We address scaling in inhomogeneous and anisotropic turbulent flows by decomposing structure functions into their irreducible representation of the SO(3) symmetry group which are designated by j, m indices. Employing simulations of channel flows with Re l ഠ 70 we demonstrate that different components characterized by different j display different scaling exponents, but for a given j these remain the same at different distances from the wall. The j 0 exponent agrees extremely well with high Re measurements of the scaling exponents, demonstrating the vitality of the SO (3) decomposition. [S0031-9007(99)09384-9] PACS numbers: 47.27.EqMost of the available data analysis and theoretical thinking about the universal statistics of the small scale structure of turbulence assume the existence of an idealized model of homogeneous and isotropic flows. In fact most realistic flows are neither homogeneous nor isotropic. Accordingly, one can analyze the data pertaining to such flows in two ways. The traditional one has been to disregard the inhomogeneity and anisotropy and proceed with the data analysis assuming that the results pertain to the homogeneous and isotropic flows. The second, which is advocated in this Letter, is to take the anisotropy explicitly into account, to carefully decompose the relevant statistical objects into their isotropic and anisotropic contributions, and to assess the degree of universality of each component separately. We analyze here direct numerical simulations (DNS) of a channel flow with Re l ഠ 70 [1-3]. The main conclusion of this Letter is that this procedure is unavoidable; in particular, it highlights the universality of the scaling exponents of the isotropic sector which are presumably those governing the universal small scale statistics at very high Reynolds numbers. In agreement with recent studies of this subject [4,5] we report that different irreducible representations of the symmetry group (characterized by indices j, m) exhibit scalar functions that scale with apparently universal exponents that differ for different j. The exponents found at low values of the Reynolds number for the j 0 (isotropic) sector are in excellent agreement with high Re results; these exponents are invariant to the position in the inhomogeneous flow, leading to reinterpretation of recent findings of position dependence as resulting from the intervention of the anisotropic sectors. The latter have nonuniversal weights that depend on the position in the flow.We consider here channel flow simulations on a grid of 256 points in the streamwise directionx, and ͑128 3 128͒ in the other two directions,ŷ,ẑ. We denote bŷ z the direction perpendicular to the walls and byŷ the spanwise direction in planes parallel to the walls. We employ periodic boundary conditions in the spanwise and streamwise directions and no-slip boundary conditions on the walls. The Reynolds number based on the Taylor scale is Re l ഠ 70 in the center of the channel ͑z 64͒. The simulation is fully symmetric with respect to the central plane. The f...

The motion and the action of microbubbles in homogeneous and isotropic turbulence are investigated through (three-dimensional) direct numerical simulations of the Navier–Stokes equations and applying the Lagrangian approach to track the bubble trajectories. The forces acting on the bubbles are added mass, drag, lift, and gravity. The bubbles are found to accumulate in vortices, preferably on the side with downward velocity. This effect, mainly caused by the lift force, leads to a reduced average bubble rise velocity. Once the reaction of the bubbles on the carrier flow is embodied using a point-force approximation, an attenuation of the turbulence on large scales and an extra forcing on small scales is found.

Microbubble-laden homogeneous and isotropic turbulent flow is investigated by using direct numerical simulation of the three-dimensional Navier–Stokes equations and computing the bubble trajectories with Lagrangian tracking. The bubble motion is calculated by taking into account the effect of fluid acceleration plus added mass, drag, gravity, and in particular the lift force, which had been neglected in many previous simulations. By comparing the results from simulations with and without lift, we find the effect of the lift force to be crucial: for passive bubbles, i.e. bubbles without backreaction on the flow (one-way coupling), the lift enhances the accumulation of bubbles on the downward flow side of vortices, resulting in a considerably reduced rise velocity of bubbles in turbulent flow, compared to still water. This also has consequences for the active bubble case, i.e. for bubbles with backreaction on the flow (two-way coupling): the energy spectrum of the turbulence is modified {non-uniformly}. Because of the combined effect of preferential bubble clustering in downflow zones and the local buoyant transfer, which reduces the vertical fluid velocity fluctuations, large-scale motions (small wavenumbers $k$) are suppressed. In contrast, small-scale motions (large wavenumbers $k$) are enhanced due to the local bubble forcing. The net effect turns out to be a reduction of the energy dissipation rate.

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We investigate by direct numerical simulations the flow that rising bubbles cause in an originally quiescent fluid. We employ the Eulerian-Lagrangian method with two-way coupling and periodic boundary conditions. In order to be able to treat up to 288000 bubbles, the following approximations and simplifications had to be introduced, as done before, e.g., by Climent and Magnaudet, Phys. Rev. Lett. 82, 4827 (1999). (i) The bubbles were treated as point particles, thus (ii) disregarding the near-field interactions among them, and (iii) effective force models for the lift and the drag forces were used. In particular, the lift coefficient was assumed to be 1/2, independent of the bubble Reynolds number and the local flow field. The results suggest that large-scale motions are generated, owing to an inverse energy cascade from the small to the large scales. However, as the Taylor-Reynolds number is only in the range of 1, the corresponding scaling of the energy spectrum with an exponent of -5/3 cannot develop over a pronounced range. In the long term, the property of local energy transfer, characteristic of real turbulence, is lost and the input of energy equals the viscous dissipation at all scales. Due to the lack of strong vortices, the bubbles spread rather uniformly in the flow. The mechanism for uniform spreading is as follows. Rising bubbles induce a velocity field behind them that acts on the following bubbles. Owing to the shear, those bubbles experience a lift force, which makes them spread to the left or right, thus preventing the formation of vertical bubble clusters and therefore of efficient forcing. Indeed, when the lift is artificially put to zero in the simulations, the flow is forced much more efficiently and a more pronounced energy that accumulation at large scales (due to the inverse energy cascade) is achieved.