In this paper, we extend Grossmann and Lohse's (GL) model [Phys. Rev. Lett. {\bf 86}, 3316 (2001)] for the predictions of Reynolds number (Re) and Nusselt number (Nu) in turbulent Rayleigh-Bénard convection (RBC). Towards this objective, we use functional forms for the prefactors of the dissipation rates in the bulk and the boundary layers. The functional forms arise due to inhibition of nonlinear interactions in the presence of walls and buoyancy compared to free turbulence, along with a deviation of viscous boundary layer profile from Prandtl-Blasius theory. We perform 60 numerical runs on a three-dimensional unit box for a range of Rayleigh numbers (Ra) and Prandtl numbers (Pr) and determine the aforementioned functional forms using machine learning. The revised predictions are in better agreement with the past numerical and experimental results than those of the GL model, especially for extreme Prandtl numbers.
Technische Universität Ilmenau（伊尔梅瑙工业大学）；Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）
In this paper, we develop a multivariate regression model and a neural network model to predict the Reynolds number (Re) and Nusselt number in turbulent thermal convection. We compare their predictions with those of earlier models of convection: Grossmann-Lohse~[Phys. Rev. Lett. \textbf{86}, 3316 (2001)], revised Grossmann-Lohse~[Phys. Fluids \textbf{33}, 015113 (2021)], and Pandey-Verma [Phys. Rev. E \textbf{94}, 053106 (2016)] models. We observe that although the predictions of all the models are quite close to each other, the machine learning models developed in this work provide the best m…
At the beginning of the second half of the twentieth century, Proudman and Pearson (J. Fluid. Mech.,2(3), 1956, pp.237-262) suggested that the functional form of the drag coefficient ($C_D$) of a single sphere subjected to uniform fluid flow consists of a series of logarithmic and power terms of the Reynolds number ($Re$).\ In this paper, we will explore the validity of the above statement for Reynolds numbers up to $10^{6}$ by using a symbolic regression machine learning method.\ The algorithm is trained by available experimental data and data from well-known correlations from the literature …
The vaporization rate of aluminum droplets in shocked flows plays a crucial role in determining the energy release rate during the combustion of the aluminized energetic materials. In this paper, the physics of the vaporization of aluminum droplets in shocked flows is numerically investigated. Surrogate models for the temporally averaged Sherwood number and Nusselt number, cast as functions of shock Mach number and Reynolds number, are developed from the simulation-based data. The results show that the Sherwood number and the Nusselt number of the droplet increase monotonically with the Reynol…
We consider laminar, fully-developed, Poiseuille flows of liquid in the Cassie state through diabatic, parallel-plate microchannels symmetrically textured with isoflux ridges. Through the use of matched asymptotic expansions we analytically develop expressions for (apparent hydrodynamic) slip lengths and variously-defined Nusselt numbers. Our small parameter ($ε$) is the pitch of the ridges divided by the height of the microchannel. When the ridges are oriented parallel to the flow, we quantify the error in the Nusselt number expressions in the literature and provide a new closed-form result. …
We study the melting dynamics of large ice balls in a turbulent von Karman flow at very high Reynolds number. Using an optical shadowgraphy setup, we record the time evolution of particle sizes. We study the heat transfer as a function of the particle scale Reynolds number for three cases: fixed ice balls melting in a region of strong turbulence with zero mean flow, fixed ice balls melting under the action of a strong mean flow with lower fluctuations, and ice balls freely advected in the whole flow. For the fixed particles cases, heat transfer is observed to be much stronger than in laminar f…
In this work, the Nusselt number is examined for periodically developed heat transfer in micro- and mini-channels with arrays of offset strip fins, subject to a constant heat flux. The Nusselt number is defined on the basis of a heat transfer coefficient which represents the spatially constant macro-scale temperature difference between the fluid and solid during conjugate heat transfer. Its values are determined numerically on a single unit cell of the array for Reynolds numbers between 1 and 600. Two combinations of the Prandtl number and the thermal conductivity ratio are selected, correspon…
We explore the potential of a formulation of the Navier-Stokes equations incorporating a random description of the small-scale velocity component. This model, established from a version of the Reynolds transport theorem adapted to a stochastic representation of the flow, gives rise to a large-scale description of the flow dynamics in which emerges an anisotropic subgrid tensor, reminiscent to the Reynolds stress tensor, together with a drift correction due to an inhomogeneous turbulence. The corresponding subgrid model, which depends on the small scales velocity variance, generalizes the Bouss…
