R. Verzicco

Direct simulation of transition in an oscillatory boundary layer

Numerical simulations of Navier-Stokes equations are performed to study the flow originated by an oscillating pressure gradient close to a wall characterized by small imperfections. The scenario of transition from the laminar to the turbulent regime is investigated and the results are interpreted in the light of existing analytical theories. The disturbed-laminar and the intermittently turbulent regimes detected experimentally are reproduced by the present simulations. Moreover it is found that imperfections of the wall are of fundamental importance in causing the growth of two-dimensional disturbances which in turn trigger turbulence in the Stokes boundary layer. Finally, in the intermittently turbulent regime, a description is given of the temporal development of turbulence characteristics

Prandtl number effects in convective turbulence

The effect of Prandtl number on the dynamics of a convective turbulentn flow is studied by numerical experiments. In particular, three series of experimentsn have been performed; in two of them the Rayleigh number spanned about two decadesn while the Prandtl number was set equal to 0.022 (mercury) and 0.7 (air). In the thirdn series, in contrast, we fixed the Rayleigh number at 6×10 5 and then Prandtl number was varied from 0.0022 up to 15. The results have shown that, depending on the Prandtln number, there are two distinct flow regimes; in the first ( Pr [lsim ]0.35)n the flow is dominated by the large-scale recirculation cell that is the most important ‘engine’n for heat transfer. In the second regime, on the other hand, the large-scale flow plays a negligiblen role in the heat transfer which is mainly transported by the thermal plumes. For the low- Pr regime a model for the heat transfer is derivedn and the predictions are in qualitative and quantitative agreement with the results of the numericaln simulations and of the experiments. All the hypotheses and the consequencesn of the model are directly checked and all the findings are consistent with the predictionsn and with experimental observations performed under similar conditions. Finally,n in order to stress the effects of the large-scale flow some counter examples are shownn in which the large-scale motion is artificially suppressed.

A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage

Finite-difference calculations with random and single-mode perturbations are used to study the three-dimensional instability of vortex rings. The basis of current understanding of the subject consists of a heuristic inviscid model (Widnall, Bliss & Tsai 1974) and a rigorous theory which predicts growth rates for thin-core uniform vorticity rings (Widnall & Tsai 1977). At sufficiently high Reynolds numbers the results correspond qualitatively to those predicted by the heuristic model: multiple bands of wavenumbers are amplified, each band having a distinct radial structure. However, a viscous correction factor to the peak inviscid growth rate is found. It is well described by the first term, 1 – α 1 (β)/ Re s , for a large range of Re s . Here Re s is the Reynolds number defined by Saffman (1978), which involves the curvature-induced strain rate. It is found to be the appropriate choice since then α 1 (β) varies weakly with core thickness β. The three most nonlinearly amplified modes are a mean azimuthal velocity in the form of opposing streams, an n = 1 mode ( n is the azimuthal wavenumber) which arises from the interaction of two second-mode bending waves and the harmonic of the primary second mode. When a single wave is excited, higher harmonics begin to grow successively later with nonlinear growth rates proportional to n . The modified mean flow has a doubly peaked azimuthal vorticity. Since the curvature-induced strain is not exactly stagnation-point flow there is a preference for elongation towards the rear of the ring: the outer structure of the instability wave forms a long wake consisting of n hairpin vortices whose waviness is phase shifted π/ n relative to the waviness in the core. Whereas the most amplified linear mode has three radial layers of structure, higher radial modes having more layers of radial structure (hairpins piled upon hairpins) are excited when the initial perturbation is large, reminiscent of visualization experiments on the formation of a turbulent ring at the generator.

Transitional regimes of low-Prandtl thermal convection in a cylindrical cell

The transitions from the onset of convection to fully developed turbulence of a Rayleigh–Benard flow, in a low-aspect-ratio cell and in mercury, are studied through three-dimensional numerical simulation of the Navier–Stokes equations. The calculation of the growth rate of the azimuthal energy modes permitted the accurate determination of the critical Rayleigh number for the establishment of the convective regime (Rac=3750) which is in good agreement with analytical and other numerical results. Increasing the Rayleigh number, the flow remained steady up to Ra≃2.11×104 when an oscillatory instability was observed. Further increases in the Rayleigh produced a chaotic state through the period doubling mechanism and finally the turbulent state was achieved. It is shown that for Ra⩾Rac the mean flow consists of a large-scale convective cell which persists in the whole range of studied Rayleigh numbers (Ra⩽106). The dependence of the Nusselt number over the Rayleigh number is also analyzed and, for Ra⩾3.75×104,...

DIRECT SIMULATIONS OF THE TRANSITIONAL REGIME OF A CIRCULAR JET

Accurate numerical simulations of temporal evolving round jets at a low Reynolds number have revealed the same features observed in experiments and vortex filament simulations. The initial layer of azimuthal vorticity, by the Kelvin–Helmholtz instability, produces vortex rings undergoing successive pairings leading to larger rings. Axisymmetric simulations have shown that the initial roll‐up is not affected by the Reynolds number, consequently insights of practical importance on the transitional regime, can be obtained from low Reynolds number simulations affordable by numerics. The 3‐D simulations displayed the formation of longitudinal structures, and their role in the spreading of the jet is described. Streamwise rib vortices develop in the braid region and these vortices are responsible for the creation of small scales, premonitory of turbulence. In analogy to the plane mixing layer, the pairing reduces the growth of longitudinal and radial vorticity components and triggers the transition to turbulence. Finally comparisons between azimuthal vorticity and passive scalar surfaces have revealed that the latter collects in fat structures while the vorticity is found in thin regions where it is augmented by stretching.

