Parallelepipedic Cavity Research Articles

Coupled heat and mass transfer in shallow caves: Interactions between turbulent convection, gas radiative transfer and moisture transport

Understanding and predicting the microclimate of shallow caves is a key issue for the conservation of parietal art. In order to determine the dominant mechanisms of heat transfer in a configuration close to that of painted caves, we performed numerical simulations of a parallelepiped cavity whose dimensions and depth are of the order of 10m. This simple geometry allowed us to use a detailed model including turbulent natural convection, gas radiation, along with vapor transport and latent heat fluxes resulting from condensation and evaporation on the walls.Gas radiation increases the flow circulation in the cavity through wall–gas radiative exchanges and significantly modifies the wall radiative flux. Conversely, the wall conductive flux remains unchanged (a non trivial behavior reported in the literature about the differentially heated cavity). In the energy balance at the walls, the radiative flux overcomes the conductive flux. However, the addition of conduction and latent heat fluxes, both driven by convection, prevails over radiation in some regions of the cavity.Heat and mass fluxes are maximum in areas of the cavity roof where the distance from the ground is the shortest. Due to the asymmetry induced by the inversion of the vertical temperature gradient twice a year, net condensation resulting in limestone dissolution is expected in these areas, whereas the other regions of the cave undergo net evaporation resulting in limestone deposition. The orders of magnitude of the condensation flux (a few microns per day) and of the retreat velocity of the wall (a few tenth of a micron per year) are in line with the field data available in the literature.

Heat transfer by natural convection and radiation in three dimensional differentially heated tall cavities

Natural convection coupled with radiation is numerically studied in a tall rectangular parallelepiped cavity filled with air and constant differentially heated opposite vertical walls. This type of tall cavity is found in walls built with hollow blocks. Three dimensional steady state numerical simulations are performed. Numerical models for pure natural convection and natural convection coupled with radiation are validated with experimental results. In this work, the tall cavity studied has a 0.06 m width, 0.155 m depth and a 2.56 m height, giving a vertical aspect ratio of A=42.7 and a horizontal aspect ratio of A′=2.6. The cold wall temperature, Tc, is fixed. The hot wall temperature, Th, is varied, obtaining the Rayleigh numbers: RaW, 1.01×104, 1.07×105 and 7.20×105. The emissivity, ϵ, varies from 0.0 to 1.0. The results show that with a given Th, the convective Nusselt number, Nuc, depends on ϵ. For the emissivity value of hollow block materials (ϵ=0.8) the radiative heat transfer represents 83% for the lowest Th. For pure natural convection, a Nuc correlation as a function of RaW is obtained. A simplified calculation of the total heat transfer, Nut, is proposed.

Chaotic mixing in an acoustically driven cavity flow

We investigate the mixing properties of an acoustic driven flow in a parallelepipedic cavity with square basis. Relying on nonnormal reflections, an acoustic beam with a square path is obtained, generating a global complex flow in the cavity which enables chaotic advection. For a moderate power of the source, the chaotic region invades the whole cavity, illustrating that acoustic streaming can be used successfully as a nonintrusive tool to mix efficiently.

Tilt Angle’s Effects on Free Convection Heat Transfer Coefficient inside a Water-Filled Rectangular Parallelepiped Enclosure

The effect of tilt angle on free convection heat transfer is investigated experimentally inside a rectangular parallelepiped cavity filled with water. The cavity had the dimensions S × S × H (m3), where S was the inside length, and H was the inside height of the cavity. The lower surface was subject to constant heat flux, and the upper surface was cooled by a stream of ambient air. The free convection heat transfer data were generated using different uniform heat fluxes. Four tilt angles were considered: 0°, 30°, 60°, and 90°. The surface temperature measurements were collected 10 h after the experimental run to ensure that a steady-state was reached. It was noticeable that the free convection heat transfer strongly depended on the tilt angle and the modified Rayleigh numbers. The 3D results showed that the Nusselt number reached a maximum at 60° at a fixed modified Rayleigh number. An enhancement in the Nusselt number at any tilt angle was observed over that of a zero tilt angle, and the percent of enhancement was 7.92–62.38%, depending on the modified Rayleigh numbers and the tilt angle. It was also observed that as the modified Rayleigh number increased, the temperature uniformity on both the hot and cold surfaces was disturbed. Furthermore, an empirical correlation between the modified Rayleigh numbers and Nusselt numbers was obtained for each angle. Moreover, two overall general correlations are obtained to cover the four tilt angles (i.e., 0°, 30°, 60°, and 90°) and the modified Rayleigh numbers, which should be helpful for engineering applications.