The convective heat transfer and flow behavior of graphene-water nanofluids are studied experimentally by focusing on transitional flow. Graphene-water nanofluids with different particle mass fractions (0.025, 0.1 and 0.2%) are produced following two-step method and using PVP as a surfactant. Thermo-physical characterization is performed by measuring viscosity and thermal conductivity of the nanofluids. Convection characteristics are experimentally studied from laminar to turbulent flow regimes. It is seen that pressure drop increases dramatically in the transition region, and laminar to turbu…
New insight into the contribution of the microscale vortex evolution towards convection heat transfer in porous media is presented in this paper. The objective is to determine how the microscale vortices influence convection heat transfer in turbulent flow inside porous media. The microscale temperature distribution is analyzed using flow visualization in 2D using streamlines and in 3D using Q structures. The pertinent observations are supplemented with the comparison of surface skin friction and heat transfer using: (1) surface skin friction lines and (2) joint PDF of pressure and skin fricti…
Previous numerical studies on homogeneous Rayleigh-Bénard convection, which is Rayleigh-Bénard convection (RBC) without walls, and therefore without boundary layers, have revealed a scaling regime that is consistent with theoretical predictions of bulk-dominated thermal convection. In this so-called asymptotic regime, previous studies have predicted that the Nusselt number ($Nu$) and the Reynolds number ($Re$) vary with the Rayleigh number ($Ra$) according to $Nu\sim Ra^{1/2}$ and $Re\sim Ra^{1/2}$ at small Prandtl number ($Pr$). In this study, we consider a flow that is similar to RBC but wit…
Thermal plumes are the energy containing eddy motions that carry heat and momentum in a convective boundary layer. The detailed understanding of their structure is of fundamental interest for a range of applications, from wall-bounded engineering flows to quantifying surface-atmosphere flux exchanges. We address the aspect of Reynolds stress anisotropy associated with the intermittent nature of heat transport in thermal plumes by performing an invariant analysis of the Reynolds stress tensor in an unstable atmospheric surface layer flow, using a field-experimental dataset. Given the intermitte…
Penetrative turbulent Rayleigh-Bénard convection which depends on the density maximum of water near $4^\circ\rm{C}$ is studied using two-dimensional (2D) and three-dimensional (3D) direct numerical simulations (DNS). The working fluid is water near $4^\circ\rm{C}$ with Prandtl number $Pr=11.57$. The considered Rayleigh numbers $Ra$ range from $10^7$ to $10^{10}$. The density inversion parameter $θ_m$ varies from 0 to 0.9. It is found that the ratio of the top and bottom thermal boundary-layer thickness ($F_λ=λ_t^θ/λ_b^θ$) increases with increasing $θ_m$, and the relationship between $F_λ$ and …
We present measurements of the orientation $θ_0$ and temperature amplitude $δ$ of the large-scale circulation in a cylindrical sample of turbulent Rayleigh-Benard convection (RBC) with aspect ratio $Γ\equiv D/L = 1.00$ ($D$ and $L$ are the diameter and height respectively) and for the Prandtl number $Pr \simeq 0.8$. Results for $θ_0$ revealed a preferred orientation with upflow in the West, consistent with a broken azimuthal invariance due to Earth's Coriolis force [see \cite{BA06b}]. They yielded the azimuthal diffusivity $D_θ$ and a corresponding Reynolds number $Re_θ$ for Rayleigh numbers o…
The hydrodynamic and thermal characteristics of a freely-vibrating circular cylinder in mixed convection flow are numerically investigated at low Reynolds numbers. The numerical investigations are conducted for a range of parameters, Ur = [2.0, 10], Pr = [0.7, 10] and Ri = [0.5, 2.0]. Whereas the Reynolds number, the mass ratio and the damping ratio are fixed. A secondary VIV lock-in region is found in the cases of high Richardson number Ri=2.0 for high reduced velocity values, in which the buoyancy-driven flow is non-trivial. A wide VIV lock-in region is formed with tremendous energy transfer…
Non-Oberbeck-Boussinesq (NOB) effects on the flow organization in two-dimensional Rayleigh-Benard turbulence are numerically analyzed. The working fluid is water. We focus on the temperature profiles, the center temperature, the Nusselt number, and on the analysis of the velocity field. Several velocity amplitudes (or Reynolds numbers) and several kinetic profiles are introduced and studied; these together describe the various features of the rather complex flow organization. The results are presented both as functions of the Rayleigh number Ra (with Ra up to 10^8) for fixed temperature differ…
Turbulent Rayleigh-Bénard convection in slender cylindrical cells exhibits rich dynamics of the large-scale circulation (LSC), with several rolls stacked on top of each other. We propose that the elliptical instability is the mechanism which causes the twisting and breaking of the LSC into multiple rolls and that the volume-averaged heat and momentum transport, represented by the Nusselt number and Reynolds number, is generally weaker for larger number $n$ of the LSC rolls. This is supported by direct numerical simulations for $Ra=5\times10^5$, $Pr=0.1$, $H=5D$ and $1\leq~n\leq4$.
