Heat Flux Vector Research Articles

Particle resolved DNS and URANS simulations of turbulent heat transfer in packed structures

An accurate prediction of turbulent heat transfer in packed-bed reactors is fundamental for the design and performance of such reactors. Various classes of Reynolds-Averaged Navier-Stokes models are available in the literature for predicting turbulent heat transfer via closing the Reynolds stress and Turbulent Heat Flux terms. In this work, we investigate the accuracy of representative unsteady RANS turbulence models in modeling heat transfer characteristics in packed-bed reactors. For this accuracy assessment, the performance of the examined unsteady RANS models is compared with particle-resolved Direct Numerical Simulations including heat transfer. The simulations are conducted in a three dimensional periodic domain consisting of randomly arranged cylindrical particles for Reynolds number Rep=1000. The extracted mean quantities (e.g. velocity and temperature), turbulent heat flux components and turbulent kinetic energy are used as a reference for the RANS models' accuracy assessment. The SST k−ω and Spalart-Allmaras models performed well in predicting mean fields. However, all models tended to mispredicted the turbulent quantities including the turbulent viscosity and heat flux vector. The thermal field calculations rely on the turbulent viscosity based Simple Gradient Diffusion Hypothesis for modeling turbulent heat flux, which introduce challenges in capturing the coupling between thermal and turbulent fields.

Three-phase lag model for thermal conductivity of a thermo-viscoelastic porous medium

This paper investigates the propagation of waves and the thermal behavior in a thermo-viscoelastic porous medium using the three-phase-lag (TPL) model which accounts for phase lags in the heat flux vector, temperature gradient and thermal displacement gradient. The study aims to capture the interactions between thermal, mechanical, and structural properties of the thermo-viscoelastic porous, isotropic, homogeneous medium. The governing equations, incorporating the TPL heat conduction law and constitutive relations for the viscoelastic porous material, are derived and analyzed. The impact of relaxation times, porosity, and viscoelastic parameters on heat propagation is investigated through analytical solutions by normal modal analysis. Analytical representations of various physical quantities (stresses components, displacements, temperature,…) in the material's domain are derived. These expressions are then assessed numerically for a specific material, with the outcomes depicted graphically. A comparison is made between the predictions of the three-phase-lag (TPL), the dual-phase-lag (DPL) and Lord–Shulman (L–S) theories in the absence and presence of voids.

Characteristics of Deterministic and Stochastic Unsteadiness of Trailing Edge Cutback Film Cooling Flows

Abstract Trailing edge cutback film cooling flows are ubiquitous in small and medium gas turbines, but they are difficult to predict accurately due to the inherent deterministic and stochastic unsteadiness that controls the effectiveness of the cooling system. To help develop accurate closure models for such flows, the characteristics of both types of unsteadiness and their effects on the mean flows are analyzed in this research. Zonal detached eddy simulation (ZDES) is performed on a trailing edge cutback flow model, and the numerical results are validated against the measured data. Then, by using spectral proper orthogonal decomposition (SPOD) reconstruction, the original dataset is segregated into deterministic and stochastic unsteadiness. The characteristics of the stress tensor and the heat flux of each type of unsteadiness are analyzed in detail, and notable differences between the two unsteadiness are identified in terms of the stress tensor anisotropy and distribution of unsteady kinetic energy and heat flux. By propagating the unsteadiness through the Reynolds-averaged Navier–Stokes (RANS) equations, the effect of different unsteadiness on the mean flow prediction is quantified. An accurate prediction of the total stress tensor reduces the prediction error in the velocity field by 79% and cooling effectiveness by 55%. An accurate prediction of the total heat flux vector reduces the prediction error in cooling effectiveness further by 37%. These findings provide valuable knowledge for the physical understanding, turbulence modeling, and aerothermal design of cutback trailing edge flows.

