Direct numerical simulations of the incompressible Navier-Stokes equations are not feasible yet for most practical turbulent flows. Therefore, dynamically less complex mathematical formulations are necessary for coarse-grained simulations. In this regard, eddy-viscosity models for Large-Eddy Simulation (LES) are probably the most popular example thereof. This type of models requires the calculation of a subgrid characteristic length which is usually associated with the local grid size. For isotropic grids, this is equal to the mesh step. However, for anisotropic or unstructured grids, such as the pancake-like meshes that are often used to resolve near-wall turbulence or shear layers, a consensus on defining the subgrid characteristic length has not been reached yet despite the fact that it can strongly affect the performance of LES models. In this context, a new definition of the subgrid characteristic length is presented in this work. This flow-dependent length scale is based on the turbulent, or subgrid stress, tensor and its representations on different grids. The simplicity and mathematical properties suggest that it can be a robust definition that minimizes the effects of mesh anisotropies on simulation results. The performance of the proposed subgrid characteristic length is successfully tested for decaying isotropic turbulence and a turbulent channel flow using artificially refined grids. Finally, a simple extension of the method for unstructured meshes is proposed and tested for a turbulent flow around a square cylinder. Comparisons with existing subgrid characteristic length scales show that the proposed definition is much more robust with respect to mesh anisotropies and has a great potential to be used in complex geometries where highly skewed (unstructured) meshes are present.
The outcome of the collision between two liquid jets depends on the liquid properties, jet velocity, and impact angle. So far studies on liquid jet impingement have been carried out in normal gravity conditions. In microgravity, jets are not accelerated and can show a different behavior than on ground. We perform an experimental analysis of the injection of liquid jets in microgravity, focusing in the jet impingement at different velocities and impact angles at low Weber numbers. Several regimes are obtained, some of which are not observable on ground. Other regimes take place at different parameter ranges than in normal gravity. A map of the observed regimes is proposed in terms of the Weber number and the impact angle.
At the crossroad between flow topology analysis and turbulence modeling, a priori studies are a reliable tool to understand the underlying physics of the subgrid-scale (SGS) motions in turbulent flows. In this paper, properties of the SGS features in the framework of a large-eddy simulation are studied for a turbulent Rayleigh-Bénard convection (RBC). To do so, data from direct numerical simulation (DNS) of a turbulent air-filled RBC in a rectangular cavity of aspect ratio unity and p spanwise open-ended distance are used at two Rayleigh numbers Ra € (108, 1010) [Dabbagh et al., "On the evolution of flowtopology in turbulent Rayleigh-Bénard convection," Phys. Fluids 28, 115105 (2016)]. First, DNS at Ra = 108 is used to assess the performance of eddy-viscosity models such as QR,Wall-Adapting Local Eddy-viscosity (WALE), and the recent S3PQR-models proposed by Trias et al. ["Building proper invariants for eddy-viscosity subgrid-scale models," Phys. Fluids 27, 065103 (2015)]. The outcomes imply that the eddy-viscosity modeling smoothes the coarse-grained viscous straining and retrieves fairly well the effect of the kinetic unfiltered scales in order to reproduce the coherent large scales. However, these models fail to approach the exact evolution of the SGS heat flux and are incapable to reproduce well the further dominant rotational enstrophy pertaining to the buoyant production. Afterwards, the key ingredients of eddy-viscosity, vt, and eddy-diffusivity, kt , are calculated a priori and revealed positive prevalent values to maintain a turbulent wind essentially driven by the mean buoyant force at the sidewalls. The topological analysis suggests that the effective turbulent diffusion paradigm and the hypothesis of a constant turbulent Prandtl number are only applicable in the large-scale strain-dominated areas in the bulk. It is shown that the bulk-dominated rotational structures of vortex-stretching (and its synchronous viscous dissipative structures) hold the highest positive values of vt ; however, the zones of backscatter energy and counter-gradient heat transport are related to the areas of compressed focal vorticity. More arguments have been attained through a priori investigation of the alignment trends imposed by existing parameterizations for the SGS heat flux, tested here inside RBC. It is shown that the parameterizations based linearly on the resolved thermal gradient are invalid in RBC. Alternatively, the tensor-diffusivity approach becomes a crucial choice of modeling the SGS heat flux, in particular, the tensorial diffusivity that includes the SGS stress tensor. This and other crucial scrutinies on a future modeling to the SGS heat flux in RBC are sought.