The present study numerically explores the mixed convection phenomena in a differentially heated ventilated square cavity with active flow modulation via a rotating plate. Forced convection flow in the cavity is attained by maintaining an external fluid flow through an opening at the bottom of the left cavity wall while leaving it through another opening at the right cavity wall. A counter-clockwise rotating plate at the center of the cavity acts as active flow modulator. Moving mesh approach is used for the rotation of the plate and the numerical solution is achieved using Arbitrary Lagrangia…
The Lie-group-based symmetry analysis, as first proposed in Avsarkisov et al. (2014) and then later modified in Oberlack et al. (2015), to generate invariant solutions in order to predict the scaling behavior of a channel flow with uniform wall transpiration, is revisited. By focusing first on the results obtained in Avsarkisov et al. (2014), we failed to reproduce two key results: (i) For different transpiration rates at a constant Reynolds number, the mean velocity profiles (in deficit form) do not universally collapse onto a single curve as claimed. (ii) The universally proposed logarithmic…
We experimentally and numerically investigate the angular momentum transport in turbulent Taylor-Couette flow for independently rotating cylinders at a small radius ratio of $η=0.357$ for various shear Reynolds numbers ($4.5\times 10^3 \leq Re_S \leq 1.2 \times 10^5$) and ratios of angular velocities ($-0.5 \leq μ\leq 0.2$). \red{The momentum transport in terms of the pseudo-Nusselt number ${Nu}_ω$ does not show a pure power law scaling with the forcing $Re_S$ and features non-constant effective scaling between $1.3\times 10^4 \leq Re_S \leq 4 \times 10^4$. This transition lies in the classica…
What is the final state of turbulence when the driving parameter approaches to infinity? For thermal turbulence, in 1962, Kraichnan proposed a so-called ultimate scaling dependence of the heat transport (quantified by the Nusselt number $\text{Nu}$) on the Rayleigh number ($\text{Ra}$), which can be extrapolated to arbitrarily high $\text{Ra}$. The existence of Kraichnan ultimate scaling has been intensively debated in the past decades. In this work, using a supergravitational thermal convection system, with an effective gravity up to 100 times the Earth's gravity, both Rayleigh number and she…
Taylor--Couette (TC) flow is the shear-driven flow between two coaxial independently rotating cylinders. In recent years, high-fidelity simulations and experiments revealed the shape of the streamwise and angular velocity profiles up to very high Reynolds numbers. However, due to curvature effects, so far no theory has been able to correctly describe the turbulent streamwise velocity profile for given radius ratio, as the classical Prandtl--von Kármán logarithmic law for turbulent boundary layers over a flat surface at most fits in a limited spatial region. Here we address this deficiency by a…
Direct numerical simulations of turbulent heat transfer in fully-developed channel flows have been performed in a range of friction Reynolds number between 60 and 180, based on the friction velocity and the channel half width $δ$, with emphasis on a puff-like structure, large-scale spatial intermittency. For the Reynolds numbers lower than 80 with a large computational domain of 51.2 x 2 x 22.5, the turbulent puff was observed and its significant influences on the momentum and heat transports were found. The spatial structure of the equilibrium puff, or the localized turbulence, was examined w…
Turbulent signals are known to exhibit burst-like activities, which affect the turbulence statistics at both large and small scales of the flow. In our study, we pursue this problem from the perspective of an event-based framework, where bursting events are studied across multiple scales in terms of both their size and duration. To illustrate our method and assess any dependence on the Reynolds number, we use two datasets - from the Melbourne wind tunnel and from SLTEST - an atmospheric surface layer experiment. We show that an index, namely the burstiness index, can be used successfully to de…
The unifying theory of scaling in thermal convection (Grossmann & Lohse (2000)) (henceforth the GL theory) suggests that there are no pure power laws for the Nusselt and Reynolds numbers as function of the Rayleigh and Prandtl numbers in the experimentally accessible parameter regime. In Grossmann & Lohse (2001) the dimensionless parameters of the theory were fitted to 155 experimental data points by Ahlers & Xu (2001) in the regime $3\times 10^7 \le Ra \le 3 \times 10^{9}$ and $4\le Pr \le 34$ and Grossmann & Lohse (2002) used the experimental data point from Qiu & Tong (2001) and the fact th…
The flow field and the heat transfer around six in-line iso-thermal circular cylinders has been studied by mean of numerical simulations. Two values of the center to center spacing ($s=3.6d$ and $4d$, where $d$ is the cylinder diameter) at Reynolds number of $100$ and Prandtl number of $0.7$ has been investigated. Similarly to the in-line two cylinder configuration, in this range a transition in the flow and in the heat transfer occurs. Two different flow patterns have been identified: the stable shear layer (SSL) mode and the shear layer secondary vortices (SLSV) mode, at $3.6$ and $4$ spacin…
