Partial cavities that formed on the vertices of wedges and on the leading edge of stationary hydrofoils were examined experimentally. The geometry of these test objects did not vary in the spanwise direction (i.e. two-dimensional). Open partial cavities formed on a series of two-dimensional wedges and on a plano-convex hydrofoil. These cavities terminated near the point of maximum cavity thickness, and small vapour-filled vortices were shed in the turbulent cavity wake. The turbulent flow in the wake of the open cavity was similar to the turbulent shear flow downstream of a rearward-facing step. Re-entrant flow was not observed in the cavity closure of open cavities, although recirculating flow associated with a region of flow separation was detected for some cases. Predictions of a two-dimensional free-streamline model of the cavitating wedge flows were compared to the experimentally observed cavities. The model predicted the profile of the open cavity only to the point of maximum cavity thickness. Examination of the flow field near the closure of the open cavities revealed adverse pressure gradients near the cavity closure. The pressure gradients around the open cavities were sufficient to cause large-scale condensation of the cavity. Unsteady re-entrant partial cavities formed on a two-dimensional NACA0009 hydrofoil. The interface of the unsteady closed cavities smoothly curved to form a re-entrant jet at the cavity terminus, and the re-entrant flow was directed upstream. The re-entrant flow impinged on the cavity interface and led to the periodic production of cloud cavitation. These cavities exhibited a laminar flow reattachment. The flow around the closed cavity was largely irrotational, while vorticity was created when the cloud cavitation collapsed downstream of the cavity. Examination of the flow field near closure of these cavities also revealed adverse pressure gradients near the partial cavity closure, but the rise in pressure did not lead to the premature condensation of the cavity.

Partial cavitation forming on the vertex of a wedge and on the leading edge of a stationary hydrofoil was experimentally examined. The geometry of these test objects varied in the spanwise direction (i.e. three-dimensional test objects). Closed cavities formed on these test objects. The interface of the closed cavities curved smoothly to form a re-entrant jet at the cavity terminus, and the re-entrant flow was directed spanwise, thus preventing its impingement on the cavity interface. The cavity shape and the pressure gradients near the closure of the closed cavities were qualitatively similar to those predicted with the two-dimensional free-streamline theory. These cavities had a steady, laminar flow reattachment. The flow around the closed cavity was largely irrotational.

This is a study on the tangential flow and advective mixing of viscoplastic fluids (Bingham plastics) between two eccentric, alternately rotating cylinders. Two geometrical configurations and various rotation modes are considered for a relatively large range of the yield stress. The hp-type finite element method with the mixed formulation is used to solve for the steady velocity and pressure fields. The bi-viscosity and the Papanastasiou models agree quantitatively with each other in predicting the velocity fields and the practically unyielded zones. However, the Papanastasiou model is more robust and economic than the bi-viscosity model in the computation using Newton iteration. In the steady flows, in addition to the motionless zones, we have discovered some plugs with rigid rotation, including rotating plugs stuck onto the outer cylinder and rotating, even counter-rotating, plugs disconnected from both cylinders. The unsteady, periodic flow is composed of a sequence of the steady flows, which is valid in the creeping flow regime. The characteristics of advective mixing in these flows have been studied by analysing the asymptotic coverages of a passive tracer, the distributions of the lineal stretching in the flow and the variations of the mean stretching of the flow with time. The tracer coverage is intuitive but qualitative and, occasionally, it depends on the initial location of the tracer. On the other hand, the distribution of stretching is quantitative and more reliable in reflecting the mixing characteristics. Interestingly, the zones of the lowest stretching in the distribution graphs are remarkably well matched with the regular zones in the tracer-coverage graphs. Furthermore, the mixing efficiency proposed by Ottino (1989) is used to characterize the advective mixing in the two geometrical configurations with various rotation modes. It is important to realize that, for plastic fluids, a major barrier to effective mixing is the unyielded fluid plugs which are controlled by the yield stress and geometrical configurations. Therefore, when designing an eccentric helical annular mixer it is important to pay attention first to the geometric issues then to the operating issues.

