Experimental investigations of steady and oscillating annular liquid curtains with different surface tensions

Experiments were performed on laminar, vertical, annular, liquid curtains to study the dynamics of steady curtains, and the onset and frequency of oscillating curtains. The experiments were conducted to observe the effects of inertia and pressure on liquid curtains with different surface tensions. For steady curtains, convergence lengths were measured as functions of Froude number and pressure differential for three different surface tensions. The factors causing the onset of oscillations in a pressurized curtain were observed and the frequency of the internal pressure fluctuations were measured for various Froude numbers and two surface tensions.List of symbols b local thickness of curtain sheet - b ₀ initial thickness of curtain or nozzle gap thickness (0.5 mm) - C _P pressure coefficient - Fr Froude number (V ₀ ² /g R ₀) - g gravitational acceleration - g gravitational acceleration - L convergence length of curtain - L ^* dimensionless convergence length (L/R ₀) - N _c convergence number (g ² R ₀ ² b ₀ /2v ₀ ² ) - P _e pressure outside the curtain (ambient) - P _i pressure inside the curtain - P pressure differential (P _i– P _e) - P _cr pressure differential at which curtain begins to oscillate - R local radius of curvature in the horizontal plane - R ₀ initial curtain radius or radius of nozzle exit (50 mm) - r _v local radius of curvature in the vertical plane - V local liquid velocity - V ₀ initial liquid velocity - V ^* dimensionless local liquid velocity (V/V ₀) - z axial distance from the nozzle - z ^* dimensionless axial distance from the nozzel (z/R ₀) - s differential length of curtain - differential angle in the horizontal plane - angle between the direction of the surface tension force in the vertical plane and the direction of r _v - deangle between the direction of the surface tension force in the horizontal plane and the direction of R - angle between r _vand R in the vertical plane - ₀ nozzle exit angle (zero degrees) - surface tension of liquid - liquid density (1.0 gm/cm³)     

The load-bearing capacity of a continuous-flow squeeze film of liquid

A new form of squeeze film system is described in which the movement of one plate towards the other is simulated by the continuous volume generation of liquid over the plate area. The liquid exudes from 1580 holes distributed uniformly over the lower plate surface. An advantage of the system is that there are no moving parts, but it is important to evaluate the device using Newtonian liquids in order to compare the load bearing capacity with that predicted by equations developed for orthodox squeeze film systems. Liquid maldistribution is shown to be a problem which may be solved in various ways, one of which is to ensure that the pressure drop through the plate is high relative to that in the squeeze film.Results obtained using Newtonian liquids make satisfactory comparison with theoretical predictions, though liquid inertia probably makes a lower contribution to load bearing than is the case for an orthodox squeeze film. Liquid maldistribution is allowed for on a theoretical basis or corrected by the use of a distributor plate placed below the perforated surface.Preliminary tests using viscoelastic solutions (based on polyacrylamide of high molecular weight) suggest that the load bearing properties of the squeeze film are significantly enhanced. A load 600 per cent greater than the theoretical load is obtained in one case, the suggestion being made that this is due to stress of viscoelastic origin.Nomenclature D Exit diameter of holes in spinnerette - F ₁ to F ₆ Vertical force on top plate due to flow in squeeze film, defined by (1), (8), (11), (12), (13) and (14) respectively - h Plate separation - h _L Distance of distributor plate from lower surface of spinnerette (function of r) - I ₀ Modified Bessel function of first kind, order 0 - I ₁ Modified Bessel function of first kind, order 1 - K ₀ Modified Bessel function of second kind, order 0 - L Length of hole, based on diameter D, giving same pressure drop as actual spinnerette holes - dm/dt Mass flowrate of liquid - N Total number of holes in spinnerette (1580) - p Isotropic pressure in squeeze film - P _RES Isotropic pressure in reservoir behind lower plate of spinnerette - p–P _RES - (dp/dr)_s Pressure gradient in squeeze film - (dp/dr)_L Pressure gradient in lower film below spinnerette when distributor plate is used - Q Total liquid volume flowrate - q _s Volume flowrate through squeeze film at radius r - q _L Volume flowrate through lower film at radius r - r radial coordinate - R radius of upper disc - - v Velocity of upper disc relative to lower one (simulated by Q/R ² in continuous flow system) - V _R Average radial liquid velocity at radius R - V _S Liquid exit velocity from single hole - V _r V V _z Point velocity components in r, and z directions respectively - z Axial coordinate - Parameter in (8) (3ND ⁴/32LR ² h ³) - Viscosity of liquid - Density of liquid - _rz Shear stress     

