We have generalized the exact solution of the Riemann problem in special relativistic hydrodynamics (Martí & Müller 1994) for arbitrary tangential flow velocities. The solution is obtained by solving the jump conditions across shocks plus an ordinary differential equation arising from the self-similarity condition along rarefaction waves, in a similar way as in purely normal flow. The dependence of the solution on the tangential velocities is analysed, and the impact of this result on the development of multi-dimensional relativistic hydrodynamic codes (of Godunov type) is discussed.

It is commonly assumed that heat flux and temperature diffusivity coefficients obtained in steady-state measurements can be used in the derivation of the heat conduction equation for fluid flows. Meanwhile it is also known that the steady-state heat flux as a function of temperature gradient in stably stratified turbulent shear flow is not monotone: at small values of temperature gradient the flux is increasing, whereas it is decreasing after a certain critical value of the temperature gradient. Therefore the problem of heat conduction for large values of temperature gradient becomes mathematically ill-posed, so that its solution (if it exists) is unstable.

In the present paper it is shown that a well-posed mathematical model is obtained if the finiteness of the adjustment time of the turbulence field to the variations of temperature gradient is taken into account. An evolution-type equation is obtained for the temperature distribution (a similar equation can be derived for the concentration if the stratification is due to salinity or suspended particles). The characteristic property which is obtained from a rigorous mathematical investigation is the formation of stepwise distributions of temperature and/or concentration from continuous initial distributions.

Finite-amplitude, forced and free oscillations of capillary bridges are studied. They are characterized by a resonant frequency and a damping rate which, in turn, depend on fluid properties, dimensions of the bridge, gravitational force relative to surface tension and amplitude of the external disturbance. The Navier–Stokes equations are solved numerically using the Galerkin/finite-element methodology for discretization in space and implicit finite differences with adaptive time stepping for discretization in time. It is found that the resonant frequency decreases and the damping rate increases almost linearly with the oscillation amplitude. Their relative changes from their corresponding values at infinitesimal amplitude depend on fluid properties and dimensions of the bridge. Moreover, careful measurement of the resonant frequency and damping rate in a well-controlled experiment may provide quite accurate values for properties of the liquid over a wide range of modified Reynolds numbers.

A parabolized set of linear equations is derived, which, in combination with the proposed solution procedure, allows for the study of both non-modal and modal disturbance growth in three-dimensional boundary layers. The method is applicable to disturbance waves whose lines of constant phase are closely aligned with the external streamline. Moreover, strongly growing disturbances may fall outside the scope of application. These equations are used in conjunction with a variational approach to compute optimal disturbances in Falkner–Skan–Cooke boundary layers subject to adverse and favourable pressure gradients. The disturbances associated with maximum energy growth initially take the form of streamwise vortices which are tilted against the mean crossflow shear. While travelling downstream these vortical structures rise into an upright position and evolve into bent streaks. The physical mechanism responsible for non-modal growth in three-dimensional boundary layers is therefore identified as a combination of the lift-up effect and the Orr mechanism. Optimal disturbances smoothly evolve into crossflow modes when entering the supercritical domain of the flow. Non-modal growth is thus found to initiate modal instabilities in three-dimensional boundary layers. Optimal growth is first studied for stationary disturbances. Influences of parameters such as sweep angle, spanwise wavenumber and position of inception are studied, and the initial optimal amplification of stationary crossflow modes because of non-modal growth is investigated. Finally, general disturbances are considered, and envelopes yielding the maximum growth at each position are computed. In general, substantial growth is already found upstream of the first neutral point. The computations show that at supercritical conditions, maximum growth of optimal disturbances in accelerated boundary layers can exceed the growth predicted for modal instabilities by several orders of magnitude.

Simultaneous measurements of the longitudinal (sx) and transverse (sz) components of the turbulent fluctuating velocity gradient at the wall of a pipe are presented for ten locations in the transverse direction. Separate measurements of sx and of sz are also presented for twenty locations. The sz pattern reveals alternating positive and negative values over lengths Λ of the magnitude suggested by the spacing of the streaky wall structure observed in a number of laboratories. Of particular interest is the observation of regular eddies for which the sz pattern is accompanied by a spatial variation of sx out of phase by about 1/4Λ. These regular eddies are associated with intermittent production of Reynolds stress and with the occurrence of large negative values of sx which could be identified with wall bursts previously reported. The time response of the axial flow appears to influence the development of these regular eddies in that the development of an sx pattern lags behind that of an sz pattern. Measurements of spatial correlations of sx and sz with different time delays, as well as direct observations of the temporal variation of sx and sz patterns, support these notions.

