Velocity-temperature Correlations Research Articles

Large-Eddy vs. Reynolds-Averaged Navier–Stokes Simulations of Flow and Heat Transfer in a U-Duct with Unsteady Flow Separation

Large-eddy simulation (LES) and Reynolds-Averaged Navier–Stokes (RANS) equations were used to study incompressible flow and heat transfer in a U-duct with a high-aspect-ratio trapezoidal cross section. For the LES, the WALE subgrid-scale model was employed, and its inflow boundary condition was provided by a concurrent LES of incompressible fully-developed flow in a straight duct with the same cross section and flow conditions as the U-duct. LES results are presented for turbulent kinetic energy, Reynolds stresses, pressure–strain rate, turbulent diffusion, turbulent transport, and velocity–temperature correlations, with a focus on how they are affected by the U-turn region of the U-duct. The LES results were also used to assess three commonly used RANS models: the realizable k-ε with the two-layer model in the near-wall region, the two-equation shear-stress transport model, and the seven-equation stress-omega Reynolds stress model. Results obtained show steady and unsteady RANS to incorrectly predict the effects of unsteady flow separation. The results obtained also identified the terms in the RANS models that need to be modified and suggested how turbulent diffusion should be modeled when there is unsteady flow separation.

Direct numerical simulations of high-enthalpy supersonic turbulent channel flows including finite-rate reactions

Direct numerical simulations of temporally evolving high-enthalpy supersonic turbulent channel flows are conducted at a Mach number of 3.0 and Reynolds number of 4880 under isothermal wall conditions. Air is assumed to behave as a five-species mixture, and chemical non-equilibrium and equilibrium assumptions are adopted to investigate the influence of finite-rate reactions on the turbulent statistics and large-scale structures. The two wall temperatures of 1733.2 and 3500 K are such that the mixture components undergo strong dissociation and recombination reactions along the channel. Investigation shows that the turbulent intensity is weakened and the mean and fluctuating temperatures are smaller when finite-rate reactions are considered. The mean dissociation degree is a quadratic function of the normal position, and its curvature under the chemical non-equilibrium assumption is greater than that under the chemical equilibrium assumption. The fluctuating mass fractions of the generated species seem to decrease slightly in the near-wall region, and their distributions are obviously different from those of the fluctuating velocity and fluctuating temperature. Finite-rate reactions increase the proportion of turbulent kinetic energy production in the skin friction decomposition, especially when the wall temperature is 3500 K. The large-scale structures visualized by the cross correlation between temperature and species mass fraction become stronger in the normal direction. The turbulent Schmidt number and several velocity–temperature correlations, including the recovery enthalpy and strong Reynolds analogy, are insensitive to the chemical reaction rate and wall temperature.

Large eddy simulations of a turbulent flow with hybrid nanofluid subjected to symmetric and asymmetric heating

The present study includes a comprehensive numerical analysis of the usage of nanofluids inside a thermal receiver to assess the possibility of enhancing the efficiency of this device. All simulations are performed on a small portion of a thermal receiver, similar to a bi-periodic channel subjected to symmetric and asymmetric heating. To generate conventional and hybrid nanofluids, nanoparticles of Al2O3 and Al2O3-Cu are dispersed in the base fluid H2O. The values of nanoparticles' diameter is 20nm and volume fraction of the nanoparticles is 2%. The Reynolds numbers used in this study are 12 000 and 18 000, which respectively correspond to mean friction Reynolds numbers of 140 and 260. The thermal Large Eddy Simulation (T-LES) approach is used to simulate the fluid flow and heat transfer inside the computational domain. The results are compared with those of prior numerical and experimental studies to validate the method and the numerical setup. A high level of agreement is achieved. In the case of symmetric and asymmetric heating, the dimensionless findings reveal that adding nanoparticles to the base fluid has a negligible impact on the velocity fluctuations. Nanoparticles make the base fluid denser and more viscous but also more conductive, allowing it to minimize kinetic energy while boosting heat transfer rate. Results also show that using hybrid nanofluids contributes better to improving the heat transfer rate compared to the base fluid and the conventional nanofluid. Introducing nanoparticles into the base fluid decreases its temperatures slightly, especially when Al2O3-Cu nanoparticles are used instead of Al2O3 nanoparticles. In the asymmetric case, water's turbulent heat flux values are greater than nanofluids and hybrid nanofluid, implying that more conductive and viscous liquids have the weakest turbulent heat flux. In the circumstances of symmetric and asymmetric heating, the variation of the velocity fluctuations is almost the same, but the behavior of the temperature–velocity correlation is entirely different. Finally, results show that symmetric heating leads to a higher heat transfer rate than asymmetric heating. The suggested research's findings may be utilized to promote the usage of nanofluids in Concentrated Solar Power (CSP) systems.