Rayleigh–Bénard flow for a Carreau fluid in a parallelepiped cavity

A continuation method is used to study Rayleigh–Bénard convection in a non-Newtonian fluid inside a parallelepiped cavity. The cavity has its length equal to twice the side of the square cross-section. Shear-thinning and shear-thickening Carreau fluids are considered. The focus is put on the two stable branches which exist in the Newtonian case, a stable primary branch of transverse rolls $B_1$ and a primary branch of longitudinal roll $B_2$ , stabilized beyond a secondary bifurcation point $S_2$ . Although the primary bifurcation points are unchanged, the non-Newtonian properties strongly modify the bifurcation diagram. Indeed, for a shear-thinning fluid, the stable solutions can exist at much smaller Rayleigh numbers ${Ra}$ , on subcritical branches beyond saddle-node points $SN_1$ and $SN_2$ , and small perturbations can be sufficient to reach them. In agreement with Bouteraa et al. (J. Fluid Mech., vol. 767, 2015, pp. 696–734), the change of the primary bifurcations from supercritical to subcritical occurs at given values of what they define as the degree of shear-thinning parameter $\alpha$ . Moreover, the value of the Rayleigh number at the saddle-node points can be approximated by a simple expression, as proposed by Jenny et al. (J. Non-Newtonian Fluid Mech., vol. 219, 2015, pp. 19–34). In the case of a shear-thickening fluid, the branches remain supercritical, but the secondary point $S_2$ is strongly moved towards larger ${Ra}$ , making it more difficult to reach the longitudinal roll solution. Energy analyses at the bifurcations $SN_1$ , $SN_2$ and $S_2$ show that the changes of the corresponding critical thresholds ${Ra}_c$ are connected with the changes of the viscous properties, but also with changes of the buoyancy effect.

Three-Dimensional Lattice Boltzmann Model for Acoustic Waves Emitted by a Source

This paper implements the lattice Boltzmann method to simulate the propagation of sound waves in three dimensions. The numerical model is exercised on the lid-driven cavity flow. Tests are then proposed on acoustic situations. The results are first confronted with analytical solutions of the spherical waves emitted by a single point source inside a cubic cavity. Then, we studied the case where the waves are emitted from a circular sound source placed at the center of the left boundary of a parallelepipedic cavity filled with water. With the circular source discretized as a set of point sources, we were able to simulate the wave propagation in 3D and calculate the sound pressure amplitude. Tests using different emission conditions and LBM relaxation times finally allowed us to get good comparisons with analytical expressions of the pressure amplitude along the source axis, highlighting the performance of the lattice Boltzmann simulations in acoustics.

Heat transfer in shallow caves: influence of turbulent convection

The enhancement of the conservation conditions of cave paintings requires a detailed understanding of heat transfer in such cavities. This article presents a numerical investigation of turbulent free convection in a parallelepipedic cavity. Non-homogeneous wall temperatures are prescribed from a large-scale model taking into account external temperature fluctuations damped by heat diffusion in the rock massif above the cavity. Large Eddy Simulation is performed to solve the turbulent flow fields for a given wall temperature field corresponding to a Rayleigh number of 8.5 x 109. The outcomes of the model are analysed in terms of statistical mean. Results show complex large scale flow patterns with regions of high turbulent intensity. The Q-criterion is used to identify turbulent structures for an instantaneous flow field. Then we analyse the spatial distribution of the conductive heat flux at the walls to locate the regions with intense convection. We show that the conductive flux smaller than the wall-to-wall radiative flux in the major part of the cavity, and close to its value at some spots.

Experimental study on influence of wall acoustic materials of 3D cavity for targeted energy transfer of a nonlinear membrane absorber

In view of applying the membrane nonlinear energy sink (NES) to reduce the low-frequency noise inside vehicle interior with acoustic absorption materials, a simplified acoustic cavity is used to model vehicle interior cavity. The system of one acoustic mode of the parallelepiped cavity and one membrane is established and the experimental set-up is built. The forced vibration responses in frequency-domain and time-domain are analyzed by using the set-up with and without the acoustic materials. Under certain excitation amplitude and frequency range, the strongly modulated responses (SMR) occurs and the plateau of suppressed resonance peaks appears. Meanwhile, the energy of the system is studied when the SMR exists. The targeted energy transfer (TET) phenomenon and the theoretical model of the system is verified by the experimental method. The wall acoustic materials of 3D cavity could affect the TET of the system, where the excitation amplitude for the beginning threshold of the desired working zone becomes larger and the excitation amplitude interval of TET becomes larger simultaneously. The ending threshold of the desired working zone of the NES is redefined. The numerical results are in good agreement with the experimental results. The application of membrane NES in 3D acoustic cavity proposed in this paper provides a new technical idea for vehicle interior low-frequency noise reduction in the future.