Solid-body collisions between smooth particles in a gas would not occur if the lubrication force for a continuum incompressible fluid were to hold at all particle separations. When the gap between the particles is of the order of the mean free path λ0 of the gas, the discrete molecular nature of the gas becomes important. For particles of radii a smaller than about 50 μm colliding in air at a relative velocity comparable to their terminal velocity, the effects of compressibility of the gas in the gap are not important.

The nature of the flow in the gap depends on the relative magnitudes of the minimum gap thickness h0 ≡ aε, the mean-free path λ0, and the distance aε1/2 over which the effects of curvature become important. The slip-flow regime, a[Gt ]λ0, was analysed by Hocking (1973) using the Maxwell slip boundary condition at the particle surface. To find the lubrication force in the transition regime (aε ∼ O(λ0)), we use the results of Cercignani & Daneri (1963) for the flux as a function of the pressure gradient in a Poiseuille channel flow. When aε[Lt ]λ0[Lt ]aε1/2, one might expect the local flow in the gap to be governed by Knudsen diffusion. However, an attempt to calculate the Knudsen diffusivity between parallel plates leads to a logarithmic divergence, which is cut off by intermolecular collisions, and the flux is therefore proportional to h0c log(λ0/h0), where c is the mean molecular speed. The non-continuum lubrication force is shown to have a weak, log - log divergence as the particle separation goes to zero. As a result, the energy dissipated in the collision is finite. In the limit of large particle inertia, the energy dissipated is 6πμU0a2(log h0/λ0 – 1.28), where 2U0 is the relative velocity of the particles.

When λ0[Gt ]aε1/2, we have a free molecular flow in the gap. In this case, owing to the curvature of the particles, the flux versus pressure gradient relation is non-local. We analyse the free molecular flow between two cylinders and obtain scalings for the lubrication force.

A theory of finite-amplitude secondary flow between concentric rotating cylinders has been published by Davey (1962). A necessary feature of the theory is the generation of harmonics of the spatial periodicity in the axial direction of the velocity field. A method has been devised to measure the amplitude of each harmonic separately and experimental results for the fundamental and first three harmonics are presented here for Taylor numbers up to 100 times the critical value. The agreement with Davey's theory is excellent, and the agreement extends far beyond the range where the theory is expected to be valid. It is shown that all the harmonics are in phase with the fundamental. This result requires that jets and shock-like structure must be present in the velocity field.

The present paper deals with the temperature field past an isothermal sphere. Acrivos & Taylor (1962) have obtained results for the case R [les ] 1 with no restriction on the Prandtl number. The results of the present paper are for the case R < 1 in which the Prandtl number is of order unity. An expansion for the Nusselt number is given up to and including the term of order R2.

Certain two-dimensional, laminar bounday layers are considered whose streamlines are closed. The speed at the solid boundary is supposed uniform, the bounday outline being stationary, and the speed in the boundary layer is supposed to differ only slightly from that of the boundary. A formal solution is then obtained for the motion in the boundary layer. The analysis confirms that a closed boundary layer may exist and yields a condition needed to determine the inviscid motion. The condition is extracted in a simplified but approximate form and two examples of its use are given. A further class of closed boundary layers, namely those for which the pressure is uniform, is also considered. For this class the condition needed to determine the inviscid motion may be derived in a form both simple and exact.

The large-amplitude oscillatory flow of a suspension of spherical particles in a pipe is studied at low Reynolds number. Particle volume fraction and velocity are examined through refractive index matching techniques. The particles migrate toward the centre of the pipe, i.e. toward regions of lower shear rate, for bulk volume fractions larger than 10 %. Steady results are in agreement with available experimental results and discrete-particle simulations for similar geometries. The dynamics of the shear-induced migration process are analysed and compared against the predictions of the suspension balance model using realistic rheological laws.

