Thermal Eddy Diffusivity Research Articles

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.

Parameterizing the nonlinear feedback on ENSO from tropical instability waves (TIWs) by nonlinear eddy thermal diffusivity

Parameterizing the nonlinear feedback on ENSO from tropical instability waves (TIWs) by nonlinear eddy thermal diffusivity

Numerical simulation of a low Prandtl number flow over a backward facing step with an anisotropic four-equation turbulence model

In recent years the use of liquid metals has become more and more popular for heat transfer applications in many fields ranging from IV generation fast nuclear reactors to solar power plants. Due to their low Prandtl number values, the similarity between dynamical and thermal fields cannot be assumed and sophisticated heat turbulence models are required to take into account the anisotropy of the turbulent heat transfer involving liquid metals. In the present work, we solve an anisotropic four-equation turbulence model coupled with the Reynolds Averaged Navier Stokes system of equations to simulate a turbulent flow of liquid sodium over a vertical backward-facing step. We implement an explicit algebraic model for Reynolds stress tensor and turbulent heat flux that takes into account flow anisotropic behavior. We study forced and mixed convection regimes when a uniform heat flux is applied on the wall behind the step. Linear isotropic approximations for eddy viscosity and eddy thermal diffusivity underestimate the turbulent heat flux components while this anisotropic model shows a better agreement with DNS results.

Direct numerical simulation of thermal channel flow for medium–high Prandtl numbers up to [formula omitted

A new set of DNS of a thermal channel flow have been conducted for friction Reynolds and Prandtl numbers up to 2000 and 10, respectively, reaching the Prandtl number of water, 7, for new Reynolds numbers, never simulated before. The Mixed Boundary Condition has been used as the thermal boundary condition. A new scaling of the thickness of the conductive sublayer is presented for medium–high Prandtl number values. The maximum of the intensity of the thermal field does not increase with the Reynolds number for the highest Prandtl numbers. This entails a good scaling near the wall of the viscous diffusion and the dissipation terms in the budgets of the temperature variance. The Nusselt number shows a power function behaviour with respect to the Prandtl number in a certain range of the friction Péclet number. Finally, the turbulent Prandtl number presents an increase near the wall for highest Prandtl numbers due to the reduction of the thermal eddy diffusivity. The statistics of all simulations can be downloaded from the web page of our group: http://personales.upv.es/serhocal/.

Assessment of Shear Stress Transport Model with Its Variant for Heat Transfer Problems

Merits of shear-stress transport and its wall-distance-free variant are evaluated by accommodating comparisons with direct numerical simulation and experimental data for heat and fluid flow problems. To improve the temperature field predictions in turbulent boundary layers and around the stagnation region accompanied by flow separation and reattachment, the thermal eddy diffusivity is approximated via the turbulent eddy viscosity using a variable turbulent Prandtl number . A new eddy-viscosity bound that responds to both rotational and irrotational strains to better predict nonequilibrium flows is incorporated. Results show substantial improvement by both models (shear-stress-transport and wall-distance-free shear-stress transport) and demonstrate the efficacy to predict heat transfer problems with consistency and commendable accuracy. Unlike the original shear-stress-transport model, the wall-distance-free version of the shear-stress-transport model requires no wall function/distance parameter to bridge the near-wall integration. Both models produce almost identical results and are applicable to flows with an anisotropic , having much higher generality than the traditional constant Prandtl number models. The current approach serves as a guidance to improve heat transfer characteristics of an isotropic or simple gradient-type heat flux model in the wall vicinity and boundary layers.

Numerical simulation of a turbulent Lead Bismuth Eutectic flow inside a 19 pin nuclear reactor bundle with a four logarithmic parameter turbulence model

Computational Fluid Dynamic allows scientists and engineers to investigate fluid flow in complex geometries and evaluate heat transfer between a solid body and a fluid. In the present paper we study the heat transfer of a Lead Bismuth Eutectic (LBE) turbulent flow in a bare 19 pin nuclear reactor bundle. When dealing with low Prandtl number fluids, like LBE for which Pr = 0.025, proper turbulence models are needed to improve the prediction of heat transfer. We use here a four logarithmic parameter turbulence model in order to calculate Reynolds stresses and turbulent heat flux. In particular, an equation for temperature fluctuations and one for their dissipation are solved. These variables are used to model thermal characteristic time scales. The results are reported for different values of the Peclet number and a fixed value of the pitch to diameter ratio. The obtained values of the Nusselt number are compared with experimental correlations, that can be found in literature, and with the ones obtained using a Simple Eddy Diffusivity model, where the eddy thermal diffusivity is calculated as proportional to eddy viscosity through a modeled turbulent Prandtl number.