This paper investigates the receptivity of boundary layers due to distributed roughness interacting with free-stream disturbances. Both acoustic and vortical perturbations are considered. An asymptotic approach based on the triple-deck formulation has been developed to determine the initial amplitude of the Tollmien–Schlichting wave to the O(R−1/8) accuracy, where R is the global Reynolds number. In the case of vortical disturbances, we show that the dominant contribution to the receptivity comes from the upper deck as well as from the so-called edge layer centred at the outer reach of the boundary layer. It is found that for certain forms of disturbances, the receptivity is independent of their vertical structure and can be fully characterized by their slip velocity at the edge of the boundary layer. A typical case is the vortical disturbance in the form of a convecting wake, for which the same conclusion as above has been reached by Dietz (1999) on the basis of measurements. Our theoretical predictions are compared with the experimental data of Dietz (1999), and a good quantitative agreement has been found. Such a comparison is the first to be made for distributed vortical receptivity. Further calculations indicate that the vortical receptivity in general is much stronger than was suggested previously. In the case of acoustic disturbances, it is found that our first-order theory is in good agreement with experiments as well as with previous theoretical results. But the second-order theory over-predicts, and the possible reasons for this are discussed.

We present a theory for the three-dimensional flow of a Bingham-plastic fluid in a shallow and wide channel. Focusing attention on slow flows appropriate for gentle slopes, low discharge rates or the final stage of deposition, we ignore inertia and apply the long-wave approximation. For steady flows, the velocity distribution, total discharge, and section-averaged flux are obtained analytically in terms of the fluid property and the geometry of the channel cross-section. Nonlinear stationary waves, which connect a uniform depth upstream to another uniform depth downstream, are then investigated, for both wet and dry beds. A numerical scheme is applied to calculate the transient flow evolution. The final development of the stationary wave due to steady discharge upstream is obtained numerically and the relation between the tongue-like shape of the wave front and the fluid property is discussed. The phase speed of the stationary wave is also derived analytically. Finally, the transient spreading of a finite fluid mass released from a reservoir after a dam break is simulated numerically. The transient development of the front and the final extent of deposition are examined.

The steepening of a normal compression wave into a shock in a homentropic flow field is understood well through the method of characteristics. In a non-homentropic flow field, however, shock formation from a compression wave is more complex. The effects of entropy (or sound speed) gradients on shock formation from a compression wave are determined using a wave front expansion in Cartesian and in spherical polar coordinates. The latter problem has application to the intense energy focusing of sonoluminescence, particularly when applied to a spherically collapsing gas. The principal result is an analytical criterion for the time and place of shock formation, for a wave propagating into a field of smoothly varying entropy.

In the present study, we use particle image velocimetry (PIV) to obtain planar, two-component velocity fields in two-dimensional, turbulent mixing layers at convective Mach numbers Mc of 0.25, 0.63, and 0.76. The experiments are performed in a large-scale blowdown wind tunnel, with high-speed free-stream Mach numbers up to 2.25 and shear-layer Reynolds numbers up to 106. Specific issues relating to the application of PIV to supersonic flows are addressed. The instantaneous data are analysed to produce maps of derived quantities such as vorticity, and ensemble averaged to provide turbulence statistics. The results show that compressibility introduces marked changes in the disposition of instantaneous velocity gradients within the layer, and hence in the vorticity field. In particular, peak transverse vorticity values are seen to be confined to thin streamwise sheets under compressible conditions, with little transverse communication. The location of these sheets near the lab-frame sonic velocity suggests a sensitivity of the compressible layer to stationary disturbances. Turbulence statistics derived from the planar velocity measurements confirm previous observations of strong suppression of transverse velocity fluctuations and primary Reynolds stress as Mc increases between 0.25 and 0.76.

A sound wave propagating in an inhomogeneous duct consisting of two semi-infinite uniform ducts with a smooth transition region in between and which carries a steady flow is considered. The duct walls may be rigid or compliant. For an irrotational sound wave it is shown that the three properties of the title are closely related, such that the validity of any two implies the validity of the third. Furthermore it is shown that the three properties are fulfilled for lossless locally reacting duct walls provided the impedance varies at most continuously. For piecewise-continuous wall properties edge conditions are essential. By an analytic continuation argument it is shown that reciprocity remains true for walls with loss. For rotational flow, energy conservation theorems have been derived only with the help of additional potential-like variables. The inter-relation between the three properties remains valid if one considers these additional variables to be known. If only the basic gasdynamic variables in both half-ducts are known, one cannot formulate an energy conservation equation; however, reciprocity is fulfilled.

A compressible multiphase unconditionally hyperbolic model is proposed. It is able to deal with a wide range of applications: interfaces between compressible materials, shock waves in condensed multiphase mixtures, homogeneous two-phase flows (bubbly and droplet flows) and cavitation in liquids. Here we focus on the generalization of the formulation to an arbitrary number of fluids, and to mass and energy transfers, and extend the associated Godunov method.

We first detail the specific problems involved in the computation of thermodynamic interface variables when dealing with compressible materials separated by well-defined interfaces. We then address one of the major problems in the modelling of detonation waves in condensed energetic materials and propose a way to suppress the mixture equation of state. We then consider another problem of practical importance related to high-pressure liquid injection and associated cavitating flow. This problem involves the dynamic creation of interfaces. We show that the multiphase model is able to solve these very different problems using a unique formulation.