The paper investigates experimentally the global wake dynamics of a simplified three-dimensional ground vehicle at a Reynolds number of $Re\simeq 4.0\times 10^{5}$. The after-body has a blunt rectangular trailing edge leading to a massive flow separation. Both the inclination (yaw and pitch angles) and the distance to the ground (ground clearance) are accurately adjustable. Two different aspect ratios of the rectangular base are considered; wider than it is tall (minor axis perpendicular to the ground) and taller than it is wide (major axis perpendicular to the ground). Measurements of the spatial distribution of the pressure at the base and velocity fields in the wake are used as topological indicators of the flow. Sensitivity analyses of the base pressure gradient expressed in polar form (modulus and phase) varying ground clearance, yaw and pitch are performed. Above a critical ground clearance and whatever the inclination is, the modulus is always found to be large due to the permanent static symmetry-breaking instability, and slightly smaller when aligned with the minor axis of the base rather than when aligned with the major axis. The instability can be characterized with a unique wake mode, quantified by this modulus (asymmetry strength) and a phase (wake orientation) which is the key ingredient of the global wake dynamics. An additional deep rear cavity that suppresses the static instability allows a basic flow to be characterized. It is shown that both the inclination and the ground clearance constrain the phase dynamics of the unstable wake in such way that the component of the pressure gradient aligned with the minor axis of the rectangular base equals that of the basic flow. Meanwhile, the other component related to the major axis adjusts to preserve the large modulus imposed by the instability. In most cases, the dynamics explores only two possible opposite values of the component along the major axis. Their respective probability depends on the geometrical environment of the wake: base shape, body inclination, ground proximity and body supports. An expression for the lateral force coefficients taking into account the wake instability is proposed.

A detailed analysis of the ‘turbulent inertia’ (TI) term (the wall-normal gradient of the Reynolds shear stress, $\mathrm{d} \langle -uv\rangle /\mathrm{d} y $), in the axial mean momentum equation is presented for turbulent pipe flows at friction Reynolds numbers $\delta ^{+} \approx 500$, 1000 and 2000 using direct numerical simulation. Two different decompositions for TI are employed to further understand the mean structure of wall turbulence. In the first, the TI term is decomposed into the sum of two velocity–vorticity correlations ($\langle v \omega _z \rangle + \langle - w \omega _y \rangle $) and their co-spectra, which we interpret as an advective transport (vorticity dispersion) contribution and a change-of-scale effect (associated with the mechanism of vorticity stretching and reorientation). In the second decomposition, TI is equivalently represented as the wall-normal gradient of the Reynolds shear stress co-spectra, which serves to clarify the accelerative or decelerative effects associated with turbulent motions at different scales. The results show that the inner-normalised position, $y_m^{+}$, where the TI profile crosses zero, as well as the beginning of the logarithmic region of the wall turbulent flows (where the viscous force is leading order) move outwards in unison with increasing Reynolds number as $y_m^{+} \sim \sqrt{\delta ^{+}}$ because the eddies located close to $y_m^{+}$ are influenced by large-scale accelerating motions of the type $\langle - w \omega _y \rangle $ related to the change-of-scale effect (due to vorticity stretching). These large-scale motions of $O(\delta ^{+})$ gain a spectrum of larger length scales with increasing $\delta ^{+}$ and are related to the emergence of a secondary peak in the $-uv$ co-spectra. With increasing Reynolds number, the influence of the $O(\delta ^{+})$ motions promotes viscosity to act over increasingly longer times, thereby increasing the $y^{+}$ extent over which the mean viscous force retains leading order. Furthermore, the TI decompositions show that the $\langle v \omega _z \rangle $ motions (advective transport and/or dispersion of vorticity) are the dominant mechanism in and above the log region, whereas $\langle - w \omega _y \rangle $ motions (vorticity stretching and/or reorientation) are most significant below the log region. The motions associated with $\langle - w \omega _y \rangle $ predominantly underlie accelerations, whereas $\langle v \omega _z \rangle $ primarily contribute to decelerations. Finally, a description of the structure of wall turbulence deduced from the present analysis and our physical interpretation is presented, and is shown to be consistent with previous flow visualisation studies.

An experimental and theoretical investigation of the air trapping by a blunt, locally spherical body impacting onto the free surface of water is conducted. In the parameter regime previously studied theoretically by Hicks & Purvis (J. Fluid Mech., vol. 649, 2010, pp. 135–163), excellent agreement between experimental data and theoretical modelling is obtained. Earlier predictions of the radius of the trapped air pocket are confirmed. A boundary element method is used to consider air cushioning of an impact of an axisymmetric body into water. Efficient computational methods are obtained by analytically integrating the boundary integral equation over the azimuthal variable. The resulting numerically computed free-surface profiles predict an annular touchdown region in excellent agreement with the experiments.