Entropy optimization of a FENE-P viscoelastic model: a numerically guided comprehensive analysis

AbstractThe influence of polymers on entropy generation processes is substantial, particularly in the fields of fluid dynamics and rheology. The FENE-P (Finitely Extensible Nonlinear Elastic-Peterlin) model describes the polymer’s dynamics as a result of the interaction between the stretching caused by the velocity gradient and the elastic force that restores the polymer to its equilibrium position. Models such as FENE-P aid in understanding and predicting polymer flow behaviour allowing for the reduction of entropy generation by optimizing system designs. A continuum approach is employed to express the heat flux vector and polymer confirmation tensor of the model. The study investigates the complex relationship between polymer conformation, flow dynamics, and heat transfer taking into account the thermophoresis (Soret effect) and mass diffusion-thermal diffusion coupling (Dufour effect) phenomena to optimize processes by reducing entropy. This study illuminates polymer’s significance in entropy minimization, improving engineering design methodologies and applications in materials science, chemical engineering, and fluid dynamics. As result, the presence of polymers leads to a substantial decrease in the total entropy of the system. This understanding provides opportunities for enhancing heat transfer systems, thereby facilitating the development of more efficient and sustainable technology.

On the dependence of the thermal module of elasticity of a two-component magnetic fluid on frequency, concentration and magnetic field

Purpose. Study of the dependence of the thermal modulus of elasticity of a two-component magnetic fluid on the magnitude of the magnetic field strength, the frequency of external disturbance and the volumetric concentration of magnetic particles.Method. The research method is based on the kinetic theory of liquid systems. Based on previously constructed kinetic equations for one-particle and two-particle distribution functions and a microscopic expression for the heat flux vector, an explicit dynamic expression for the thermal modulus of elasticity of magnetic fluids is obtained. In fast processes in liquids, heat transfer occurs in waves and their propagation is similar to the propagation of second sound in helium II. The thermal modulus of elasticity in liquids appears at high frequencies and ensures the propagation of second sound. The expression for the thermal modulus of elasticity consists of potential and kinetic parts, taking into account structural and translational relaxation processes, respectively. To study the thermoelastic properties of magnetic fluids, appropriate expressions of potential interaction energies were selected for each subsystem, allowing for numerical calculations.Results. Numerical calculations of the frequency and concentration dependence of the dynamic thermal modulus of elasticity in the presence of an external magnetic field in a kerosene-based magnetic fluid were carried out. The calculation results show that an increase in the influence of external disturbances leads to a nonlinear increase in the thermal modulus of elasticity in the magnetic fluid. An increase in the volume concentration of magnetic particles and an increase in the magnetic field strength also led to a nonlinear increase in the thermal modulus of elasticity in the magnetic fluid.Conclusion. It has been established that, due to taking into account translational and structural relaxation processes, the region of frequency dispersion of the thermal elastic modulus is wide. Numerical calculations carried out at different values of the external magnetic field and the volume concentration of magnetic particles showed that although an increase in the magnetic field and concentration of magnetic particles leads to an increase in the thermal modulus of elasticity, their increase does not affect the change in the frequency dispersion region.

Firehose instability in heat-conducting solar wind plasmas including FLR corrections and electrical resistivity

The effects of finite Larmor radius (FLR) corrections and heat-flux vector are studied on the pressure anisotropy-driven firehose instability in finitely conducting solar wind plasmas described by the double-adiabatic Chew, Goldberger and Low (CGL) fluid theory. The fluid description of collisionless plasmas is governed through modified adiabatic equations due to the heat-flux vector and finite ion Larmor radius corrections. The analytical dispersion relation of the firehose instability has been derived using the normal mode analysis and discussed in the solar wind plasmas. In the transverse mode, the dispersion relation of the Alfvénic mode is modified due to electrical resistivity and FLR corrections. In the longitudinal mode, the effects of the heat-flux parameter and electrical resistivity are observed separately. The dispersion relation of the firehose mode is modified due to the combined effects of FLR corrections and electrical resistivity. The graphical illustrations show that finite electrical resistivity and ion Larmor frequency destabilize the growth rate of the firehose instability. The results are useful for analyzing the solar mission data to study the firehose instability in the solar wind plasmas.

