Range Of Prandtl Numbers Research Articles

Flow state transition induced by emergence of orbiting satellite eddies in two-dimensional turbulent Rayleigh–Bénard convection

We report a numerical investigation of a previously noticed but less explored flow state transition in two-dimensional turbulent Rayleigh–Bénard convection. The simulations are performed in a square domain over a Rayleigh number range of $10^7 \leq Ra \leq 2 \times 10^{11}$ and a Prandtl number range of $0.25 \leq Pr \leq 20$ . The transition is characterized by the emergence of multiple satellite eddies with increasing $Ra$ , which orbit around and interact with the main vortex roll in the system. Consequently, the main roll is squeezed to a smaller size compared with the domain and wanders around in the bulk region irregularly and extensively. This is in sharp contrast to the flow state before the transition, which is featured by a domain-sized circulatory roll with its vortex centre ‘condensed’ near the domain's centre. Detailed velocity field analysis reveals that there exists an abrupt increase in the energy fluctuations of the Fourier modes during the transition. Based on this phase-transition-like signal, the critical condition for the transition is found to follow a scaling relation as $Ra_t \sim Pr^{1.41}$ where $Ra_t$ is the critical Rayleigh number for the transition. This scaling relation is quantitatively explained by a phenomenological model grounded on the bistability behaviour (i.e. spontaneous and stochastic switching between the two flow states) observed at the edge of the transition. The model can also account for the effects of aspect ratio on the transition reported in the literature (van der Poel et al., Phys. Fluids, vol. 24, 2012).

Mean temperature profiles in turbulent internal flows

We derive explicit formulas for the mean profiles of temperature (modeled as a passive scalar) in forced turbulent convection, as a function of the Reynolds and Prandtl numbers. The derivation leverages on the observed universality of the inner-layer thermal eddy diffusivity with respect to Reynolds and Prandtl number variations and across different flows, and on universality of the passive scalar defect in the core flow. Matching of the inner- and outer-layer expression yields a smooth compound mean temperature profile. We find excellent agreement of the analytical profile with data from direct numerical simulations of pipe and channel flows under various thermal forcing conditions, and over a wide range of Reynolds and Prandtl numbers.

A comprehensive numerical study on heat transfer and friction characteristics of offset-strip fins

Offset-strip fins are among the most used geometries in compact heat exchangers. The geometric and flow parameters of the fins affect their heat transfer effectiveness and head losses. Hence, accurate predictive models are needed to guide the design process. However, most of the correlations available in the literature are valid only for a limited set of geometric configurations and flow regimes. This work discusses the derivation of multivariate response surfaces for the equivalent Darcy and Colburn factors in offset-strip fins. These surfaces feature clear applicability ranges and extend over wide Reynolds and Prandtl number ranges (50≤Re≤12000, 0.71≤Pr≤190). In addition, a novel empirical model for the Prandtl number scaling exponent is proposed. The analysis is carried out through a Design of Experiment approach, performed with computational techniques. Each numerical experiment is carried out by CFD analysis of a periodic fin geometry. The obtained response surfaces approximate CFD results with a mean deviation of ±8.4%. Moreover, application of the correlations to the analysis of complete heat exchangers issued mean and maximum deviations of ±7.8% and ±20%, respectively, thus highlighting the usefulness of the proposed models for the accurate modelling of offset-strip fins in heat transfer applications.

CFD study of heat transfer in power‐law fluids over a corrugated cylinder

AbstractThe computational study of power‐law fluid flow, along with heat transfer attributes over a corrugated heated cylinder, is explored using ANSYS FLUENT (Version 18.0). Fluids power‐law indices fall in the range of 0.25 ≤ n ≤ 1.5, and the Reynolds number spans in the range of 1 ≤ ReN ≤ 40. The flow is two‐dimensional, steady, and laminar. A wide range of Prandtl numbers (0.7 ≤ PrN ≤ 500) is used to cover the most industrially applied fluids. A domain height of 135Dh is used. A grid with the smallest element size of 0.04 m and 135,914 nodes was used. Flow and heat transfer attributes were studied using streamlines, isotherms, and local and average Nusselt numbers. The average Nusselt number increases with ReN and/or PrN. The heat transfer rate is significantly lower in dilatant fluids and higher in pseudoplastic fluids than in Newtonian fluids. The onset of wake formation behind the cylinder takes place at ReN = 10. The increase in Reynolds number (ReN) and power‐law index (n) causes an increase in wake size. Heat transfer increases with the Reynolds number and/or decrease in the power‐law index. The enhancement in heat transfer due to corrugation is studied in detail in terms of average Nusselt number, which has not been studied for arched corrugated cylinder, even for Newtonian fluids in low Reynolds number range. A Nusselt number correlation is also developed for the given ranges of conditions.

