin flame front. The results show a good agreement between the computed growth rate and the theoretical value when the thickness of the flame front is much smaller than the wavelength of the disturbance. When this condition is not met, the growth rate becomes lower than the theory in agreement with the restriction under which the theory is valid.

For the computations in homogeneous turbulence, the results show an increase in the turbulent flame speed with increasing turbulent intensity at the position of the flame front. This is in good agreement with experimental data and theory. The turbulent flame speed shows also an increase as a function of the heat release parameter. This is because disturbances on the flame front, induced by the turbulence, are enhanced by the Darrieus–Landau instability.

The budgets of the turbulent kinetic energy and the enstrophy show that the expansion of the gas across the flame front suppresses the turbulence. At higher expansion rates, turbulence in the direction of the mean velocity increases and as a result turbulence becomes strongly anisotropic. The increase is due to two processes. The first is the influence of the Darrieus–Landau instability already mentioned. The second is the baroclinic production of vorticity owing to the flame front density and pressure gradients not being aligned.

A series of experiments were performed to study the absolute instability of Type I travelling crossflow modes in the boundary layer on a smooth disk rotating at constant speed. The basic flow agreed with analytic theory, and the growth of natural disturbances matched linear theory predictions. Controlled temporal disturbances were introduced by a short-duration air pulse from a hypodermic tube located above the disk and outside the boundary layer. The air pulse was positioned just outboard of the linear-theory critical radius for Type I crossflow modes. A hot-wire sensor primarily sensitive to the azimuthal velocity component, was positioned at different spatial ($r,\theta$) locations on the disk to document the growth of disturbances produced by the air pulses. Ensemble averages conditioned on the air pulses revealed wave packets that evolved in time and space. Two amplitudes of air pulses were used. The lower amplitude was verified to produced wave packets with linear amplitude characteristics. The space–time evolution of the leading and trailing edges of the wave packets were followed past the critical radius for the absolute instability, $r_{c_{A}}$. With the lower amplitudes, the spreading of the disturbance wave packets did not continue to grow in time as $r_{c_{A}}$ was approached. Rather, the spreading of the trailing edge of the wave packet decelerated and asymptotically approached a constant. This result supports previous linear DNS simulations where it was concluded that the absolute instability does not produce a global mode and that linear disturbance wave packets are dominated by the convective instability. The larger-amplitude disturbances were found to produce larger temporal spreading of the wave packets. This was accompanied by a sharp growth in the wave packet amplitude past $r_{c_{A}}$. Explanations for this are discussed.

A multiple-scale perturbation expansion is applied to extract a closed system of two equations governing the scalar descriptors of the turbulence energy spectrum from a spectral closure model. The result applies when the length scale and total energy input of a force that maintains a steady state of homogeneous isotropic turbulence are perturbed slowly and the energy spectrum consequently evolves slowly compared to the time scales of the turbulence itself.

According to parallel-flow theory based on the Orr–Sommerfeld equation, a mixing layer is unstable at all Reynolds numbers. However this is untenable from energy considerations, which demand that there exist a non-zero Reynolds number below which disturbances cannot extract net energy from the mean flow. It is shown here that a linear stability analysis of similarity solutions of the plane mixing layer, including the effects of flow non-parallelism using the minimal composite theory and the properties of adjoints, following Govindarajan & Narasimha (Theor. Comput. Fluid Dyn. vol. 19, 2005, p. 229) resolves the issue by yielding a non-zero critical Reynolds number for co-flowing streams of any velocity ratio. The critical Reynolds number for the total disturbance kinetic energy is found to be close to 30 for all velocity ratios in the range from zero to unity.

We investigate the effect of boundary roughness on the dynamical properties of the flow in laminar Ekman boundary layers. The study considers wavy boundaries having both horizontal wavelength and vertical extent comparable in size with the boundary layer width. In the case of flat boundaries, Ekman layers are known to be active, i.e. to affect significantly the dynamics of the mainstream flow. We show how the layer modelling needs to be modified to account for such wavy boundaries. In particular, nonlinear terms enter the laminar description. This model can be linearized in the limit of small Reynolds numbers. The resulting equations are studied using both asymptotic expansions and full numerical simulations. We find that small-scale roughness significantly alters energy dissipation in the boundary layer. This can result in either a reduction or an increase of dissipation, depending on, in particular, the orientation of the mainstream flow with respect to boundary modulation. Agreement is obtained between theoretical and computational results.

