We study the gas flow processes in ultra-tight porous media in which the matrix pore network is composed of nanometre- to micrometre-size pores. We formulate a pressure-dependent permeability function, referred to as the apparent permeability function (APF), assuming that Knudsen diffusion and slip flow (the Klinkenberg effect) are the main contributors to the overall flow in porous media. The APF predicts that in nanometre-size pores, gas permeability values are as much as 10 times greater than results obtained by continuum hydrodynamics predictions, and with increasing pore size (i.e. of the order of the micrometre), gas permeability converges to continuum hydrodynamics values. In addition, the APF predicts that an increase in the fractal dimension of the pore surface leads to a decrease in Knudsen diffusion. Using the homogenization method, a rigorous analysis is performed to examine whether the APF is preserved throughout the process of upscaling from local scale to large scale. We use the well-known pulse-decay experiment to estimate the main parameter of the APF, which is Darcy permeability. Our newly derived late-transient analytical solution and the late-transient numerical solution consistently match the pressure decay data and yield approximately the same estimated value for Darcy permeability at the typical core-sample initial pressure range and pressure difference. Other parameters of the APF may be determined from independent laboratory experiments; however, a pulse-decay experiment can be used to estimate the unknown parameters of the APF if multiple tests are performed and/or the parameters are strictly constrained by upper and lower bounds.

The formation of singularities in two-dimensional magnetohydrodynamic flow is investigated by direct numerical simulation. It is shown that two-dimensional magnetohydrodynamic turbulence is not as singular as three-dimensional hydrodynamic turbulence (in the sense that it has a less highly excited small-scale structure) but that it is more singular than two-dimensional hydrodynamic turbulence.

A theory describing two-phase displacement in the gap between closely spaced planes is developed. The main assumptions of the theory are that the displaced fluid wets the walls, and that the capillary number Ca and the ratio of gap width to transverse characteristic length ε are both small. Relatively mild restrictions apply to the ratio M of viscosities of displacing to displaced fluids; in particular the theory holds for M = o(Ca−1/3). We formulate the theory as a double asymptotic expansion in the small parameters ε and Ca1/3. The expansion in ε is uniform while that in Ca1/3 is not, necessitating the use of matched asymptotic expansions. The previous work of Bretherton (1961) is clarified and extended, and both the form and the constants in the effective boundary condition of Chouke, van Meurs & van der Poel (1959) and of Saffman & Taylor (1958) are determined.

The displacement height appears in the logarithmic velocity profile for rough-wall boundary layers as a reference height for the vertical co-ordinate. It is shown that this height should be regarded as the level at which the mean drag on the surface appears to act. The equations of motion then show that this also coincides with the average displacement thickness for the shear stress.

A simple analytical model, experimental results and dimensional analysis are all used to indicate how the displacement height depends upon the detailed geometry of the roughness elements.

A nominally-zero-pressure-gradient incompressible boundary layer over a smooth flat plate was simulated for a continuous momentum thickness Reynolds number range of 80 ≤ Reθ ≤ 940. Transition which is completed at approximately Reθ = 750 was triggered by intermittent localized disturbances arising from patches of isotropic turbulence introduced periodically from the free stream at Reθ = 80. Streamwise pressure gradient is quantified with several measures and is demonstrated to be weak. Blasius boundary layer is maintained in the early transitional region of 80 < Reθ < 180 within which the maximum deviation of skin friction from the theoretical solution is less than 1%. Mean and second-order turbulence statistics are compared with classic experimental data, and they constitute a rare DNS dataset for the spatially developing zero-pressure-gradient turbulent flat-plate boundary layer. Our calculations indicate that in the present spatially developing low-Reynolds-number turbulent flat-plate boundary layer, total shear stress mildly overshoots the wall shear stress in the near-wall region of 2–20 wall units with vanishing normal gradient at the wall. Overshoots as high as 10% across a wider percentage of the boundary layer thickness exist in the late transitional region. The former is a residual effect of the latter. The instantaneous flow fields are vividly populated by hairpin vortices. This is the first time that direct evidence (in the form of a solution of the Navier–Stokes equations, obeying the statistical measurements, as opposed to synthetic superposition of the structures) shows such dominance of these structures. Hairpin packets arising from upstream fragmented Λ structures are found to be instrumental in the breakdown of the present boundary layer bypass transition.

