Probability Density Function Of Temperature Research Articles

Dominant heat transfer mechanism with conical roughness in a cubical box in turbulent Rayleigh–Bénard convection

In the present study, we numerically investigate the effect of Prandtl number on the heat transfer mechanism in turbulent Rayleigh–Bénard convection inside a cubical box endowed with conical roughness elements. The Rayleigh number is kept fixed at Ra=108, while the Prandtl number (Pr) varies from 1 to 50. In contrast to the monotonic increasing trend of the Nusselt number (Nu∼Pr0.27) in the two-dimensional (2D) roughness explored previously [Sharma et al., “Influence of Prandtl number in turbulent Rayleigh-Bénard convection over rough surfaces,” Phys. Rev. Fluids 7, 104609 (2022)], it assumes an invariant behavior (∼Pr0.012) for three dimensional (3D), though it is approximately 50% higher than its smooth counterpart. Flow intensity, measured in terms of Reynolds number (Re), drops with increasing Pr showing consistently lower magnitude for the 3D configuration. The addition of roughness elements is observed to disrupt the preferred orientation of large-scale circulation (LSC). The effect is predominant for lower Pr, where the roughness interferes most with the natural bias of LSC toward the diagonal planes of the cubical box. The analysis of plume statistics reveals that both coverage and intensity of plumes are augmented for the roughened cell. Increased homogeneity in the flow at higher Pr is reflected by the emergence of a more pronounced and distinguishable peak in probability density functions of temperature and velocity. Temporal spectra and variance data substantiate augmented intensity of fluctuations in the rough cell, while behavioral differences in the flow at different Pr are elucidated by using cross correlation of vertical velocity and temperature fluctuations.

Effects of Thermal Expansion and Pressure on Far-Field Fluctuations of Heterogeneous Propellants

In this work, we carry out three-dimensional mesoscale simulations of heterogeneous solid propellant combustion. We solve the reactive low-Mach-number equations in the gas phase with complete coupling to the solid phase. The model takes into account thermal expansion and deformation in the solid phase by using a hypoelastic law in the quasi-static limit. To account for morphology, we select two different propellant formulations with different particle size distributions and present the results as a function of pressure. The thermomechanical behavior is accessed by examining quantities such as strain and stress in the propellant as a function of pressure and propellant morphology. We also show that temperature and velocity fluctuations exist in the far field above the propellant surface and that these fluctuations can be significant. To better understand the nature of these fluctuations, we vary the pressure and make relevant plots of normal velocity and temperature probability density functions, as well as time autocorrelations. Such descriptions are necessary to account for the coupling between the mesoscale and the macroscale, where the fluctuations at the mesoscale can affect quantities at the macroscale, such as head-end pressure, trigger parietal vortex shedding, and aeroacoustics.

Low-Prandtl-number effects on global and local statistics in two-dimensional Rayleigh–Bénard convection

We present global and local statistical properties of turbulent Rayleigh–Bénard (RB) convection at low Prandtl numbers in this work. A series of high resolution two-dimensional (2D) direct numerical simulations are carried out in a square box for the Prandtl number ranges 0.005≤Pr≤0.07 and 0.01≤Pr≤0.15 at Rayleigh numbers Ra=107 and Ra=108, respectively. The global heat and momentum transport expressed as Nusselt number Nu and Reynolds number Re are found to scale as Nu∼Pr0.14 and Re∼Pr−0.82 for Ra=107, and Nu∼Pr0.11Re∼Pr−0.93 for Ra=108. The local velocity fluctuation at the cell center shows larger amplitudes at lowered Pr, indicating a stronger turbulence in the bulk. The magnitudes of kinetic and thermal energy dissipation rates in the bulk also increase with the decreasing of Pr, due to the intensified velocity gradient and larger thermal diffusivity, respectively. In the cell central region, probability density functions (PDFs) of velocity show a bimodal distribution, and it approaches the Gaussian distribution at higher Pr, while the PDFs of temperature display a stretched exponential shape with intermittent behavior. The kinetic energy spectra further reveal that the velocity cascade follows the Bolgiano–Obukhov scaling in the bulk of the convective flow.

