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Register a new userMixing in modulated turbulence. Analytical results
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Recent numerical results show that if a scalar is mixed by periodically forced turbulence, the average mixing rate is directly affected for forcing frequencies small compared to the integral turbulence frequency. We elucidate this by an analytical study using simple turbulence models for spectral transfer. Adding a large amplitude modulation to the forcing of the velocity field enhances the energy transfer and simultaneously diminishes the scalar transfer. Adding a modulation to a random stirring protocol will thus negatively influence the mixing rate. We further derive the asymptotic behaviour of the response of the passive scalar quantities in the same flow for low and high forcing frequencies.
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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 tensor of turbulent stress as a function of the time average of the 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, a kind of extension of classical Boussinesq theory of turbulent viscosity. This leads to rather complex nonlinear integral equation(s) for the time averaged velocity field. This one satisfies some symmetries of the Euler equations. Such symmetries were used by Prandtl and Landau to make various predictions about the shape of the turbulent domain in simple geometries. We explore specifically the case of mixing layer for which the average velocity field only depends on the angle in the wedge behind the splitter plate. This solution yields a pressure difference between the two sides of the splitter which contributes to the lift felt by the plate. Moreover, because of the structure of the equations for the turbulent stress, one can satisfy the Cauchy-Schwarz inequalities, also called the realizability conditions, for this turbulent stress. Such realizability conditions cannot be satisfied with a simple turbulent viscosity.
We investigate the large-scale intermittency of vertical velocity and temperature, and the mixing properties of stably stratified turbulent flows using both Lagrangian and Eulerian fields from direct numerical simulations, in a parameter space relevant for the atmosphere and the oceans. Over a range of Froude numbers of geophysical interest ($approx 0.05-0.3$) we observe very large fluctuations of the vertical components of the velocity and the potential temperature, localized in space and time, with a sharp transition leading to non-Gaussian wings of the probability distribution functions. This behavior is captured by a simple model representing the competition between gravity waves on a fast time-scale and nonlinear steepening on a slower time-scale. The existence of a resonant regime characterized by enhanced large-scale intermittency, as understood within the framework of the proposed model, is then linked to the emergence of structures in the velocity and potential temperature fields, localized overturning and mixing. Finally, in the same regime we observe a linear scaling of the mixing efficiency with the Froude number and an increase of its value of roughly one order of magnitude.
Since the introduction of the logarithmic law of the wall more than 80 years ago, the equation for the mean velocity profile in turbulent boundary layers has been widely applied to model near-surface processes and parameterise surface drag. Yet the hypothetical turbulent eddies proposed in the original logarithmic law derivation and mixing length theory of Prandtl have never been conclusively linked to physical features in the flow. Here, we present evidence that suggests these eddies correspond to regions of coherent streamwise momentum known as uniform momentum zones (UMZs). The arrangement of UMZs results in a step-like shape for the instantaneous velocity profile, and the smooth mean profile results from the average UMZ properties, which are shown to scale with the friction velocity and wall-normal distance in the logarithmic region. These findings are confirmed across a wide range of Reynolds number and surface roughness conditions from the laboratory scale to the atmospheric surface layer.
We use direct numerical simulations to compute structure functions, scaling exponents, probability density functions and turbulent transport coefficients of passive scalars in turbulent rotating helical and non-helical flows. We show that helicity affects the inertial range scaling of the velocity and of the passive scalar when rotation is present, with a spectral law consistent with $sim k_{perp}^{-1.4}$ for the passive scalar variance spectrum. This scaling law is consistent with the phenomenological argument presented in cite{imazio2011} for rotating non-helical flows, wich states that if energy follows a $E(k)sim k^{-n}$ law, then the passive scalar variance follows a law $V(k) sim k^{-n_{theta}}$ with $n_{theta}=(5-n)/2$. With the second order scaling exponent obtained from this law, and using the Kraichnan model, we obtain anomalous scaling exponents for the passive scalar that are in good agreement with the numerical results. Intermittency of the passive scalar is found to be stronger than in the non-helical rotating case, a result that is also confirmed by stronger non-Gaussian tails in the probability density functions of field increments. Finally, Ficks law is used to compute the effective diffusion coefficients in the directions parallel and perpendicular to the rotation axis. Calculations indicate that horizontal diffusion decreases in the presence of helicity in rotating flows, while vertical diffusion increases. We use a mean field argument to explain this behavior in terms of the amplitude of velocity field fluctuations.
The mixing effectiveness, i.e., the enhancement of molecular diffusion, of a flow can be quantified in terms of the suppression of concentration variance of a passive scalar sustained by steady sources and sinks. The mixing enhancement defined this way is the ratio of the RMS fluctuations of the scalar mixed by molecular diffusion alone to the (statistically steady-state) RMS fluctuations of the scalar density in the presence of stirring. This measure of the effectiveness of the stirring is naturally related to the enhancement factor of the equivalent eddy diffusivity over molecular diffusion, and depends on the Peclet number. It was recently noted that the maximum possible mixing enhancement at a given Peclet number depends as well on the structure of the sources and sinks. That is, the mixing efficiency, the effective diffusivity, or the eddy diffusion of a flow generally depends on the sources and sinks of whatever is being stirred. Here we present the results of particle-based simulations quantitatively confirming the source-sink dependence of the mixing enhancement as a function of Peclet number for a model flow.
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