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Register a new userGlobal energy budgets in turbulent Couette and Poiseuille flows
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Turbulent plane Poiseuille and Couette flows share the same geometry, but produce their flow rate owing to different external drivers, pressure gradient and shear respectively. By looking at integral energy fluxes, we pose and answer the question of which flow performs better at creating flow rate. We define a flow {em efficiency}, that quantifies the fraction of power used to produce flow rate instead of being wasted as a turbulent overhead; {em effectiveness}, instead, describes the amount of flow rate produced by a given power. The work by Gatti emph{et al.} (emph{J. Fluid Mech.} vol.857, 2018, pp. 345--373), where the constant power input (CPI) concept was developed to compare turbulent Poiseuille flows with drag reduction, is here extended to compare different flows. By decomposing the mean velocity field into a laminar contribution and a deviation, analytical expressions are derived which are the energy-flux equivalents of the FIK identity. These concepts are applied to literature data supplemented by a new set of direct numerical simulations, to find that Couette flows are less efficient but more effective than Poiseuille ones. The reason is traced to the more effective laminar component of Couette flows, which compensates for their higher turbulent activity. It is also observed that, when the fluctuating fields of the two flows are fed with the same total power fraction, Couette flows dissipate a smaller percentage of it via turbulent dissipation. A decomposition of the fluctuating field into large and small scales explains this feature: Couette flows develop stronger large-scale structures, which alter the mean flow while contributing less significantly to dissipation.
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We explore the effect of forcing on the linear shear flow or plane Couette flow, which is also the background flow in the very small region of the Keplerian accretion disk. We show that depending on the strength of forcing and boundary conditions suitable for the systems under consideration, the background plane shear flow and, hence, accretion disk velocity profile modifies to parabolic flow, which is plane Poiseuille flow or Couette-Poiseuille flow, depending on the frame of reference. In the presence of rotation, plane Poiseuille flow becomes unstable at a smaller Reynolds number under pure vertical as well as threedimensional perturbations. Hence, while rotation stabilizes plane Couette flow, the same destabilizes plane Poiseuille flow faster and forced local accretion disk. Depending on the various factors, when local linear shear flow becomes Poiseuille flow in the shearing box due to the presence of extra force, the flow becomes unstable even for the Keplerian rotation and hence turbulence will pop in there. This helps in resolving a long standing problem of sub-critical transition to turbulence in hydrodynamic accretion disks and laboratory plane Couette flow.
The ultimate goal of a sound theory of turbulence in fluids is to close in a rational way the Reynolds equations, namely to express the time averaged turbulent stress tensor as a function of the time averaged velocity field. This closure problem is a deep and unsolved problem of statistical physics whose solution requires to go beyond the assumption of a homogeneous and isotropic state, as fluctuations in turbulent flows are strongly related to the geometry of this flow. This links the dissipation to the space dependence of the average velocity field. Based on the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations, it has been recently observed that the closure problem is strongly restricted, and that it implies that the turbulent stress is a non local function (in space) of the average velocity field, an extension of classical Boussinesq theory of turbulent viscosity. The resulting equations for the turbulent stress are derived here in one of the simplest possible physical situation, the turbulent Poiseuille flow between two parallel plates. In this case the integral kernel giving the turbulent stress, as function of the averaged velocity field, takes a simple form leading to a full analysis of the averaged turbulent flow in the limit of a very large Reynolds number. In this limit one has to match a viscous boundary layer, near the walls bounding the flow, and an outer solution in the bulk of the flow. This asymptotic analysis is non trivial because one has to match solution with logarithms. A non trivial and somewhat unexpected feature of this solution is that, besides the boundary layers close to the walls, there is another inner boundary layer near the center plane of the flow.
Phoresis, the drift of particles induced by scalar gradients in a flow, can result in an effective compressibility, bringing together or repelling particles from each other. Here, we ask whether this effect can affect the transport of particles in a turbulent flow. To this end, we study how the dispersion of a cloud of phoretic particles is modified when injected in the flow, together with a blob of scalar, whose effect is to transiently bring particles together, or push them away from the center of the blob. The resulting phoretic effect can be quantified by a single dimensionless number. Phenomenological considerations lead to simple predictions for the mean separation between particles, which are consistent with results of direct numerical simulations. Using the numerical results presented here, as well as those from previous studies, we discuss quantitatively the experimental consequences of this work and the possible impact of such phoretic mechanisms in natural systems.
Data from Direct Numerical Simulations of disperse bubbly flows in a vertical channel are used to study the effect of the bubbles on the carrier-phase turbulence. A new method is developed, based on the barycentric map approach, that allows to quantify the anisotropy and componentiality of the flow at any scale. Using this the bubbles are found to significantly enhance flow anisotropy at all scales compared with the unladen case, and for some bubble cases, very strong anisotropy persists down to the smallest flow scales. The strongest anisotropy observed was for the cases involving small bubbles. Concerning the inter-scale energy transfer, our results indicate that for the bubble-laden cases, the energy transfer is from large to small scales, just as for the unladen case. However, there is evidence of an upscale transfer when considering the transfer of energy associated with particular components of the velocity field. Although the direction of the energy transfer is the same with and without the bubbles, the transfer is much stronger for the bubble-laden cases, suggesting that the bubbles play a strong role in enhancing the activity of the nonlinear term in the flow. The normalized forms of the fourth and sixth-order structure functions are also considered, and reveal that the introduction of bubbles into the flow strongly enhances intermittency in the dissipation range, but suppresses it at larger scales. This strong enhancement of the dissipation scale intermittency has significant implications for understanding how the bubbles might modify the mixing properties of turbulent flows.
We report the onset of elastic turbulence in a two-dimensional Taylor-Couette geometry using numerical solutions of the Oldroyd-B model, also performed at high Weissenberg numbers with the program OpenFOAM. Beyond a critical Weissenberg number, an elastic instability causes a supercritical transition from the laminar Taylor-Couette to a turbulent flow. The order parameter, the time average of secondary-flow strength, follows the scaling law $Phi propto (mathrm{Wi} -mathrm{Wi}_c)^{gamma}$ with $mathrm{Wi}_c=10$ and $gamma = 0.45$. The power spectrum of the velocity fluctuations shows a power-law decay with a characteristic exponent, which strongly depends on the radial position. It is greater than two, which we relate to the dimension of the geometry.
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