Statistical Analyses of the Non-Uniform Longitudinal Temperature Distribution in Steel Box Girder Bridge

The frequently conventional assumption that bridge temperature is uniformly distributed on long-span bridges could lead to uncertainty when analyzing temperature effects. This study investigated the surface temperature of steel box girders on a long-span suspension bridge, emphasizing the distribution characteristics in the longitudinal (spanwise) direction. The girder surface temperature distribution was monitored using the long-term structural health monitoring system (SHMS). First, the probability density functions (PDF) of the girder surface temperature were analyzed. The results showed that the PDFs had bimodal characteristics and could be well-fitted using the weighted superposition of two normal distributions. Meanwhile, there was an obvious difference between the PDFs of the measuring points at different longitudinal sections of the bridge, which is inconsistent with the assumption that the temperature was uniformly distributed in the longitudinal direction. Subsequently, the longitudinal distributions of the girder surface temperature were statistically analyzed, and polynomial functions were introduced to fit the distribution curves along the left and right sides of the mid-span. A correlation analysis was then performed, highlighting the variability in temperature in the longitudinal direction. Additionally, the longitudinal temperature distribution pattern could be summarized as (i) the highest in the mid-span, the lowest in the tower, and increasing along the side span; (ii) there were also significant differences between the left and right sides of the mid-span. Finally, the time- and space- distributions of the temperature were studied, and a contour map was displayed. The results showed that the girder surface temperature had significant three-dimensional spatial characteristics and was not only non-uniformly distributed in space but also in time. This work is useful for a more accurate analysis of temperature effects on long-span bridges.

Generalized model of laser-induced peak asymmetry in Raman lines

The rate and precision at which samples can be scanned by Raman spectroscopy strongly depend on laser and material parameters. In this article, we describe the trade-off between parameters that increased laser intensities to improve resolution and reduce integration times, and its effect on thermally induced shift and asymmetric broadening of the line profile, especially in the case of resonant Raman. We present an analytical approximation to describe this phenomenon for all volumetrically absorbing materials and a wide range of laser parameters. This allows the determination of an optimal scan rate for the sample material and the required optical resolution, or vice versa, the determination and accurate correction for thermally induced shifts and asymmetries. This study provides an analytical quantification of this often-neglected line asymmetry and allows us to correct for its impact on the signal with few material properties and laser parameters. It may, in particular, allow us to discriminate this effect against other sources of peak asymmetry due to intrinsic properties. We obtain this analytical expression by condensing a parametrized finite element method model into a heuristic probability density function of temperature that describes the full parameter space. This function can be applied to any thermally undistorted line shape by convolution to determine a corrected line profile. This profile then provides a parameter-dependent optimized fitting function for an optimal determination of Raman signal parameters.

Passage Arrangement Optimization of Plate-Fin Heat Exchanger under Uncertain Operating Conditions

As the total flow rate and passage number increase, the quality of the passage arrangement has an important influence on the thermal efficiency of a multi-stream plate-fin heat exchanger (PFHE). The operating conditions are uncertain due to changes in environmental and product requirements, which affect the design of passage arrangements. If this influence is not considered enough in design, it will lead to the loss of heat transfer efficiency of the PFHE. Therefore, an optimization design method for passage arrangement is developed considering the influences of uncertain operating conditions. First, the uncertain operating conditions of the PFHE are represented using probability density functions of mass flow rates and temperature. Next, the optimization goals are chosen through the correlation calculation between the passage arrangement and the heat transfer indexes by the integral-mean temperature difference method. The particle swarm optimization algorithm is adaptively improved for the passage arrangement calculation. Then, the result under uncertain operating conditions is compared with the result under a single condition. The total heat transfer rate is increased, and the dimensionless mean square error of the cumulative heat load is decreased, so the method of designing the passage arrangement under uncertain operating conditions is effective.