Numerical modeling of a benchmark experiment on equiaxed solidification of a Sn–Pb alloy with electromagnetic stirring and natural convection

A three-phase volume averaged equiaxed model is applied to simulation of an experiment on solidification of the binary alloy Sn-10 wt.% Pb subjected to the electromagnetic stirring. The experiment, whose description was published earlier, was performed in a parallelepiped cavity under controlled cooling conditions and with real-time two-dimensional temperature measurement over a lateral surface of the cavity co-planar with direction of solidification. Applied numerical model treats motion of the liquid and equiaxed grains whose growth kinetics is taken into account and uses a double time step scheme to accelerate solution. Growth of columnar dendrite is not considered. It is shown that electromagnetic force acting in a direction opposite to the natural convection flow creates moderately turbulent flow in pure liquid which is treated with a realizable kε−ε model. It is demonstrated that calculated temperature distribution in the cavity well reproduces temperature maps reconstructed from thermocouples measurements throughout the experiment. Final macrosegregation map and distribution of density grain number are qualitatively similar to those obtained in the experiment. Variation of intensity of electromagnetic stirring in numerical model shows that this affects shape and localization of positive segregation region at the bottom of the cavity.

Effect of Diffusion Length in Modeling of Equiaxed Dendritic Solidification under Buoyancy Flow in a Configuration of Hebditch–Hunt Experiment

Modeling of equiaxed solidification is vital for understanding the solidification process of metallic alloys. In this work, an extended literature review is given for the models currently used for equiaxed solidification simulations. Based on this analysis, we present a three-phase multiscale equiaxed solidification model in which some approximations regarding solute transport at microscopic scale are put together in a new way and incorporated into macroscopic transport equations. For the latter, a term relating to the momentum exchange between the two phases is revised, and a modification for the grain packing algorithm is proposed. A modernized model for equiaxed dendrite growth is tested using a case of solidification of Sn-5 wt pct Pb alloy in a parallelepiped cavity that mimics the Hebditch–Hunt experiment. The results obtained using two approaches to calculate diffusion length are presented and compared both with each other and with numerical results from elsewhere. It is demonstrated that diffusion length has a crucial effect on the final segregation pattern.

Performance Comparison and Critical Examination of the Most Popular Stirring Techniques in Reverberation Chambers Using the “Well-Stirred” Condition Method

This paper presents a performance comparison and a critical examination of the most popular stirring techniques proposed in the literature for reverberation chambers, implemented in the same parallelepipedic cavity (excluding naturally vibrating intrinsic reverberation chambers). This yields a fair comparison (in our experimental conditions) of these techniques in terms of performance and, in particular, from which frequency, each stirring process can lead to a well-stirred reverberation chamber. This analysis is performed, thanks to a recently published method based on $S_{11}$ measurements of an antenna within the chamber for different stirring configurations (i.e., different positions of a mode stirrer for a mechanically stirred chamber). The method is applied (including some adjustments when necessary) for the three main families of stirring techniques. In order to exhaustively assess the performance of each implemented technique, experiments are performed for different quantities of absorbers inserted in the cavity in order to study the effect of the average composite qualify factor of the chamber on each stirring process.

Effect of reboosting manoeuvres on the determination of the Soret coefficients of DCMIX ternary systems

Experiments for measuring the thermodiffusion and diffusion coefficients in ternary liquid mixtures need to be performed in a gravity free environment to avoid convective flows. If convection can be neglected, the resulting concentration field can be used to fit the analytical solution and subsequently obtain the values of the two Soret coefficients. The so-callled Diffusion Coefficients Measurements in ternary mIXtures (DCMIX) experiments carried out onboard the International Space Station use a parallelepipedic cavity (10×10×5mm) called Soret cell, with two opposite heated walls at different temperatures to attain a linear concentration between them. When a reboosting (to keep, for instance, the station in the correct orbit) occurs, the acceleration level increases considerably and, thus, the induced convection can potentially distort the experimental results. In the present study numerical simulations have been carried out using five different reboosting scenarios (reboosting occurring at equally distributed times from beginning up to the end of the thermodiffusive step), and the experimental procedure for measuring the Soret coefficient has been reproduced. Three different mixtures have been considered, corresponding to the DCMIX1, DCMIX2 and DCMIX3 campaigns. It has been found that the thermal Grashof number determines the sensitivity of the system to the reboosting in these three cases. The reboosting maneuvers can strongly affect the concentration field, and therefore, the calculation of the Soret coefficients. The closer to the end of the thermodiffusive step the reboosting occurs, the larger the error is. The DCMIX2 is the most distorted mixture reaching an approximately 100% of error when reboosting acts at the end. But if the reboosting takes place at the beginning, the above-mentioned error is smaller than 1%. Also, the results depend on the method used to measure the concentration field. When the concentration difference averaged at the hot and cold walls method is used, the error is generally lower.

