A linear non-homogeneous analysis is presented for the standing waves produced on the hollow core of an irrotational vortex by an arbitrary obstacle on the wall of the tube containing the vortex. The group-velocity criterion based upon Kelvin's corresponding dispersion relation predicts whether a certain asymptotic wave pattern appears upstream or downstream of the obstacle. The analysis leads to amplitude singularities for the standing waves at certain critical radii of the core. The particularly interesting case of a counter-helix for which the wave energy is propagating upstream appears for a first-mode angular disturbance. For this situation it seems to be possible that the helix ends in a hydraulic jump and is continued by a counter-helix downstream, as the core size gradually diminishes due to the deceleration of the flow caused by viscous effects (not included in the analysis). The capillary-wave pattern produced by surface tension is also considered. A brief outline for the analogous wave problem is given for the case where the fluid rotates like a rigid body.

Photographic observations of hollow-core vortices in water flow are presented which confirm the qualitative predictions of the analysis, both for the response to an axisymmetric area contraction and also to a 90° bend at the downstream end of the vortex tube.

Calculations are presented of the development of the turbulent boundary layer on a rotating cylinder with its axis along the stream, and the results are compared with measurements made by Parr. With the choice of a suitable co-ordinate system, the boundary layer downstream of the nose of the cylinder approximates closely to a condition of two-dimensionality, and a simple integral method of solution can be applied. The only evidence of three-dimensionality lies in the destabilizing effect of rotation on the turbulence structure of the layer and an analysis of this instability has been made which relates changes in mixing length and entrainment to an instability parameter in the form of a Richardson number.

In spite of the changes in shear stress distribution and entrainment brought about by rotation, mean velocity profiles and skin friction values are found to be related to H and Rθ in the same way as for the stationary flat plate.

We present here a new stochastic modelling approach in the constitution of fluid flow reduced-order models. This framework introduces a spatially inhomogeneous random field to represent the unresolved small-scale velocity component. Such a decomposition of the velocity in terms of a smooth large-scale velocity component and a rough, highly oscillating component gives rise, without any supplementary assumption, to a large-scale flow dynamics that includes a modified advection term together with an inhomogeneous diffusion term. Both of those terms, related respectively to turbophoresis and mixing effects, depend on the variance of the unresolved small-scale velocity component. They bring an explicit subgrid term to the reduced system which enables us to take into account the action of the truncated modes. Besides, a decomposition of the variance tensor in terms of diffusion modes provides a meaningful statistical representation of the stationary or non-stationary structuration of the small-scale velocity and of its action on the resolved modes. This supplies a useful tool for turbulent fluid flow data analysis. We apply this methodology to circular cylinder wake flow at Reynolds numbers $Re=100$ and $Re=3900$. The finite-dimensional models of the wake flows reveal the energy and the anisotropy distributions of the small-scale diffusion modes. These distributions identify critical regions where corrective advection effects, as well as structured energy dissipation effects, take place. In providing rigorously derived subgrid terms, the proposed approach yields accurate and robust temporal reconstruction of the low-dimensional models.

Kinetic theory and acoustic measurements have proven that the bulk viscosity associated with the expansion or compression effect cannot be ignored in compressible fluids except for monatomic gases. A new theoretical formula for the bulk viscosity coefficient (BVC) $\unicode[STIX]{x1D701}$ is derived by the continuum medium methodology, which provides a further understanding of the bulk viscosity, i.e. $\unicode[STIX]{x1D701}$ is equal to the product of the bulk modulus $K$ and the relaxation time $\unicode[STIX]{x1D70F}$ ($\unicode[STIX]{x1D701}=K\unicode[STIX]{x1D70F}$). The continuum and kinetic theories present consistent results from macro- and microperspectives respectively, only differing in terms of a coefficient. The theoretical predictions of the BVC in diatomic molecules, such as $\text{N}_{2}$, $\text{O}_{2}$ and CO, show good agreement with the experimental data over a wide range of temperature. In addition, the vibrational contributions to $\unicode[STIX]{x1D701}$ are controlled by a rapid exponential decrease at high temperatures, while at low temperatures a slow linear increase proceeds for the rotational cases. The relaxation time $\unicode[STIX]{x1D70F}$, collision number $Z$, BVC $\unicode[STIX]{x1D701}$ and ratio of bulk-to-shear viscosities $\unicode[STIX]{x1D701}/\unicode[STIX]{x1D707}$ in the vibrational mode are found to be several orders of magnitude larger than those in the rotational mode.