Small-scale dynamics is the spirit of turbulence physics. It implicates many attributes of flow topology evolution, coherent structures, hairpin vorticity dynamics, and mechanism of the kinetic energy cascade. In this work, several dynamical aspects of the small-scale motions have been numerically studied in a framework of Rayleigh-Benard convection (RBC). To do so, direct numerical simulations have been carried out at two Rayleigh numbers Ra = 10(8) and 10(10), inside an air-filled rectangular cell of aspect ratio unity and pi span-wise open-ended distance. As a main feature, the average rate of the invariants of the velocity gradient tensor (Q(G), R-G) has displayed the so-called "teardrop" spiraling shape through the bulk region. Therein, the mean trajectories are swirling inwards revealing a periodic spin around the converging origin of a constant period that is found to be proportional to the plumes lifetime. This suggests that the thermal plumes participate in the coherent large-scale circulation and the turbulent wind created in the bulk. Particularly, it happens when the plumes elongate substantially to contribute to the large-scale eddies at the lower turbulent state. Supplementary small-scale properties, which are widely common in many turbulent flows have been observed in RBC. For example, the strong preferential alignment of vorticity with the intermediate eigenstrain vector, and the asymmetric alignment between vorticity and the vortex-stretching vector. It has been deduced that in a hard turbulent flow regime, local self-amplifications of straining regions aid in contracting the vorticity worms, and enhance the local interactions vorticity/strain to support the linear vortex-stretching contributions. On the other hand, the evolution of invariants pertained to the traceless part of velocity-times-temperature gradient tensor has also been considered in order to determine the role of thermals in the fine-scale dynamics. These new invariants show an incorporation of kinetic and thermal gradient dynamics that indicate directly the evolution and lifetime of thermal plume structures. By applying an identical approach, the rates of the new invariants have shown a symmetric cycling behaviour decaying towards two skew-symmetric converging origins at the lower Ra number. The trajectories near origins address the hot and cold coherent plumes that travel as an average large-scale heat flux in the sidewall vicinities, and denote a periodic spin period close to the plumes lifetime. At the hard turbulent case, the spiraling trajectories travel in shorter tracks to reveal the reduced lifetime of plumes under the dissipative and mixing effects. The turbulent background kinetic derivatives get self-amplified and the trajectories converge to a zero-valued origin indicating that there is no contribution from the plumes to the average coherent large scales of heat flux. These and other peculiar scrutinies on the small-scale motions in RBC have been enlightened, and may have a fruitful consequence on modelling approaches of buoyancy-driven turbulence.
Copyright 2016 AIP Publishing. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing.
Small-scale dynamics is the spirit of turbulence physics. It implicates many attributes of flow topology evolution, coherent structures, hairpin vorticity dynamics, and mechanism of the kinetic energy cascade. In this work, several dynamical aspects of the small-scale motions have been numerically studied in a framework of Rayleigh-Benard convection (RBC). To do so, direct numerical simulations have been carried out at two Rayleigh numbers Ra = 10(8) and 10(10), inside an air-filled rectangular cell of aspect ratio unity and pi span-wise open-ended distance. As a main feature, the average rate of the invariants of the velocity gradient tensor (Q(G), R-G) has displayed the so-called
We present an experimental study on the characteristics of liquid jets in different configurations. We consider jets injected perpendicular to gravity, jets injected parallel to gravity, and jets injected in a microgravity environment. We study the role played by gravity in the jet breakup length and in the dynamics of the droplets generated after breakup. We analyze droplets obtained in the dripping and jetting regimes, focusing the study on their size, trajectory, oscillation, and rotation. The particularities of the considered injection configurations are analyzed. In normal gravity conditions, in the dripping and jetting regimes, the breakup length increases with the Weber number. The transition between these regimes occurs at Wecr ˜ 3.2. Droplets are notably larger in the dripping regime than in the jetting one. In the latter case, droplet mean size decreases as the liquid flow rate is increased. In microgravity conditions, droplet trajectories form a conical shape due to droplet bouncing after collision. When a collision takes place, coalescence tends to occur at low modified Weber numbers (We m < 2) while bouncing is observed at higher values (We m > 2). The surface of a droplet oscillates after bouncing or coalescence events, following a damped oscillator behavior. The observed oscillation frequency agrees with theoretical predictions.