Numerical study of magneto-hydrodynamics natural convective flow in a porous-corrugated enclosure using a higher-order compact finite difference scheme

This study explores the numerical investigation of two-dimensional natural convection in a porous corrugated enclosure with heat sources of different heights and electrically conducting fluids. The governing equations are solved using a higher-order compact finite difference scheme. Boundaries on the top and sides are subject to thermal cooling and adiabatic conditions, respectively. Numerical results are presented for wide range of Rayleigh (103 to 106), Darcy (10−5 to 10−1), Hartmann (25 to 150), and Prandtl (0.015 to 10) numbers. For fixed high Rayleigh-Darcy values, the porous medium has higher permeability. Most significant possible multiple steady solutions can be found in the case 1 at Ra=106,Da=10−2,Ha=50, and Pr=10.0. In the low to moderate critical values of Prandtl numbers, it encompasses a wide range of fluid flow phenomena, from stable to unstable. The maximum enhancement of the averaged Nusselt numbers on the boundaries in increasing and decreasing trends are viz. Ra(139.13% and 54.37%), Da(46.14% and 74.25%), Ha(633.04% and 74.73%), and Pr(142.52% and 45.92%) for case 1, Ra(129.64% and 0.32%), Da(40.68% and 69.66%), Ha(46.65% and 80.85%), and Pr(18.41% and 18.23%) for case 2, and Ra(163.98% and 77.20%), Da(8.25% and 58.68%), Ha(81.38% and 59.17%), and Pr(30.60% and 22.46%) for case 3.

Integral solution method of the laminar conjugated natural convection heat transfer from a hollow cylinder

The laminar conjugated natural convection heat transfer from a hollow cylinder is a common problem in engineering. The accurate calculation of the hot-spot temperature helps to monitor and evaluate the operating status of the equipment. In this paper, first, a mathematical model of the conjugated natural convection from a hollow cylinder is established. The boundary layer integral equations under arbitrary heat flux distribution on the surface are derived, and the optimal velocity and temperature profiles corresponding to different Prandtl numbers are determined. By combining the integral equations and the thermal network model, the hot-spot temperature and the average surface temperature of a hollow cylinder are solved iteratively. Subsequently, a computational fluid dynamic (CFD) simulation based on finite volume method is carried out to verify the accuracy of the integral solution method. The maximum relative error of the computed results of the two methods is 1.35%. The integral solution method performs much better in computation speed, increasing by 90 times. It is also found that as the thermal conductivity of solid materials decreases, although the hot-spot temperature increases, the average surface temperature remains basically unchanged. Finally, based on the lumped thermal equivalent circuit, the correlation of the Nusselt number corresponding to the hot-spot temperature is predicted. The parameters in the correlation are estimated by regression orthogonal design. The results show that under a wide range of solid thermal conductivity and fluid Prandtl numbers, the maximum error of the correlation is 6.02%. Therefore, this novel method is feasible and significant for calculating hot-spot temperature.

Formation and evolution of laminar thermal structures: correlation to the thermal boundary layer and effects of heating time