Heat‐Engine and Entropy‐Production Analyses of the World Ocean

AbstractA heat‐engine analysis is performed in order to determine the ocean's Carnot efficiency, the maximum thermal energy (i.e., the Carnot work) available to drive the ocean circulation, and the ocean's irreversible entropy production. The analysis is based on a 36‐year monthly climatology of the surface heat flux; estimates of geothermal input, mechanical energy input, and heat storage; and a detailed treatment of freshwater fluxes. The ocean's Carnot efficiency is estimated to be 0.86%. The Carnot work is equal to 110 TW, which exceeds the mechanical energy input by the winds and tides. The net entropy flux from the ocean to its surroundings is found to be 0.617 TW/K, in good agreement with an independent estimate of the irreversible entropy production from viscous dissipation, thermal mixing, and haline mixing, demonstrating a balanced entropy budget of the ocean. Thermal mixing accounts for 56% of this entropy production, with most of the remaining production due to viscous dissipation and only small contributions from haline mixing and sea ice melting. Two thirds of the thermal entropy production is due to isopycnal mixing. The effect of thermal diffusion is to reduce the Carnot work significantly. The actual reduction is proportional to the amplitude of the (poorly known) eddy thermal diffusivity.

Mean temperature profiles in turbulent thermal convection

To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al., Phys. Rev. Lett. 114 (2015), the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated respectively as a cubic and a linear function of the distance from the plate. In the present work we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from 10^7 to 10^9.

Scaling of thermal stratification or mixing in outlet plena of SFRs

Sodium-cooled fast reactors (SFRs) have a tendency for thermal stratification in the hot or outlet plena especially during shut-down and loss of forced convection transients. The formation of thermally stratified zones in the outlet plena can: lead to thermal fatigue in adjoining solid structures; and impact passive heat removal capabilities. There are limited number of experimentally validated models in literature for thermal stratification or mixing in liquid metal pools. A high fidelity scaled-down experimental facility is required to address these needs and capture the physical phenomenon close to what is expected in a real system. The design of experimental facility and instrumentation becomes complicated with use of liquid sodium, so a surrogate fluid can simplify the design and operation considerably, providing flexibility to obtain high quality measurements. Similarity analysis based scaling methodology is used to design a scaled-down model of outlet plenum with liquid gallium (Pr∼ 0.025) as a surrogate for liquid sodium (Pr∼0.005). However, reduction in size and different choice of coolant must be handled carefully to maintain relevance and reliability of the experimental facility. The possible differences in thermal stratification or mixing physics in the scaled-down model and the prototype SFR are evaluated by quantifying eddy thermal diffusivity and mixing efficiency. Empirical models are used to correlate these quantities as a function of Richardson number and turbulent Reynolds number, and verified using computational fluid dynamics simulations. The verified empirical model is used to construct mixing efficiency for the prototypical conditions and the scaled-down model conditions. For low flow rates in both geometries, the results show thermal profiles dominated by molecular thermal diffusivity resulting into a fully stratified interface. The full flow cases for both geometries show energetic mixing, characteristic of eddy thermal diffusivity dominated, mixed temperature profiles. These results provide evidence of accurate scaling analysis by capturing the thermal stratification or mixing physics representing the dominant physical modes of those of the prototype SFR.