Green-Kubo paralell tempering molecular dynamics determination of the thermal conductivity for the wurtzite ZnO: statistical problems of the method

The thermal conductivity for the wurtzite ZnO is determined in the temperature range from 300 to 1100 K by using parallel tempering molecular dynamics within the Green-Kubo approach and a classical Morse-Born-Mayer-Coulomb hybrid interaction potential. Compared to other previous calculations for the thermal conductivity of common crystals within the same Green-Kubo and molecular dynamics approach, the used parallel tempering scheme shows some appealing improvements in the calculation of the time self-correlation of the heat flux vector, although at the price of using a relatively large number of total computational steps. However, in spite of the found improvements for the calculation of the self-correlation of the heat flux vector, some statistical problems on this point remain on the particular application of the method. Finally, even with the presence of a clear statistical noise, the obtained values and temperature trend of the calculated thermal conductivity shows the classical 1/T decaying behavior reported in previous works for wurtzite ZnO and other related semiconductor systems using the alternative Boltzmann transport equation theory.

Application of Extended Eddy-viscosity and Elliptic-Relaxation Approaches to Turbulent Convective Flow in a Partially Divided Cavity

The paper reports on the numerical turbulence model in predicting mass, momentum and heat transfer in a partially divided cavity heated from the side using buoyancy-extended eddy-viscosity and elliptic relaxation approach with the algebraic expressions for the Reynold stress tensor and turbulent heat flux vector. The CDS (central differencing scheme) and LUDS (linear upwind differencing scheme) were used as the discretization method and the governing equations were solved using the finite volume method and Navier-Stokes solver. Validation of the model has been carried out by experimental data of convective flow in the cavity as well as by numerical data DNS (direct numerical simulation). The model agrees very well with the experiment and DNS and it is also able to demonstrate the performance which is comparable to that of the previous advanced second-moment closure model (SMC) in the literature. The results show that the model is suitable for use in simulations of the turbulent convective flow in a cavity with partition and it has the potential to be applied to more complex cavities and a wide range of turbulence levels.

Modeling of heat conduction through rate equations

AbstractStarting from a classical thermodynamic approach, we derive rate-type equations to describe the behavior of heat flow in deformable media. Constitutive equations are defined in the material (Lagrangian) description where the standard time derivative satisfies the principle of objectivity. The statement of the Second Law is formulated in the classical form and the thermodynamic restrictions are then developed following a variant of the Coleman-Noll procedure where the entropy production too is given by a non-negative constitutive equation. Both the free energy and the entropy production are assumed to depend on a common set of independent variables involving, in addition to temperature, both temperature gradient and heat-flux vector together with their time derivatives. This approach results in rate-type constitutive function for the heat flux that are intrinsically consistent with the Second Law and easily amenable to analysis. In addition to providing already known models (e.g., Maxwell-Cattaneo-Vernotte’s and Jeffreys-like heat conductors), this scheme allows the formulation of new models of heat transport that are likely to apply also in nanosystems. This is consistent with the fact that higher-order time derivatives of the heat flux are in order when high-rate regimes occur.

Fundamental heat and entropy transfer mechanisms of convection underlying the first and second laws for an unsteady multiport system and their applications in parallel flow heat exchangers

Although thermal convection is ubiquitous in nature and technology, it has long been argued whether convection can be considered as an independent mode of heat transfer. Here, we show that pure convection mode is generally recognized from the classical first and second laws via mass flow by rooting the unsteady mass conservation relation through an equivalent multiport system. The fundamental distinction is made between convection driven by the thermal and entropy potential differences and mass flow accompanied by zero potential difference. The constitutive relations of the convective heat and entropy fluxes and their thermodynamic relationship are developed in terms of their potential differences with the exponential distribution of temperature difference in a compressible flow. In the absence of work transfer, the first and second laws are simply expressed by the standard Maxwell’s electromagnetic type equations via the total heat and entropy flux vectors. The pure convection theory is applied to a parallel-flow heat exchanger and the linear temperature difference distribution is obtained for both streams via the first-law definitions of the convective and overall heat transfer coefficients in an internal flow. The entropy generation rate inside a heat exchanger is recalculated and indicated to be driven by the difference between the reciprocal of outlet temperatures of two streams, from which the entropy generation number between outlets is developed to eliminate the entropy generation paradox. The present linear temperature difference profiles agree well with the experimental data, and the outlet entropy generation number is also justified by comparing with the previous nondimensionalizing entropy generation numbers.