The inertial response of a particle to turbulent flows is a problem of relevance to a wide range of environmental and engineering problems. The equation most often used to describe the force balance is the Maxey-Riley equation, which includes in addition to buoyancy and steady drag forces, an unsteady Basset drag force related to past particle acceleration. Here we provide a historical review of how the Maxey-Riley equation was developed and how it is only suited for studies where the Reynolds number is less than unity. Revisiting the innovative mathematical methods employed by Basset (1888), …
In this thesis, a variable speed heat conducting cylinder is positioned in the middle of a rectangular channel with an active flow modification system to demonstrate a numerical study of steady two-dimensional mixed convention heat transfer phenomena. In the current study, the lower wall has a discrete isoflux heater installed while the upper wall is kept at an isothermal low temperature. The Prandtl number of air passing through the channel is held constant at 0.71 while the Reynolds and Grashof numbers are greatly varied for four different cylinder configurations, including (a) channels with…
In this paper we found, by means of numerical simulations, a transition in the oscillatory character of the flow field for a particular combination of buoyancy and spacing in an array of six circular cylinders at a Reynolds number of 100 and Prandtl number of 0.7. The cylinders are iso-thermal and they are aligned with the Earth acceleration (g). According to the array orientation, an aiding or an opposing buoyancy is considered. The effect of natural convection with respect to the forced convection is modulated with the Richardson number, Ri, ranging between -1 and 1. Two values of center to …
In this paper, we experimentally study the influence of large-scale Taylor rolls on the small-scale statistics and the flow organization in fully turbulent Taylor-Couette flow {for Reynolds numbers up to $\text{Re}_S=3\times 10^5$}. The velocity field in the gap confined by coaxial and independently rotating cylinders at a radius ratio of $η=0.714$ is measured using planar {particle image velocimetry} in horizontal planes at different cylinder heights. Flow regions with and without prominent Taylor vortices are compared. We show that the local angular momentum transport (expressed in terms of …
In this paper, we have proposed a modified Marker-And-Cell (MAC) method to investigate the problem of an unsteady 2-D incompressible flow with heat and mass transfer at low, moderate, and high Reynolds numbers with no-slip and slip boundary conditions. We have used this method to solve the governing equations along with the boundary conditions and thereby to compute the flow variables, viz. $u$-velocity, $v$-velocity, $P$, $T$, and $C$. We have used the staggered grid approach of this method to discretize the governing equations of the problem. A modified MAC algorithm was proposed and used to…
We report an experimental study aiming to clarify the role of boundary conditions (BC) in high Rayleigh number $10^8 < {\rm{Ra}} < 3 \times 10^{12}$ turbulent thermal convection of cryogenic helium gas. We switch between BC closer to constant heat flux (CF) and constant temperature (CT) applied to the highly conducting bottom plate of the aspect ratio one cylindrical cell 30 cm in size, leading to dramatic changes in the temperature probability density function and in power spectral density of the temperature fluctuations measured at the bottom plate, while the dynamic thermal behaviour of the…
The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large-scale convection-roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.
A numerical study of convection with stress-free boundary conditions in the presence of an imposed magnetic field that is tilted with respect to the direction of gravity is carried out in the limit of small magnetic Reynolds number. The dynamics are investigated over a range of Rayleigh number $Ra$ and Chandrasekhar numbers up to $Q = 2\times10^6$, with the tilt angle between the gravity vector and imposed magnetic field vector fixed at $45^{\circ}$. For a fixed value of $Q$ and increasing $Ra$, the convection dynamics can be broadly characterized by three primary flow regimes: (1) quasi-two-d…
Results from direct numerical simulations of vertical natural convection at Rayleigh numbers $1.0\times 10^5$-$1.0\times 10^9$ and Prandtl number $0.709$ support a generalised applicability of the Grossmann-Lohse (GL) theory, which was originally developed for horizontal natural (Rayleigh-B{é}nard) convection. In accordance with the GL theory, it is shown that the boundary-layer thicknesses of the velocity and temperature fields in vertical natural convection obey laminar-like Prandtl-Blasius-Pohlhausen scaling. Specifically, the normalised mean boundary-layer thicknesses scale with the $-1/2$…
A solvable turbulent model is used to predict both the structure of the boundary layer and the scaling laws in thermal convection. The transport of heat depends on the interplay between the thermal, viscous and integral scales of turbulence, and thus, on both the Prandtl number and the Reynolds numbers. Depending on their values, a wide variety of possible regimes is found, including the classical 2/7 and 1/3 law, and a new $4/13=0.308$ law for the Nusselt power law variation with the Rayleigh number.