We then develop the Godunov method for this model. We show how the non-conservative terms must be discretized in order to fulfil the interface conditions. Numerical resolution of interface conditions and partial equilibrium multiphase mixtures also requires the introduction of infinite relaxation terms. We propose a way to solve them in the context of an arbitrary number of fluids. This is of particular importance for the development of multimaterial reactive hydrocodes. We finally extend the discretization method in the multidimensional case, and show some results and validations of the model and method.

We report on calculations and experiments with strong shocks diffracting over rigid ramps in argon. The numerical results were obtained by integrating the conservation equations that included the Navier–Stokes equations. The results predict that if the ramp angle θ is less than the angle θe that corresponds to the detachment of a shock, θ < θe, then the onset of Mach reflection (MR) will be delayed by the initial appearance of a precursor regular reflection (PRR). The PRR is subsequently swept away by an overtaking corner signal (cs) that forces the eruption of the MR which then rapidly evolves into a self-similar state. An objective was to make an experimental test of the predictions. These were confirmed by twice photographing the diffracting shock as it travelled along the ramp. We could get a PRR with the first exposure and an MR with the second. According to the von Neumann perfect gas theory, a PRR does not exist when θ < θe. A viscous length scale xint is a measure of the position on the ramp where the dynamic transition PRR → MR takes place. It is significantly larger in the experiments than in the calculations. This is attributed to the fact that fluctuations from turbulence and surface roughness were not modelled in the calculations. It was found that xint → ∞ as θ → θe. Experiments were done to find out how xint depended on the initial shock tube pressure p0. The dependence was strong but could be greatly reduced by forming a Reynolds number based on xint. Finally by definition, regular reflection (RR) never interacts with a boundary layer, while PRR always interacts; so they are different phenomena.

Drag reduction in two-dimensional flow over a circular cylinder, achieved using rotary oscillation, was investigated with computational simulations. In the experiments of Tokumaru & Dimotakis (1991), this mechanism was observed to yield up to 80% drag reduction at Re = 15 000 for certain ranges of frequency and amplitude of sinusoidal rotary oscillation. Simulations with a high-resolution viscous vortex method were carried out over a range of Reynolds numbers (150–15 000) to explore the effects of oscillatory rotational forcing. Significant drag reduction was observed for a rotational forcing which had been very effective in the experiments. The impact of the forcing is strongly Reynolds number dependent. The cylinder oscillation appears to trigger a distinctive shedding pattern which is related to the Reynolds number dependence of the drag reduction. It appears that the source of this unusual shedding pattern and associated drag reduction is vortex dynamics in the boundary layer initiated by the oscillatory cylinder rotation. The practical efficiency of the drag reduction procedure is also discussed.

A direct numerical simulation of the spherical Couette flow between two spheres with the inner sphere rotating was performed to investigate the detailed structure, formation process and mechanism of the spiral Taylor–Görtler (TG) vortices. For comparison with our previous experiments, a moderate gap case with clearance ratio β = 0.14 is chosen in the present numerical study. With adequate initial and boundary conditions, we have sucessfully simulated the supercritical spiral TG vortex flow in this system. Analysis of the numerical results reveals the structure and features of the spiral TG vortices. The flow consists of one toroidal TG vortex, one toroidal vortex cell, three spiral TG vortices and a secondary flow circulation in each hemisphere, and this supercritical flow solution features rotational and equatorial asymmetries. It is found that the spiral TG vortices are composed of a pair of counter-rotating, unequal spiral vortices with essentially different structural forms. One begins in the secondary flow circulation at higher latitude and ends with a connection to the toroidal vortex cell at lower latitude while the other one starts on the inner rotating spherical surface at lower latitude and ends on the outer stationary spherical surface at higher latitude. Through sucessive visualizations which display the transient features of the spiral TG vortices, we observe that vortex tearing, splitting, tilting, reconnecting, stretching and compressing occur in the formation of the spiral TG vortices. Pairing of two alternating helical vortices is the key process in their evolution. To understand the formation mechanism, we consider the vorticity production in the azimuthal vorticity component equation. The important vorticity tilting and stretching terms play different roles in the formation process of these two counter-rotating spiral vortices. The vorticity tilting term is responsible for generating both of the spiral vortices. The vorticity stretching term acts to stretch one of the spiral vortices from the inner sphere to the outer sphere while suppressing the stretching of the other in the azimuthal direction. The different formation mechanisms for these two counter-rotating spiral vortices lead to the structure of the spiral TG vortices.