Association between El Niño and extreme temperatures in southern South America in CMIP5 models. Part 2: future climate projections

This study provides an overview of projected future changes in the probability density function (PDF) and variability of the equatorial Pacific sea surface temperature in the El Niño 3.4 region (SST3.4). We analyzed how these changes affect the relationship with extreme temperature events in southern South America under global warming. We used 2 different emission scenarios for climate change simulations in the Coupled Model Intercomparison Project Phase 5 (CMIP5): RCP4.5 and 8.5, and for 2 future periods: 2031-2060 and 2071-2100. The Kolmogorov-Smirnov test was applied to compare the PDFs of SST3.4 between historical and future periods. We found that the PDFs of SST3.4 for both future periods differed significantly from the PDFs of the historical period for both scenarios. Changes in SST3.4 under greenhouse warming will alter the intensity and position of the teleconnection patterns. In particular, we observed that the mean difference in sea level pressure anomalies between El Niño and La Niña will weaken in the future. This result could explain why, in general, no significant correlations or quantile regressions were projected between SST3.4 and extreme temperature indices in southern South America for both the near and far future. The projections of the best-performing models in the historical period did not provide much clarity because they differ from each other but showed us that the individual models simulate changes in the relationships between SST3.4 and extremes temperature according to the period and scenario.

Effect of boundary conditions in turbulent thermal convection (a) (a)Contribution to the Focus Issue Turbulent Thermal Convection edited by Mahendra Verma and Jörg Schumacher.

We report an experimental study aiming to clarify the role of boundary conditions (BC) in high Rayleigh number turbulent thermal convection of cryogenic helium gas. We switch between BC closer to constant heat flux (CF) and constant temperature (CT) applied to the highly conducting bottom plate of the aspect-ratio-one cylindrical cell 30 cm in size, leading to dramatic changes in the temperature probability density function and in power spectral density of the temperature fluctuations measured at the bottom plate, while the dynamic thermal behaviour of the top plate and bulk convective flow remain unaffected. Within our experimental accuracy, we find no appreciable changes in Reynolds number Re(Ra) scaling, in the dimensionless heat transfer efficiency expressed via Nusselt number Nu(Ra) scaling, nor in the rate of direction reversals of large scale circulation.

Electrical conductivity and temperature of the Earth's mantle inferred from Bayesian inversion of Swarm vector magnetic data

Study of induced magnetic field is a powerful way to sound Earth internal structure. This work presents a full analysis and interpretation in terms of electrical conductivity and temperature of vector magnetic field measurements from Swarm Level1b data product from 26/11/2013 to 31/12/2019. Time series of the Gauss coefficients associated with the induced and inducing magnetic field are obtained from the data after removal of the core and lithospheric fields models and data selection. A Bayesian inversion of the induced field Gauss coefficients is then performed to obtain a new estimate of Earth's 1D mantle electrical conductivity down to 2000 km depth. This profile is fully compatible with the profiles derived from satellite and ground magnetic observatories data but does not present in the lower mantle the increase predicted by laboratory-based conductivity profile associated to classical mantle composition and temperature profile. Using the most recent database to model the electrical conductivity of all mineral mantle phases, two different methods are used to interpret Swarm data in terms of temperature for a given composition and water content. The first one is based on an interpretation of the conductivity estimates in terms of temperature by classical numerical root search. The second one consists in inferring a temperature probability density function from a Bayesian inversion of the Gauss coefficients associated to the induced magnetic field. Our results show that the later provide more reliable estimates of mantle temperatures, in relation to more physically grounded prior values. This second method provides also tighter constraints on the electrical conductivity estimates of the lower mantle.

Probability density functions for fluctuations in turbulent two-phase flames

Probability density function methods for characterizing various fluctuating quantities in the reacting gas phase of turbulent spray flames are presented, for the development of which O'Brien has been a pioneer. Balance equations for two-phase flows considered as a piecewise continuous medium are first presented with the interface conditions containing the vaporization–condensation equilibrium. From these, the equation for the joint probability density function of temperature and composition in the gas phase is derived, in which specific terms related to vaporization are identified. The terms that require modeling are then highlighted. The first one is the classical micromixing term, which has to include all the features already discussed for single phase reacting flows, but adaptation to two-phase flows has to be questioned. Mainly, several unclosed terms deal with vaporization, requiring specific attention. It is shown that proposals for their modeling can be obtained on the same basis as in the so-called Eulerian–Lagrangian models, which will directly induce changes in the shape of the probability density function. We finally discuss the model presented herein and comment on its connection with recently published literature for this subject. Specific attention is devoted to the combustion regimes and to the presence of diffusion flamelets surrounding fuel droplets or groups of droplets. The proposals presented here go one step further compared to the recent studies, and now, they have to be compared with well-defined experiments, numerical experiments with direct numerical simulations, or real experiments in conveniently controlled conditions.

