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Register a new userTurbulent Prandtl number and characteristic length scales in stably stratified flows: steady-state analytical solutions
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In this study, the stability dependence of turbulent Prandtl number ($Pr_t$) is quantified via a novel and simple analytical approach. Based on the variance and flux budget equations, a hybrid length scale formulation is first proposed and its functional relationships to well-known length scales are established. Next, the ratios of these length scales are utilized to derive an explicit relationship between $Pr_t$ and gradient Richardson number. In addition, theoretical predictions are made for several key turbulence variables (e.g., dissipation rates, normalized fluxes). The results from our proposed approach are compared against other competing formulations as well as published datasets. Overall, the agreement between the different approaches is rather good despite their different theoretical foundations and assumptions.
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The energy and flux budget (EFB) closure theory for a passive scalar (non-buoyant and non-inertial particles or gaseous admixtures) is developed for stably stratified turbulence. The physical background of the EFB turbulence closures is based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, as well as the turbulent flux of particles. The EFB turbulence closure is designed for stratified geophysical flows from neutral to very stable stratification and it implies that turbulence is maintained by the velocity shear at any stratification. In a steady-state, expressions for the turbulent flux of passive scalar and the anisotropic non-symmetric turbulent diffusion tensor are derived, and universal flux Richardson number dependencies of the components of this tensor are obtained. The diagonal component in the vertical direction of the turbulent diffusion tensor is suppressed by strong stratification, while the diagonal components in the horizontal directions are not suppressed, and they are dominant in comparison with the other components of turbulent diffusion tensor. This implies that any initially created strongly inhomogeneous particle cloud is evolved into a thin pancake in horizontal plane with very slow increase of its thickness in the vertical direction. The turbulent Schmidt number increases linearly with the gradient Richardson number. Considering the applications of these results to the atmospheric boundary-layer turbulence, the theoretical relationships are derived which allow to determine the turbulent diffusion tensor as a function of the vertical coordinate measured in the units of the local Obukhov length scale. The obtained relations are potentially useful in modelling applications of particle dispersion in the atmospheric boundary-layer turbulence and free atmosphere turbulence.
One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iteration scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the near-continuum regime, hundreds of thousands iterations are needed, and yet the converged solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implicit unified gas kinetic scheme (UGKS) and its variants have significantly reduced the iterations in the near-continuum flow regime, but still much higher than that of the highly rarefied gas flows. In this paper, we put forward a general synthetic iteration scheme (GSIS) to find the steady-state solutions of general rarefied gas flows within dozens of iterations at any Knudsen number. As the GSIS does not rely on the specific kinetic model/collision operator, it can be naturally extended to quickly find converged solutions for mixture flows and even flows involving chemical reactions. These two superior advantages are also expected to accelerate the slow convergence in simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version.
(abridged) Context: Turbulent diffusion of large-scale flows and magnetic fields play major roles in many astrophysical systems. Aims: Our goal is to compute turbulent viscosity and magnetic diffusivity, relevant for diffusing large-scale flows and magnetic fields, respectively, and their ratio, the turbulent magnetic Prandtl number, ${rm Pm}_{rm t}$, for isotropically forced homogeneous turbulence. Methods: We use simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed large-scale sinusoidal shear flow. Turbulent viscosity is computed either from the resulting Reynolds stress or from the decay rate of the large-scale flow. Turbulent magnetic diffusivity is computed using the test-field method. The scale dependence of the coefficients is studied by varying the wavenumber of the imposed sinusoidal shear and test fields. Results: We find that turbulent viscosity and magnetic diffusivity are in general of the same order of magnitude. Furthermore, the turbulent viscosity depends on the fluid Reynolds number (${rm Re}$) and scale separation ratio of turbulence. The scale dependence of the turbulent viscosity is found to be well approximated by a Lorentzian. The results for the turbulent transport coefficients appear to converge at sufficiently high values of ${rm Re}$ and the scale separation ratio. However, a weak decreasing trend is found even at the largest values of ${rm Re}$. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large ${rm Re}$ whereas for small ${rm Re}$, we find values between 0.5 and 0.6. Conclusions: The turbulent magnetic diffusivity is in general consistently higher than the turbulent viscosity. The actual value of ${rm Pm}_{rm t}$ found from the simulations ($approx0.9ldots0.95$) at large ${rm Re}$ and scale separation ratio is higher than any of the analytic predictions.
In this paper, we investigate the upward overshooting by three-dimensional numerical simulations. We find that the above convectively stable zone can be partitioned into three layers: the thermal adjustment layer (mixing both entropy and material), the turbulent dissipation layer (mixing material but not entropy), and the thermal dissipation layer (mixing neither entropy nor material). The turbulent dissipation layer is separated from the thermal adjustment layer and the thermal dissipation layer by the first and second zero points of the vertical velocity correlation. The simulation results are in good agreement with the prediction of the one-dimensional turbulent Reynolds stress model. First, the layer structure is similar. Second, the upper boundary of the thermal adjustment layer is close to the peak of the magnitude of the temperature perturbation. Third, the Peclet number at the upper boundary of the turbulent dissipation layer is close to 1. In addition, we have studied the scalings of the overshooting distance on the relative stability parameter $S$, the Prandtl number $rm Pr$, and the Peclet number $rm Pe$. The scaling on $S$ is not unique. The trend is that the overshooting distance decreases with $S$. Fitting on $rm Pr$ shows that the overshooting distance increases with $rm Pr$. Fitting on $rm Pe$ shows that the overshooting distance decreases with $rm Pe$. Finally, we calculate the ratio of the thickness of the turbulent dissipation layer to that of the thermal adjustment layer. The ratio remains almost constant, with an approximate value of 2.4.
In this paper we advance physical background of the energy- and flux-budget turbulence closure based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time-scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth rotation. In accordance to modern experimental evidence, the closure implies maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the two principally different regimes: strong turbulence at Ri << 1 typical of boundary-layer flows and characterised by the practically constant turbulent Prandtl number; and weak turbulence at Ri > 1 typical of the free atmosphere or deep ocean, where the turbulent Prandtl number asymptotically linearly increases with increasing Ri (which implies very strong suppression of the heat transfer compared to the momentum transfer). For use in different applications, the closure is formulated at different levels of complexity, from the local algebraic model relevant to the steady-state regime of turbulence to a hierarchy of non-local closures including simpler down-gradient models, presented in terms of the eddy-viscosity and eddy-conductivity, and general non-gradient model based on prognostic equations for all basic parameters of turbulence including turbulent fluxes.
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