Direct simulations of the incompressible Navier-Stokes equations are limited to relatively low-Reynolds numbers. Hence, dynamically less complex mathematical formulations are necessary for coarse-grain simulations. Eddy-viscosity models for large-eddy simulation is probably the most popular example thereof: they rely on differential operators that should properly detect different flow configurations (laminar and 2D flows, near-wall behavior, transitional regime, etc.). Most of them are based on the combination of invariants of a symmetric tensor that depends on the gradient of the resolved velocity field, . In this work, models are presented within a framework consisting of a 5D phase space of invariants. In this way, new models can be constructed by imposing appropriate restrictions in this space. For instance, considering the three invariants P GG T , Q GG T , and R GG T of the tensorGG T , and imposing the proper cubic near-wall behavior, i.e., , we deduce that the eddy-viscosity is given by . Moreover, only R GG T -dependent models, i.e., p > - 5/2, switch off for 2D flows. Finally, the model constant may be related with the Vreman’s model constant via ; this guarantees both numerical stability and that the models have less or equal dissipation than Vreman’s model, i.e., . The performance of the proposed models is successfully tested for decaying isotropic turbulence and a turbulent channel flow. The former test-case has revealed that the model constant, C s3pqr , should be higher than 0.458 to obtain the right amount of subgrid-scale dissipation, i.e., C s3pq = 0.572 (p = - 5/2), C s3pr = 0.709 (p = - 1), and C s3qr = 0.762 (p = 0).
Copyright 2015 AIP Publishing. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing.
Experiments with vortex rings impinging permeable and solid boundaries are presented in order to investigate the influence of permeability. Utilizing Particle Image Velocimetry, we compared the behaviour of a vortex ring impinging four different reticulated foams (with permeability k ~ 26 - 85 × 10-8 m2) and a solid boundary. Results show how permeability affects the stretching phenomena of the vortex ring and the formation and evolution of the secondary vortex ring with opposite sign. Moreover, permeability also affects the macroscopic no-slip boundary condition found on the solid boundary, turning it into an apparent slip boundary condition for the most permeable boundary. The apparent slip-boundary condition and the flux exchange between the ambient fluid and the foam are jointly responsible for both the modified formation of the secondary vortex and changes on the vortex ring diameter increase.
It is well known that the flow past a circular cylinder at critical Reynolds number combines flow separation, turbulence transition, reattachment of the flow, and further turbulent separation of the boundary layer. The transition to turbulence in the boundary layer causes the delaying of the separation point and an important reduction of the drag force on the cylinder surface known as the drag crisis. In the present work, large-eddy simulations of the flow past a cylinder at Reynolds numbers in the range 2.5 × 105-6.5 × 105 are performed. It is shown how the pressure distribution changes as the Reynolds number increases in an asymmetric manner, occurring first on one side of the cylinder and then on the other side to complete the drop in the drag up to 0.23 at Re = 6.5 × 105. These variations in the pressure profile are accompanied by the presence of a small recirculation bubble, observed as a small plateau in the pressure, and located around ¿ = 105° (measured from the stagnation point). This small recirculation bubble anticipated by the experimental measurements is here well captured by the present computations and its position and size measured at every Reynolds number. The changes in the wake configuration as the Reynolds number increases are also shown and their relation to the increase in the vortex shedding frequency is discussed. The power spectra for the velocity fluctuations are computed. The analysis of the resulting spectrum showed the footprint of Kelvin-Helmholtz instabilities in the whole range. It is found that the ratio of these instabilities frequency to the primary vortex shedding frequency matches quite well the scaling proposed by Prasad and Williamson [“The instability of the separated shear layer from a bluff body,” Phys. Fluids 8, 1347 (1996); “The instability of the shear layer separating from a bluff body,” J. Fluid Mech. 333, 375–492 (1997)] (f KH /fvs ¿ Re 0.67).
The transition to three-dimensional and unsteady flow in an annulus with a discrete heat source on the inner cylinder is studied numerically. For large applied heat flux through the heater (large Grashof number Gr), there is a strong wall plume originating at the heater that reaches the top and forms a large scale axisymmetric wavy structure along the top. For Gr approximate to 6 x 109, this wavy structure becomes unstable to three-dimensional instabilities with high azimuthal wavenumbers m similar to 30, influenced by mode competition within an Eckhaus band of wavenumbers. Coexisting with some of these steady three-dimensional states, solution branches with localized defects break parity and result in spatio-temporal dynamics. We have identified two such time dependent states. One is a limit cycle that while breaking spatial parity, retains spatio-temporal parity. The other branch corresponds to quasi-periodic states that have globally broken parity. (C) 2014 AIP Publishing LLC.