Heat Conduction of Micropolar Solids Sensitive to Mirror Reflections of Three-Dimensional Space

This article considers a variant of the heat conduction theory of thermal conductivity, in which the heat flux pseudovector has a weight of 1. The pseudoinvariants associated to the heat flux pseudovector are sensitive to mirror reflections and inversions of threedimensional space. The primary purpose of the study was to find a heat flux vector that is algebraically equivalent to the microrotation pseudovector and to measure elementary volumes and areas using pseudoinvariants that are sensitive to mirror reflections. To represent spinor displacements, a contravariant microrotation pseudovector with a weight of +1 was selected. Thus, the heat flux and mass density were expressed as odd-weight pseudotensors. The Helmholtz free energy per unit doublet pseudoinvariant volume was employed as the thermodynamic state potential of the following functional arguments: absolute temperature, symmetric parts, and accompanying vectors for the linear asymmetric strain tensor and the wryness pseudotensor. The results obtained show that the thermal conductivity coefficient and heat capacity of elastic micropolar solids are pseudoscalars of odd weight, indicating their sensitivity to mirror reflections.

Non-Maxwellian electron effects on the macroscopic response of a Hall thruster discharge from an axial–radial kinetic model

A 2D axial–radial particle-in-cell (PIC) model of a Hall thruster discharge has been developed to analyze (mainly) the fluid equations satisfied by the azimuthally-averaged slow dynamics of electrons. Their weak collisionality together with a strong interaction with the thruster walls lead to a non-Maxwellian velocity distribution function (VDF). Consequently, the resulting macroscopic response differs from a conventional collisional fluid. First, the gyrotropic (diagonal) part of the pressure tensor is anisotropic. Second, its gyroviscous part, although small, is relevant in the azimuthal momentum balance, where the dominant contributions are orders of magnitude lower than in the axial momentum balance. Third, the heat flux vector does not satisfy simple laws, although convective and conductive behaviors can be identified for the parallel and perpendicular components, respectively. And fourth, the electron wall interaction parameters can differ largely from the classical sheath theory, based on near Maxwellian VDF. Furthermore, these effects behave differently in the near-anode and near-exit regions of the channel. Still, the profiles of basic plasma magnitudes agree well with those of 1D axial fluid models. To facilitate the interpretation of the plasma response, a quasiplanar geometry, a purely-radial magnetic field, and a simple empirical model of cross-field transport were used; but realistic configurations and a more elaborate anomalous diffusion formulation can be incorporated. Computational time was controlled by using an augmented vacuum permittivity and a stationary depletion law for neutrals.

Распространение магнитогидродинамических волн и гравитационная неустойчивость в астрофизической анизотропной и теплопроводной плазме

The hydrodynamic instability of a magnetised, self-gravitating rotating anisotropic plasma is analysed in the collisionless approximation and taking into account the heat flux vector on the basis of the modified Chew-Goldberger-Low equations. The dispersion relation is derived and simplified cases of propagation of small-amplitude perturbation waves are discussed, with modified analytical criteria for hydrodynamic instability obtained. In accordance with the general dispersion relation, three cases when the propagation of the perturbation wave passes across, along, and obliquely to the magnetic field vector are specifically considered. It is shown that anisotropic pressure and heat flux not only modify the classical Jeans instability criterion but also induce new unstable regions. It is found that the presence of uniform plasma rotation reduces the critical wave number and has a stabilising effect on the gravitational instability criterion for transverse propagation of the perturbation wave and has no effect in the case of its longitudinal propagation. The inclusion of thermal flows leads to the appearance of two additional wave modes. The results obtained are important for the construction of evolutionary magnetohydrodynamic models of collisionless astrophysical plasma.