A proposal of turbulence model including the transport equations for eddy viscosity and eddy thermal diffusivity

A proposal of turbulence model including the transport equations for eddy viscosity and eddy thermal diffusivity

Influence of texture on thermal transport in streamwise-aligned superhydrophobic turbulent channels

A series of Direct Numerical Simulations (DNS) is performed for a systematic study of thermal transport in a periodic turbulent channel bounded by superhydrophobic surface (SHS) walls at a friction Reynolds number of Reτ=180. The SHS examined here is comprised of interspersed ridges and cavities aligned along the mean streamwise flow direction. The SHS is modelled as a planar surface consisting of spanwise-alternating regions of free-shear and no-slip boundary conditions. Water is considered as the bulk fluid where the SHS surface is assumed to be in a Cassie-Baxter state with non-wetting cavities containing air. The ridges are maintained at a constant temperature while heat transfer through the air/water interface supported by the ridges is assumed negligible and is modelled as adiabatic. Phase averaged statistics were obtained for a range of relative periodicity widths. Reorganisation of secondary flow structures owing to an increase in SHS feature width and its effect on thermal transport are analyzed with a particular focus on phase-averaged statistics. The relative feature width or periodicity length is found to influence strongly on thermal performance, or the average Nusselt number. A reduction in the turbulent Prandtl number is observed in the buffer region due to an increase in thermal eddy diffusivity relative to the momentum eddy diffusivity. The amount of heat transfer and mixing depend upon the periodicity length, and there is a reduction in the heat transfer along with the drag-reduction. A substantial decrease in the temperature fluctuations is observed in the vicinity of the wall as the production of temperature variance is diminished with an increase in the periodicity length, primarily due to a reduction in the mean temperature gradient in the wall-normal direction.

Heat transfer in a turbulent channel flow with square bars or circular rods on one wall

Direct numerical simulations (DNS) are carried out to study the passive heat transport in a turbulent channel flow with either square bars or circular rods on one wall. Several values of the pitch (${\it\lambda}$) to height ($k$) ratio and two Reynolds numbers are considered. The roughness increases the heat transfer by inducing ejections at the leading edge of the roughness elements. The amounts of heat transfer and mixing depend on the separation between the roughness elements, an increase in heat transfer accompanying an increase in drag. The ratio of non-dimensional heat flux to the non-dimensional wall shear stress is higher for circular rods than square bars irrespectively of the pitch to height ratio. The turbulent heat flux varies within the cavities and is larger near the roughness elements. Both momentum and thermal eddy diffusivities increase relative to the smooth wall. For square cavities (${\it\lambda}/k=2$) the turbulent Prandtl number is smaller than for a smooth channel near the wall. As ${\it\lambda}/k$ increases, the turbulent Prandtl number increases up to a maximum of 2.5 at the crests plane of the square bars (${\it\lambda}/k=7.5$). With increasing distance from the wall, the differences with respect to the smooth wall vanish and at three roughness heights above the crests plane, the turbulent Prandtl number is essentially the same for smooth and rough walls.

Large-eddy simulation of turbulent flows in a heated streamwise rotating channel

In this paper, large-eddy simulation has been performed to investigate the impacts of the centrifugal and Coriolis forces on a heated plane channel flow subjected to streamwise system rotations. In order to study the rotation effects, a variety of rotation numbers Roτ ranging from 0 to 15 have been tested in conjunction with a fixed low Reynolds number Reτ=300. The subgrid-scale effects on the budget balance of turbulent stresses and heat fluxes have been investigated, and the physical mechanisms underlying the transport of resolved turbulent stresses under the influence of streamwise system rotations have been thoroughly analyzed. Numerical simulations were performed using two dynamic subgrid-scale stress models and two dynamic subgrid-scale heat flux models; namely, the conventional dynamic model (DM) and an advanced dynamic nonlinear model (DNM) for closure of the filtered momentum equation, and the conventional dynamic eddy thermal diffusivity model (DEDM) and an advanced dynamic full linear tensor thermal diffusivity model (DFLTDM) for closure of the filtered thermal energy equation. The predictive performances of the DM and DNM have been carefully compared in conjunction with the same subgrid-scale heat flux model DFLTDM. In order to validate the numerical approach, turbulent statistics obtained from the simulations have been compared with the available direct numerical simulation data.

