Stratified Turbulent Flows Research Articles

A Cartesian grid refinement method for simulating thermally stratified urban environments

The capability to accurately simulate thermally stratified environments highly depends on the numerical model which governs the physical problem, but also on the temporal and spatial discretization resolution. This paper presents a multi-resolution Cartesian mesh refinement method for solving the incompressible Navier–Stokes equations, while the standard k–ε turbulence model is employed to resolve stratified turbulent flows. Hanging nodes are used to facilitate the implementation of high order discretization schemes and the algebraic system of the discretized differential equations is solved with a generic modified Alternating Direction Implicit (m-ADI) method developed to resolve locally refined grids. The overall performance of the new flow solver is initially evaluated against experimental data related to the laminar flow past a square cylinder for Reynolds numbers, defined based on the mean ambient wind speed and the characteristic length scale, of 87≤Re≤250 and neutral thermal stratification. The unsteady character of the flow field is successfully predicted and the grid refinement approach enables accurate low cost and low blockage simulations. The flow solver is applied to simulate the flow over thermally stratified idealized urban areas. Good agreement with experimental measurements, where the bulk Richardson number was defined based on the mean wind speed at the top of the reference street canyon, is obtained for −0.21≤Rb≤0.78 while the grid refinement method allowed simulation of the experimental setup in detail, including all the roughness elements and the idealized street canyons. The numerical calculations indicate that even weak thermal stability can lead to laminarization of the induced flow field; reducing at the same time the potential for air mixing and results to air degradation. Even though the recirculation patterns are influenced under weak thermal stratification, extreme thermal instability within the region of −16≤Rb≤−64 with Re=6800 results to the development of two counter-rotating vortices of equivalent kinetic energy. However, the linear correlation of the average kinetic energy at the roof-level with the Rb number indicates that the buoyancy forces determine the air mixing levels.

Lock-exchange flows in inclined pipes: the relevance of the Prandtl mixing length model

We examine the applicability of the Prandtl mixing length model to transverse momentum and mass flux in strongly confined, stably stratified turbulent shear flows. These fluxes were measured in the vertical diametral plane of lock-exchange flows in an inclined pipe by the simultaneous use of planar laser-induced fluorescence and particle image velocimetry at local Reynolds numbers ranging from Re = 580 to 1770 and Richardson numbers ranging from Ri = 0.26 and 1.6. Measurements indicate that the eddy diffusivities of mass and momentum are symmetric about the pipe axis, with their maximum at the axis. The corresponding Prandtl mixing lengths decrease with increasing distance from the pipe axis within the central 60% of the pipe cross-section. Within the range of experimental conditions, the mixing lengths at the axis increase linearly with Ri so that the corresponding turbulent Prandtl number Prt decreases with Ri. In contrast, Prt and the mixing lengths do not display a systematic dependence on Re. Comparison with unbounded and semi-bound shear flows suggests that the strong confinement imposed by the pipe wall may be constraining the integral length scale and Prandtl mixing lengths.

Features of eddy diffusion of momentum and heat in stably stratified flows of environment

The study was performed on features of eddy transport of momentum and heat in the lower atmosphere, which includes the planetary boundary layer as well as upper troposphere and lower stratosphere; the study used a three-parametric method for modeling of stratified turbulent flows. A focus was put on analysis of behavior of vertical diffusivities of momentum and heat: data were obtained through direct measurements and through simulation with three-parametric turbulence model with account for effects of internal gravitational waves which support momentum transfer under condition of very stable stratification (while possessing considerable wavy dynamics). It was demonstrated that the profile of vertical diffusivity for momentum, which was calculated using three-parametric turbulence model, is in agreement with data on direct measurements when measured either within a stable stratified planetary boundary layer or beyond this layer, in free atmosphere.