The Role of Particle Inertia and Thermal Inertia in Heat Transfer in a Non-Isothermal Particle-Laden Turbulent Flow

We present an analysis of the effect of particle inertia and thermal inertia on the heat transfer in a turbulent shearless flow, where an inhomogeneous passive temperature field is advected along with inertial point particles by a homogeneous isotropic velocity field. Eulerian–Lagrangian direct numerical simulations are carried out in both one- and two-way coupling regimes and analyzed through single-point statistics. The role of particle inertia and thermal inertia is discussed by introducing a new decomposition of particle second-order moments in terms of correlations involving Lagrangian acceleration and time derivative of particles. We present how particle relaxation times mediate the level of particle velocity–temperature correlation, which gives particle contribution to the overall heat transfer. For each thermal Stokes number, a critical Stokes number is individuated. The effect of particle feedback on the attenuation or enhancement of fluid temperature variance is presented. We show that particle feedback enhances fluid temperature variance for Stokes numbers less than one and damps is for larger than one Stokes number, regardless of the thermal Stokes number, even if this effect is amplified by an increasing thermal inertia.

Heat transport in a non-homothermal turbulent particle-laden flow in the collisional regime

In this investigation, we explore the impact of particle collisions on turbulent heat flux within a temporally developing thermal mixing layer arising from the interaction of two homothermal regions driven by a statistically homogeneous and isotropic velocity field. Employing two-way thermally coupled Eulerian-Lagrangian Direct Numerical Simulations (DNSs) at a Taylor microscale Reynolds number up to 124, we examine the influence of particle collisions for a Stokes number from 0.2 to 3, maintaining a fixed thermal to kinetic relaxation times ratio of 4.43. Our study quantifies the reduction in average heat transport induced by particle-to-particle collisions, which disrupt the temperature-velocity correlation created by the initial temperature difference. Notably, while collisions diminish this correlation, our analysis reveals their overall impact remains minor, even at the highest simulated Stokes number. Additionally, we present statistics of the temperature difference among colliding particles across various flow conditions.

Turbulent diffusion of scalar and heat in an off-source heated steady round jet

We study, using direct numerical simulation, the characteristics of an off-source heated jet (OSHJ), the rate of heat addition being proportional to the local concentration of a scalar. The heating disturbs the self-similar state of the unheated jet (UJ) entering the heat injection zone (HIZ). Based on the characteristics of the velocity and scalar statistics, the evolution of the OSHJ is demarcated into three axial zones. Zone I, which is dynamically the most active region, exhibits rapid variations in quantities like the mean radial velocity, the scalar-to-velocity width ratio and the second-order correlations. Zone II represents the relaxation of these quantities towards a new self-similar state which is realized in Zone III above the HIZ. We find that the temperature–velocity correlation is negative in the core of the OSHJ in Zone I, which becomes positive for all radial locations in Zones II and III. We interpret this behavior by decomposing the effect of heat addition into “heating” and “buoyancy” factors; the former describes the effect of axial velocity on heating while the latter describes the effect of added buoyancy on flow acceleration. The heating factor dominates in Zone I leading to the negative temperature–velocity correlation, which we conjecture to be the reason for the “disruption” of coherent structures in the OSHJ reported in the literature. Based on wavelet analysis of the scalar field we identify the length scale of the ring-like coherent structure that gets disrupted upon heat addition, which enhances the scalar homogeneity in the core of the OSHJ near the base of the HIZ. Where possible, we compare our results with the literature on heated jets within the context of the three demarcated zones. We find that external heating decreases the scalar-to-velocity flow widths relative to the UJ to a value in Zone III that is largely insensitive to the precise details of heat addition , such as the amount of heat added and the extent of the HIZ. We also present the effect of heat addition on turbulent Schmidt and Prandtl numbers. Our findings are relevant to the dynamics of cumulus clouds which are governed by a similar interplay of scalar, velocity and temperature fields as reported here.