Well-resolved large-eddy simulations (LES) are performed in order to investigate flow phenomena and turbulence structure of the boundary layer along a supersonic compression ramp. The numerical simulations directly reproduce an available experimental result. The compression ramp has a deflection angle of $\beta\,{=}\,25^\circ$. The mean free-stream Mach number is $M_\infty\,{=}\,2.95$. The Reynolds number based on the incoming boundary-layer thickness is $Re_{\delta_0}\,{=}\,63\,560$ in accordance with the reference experiment. These simulations overcome deficiencies of earlier direct numerical simulations (DNS) and LES in terms of ramp-deflection angle, Reynolds number and spanwise size of the computational domain which is required for capturing the essential flow phenomena. The filtered conservation equations for mass, momentum and energy are solved with a high-order finite-difference scheme. The effect of subgrid scales is modelled by the approximate deconvolution model. About $18.5\,{\times}\,10^6$ grid points are used for discretizing the computational domain. To obtain mean flow and turbulence structure the flow is sampled 1272 times over 703 characteristic time scales of the incoming boundary layer. Statistical data are computed from these samples. An analysis of the data shows good agreement with the experiment in terms of mean quantities such as shock position, separation and reattachment location, skin-friction and surface-pressure distributions, and turbulence structure. The computational data confirm theoretical and experimental results on fluctuation amplification across the interaction region. In the wake of the main shock a shedding of shocklets is observed. The temporal behaviour of the coupled shock–separation system agrees well with experimental data. Unlike previous DNS the present simulation data provide indications of a large-scale shock motion. Also, evidence for the existence of three-dimensional large-scale streamwise structures, commonly referred to as Görtler-like vortices, is found.

It has been shown recently that the instability of a two-dimensional wake increases when it is confined in the transverse direction by two flat plates. Confinement causes the transition from convective to absolute instability to occur at lower values of shear. This paper examines this effect comprehensively and concludes that it is due to the constructive interaction of modes with zero group velocity in the wake (or jet) and in the surrounding flow. Maximum instability occurs when the wavenumber of the fundamental mode in the wake (or jet) matches that of the fundamental mode in the surrounding flow. Other regions of high instability occur when the harmonics of one mode interact with the fundamental of the other. This effect is examined at density ratios from 0.001 to 1000. At each density ratio, the confinement which causes maximum absolute instability can be predicted. This study also shows that it is vital to examine the wavenumber of absolutely unstable modes in order to avoid over-predicting the absolute instability. In some situations this wavenumber is vanishingly small and the mode must be discounted on physical grounds.

We propose a diagnostic technique to detect instability waves in a subsonic round jet using a phased microphone array. The detection algorithm is analogous to the beam-forming technique, which is typically used with a far-field microphone array to localize noise sources. By replacing the reference solutions used in the conventional beam-forming with eigenfunctions from linear stability analysis, the amplitudes of instability waves in the axisymmetric and first two azimuthal modes are inferred. Experimental measurements with particle image velocimetry and a database from direct numerical simulation are incorporated to design a conical array that is placed just outside the mixing layer near the nozzle exit. The proposed diagnostic technique is tested in experiments by checking for consistency of the radial decay, streamwise evolution and phase correlation of hydrodynamic pressure. The results demonstrate that in a statistical sense, the pressure field is consistent with instability waves evolving in the turbulent mean flow from the nozzle exit to the end of the potential core, particularly near the most amplified frequency of each azimuthal mode. We apply this technique to study the effects of jet Mach number and temperature ratio on the azimuthal mode balance and evolution of instability waves. We also compare the results from the beam-forming algorithm with the proper orthogonal decomposition and discuss some implications for jet noise.

The previously unstudied problem of an axisymmetric gravity current released on a cone is investigated. A formulation of the problem based on the shallow-water equations, with and without entrainment and bottom drag, is given. Analytical asymptotic solutions are found and compared to numerical solutions of the equations. The inclusion of entrainment and drag is seen to play a significant role in limiting the propagation speed of the gravity current and also in altering the shape of the current. These theoretical predictions are compared to laboratory experiments and to two-dimensional numerical simulations. The shallow-water solutions including entrainment are found to provide a much better comparison with the experiments than the solutions without entrainment. In particular, the observed front speed and dilution of the head are in good agreement. Some applications to industrial and environmental problems such as dense gas dispersion are briefly discussed.