Transition to stably stratified states in open channel flow with radiative surface heating

AbstractDirect numerical simulations (DNS) of turbulent stratified flow in an open channel with an internal heat source following the Beer–Lambert law from the surface are used to investigate the transition from neutral to strongly stable flow. Our buoyancy bulk parameter is defined through the ratio of the domain height ${\it\delta}$ to $\mathscr{L}$, a bulk Obukhov length scale for the flow. We cover the range ${\it\lambda}={\it\delta}/\mathscr{L}=0{-}2.0$, from neutral conditions to the onset of the stable regime, with the Reynolds number range $Re_{{\it\tau}}=200{-}800$, at a Prandtl number of 0.71. The result is a boundary layer flow where the effects of stratification are weak in the wall region but progressively stronger in the outer layer up to the free surface. At ${\it\lambda}\simeq 1$ the turbulent kinetic energy (TKE) budget is in local equilibrium over a region extending from the near-wall region to a free-surface affected region a distance $l_{{\it\nu}}$ from the surface, with $l_{{\it\nu}}/{\it\delta}\sim Re^{-1/2}$. In this equilibrium region the flow can be characterised by the flux Richardson number $R_{f}$ and the local Obukhov length scale ${\it\Lambda}$. At higher ${\it\lambda}$ local mixing limit conditions are observed over an extended region. At ${\it\lambda}=2$ the flux Richardson number approaches critical limit values of $R_{f,c}\simeq 0.18$ and gradient Richardson number $Ri_{c}\simeq 0.2$. At high ${\it\lambda}$, we obtain a flow field where buoyancy interacts with the smallest scales of motion and the turbulent shear stress and buoyancy flux are suppressed to molecular levels. We find that this regime can be identified in terms of the parameter $Re_{\mathscr{L},c}=\mathscr{L}u_{{\it\tau}}/{\it\nu}\lesssim 200{-}400$ (where $u_{{\it\tau}}$ is the friction velocity and ${\it\nu}$ the kinematic viscosity), which is related to the $L_{\ast }$ parameter of Flores and Riley (Boundary-Layer Meteorol., vol. 139 (2), 2011, pp. 241–259) and buoyancy Reynolds number $\mathscr{R}$. With energetic equilibrium attained, the local buoyancy Reynolds number, $Re_{{\it\Lambda}}={\it\Lambda}\langle u^{\prime }w^{\prime }\rangle ^{1/2}/{\it\nu}$, is directly related to the separation of the Ozmidov ($l_{O}$) and Kolmogorov (${\it\eta}$) length scales in the outer boundary layer by $Re_{{\it\Lambda}}\simeq \mathscr{R}\equiv (l_{O}/{\it\eta})^{4/3}$. The inner wall region has the behaviour $\mathscr{R}\sim Re_{\mathscr{L}}Re_{{\it\tau}}$, in contrast to stratified boundary layer flows where the buoyancy flux is non-zero at the wall and $\mathscr{R}\sim Re_{\mathscr{L}}$.

CFD based approach for modeling direct contact condensation heat transfer in two-phase turbulent stratified flows

This paper describes a CFD based strategy for the modeling of stratified two-phase flows with heat and mass transfer across a moving steam-water interface due to direct contact condensation. Such flows have been of major importance for example in connection with the analysis of nuclear reactor safety systems, in particular during two-phase Pressurized Thermal Shock (PTS) scenarios. The approach is based on the two-fluid phase-average model. The interfacial friction was modeled by using an Algebraic Interfacial Area Density (AIAD) framework where the drag coefficient is a function of the local flow characteristics. To show the impact of the modeling of interfacial friction the simulation with the AIAD model was compared with a simulation where a constant drag coefficient of 0.44 was used in the whole domain. For the modeling of interfacial heat and mass transfer two correlations for the water heat transfer coefficient based on the penetration theory were utilized. The CFD simulations were validated against a steady-state TOPFLOW-PTS steam/water experiment. In the experiment, very detailed temperature measurements were conducted using special thermocouple lances and infrared thermography. Total condensation rate was determined indirectly by using three different methods. The simulations have shown that the results obtained with the AIAD model are considerably closer to the experimental observations than the results obtained with the constant drag coefficient. The condensation models used in the current study predict quite different total condensation rates. That caused significant differences in the temperature field. The simulations of the TOPFLOW-PTS steam/water experiment with condensation have shown that the proposed CFD modeling approach can be successfully applied for the prediction of temperature field and condensation rate during two-phase Pressurized Thermal Shock scenarios. However, the modeling of turbulent interfacial heat transfer should be improved.