Capillary-zone electrophoresis (CZE) is an efficient separation method in analytical chemistry. It exploits the difference in electrophoretic migration speeds between charged molecular species in aqueous solution when an external electric field is applied to achieve separation. In most cases the electrophoretic migration of species is also accompanied by a bulk electro-osmotic flow in the capillary due to the presence of a zeta-potential at the capillary wall. Adsorption of charged species at the wall could modify this zeta-potential in a non-uniform manner. This induces axial pressure gradients, so that the flow is no longer uniform over the capillary cross-section. The resulting shear-induced dispersion of the sample is a serious cause of band broadening in CZE particularly for species such as proteins and peptides which adsorb strongly on capillary walls. The problem of the spatio-temporal evolution of the sample concentration is studied in the presence of such wall interactions. An asymptotic theory is developed that is valid provided axial variations have characteristic length scales that are much larger than the capillary radius and temporal variations have a characteristic time scale much larger than the characteristic diffusion time over a capillary radius. These conditions are normally satisfied in CZE, except when the sample is close to the inlet, on account of the capillary length being very much larger than its radius. It is shown that the cross-sectionally averaged sample concentration obeys a one-dimensional partial differential equation. Further, the full three-dimensional concentration field may be calculated once the cross-sectionally averaged concentration field is known. The reduced system is integrated numerically and is shown to lead to predictions consistent with known observations on CZE in the presence of wall interactions.

The effect of an insoluble surfactant on the gravity-driven flow of a liquid film down an inclined wall with periodic undulations or indentations is investigated in the limit of vanishing Reynolds number. A perturbation analysis for walls with small-amplitude sinusoidal corrugations reveals that the surfactant amplifies the deformation of the film surface, though it also renders the film thickness more uniform over the inclined surface. The effect of the surfactant is most significant when the film thickness is less than half the wall period. To explain the deforming influence of the surfactant, a linear stability analysis of film flow down an inclined plane is undertaken for two-dimensional perturbations. The results reveal the occurrence of a Marangoni normal mode whose rate of decay is lower than that of the single mode occurring in the absence of surfactants. Numerical methods based on a combined boundary-element/finite-volume method are implemented to compute flow down a periodic wall with large-amplitude corrugations or semi-circular depressions. In the case of a wavy wall, it is found that the shape of the film surface is described well by the linear perturbation expansion for small and moderate wave amplitudes. Streamline patterns reveal that, although the effect of the surfactant on the shape of the film surface is generally small, Marangoni tractions may have a profound influence on the kinematics by causing the onset of regions of recirculating flow.

A modification of Homann’s axisymmetric outer potential stagnation-point flow of strain rate is obtained by adding periodic radial and azimuthal velocities of the form and , respectively, where is a shear rate. This leads to the discovery of a new family of asymmetric viscous stagnation-point flows depending on the shear-to-strain-rate ratio that exist over the range . Numerical solutions for the wall shear stress parameters and the displacement thicknesses are given and compared with their large- asymptotic behaviours. Sample similarity velocity profiles are also presented.

The identification of uniform momentum zones in wall-turbulence, introduced by Adrian, Meinhart & Tomkins (J. Fluid Mech., vol. 422, 2000, pp. 1–54) has been applied to turbulent channel flow, revealing a large ‘core’ region having high and uniform velocity magnitude. Examination of the core reveals that it is a region of relatively weak turbulence levels. For channel flow in the range $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}Re_{\tau } = 1000\text {--}4000$, it was found that the ‘core’ is identifiable by regions bounded by the continuous isocontour lines of the streamwise velocity at $0.95U_{CL}$ (95 % of the centreline velocity). A detailed investigation into the properties of the core has revealed it has a large-scale oscillation which is predominantly anti-symmetric with respect to the channel centreline as it moves through the channel, and there is a distinct jump in turbulence statistics as the core boundary is crossed. It is concluded that the edge of the core demarcates a shear layer of relatively intense vorticity such that the interior of the core contains weakly varying, very low-level turbulence (relative to the flow closer to the wall). Although channel flows are generally referred to as ‘fully turbulent’, these findings suggest there exists a relatively large and ‘quiescent’ core region with a boundary qualitatively similar to the turbulent/non-turbulent interface of boundary layers, jets and wakes.