Thermal large-eddy simulation methods to model highly anisothermal and turbulent flows

Thermal large-eddy simulations (T-LES) of highly anisothermal and turbulent channel flows are assessed using direct numerical simulations (DNS). The investigated conditions are representative of solar receivers used in concentrated solar power towers. Four thermal operating conditions are considered. They aim to study several locations in the solar receiver. They are distinguished by different temperature profiles and thus different wall heat fluxes. The mean friction Reynolds number is close to 800 for all the simulations. The Navier–Stokes equations are solved under the low-Mach-number approximation. The nonlinear terms corresponding to the velocity–velocity and the velocity–temperature correlations are modeled. Functional, structural, and mixed models are investigated. An extension of the anisotropic minimum dissipation (AMD) model to compressible case and two-layer mixed models are proposed and assessed. Fourth-order and second-order centered schemes are tested for the discretization of the momentum convection term. First, a global assessment of 16T-LES approaches on mean quantities and correlations for three different meshes is performed in reference conditions. Then, three of the T-LES are selected for more detailed analyses. The mesh effect and the influence of the thermal conditions on the model accuracy are investigated. These detailed studies consist of the comparison of the relative error of the T-LES on mean quantities and correlations and the visualization of the normalized profiles as functions of the wall-normal distance. The results highlight the good agreement of two-layer mixed models consisting of the combination of the Bardina and the AMD models with the DNS for the three tested meshes.

Strongly Heated Turbulent Flow in a Channel with Pin Fins

Large-eddy simulations (LES) were performed to study the turbulent flow in a channel of height H with a staggered array of pin fins with diameter D = H/2 as a function of heating loads that are relevant to the cooling of turbine blades and vanes. The following three heating loads were investigated—wall-to-coolant temperatures of Tw/Tc = 1.01, 2.0, and 4.0—where the Reynolds number at the channel inlet was 10,000 and the back pressure at the channel outlet was 1 bar. For the LES, two different subgrid-scale models—the dynamic kinetic energy model (DKEM) and the wall-adapting local eddy-viscosity model (WALE)—were examined and compared. This study was validated by comparing with data from direct numerical simulation and experimental measurements. The results obtained show high heating loads to create wall jets next to all heated surfaces that significantly alter the structure of the turbulent flow. Results generated on effects of heat loads on the mean and fluctuating components of velocity and temperature, turbulent kinetic energy, the anisotropy of the Reynolds stresses, and velocity-temperature correlations can be used to improve existing RANS models.

Spectral Analysis on Transport Budgets of Turbulent Heat Fluxes in Plane Couette Turbulence

In recent years, scale-by-scale energy transport in wall turbulence has been intensively studied, and the complex spatial and interscale transfer of turbulent energy has been investigated. As the enhancement of heat transfer is one of the most important aspects of turbulence from an engineering perspective, it is also important to study how turbulent heat fluxes are transported in space and in scale by nonlinear multi-scale interactions in wall turbulence as well as turbulent energy. In the present study, the spectral transport budgets of turbulent heat fluxes are investigated based on direct numerical simulation data of a turbulent plane Couette flow with a passive scalar heat transfer. The transport budgets of spanwise spectra of temperature fluctuation and velocity-temperature correlations are investigated in detail in comparison to those of the corresponding Reynolds stress spectra. The similarity and difference between those scale-by-scale transports are discussed, with a particular focus on the roles of interscale transport and spatial turbulent diffusion. As a result, it is found that the spectral transport of the temperature-related statistics is quite similar to those of the Reynolds stresses, and in particular, the inverse interscale transfer is commonly observed throughout the channel in both transport of the Reynolds shear stress and wall-normal turbulent heat flux.

Possible Evidence for Shear-driven Kelvin–Helmholtz Instability along the Boundary of Fast and Slow Solar Wind in the Corona

This paper reports the first possible evidence for the development of the Kelvin–Helmholtz (KH) instability at the border of coronal holes separating the associated fast wind from the slower wind originating from adjacent streamer regions. Based on a statistical data set of spectroscopic measurements of the UV corona acquired with the UltraViolet Coronagraph Spectrometer on board the SOlar and Heliospheric Observatory during the minimum activity of solar cycle 22, high temperature–velocity correlations are found along the fast/slow solar wind interface region and interpreted as manifestations of KH vortices formed by the roll-up of the shear flow, whose dissipation could lead to higher heating and, because of that, higher velocities. These observational results are supported by solving coupled solar wind and turbulence transport equations including a KH-driven source of turbulence along the tangential velocity discontinuity between faster and slower coronal flows: numerical analysis indicates that the correlation between the solar wind speed and temperature is large in the presence of the shear source of turbulence. These findings suggest that the KH instability may play an important role both in the plasma dynamics and in the energy deposition at the boundaries of coronal holes and equatorial streamers.