RANS Modeling of Stably Stratified Turbulent Boundary Layer Flows in OpenFOAM®

Quantifying mixing processes relating to the transport of heat, momentum, and scalar quantities of stably stratified turbulent geophysical flows remains a substantial task. In a stably stratified flow, such as the stable atmospheric boundary layer (SABL), buoyancy forces have a significant impact on the flow characteristics. This study investigates constant and stability-dependent turbulent Prandtl number ( Pr t ) formulations linking the turbulent viscosity ( ν t ) and diffusivity ( κ t ) for modeling applications of boundary layer flows. Numerical simulations of plane Couette flow and pressure-driven channel flow are performed using the Reynolds-averaged Navier-Stokes (RANS) framework with the standard k - e turbulence model. Results are compared with DNS data to evaluate model efficacy for predicting mean velocity and density fields. In channel flow simulations, a Prandtl number formulation for wall-bounded flows is introduced to alleviate overmixing of the mean density field. This research reveals that appropriate specification of Pr t can improve predictions of stably stratified turbulent boundary layer flows.

A priori study for the modelling of velocity–interface correlations in the stratified air–water flows

This work concerns the modelling of stratified two-phase turbulent flows with interfaces. We consider an equation for an intermittency function α(x,t) which denotes the probability of finding an interface at a given time t and a given point x. In Wacławczyk and Oberlack (2011) a model for the unclosed terms in this equation was proposed. Here, we investigate the performance of this model by a priori tests, and finally, based on the a priori data discuss its possible modification and improvements.

On the interfacial roughness scale in turbulent stratified two-phase flow: 3D lattice Boltzmann numerical simulations with forced turbulence and surfactant

Numerical 3D simulations of turbulent, stratified two-phase shear flow with a surfactant laden interface were used to test and develop a phenomenological interfacial roughness scale model where the energy required to deform the interface (buoyancy, interfacial tension, and viscous work) is proportional to the turbulent kinetic energy adjacent to the interface.The turbulence was forced in the upper and lower liquids in the simulations, to emulate the interfacial dynamics without requiring (prohibitively) large simulation domains and Reynolds numbers. The addition of surfactant lead to an increased roughness scale (for the same turbulent kinetic energy) due to the introduction of interfacial dilatational elasticity that suppressed horizontal motion parallel to the interface, and enhanced the vertical motion.The phenomenological roughness scale model was not fully developed for dilatational elasticity in this work, but we proposed a source term that represents surfactant induced pressure fluctuations near the interface. This source term should be developed further to account for the relation between surfactant density fluctuations and turbulence adjacent to the interface. We foresee that the roughness scale model can be used as a basis for more general interfacial closure relations in Reynolds averaged turbulence models, where also mobile surfactant is accounted for.

Sheared stably stratified turbulence and large-scale waves in a lid driven cavity

We investigated experimentally stably stratified turbulent flows in a lid driven cavity with a non-zero vertical mean temperature gradient in order to identify the parameters governing the mean and turbulent flows and to understand their effects on the momentum and heat transfer. We found that the mean velocity patterns (e.g., the form and the sizes of the large-scale circulations) depend strongly on the degree of the temperature stratification. In the case of strong stable stratification, the strong turbulence region is located in the vicinity of the main large-scale circulation. We detected the large-scale nonlinear oscillations in the case of strong stable stratification which can be interpreted as nonlinear internal gravity waves. The ratio of the main energy-containing frequencies of these waves in velocity and temperature fields in the nonlinear stage is about 2. The amplitude of the waves increases in the region of weak turbulence (near the bottom wall of the cavity), whereby the vertical mean temperature gradient increases.