The turbulent–non-turbulent interface (TNTI) of supersonic turbulent boundary layers is a fundamental but relatively unexplored physics problem. In this study, we present experimental results from fractal analysis on the TNTI of supersonic turbulent boundary layers, and test the applicability of the additive law for these flows. By applying the nanoparticle-tracer planar laser scattering (NPLS) technique in a supersonic wind tunnel, we obtain data covering nearly three decades in scale. The box-counting results indicate that the TNTI of supersonic turbulent boundary layers is a self-similar fractal with a fractal dimension of 2.31. By comparing data sets acquired from two orthogonal planes, we find that the scaling exponent does not depend on direction, consistent with the validity of the additive law for the TNTI of turbulent boundary layers in a scale range with the large-scale limit not exceeding approximately $0.05\unicode[STIX]{x1D6FF}$.

The interactions among pressure, density, vorticity and their gradients in compressible turbulent channel flows (TCF) are studied using direct numerical simulations (DNS). DNS of three isothermal-wall TCF for Mach number Ma = 0.2, 0.7, and 1.5, respectively are performed using a discontinuous Galerkin method (DGM). The Reynolds numbers of these three cases are ≈2800, based on the bulk velocity, bulk density, half channel width and dynamic viscosity at the wall. A high cross-correlation between density and spanwise vorticity occurs at y+≈4, which is coincident with the peak mean spanwise baroclinicity. The relationship between the spanwise baroclinicity and the correlation is analysed. The difference between the evolution of density and spanwise vorticity very near the wall is discussed. The transport equation for the mean product of density and vorticity fluctuations 〈ρ′ω′i〉 is presented and the distributions of terms in the 〈ρ′ω′z〉 transport equation indicate that the minima and maxima of the profiles are located around y+≈5. The connection between pressure gradients and vorticity fluxes for compressible turbulent flows with variable viscosity has been formulated and verified. High correlations (0.7–1.0) between pressure gradient and vorticity flux are found very close to the wall (y+<5). The correlation coefficients are significantly influenced by Ma and viscosity in this region. Turbulence advection plays an important role in destroying the high correlations between pressure gradient and vorticity flux in the region away from the wall (y+ > 5).

This experimental study examines ventilated supercavity formation in a free-surface bounded environment where a body is in motion and the fluid is at rest. For a given torpedo-shaped body and water depth ($H$), depending on the cavitator diameter ($d_{c}$) and the submergence depth ($h_{s}$), four different cases are investigated according to the blockage ratio ($B=d_{c}/d_{h}$, where $d_{h}$ is the hydraulic diameter) and the dimensionless submergence depth ($h^{\ast }=h_{s}/H$). Cases 1–4 are, respectively, no cavitator in fully submerged ($B=0$, $h^{\ast }=0.5$), small blockage in fully submerged ($B=1.5\,\%$, $h^{\ast }=0.5$), small blockage in shallowly submerged ($B=1.5\,\%$, $h^{\ast }=0.17$) and large blockage in fully submerged ($B=3\,\%$, $h^{\ast }=0.5$) cases. In case 1, no supercavitation is observed and only a bubbly flow (B) and a foamy cavity (FC) are observed. In non-zero blockage cases 2–4, various non-bubbly and non-foamy steady states are observed according to the cavitator-diameter-based Froude number ($Fr$), air-entrainment coefficient ($C_{q}$) and the cavitation number ($\unicode[STIX]{x1D70E}_{c}$). The ranges of $Fr$, $C_{q}$ and $\unicode[STIX]{x1D70E}_{c}$ are $Fr=2.6{-}18.2$, $C_{q}=0{-}6$, $\unicode[STIX]{x1D70E}_{c}=0{-}1$ for cases 2 and 3, and $Fr=1.8{-}12.9$, $C_{q}=0{-}1.5$, $\unicode[STIX]{x1D70E}_{c}=0{-}1$ for case 4. In cases 2 and 3, a twin-vortex supercavity (TV), a reentrant-jet supercavity (RJ), a half-supercavity with foamy cavity downstream (HSF), B and FC are observed. Supercavities in case 3 are not top–bottom symmetric. In case 4, a half-supercavity with a ring-type vortex shedding downstream (HSV), double-layer supercavities (RJ inside and TV outside (RJTV), TV inside and TV outside (TVTV), RJ inside and RJ outside (RJRJ)), B, FC and TV are observed. The cavitation numbers ($\unicode[STIX]{x1D70E}_{c}$) are approximately 0.9 for the B, FC and HSF, 0.25 for the HSV, and 0.1 for the TV, RJ, RJTV, TVTV and RJRJ supercavities. In cases 2–4, for a given $Fr$, there exists a minimum cavitation number in the formation of a supercavity while the minimum cavitation number decreases as the $Fr$ increases. In cases 2 and 3, it is observed that a high $Fr$ favours an RJ and a low $Fr$ favours a TV. For the RJ supercavities in cases 2 and 3, the cavity width is always larger than the cavity height. In addition, the cavity length, height and width all increase (decrease) as the $\unicode[STIX]{x1D70E}_{c}$ decreases (increases). The cavity length in case 3 is smaller than that in case 2. In both cases 2 and 3, the cavity length depends little on the $Fr$. In case 2, the cavity height and width increase as the $Fr$ increases. In case 3, the cavity height and width show a weak dependence on the $Fr$. Compared to case 2, for the same $Fr$, $C_{q}$ and $\unicode[STIX]{x1D70E}_{c}$, case 4 admits a double-layer supercavity instead of a single-layer supercavity. Connected with this behavioural observation, the body-frontal-area-based drag coefficient for a moving torpedo-shaped body with a supercavity is measured to be approximately 0.11 while that for a cavitator-free moving body without a supercavity is approximately 0.4.