Measurements of heat transfer coefficients between turbulent liquid jets and a radially finned heated disk placed at the bottom of an open vertical circular cavity

 Experiments have been performed to assess the impact of an extended surface on the heat transfer enhancement for axisymmetric, turbulent liquid jet impingement on a heated round disk. The disk, with an array of integral radial fins mounted on its surface, is placed at the bottom of an open vertical circular cavity. Hydrodynamic and heat transfer data were obtained for a dielectric fluorocarbon liquid FC-77. For a fixed circular heater of diameter D=22.23 mm, several geometric parameters were tested: the nozzle diameter (4.42≤d≤9.27 mm), the confining wall diameter of the vertical cavity (22.23≤D _c≤30.16 mm), and the nozzle-to-heater spacing (0.5≤S/d≤5.0). The FC-77 flow rates varied from =0.2 to 11.0 l/min producing Reynolds numbers in the wide interval 700≤Re _d ≤44,000. For d=4.42 mm, the heat transfer response to the separation distance S/d was small but increased gradually with increasing nozzle diameter up to d=9.27 mm. The thermal resistance R _th increased with the confining wall diameter D _c and also with the nozzle diameter d. A minimum value of the thermal resistance of R _th,min=0.4 cm² K/W was attained for a combination of d=4.42 mm, D _c=22.23 mm, S/d=1, and =7.5 l/min. Based on a simplified heat transfer model, reasonable agreement was obtained between measured values of the thermal resistance and the R _th-predictions. The total fin effectiveness ɛ_f was shown to increase with increasing nozzle diameter, but was invariant with the flow rate (or the jet exit velocity). More than a three-fold heat transfer enhancement was realized through the addition of the array of integral radial fins on the heated round disk. Received on 30 August 2000 / Published online: 29 November 2001     

The hydraulic jump in circular jet impingement and in other thin liquid films

The circular hydraulic jump exhibits behavior quite different from that commonly observed in planar jumps. Here we examine experimentally some of the causes and consequences of those differences. We suggest that surface tension plays a dominant role in establishing the shape of the circular jump for impinging jets. The importance of surface tension is a direct result of the thinness of the liquid films normally encountered in circular jump configurations. A sequence of instabilities appears in the jump's structure as the subcritical liquid film becomes thicker and surface tension effects decrease. These conclusions are corroborated by experiments on thin planar films which result in unusual jump structures, like those seen in circular jumps. In addition, we show that the standard momentum balance for the circular jump is effective only at relatively low supercritical Froude numbers or at low ratios of downstream to upstream depth. Typical values of those parameters for circular jumps are often quite large relative to the usual values for planar open-channel flows.List of Symbols d jet diameter - D fictitious downstream drag force - Fr _d jet Froude number, u _f / gd - Fr _h supercritical film Froude number, u _f d ²/8r _j gh ³ - Fr _s subcritical film Froude number, u _f d ²/8r _s gs ³ - g gravitational body force - h local thickness of liquid sheet - p hydrostatic pressure - r radius measured from jet stagnation point - r _j radius at which hydraulic jump begins - r _s radius at which subcritical depth equals s - R radius of curvature of jump interface - Re _d Reynolds number of the jet, u _f d/v. - s liquid sheet thickness after hydraulic jump - u(r,y) radial velocity distribution in liquid film - u _f velocity of impinging jet - _h depth average velocity for sheet of thickness h, u _f d ²/8rh - y distance normal to the wall - We Weber number of jump, s pg/ Greek letters v liquid kinematic viscosity - liquid density - surface tension     

Magnetic resonance imaging of velocity fields, the void fraction and gas dynamics in a cavitating liquid

In acoustic cavitation, the relationship between the bubble dynamics on the microscale and the flow properties on the macroscale is critical in determining sonochemical reaction kinetics. A new technique was developed to measure the void fraction and estimate water mobility in the vicinity of cavitating bubbles using phase-encoded magnetic resonance imaging with short characteristic measurement timescales (0.1–1 ms). The exponential behavior of the NMR signal decay indicated the fast diffusion regime, with the relationship between local mechanical dispersion D _mix and the average bubble radius R, D_mix >> \frac2R²10^-4s, D_{\rm mix}\gg \frac{2R^2}{10^{-4}\hbox{s}}, resulting in dispersion of orders of magnitude greater than diffusion in quiescent water. For two different samples (water and a surfactant solution), the independent measurements of three-dimensional void fraction and velocity fields permitted the calculation of compressibility, divergence and vorticity of the cavitating medium. The measured dynamics of the dissolved gas, compared with that of the surrounding liquid, reflected the difference in the bubble coalescence and lifetimes and correlated with the macroscopic flow parameters.     