We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien–Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter $E$ for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of $E$ on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of $E$, which are skewed with their peaks increasing towards larger values of $E$ as the Reynolds number increases. We then study the effect of multiple frequencies described by a periodically correlated random process. We find that the mean value of $E$ is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing ‘smart’ PDFs of the stochastic input that can result in improved heat transfer enhancement rates.

In this paper we consider the spectrum and stability properties of small-amplitude waves in three-dimensional inviscid compressible swirling flow with non-zero mean vorticity, contained in an infinitely long annular circular cylinder. The mean flow has swirl and sheared axial components which are general functions of radius. We describe the form of the spectrum, in particular the three distinct types of disturbance: sonic (or acoustic) modes; nearly-convected modes; and the non-modal continuous spectrum. The phenomenon of accumulation of infinitely many eigenvalues of the nearly-convected type in the complex wavenumber-plane is classified carefully: we find two different regimes of accumulating neutral modes and one regime of accumulating instability modes, and analytic conditions for the occurrence of each type of behaviour are given. We also discuss the Green's function for the unsteady field, and in particular the contribution made by the continuous spectrum. We show that this contribution can grow algebraically downstream, and is responsible for a new type of convective instability. The algebraic growth rate of this instability is a complicated function of the mean flow parameters, and can be arbitrarily large as a function of radius in cases in which the local convected wavenumber has a local extremum. The algebraic instability we describe is additional to any conventional modal instability which may be present, and indeed we exhibit cases which are convectively stable to modes, but which nevertheless grow algebraically downstream.

The transport of fluid from a recirculation region adjacent to a free surface is studied using a numerical method validated with experimental flow visualization. The flow is an example of a liquid film coating process, and consists of two counter-rotating rolls placed side-by-side and half-submerged in a bath of fluid. In the gap between the rolls a recirculation zone exists just below the free surface, around which the flow splits into two films. Fluid recirculating for long periods has been identified as a source of coating defects, so this paper considers a possible method of inducing stirring. The flow is modulated by driving one of the rolls through a Hooke's joint, which delivers a well-characterized periodic perturbation to the roll speed. In response to this speed modulation, the free surface undergoes a periodic change in position and shape which drives an exchange of fluid between the recirculation region and the surrounding flow. The amplitude of the free-surface motion is strongly dependent on modulation frequency.

The dynamics of the free surface preclude a quasi-steady approach, even in the small-frequency limit, and so a fully time-dependent analysis based on the finite element method is employed. Trigonometric temporal interpolation of the finite element data is used to make passive tracer advection calculations more efficient, and excellent agreement is seen between simulation and experiment. Computations of the stable and unstable invariant manifolds associated with periodic points on the free surface reveal that the exchange of fluid is governed by a self-intersecting turnstile mechanism, by which most fluid entrained during a modulation cycle is ejected later in the same cycle.

Transport over several cycles is explored by observation of the evacuation of passive tracers initially distributed uniformly in the recirculation zone. Results demonstrate the persistence of unmixed cores whose size is dependent on the modulation frequency. By considering the percentage of tracers remaining after a fixed number of cycles, contours in frequency–amplitude space show that for each modulation amplitude there is a frequency which produces the most effective transport, with up to 80 % of tracers removed by a modulation which produces only a 5 % change in film thickness. Finally it is shown how modulation of both rolls at slightly different phases can reduce the film thickness variation to about 1 % while maintaining the level of transport.

Nonlinear travelling waves that are precursors to laminar–turbulent transition and capture the main structures of the turbulent buffer layer have recently been found to exist in all the canonical parallel flow geometries. We study the effect of polymer additives on these ‘exact coherent states’ (ECS) in the plane Poiseuille geometry, focusing on Reynolds numbers slightly above transition. Many key aspects of the turbulent drag reduction phenomenon are found, including delay in transition to turbulence, drag reduction onset threshold, and diameter and concentration effects. Furthermore, examination of the ECS existence region leads to a distinct prediction, consistent with experiments, regarding the nature of the maximum drag reduction regime: at sufficiently high wall shear rates, viscoelasticity is found to completely suppress the normal (i.e. streamwise-vortex-dominated) dynamics of the near-wall region, suggesting that the maximum drag reduction regime is dominated by a distinct class of flow structures.