We report an experimental study of the formation and evolution of laminar thermal structures generated by a small heat source, with a focus on their correlation to the thermal boundary layer and effects of heating time $t_{heat}$ . The experiments are performed over the flux Rayleigh number ( $Ra_f$ ) range $2.1\times 10^6 \leq Ra_f \leq 3.6\times 10^{7}$ and the Prandtl number ( $Pr$ ) range $28.6 \leq Pr \leq 904.7$ . The corresponding Rayleigh number ( $Ra= t_{heat}\,Ra_{f}/\tau _0\,Pr$ ) range is $900 \leq Ra \leq 4\times 10^{4}$ , where $\tau _0$ is a diffusion time scale. For thermal structures generated by continuous heating (i.e. starting plumes), their formation process exists three characteristic times that are well reflected by changes in the thermal boundary layer thickness. These characteristic times, denoted as $t_{emit}$ , $t_{recover}$ and $t_{static}$ , correspond to the moments when the plume emission begins and completes, and when the thermal boundary layer becomes quasi-static, respectively. Their $Ra_f$ – $Pr$ dependencies are found to be $t_{emit}/\tau _0\sim Ra_f^{-0.41}\,Pr^{0.41}$ , $t_{recover}/\tau _0\sim Ra_f^{-0.48}\,Pr^{0.48}$ and $t_{static}/\tau _0\sim Ra_f^{-0.49}\,Pr^{0.33}$ , respectively. Thermal structures generated by finite $t_{heat}$ exhibit similar evolution dynamics once $t_{heat} \ge t_{emit}$ , with the accelerating stage behaving like starting plumes and the decay stage like thermals (i.e. a finite amount of buoyant fluids). It is further found that their maximum rising velocity experiences a transition in the $Ra$ -dependence from $Ra$ to $(Ra\ln Ra)^{0.5}$ at $Ra \simeq 6000$ ; and their maximum acceleration reaches the value of starting plumes at $t_{heat}\simeq t_{recover}$ , and remains unchanged for larger $t_{heat}$ . In particular, the maximum rising velocity for the cases with $t_{heat} = t_{recover}$ follows a scaling relation $Ra_f^{0.37}\,Pr^{-0.37}$ , in contrast to the relation $Ra_f^{0.48}\,Pr^{-0.48}$ for starting plumes. This study provides a more comprehensive understanding of laminar thermal structures, which are relevant to a range of processes in nature and laboratory systems such as Rayleigh–Bénard convection.

Thermochemical convection in a slowly rotating spherical shell: a transition between prograde and retrograde flows

We numerically investigate thermochemically-driven convection in a spherical shell at a high Ekman number for a range of Rayleigh numbers, Prandtl numbers and buoyancy ratios. Rotating convection displays a nearly axisymmetric large-scale structure and appears mostly unaffected by the nature (thermal or chemical) of the driving buoyancy source. We observe both precession directions, prograde and retrograde, of convection structures and identify a snap-through transition between the prograde and retrograde flows in the parameter space spanned by the Rayleigh and Prandtl numbers. Equatorially wall-attached and spiralling columnar convection structures, which typically emerge in rapidly rotating shells, are not observed.

A Comprehensive Evaluation of Turbulence Models for Predicting Heat Transfer in Turbulent Channel Flow across Various Prandtl Number Regimes

Turbulent heat transfer in channel flows is an important area of research due to its simple geometry and diverse industrial applications. Reynolds-Averaged Navier–Stokes (RANS) models are the most-affordable simulation methodology and are often the only viable choice for investigating industrial flows. However, accurate modelling of wall-bounded flows is challenging in RANS, and the assessment of the performance of RANS models for heated turbulent channel flow has not been sufficiently investigated for a wide range of Reynolds and Prandtl numbers. In this study, five RANS models are assessed for their ability to predict heat transfer in channel flows across a wide range of Reynolds and Prandtl numbers (Pr) by comparing the RANS results with respect to the corresponding Direct Numerical Simulation data. The models include three Eddy Viscosity Models (EVMs): standard k−ϵ, low Reynolds number k−ϵLS, and k−ωSST, as well as two Reynolds Stress Models (RSMs): Launder–Reece–Rodi and Speziale–Sarkar–Gatski models. The study analyses the Reynolds number effects on turbulent heat transfer in a channel flow at a Pr of 0.71 for friction Reynolds number values of 180,395,640, and 1020. The results show that all models accurately predict velocity across all Reynolds numbers, but the accuracy of mean temperature prediction drops with increasing Reynolds number for all models, except for the k−ωSST model. The study also analyses the Pr effects on turbulent heat transfer in a channel flow with Pr values between 0.025 and 10.0. An error analysis is performed on the results obtained from different turbulence models, and it is shown that the k−ωSST model has the smallest error for the predictions of the mean temperature and Nusselt number for high-Prandtl-number flows, while the low Reynolds number k−ϵLS model shows the smallest errors for low-Prandtl-number flows at different Reynolds numbers. An analytical solution is utilised to identify Pr effects on forced convection in a channel flow into three different regimes: analytical region, transitional region, and turbulent diffusion-dominated region. These regimes are helpful to discuss the validity of the models in relation to the Pr. The findings of this paper provide insights into the performance of different RANS models for heat transfer predictions in a channel flow.

