Clet Number Research Articles

IMPACT OF AN ANISOTROPIC POROUS MEDIA ON THERMOBIOCONVECTION INSTABILITY IN THE PRESENCE OF GYROTACTICMICROORGANISMS AND HEATING FROM BELOW

The onset of thermobioconvection in a horizontal fluid layer saturated by gyrotactic microorganisms into an anisotropic porous medium is examined. The modeling of the governing equations considers heating from below, Darcy flow, and Boussinesq approximations along with the presence of gyrotactic microorganisms. The system of ordinary differential equations is obtained using linear stability analysis and the normal mode technique. The single-term Galerkin method casts the analytical solutions while the higher-order Galerkin technique is employed to compute the numerical solutions. The influence of the mechanical and thermal anisotropy parameters along with all bioconvection parameters on the onset of thermobioconvection are analytically as well as numerically discussed. It is perceived that bioconvection Rayleigh-Darcy number <i>Rb</i>, the gyrotactic number<i> G</i>, and Péclet number <i>Q</i> are to fast-forward the onset of convective motion. On the other hand, thermal anisotropy produces a stable system and acts to postpone the bioconvection pattern formulation. Mechanical anisotropy is found to have a destabilizing impact on the stability of the suspension and helps in the development of bioconvection. The enhancement of mechanical and thermal anisotropy is found to increase the size of bioconvective cells whereas the augmentation of the concentration of gyrotactic specie reduces the convective cell size.

ANALYSIS OF THE HEAT TRANSFER BY LAMINAR FLOW IN THERMAL ENTRANCE REGION OF A CONDUIT WITH WAVY SURFACE SUBJECTED TO A UNIFORM WALL HEAT FLUX

Thermal developing laminar Poiseuille flow via a conduit with wavy surface is studied taking plane and cylindrical geometries as a case in point. A transformation procedure from the original system with wavy walls heated by a type of heat change to the one of straight walls subjected to periodic heat flux of the same period is proposed. Then, by the functional analysis method the solution in thermal entry area is obtained with regard to uniform inlet temperature and neglecting both viscous dissipation and the axial conduction. Mean and local Nusselt numbers from Nusselt as well as temperature in the entrance region are obtained for a conduit with small amplitude of the wavy walls. The principal results presented show that the solution regarding the two geometries has the same form and describe completely the original problem, and the optimization of the system for walls with low amplitude corresponds to large dimensionless pulsation for great values of Péclet number.

NONLINEAR FORCED CONVECTIVE HEAT TRANSFER IN A COMPOSITE AIR/POROUS LAYER WITH PERMEABLE TOP AND BOTTOM BOUNDARIES AND ENERGY SOURCE: DIFFERENT PENETRATION DEPTH OF PATTERNS

This paper deals with nonlinear convective heat transfer in a composite heat-generating air/porous system bounded by the top and bottom solid permeable planes of equal temperature and forced by a vertical throughflow. In the presence of a uniform energy source, penetrative convection originates over this throughflow. It is numerically simulated by the finite difference method and Newton's method. The paper extends previous studies by considering the evolution of the supercritical and subcritical nonlinear convective patterns of a different penetration depth, increasing the supercriticality <i>r</i> from 0 to 2, Pèclet number Pefrom - 6 to 5, and depth ratio <i>d</i> from 0.05 to 0.20. The convection together with the basic throughflow contributes to the total heat transfer rate and improves thermal performance of the partial porous system. The case can find its application, for example, in storing the biologically active plant products releasing heat. The short-wave convection localized in the upper air layer appears over the upward throughflow with Pe > 0. Its relative contribution to the total heat flux is much smaller than that of the basic throughflow and decreases with increases in the values of Pe and <i>d</i>. Both local and large-scale convective patterns can initiate over the downward throughflow with Pe < 0. Their relative contributions to the total heat flux grow with increasing the absolute values of Pe and <i>d</i>. The large-scale convection that covers the air and porous layers at Pe = -6, <i>d </i>= 0.05 enhances heat transfer more effectively than does the local one occurring within the air layer.