The exchange of energy for an inviscid gravity current which is released from a lock and then propagates over a horizontal boundary is considered. Attention is focused on effects due to stratification in the ambient. The investigation uses both a one-layer shallow-water model and Navier–Stokes finite-difference simulations. There is good agreement between these two approaches for the energy of the dense fluid (the current). The results indicate that with respect to the behaviour of energy as a function of time we can distinguish between: ($a$) currents propagating at supercritical speed (with respect to the fastest internal wave in the ambient), including a nose propagating into an unstratified ambient; and ($b$) currents propagating at subcritical speed, including the strongest effective stratification for which the density at the base of the ambient is equal to that of the current. The stratification enhances the accumulation of potential energy in the ambient and reduces the energy decay (dissipation) of the two-fluid system. The interaction of the internal waves with the head of the current in the subcritical case has no significant influence on the energy balance of the current.

A generalization of criticality – called secondary criticality – is introduced and applied to finite-amplitude Stokes waves. The theory shows that secondary criticality signals a bifurcation to a class of steady dark solitary waves which are biasymptotic to a Stokes wave with a phase jump in between, and synchronized with the Stokes wave. We find the that the bifurcation to these new solitary waves – from Stokes gravity waves in shallow water – is pervasive, even at low amplitude. The theory proceeds by generalizing concepts from hydraulics: three additional functionals are introduced which represent non-uniformity and extend the familiar mass flux, total head and flow force, the most important of which is the wave action flux. The theory works because the hydraulic quantities can be related to the governing equations in a precise way using the multi-symplectic Hamiltonian formulation of water waves. In this setting, uniform flows and Stokes waves coupled to a uniform flow are relative equilibria which have an attendant geometric theory using symmetry and conservation laws. A flow is then ‘critical’ if the relative equilibrium representation is degenerate. By characterizing successively non-uniform flows and unsteady flows as relative equilibria, a generalization of criticality is immediate. Recent results on the local nonlinear behaviour near a degenerate relative equilibrium are used to predict all the qualitative properties of the bifurcating dark solitary waves, including the phase shift. The theory of secondary criticality provides new insight into unsteady waves in shallow water as well. A new interpretation of the Benjamin–Feir instability from the viewpoint of hydraulics, and the connection with the creation of unsteady dark solitary waves, is given in Part 2.

The theory for criticality presented in Part 1 is extended to the unsteady problem, and a new formulation of the Benjamin–Feir instability for Stokes waves in finite depth coupled to a mean flow, which takes the criticality matrix as an organizing centre, is presented. The generation of unsteady dark solitary waves at points of stability changes and their connection with the steady dark solitary waves of Part 1 are also discussed.

This paper describes steady flow through a porous medium in a shallow two-dimensional cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated. The main emphasis is on the situation where the Darcy–Rayleigh number $R$ is large and the aspect ratio of the cavity $L$ (length/depth) is of order $R^{1/2}$. For a monotonic temperature distribution at the upper surface, the leading-order problem consists of an interaction involving the horizontal boundary-layer equations, which govern the flow throughout most of the cavity, and the vertical boundary-layer equations which govern the flow near the colder sidewall. This problem is solved using numerical and asymptotic methods. The limiting cases where $L\,{\gg}\,R^{1/2}$ and $L\,{\ll}\,R^{1/2}$ are also discussed.

We study the linear temporal hydrodynamic stability in the Rayleigh–Bénard problem for a compressible fluid (a perfect gas) under marginally super-adiabatic conditions, i.e. when the ambient temperature gradient only slightly exceeds the adiabatic gradient and then only within the fluid adjacent to the upper (cold) wall. The onset of convection in this limit demonstrates some unique features which differ qualitatively from those of the familiar Boussinesq approximation. Thus, the ensuing convection is effectively confined to a narrow domain of the fluid close to the upper wall and is characterized by large wavenumbers. Furthermore, these distinct attributes persist with diminishing temperature difference, implying that the prevailing generalized Boussinesq approximation (based on the use of the potential temperature gradient) is non-uniform in the present limit. This non-uniformity is resolved in terms of the small yet significant variations of fluid properties (which are commonly neglected). We comment on the analogy between the present problem and the Taylor–Couette problem for a viscous incompressible fluid within a narrow gap between counter-rotating cylinders. We briefly discuss the potential relevance of the present limit to some recent observations of the onset of convection within near-critical fluids.