The mean wall-normal gradients of the Reynolds shear stress and the turbulent kinetic energy have direct connections to the transport mechanisms of turbulent-boundary-layer flow. According to the Stokes–Helmholtz decomposition, these gradients can be expressed in terms of velocity–vorticity products. Physical experiments were conducted to explore the statistical properties of some of the relevant velocity–vorticity products. The high-Reynolds-number data (Rθ ≃ O(106), where θ is the momentum thickness) were acquired in the near neutrally stable atmospheric-surface-layer flow over a salt playa under both smooth- and rough-wall conditions. The low-Rθ data were from a database acquired in a large-scale laboratory facility at 1000 > Rθ > 5000. Corresponding to a companion study of the Reynolds stresses (Priyadarshana & Klewicki, Phys. Fluids, vol. 16, 2004, p. 4586), comparisons of low- and high-Rθ as well as smooth- and rough-wall boundary-layer results were made at the approximate wall-normal locations yp/2 and 2yp, where yp is the wall-normal location of the peak of the Reynolds shear stress, at each Reynolds number. In this paper, the properties of the vωz, wωy and uωz products are analysed through their statistics and cospectra over a three-decade variation in Reynolds number. Here u, v and w are the fluctuating streamwise, wall-normal and spanwise velocity components and ωy and ωz are the fluctuating wall-normal and spanwise vorticity components. It is observed that v–ωz statistics and spectral behaviours exhibit considerable sensitivity to Reynolds number as well as to wall roughness. More broadly, the correlations between the v and ω fields are seen to arise from a ‘scale selection’ near the peak in the associated vorticity spectra and, in some cases, near the peak in the associated velocity spectra as well.

A physical model for compressible flows with phase transition is described, in which all the processes of phase transition, i.e. nucleation, droplet growth, droplet evaporation and de-nucleation, are incorporated. The model is focused on dilute mixtures of vapour and droplets in a carrier gas with typical maximum liquid mass fraction smaller than 0.02. The new model is based on a reinterpretation of Hill's method of moments of the droplet size distribution function. Starting from the general dynamic equation, it is emphasized that nucleation or de-nucleation correspond to the rates at which droplets enter or leave droplet size space, respectively. Nucleation and de-nucleation have to be treated differently in agreement with their differences in physical nature. Attention is given to the droplet growth model that takes into account Knudsen effects and temperature differences between droplets and gas. The new phase transition model is then combined with the Euler equations and results in a new numerical method: ASCE2D. The numerical method is first applied to the problem of shock/expansion wave formation in a closed shock tube with humid nitrogen as a driver gas. Nucleation and droplet growth are induced by the expansion wave, and in turn affect the structure of the expansion wave. When the main shock, reflected from the end wall of the low-pressure section, passes the condensation zone, evaporation and de-nucleation occur. As a second example, the problem of the flow of humid nitrogen in a pulse-expansion wave tube, designed to study nucleation and droplet growth in monodisperse clouds, is investigated experimentally and numerically.

Starting with expressions for viscous stress, heat flux and diffusion flux, we formulate a continuum theory for steady flow of a binary mixture of chemically inert perfect gases through a normal shock of arbitrary strength. For shocks of vanishing strength, a solution by series expansion in Grad's (1952) shcok-strength parameter gives a result essentially the same as found previously by Dyakov (1954). For stronger shocks a straightforward numerical integration, quite analogous to that useful in the simpler pure-gas problem, is laid out.

The resulting problem has eight parameters: shock strength, ratio of specific heats, ration of bulk viscosity to shear viscosity, Prandtl number, Schmidt number, thermal diffusion factor, molecular mass ratio, and initial mixture concentration. A dozen examples were worked out on a simple desk calculator to exhibit the influence of some of these parameters. They involve the gas pairs argon40–argon36, argon–neon, argon–helium, and xenon–helium.

In discussing the results, special attention is paid to the degree of success with which the weak-shock theory may be extrapolated to arbitrary shock strength, and to the question of the accuracy of the Navier-Stokes approximation for a mixture of gases of very different molecular weights.