Some characteristic properties of the solutions in the three-phase-lag heat conduction

In this paper we consider the three-phase-lag model of heat conduction that involves second-order effects in phase lag of the heat flux vector. This model leads to a fourth-order in time equation of Moore–Gibson–Thompson type. We use the thermodynamic restrictions derived from the compatibility of the constitutive equation with the Second Law of Thermodynamics to study the properties of the solutions of the initial boundary value problems associated with the model in concern. In this connection we establish a series of well-posedness results concerning the related solutions like: uniqueness, continuous data dependence, exponentially stability or domain of influence. Furthermore, based on the thermodynamic restrictions, we show that the thermal model in question admits damped in time propagating waves as well as exponentially decaying standing modes. We also show that when the thermodynamic restrictions are not fulfilled, then wave solutions appear that cause the energy blows up as time goes to infinity.

NccrFOAM suite: Nonlinear coupled constitutive relation solver in the OpenFOAM framework for rarefied and microscale gas flows with vibrational non-equilibrium

The nccrFOAM suite is a collection of nonlinear coupled constitutive relation (NCCR) solvers for rarefied and microscale gas flows with vibrational non-equilibrium, in conjunction with the conservation laws implemented in the foam-extend framework which is an open-source solver with a General Public License (GPL 3). nccrFOAM solvers are developed as an extension to the dbnsTurbFoam solver by implementing additional algebraic constitutive relations for the non-conserved quantities of the stress tensor and heat flux vector. In contrast to Navier-Stokes-Fourier (NSF) solvers that employ first-order constitutive relations to calculate non-conserved quantities and consequently suffer from obvious shortcomings when simulating gas flows in high non-equilibrium, the second-order NCCR framework presents a novel alternative to simulate rarefied and microscale gas flows in a better and a more intuitive manner. In addition to the solver for monoatomic gases, the solvers are for the first time implemented for diatomic and polyatomic gases with translational-rotational and vibrational degrees of freedom based on second-order constitutive models in a three-dimensional framework. Towards this, a new foam-extend library has been written based on the two-temperature formulation to handle the translational-rotational and vibrational modes. The new foam-extend solver was validated for several representative problems, and an exhaustive list of tutorials is documented. The solver will certainly benefit the rarefied and microscale gas dynamics and hypersonics communities at large interested in flows involving high degrees of rarefaction, speed, and temperature variations.

Boundary effect and heat vortices of hydrodynamic heat conduction in graphene

A SIMPLE like algorithm is proposed for numerically solving the Guyer-Krumhansl equations. Results show that thermal conductivity of suspended graphene depends on the heat flux boundary conditions. If the strength of the point source is fixed, only when the strength of the point sink exceeds a critical value can the heat vortices occur in the heat flux vector field. We also find that heat vortices only appear when the side length perpendicular to the dominant heat flow direction exceeds a critical value. Furthermore, the no-slip boundary condition is beneficial for the generation of heat vortices, while the specular phonon-boundary interactions can disrupt the heat vortices. These findings will contribute to experiments on heat vortex in the heat flux field of suspended graphene. Finally, it is found that the heat can flow from some cold regions to hot regions for the hydrodynamic heat conduction in suspended single-layer graphene.

Implementing the direct relaxation process in the stochastic particle method for flexible molecular collisions

Multi-scale phenomena are prevalent and significant across various disciplines. For multi-scale flow physics in the gas-kinetic theory based on Boltzmann equation or its simplified mathematical models (called Boltzmann model equations), the multi-scale mechanism can be modeled by the philosophy of unified modeling, where the free transport behaviors of gas particles and their collision behaviors are coupled by the temporal integral solutions (or characteristic line solutions) of Boltzmann model equations, which leads to a unified/multi-scale property in all scales. Also, the stochastic particle methods are based on these Boltzmann model equations. The corresponding numerical methods are, thus, limited by these model equations. This paper aims to overcome this restriction by replacing these modeled collision operators with a simple direct relaxation (DR) process. Since the collision term of Boltzmann model equation should fulfill the correct relaxation rates of non-equilibrium macro-variables, such as stress tensor and heat flux vector, along with other basic properties, such as conservation and H theorem, the DR process is designed to be directly based on these crucial relaxation rates. Therefore, with the DR strategy for calculating particle collisions, the numerical method can be established without constructing collision operator. Furthermore, the DR has the flexibility and simplicity to recover various models. In this work, Xu's and Yuan's new models are recovered in to illustrate the validation and performance of DR. Moreover, since at the inlet/outlet boundaries, subsonic, supersonic, and hypersonic flows can simultaneously exist, a generalized numerical boundary condition is also considered in the particle methods developed in this paper. Finally, the validation and accuracy of the present method are examined with a series of test cases.