Laminar Flow Heat Transfer Studies in a Twisted Square Duct for Constant Wall Temperature

Abstract: Three dimensional analysis of steady fully developed laminar flow inside twisted duct of square cross section flow area is studied for Reynolds number range of 100 – 2000 using a commercially available software. Twist ratio used are 2.5, 5, 10 and 20. Heat transfer results are generated for a uniform wall temperature case for Prandtl number range of 0.7 to 20. Maximum values for the product of friction factor and Reynolds number is observed for a twist ratio of 2.5 and Reynold number of 2000. Maximum Nusselt number is observed for the same values along with Prandtl number of 20. Correlations for friction factor and Nusselt number are developed involving swirl parameter. Local distribution of friction factor ratio and Nusselt number across a cross-section is presented. Heat transfer enhancement for Reynold number of 2000 and Prandtl number of 0.7 for twist ratio of 2.5, 5, 10, and 20 are 33 %, 18 %, 10 % and 7 % respectively. The results are significant because it will contribute to development of compact heat exchanger.

Exact and analytical solutions for self‐similar thermal boundary layer flows over a moving wedge

AbstractThis article aims to achieve exact and analytical solutions for the classical Falkner–Skan equation with heat transfer (FSE‐HT). Specifically, when the pressure gradient parameter , there already exists a closed‐form solution in the literature for the Falkner–Skan flow equation. The main purpose here is to extend this case to obtain a closed‐form solution to the heat transport equation with the solubility condition . An algorithm is presented and is found to be new to the literature that enriches the physical properties of FSE‐HT. It is shown that for the moving wedge parameter , the momentum and temperature equations show multiple solutions analytically. The skin friction coefficient and the heat transfer rate are also obtained in analytical form. The thus‐obtained solution is then adapted to derive an analytical solution applicable to a wide range of pressure gradient parameters and Prandtl numbers . Furthermore, an asymptotic analysis is conducted, focusing on scenarios where the moving wedge parameter becomes significantly large (). Nevertheless, in all the above‐mentioned cases, the skin friction coefficient () and the heat transfer rate () are compared with the direct numerical solutions of the boundary layer equations, and it is found that the results are in good agreement. These solutions provide a benchmark and shed light on further studies on the families of FSE‐HT.

Numerical investigation on the thermal-hydraulic performance of high Prandtl number fluid in offset strip fin with different offset

To rigorously analyze the thermo-hydraulic performance of offset strip fin operating with high Prandtl number fluid and variable fin offset values, numerical simulation was employed to investigate changes in the Colburn j and the Fanning friction f under varying fin offsets. The results, obtained using a reliable numerical model, indicated that an increase in offset led to a 33.08% and 33.97% rise in j and f, respectively, demonstrating the offset's influence on the thermal-hydraulic performance of offset strip fins. By employing orthogonal design techniques, a set of 49 design points for fin pitch, fin thickness, fin height, interrupted length, fin offset, Reynolds number, and Prandtl number were obtained. Regression analysis was conducted on the simulation results of these 49 design points, ultimately deriving correlations for j and f within the Reynolds number range of 100–1000 and the Prandtl number range of 2–13. A comparison between the predicted and simulated data revealed that j and f fell within the error range of 20% for 91.84% and 93.88% of the data, respectively, confirming the accuracy of the proposed correlations. Furthermore, comparing the correlations from this study with other previously established correlations demonstrated that the proposed correlations were more suitable for predicting the thermal-hydraulic performance of offset strip fins with high Prandtl number fluids. For fluids with Prandtl number range of 30–150, the predicted data of j and f exhibited errors within 15% and 20% of the experimental data, respectively, further validating the applicability of these correlations for predicting the thermo-hydraulic performance of offset strip fins with higher Prandtl number fluids as the working medium.