A novel strategy to enhance the cooling effectiveness in a confined porous jet impingement using non-Newtonian fluids

Jet impingement configuration is commonly employed in the thermal management of the gas turbine engines, rocket launchers, and high-density electrical equipment cooling to remove a large amount of heat. Owing to such overwhelming applications, significant research efforts have been devoted to enhance the cooling performance of jet impingement configuration. In this work, the influence of shear-thinning behaviour in overall heat transfer characteristics have been investigated as a function of pertinent dimensionless numbers as 10−6 ≤ Da ≤ 10−2 (Darcy number), 1 ≤ Pe ≤ 102 (Péclet number), 0 ≤ Ri ≤ 4 (Richardson number), 0.4 ≤ n ≤ 1 (shear-thinning index). The numerical results on local thermal non-equilibrium (LTNE) parameter, streamlines and isotherms, local and average Nusselt number are reported for a range of power-law index, Péclet number and Richardson number. All in all, shear-thinning fluids offer 50–60% enhancement in Nusselt number with reference to the Newtonian fluid at a higher Darcy number. A correlation for average Nusselt number is put forward to enable its use for process design calculations.

Numerical Analysis of Nonlocal Convection—Comparison with Three-dimensional Numerical Simulations of Efficient Turbulent Convection

We compare 1D nonlocal turbulent convection models with 3D hydrodynamic numerical simulations. We study the validity of closure models and turbulent coefficients by varying the Prandtl number, the P$\acute{e}$clet number, and the depth of the convection zone. Four closure models of the fourth-order moments are evaluated with the 3D simulation data. The performance of the closure models varies among different cases and different fourth-order moments. We solve the dynamic equations of moments together with equations of the thermal structure. Unfortunately, we cannot obtain steady-state solutions when these closure models of fourth-order moments are adopted. The numerical solutions of the down-gradient approximations of the third-order moments, on the other hand, are robust. We calibrate the coefficients of the 1D down-gradient model from the 3D simulation data. The calibrated coefficients are more robust in the cases of deep convection zones. Finally we have compared the 1D steady-state solutions with the 3D simulation results. The 1D model has captured many features appearing in the 3D simulations : (1) $\nabla-\nabla_{a}$ has a U-shape with a minimum value at the lower part of the convection zone. (2) There exists a bump for $\nabla-\nabla_{a}$ near the top of the convection zone when the P$\acute{e}$clet number is large. (3) The temperature gradient can be sub-adiabatic due to the nonlocal effect. Apart from these similarities, the prediction on the kinetic energy flux, however, is unsatisfactory.

Hydraulic dispersion of diurnal reactive constituents in an open channel eutrophic flow

Summary Closely related to the solar photocycle, plankton growth in eutrophic waters displays a diurnal variation because of photosynthesis and respiration. Presented in this paper is an analytical study of the diurnal variation of mean concentration of plankton and nutrient in an open channel eutrophic flow initiated by an instantaneous emission. The evolution of the concentration is shown driven by the combination of hydraulic dispersion and diurnal reaction between plankton and nutrient. The analytical solution for longitudinal distribution of concentration is rigorously derived and illustrated, based on the time dependent hydraulic dispersivity. Numerical results are presented and characterized by the reaction rate, yield factor and period for the diurnal reaction and the P e clet number of the flow. For typical applications such as ecological risk assessment and environmental impact assessment, the upper and lower limits of critical length and duration of five typical pollutant concentrations are concretely illustrated for given concentration criterions. Remarkable diurnal variations are revealed up to around one third in the critical length and duration for plankton, and about ten percent for nutrient.

Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell.

The influence of fluid dispersion on the Saffman-Taylor instability in miscible fluids has been investigated in both the linear and the nonlinear regimes. The convective characteristic scales are used for the dimensionless formulation that incorporates the Péclet number (Pe) into the governing equations as a measure for the fluid dispersion. A linear stability analysis (LSA) is performed in a similarity transformation domain using the quasi-steady-state approximation. LSA results confirm that a flow with a large Pe has a higher growth rate than a flow with a small Pe. The critical Péclet number (Pec) for the onset of instability for all possible wave numbers and also a power-law relation of the onset time and most unstable wave number with Pe are observed. Unlike the radial source flow, Pec is found to vary with t0. A Fourier spectral method is used for direct numerical simulations (DNS) of the fully nonlinear system. The power-law dependence of the onset of instability ton on Pe is obtained from the DNS and found to be inversely proportional to Pe and it is in good agreement with that obtained from the LSA. The influence of the anisotropic dispersion is analyzed in both the linear and the nonlinear regimes. The results obtained confirm that for a weak transverse dispersion merging happens slowly and hence the small wave perturbations become unstable. We also observ that the onset of instability sets in early when the transverse dispersion is weak and varies with the anisotropic dispersion coefficient, ε, as ∼√[ε], in compliance with the LSA. The combined effect of the Korteweg stress and Pe in the linear regime is pursued. It is observed that, depending on various flow parameters, a fluid system with a larger Pe exhibits a lower instantaneous growth rate than a system with a smaller Pe, which is contrary to the results when such stresses are absent.