A simultaneous technique employing particle image velocimetry (PIV) and shadowgraphy was used to study vertical slug flow in non-Newtonian fluids. Two aqueous solutions of 0.8 and 1.0 wt% carboxymethylcellulose (CMC) were studied and the flow field around individual Taylor bubbles fully characterized. The rheological fluid properties and pipe dimension yielded Reynolds numbers of 8 and 4 and Deborah numbers of 0.2 and 0.4. A negative wake was found downstream of the Taylor bubbles in both fluids. Below the bubble trailing edge, along the axis region, the fluid flows in the opposite direction to the bubble (negative wake), originating rotational liquid movements in adjacent regions. Even far downward from the bubble, rotational liquid movements are clearly seen and measured. In the 1.0 wt% CMC solution, the bubble trailing edge has the shape of a two-dimensional cusp. This two-dimensional cusp, of small dimensions, is seen in different orientations during the bubble rise-indicating a fast rotational movement. The asymmetrical shape of the trailing edge is responsible for small asymmetries in the flow in the wake region (three-dimensional flow). The asymmetrical shape associated with the rotational movement is responsible for an unsteady flow of small amplitude. In the 0.8 wt% CMC solution, the shape of the trailing edge changes during the bubble rise. An axisymmetric axial oscillation a continuous expansion and contraction of the trailing edge, is the origin of this behaviour. This oscillatory movement is responsible for an unsteady flow of small amplitude in the wake region.

The viscometer investigations of Donnelly & Ozima (1962) on the stability of flow between rotating cylinders in the presence of a magnetic field have been extended by making careful torque measurements in the subcritical range using improved mercury. The results show that a small difference in stability can be observed with conducting cylinders as compared with insulating cylinders. The results also suggest that a heretofore unnoticed instability begins well before the instability studied earlier. This relatively weak instability accounts for some of the anomalous effective viscosity observed by Donnelly & Ozima.

The linear stability of the developing flow of an incompressible fluid in the entrance region of a circular tube is investigated. The case of non-axisymmetric small disturbances is considered in the analysis. The main-flow velocity distribution used in the stability calculations is that from the solution of the linearized momentum equation. The eigenvalue problem consisting of the disturbance equations and the boundary conditions is solved by a direct numerical integration scheme along with an iteration procedure. An orthonormalization method is employed to remove the ‘parasitic errors’ inherent in the numerical integration of the coupled disturbance equations. The flow is found to be unstable to non-axisymmetric disturbances with an azimuthal wavenumber of one. Neutral-stability curves and critical Reynolds numbers at various axial locations are presented. A comparison of these results is made with those for axisymmetric disturbances reported by Huang & Chen. It is found that the first instability of the flow is due to non-axisymmetric disturbances and occurs in the entrance region of the pipe with a minimum critical Reynolds number of 19 780.

We examine the statistical properties of extreme and rogue wave activity in crossing directional seas, to constrain the probabilistic distributions of wave heights and wave crests in complex sea states; such crossing seas alter the statistical structure of surface waves and are known to have been involved in several marine accidents. Further, we examine the relationship between the kurtosis as an indicator of nonlinearity in the spectrum and the directionality and crossing angles of the sea-state components. Experimental tests of two-component directionally spread irregular waves with varying frequency, directional spreading and component crossing angles were carried out at the Ocean Basin Laboratory in Trondheim, Norway. The results from the experiments show that wave heights are well described by a first-order (linear) statistical distribution, while for the wave crest heights several cases exceed a second-order distribution. The number of rogue waves is relatively low overall, which agrees with previous findings in directionally spread seas. The kurtosis and wave and crest height exceedance probabilities were more affected by varying the directional spreading of the components than by varying the crossing angles between components; reducing the component directional spreading increases the kurtosis and increases the exceedance probabilities. The kurtosis can be estimated quite well for two-component seas from the directional spreading using an empirical relationship based on the two-dimensional Benjamin–Feir index when the effects of bound modes are included. This result may allow forecasting of the probability of extreme waves from the directional spreading in complex sea states.