Numerical simulation of large dunes in meandering streams and rivers with in-stream rock structures

The evolution and migration of large dunes in a realistic intermediate-size experimental stream, the Saint Anthony Falls Laboratory (SAFL) Outdoor StreamLab (OSL), and two large-scale meandering rivers with in–stream rock structures are studied numerically using the SAFL Virtual StreamLab hydro-morphodynamic (VSL3D) model. Due to the challenges arising from mesh quality and large disparity in time-scales, coupled morpho- and hydro-dynamics simulations of bed forms has, for the most part, been restricted to sand wave amplitudes of few centimeters. In this work, we overcome such difficulties by employing the immersed boundary approach and a dual time–stepping technique of the VSL3D model [63]. The VSL3D employs the curvilinear immersed boundary (CURVIB) method along with a suspended sediment load module and is capable of simulating turbulent stratified flows coupled with bed morphodynamic evolution in realistic riverine environments with arbitrarily complex hydraulic structures. Turbulence is handled either via large-eddy simulation (LES) with the dynamic Smagorinski subgrid scale model or unsteady Reynolds-averaged Navier Stokes (URANS) equations closed with the k-ω turbulence model. Simulations in the intermediate-scale OSL channel, in which we also collected experimental morphodynamic data, show that LES can capture the evolution and migration of bed forms with characteristics that are in good agreement with experimental measurements. The URANS model, however, fails to excite the bed instability in the OSL channel but captures realistic dune evolution in the two large-scale meandering rivers. This finding is especially important as it demonstrates the potential of the VSL3D model as a powerful tool for simulating morphodynamic evolution under prototype conditions. To our knowledge, our work is the first attempt to simulate large-scale bed forms in waterways with an order of magnitude disparity in spatial scales, from the ∼2.7m wide OSL channel to the 27m wide rivers. Accordingly, the height of the simulated dunes ranges from ∼0.2m to 2.0m and the wavelength ranges from ∼0.1m to 50m for the OSL and large-scale rivers, respectively. For all cases the statistical properties of the simulated bed forms are shown to agree well with those of bed forms observed in nature.

Ignition Probability in a Stratified Turbulent Flow With a Sunken Fire Igniter

The evolution of a spark kernel ejected by a sunken fire igniter into a turbulent, fuel–air stratified crossflow was studied both experimentally and using a model in a configuration that is similar to the conditions found in turbine engine combustors. This study allows for variations in the transit time of the kernel across a uniform nonflammable region, before entering a second stream containing a flammable fuel–air mixture. High speed schlieren and emission imaging systems are used to visualize the evolution of the kernel and determine the probability of ignition based on measurements over many spark events. Experiments are performed for a range of mean velocities, transit times, inlet (preheat) temperatures, flammable zone equivalence ratios, and nonflammable zone equivalence ratios. In addition to the typical dependence of ignition on the equivalence ratio of the flammable mixture, the results indicate the strong influence of the kernel transit time and the inlet flow temperature on the probability of ignition. The entrainment between the kernel and the surrounding flow appears to be primarily controlled by the kernel ejection-induced flowfield. Reduced-order modeling suggests that the lowering of the kernel temperature associated with entrainment of the nonflammable mixture significantly reduces the ignition probability, and leads to the conclusion that the presence of fuel close to the igniter is necessary to ensure reliable ignition under adverse conditions.

Numerical simulation of stably stratified turbulent flows over an urban surface: Spectra and scales and parameterization of temperature and wind-velocity profiles

Stably stratified turbulent flows over surfaces with explicit roughness elements have been calculated using an LES model. The results of calculations for different height distributions of external dynamic forcing are compared and discussed. The spatial spectra and cospectra of turbulent wind-velocity fluctuations have been calculated and different methods of normalizing them have been studied. A parameterization allowing mean wind-velocity and temperature profiles to be approximated has been proposed for turbulence scale.