Numerical study of thermally developing turbulent internal flows

The effect of fluid Prandtl number on the thermal characteristics of the flow at the thermal-entry region of a hydrodynamically fully developed turbulent channel flow at a friction Reynolds number of Reτ=180 is investigated using direct numerical simulation. Four test cases with molecular Prandtl numbers of Pr = 0.71, 1, 3, 7 are considered. The numerical results confirm that in contrast to laminar flows, where the thermal entry length increases with the increase of the Prandtl number, for turbulent flows the increase of the Prandtl number accelerates the thermal development of the flow toward its asymptotic fully developed state. The variations of the Nusselt number, and the mean and rms temperature profiles are reported in the streamwise direction. It is shown that at highest Prandtl number considered in this study (Pr =7 ), while the mean temperature profile in the logarithmic region is still developing, the temperature profile in the conductive sublayer and the Nusselt number approach their fully developed values, confirming the dependence of the Nusselt number of turbulent flows to the development of the conductive sublayer. The development of the rms temperature profiles in the streamwise direction is also shown to be a good indicator for the flow reaching its thermally fully developed condition. The temperature-velocity correlations, temperature variance budgets, and contours of temperature are also reported at different streamwise locations for different Prandtl numbers.

Kinetic-energy-flux-constrained model using an artificial neural network for large-eddy simulation of compressible wall-bounded turbulence

Kinetic energy flux (KEF) is an important physical quantity that characterizes cascades of kinetic energy in turbulent flows. In large-eddy simulation (LES), it is crucial for the subgrid-scale (SGS) model to accurately predict the KEF in turbulence. In this paper, we propose a new eddy-viscosity SGS model constrained by the properly modelled KEF for LES of compressible wall-bounded turbulence. The new methodology has the advantages of both accurate prediction of the KEF and strong numerical stability in LES. We can obtain an approximate KEF by the tensor-diffusivity model, which has a high correlation with the real value. Then, using the artificial neural network method, the local ratios between the real KEF and the approximate KEF are accurately modelled. Consequently, the SGS model can be improved by the product of that ratio and the approximate KEF. In LES of compressible turbulent channel flow, the new model can accurately predict mean velocity profile, turbulence intensities, Reynolds stress, temperature–velocity correlation, etc. Additionally, for the case of a compressible flat-plate boundary layer, the new model can accurately predict some key quantities, including the onset of transitions and transition peaks, the skin-friction coefficient, the mean velocity in the turbulence region, etc., and it can also predict the energy backscatters in turbulence. Furthermore, the proposed model also shows more advantages for coarser grids.

Numerical study of the coherent structures in a transitional vertical channel natural convection flow

Numerical simulations of a spatially developing transitional flow in a vertical channel with one side uniformly heated and subjected to random velocity fluctuations at the inlet have been performed. Two characteristic frequency bands are observed in the flow, near the heated wall. The ability of the Proper Orthogonal Decomposition and the time-domain Spectral Proper Orthogonal Decomposition (SPOD) to decompose the flow is assessed, and SPOD is shown to be a powerful tool, as it is capable of separating the most energetic modes into two great families whose frequency content matches the frequency bands previously identified. The spatial structure of the modes is described, and their contribution to the turbulent heat transfer and velocity-temperature correlation is evaluated. Finally, the modes are linked to coherent structures that are observed in instantaneous visualizations of the flow, and a scenario of the development of the coherent structures in the laminar-turbulent transitional process is proposed.

Direct numerical simulation of turbulent particle-laden flows: effects of Prandtl and Stokes numbers

The effect of molecular Prandtl number on the turbulent heat transfer of a non-isothermal turbulent channel flow laden with particles with four different Stokes number is investigated using direct numerical simulation coupled with a Lagrangian particle tracking. The flow configuration is a downward vertical channel with side walls kept at constant temperature. A total of 16 test cases with four different Prandtl numbers of Pr = 0.71, 1, 3, 7, and four particle sizes with the Stokes numbers of St= 40, 60, 100, 190 are considered. The mass loading, particle-to-fluid density ratio, and specific heat ratio are constant for all cases and they are equal to 0.57, 7298, and 0.39, respectively. It is shown that by the addition of particles, the heat transfer rate decreases and the thickness of the conductive sublayer thickness increases for all cases. The percentage of heat transfer modulation resulting from the addition of particles is almost independent of the Prandtl number but it slightly depends on the size of the particles, with the maximum being for smaller particles. The mean and rms values of temperature, temperature-velocity correlations, mean energy equation, and temperature variance budgets are presented. Contours of temperature in planes parallel to the wall and normal to the streamwise direction are also reported. At higher Prandtl numbers, the temperature contours clearly visualize the mushroom shaped structures formed as a result of the ejection of low-speed fluid from the wall.