Dynamic mode decomposition of thermocapillary convection in GaAs melt liquid bridge between unequal ends

The effects of the radius ratio on the stability of thermocapillary convection in the GaAs melt (Pr=0.068) liquid bridge, situated between unequal ends, are studied by three-dimensional direct numerical simulation. The simulated results reveal that, for the first time, the GaAs melt flow transforms from a stable axisymmetric flow into a three-dimensional steady flow through the first bifurcation at 0.5≤Γr≤0.9. The bifurcation phenomenon was previously common in the range of small Prandtl numbers(0.001≤Pr≤0.057). Additionally, we observe that the stability of the thermocapillary liquid bridge is initially weakened and then slowly improved as the radius ratio Γr decreases. The second bifurcation occurs with the further increase of Ma, and the flow becomes oscillatory. Remarkably, two different oscillation forms are identified, depending on the difference in the degree of symmetry of the flow field structure during its oscillation process. Once the flow reaches a stable periodic oscillation, the Dynamic Mode Decomposition (DMD) technique is used to study the inherent spatio-temporal evolution of the temperature field. By combining the spectrum obtained from the Fast Fourier Transform (FFT), we provide evidence for the existence of mixed forms with different azimuthal wavenumbers from a dynamic perspective, intuitively displaying the different components of the oscillation and examining the contribution of different modes to the oscillation. The dynamic model analysis demonstrates that oscillation form I is a mixed form with m = 1 and m = 2, while oscillation form II belongs to the form where m = 2 is absolutely dominant.

Improved asymptotic expansion method for laminar fluid flow and heat transfer in conical gaps with disks rotating

PurposeThe purpose of this paper was to study laminar fluid flow and convective heat transfer in a conical gap at small conicity angles up to 4° for the case of disk rotation with a fixed cone.Design/methodology/approachIn this paper, the improved asymptotic expansion method developed by the author was applied to the self-similar Navier–Stokes equations. The characteristic Reynolds number ranged from 0.001 to 2.0, and the Prandtl numbers ranged from 0.71 to 10.FindingsCompared to previous approaches, the improved asymptotic expansion method has an accuracy like the self-similar solution in a significantly wider range of Reynolds and Prandtl numbers. Including radial thermal conductivity in the energy equation at small conicity angle leads to insignificant deviations of the Nusselt number (maximum 1.23%).Practical implicationsThis problem has applications in rheometry to experimentally determine viscosity of liquids, as well as in bioengineering and medicine, where cone-and-disk devices serve as an incubator for nurturing endothelial cells.Social implicationsThe study can help design more effective devices to nurture endothelial cells, which regulate exchanges between the bloodstream and the surrounding tissues.Originality/valueTo the best of the authors’ knowledge, for the first time, novel approximate analytical solutions were obtained for the radial, tangential and axial velocity components, flow swirl angle on the disk, tangential stresses on both surfaces, as well as static pressure, which varies not only with the Reynolds number but also across the gap. These solutions are in excellent agreement with the self-similar solution.

An explicit representation for mean profiles and fluxes in forced passive scalar convection

We derive explicit formulae for the mean profiles of passive scalars (either temperature or concentration of a diffusing substance), and their respective wall fluxes (either heat or mass fluxes), in forced turbulent convection, as a function of the Reynolds and Prandtl numbers. Direct numerical simulation data for turbulent flow within a smooth straight pipe of circular cross-section, at friction Reynolds number ${{Re}}_{\tau }=1140$ , in the range of Prandtl numbers from ${{Pr}}=0.00625$ to ${{Pr}}=16$ , are used to infer the proper analytical form of the eddy diffusivity. This is leveraged to derive accurate predictive formulae for the mean passive scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling laws result for the thickness of the conductive (diffusive) layer, and for the Nusselt number, which significantly extend the predictive envelope of classical formulae.

Numerical investigation of effect of surface pattern and rotation on power-law fluid flow and heat transfer around a cylinder in laminar flow regime