Numerical and experimental study of the interaction between a vortex dipole and a circular cylinder.

This paper describes a study of the centred collision between a dipolar vortex and a solid circular cylinder. The flow was analysed experimentally by using dye visualizations and streak photography. Flow characteristics such as vorticity fields and the transport of passive tracers were compared with numerical simulations. Observations revealed that thin layers of vorticity, created at the cylinder wall are advected by the primary dipole halves, which, while rolling up into compact patches, give rise to the formation of two new asymmetric dipoles that move away along curved trajectories. The structure of the vorticity distribution inside the dipole, before and after the collision, has been investigated. Both the numerical and the experimental results indicate that the vorticity patches originating from the original primary dipole approximately preserve their original functional relationship ω=f(ψ), while the secondary vorticity patches show a tendency to organize into structures attaining a similar relationship.

DYNAMICS OF BAROCLINIC VORTICES IN A ROTATING, STRATIFIED FLUID : A NUMERICAL STUDY

This study deals with the instabilities that arise in the flow generated in a rotating tank by the evolution of a two-layer density stratified fluid. Numerical investigations have been performed by direct simulation of the Navier-Stokes equations for axisymmetric and fully three-dimensional flows. In the former case results have shown the attainment, in a very short time, of an equilibrium position and the formation of an anticyclonic structure in the upper light layer and a cyclonic one in the lower layer, consistently with the observation of Griffiths and Linden. In the long term, however, the Ekman layer at the bottom damps out the cyclone and a steady state with only an anticyclone in the upper layer is reached. In three-dimensions the flow is unstable to azimuthal disturbances and the steady state is no longer achieved. In particular a ring of cyclonic vorticity, surrounding the anticyclone, by the combined effects of baroclinic and barotropic processes, breaks, entrains vorticity from the anticyclon...

Direct simulation of transition in Stokes boundary layers

Numerical simulations of the Stokes boundary layer over a three‐dimensional wavy wall are performed in order to investigate the role played by infinitesimal wall imperfections in triggering transition to turbulence. Our results show flow patterns qualitatively similar to those experimentally detected. In particular the laminar, disturbed‐laminar and intermittent turbulent regimes are recovered. The characteristics of the above flow regimes are analyzed.

Normal and oblique collisions of a vortex ring with a wall

The oblique collision of a vortex ring with a solid wall, atRe=Γ/ν=1389, has been analysed by the direct simulation of the Navier-Stokes equations in Cartesian coordinates. In accordance with a previous experimental study [1], the secondary vorticity produced at the wall is organized into a loop-like vortex in the region of the ring furthest away from the wall. As the ring approaches the wall, the region closest is subjected to a high rate of stretching which increases the vorticity in the core. The vorticity gradients along the core generate bi-helical vortex lines continually displaced towards the region of the ring furthest away from the wall. The analysis of the vorticity and straining fields revealed that the pressure gradient along the core is responsible for the convective motion that displaces these vortex lines and accumulates secondary vorticity in the region far from the wall. This vorticity rolls up and forms a secondary structure which by self-induction moves away from the wall.The fundamental role of the differential stretching has been demonstrated by comparing the case of oblique collision with that of normal collision and with the collision of a two-dimensional vortex pair with an oblique wall.SommarioLinterazione di un vortice ad anello con una parete obliqua, aRe=1389, è stata analizzata mediante la simulazione diretta delle equazioni di Navier-Stokes in coordinate cartesiane. In accordo con un precedente esperimento [1] è stato evidenziato che la vorticità secondaria, prodotta alla parete, si organizza in una strutura vorticosa a ‘loop’ nella regione dellanello più lontana dalla parete. Quando il vortice si avvicina alla parete, la parte più vicina è soggetta ad unelevata deformazione che aumenta il valore della vorticità nel ‘core’. La distribuzione non uniforme di vorticità lungo il ‘core’ del vortice genera delle linee di vorticità elicoidali che vengono transportate verso la regione dellanello più lontana dalla parete. Lanalisi dei campi di vorticità e di deformazione ha rivelato che il gradiente di pressione, dovuto al campo di deformazione non uniforme lungo il ‘core’ del vortice, è responsabile di un moto convettivo che trasporta le linee di vorticità ed accumula la vorticità secondaria nella regione del vortice più lontana dalla parete, dove la struttura secondaria viene generata.Il ruolo fondamentale della deformazione non uniforme è stato evidenziato mediante il confronto della collisione obliqua coni casi di collisione normale e di collisione di una coppia di vortici bidimensionali con una parete obliqua.

On steady columnar vortices under local compression

Numerical simulations are presented of the long time behaviour of viscous columnar vortices subject to non-uniform axial stretching. The relevant result is that the vortices reach a steady state even when the axial average of the strain is zero, such that they are being compressed during half of their extent. The structure of the flow is analysed and shown to range from local Burgers equilibrium to massive separation. For an intermediate range of Reynolds numbers the vortices are more or less uniform and compact, and it is suggested that this condition is related to the strong vortices observed in turbulent flows. The reason for the survival of the vortices under compression is traced to induced axial pressure gradients and to the viscous cancellation of outgoing vorticity. Theoretical analyses of the linear Burgers’ regime and of the onset of separation are presented and compared to the numerical experiments. The results are related to the observation of intermittency in turbulence, and shown to be consistent both with the observed scaling of vortex diameter, and with the lack of intermittency of the velocity signal.