The stability of cellular convection flow in a layer heated from below is discussed for Rayleigh number R close to the critical value Rc. It is shown that in this region the stable stationary solution is determined by a minimum of the integral \[ \int_0^{H_0}R(H)\,dH, \] where R(H) is a functional of arbitrary convective velocity fields which satisfy the boundary conditions. For the stationary solutions R(H) is equal to the Rayleigh number. H0 is a given value of the convective heat transport. In a second part of the paper explicit results are derived for the convection problem with deviations from the Boussinesq approximation owing to the temperature dependence of the material properties.

An analytical theory is developed for the stability properties of planar fronts of premixed laminar flames freely propagating downwards in a uniform reacting mixture. The coupling between the hydrodynamics and the diffusion process is described for an arbitrary expansion of the gas across the flame. Viscous effects are included with an arbitrary Prandtl number. The flame structure is described for a large value of the reduced activation energy and for a Lewis number close to unity. The flame thickness is assumed to be small compared with the wavelength of the wrinkles of the front, this wavelength being also the characteristic lengthscale of the perturbations of the flow field outside the flame. A two-scale method is then used to solve the problem. The results show that the acceleration of gravity associated with the diffusion mechanisms inside the front can counterbalance the hydrodynamical instability when the laminar-flame velocity is low enough. The theory provides predictions concerning the instability threshold. In particular, the dimensions of the cells are predicted to be large compared with the flame thickness, and thus the basic assumption of the theory is verified. Furthermore, the quantitative predictions are in good agreement with the existing experimental data.

The bifurcation is shown to be of a different nature than predicted by the purely diffusive–thermal model.

The viscous diffusivities are supposed to be independent of the temperature, and then the viscosity is proved to have no effect at all on the dynamical properties of the flame front.

This paper is a contribution to the mathematical analysis of the stability of steady axisymmetric parallel flows of uniform fluid in the absence of rigid boundaries. A jet at sufficiently high Reynolds number for the angle of viscous spreding to be small is the typical example of the primary flows considered, and the theoretical velocity profile far downstream in such a jet is kept in mind continually. It is obvious from experience that such jets are unstable, presumably to infinitesimal disturbances, but there is little observational data about the critical Reynolds number or the mode of disturbance that grows most rapidly at a given Reynolds number.

The typical small disturbance considered is a Fourier component with sinusoidal dependence on both ax and nϕ (x, r, ϕ are cylindrical polar co-ordinates). There is no analogue of Squire's theorem for two-dimensional primary flows, and both α and n are essential parameters of the disturbance. We have concentrated on the stability characteristics in the limit of large Reynolds number, and have aimed in particular at determining the (integral) value of n at which the growth rate is a maximum in these simpler circumstances.

A number of general results for inviscid fluid are established, many of them analogues of corresponding results for two-dimensional primary flow. A necessary condition for the existence of amplified disturbances is that $Q(r)=rU^{\prime}|(n^2+\alpha ^2 r^2)$ should have a numerical maximum a t some point in the fluid; this condition is satisfied for all n in the case of a cylindrical shear layer or ‘top-hat’ jet profile (for which a complete solution of the disturbance equation can be obtained), and for n > 1 in the case of a ‘far-downstream’ jet profile. The wave speed cr of a neutral disturbance is equal to the value of U either a t the point where dQ/dr = 0 or at r = 0. In the latter case the eigen-function (if one exists) is singular at the axis in general; the former case is presumably relevant to the ‘upper branch’ of the curve of neutral stability (for given n). The Reynolds stress due to the disturbance acting across a cylindrical surface is examined. Here, as in some other contexts, it is useful to consider components of velocity parallel and perpendicular to a circular helix on which the phase of the disturbance wave is constant. For a neutral disturbance the component of disturbance velocity parallel to the local wave helix is infinite a t the critical point where U = cr, (corresponding to the known singularity for a three-dimensional disturbance to two-dimensional flow), and there is a peak in the Reynolds stress there.

It is shown from the form of the disturbance equation that there is an upper limit to the value of n (≠ 0) for a neutral (inviscid) disturbance with cr equal to the value of U at the point where Q′ = 0. In the case of a jet with a ‘far-downstream’ profile, only the value n = 1 satisfies this restriction; thus only the sinuous mode n = 1 can yield amplified disturbances in an inviscid fluid. A numerical investigation shows that for this profile the wave-number of the neutral disturbance with n = 1 is α = 1·46.