An improved theory of thermoelasticity for ultrafast heating of materials using short and ultrashort laser pulses

This paper introduces an improved thermoelastic theory designed specifically to address the challenges posed by fast and ultrafast heating of materials. The proposed theory incorporates a time delay in the temperature gradient and introduces the rate of temperature gradient as an additional term in the heat equation. This term accounts for the phase lag and the precedence of the temperature gradient and heat flux vector. By leveraging the sign of the temperature gradient rate, the theory automatically determines the precedence of either the temperature gradient or the heat flux vector. To verify the effectiveness of the proposed theory, experiments reported in the literature were simulated using several theories, namely the classical, Lord-Schulman (LS), Green-Lindsay (GL), dual-phase-lag (DPL), and the presented theory. The simulations considered the effects of relaxation times and pulse duration on the results. The findings reveal that for pulse durations shorter than 50 ps, the second-order time derivative of the temperature gradient becomes a crucial term in the energy equation for accurately predicting temperature profiles. Additionally, as the pulse duration decreases, the sensitivity to relaxation times increases. Regarding the surface temperature corresponding to the material ablation threshold, it is observed that for an 80 fs laser pulse, the ablation temperature falls between the boiling and critical temperatures, significantly higher than the boiling temperature associated with the ablation by a 270 ps pulse duration. Lastly, the simulation results exhibit a high level of agreement with the experimental data, particularly in terms of displacements, further validating the accuracy of the presented simulations.

Technique for calculating streamwise local fluxes in general flows

This paper presents a new technique for calculating microscopic local fluxes that provides a robust and accessible way to obtain the streamwise fluxes in general flows. Flux calculation is one of the fundamental elements in molecular simulations. Several-decades-long research efforts on the subject made significant progresses since the pioneering work of Irving and Kirkwood [J. Chem. Phys, (1950) 18, 817–829]. However, some issues remain regarding the fluxes in the flow direction. The streaming velocity (SV) and the fluxes independent of SV are difficult to manage during computation. For example, the local fluxes and SV that vary in more than one direction are not well represented despite being prevalent in the microscopic phenomena such as nanofluidics, biotechnologies, and energy conversions. The technique presented in this paper is based on the instantaneous microscopic conservation equations and depends only on the total fluxes that include the convective contributions. This allows a calculation of all the fluxes at the end of a simulation run, with no complication arising in terms of defining and collecting SV, pressure tensor, and heat flux vector. Widely-used methods such as the method of planes and volume-average method were adapted, respectively. This study then investigated a planar shear flow in a highly confined channel, where a flow is unknown at the beginning of a simulation, and the controversial streamwise heat flux with no temperature gradient was present. The flow and fluxes were structured, and the streamwise heat flux evolved significantly and slowly over a few tens of nano-seconds even with steady boundary conditions.

Relaxation-time model for the post-Newtonian Boltzmann equation

The non-equilibrium contributions to the post-Newtonian hydrodynamic equations are determined from a relaxation-time model of the post-Newtonian Boltzmann equation. The Chapman- Enskog method is used to calculate the non-equilibrium distribution function. The components of the energy–momentum tensor are found from the knowledge of the non-equilibrium and the post-Newtonian equilibrium Maxwell–Jüttner distribution functions. The linearized field equations for the mass, momentum and internal energy densities coupled with the three Poisson equations of the post-Newtonian approximation are investigated by considering a plane wave representation of the fields. The constitutive equations for the viscous stress and heat flux vector are obtained and it is shown that the transport coefficients of shear viscosity and heat conductivity do depend on the Newtonian gravitational potential.