Climate change and thermo-solar patterns of office buildings with/without window films in extreme hot-arid climate of Kuwait

This research attempts to develop a systematic method to address climate changes by studying temperature-humidity patterns as the two major meteorological parameters and to moderate these effects using a commercially available window film on interior windows of an office building with double-glazing. A three-floor educational building at Australian College of Kuwait (ACK) was equipped with temperature, humidity and illuminance sensors for two similar size offices, one with 3 M Neutral 20 window films. Total readings of 50,000 entries from each sensor was recorded every 3 min by a home-designed microprocessor-based logging system for three months of June, July, and August 2019. In parallel, the ACK building was simulated in EnergyPlus and DesignBuilder software and calibrated with the experimental measurements. Histograms and a probability density function (PDF) of temperature and humidity are built by a representative rational function as a model of main climate parameters variations. By examining min-mean-max values of data, it is observed that the office with window films has increased indoor humidity and, in most instances, reduced temperature by 2–5 °C compared to double-glazed bare windows which can significantly reduce cooling loads in extreme hot-arid climates. Simulation results are presented on the energy saved and the reduced CO2 footprints by applying the solar window films in the ACK building.

Evaluation of extreme temperature in multi-RCM simulations over CORDEX-East Asia phase II domain

In this study, the extreme temperature in high-resolution hindcast simulations conducted by three reanalysis-driven regional climate models (WRF, RegCM4 and CCLM) is evaluated during the period of 1989–2009. We select 10 extreme temperature indices recommended by the Expert Team on Climate Change Detection and Indices to compare the simulations with the CN05.1 dataset. In addition, apparent temperature (AT) combining relative humidity and air temperature is applied to define the humid heat wave indices. The results show that the regional climate models (RCMs) can skillfully reproduce the spatial distribution and the probability density function of daily maximum temperature (Tmax) and minimum temperature (Tmin) in spite of some biases in certain ranges and regions. For the intensity indices (TXx, TNn, TXn, TNx), the RCMs demonstrate very high spatial correlations with the observation, but they have difficulties in reproducing the interannual variations of these indices. CCLM best simulates the interannual variabilities of the percentile-based indices, while WRF shows better skill in reproducing their spatial patterns (TX10p, TX90p). In regard to the duration indices, RegCM4 outperforms WRF and CCLM in simulating the consecutive cold events (CSDI), while WRF performs best for the continuous hot days (WSDI). The humid heat wave, which is characterized by both high temperature and humidity, is skillfully reproduced by the RCMs as well. The overestimation of the magnitude and intensity of humid heat waves in WRF can be attributed to the favorable large-scale circulation and water vapor condition.

Temperature measurement of a turbulent buoyant ethylene diffusion flame using a dual-thermocouple technique

High-frequency temperature measurements were carefully conducted for a 15 kW buoyant turbulent ethylene diffusion flame over a 15.2 cm diameter gas burner with air co-flow. A dual-thermocouple probe, consisting of two fine-wire thermocouples with 25 μm and 50 μm wire diameters, was used to determine a compensated turbulent gas temperature. A sensitivity analysis shows that temperatures resolved using this dual-thermocouple technique are less sensitive to changes in thermocouple bead size, therefore, uncertainty is greatly reduced even when soot deposition on the thermocouple bead occurs in sooty flames. Mean and root-mean square (rms) fluctuations of gas temperature were recorded in a two-dimensional plane across the flame centerline. The mean gas temperature monotonically decreases away from the flame centerline at most flame heights, except for 1 diameter above the burner, where a temperature dip is observed. The rms temperature peaks shift from the edge of the flame to the center as the height increases. This is due to the enhanced mixing between fuel and air, which is further shown using probability density functions of the local gas temperature. A systematic temperature dataset with high spatial resolution is established for sooty flames, which is valuable for future soot and radiation model validation.