The outer intermittent region of a fully developed turbulent boundary layer with zero pressure gradient was extensively explored in the hope of shedding some light on the shape and motion of the interface separating the turbulent and non-turbulent regions as well as on the nature of the related large-scale eddies within the turbulent regime. Novel measuring techniques were devised, such as conditional sampling and conditional averaging, and others were turned to new uses, such as reorganizing in map form the space-time auto- and cross-correlation data involving both the U and V velocity components as well as I, the intermittency function. On the basis of the new experimental results, a conceptual model for the development of the interface and for the entrainment of new fluid is proposed.

A passive technique of increasing entrainment was found by using a small-aspect-ratio elliptic jet. The entrainment ratio of an elliptic jet was several times greater than that of a circular jet or a plane jet. The self-induction of the asymmetric coherent structure caused azimuthal distortions which were responsible for engulfing large amounts of surrounding fluid into the jet. In an elliptic jet, an interesting feature in the initial stability process is that the thickness of the shear layer varies around the nozzle. The data indicated that instability frequency was scaled with the thinnest initial momentum thickness which was associated with the maximum vorticity. Turbulence properties were also examined and were found to be significantly different in the major- and minor-axis planes.

Analysis of combustion instabilities relies in most cases on linear analysis but most observations of these processes are carried out in the nonlinear regime where the system oscillates at a limit cycle. The objective of this paper is to deal with these two manifestations of combustion instabilities in a unified framework. The flame is recognized as the main nonlinear element in the system and its response to perturbations is characterized in terms of generalized transfer functions which assume that the gain and phase depend on the amplitude level of the input. This ‘describing function’ framework implies that the fundamental frequency is predominant and that the higher harmonics generated in the nonlinear element are weak because the higher frequencies are filtered out by the other components of the system. Based on this idea, a methodology is proposed to investigate the nonlinear stability of burners by associating the flame describing function with a frequency-domain analysis of the burner acoustics. These elements yield a nonlinear dispersion relation which can be solved, yielding growth rates and eigenfrequencies, which depend on the amplitude level of perturbations impinging on the flame. This method is used to investigate the regimes of oscillation of a well-controlled experiment. The system includes a resonant upstream manifold formed by a duct having a continuously adjustable length and a combustion region comprising a large number of flames stabilized on a multipoint injection system. The growth rates and eigenfrequencies are determined for a wide range of duct lengths. For certain values of this parameter we find a positive growth rate for vanishingly small amplitude levels, indicating that the system is linearly unstable. The growth rate then changes as the amplitude is increased and eventually vanishes for a finite amplitude, indicating the existence of a limit cycle. For other values of the length, the growth rate is initially negative, becomes positive for a finite amplitude and drops to zero for a higher value. This indicates that the system is linearly stable but nonlinearly unstable. Using calculated growth rates it is possible to predict amplitudes of oscillation when the system operates on a limit cycle. Mode switching and instability triggering may also be anticipated by comparing the growth rate curves. Theoretical results are found to be in excellent agreement with measurements, indicating that the flame describing function (FDF) methodology constitutes a suitable framework for nonlinear instability analysis.

In the outer region of fully developed turbulent pipe flow very large-scale motions reach wavelengths more than 8$R$–16$R$ long (where $R$ is the pipe radius), and large-scale motions with wavelengths of $2R$–$3R$ occur throughout the layer. The very-large-scale motions are energetic, typically containing half of the turbulent kinetic energy of the streamwise component, and they are unexpectedly active, typically containing more than half of the Reynolds shear stress. The spectra of the $y$-derivatives of the Reynolds shear stress show that the very-large-scale motions contribute about the same amount to the net Reynolds shear force, d$\overline{-u'v'}/{\rm d}y$, as the combination of all smaller motions, including the large-scale motions and the main turbulent motions. The main turbulent motions, defined as the motions small enough to be in a statistical equilibrium (and hence smaller than the large-scale motions) contribute relatively little to the Reynolds shear stress, but they constitute over half of the net Reynolds shear force.

An analytic model of free-surface flow over an erodible bed is developed and used to investigate the stability of the fluid-bed interface and the characteristics of the bed features. The model is based on the potential flow over a two-dimensional, moving, wavy bed with a sinusoidal profile of varying amplitude, and a sediment transport relation in which the transport rate is proportional to a power of the fluid velocity at the level of the bed. By assuming that the dominant wavelength is that for which the rate of amplitude growth is the greatest, expressions are obtained for the wavelength and velocity of the bed features. In addition, conditions for the occurrence of the different configurations, dunes, flat bed, and antidunes, are found from the model. The predicted wavelengths of antidunes and ranges of wavelengths of dunes, and the predicted conditions for change of bed configuration are found to be in good agreement with experimental data. Finally, brief consideration is given to the factors involved in determining the maximum heights of the bed features and surface waves.