Bifurcations from steady to quasi-periodic flows in a laterally heated cavity filled with low Prandtl number fluids

This study deals with the transition toward quasi-periodicity of buoyant convection generated by a horizontal temperature gradient in a three-dimensional parallelepipedic cavity with dimensions $4\times 2\times 1$ (length $\times$ width $\times$ height). Numerical continuation techniques, coupled with an Arnoldi method, are used to locate the steady and Hopf bifurcation points as well as the different steady and periodic flow branches emerging from them for Prandtl numbers ranging from 0 to 0.025 (liquid metals). Our results highlight the existence of two steady states along with many periodic cycles, all with different symmetries. The bifurcation scenarios consist of complex paths between these different solutions, giving a succession of stable flow states as the Grashof number is increased, from steady to periodic and quasi-periodic. The change of these scenarios with the Prandtl number, in connection with the crossing of bifurcation points, was carefully analysed.

Onset of thermal convection in a rectangular parallelepiped cavity of small aspect ratios

Onset of thermal convection of a fluid in a rectangular parallelepiped cavity of small aspect ratios is examined both numerically and analytically under the assumption that all walls are rigid and of perfect thermal conductance exposed to a vertically linear temperature field. Critical Rayleigh number Rc and the steady velocity and temperature fields of most unstable modes are computed by a Galerkin spectral method of high accuracy for aspect ratios Ax and Ay either or both of which are small. We find that if Ax is decreased to 0 with Ay being kept constant, Rc increases proportionally to , the convection rolls of most unstable mode whose axes are parallel to the shorter side walls become narrower, and their number increases proportionally to . Moreover, as Ax is decreased, we observe the changes of the symmetry of most unstable mode that occur more frequently for smaller Ax. However, if is decreased to 0, although we again observe the increase in Rc proportional to , we obtain only one narrow convection roll as the velocity field of most unstable mode for all A. The expressions of Rc and velocity fields in the limit of or are obtained by an asymptotic analysis in which the dependences of Rc and the magnitude and length scale of velocity fields of most unstable modes on Ax and Ay in the numerical computations are used. For example, Rc is approximated by and in the limits of and , respectively. Moreover, analytical expressions of some components of velocity fields in these limits are derived. Finally, we find that for small Ax or A the agreement between the numerical and analytical results on Rc and velocity field is quite good except for the velocity field in thin wall layers near the top and bottom walls.

Parameters design of a nonlinear membrane absorber applied to 3D acoustic cavity based on targeted energy transfer (TET)

The targeted energy transfer (TET) phenomenon is demonstrated and analyzed between an acoustic medium inside a parallelepiped cavity and a thin viscoelastic membrane that is mounted on one wall of the cavity and that is working as a nonlinear absorber or a nonlinear energy sink (NES). Based on a two degrees-of-freedom (DOF) system, which is comprised by an acoustic mode of 3D acoustic cavity and a nonlinear membrane NES, the desired working zone for the membrane NES is defined and the two thresholds of the zone are obtained. In this paper, we investigate a parameter analysis of a nonlinear membrane absorber based on TET technique to design the NES. The physical parameters of the membrane and the place of the membrane on the wall of the cavity are studied. It can finally allow us to determine where is the better place for the membrane NES and which parameter affects mainly the desired working zone for the membrane NES that has to be used for reducing low frequency resonances inside a given acoustic cavity. Numerical simulations are preformed to validate the TET phenomenon and the results of the parameter analysis.

Experimental thermal time estimation for Mach‐Zehnder interferometer to study coupled heat and mass transfer

In the present investigation, a theoretical approach and experimental models are discussed for estimating the experimental thermal time of an optical digital interferometry experiment. The investigation deals with coupling between heat and mass transfer phenomena. Mach‐Zehnder Interferometry has been found to be an accurate and precise experimental tool for visualizing the thermodiffusion phenomenon inside a parallelepiped cavity, when a thermal gradient is applied to its two opposite sides. Processing the results of this experiment requires distinguishing the two experimental phases: 1) refractive index changes due to change of temperature and 2) refractive index changes due to the separation of the components. We can separate these two phases according to the thermal time of the liquid mixture. In previous studies L2/χ (L = distance between hot and cold sides, χ = thermal diffusivity of the mixture) was used as the thermal time of the liquid. However, due to a high separation rate at the beginning of the thermodiffusion process, an accurate estimation of the thermal time can significantly affect the final result of the interferometry measurements. Here, a theoretical approach and an experimental method were developed to estimate the proper thermal time for this experiment. We also discuss the importance of precisely estimating the thermal time on the experimental results. This study shows that assuming thermal time equals L2/χ may cause noticeable underestimation when measuring the Soret coefficients and maximum separation values.

