We present a variational data assimilation method in order to improve the accuracy of velocity fields $\tilde{\boldsymbol{v}}$, that are measured using particle image velocimetry (PIV). The method minimises the space–time integral of the difference between the reconstruction $\boldsymbol{u}$ and $\tilde{\boldsymbol{v}}$, under the constraint, that $\boldsymbol{u}$ satisfies conservation of mass and momentum. We apply the method to synthetic velocimetry data, in a two-dimensional turbulent flow, where realistic PIV noise is generated by computationally mimicking the PIV measurement process. The method performs optimally when the assimilation integration time is of the order of the flow correlation time. We interpret these results by comparing them to one-dimensional diffusion and advection problems, for which we derive analytical expressions for the reconstruction error.

The azimuthal structure and the generation process of steepened acoustic waves are investigated in the near field of temporal round jets at Mach numbers of 2 and 3. Initially, the shear layers of the jets are in a laminar state and display instability waves whose main properties are close to those predicted from linear temporal analysis. Then, they transition to a turbulent state and generate high-intensity Mach waves displaying sharp compressions typical of those recorded for jets producing crackle noise. These waves are first shown to be poorly reproduced when only the axisymmetric mode is considered, but to be well captured with the first five azimuthal modes. Their generation process is investigated by performing conditional averages of the flow and acoustic fields triggered by the detection of intense positive pressure peak close to the jets. No steepened waves are visible in the conditionally averaged pressure profiles when the procedure involves only one azimuthal mode at a time. However, sharp compressions are obtained based on the first five modes taken together. In that case, the steep compressions are correlated over a limited portion of the jet circumference and are steeper as more azimuthal modes are considered. Moreover, a direct link is established between the steepened waves and the supersonic convection of large-scale coherent flow structures located in the supersonic core of the jets. This indicates that these waves constitute an extreme, nonlinear case of Mach wave radiation by these structures. In addition, the capacity of flow structures to generate sharp, steepened waves is related to their shapes. More particularly, flow structures with a large extent in the radial direction are shown to produce stronger and steeper Mach waves than those that are elongated in the flow direction.

Presented here are two results concerning the interaction between vorticity and strain. Both are obtained experimentally by investigating the hyperbolic flow created in Taylor's four-roll mill. It is first shown that this pure straining flow becomes intrinsically unstable through a supercritical bifurcation to form an array of counter-rotating vortices aligned in the stretching direction. The dimensionless parameter characterizing the flow is the internal Reynolds number Re = γΔ2/v based on the velocity gradient γ and on the gap between the rollers Δ, and the threshold value is Rec = 17. Near the threshold, the transverse velocity profiles of these vortices are in excellent agreement with those predicted by the theory of Kerr & Dold (1994) in the case of an infinite hyperbolic flow. A second result is obtained at high Reynolds number. Measurements of the velocity profile in the direction parallel to the vortices show that the velocity gradient (the stretching) is systematically weaker inside the vortices than elsewhere. This demonstrates experimentally the existence of a negative feedback of rotation on stretching. This effect is ascribed to the two-dimensionalization due to the vortex fast rotation. An implication of these results for turbulent flows is a nonlinear limitation of the vorticity stretching, an effect characterized recently by Ohkitani 1998.

Turbulent convection induced by heating the bottom boundary of a horizontally homogeneous, linearly (temperature) stratified, rotating fluid layer is studied using a series of laboratory experiments. It is shown that the growth of the convective mixed layer is dynamically affected by background rotation (or Coriolis forces) when the parameter R = (h2Ω3/q0)2/3 exceeds a critical value of Rc ≈ 275. Here h is the depth of the convective layer, Ω is the rate of rotation, and q0 is the buoyancy flux at the bottom boundary. At larger R, the buoyancy gradient in the mixed layer appears to scale as (db/dz)ml = CΩ2, where C ≈ 0.02. Conversely, when R < Rc, the buoyancy gradient is independent of Ω and approaches that of the non-rotating case. The entrainment velocity, ue, for R > Rc was found to be dependent on Ω according to E = [Ri(1 + CΩ2/N2)]−1, where E is the entrainment coefficient based on the convective velocity w∗ = (q0h)1/3, E = ue/w∗, Ri is the Richardson number Ri = N2h2/w2∗, and N is the buoyancy frequency of the overlying stratified layer. The results indicate that entrainment in this case is dominated by non-penetrative convection, although the convective plumes can penetrate the interface in the form of lenticular protrusions.