We have studied the effect of surface topography, fluid behavior, and rotation on fluid flow and heat transfer phenomena over a cylinder. In this study, we have incorporated a sinusoidal surface topography to account for the impact of surface patterns and a power-law model to include fluid behavior. The governing equations are solved numerically for a range of pattern frequency (ω=5 and 11), pattern amplitude (δ=0.01 and 0.1), power-law index (0.4≤n≤1.6), Prandtl number (1≤Pr≤100), rotational speed (0.5≤α≤2), and Reynolds number (5≤Re≤40). In particular, the study aims to determine the degree to which various macroscopic parameters, such as the drag and lift coefficients, the average Nusselt number in relation to the Reynolds number, the Prandtl number, the rotating speed, and the power-law index, vary. In this range of Reynolds number, the flow around a circular cylinder is steady, with two symmetric vortices in the rear side. The sliding mesh method is used to deal with dynamic interface between solid and fluid. The streamlines are drawn to visualize the flow field around the patterned cylinder. For a non-rotating patterned cylinder, small recirculation zones are observed over the trough, which are absent in circular cylinders. The size of these recirculation regions increases on increase in Reynolds number and power-law index. On adding rotation to the cylinder, these recirculation zones move away from the cylinder and appear over the crest. On increasing the rotating speed of the cylinder, the frontal vortices disappear. The enveloping vortex gets larger with increase in rotating speed and power-law index. The size of the rear detached vortex increases with power-law index and Reynolds number and decreases with rotating speed. It is observed that the results are contrasted with the previous studies on smooth circular cylinder. The drag force acting on the patterned cylinder is seen to be reduced. Compared to a circular cylinder, a significant reduction in drag can be achieved by choosing suitable value of pattern frequency (ω) and amplitude (δ). Overall, there is a decrease in the amount of drag reduction as the Reynolds number increases. The behavior of the fluid has a considerable influence on the reduction of drag. It has been observed that shear-thickening fluid significantly contributes to the enhancement of drag reduction. For a higher value of pattern frequency and amplitude (ω=11,δ=0.1), the drag force reduces significantly for Newtonian and shear-thickening fluids at higher rotating speed (α=2). Also, the pattern frequency and amplitude substantially impact the average Nusselt number. On increase in pattern frequency and amplitude, a progressive decrease in the average Nusselt number is observed. Compared to shear-thinning and Newtonian fluid, shear-thickening fluid exhibits a greater reduction in average Nusselt number. One correlation is provided at the end to show the relationship between the average Nusselt number, the Prandtl number, the Reynolds number, the rotating speed, and the power-law index.

Effect of horizontal magnetic field on Küppers–Lortz instability

We investigate the effect of an external horizontal magnetic field on the Küppers–Lortz instability (KLI) in rotating Rayleigh–Bénard convection of Boussinesq fluids using weakly nonlinear theory along with linear theory. By the KLI, we mean the instability where the two-dimensional roll solutions of the system occurring at the onset of convection become unstable against the perturbations by rolls oriented at different angles with the previous one as the rotation rate exceeds a critical value. The governing parameters, namely, the Prandtl number (Pr), the Taylor number (Ta), and the Chandrasekhar number (Q), are varied in the ranges 0.8≤Pr<∞, 0<Ta≤104, and 0≤Q≤104, respectively, by considering the vanishingly small magnetic Prandtl number limit. In the Pr→∞ limit, magnetic field is found to inhibit the KLI by enhancing the critical Taylor number (Tac) for its onset. On the other hand, for finite Prandtl number fluids, the KLI is favored for lower Q, and it is inhibited for higher Q. Interestingly, in the finite Prandtl number range, both KLI and small angle instability are manifested depending on the Prandtl number. No small-angle instability is observed for Pr≥50, and the rotation-induced KLI is inhibited predominantly by the magnetic field, while, for Pr<50, along with the Küppers–Lortz instability, small-angle instability is also observed. However, in this case, the KLI is favored for lower Q, while it is inhibited for higher Q.

Prandtl number effects on passive scalars in turbulent pipe flow

We study the statistics of passive scalars (either temperature or concentration of a diffusing substance) at friction Reynolds number ${Re}_{\tau }=1140$ , for turbulent flow within a smooth straight pipe of circular cross-section, in the range of Prandtl numbers from ${Pr}=0.00625$ , to ${Pr}=16$ , using direct numerical simulations (DNS) of the Navier–Stokes equations. Whereas the organization of passive scalars is similar to the axial velocity field at ${Pr} = O(1)$ , similarity is impaired at low Prandtl number, at which the buffer-layer dynamics is filtered out, and at high Prandtl number, at which the passive scalar fluctuations become confined to the near-wall layer. The mean scalar profiles at ${Pr} \gtrsim 0.0125$ are found to exhibit logarithmic overlap layers, and universal parabolic distributions in the core part of the flow. Near-universality of the eddy diffusivity is exploited to derive accurate predictive formulas for the mean scalar profiles, and for the corresponding logarithmic offset function. Asymptotic scaling formulas are derived for the thickness of the conductive (diffusive) layer, for the peak scalar variance, and its production rate. The DNS data are leveraged to synthesize a modified form of the classical predictive formula of Kader & Yaglom (Intl J. Heat Mass Transfer, vol. 15, 1972, pp. 2329–2351), which is capable of accounting accurately for the dependence on both Reynolds and Prandtl numbers, for ${Pr} \gtrsim 0.25$ .