Near-field acoustic streaming jet.

A numerical and experimental investigation of the acoustic streaming flow in the near field of a circular plane ultrasonic transducer in water is performed. The experimental domain is a parallelepipedic cavity delimited by absorbing walls to avoid acoustic reflection, with a top free surface. The flow velocities are measured by particle image velocimetry, leading to well-resolved velocity profiles. The theoretical model is based on a linear acoustic propagation model, which correctly reproduces the acoustic field mapped experimentally using a hydrophone, and an acoustic force term introduced in the Navier-Stokes equations under the plane-wave assumption. Despite the complexity of the acoustic field in the near field, in particular in the vicinity of the acoustic source, a good agreement between the experimental measurements and the numerical results for the velocity field is obtained, validating our numerical approach and justifying the planar wave assumption in conditions where it is a priori far from obvious. The flow structure is found to be correlated with the acoustic field shape. Indeed, the longitudinal profiles of the velocity present a wavering linked to the variations in acoustic intensity along the beam axis and transverse profiles exhibit a complex shape strongly influenced by the transverse variations of the acoustic intensity in the beam. Finally, the velocity in the jet is found to increase as the square root of the acoustic force times the distance from the origin of the jet over a major part of the cavity, after a strong short initial increase, where the velocity scales with the square of the distance from the upstream wall.

Velocity field and parametric analysis of a subsonic, medium-Reynolds number cavity flow

Cavity flows are a class of flows bounded by material structures, where a recirculation region is present, and they are found in many practical applications. In the present study, the interaction between a boundary layer and an open parallelepipedic cavity develops a Kelvin–Helmholtz-like instability coupled with the cavity recirculation. PIV measurements of the flow are carried out in two orthogonal planes inside the cavity, for different aspect ratios, incompressible flow conditions, and Reynolds numbers in the range 1,900–12,000. Mean velocity and second-order moments of velocity fluctuations reveal the flow morphology. For particular conditions, centrifugal instabilities appear that are induced by flow curvature due to wall confinement. The use of an identification criterion indicates the presence of pairs of counter-rotating vortices winded around the recirculation. A parametric analysis is conducted, and the inviscid Rayleigh discriminant provides the potentially unstable flow regions inside the cavity. Finally, a stability parameter considering the ratio between centrifugal destabilizing effects and stabilizing viscous effects is carried out and gives thresholds for the emergence of the centrifugal instability. The study draws to an end with a comparison with a well-documented lid-driven cavity flow.

Scaling and direct stability analyses of natural convection induced by absorption of solar radiation in a parallelepiped cavity

The present study considers the early stage of the flow development in a water-filled, three-dimensional parallelepiped cavity subject to solar radiation. The thermal structure consists of an upper stable stratification due to internal heating, provided by the direct absorption of radiation, and a bottom potentially unstable boundary layer due to the absorption and re-emission of the residual radiation by the bottom boundary. The growth of the competing thermal layers and the stability properties of the bottom thermal boundary layer are investigated by means of a scaling analysis and a direct stability analysis based on three-dimensional numerical simulation. The scaling relations are benchmarked against the numerical simulation, and the effects of the normalised water depth, the Rayleigh number and the box aspect ratio on the thermal instability are studied.

Theoretical and numerical study of targeted energy transfer inside an acoustic cavity by a non-linear membrane absorber

This paper investigates the Targeted Energy Transfer (TET) between an acoustic medium inside a parallelepiped cavity and a thin viscoelastic membrane that is mounted on one wall of the cavity and that is working as a Non-linear Energy Sink (NES). Applying a harmonic source excitation inside the cavity, we look for reducing the resonance peaks of the acoustic cavity by mean of the so-called “Strongly Modulated Response” (SMR). Using a single term harmonic balance approach, we determine analytical simplified expressions for the non-linear normal modes (NNMs) and the periodic forced responses of this coupled system. We define the “desired working zone” for the NES as the forcing level interval between the first destabilization of the resonance peak (TET beginning) and the appearance of an additional branch of periodic regimes. An analytical formula is derived for the beginning of TET while the appearance of the undesired periodic regimes is obtained numerically. These results prove to be useful for the practical design of NES to be mounted on an acoustic cavity.