Numerical analysis of the rotating viscous flow approaching a solid sphere

A numerical simulation is performed to investigate the flow induced by a sphere moving along the axis of a rotating cylindrical container filled with the viscous fluid. Three‐dimensional incompressible Navier–Stokes equations are solved using a finite element method. The objective of this study is to examine the feature of waves generated by the Coriolis force at moderate Rossby numbers and that to what extent the Taylor–Proudman theorem is valid for the viscous rotating flow at small Rossby number and large Reynolds number. Calculations have been undertaken at the Rossby numbers (R_o) of 1 and 0.02 and the Reynolds numbers (R_e) of 200 and 500. When R_o=O(1), inertia waves are exhibited in the rotating flow past a sphere. The effects of the Reynolds number and the ratio of the radius of the sphere and that of the rotating cylinder on the flow structure are examined. When R_o ? 1, as predicted by the Taylor–Proudman theorem for inviscid flow, the so‐called ‘Taylor column’ is also generated in the viscous fluid flow after an evolutionary course of vortical flow structures. The initial evolution and final formation of the ‘Taylor column’ are exhibited. According to the present calculation, it has been verified that major theoretical statement about the rotating flow of the inviscid fluid may still approximately predict the rotating flow structure of the viscous fluid in a certain regime of the Reynolds number. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical study of convection of a liquid heated from below

The equilibrium of a liquid heated from below is stable only for small values of the vertical temperature gradient. With increase of the temperature gradient a critical equilibrium situation occurs, as a result of which convection develops. If the liquid fills a closed cavity, then there is a discrete sequence of critical temperature gradients (Rayleigh numbers) for which the equilibrium loses stability with respect to small characteristic disturbances. This sequence of critical gradients and motions may be found from the solution of the linear problem of equilibrium stability relative to small disturbances. If the temperature gradient exceeds the lower critical value, then (for steady-state heating conditions) there is established in the liquid a steady convective motion of a definite amplitude which depends on the magnitude of the temperature gradient. Naturally, the amplitude of the steady convective motion cannot be determined from linear stability theory; to find this amplitude we must solve the problem of convection with heating from below in the nonlinear formulation. A nonlinear study of the steady motion of a liquid in a closed cavity with heating from below was made in [1]. In that study it was shown that for Rayleigh numbers R which are less than the lower critical value R_c steady-state motions of the liquid are not possible. With R>R_c a steady convection arises, whose amplitude near the threshold is small and proportional to (R–R_c)^1/2 (the so-called soft instability)-this is in complete agreement with the results of the phenom-enological theory of Landau [2, 3].Primarily, various versions of the method of expansion in powers of the amplitude [4–8] have been used, and, consequently, the results obtained in those studies are valid only for values of R which are close to R_c, i. e., near the convection threshold.It is apparent that the study of developed convective motion far from the threshold can be carried out only numerically, with the use of digital computers. In [9, 10] the numerical methods have been successfully used for the study of developed convection in an infinite plane horizontal liquid layer.The present paper undertakes the numerical study of plane convective motions of a liquid in a closed cavity of square section. The complete nonlinear system of convection equations is solved by the method of finite differences on a digital computer for various values of the Rayleigh number, the maximal value exceeding by a factor of 40 the minimal critical value R_c. The numerical solution permits following the development of the steady motion which arises with R>R_c in the course of increase of the Rayleigh number and permits study of the oscillatory motions which occur at some value of the parameter R. The heat transfer through the cavity is studied. The corresponding linear problem on equilibrium stability is solved approximately by the Galerkin method.     

Numerical studies of the fingering phenomena for the thin film equation

We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is h_t + (h²?h³)_x = ??·(h³?Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h²?h³)_x, we use an implicit essentially non‐oscillatory (ENO)‐type scheme and get a good stability property. For the diffusion part ??·(h³?Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena. Copyright © 2010 John Wiley & Sons, Ltd.     