Effect of shear on local boundary layers in turbulent convection

In Rayleigh Bénard convection, for a range of Prandtl numbers $4.69 \leqslant Pr \leqslant 5.88$ and Rayleigh numbers $5.52\times 10^5 \leqslant Ra \leqslant 1.21\times 10^9$ , we study the effect of shear by the inherent large-scale flow (LSF) on the local boundary layers on the hot plate. The velocity distribution in a horizontal plane within the boundary layers at each $Ra$ , at any instant, is (A) unimodal with a peak at approximately the natural convection boundary layer velocities $V_{bl}$ ; (B) bimodal with the first peak between $V_{bl}$ and $V_{L}$ , the shear velocities created by the LSF close to the plate; or (C) unimodal with the peak at approximately $V_{L}$ . Type A distributions occur more at lower $Ra$ , while type C occur more at higher $Ra$ , with type B occurring more at intermediate $Ra$ . We show that the second peak of the bimodal type B distributions, and the peak of the unimodal type C distributions, scale as $V_{L}$ scales with $Ra$ . We then show that the areas of such regions that have velocities of the order of $V_{L}$ increase exponentially with increase in $Ra$ and then saturate. The velocities in the remaining regions, which contribute to the first peak of the bimodal type B distributions and the single peak of type A distributions, are also affected by the shear. We show that the Reynolds number based on these velocities scale as $Re_{bs}$ , the Reynolds number based on the boundary layer velocities forced externally by the shear due to the LSF, which we obtained as a perturbation solution of the scaling relations derived from integral boundary layer equations. For $Pr=1$ and aspect ratio $\varGamma =1$ , $Re_{bs} \sim Ra^{0.375}$ for small shear, similar to the observed flux scaling in a possible ultimate regime. The velocity at the edge of the natural convection boundary layers was found to increase with $Ra$ as $Ra^{0.35}$ ; since $V_{bl}\sim Ra^{1/3}$ , this suggests a possible shear domination of the boundary layers at high $Ra$ . The effect of shear, however, decreases with increase in $Pr$ and with increase in $\varGamma$ , and becomes negligible for $Pr\geqslant 100$ at $\varGamma =1$ or for $\varGamma \geqslant 20$ at $Pr=1$ , causing $Re_{bs}\sim Ra_w^{1/3}$ .

Turbulent disruption of density staircases in stratified shear flows

The formation of step-like ‘density staircase’ distributions induced by stratification and turbulence has been widely studied, and can be explained by the ‘instability’ of a sufficiently strongly stably stratified turbulent flow due to the decrease of the turbulent density flux with increasing stratification via the ‘Phillips mechanism’ (Phillips, Deep Sea Research and Oceanographic Abstracts, vol. 19, 1972, pp. 79–81. Elsevier). However, such density staircases are not often observed in ocean interiors, except in regions where double-diffusion processes are important, leading to thermohaline staircases. Using reduced-order models for the evolution of velocity and density gradients, we analyse staircase formation in stratified and sheared turbulent flows. Under the assumption of inertial scaling $\epsilon \sim U^3/L$ for the kinetic energy dissipation rate $\epsilon$ , where $U$ and $L$ are characteristic velocity and length scales, we determine ranges of bulk Richardson numbers ${Ri}_{b}$ and turbulent Prandtl numbers ${Pr}_{T}$ for which staircases can potentially form and show that the Phillips mechanism only survives in the limit of sufficiently small turbulent Prandtl numbers. For relevant oceanic parameters, a range of turbulent Prandtl numbers above which the system is not prone to staircases is found to be ${Pr}_T \simeq 0.5\unicode{x2013}0.8$ . Since several studies indicate that the turbulent Prandtl number in stably stratified turbulence and in ocean interiors is usually above this threshold, this result supports the empirical observation that staircases are not favoured in ocean interiors in the presence of relatively homogeneous and sustained turbulence. We also show that our analysis is robust to other scalings for $\epsilon$ (such as the more strongly stratified scaling $\epsilon \sim U^{2}N_{c}$ , where $N_{c}$ is a characteristic value of the buoyancy frequency), supporting our results in both shear-dominated and buoyancy-dominated turbulent regimes as well as in weakly and strongly stratified regimes.