The effect of anisotropic diffusivity on rotating magnetoconvection in the Earth's core

A thermal diffusive process in the Earth's core is principally enhanced by small-scale flows that are highly anisotropic because of the Earth's rapid rotation and a strong magnetic field. This means that a thermal eddy diffusivity should not be a scalar but a tensor. The effect of such anisotropic tensor diffusivity, which is to be prescribed, on dynamics in the Earth's core is investigated through numerical simulations of magnetoconvection in a rapidly rotating system. A certain degree of anisotropy has an insignificant effect on the character, like kinetic and magnetic energies, of magnetoconvection in a small region with periodic boundaries in the three directions. However, in a region with top and bottom rigid boundary surfaces, kinetic and magnetic energies of magnetoconvection can be altered by the same degree of anisotropy. This implies that anisotropic tensor diffusivity affects on dynamics in the core, in particular near the boundary surfaces.

Heat transfer analysis in thermal entrance region under turbulent flow conditions

ABSTRACTTurbulent heat transfer in the thermal entrance region of the pipe was studied by numerical solution of the energy equation using finite difference technique under different operating conditions. The turbulence effect was accounted for by introducing thermal eddy diffusivity term in the energy equation in both axial and radial directions. The axial and radial temperature profiles, the local heat transfer coefficient in the entrance and fully developed regions, and the entrance lengths were obtained and discussed. In addition, the role of thermal eddy diffusivity in the thermal entrance region and fully developed region was investigated. The finite difference numerical solution was also commented on. The theoretical results were compared with previous experimental works, and a reasonably good agreement was found. It was found that the temperature gradient in axial direction is very steep, close to the leading edge near the pipe's surface and becomes almost negligible after a distance equals few pipe diameters, depending on Reynolds number (Re), Prandtl number (Pr), and heat flux (q). The radial temperature profile was very steep, close to the pipe's surface near the leading edge and becomes almost constant at a very short distance from the pipe surface, depending on Re, Pr, and q. The entrance length (Lt) was found to be reached after 0.5–3.5 pipe diameters distance for the investigated ranges of Re, Pr, and q. Thermal eddy diffusivity plays an important role in axial and radial transport. Ignoring it, causes underestimation in the heat transfer coefficient and overestimation in the entrance length. © 2012 Curtin University of Technology and John Wiley & Sons, Ltd.

Farmland microclimate and yield of winter wheat under different row spacing

To gain new insights into the underlying mechanisms responsible for farmland microclimate and yield of winter wheat under different row spacings [row spacing: 7 cm (RS7); 14 cm (RS14); 24.5 cm (RS24.5); 49 cm (RS49)], field experiments were conducted at Taian, China. Between the hours 13:00 and 14:00, low values of relative humidity (RH) and air temperature at 5 cm above the ground for the various row spacings were obtained. The average RH of RS7, RS14, RS24.5, and RS49 was 91.0%, 90.3%, 85.7%, and 76.4%, respectively; the average air temperature were 22.6°C, 23.2°C, 24.3°C and 25.3°C, respectively. The soil temperature of RS49 was higher than those of RS7 and RS14. The maximum values of eddy thermal diffusivity was 0.07 m2 s-1, as well as sensible and soil heat fluxes, were obtained at 12:00 o’clock. The order of these parameters was similar to that of the air temperature, but opposite to that of the latent heat flux. The yield of RS49 was significantly lower than those of the other treatments (P < 0.05). The present study indicates that excessive row spacing may result in severe dissipation of energy. RS14 is the optimum condition under uniform planting density.

Elevated mixing at a front

The mesoscale, submesoscale, and microscale structure of a front in the California Current was observed using a towed vehicle outfitted with microconductivity sensors. Thirteen &gt;60 km cross‐front sections from 0 to 350 m in depth were covered in 3.5 days. Objectively mapped data are fit via the Omega (ω) equation to obtain vertical velocity. A composite cross‐front section shows elevated mixing on the dense side within 10–20 km of the front. Water downwells and gradients are elevated there: i.e., Rossby number (Ro), horizontal strain (α), spice gradients, and microscale thermal dissipation (χ). Thermal eddy diffusivity (KT) reaches 10−3 m2 s−1 and increases 3–10× from the anticyclonic to the cyclonic side with a depth mean of ∼10−4 m2 s−1. The spatial structure of KT, Ro, and α are similar on the dense side, suggesting an energy cascade from the mesoscale via the submesoscale to the microscale. However, it is unclear whether frontogenesis, internal wave blocking by elevated vorticity, or internal wave trapping by large α produces the elevated mixing. The mean turbulent heat flux opposes the mean restratifying, mesoscale heat flux of 10 W m−2 and may allow the front to persist. Turbulent nitrate fluxes are 0.1–0.3 mmol m−2 s−1. Chlorophyll fluorescence and beam transmission reveal a &lt;6 km wide, ∼100 km long alongfront streamer which is a deep biomass maximum. Time scales for mixing and nutrient fluxes are 0.3–3 days, which are similar to phytoplankton growth rates and the time scale for frontal evolution.