We present a numerical study of Rayleigh-Bénard convection disturbed by a longitudinal wind. Our results show that under the action of the wind, the vertical heat flux through the cell initially decreases, due to the mechanism of plumes-sweeping, and then increases again when turbulent forced convection dominates over the buoyancy. As a result, the Nusselt number is a non-monotonic function of the shear Reynolds number. We provide a simple model that captures with good accuracy all the dynamical regimes observed. We expect that our findings can lead the way to a more fundamental understanding …
Scalings of the streamwise velocity energy spectra in turbulent boundary layers were considered in Part 1. A spectral decomposition analysis provided a means to separate out attached and non-attached eddy contributions and was used to generate three spectral sub-components, one of which is a close representation of the spectral signature induced by self-similar, wall-attached turbulence. Since sub-components of the streamwise turbulence intensity $\overline{u^2}$ follow from an integration of the velocity energy spectra, we here focus on the scaling of the former. Wall-normal profiles and Reyn…
We present a theory for the self-propulsion of symmetric, half spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air-water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion-advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary cond…
Three-dimensional miscible Rayleigh--Taylor (RT) turbulence at small Atwood number and at Prandtl number one is investigated by means of high resolution direct numerical simulations of the Boussinesq equations. RT turbulence is a paradigmatic time-dependent turbulent system in which the integral scale grows in time following the evolution of the mixing region. In order to fully characterize the statistical properties of the flow, both temporal and spatial behavior of relevant statistical indicators have been analyzed. Scaling of both global quantities ({\it e.g.}, Rayleigh, Nusselt and Reynold…
In turbulent Rayleigh-Bénard (RB) convection, a transition to the so-called ultimate regime, in which the boundary layers (BL) are of turbulent type, has been postulated. Indeed, at very large Rayleigh number $Ra \approx 10^{13}-10^{14}$ a transition in the scaling of the global Nusselt number $Nu$ (the dimensionless heat transfer) and the Reynolds number with $Ra$ has been observed in experiments and very recently in direct numerical simulations (DNS) of two-dimensional (2D) RB. In this paper we analyse the local scaling properties of the lateral temperature structure functions in the BLs of …
We analyze the global transport properties of turbulent Taylor-Couette flow in the strongly turbulent regime for independently rotating outer and inner cylinders, reaching Reynolds numbers of the inner and outer cylinders of Re_i = 2 x 10^6 and Re_o = 1.4 x 10^6, respectively. For all Re_i, Re_o, the dimensionless torque G scales as a function of the Taylor number Ta (which is proportional to the square of the difference between the angular velocities of the inner and outer cylinders) with a universal effective scaling law G \propto Ta^{0.88}, corresponding to Nu_omega \propto Ta^{0.38} for th…
Rayleigh-Benard convection and Taylor-Couette flow are two canonical flows that have many properties in common. We here compare the two flows in detail for parameter values where the Nusselt numbers, i.e. the thermal transport and the angular momentum transport normalized by the corresponding laminar values, coincide. We study turbulent Rayleigh-Benard convection in air at Rayleigh number Ra=1e7 and Taylor-Couette flow at shear Reynolds number Re_S=2e4 for two different mean rotation rates but the same Nusselt numbers. For individual pairwise related fields and convective currents, we compare …
We report the observation of superstructures, i.e.\ very large-scale and long living coherent structures in highly turbulent Rayleigh-Bénard convection up to Rayleigh $Ra=10^9$. We perform direct numerical simulations in horizontally periodic domains with aspect ratios up to $Γ=128$. In the considered $Ra$ number regime the thermal superstructures have a horizontal extend of six to seven times the height of the domain and their size is independent of $Ra$. Many laboratory experiments and numerical simulations have focused on small aspect ratio cells in order to achieve the highest possible $Ra…