Temperature distribution analysis of steel box-girder based on long-term monitoring data

Temperature may have more significant influences on structural responses than operational loads or structural damage. Therefore, a comprehensive understanding of temperature distributions has great significance for proper design and maintenance of bridges. In this study, the temperature distribution of the steel box girder is systematically investigated based on the structural health monitoring system (SHMS) of the Sutong Cable-stayed Bridge. Specifically, the characteristics of the temperature and temperature difference between different measurement points are studied based on field temperature measurements. Accordingly, the probability density distributions of the temperature and temperature difference are calculated statistically, which are further described by the general formulas. The results indicate that: (1) the temperature and temperature difference exhibit distinct seasonal characteristics and strong periodicity, and the temperature and temperature difference among different measurement points are strongly correlated, respectively; (2) the probability density of the temperature difference distribution presents strong non-Gaussian characteristics; (3) the probability density function of temperature can be described by the weighted sum of four Normal distributions. Meanwhile, the temperature difference can be described by the weighted sum of Weibull distribution and Normal distribution.

Transported PDF simulation of auto-ignition of a turbulent methane jet in a hot, vitiated coflow

In the current work, the auto-ignition of a turbulent round methane jet is studied numerically by means of a transported probability density function (PDF) method. The methane jet is issued into a hot, vitiated coflow, where it ignites to form a steady lifted flame. For this flame, experimental data of hydroxyl, temperature and mixture fraction are provided in the area where the fuel auto-ignites. To model this experiment, the transport equation for the thermochemical PDF is solved using a hybrid finite volume / Lagrangian Monte-Carlo method. Turbulence is modelled using the k-ϵ turbulence model including a jet-correction. Computational results are compared to experimental data in terms of mean quantities, variances and lift-off height. Moreover, the structure of the one-point, one-time marginal PDF of temperature is analysed and compared to experimental data which are provided in this work. It is found that the transported PDF method in conjunction with the k-ϵ model is capable of reproducing these statistical data very well. In particular the effect of ignition on the marginal PDF of temperature can be well reproduced with this approach. To further analyse the relevant processes in the evolution of the temperature PDF, a statistically homogeneous system is studied both numerically and analytically.

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.

Turbulent temperature fluctuations in a closed Rayleigh–Bénard convection cell

We report a systematic study of spatial variations of the probability density function (PDF) $P(\unicode[STIX]{x1D6FF}T)$ for temperature fluctuations $\unicode[STIX]{x1D6FF}T$ in turbulent Rayleigh–Bénard convection along the central axis of two different convection cells. One of the convection cells is a vertical thin disk and the other is an upright cylinder of aspect ratio unity. By changing the distance $z$ away from the bottom conducting plate, we find the functional form of the measured $P(\unicode[STIX]{x1D6FF}T)$ in both cells evolves continuously with distinct changes in four different flow regions, namely, the thermal boundary layer, mixing zone, turbulent bulk region and cell centre. By assuming temperature fluctuations in different flow regions are all made from two independent sources, namely, a homogeneous (turbulent) background which obeys Gaussian statistics and non-uniform thermal plumes with an exponential distribution, we obtain the analytic expressions of $P(\unicode[STIX]{x1D6FF}T)$ in four different flow regions, which are found to be in good agreement with the experimental results. Our work thus provides a unique theoretical framework with a common set of parameters to quantitatively describe the effect of turbulent background, thermal plumes and their spatio-temporal intermittency on the temperature PDF $P(\unicode[STIX]{x1D6FF}T)$.

Direct Numerical Simulation of a Low Prandtl Number Rayleigh–Bénard Convection in a Square Box

Direct numerical simulations for low Prandtl number fluid (Pr = 0.0216) are used to study the steady-state Rayleigh–Bénard convection (RB) in a two-dimensional unit aspect ratio box. The steady-state RB convection is characterized by analyzing the time-averaged temperature-field, and flow field for a wide range of Rayleigh number (2.1 × 105 ⩽ Ra ⩽ 2.1 × 108). It is seen that the time-averaged and space-averaged Nusselt number (Nuh¯) at the hot-wall monotonically increases with the increase in Rayleigh number (Ra) and the results show a power law scaling Nuh¯∝Ra0.2593. The current Nusselt number results are compared with the results available in the literature. The complex flow is analyzed by studying the frequency power spectra of the steady-state signal of the vertical velocity at the midpoint of the box for different Ra and probability density function of dimensionless temperature at various locations along the midline of the box.