A laboratory investigation was performed to study distinctive features of the laminar-to-turbulent transition over distributed roughness characterized by two-dimensional (2D) and three-dimensional (3D) periodic, low-order topographies at roughness Reynolds number $Re_{k}\approx 300$. Systematic experiments were performed using high-spatial-resolution planar particle image velocimetry (PIV) in a refractive-index-matching (RIM) channel, where the roughness covered the entire length of the test section. The results show that the flow over the 2D roughness becomes turbulent much sooner than its 3D counterpart ($Re_{x}=50\,000$ versus 120 000). This is attributed to the presence of a velocity inflection point resulting from flow separation within the troughs of the 2D roughness. In the transitional region, unsteady disturbances above the two roughnesses appear upstream of near-roughness disturbances. The above-roughness disturbances are associated with the inflection point in the vertically displaced boundary layer for the 2D case, and with the mean velocity deficit resulting from the interaction of the wakes of upstream elements for the 3D case. The near-roughness fluctuations are associated with the shear layer present behind the crests of both roughnesses. The transitional region is characterized by the interaction between above- and near-roughness disturbances, which merge, leading to a rapid vertical growth of the turbulent fluctuations.

This paper explores the phenomena associated with the emergence of gas bubbles from a submerged granular bed. While there are many natural and industrial applications, we focus on the particular circumstances and consequences associated with the emergence of methane bubbles from the beds of lakes and reservoirs since there are significant implications for the dynamics of lakes and reservoirs and for global warming. This paper describes an experimental study of the processes of bubble emergence from a granular bed. Two distinct emergence modes are identified, mode 1 being simply the percolation of small bubbles through the interstices of the bed, while mode 2 involves the cumulative growth of a larger bubble until its buoyancy overcomes the surface tension effects. We demonstrate the conditions dividing the two modes (primarily the grain size) and show that this accords with simple analytical evaluations. These observations are consistent with previous studies of the dynamics of bubbles within porous beds. The two emergence modes also induce quite different particle fluidization levels. The latter are measured and correlated with a diffusion model similar to that originally employed in river sedimentation models by Vanoni and others. Both the particle diffusivity and the particle flux at the surface of the granular bed are measured and compared with a simple analytical model. These mixing processes can be consider applicable not only to the grains themselves, but also to the nutrients and/or contaminants within the bed. In this respect they are shown to be much more powerful than other mixing processes (such as the turbulence in the benthic boundary layer) and could, therefore, play a dominant role in the dynamics of lakes and reservoirs.

An analysis is presented for the steady, two-dimensional, free convection around line sources of heat and heated cylinders in unbounded saturated porous media. It is extended to account also for the effects of forced convection. The study is based on the Boussinesq equations, with the velocities calculated using Darcy's law.

The analysis begins with the non-dimensional formulation and numerical solution of the problem of pure free convection around a line source of heat. When this analysis is extended to include the effects of forced convection, two parameters appear in the non-dimensional formulation: the non-dimensional value, V∞, of the free-stream velocity and its angle γ of inclination with respect to the vertical. We first describe the asymptotic form of the solution for large and small values of the distance to the source. The far-field description, which is also applicable to the flow around heated cylinders, is needed to facilitate the numerical solution of the problem. It includes a thermal wake, aligned with the free stream, and an outer irrotational flow with a sink and a vortex at the line source. The temperature distribution near the source involves a constant A0(V∞, γ), to be calculated with the numerical solution of the complete problem, which is used in the evaluation of the heat transfer from heated cylinders when the Rayleigh and Péclet numbers are small compared with unity. In this case we find an inner region where heat conduction is dominant, and an outer region where the cylinder appears as a line source of heat. The asymptotic analysis is complemented with the numerical solution of the general problem for circular cylinders with a wide range of Rayleigh numbers and some representative values of V∞ and γ. We give correlations for the Nusselt number in the limiting cases of pure free convection and pure forced convection.