Numerical simulation of stably stratified turbulent flows over flat and urban surfaces

Stably stratified turbulent flows over surfaces with explicit roughness elements simulating an urban built-up area have been calculated using an LES model. A method of conducting numerical experiments allowing turbulent flows with specified values of the Obukhov scale L to reach a quasisteady state at the surface has been proposed. It has been shown that, to calculate both temperature and wind-velocity profiles over such objects, one can use the same universal dependences that are used for a flat surface. It has been found that stable stratification does not affect the roughness parameter z0u and the displacement height D.

Experimental study of stratified turbulence forced with columnar dipoles

We present a novel experimental setup aimed at producing a forced strongly stratified turbulent flow. The flow is forced by an arena of 12 vortex pair generators in a large tank. The continuous interactions of the randomly produced vortex pairs give rise to a statistically stationary disordered flow in contrast to previous experiments where the stratified turbulence is decaying. The buoyancy frequency N is set to its highest value N = 1.7 rad/s using salt as stratifying agent so that the horizontal Froude number Fh = Ω/N is low, while the buoyancy Reynolds number \documentclass[12pt]{minimal}\begin{document}$\mathcal {R} = Re {F_h}^2$\end{document}R=ReFh2, where Re = Ωa2/ν is the classical Reynolds number, is as high as possible given the experimental constraints (Ω is the maximum angular velocity of the vortices, a their radius and ν the viscosity). PIV measurements show that the flow is not homogeneous in the horizontal plane and is organised into horizontal layers along the vertical. When \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}$\end{document}R is increased, we observe a progressive evolution from the viscosity dominated regime with smooth layers to a regime with small scales superimposed on the layers and for which the vertical Froude number is of order one. The latter regime resembles the strongly stratified turbulent regime with a downscale cascade that has been predicted for large \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}$\end{document}R. However, horizontal second order structure functions do not exhibit a clear inertial range for the largest \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}$\end{document}R achieved \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}=310$\end{document}R=310. In addition, the corresponding turbulent buoyancy Reynolds number \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}_t=P/(\nu N^2)$\end{document}Rt=P/(νN2) based on an estimation of the injection rate of energy P is only of order unity \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}_t \simeq 0.4$\end{document}Rt≃0.4 indicating that only the edge of the strongly stratified turbulent regime has been reached. However, these results suggest that sufficiently large turbulent buoyancy Reynolds numbers, \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}_t \simeq 10$\end{document}Rt≃10, could be achieved experimentally by scaling up five times this novel set-up.

Two phenomenological constants explain similarity laws in stably stratified turbulence.

In stably stratified turbulent flows, the mixing efficiency associated with eddy diffusivity for heat, or equivalently the turbulent Prandtl number (Pr(t)), is fraught with complex dynamics originating from the scalewise interplay between shear generation of turbulence and its dissipation by density gradients. A large corpus of data and numerical simulations agree on a near-universal relation between Pr(t) and the Richardson number (R(i)), which encodes the relative importance of buoyancy dissipation to mechanical production of turbulent kinetic energy. The Pr(t)-R(i) relation is shown to be derivable solely from the cospectral budgets for momentum and heat fluxes if a Rotta-like return to isotropy closure for the pressure-strain effects and Kolmogorov's theory for turbulent cascade are invoked. The ratio of the Kolmogorov to the Kolmogorov-Obukhov-Corrsin phenomenological constants, and a constant associated with isotropization of the production whose value (= 3/5) has been predicted from Rapid Distortion Theory, explain all the macroscopic nonlinearities.

Diagnosing mixing in stratified turbulent flows with a locally defined available potential energy

AbstractA local available potential energy (APE) density useful as suitable diagnostic tool in turbulent stratified flows is considered under the Boussinesq approximation. The local APE is positive, and in the limit of infinitesimal perturbation from an equilibrium state recovers the Lorenz energy cycle definition of APE. In a turbulent stratified flow, the APE can be Reynolds-decomposed into non-trivial mean and turbulent components, which are connected to the mean and turbulent kinetic energy by suitably defined fluxes. We show that the turbulent buoyancy flux $\overline{w'b'}$ and the rate of production of turbulent APE coincide only under very special circumstances. The framework is applied to derive some global bounds on the mixing efficiency of some representative flows.