Boundary layer turbulence and freestream turbulence interface, turbulent spot and freestream turbulence interface, laminar boundary layer and freestream turbulence interface

We study the boundary-layer turbulence and freestream turbulence interface (BTFTI), the turbulent spot and freestream turbulence interface (TSFTI), and the laminar boundary-layer and freestream turbulence interface (LBFTI) using direct simulation. Grid spacings in the freestream are less than 1 Kolmogorov length scale during transition. Probability density functions of temperature and its derivatives are used to select the interface identification threshold, corroborated by a vorticity-based method. The interfaces so detected are confirmed to be physical a posteriori by the distinctive quasi-step-jump behavior in the swirling strength and temperature statistics along traverses normal to the BTFTI and TSFTI. No interface-normal inflection is detected across the LBFTI for either swirling strength, temperature, vorticity magnitude, Reynolds shear stress, streamwise velocity, normal velocity, or turbulence kinetic energy. The present direct numerical simulation data thus cast serious doubts on the shear-sheltering hypothesis/theory, which asserts that a subset of freestream fluctuations is blocked by the LBFTI. In the early stage of transition, quasi-spanwise structures exist on the LBFTI. The TSFTI shape is dominated by head prints of concentrated hairpin vortices. Further downstream, the BTFTI geometry is strongly modulated by groves of hairpin vortices (the boundary layer large-scale motions) with a distinct streamwise preferential orientation. Streamwise velocity and turbulence kinetic energy only exhibit minor plateaus (rather than quasi-step-jump) across the BTFTI and the TSFTI. We emphasize that it is more meaningful and important to acquire reproducible and reliable interface-normal statistics prior to considering any plausible substructures and elusive transient dynamics of the BTFTI, TSFTI, and LBFTI.

Global turbulence features across marginality and non-local pedestal-core interactions

Spatially non-local aspects of turbulent transport in tokamak plasmas are examined with global gyrokinetic simulations using the ORB5 code. Inspired by very accurate measurements in the TCV tokamak in L-mode, we initialise plasma profiles with constant logarithmic gradients in the core and constant linear gradients in the ‘pedestal’ (). The main finding is that transport in the core is strongly affected by the presence of pedestal gradients. This non-local pedestal-core coupling appears to be correlated with the appearance of repetitive avalanches that propagate across both pedestal and core regions. Below a certain threshold value in pedestal gradient, no well-defined frequency is found for avalanches. Above this threshold, a well-defined frequency shows up, which roughly matches that of the local geodesic acoustic mode (GAM) frequency near the plasma edge and is thus well below the local GAM frequency in the core: this behaviour is very similar to the global coherent mode structure observed in TCV. Above this threshold in pedestal gradient, the core transport increases sharply: there is therefore a non-locality in marginality. The probability density functions (PDFs) of density, temperature, temperature gradient and potential are found to have nearly Gaussian distributions, whereas the heat flux can have, in the presence of avalanches, a more or less strongly positively skewed PDF, which could be fitted by a log-normal distribution. The skewness of the heat flux is found to be radially and non-locally dependent: its value in the plasma core critically depends on the presence of gradients in the pedestal. The relation flux versus gradient is examined in detail. The local instantaneous flux versus gradient relation shows a hysteresis behaviour during an avalanche but no clear correlation, unlike the flux and zonal flow (ZF) shearing rate, which are anti-correlated: flux is higher when shearing rate is lower. This leads to corrugated time-averaged radial profiles of transport, heat and temperature gradient, with heat diffusivity having local maxima where the ZF shearing rate goes to zero and temperature gradient has local minima. Finally, we show how the flux versus gradient relation can be analysed locally for series of simulations with different averaged gradients.