A new interpolation method for Lagrangian statistics in non-isothermal gas-solid flow

In this article, a new interpolation method is developed for the Lagrangian statistics in the non-isothermal gas-solid flow. The Lagrangian statistics are inseparable from interpolation, especially the optimal interpolation. The new interpolation method in this article is developed based on the smooth Dirac delta function, which is also called delta function interpolation method (DFIM). The performance of DFIM is compared with that of the spectral interpolation method (the interpolation method with the highest precision). The comparison results indicate that the performance of DFIM, especially the DFIM, is superior. The Lagrangian statistics method is one of the most effective methods for heat transfer analysis in the gas-solid flow. For the heat transfer analysis of gas-solid flow in this article, the relevant Lagrangian statistics are evaluated based on the newly developed DFIM, which include the temperature autocorrelation, the velocity autocorrelation, the second-order Lagrangian structure function, and the temperature velocity correlation function. The effect of the particle inertia on the heat transfer of the gas-solid flow is considered based on these Lagrangian statistics. The results show that the particle inertia has a great effect on the heat transfer in the gas-solid flow. As the particle inertia increases, the heat transfer in the gas-solid flow increases obviously. These results are helpful to understand the thermal behavior of the gas-solid flow.HighlightsNew interpolation method is developed for the Lagrangian statistics in non-isothermal gas-solid flow.Based on the relevant Lagrangian statistics, the heat transfer in the gas-solid flow is analyzed.The DFIM is much less computational cost than the spectral interpolation method.

Statistical analysis of temperature distribution on vortex surfaces in hypersonic turbulent boundary layer

The nonuniform temperature distribution (NUTD) on the coherent vortex surfaces of hypersonic turbulent boundary layer (TBL) is studied using the conditional sampling technique. The direct numerical simulation data of Mach 8 flat-plate TBL flows with different wall temperatures, Tw/T∞ = 10.03 and 1.9, are used for this research, and the coherent vortex surface is identified by the Ω-criterion. Two characteristic sides of the vortex are defined, which are represented by the positive and negative streamwise velocity fluctuations (±u′) of the vortex surfaces. The conditional sampling results between the mean temperature of the two sides show that there is a significant difference of up to 20% at the same wall-normal location. Furthermore, the velocity-temperature fluctuation correlations (Ru′T′ and Rv′T′) at the characteristic sides of vortex surfaces are studied. It is found that the temperature fluctuations are redistributed by the vortex rotational motion that has taken effect through Ru′T′ and Rv′T′ and then lead to the NUTD. The NUTD features are changed quantitatively by wall cooling but share the similar mechanism as that of the higher-temperature case.

Velocity–temperature correlations in high-temperature supersonic turbulent channel flows for two gas models

Velocity–temperature correlations in a high-temperature supersonic turbulent channel flows, including thermally perfect gas (TPG) and calorically perfect gas (CPG), are investigated based on the direct numerical simulation database [Chen et al., J. Turbul. 19 (2018) 365] to study the gas model effects. The results show that in fully developed turbulent channel flow, the Reynolds analogy factor remains close to 1.2 for both gas models. The “recovery enthalpy” is better than Walz’s equation to connect the mean stream-wise velocity with mean static temperature because it is independent with gas models. The modified strong Reynolds analogy for TPG is more accurate scaling than that for CPG, and the turbulent Prandtl number is insensitive to gas models. In addition, the influence of gas model on the probability density functions of stream-wise velocity and static temperature concentrate on the corresponding right tails.