In this paper we study the problem of thermal convection in a laterally heated, finite, horizontal cylinder. We consider cylinders of moderate aspect ratio (height/diameter approximate to 2) containing a small Prandtl number fluid (sigma < 0.026) typical of molten metals and molten semiconductors. We use the Navier-Stokes and energy equations in the Boussinesq approximation to calculate numerically the basic steady states, analyze their linear stability, and compute some nonlinear secondary flows originated from the instabilities. All the calculated flows and the stability analysis are characterized by their symmetry properties. Due to the confined cylindrical geometry, -presence of lateral walls and lids-, all the flows are completely three dimensional even for the basic steady states. In the range of Prandtl numbers studied, we have identified four different types of instabilities, either oscillatory or stationary. The physical mechanisms, shear or buoyancy, of the corresponding flow transitions have been analyzed. As the value of the Prandtl number approaches sigma = 0.026 the scenario of bifurcations becomes more complicated due to the existence of two different stable basic states originated in a saddle-node bifurcation; a fact that had been overlooked in previous works. (C) 2014 AIP Publishing LLC.
The presence of low-frequency fluctuations in the wake of bluff bodies have been observed in several investigations. Even though the flow past a circular cylinder at Re = 3900 (Re = UrefD/¿) has been the object of several experimental and numerical investigations, there is a large scattering in the average statistics in the near wake. In the present work, the flow dynamics of the near wake region behind a circular cylinder has been investigated by means of direct numerical simulations and statistics have been computed for more than 858 shedding cycles. The analysis of instantaneous velocity signals of several probes located in the vortex formation region, point out the existence of a low-frequency fluctuation at the non-dimensional frequency of fm = 0.0064. This large-scale almost periodic motion seems to be related with the modulation of the recirculation bubble which causes its shrinking and enlargement over the time. Two different configurations have been identified: (i) a high-energy mode with larger fluctuations in the shear-layer and in the vortex formation region (Mode H) and (ii) a low-energy mode with weaker fluctuations in the shear layer (Mode L). The influence of such a low-frequency in the wake topology has been studied not only by means of the phase-average flow field for each mode, but also by the analysis of the time-average first- and second-order statistics of each wake mode. The results are compared with the long-term averaged solution and with results in the existing literature.
Electronic version of an article published as "Physics of fluids", vol. 25, no 8, 2013. DOI: http://dx.doi.org/10.1063/1.4818641.
The presence of low-frequency fluctuations in the wake of bluff bodies have been observed in several investigations. Even though the flow past a circular cylinder at Re = 3900 (Re = U ref D/ν) has been the object of several experimental and numerical investigations, there is a large scattering in the average statistics in the near wake. In the present work, the flow dynamics of the near wake region behind a circular cylinder has been investigated by means of direct numerical simulations and statistics have been computed for more than 858 shedding cycles. The analysis of instantaneous velocity signals of several probes located in the vortex formation region, point out the existence of a low-frequency fluctuation at the non-dimensional frequency of f m = 0.0064. This large-scale almost periodic motion seems to be related with the modulation of the recirculation bubble which causes its shrinking and enlargement over the time. Two different configurations have been identified: (i) a high-energy mode with larger fluctuations in the shear-layer and in the vortex formation region (Mode H) and (ii) a low-energy mode with weaker fluctuations in the shear layer (Mode L). The influence of such a low-frequency in the wake topology has been studied not only by means of the phase-average flow field for each mode, but also by the analysis of the time-average first- and second-order statistics of each wake mode. The results are compared with the long-term averaged solution and with results in the existing literature.
The influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the known Rec computed without mixture. For positive and sufficiently large compositional gradients, Rec decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number mc decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of mc and ωc for Rc sufficiently high. Potential laws are obtained numerically from the dependence of Rec and of the critical frequency ωc on Rc, for the moderate and small Ekman numbers explored.