A study is made of the entrainment interface in a two-layer flow with one layer turbulent owing to an imposed surface stress. Turbulent and wave parameters were measured from films of an experiment carried out by Kantha, Phillips & Azad (1977) and are analysed to determine relationships between wave and turbulent length and velocity scales and bulk scales.

A measure of interface distortion is calculated and it is found to be related to the mixed-layer depth only at small overall Richardson number. The local velocity of advance of the interface is determined and is found to vary with overall Richardson number tending to the Kolmogorov scale at small Ri and to Phillips’ (1966) quasilaminar scale at large Ri.

Kelvin-Helmholtz type instabilities are observed and their wavelength, amplitude and lifetime are measured. The wavelength is not found to be related to the mixed-layer depth but some dependence on overall Richardson number is observed. An interfacial mean flow Richardson number is estimated and the variation of wave slope at maximum amplitude with this Richardson number is found to be comparable with observations of Thorpe (1973a) in a laminar two-layer flow. A relationship between the wavelength of an instability and the distortion length scale is found and is discussed.

Journal of Fluid Mechanics, vol. 443 (2001), pp. 69–99

An initial localized eddy was generated in a rotating tank by a source–sink method to study the behaviour of an isolated barotropic eddy on a β-plane. The evolution of the eddy was compared with the laboratory experiments by Firing & Beardsley (1976) and by Takematsu & Kita (1985, 1988), confirming the northwestward (southwestward) translation of a cyclonic (anticyclonic) isolated eddy due to nonlinear effects. Anticyclonic eddies were contrasted with cyclonic eddies in the tank experiment, showing a cyclonic–anticyclonic asymmetry due to the topographic β as a substitute for the planetary β. The fluid experiment was simulated well by numerical simulation based on the quasi-geostrophic vorticity equation. Numerical experiments verified the northwestward (southwestward) translation both for an initially Gaussian and initially Rankine-type isolated cyclonic (anticyclonic) eddy.

The Stokes wave is generalized to progressive waves in deep water which are periodic in two orthogonal directions, and are steady relative to a frame of reference moving in one of these directions. These doubly periodic waves are nonlinear at their lowest approximation, and are calculated from the nonlinear equations for irrotational motion in deep water. It is shown how doubly periodic waves of small but finite wave slope may be calculated also from the nonlinear Schrödinger equation. The three-dimensional paths of particles on the free surface of a doubly periodic wave are found, and the interesting property is demonstrated that the mean particle paths differ from the direction of advance of the wave crests. The upper boundary of occurrence of doubly periodic waves at the smaller wavelength ratios is identified with the stability boundary for Stokes waves. The investigation aims to provide a closer approximation than Stokes waves to local wave structures on the ocean.

The Burgers equation, in spherical and cylindrical symmetries, is studied numerically using pseudospectral and implicit finite difference methods, starting from discontinuous initial (N wave) conditions. The study spans long and varied regimes–embryonic shock, Taylor shock, thick evolutionary shock, and (linear) old age. The initial steep-shock regime is covered by the more accurate pseudospectral approach, while the later smooth regime is conveniently handled by the (relatively inexpensive) implicit scheme. We also give some analytic results for both spherically and cylindrically symmetric cases. The analytic forms of the Reynolds number are found. These give results in close agreement with those found from the numerical solutions. The terminal (old age) solutions are also completely determined. Our analysis supplements that of Crighton & Scott (1979) who used a matched asymptotic approach. They found analytic solutions in the embryonic-shock and the Taylor-shock regions for all geometries, and in the evolutionary-shock region, leading to old age, for the spherically symmetric case. The numerical solution of Sachdev & Seebass (1973) is updated in a comprehensive manner; in particular, the embryonic-shock regime and the old-age solution missed by their study are given in detail. We also study numerically the non-planar equation in the form for which the viscous term has a variable coefficient. It is shown that the numerical methods used in the present study are sufficiently versatile to tackle initial-value problems for generalized Burgers equations.