Combustion instability is investigated in the case of a multiple inlet combustor with dump. It is shown that low-frequency instabilities are acoustically coupled and occur at the eigenfrequencies of the system. Using spark-schlieren and a special phase-average imaging of the C2-radical emission, the fluid-mechanical processes involved in a vortex-driven mode of instability are investigated. The phase-average images provide maps of the local non-steady heat release. From the data collected on the combustor the processes of vortex shedding, growth, interactions and burning are described. The phases between the pressure, velocity and heat-release fluctuations are determined. The implications of the global Rayleigh criterion are verified and a mechanism for low-frequency vortex-driven instabilities is proposed.

Conditionally sampled hot-wire measurements were taken in a pipe at Reynolds numbers corresponding to the onset of turbulence. The pipe was smooth and carefully aligned so that turbulent slugs appeared naturally at Re > 5 × 104. Transition could be initiated at lower Re by introducing disturbances into the inlet. For smooth or only slightly disturbed inlets, transition occurs as a result of instabilities in the boundary layer long before the flow becomes fully developed in the pipe. This type of transition gives rise to turbulent slugs which occupy the entire cross-section of the pipe, and they grow in length as they proceed downstream. The leading and trailing ‘fronts’ of a turbulent slug are clearly defined. A unique relation seems to exist between the velocity of the interface and the velocity of the fluid by which relaminarization of turbulent fluid is prevented. The length of slugs is of the same order of magnitude as the length of the pipe, although the lengths of individual slugs differ at the same flow conditions. The structure of the flow in the interior of a slug is identical to that in a fully developed turbulent pipe flow. Near the interfaces, where the mean motion changes from a laminar to a turbulent state, the velocity profiles develop inflexions. The total turbulent intensity near the interfaces is very high and it may reach 15% of the velocity at the centre of the pipe. A turbulent energy balance was made for the flow near the interfaces. All of the terms contributing to the energy balance must vanish identically somewhere on the interface if that portion of the interface does not entrain non-turbulent fluid. It appears that diffusion which also includes pressure transport is the most likely mechanism by which turbulent energy can be transferred to non-turbulent fluid. The dissipation term at the interface is negligible and increases with increasing turbulent energy towards the interior of the slug.

Mixed laminar and turbulent flows were observed far downstream for \[ 2000 < Re < 2700 \] when a large disturbance was introduced into the inlet. The flow in the vicinity of the inlet, however, was turbulent at much lower Re. The turbulent regions which are convected downstream at a velocity which is slightly smaller than the average velocity in the pipe we shall henceforth call puffs. The leading front of a puff does not have a clearly defined interface and the trailing front is clearly defined only in the vicinity of the centre-line. The length and structure of the puff is independent of the character of the obstruction which created it, provided that the latter is big enough to produce turbulent flow at the inlet. The puff will be discussed in more detail later.

The stability properties of the flow past an infinitely long circular cylinder are studied in the context of linear theory. An immersed-boundary technique is used to represent the cylinder surface on a Cartesian mesh. The characteristics of both direct and adjoint perturbation modes are studied and the regions of the flow more sensitive to momentum forcing and mass injection are identified. The analysis shows that the maximum of the perturbation envelope amplitude is reached far downstream of the separation bubble, where as the highest receptivity is attained in the near wake of the cylinder, close to the body surface. The large difference between the spatial structure of the two-dimensional direct and adjoint modes suggests that the instability mechanism cannot be identified from the study of either eigenfunctions separately. For this reason a structural stability analysis of the problem is used to analyse the process which gives rise to the self-sustained mode. In particular, the region of maximum coupling among the velocity components is localized by inspecting the spatial distribution of the product between the direct and adjoint modes. Results show that the instability mechanism is located in two lobes placed symmetrically across the separation bubble, confirming the qualitative results obtained through a locally plane-wave analysis. The relevance of this novel technique to the development of effective control strategies for vortex shedding behind bluff bodies is illustrated by comparing the theoretical predictions based on the structural perturbation analysis with the experimental data of Strykowski & Sreenivasan (J. Fluid Mech. vol. 218, 1990, p. 71).