Shear-accelerated crystallization in a supercooled atomic liquid.

A bulk metallic glass forming alloy is subjected to shear flow in its supercooled state by compression of a short rod to produce a flat disk. The resulting material exhibits enhanced crystallization kinetics during isothermal annealing as reflected in the decrease of the crystallization time relative to the nondeformed case. The transition from quiescent to shear-accelerated crystallization is linked to strain accumulated during shear flow above a critical shear rate γ̇(c)≈0.3 s(-1) which corresponds to Péclet number, Pe∼O(1). The observation of shear-accelerated crystallization in an atomic system at modest shear rates is uncommon. It is made possible here by the substantial viscosity of the supercooled liquid which increases strongly with temperature in the approach to the glass transition. We may therefore anticipate the encounter of nontrivial shear-related effects during thermoplastic deformation of similar systems.

Flow Characterization of Vapor Phase of Geothermal Fluid in Pipe Using Isotope <sup>85</sup>Kr and Residence Time Distribution Modeling

Measurement of vapor flow in geothermal pipe faces great challenges due to fast fluids flow in high-temperature and high-pressure environment. In present study the flow rate measurement has been performed to characterization the geothermal vapor flow in a pipe. The experiment was carried out in a pipe which is connected to a geothermal production well, KMJ-14. The pipe has a 10 ” outside diameter and contains dry vapor at a pressure of 8 kg/cm 2 and a temperature of 170 o C. Krypton-85 gas isotope ( 85 Kr) has been injected into the pipe. Three collimated radiation detectors positioned respectively at 127, 177 and 227 m from injection point were used to obtain experimental data which represent radiotracer residence time distribution (RTD) in the pipe. The last detector at the position of 227 m did not respond, which might be due to problems in cable connections. Flow properties calculated using mean residence time (MRT) shows that the flow rate of the vapor in pipe is 10.98 m/s, much faster than fluid flow commonly found in various industrial process plants. Best fitting evaluated using dedicated software developed by IAEA expert obtained the P e clet number Pe as 223. This means that the flow of vapor of geothermal fluids in pipe is plug flow in character. The molecular diffusion coefficient is 0 . 45 m 2 /s, calculated from the axial dispersion model. Received: 21 November 2013; Revised: 16 May 2014; Accepted: 23 May 2014

Scaling of heat flux and energy spectrum for very large Prandtl number convection.

Under the limit of infinite Prandtl number, we derive analytical expressions for the large-scale quantities, e.g., Péclet number Pe, Nusselt number Nu, and rms value of the temperature fluctuations θ(rms). We complement the analytical work with direct numerical simulations, and show that Nu ∼ Ra(γ) with γ ≈ (0.30-0.32), Pe ∼ Ra(η) with η ≈ (0.57-0.61), and θ(rms) ∼ const. The Nusselt number is observed to be an intricate function of Pe, θ(rms), and a correlation function between the vertical velocity and temperature. Using the scaling of large-scale fields, we show that the energy spectrum E(u)(k) ∼ k(-13/3), which is in a very good agreement with our numerical results. The entropy spectrum E(θ)(k), however, exhibits dual branches consisting of k(-2) and k(0) spectra; the k(-2) branch corresponds to the Fourier modes θ[over ̂](0,0,2n), which are approximately -1/(2 nπ). The scaling relations for Prandtl number beyond 10(2) match with those for infinite Prandtl number.

Simulation of a Microfluidic Gradient Generator using Lattice Boltzmann Methods

Microfluidics provides a powerful and versatile technology to accurately control spatial and temporal conditions for cell culturing and can therefore be used to study cellular responses to gradients. Here we use Lattice Boltzmann methods (LBM) to solve both the Navier-Stokes equation (NSE) for the fluid and the coupled convection-diffusion equation (CDE) for the compounds that form the diffusion-based gradient. The design of a microfluidic chamber for diffusion-based gradients must avoid flow through the cell chamber. This can be achieved by alternately opening the source and the sink channels. The fast toggling of microfluidic valves requires switching between different boundary conditions. We demonstrate that the LBM is a powerful method for handling complex geometries, high PeIclet number conditions, discontinuities in the boundary conditions, and multiphysics coupling.