Mathematical modeling of the thermal discharge under various operational capacities of thermal power plants

This paper presents a mathematical model of the thermal discharge under various operational capacities of thermal power plants, which is solved by the equations of Navier - Stokes and temperature for an incompressible fluid in a stratified medium based on the method of splitting by physical parameters that can be discretize by the control volume method. In the first step it is assumed that the transfer of momentum carried out only by convection and diffusion. Intermediate velocity field is solved by 5-step Runge - Kutta method. At the second stage, the pressure field is solved based on the found intermediate velocity field. The algorithm is parallelized on high-performance systems. The obtained numerical results of three-dimensional stratified turbulent flow reveals to approximate qualitatively and quantitatively the basic laws of hydrothermal processes occurring in the aquatic environment.

Numerical Simulation of Stably Stratified Turbulent Flows over an Urban Surface: Spectra and Scales and Parameterization of Temperature and Wind-Velocity Profiles

Numerical Simulation of Stably Stratified Turbulent Flows over an Urban Surface: Spectra and Scales and Parameterization of Temperature and Wind-Velocity Profiles

Numerical Simulation of Stably Stratified Turbulent Flows over Flat and Urban Surfaces

Numerical Simulation of Stably Stratified Turbulent Flows over Flat and Urban Surfaces

Time-dependent, non-monotonic mixing in stratified turbulent shear flows: implications for oceanographic estimates of buoyancy flux

AbstractWe employ direct numerical simulation to investigate the efficiency of diapycnal mixing by shear-induced turbulence in stably stratified free shear layers for flows with bulk Richardson numbers in the range $0. 12\leq R{i}_{0} \leq 0. 2$ and Reynolds number $Re= 6000$. We show that mixing efficiency depends non-monotonically upon $R{i}_{0} $, peaking in the range 0.14–0.16, which coincides closely with the range in which both the buoyancy flux and the dissipation rate are maximum. By detailed analyses of the energetics of flow evolution and the underlying dynamics, we show that the existence of high mixing efficiency in the range $0. 14\lt R{i}_{0} \lt 0. 16$ is due to the emergence of a large number of small-scale instabilities which do not exist at lower Richardson numbers and are stabilized at high Richardson numbers. As discussed in Mashayek & Peltier (J. Fluid Mech., vol. 725, 2013, pp. 216–261), the existence of such a well-populated ‘zoo’ of secondary instabilities at intermediate Richardson numbers and the subsequent high mixing efficiency is realized only if the Reynolds number is higher than a critical value which is generally higher than that achievable in laboratory settings, as well as that which was achieved in the majority of previous numerical studies of shear-induced stratified turbulence. We furthermore show that the primary assumptions upon which the widely employed Osborn (J. Phys. Oceanogr. vol. 10, 1980, pp. 83–89) formula is based, as well as its counterparts and derivatives, which relate buoyancy flux to dissipation rate through a (constant) flux coefficient ($\Gamma $), fail at higher Richardson numbers provided that the Reynolds number is sufficiently high. Specifically, we show that the assumptions of fully developed, stationary, and isotropic turbulence all break down at high Richardson numbers. We show that the breakdown of these assumptions occurs most prominently at Richardson numbers above that corresponding to the maximum mixing efficiency, a fact that highlights the importance of the non-monotonicity of the dependence of mixing efficiency upon Richardson number, which we establish to be characteristic of stratified shear-induced turbulence. At high $R{i}_{0} $, the lifecycle of the turbulence is composed of a rapidly growing phase followed by a phase of rapid decay. Throughout the lifecycle, there is considerable exchange of energy between the small-scale turbulence and larger coherent structures which survive the various stages of flow evolution. Since shear instability is one of the most prominent mechanisms for turbulent dissipation of energy at scales below hundreds of metres and at various depths of the ocean, our results have important implications for the inference of turbulent diffusivities on the basis of microstructure measurements in the oceanic environment.