Particle-resolved numerical simulations of the gas–solid heat transfer in arrays of random motionless particles

Particle-resolved direct numerical simulations of non-isothermal gas–solid flows have been performed and analyzed from microscopic to macroscopic scales. The numerical configuration consists in an assembly of random motionless spherical particles exchanging heat with the surrounding moving fluid throughout the solid surface. Numerical simulations have been carried out using a Lagrangian VOF approach based on fictitious domain framework and penalty methods. The entire numerical approach (numerical solution and post-processing) has first been validated on a single particle through academic test cases of heat transfer by pure diffusion and by forced convection for which analytical solution or empirical correlations are available from the literature. Then, it has been used for simulating gas–solid heat exchanges in dense regimes, fully resolving fluid velocity and temperature evolving within random arrays of fixed particles. Three Reynolds numbers and four solid volume fractions, for unity Prandtl number, have been investigated. Two Nusselt numbers based, respectively, on the fluid temperature and on the bulk (cup-mixing) temperature have been computed and analyzed. Numerical results revealed differences between the two Nusselt numbers for a selected operating point. This outcome shows the inadequacy of the Nusselt number based on the bulk temperature to accurately reproduce the heat transfer rate when an Eulerian–Eulerian approach is used. Finally, a connection between the ratio of the two Nusselt numbers and the fluctuating fluid velocity–temperature correlation in the mean flow direction is pointed out. Based on such a Nusselt number ratio, a model is proposed for it.

Effects of dimensional wall temperature on velocity-temperature correlations in supersonic turbulent channel flow of thermally perfect gas

Direct numerical simulations of temporally evolving supersonic turbulent channel flows of thermally perfect gas are conducted at Mach number 3.0 and Reynolds number 4800 for various values of the dimensional wall temperature to study the influence of the latter on the velocity-temperature correlations. The results show that in a fully developed turbulent channel flow, as the dimensional wall temperature increases, there is little change in the mean velocity, but the mean temperature decreases. The mean temperature is found to be a quadratic function of the mean velocity, the curvature of which increases with increasing dimensional wall temperature. The concept of “recovery enthalpy” provides a connection between the mean velocity and the mean temperature, and is independent of dimensional wall temperature. The right tails of probability density function of the streamwise velocity fluctuation grows with increasing dimensional wall temperature. The dimensional wall temperature does not have a significant influence on the Reynolds analogy factor or strong Reynolds analogy (SRA). The modifications of SRA by Huang et al. and Zhang et al. provide reasonably good results, which are better than those of the modifications by Cebeci and Smith and by Rubesin.

Boundary layer fluctuations in turbulent Rayleigh–Bénard convection

We report a combined experimental and numerical study of the effect of boundary layer (BL) fluctuations on the scaling properties of the mean temperature profile$\unicode[STIX]{x1D703}(z)$and temperature variance profile$\unicode[STIX]{x1D702}(z)$in turbulent Rayleigh–Bénard convection in a thin disk cell and an upright cylinder of aspect ratio unity. Two scaling regions are found with increasing distance$z$away from the bottom conducting plate. In the BL region, the measured$\unicode[STIX]{x1D703}(z)$and$\unicode[STIX]{x1D702}(z)$are found to have the scaling forms$\unicode[STIX]{x1D703}(z/\unicode[STIX]{x1D6FF})$and$\unicode[STIX]{x1D702}(z/\unicode[STIX]{x1D6FF})$, respectively, with varying thermal BL thickness$\unicode[STIX]{x1D6FF}$. The functional forms of the measured$\unicode[STIX]{x1D703}(z/\unicode[STIX]{x1D6FF})$and$\unicode[STIX]{x1D702}(z/\unicode[STIX]{x1D6FF})$in the two convection cells agree well with the recently derived BL equations by Shishkinaet al.(Phys. Rev. Lett., vol. 114, 2015, 114302) and by Wanget al.(Phys. Rev. Fluids, vol. 1, 2016, 082301). In the mixing zone outside the BL region, the measured$\unicode[STIX]{x1D703}(z)$remains approximately constant, whereas the measured$\unicode[STIX]{x1D702}(z)$is found to scale with the cell height$H$in the two convection cells and follows a power law,$\unicode[STIX]{x1D702}(z)\sim (z/H)^{\unicode[STIX]{x1D716}}$, with the obtained values of$\unicode[STIX]{x1D716}$being close to$-1$. Based on the experimental and numerical findings, we derive a new equation for$\unicode[STIX]{x1D702}(z)$in the mixing zone, which has a power-law solution in good agreement with the experimental and numerical results. Our work demonstrates that the effect of BL fluctuations can be adequately described by the velocity–temperature correlation functions and the new BL equations capture the essential physics.