C121 LESデータに基づく熱連成解析手法の提案とピンフィン冷却への応用(OS-5:乱流伝熱の実験とCFDの融合(2))

A numerical method was proposed to solve time-mean conjugate heat transfer in complex turbulent flows. The method is based on RANS, in which eddy thermal diffusivity is estimated by the turbulence statistics obtained by large eddy simulation. The proposed method was applied to examine the effect of endwall modification on total heat transfer in inclined pin-fin channels with a flat or wavy endwall. As a result, it is found that the wavy endwall shows an effect to enhance endwall heat transfer than flat endwall, while it shows less heat transfer rate on the pin surface. Thus, it is found that the benefit of wavy endwall depends on Biot number, which determines the heat transfer contribution of pin-fin surface to the total heat transfer area.

Evaluation of momentum and thermal eddy diffusivities for turbulent flow in tubes

The present investigation deals with a differential formulation to estimate the eddy diffusivity together with the universal velocity for fully developed turbulent flows in a tube. The subsequent theoretical predictions of wall friction coefficients and Nusselt numbers are in reasonable agreement with the classical solution of Blasius wall friction coefficient and Dittus and Boelter correlation for heat transfer, Nu = 0.023 Re 0.8 Pr 1/3 . The differential form of the eddy diffusivity equation is rendered into an explicit form by regression of the computer runs for wide range of 200 < R + < 3000 and 1 < Pr < 1000: ε m ν = 0.0195 ( y + u + ) 1 - exp - y + u + 100 2 . The prediction of Nusselt numbers from the eddy diffusivity expression satisfactorily agree with the Dittus and Boelter heat transfer correlation. In addition, the analysis can be further modified by altering the correction factor to agree with the predictions from Petukhov–Gnielinski correlation. Thus the efficacy of the model is established for different correlations with a modification in the correction factor for mixing length.

The effects of turbulence on molten pool transport during melting and solidification processes in continuous conduction mode laser welding of copper–nickel dissimilar couple

The melting and solidification stages of a continuous copper–nickel dissimilar metal conduction mode laser welding have been simulated numerically in this study. The heat, mass and momentum transports in molten metal pool have been analysed using both laminar and turbulent flow models separately for the same process parameters. The phase change aspects related to solidification and melting are accounted for by a modified enthalpy–porosity technique while the turbulent transport is modelled by a high Reynolds number k– ε model. It has been observed that temperature fields obtained from both laminar and turbulent transport simulations are qualitatively similar to each other. The molecular thermal diffusivity of the molten metal mixture is found to be in the same order of magnitude as eddy thermal diffusivity, as a result of which the thermal field gets marginally affected by fluid turbulence. By contrast, eddy viscosity remains much greater than molecular viscosity, which leads to greater amount of momentum diffusion in the case of a turbulent molten metal pool, in comparison to that obtained from the corresponding laminar simulation. This is reflected in the reduction in maximum velocity magnitude in the turbulent simulation in comparison to the maximum velocity obtained from laminar simulation. In the case of species transport, the turbulent mass diffusivity is found to be about 10 7–10 8 times greater than molecular mass diffusivity. As a result, the species field in turbulent simulation shows characteristics of better mixing between two dissimilar molten metals than the species field obtained using the laminar transport model. The species distribution obtained from turbulent transport is shown to be in better agreement with experimental data reported in literature than the corresponding mass fraction distribution obtained from laminar simulation. It is also found that species distribution in the molten pool is principally determined by advective and diffusive transport during the melting stage and species transport by advection and eddy diffusion in turbulent pool increasingly weakens with decreasing temperature during the cooling following the laser melting stage.