Mercader, M.; Batiste, O.; Ramirez de La Piscina, L.; Ruiz, X.; Casademunt, J.; Rüdiger, S. Physics of fluids Vol. 17, num. 104108, p. 1-13 DOI: 10.1063/1.2107907 Data de publicació: 2005-10 Article en revista
Two-dimensional nonlinear thermal convection in a cylindrical annulus is numerically analyzed for a Boussinesq fluid of low Prandtl number s=0.025. For a fixed value of the radius ratio, ¿=0.3, different types of steady columnar patterns are found. The stability of these convection patterns and the spatial interaction between them, which result in the formation of mixed modes, are investigated by considering the full nonlinear set of Navier–Stokes equations. Special attention is paid to the strong spatial interaction of the initially unstable modes with wavenumbers n=2 and n=4, which leads, through global bifurcations, to multiple stable quasi-periodic states of the system. A detailed picture of the nonlinear dynamics until temporal chaotic patterns set in is presented and understood in terms of local and global symmetry-breaking bifurcations of the O(2)-symmetric system.
The wind-driven ocean circulation at midlatitudes is susceptible to several types of instabilities. One of the simplest models of these flows is the quasigeostrophic barotropic potential vorticity equation in an idealized ocean basin. In this model, the route to complex spatio/temporal flows is through successive bifurcations. The aim of this study is to describe the physics of the destabilization process of a periodic wind-driven flow associated with a secondary bifurcation. Although bifurcation theory has proven to be a valuable tool to determine the physical mechanisms of destabilization of fluid flows, the analysis of the stability of time-dependent (for example, periodic) flows, using this methodology, is computationally unpractical, due to the large number of degrees-of-freedom involved. The approach followed here is to construct a low-order model using numerical Galerkin projection of the full model equations onto the dynamically active eigenmodes. The resulting reduced model is shown to capture the local dynamics of the full model. The physical mechanism of the destabilization of the periodic wind-driven flow is deduced from the reduced model. While there are several stabilizing processes, notably rectification, the destabilization occurs due to time-dependent increase of the background horizontal shear in the flow.
The steady perturbation caused in a longshore flow by a bottom undulation is considered. The bedforms are assumed to be alongshore periodic, with crests in the cross-shore direction and with a small amplitude in order for linear theory to be applicable. The inviscid shallow-water equations are considered in order to investigate topographic resonance, that is, the condition under which the perturbation in the flow reaches a maximum. Since upstream edge waves held stationary by the mean flow are solutions to the homogeneous resonance equations, the existence of such flows gives rise to the existence of resonances of infinite amplitude (linear, inviscid theory). For a maximum local Froude number of the basic flow F of less than 1, the flow is found to behave subcritically according to classic channel flow theory. In addition, neither steady edge waves nor infinite amplitude resonances exist in this case. However, by numerical simulation, a finite maximum in the flow perturbation as a function of bedform wavelength is found. This topographic resonance is rather weak and wide banded. For a bedform height of 1% the local water depth, the perturbation on the flow may typically be 4% of the mean current. The resonant wavelength is between two and three times the distance of the peak longshore current to the shoreline, lV, when the current profile has a maximum at some distance offshore, or nearly four times the cross-shore length scale of the sandbars, l, for a flow profile monotonically increasing to a constant current far offshore. For F¿1 resonances of infinite amplitude are found. For every F, lV, and l, there is an infinite set of resonant modes with an increasing cross-shore complexity when the mode number increases, similarly to edge waves. The resonant wavelength increases with F and with lV. Some implications on the growth of transverse sandbar families and cuspidal coast are discussed. ¿
A theoretical analysis of the effect of a longshore mean shear flow on edge waves is performed in the framework of linear shallow water equations. A single equation describing edge waves as well as neutral shear waves is obtained. A numerical method of calculation is given for the dispersion relations and the wave pattern, accounting for any beach topography and any mean flow profile (remaining constant alongshore and with a straight shoreline). Numerical calculations are presented for a simple exponential flow profile and for a plane bottom. A Doppler shift in the frequencies and a variation in the offshore extension of the waves are found, depending on the maximum local Froude number of the current, F, defined as F = [V(x)/vgH(x)]max, where V(x) stands for the mean longshore current, H(x) for the depth, and g for the gravity. The maximum shift in frequency is for wave numbers of about Vx(0)2/gm and frequencies of about Vx(0), where Vx(0) is the shear at the shoreline and m is the beach slope. For instance, these maximum differences may reach about 40% for F=0.5. Waves of any wavelength can always propagate downstream, but they can propagate upstream only for F=Fc~0.7. For mean flows with F=Fc only waves shorter or longer than some forbidden wavelengths can propagate against the current. An analytical dispersion relation of asymptotic general validity for short waves (corresponding to the gravity range in real beaches) is given. The numerical model as well as this analytical dispersion relation is tested by means of a nonplanar real topography. ¿