We discuss mixing and transport in three-dimensional, steady, Navier–Stokes flows with the no-slip condition at the boundaries. The advective flux is related to the dynamics of the Navier–Stokes equations and a prediction is made of the scaling of the advective flux with the Reynolds number: the flux is expected to decay as the Reynolds number goes to infinity. This prediction is made via a Melnikov-type calculation together with boundary layer concepts through which the flow is split into an integrable and a small non-integrable part. The rate of decay is related to the details of viscous flow in boundary layers. The Melnikov function is related to the Bernoulli integral of the underlying Euler flow. The effects of molecular diffusivity are discussed and the effective axial diffusivity scaling predicted as a function of Reynolds and Péclet numbers. Using these ideas, we study the mass transport in the wavy vortex flow in the Taylor–Couette apparatus as a particular example. We propose an explanation of the observed non-monotonic behaviour of flux with increasing Reynolds number that was not captured in any of the previous models. It is shown that there is a Reynolds number at which the axial flux in the wavy vortex flow is maximized. At the low range of Reynolds numbers for which the wavy vortex flow is stable the flux increases, while for large Reynolds numbers it decreases. We compare these predictions with the available experimental and numerical data on the wavy vortex flow.

A computational study of three-dimensional vortex–cylinder interaction is reported for the case where the nominal orientation of the cylinder axis is normal to the vortex axis. The computations are performed using a new tetrahedral vorticity element method for incompressible viscous fluids, in which vorticity is interpolated using a tetrahedral mesh that is refit to the Lagrangian computational points at each timestep. Fast computation of the Biot-Savart integral for velocity is performed using a box-point multipole acceleration method for distant tetrahedra and Gaussian quadratures for nearby tetrahedra. A moving least-square method is used for differentiation, and a flux-based vorticity boundary condition algorithm is employed for satisfaction of the no-slip condition. The velocity induced by the primary vortex is obtained using a filament model and the Navier–Stokes computations focus on development of boundary-layer separation from the cylinder and the form and dynamics of the ejected secondary vorticity structure. As the secondary vorticity is drawn outward by the vortex-induced flow and wraps around the vortex, it has a substantial effect both on the essentially inviscid flow field external to the boundary layer and on the cylinder surface pressure field. Cases are examined with background free-stream velocity oriented in the positive and negative directions along the cylinder axis, with free-stream velocity normal to the cylinder axis, and with no free-stream velocity. Computations with no free-stream velocity and those with free-stream velocity tangent to the cylinder axis exhibit similar secondary vorticity structures, consisting of a vortex loop (or hairpin) that wraps around the primary vortex and is attached to the cylinder boundary layer at two points. Computations with free-stream velocity oriented normal to the cylinder axis exhibit secondary vorticity structure of a markedly different character, in which the secondary eddy remains close to the cylinder boundary and has a quasi-two-dimensional form for an extended time period.

This paper deals with the stability analysis of an axially symmetric liquid metal flow driven by a rotating magnetic field in a cylinder of finite dimensions. The limit of linear stability with respect to axially symmetric perturbations is found for diameter-to-height ratios between 0.4 and 1. This oscillatory instability is shown to be different from the expected Taylor–Görtler vortices. Several linearly unstable steady solutions are found close to the stable basic state. It is shown that small finite-amplitude perturbations in the form of Taylor–Görtler vortices give rise to instability in the linearly stable regime.

Using the Hopf–Doering–Constantin decomposition, we derive upper bounds on the vertical heat flux in closed containers. It is found that the original bound of Doering & Constantin (1996) for Nusselt number as a function of Rayleigh number, Nu [ges ] √R/4, holds, at the very least, asymptotically as R → ∞ under reasonably diverse experimental settings.

The structure of velocity in the outer region of turbulent channel flow (y+ [gsim ] 100) is examined statistically to determine the average flow field associated with spanwise vortical motions. Particle image velocimetry measurements of the streamwise and wall-normal velocity components are correlated with a vortex marker (swirling strength) in the streamwise–wall-normal plane, and linear stochastic estimation is used to estimate the conditional average of the two-dimensional velocity field associated with a swirling motion. The mean structure consists of a series of swirling motions located along a line inclined at 12°–13° from the wall. The pattern is consistent with the observations of outer-layer wall turbulence in which groups of hairpin vortices occur aligned in the streamwise direction. While the observational evidence for the aforementioned model was based upon both experimental and computational visualization of instantaneous structures, the present results show that, on average, the instantaneous structures occur with sufficient frequency, strength, and order to leave an imprint on the statistics of the flow as well. Results at Reτ = 547 and 1734 are presented.