A new cavity model that can explain the interaction between viscous effects including vortices and cavitation bubbles is presented in this study. This model, which is named a bubble two-phase flow (BTF) model, treats the inside and outside of a cavity as one continuum by regarding the cavity as a compressible viscous fluid whose density changes greatly. Navier–Stokes equations including cavitation bubble clusters are solved in finite-difference form by a time-marching scheme, where the growth and collapse of a bubble cluster is given by a modified Rayleigh's equation. Computation was made on a two-dimensional flow field around a hydrofoil NACA0015 at angles of attack of 8° and 20°. The Reynolds number was 3 × 105. The experiments were also performed at the same Reynolds number for comparison. The computed results by the BTF cavity model can express the feature of cloud-type cavitation shed from the trailing edge of the attached cavities when the angle of attack is 8°. It shows the mechanism of cavitation cloud generation and large-scale vortices. The boundary layer separates at the cavity leading edge. Then it rolls up and produces the cavitation cloud. In other words, the instability of the shear layer may produce the cavitation cloud. When the angle of attack is 20°, the flow was fully separated from the leading edge of the hydrofoil and vortex cavitation occurs in the separated region. The BTF cavity model can also express the generation of such vortex cavitation and the effect of cavitation nuclei in the uniform flow.

This paper is concerned with the dispersion of a dynamically neutral material quantity in a fluid flowing throuh a porous medium. The medium is regarded as an assemblage of randomly orientated straight pores, and it is assumed that the path of a marked element of the material quantity consists of a sequence of statistically independent steps whose direction and duration vary in some random manner. The probability density function for the displacement of a single marked element is calculated and values for the dispersion of a cloud of marked elements then follow.

The case is examined in which the flow satisfies Darcy's law (i.e. the mean velocity is linearly proportional to the mean pressure gradient), and the molecular diffusivity is sufficiently small for the dispersion to be primarily due to the randomness of the streamlines, but it is not assumed that effects of molecular diffusion can be altogether neglected. It is shown that the longitudinal dispersion in the direction of the mean flow may be described asymptotically by an effective diffusivity which is a function of U, l, a, K and T. (U denotes the average velocity, l the pore length, a the pore radius which is shown to be related to the permeability, k the molecular diffusivity, and T the time from the initial instant.) Expressions for the longitudinal diffusivity kl are obtained according to the relative values of l/U, T, t0 = l2/2k and t1 = a2/8k. These are given in §4, equations (4.3), (4.4) and (4.5). Speaking roughly, when t0 [Gt ] T [Gt ] l/U,kl/Ul is alogarithmic function of UT/l and increases with T; when T [Gt ] t0 [Gt ] l/U, which must eventually be the case however small k,kl/Ul is alogarithmic function of Ul/k and independent of T. The theoretical results are compared with experimental data reported in the literature and approximate agreement is obtained when E is put equal to the average diameter of the particles composing the porous medium.

The lateral dispersion in the direction perpendicular to that of the mean velocity is found to be governed asymptotically by an effective diffusivity $k_t = \frac {3}{16} Ul$. However, it is pointed out that some of the assumptions, namely that successive steps are statistically independent and that the dispersion of a cloud follows immediately from the statistical properties of the displacement of a single marked element, may not be valid for the lateral dispersion and this result is therefore suspect.

Remarks are made in §5 on the dispersion for high values of the Reynolds number Ul/v (v = kinematic viscosity) when Darcy's law is not obeyed, and it is argued that kl/Ul should decrease as Ul/v increases.

The wake of a circular cylinder is investigated near the oscillation threshold by means of a laser probe. Above the threshold the transient regime is studied and described by a Stuart-Landau law (already found to be relevant in explaining free-oscillating regimes). Below the critical point, impulse and resonant regimes are examined, so the coefficients of the Stuart-Landau equation are determined.

Moreover, in the supercritical regime, the behaviour of the (externally forced) oscillating system is described, varying parameters such as threshold deviation, forcing frequency and amplitude. The different zones of entrainment and desynchronization are given for simple or harmonic frequency.

When a smooth jet of water falls vertically on to a horizontal plane, it spreads out radially in a thin layer bounded by a circular hydraulic jump, outside which the depth is much greater. The motion in the layer is studied here by means of boundary-layer theory, both for laminar and for turbulent flow, and relations are obtained for the radius of the hydraulic jump. These relations are compared with experimental results. The analogous problems of two-dimensional flow are also treated.