Reliability of fluctuation-induced transport in a Maxwell-demon-type engine

We study the transport properties of an overdamped Brownian particle which is simultaneously in contact with two thermal baths. The first bath is modeled by an additive thermal noise at temperature $T_A$. The second bath is associated with a multiplicative thermal noise at temperature $T_B$. The analytical expressions for the particle velocity and diffusion constant are derived for this system, and the reliability or coherence of transport is analyzed by means of their ratio in terms of a dimensionless P\'{e}clet number. We find that the transport is not very coherent, though one can get significantly higher currents.

Modelling of dendritic growth in forced and natural convections

A two-dimensional lattice Boltzmann method （LBM） based model is developed for the modelling of dendritic growth during alloy solidification in the presence of forced and natural convections. Instead of conventional continuum-based Navier-Stokes solvers，the present model adopts a kinetic-based LBM for the numerical computations of transport phenomena during solidification. Three sets of distribution functions are employed to constitute the LBM evolution equations for numerically calculating fluid flow as well as solutal and thermal transports which are controlled by both diffusion and convection. By solving the LBM evolution equations， the local temperature and the composition at the solid/liquid interface can be obtained. The kinetics of dendritic growth is then investigated based on a solutal equilibrium approach proposed by Zhu and Stefanescu. The model is verified by the comparison between the simulations and theoretical predictions. The simulated upstream tip velocities and radii of the dendrite growing in a melt with natural convections are found to be in reasonabl agreement with the predictions from the modified Lipton-Glicksman-Kurz model that takes into account the effects of convection. For the convective dendritic growth in a forced flow， the simulated growth Pclet number of the upstream tip as a function of the flow Pclet number is very close to the Oseen-Ivantsov solution. It is also found that convection transports heat and solute from the upstream region to the downstream region， producing asymmetrical dendrite that grows faster in the upstream direction， whereas slower in the downstream direction.

Modelling of solutal dendritic growth in three dimensions

A three-dimensional （3D） model for the simulation of solutal dendritic growth in the low Pclet number region is presented. In the model is adopted a solutal equilibrium approach previously proposed by Zhu and Stefanescu to calculate the evolution of the solid/liquid（SL） interface， which allows the accurate simulation of dendritic growth from the initial unstable stage to the steady-state stage with a good computational efficiency. In this approach， the kinetics of dendritic growth is driven by the difference between the local equilibrium composition， calculated from the local temperature and curvature， and the local actual liquid composition，obtained by solving the solutal transport equation. To describe the specific crystallographic orientation of 3D dendritic growth， a weighted mean curvature （WMC） algorithm， which is incorporated with the anisotropy of surface tension， is proposed to calculate the local SL interface curvature. This approach is convenient to be implemented and to make the transformation for WMC calculation from two to three dimensions. The model is verified by the comparison of our numerical resuets with the analytical ones. The simulated steady-state tip velocity and radius varying with the degree of undercooling of an Al-2wt%Cu alloy are found to be close to the ones predicted by the Lipton-Glicksman-Kurz analytical model. The steady-state morphology of the needle dendrite tip is analyzed. It is found that the tip is nonaxisymmetric and deviates from a paraboloid in the manner of the fourfold symmetry. Finally， the simulated 3D multi-equiaxed dendrites with various crystallographic orientations are presented.

Phase velocity and phase diffusion in periodically driven discrete-state systems

We develop a theory to calculate the effective phase diffusion coefficient and the mean phase velocity in periodically driven stochastic models with two discrete states. This theory is applied to a dichotomically driven Markovian two-state system. Explicit expressions for the mean phase velocity, the effective phase diffusion coefficient, and the Pe clet number are analytically calculated. The latter indicates as a measure of phase-coherence forced synchronization of the stochastic system with respect to the periodic driving and exhibits a "bona fide" resonance. In a second step, the theory is applied to a non-Markovian two-state system modeling excitable systems. The results prove again stochastic synchronization to the periodic driving and are in good agreement with simulations of a stochastic FitzHugh-Nagumo system.

Heat transfer from a sphere immersed in a stream of an inviscid fluid at small P�clet number

Forced heat convection from a sphere at low Peclet number is studied using the method of matched asymptotic expansions. An inviscid theory is applied for the velocity field. The results obtained in this paper may be applicable to the flow of low Prandtl number fluids.

The Theory of Dispersal During Laminar Flow in Tubes. II.

The solution of the problem introduced in Part I is found for the case of large diffusion P�clet number (Y). The work is carried out principally in the third approximation of the Galerkin method, and the results are presented in some detail.