Experimental investigation of the creeping motion of a drop in a vertical tube

New experimental data regarding the motion of a drop along the axis of a vertical tube, filled with another highly viscous liquid, are obtained. The experiments are realised with sufficiently large drops for an internal circulation to develop and also for different pairs of fluids; the preponderant role of the gravity on the drop shape and consequently on its terminal velocity is pointed out. Moreover, by means of a visualization technique, details on the flow both inside and outside the drop are given.List of symbols g gravity acceleration - r distance from the drop center - R equivalent radius of the drop, i.e. the radius of the sphere having the same volume as the drop - R _EQ radius of the equatorial section of the drop - R _T tube radius - L _AX half length of the drop - U ₀ terminal velocity of the drop - P _s Poiseuille number= U _{0 e} /4 g R ² - Fr Fronde number = U ₀ ² _e /2 g R - Re Reynolds number = 2 U ₀ R _e / _e - E _o Eötvös number = 4g R ²/ - deformation parameter = _e U ₀/ - apparent density of the suspended liquid= | _i – _e | - _i viscosity of the suspended liquid - _e viscosity of the suspending liquid - drop-to-tube radius ratio = R/R _T - _EQ equatorial drop-to-tube radius ratio = R EQ/R _T - interfacial tension     

Numerical and Experimental Investigation of Liquid Residue in 90° Bend Microchannel Based on Gas–Liquid Two-Phase Flow

The microfluidic chip for nucleic acid detection in vitro is an essential application of microfluidic technology to the process of in vitro diagnosis. The 90° bend microchannels in chip designed for facilitating assay reagent delivery may cause reagent residues and cast mutual contamination between detection reagents, which significantly affects the detection accuracy. In this paper, a two-dimensional gas–liquid two-phase flow model is constructed to simulate the liquid residue phenomenon. Using the results of simulation, the residual liquid generation can be observed and the area of residual liquid can be obtained. The accuracy of the numerical simulation is verified by comparison with the experimental results. The effects of the fillet radius R, the diameter ratio d₁/d₂ of the vertical to horizontal sections, the flow velocity v, and the surface roughness R_a on the residual amount are studied. We find that the fillet radius is inversely proportional to the residual amount within the range v = 20–100 mm/s and there is almost no liquid residue in the channel when the radius increases to R = 1 mm. When the channel diameter ratio d₁/d₂ increases, the liquid residual amount also increases by approximately 98%. The increased surface roughness R_a significantly increases the residual amount. The results of this study provide a reference for the optimal design of microchannels on chips.

     

Wall visualization of swirling decaying flow using a dot-paint method

This paper deals with the visualization of swirling decaying flow in an annular cell fitted with a tangential inlet. A wall visualization method, the so-called dot-paint method, allows the determination of the flow direction on both cylinders of the cell. This study showed the complex structure of the flow field just downstream of the inlet, where a recirculation zone exists, the effects of which are more sensitive on the inner cylinder. The flow structure can be considered as three-dimensional in the whole entrance section. The swirl number and the entrance length were estimated using the measured angle of the streamlines. Experimental correlations of these two parameters, taking into account the Reynolds number and the axial distance from the tangential inlet, are given.List of symbols e = R ₂ – R ₁ thickness of the annular gap (m) - L _ax entrance length of axial flow on the outer cylinder (m) - L _ti length of the three-dimensional flow region on the inner cylinder (m) - L _to length of the three-dimensional flow region on the outer cylinder (m) - Q _v volumetric flowrate in the annular cell (m³s^–) - r radial position (m) - R ₁ external radius of the inner cylinder (m) - R ₂ internal radius of the outer cylinder (m) - Re=2eU _m /v Reynolds number - Sn swirl number - T time average resulting velocity (m s^–) - u time average axial velocity component (ms ^–) - average velocity in the annulus (m s^–) - w time average tangential velocity component (m s^–) - x axial location from the tangential inlet (m) - _e diameter of the tangential inlet (m) - streak angle with respect to the horizontal (degree) - angle with respect to the tangential inlet axis (degree) - gn kinematic viscosity of the working liquid (m²s^–)     

Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data

We investigate the well-posedness of (1) the heat flow of harmonic maps from \mathbb Rⁿ{\mathbb R^n} to a compact Riemannian manifold N without boundary for initial data in BMO; and (2) the hydrodynamic flow (u, d) of nematic liquid crystals on \mathbb Rⁿ{\mathbb R^n} for initial data in BMO⁻¹ × BMO.     

Flow past a square cylinder at low Reynolds numbers

Results are presented for the flow past a stationary square cylinder at zero incidence for Reynolds number, Re ? 150. A stabilized finite‐element formulation is employed to discretize the equations of incompressible fluid flow in two‐dimensions. For the first time, values of the laminar separation Reynolds number, Re_s, and separation angle, θ_s, at Re_s are predicted. Also, the variation of θ_s with Re is presented. It is found that the steady separation initiates at Re = 1.15. Contrary to the popular belief that separation originates at the rear sharp corners, it is found to originate from the base point, i.e. θ_s=180^° at Re = Re_s. For Re > 5, θ_s approaches the limit of 135 ^°. The length of the separation bubble increases approximately linearly with increasing Re. The drag coefficient varies as Re^?0.66. Flow characteristics at Re ? 40 are also presented for elliptical cylinders of aspect ratios 0.2, 0.5, 0.8 and 1 (circle) having the same characteristic dimension as the square and major axis oriented normal to the free‐stream. Compared with a circular cylinder, the flow separates at a much lower Re from a square cylinder leading to the formation of a bigger wake (larger bubble length and width). Consequently, at a given Re, the drag on a square cylinder is more than the drag of a circular cylinder. This suggests that a cylinder with square section is more bluff than the one with circular section. Among all the cylinder shapes studied, the square cylinder with sharp corners generates the largest amount of drag. Copyright © 2010 John Wiley & Sons, Ltd.     

A spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for conjugate heat transfer applications

We present a spectral‐element discontinuous Galerkin thermal lattice Boltzmann method for fluid–solid conjugate heat transfer applications. Using the discrete Boltzmann equation, we propose a numerical scheme for conjugate heat transfer applications on unstructured, non‐uniform grids. We employ a double‐distribution thermal lattice Boltzmann model to resolve flows with variable Prandtl (Pr) number. Based upon its finite element heritage, the spectral‐element discontinuous Galerkin discretization provides an effective means to model and investigate thermal transport in applications with complex geometries. Our solutions are represented by the tensor product basis of the one‐dimensional Legendre–Lagrange interpolation polynomials. A high‐order discretization is employed on body‐conforming hexahedral elements with Gauss–Lobatto–Legendre quadrature nodes. Thermal and hydrodynamic bounce‐back boundary conditions are imposed via the numerical flux formulation that arises because of the discontinuous Galerkin approach. As a result, our scheme does not require tedious extrapolation at the boundaries, which may cause loss of mass conservation. We compare solutions of the proposed scheme with an analytical solution for a solid–solid conjugate heat transfer problem in a 2D annulus and illustrate the capture of temperature continuities across interfaces for conductivity ratio γ > 1. We also investigate the effect of Reynolds (Re) and Grashof (Gr) number on the conjugate heat transfer between a heat‐generating solid and a surrounding fluid. Steady‐state results are presented for Re = 5?40 and Gr = 10⁵?10⁶. In each case, we discuss the effect of Re and Gr on the heat flux (i.e. Nusselt number Nu) at the fluid–solid interface. Our results are validated against previous studies that employ finite‐difference and continuous spectral‐element methods to solve the Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

A similarity transformation for subcooled mixed convection film boiling in a porous medium

The classical two-phase boundary layer concept was adopted to analyse the problem of combined free and forced convection film boiling over an impermeable wall embedded in a porous medium. The governing equations, namely, the equations of continuity, energy and Darcy's model, were written for both vapor and liquid layers, and were solved simultaneously by means of the similarity transformation. Similarity solutions are found for a vertical flat plate, a horizontal circular cylinder and a sphere. Numerical integrations were carried out for given sets of the parameters associated with the vapor superheating Sup, the liquid subcooling Sub, the vapor mass flow rate R and the ratio of the liquid Rayleigh number to Peclet number Ra _xf /Pe _xf . It is found that a significant simplification of the problem is possible by setting the liquid stream function at the interface to zero. This simplification also reveals that the solution of the problem virtually depends on only three parameters, namely, Sup, Ra _xf /Pe _xf and the lumped parameter Sub/R.     

Heat transfer by laminar flow of an elastico-viscous liquid in a circular cylinder with linearly varying wall temperature

Summary The heat transfer by laminar flow of elastico-viscous liquids in a circular cylinder with linearly varying wall temperature has been studied by using the constitutive equation of motion for elastico-viscous liquids and energy equation. The flow phenomenon are characterized by two parameters R _c and S. The presence of the elastic elements in the viscous liquid considerably affects the velocity and temperature distributions.     

Comparative study of Euler/Euler and Euler/Lagrange approaches simulating evaporation in a turbulent gas–liquid flow

This study deals with the Reynolds‐averaged Navier–Stokes simulation of evaporation in a turbulent gas–liquid flow in a three‐dimensional duct, focussing on the results obtained by a four‐equation turbulence model within the framework of the Euler/Euler approach for multiphase flow calculations: in addition to the two‐equation k?ε model describing the turbulence of the continuous (C) phase, the computational model employs transport equations for the turbulence kinetic energy of the disperse (D) phase and for the velocity covariance q=〈{u}^D{u}^C〉^D. In the present study, the evaporation model according to Abramzon and Sirignano (Int. J. Heat Mass Transfer 1989; 32 :1605–1618) has been extended by introducing an additional transport equation for a newly defined quantity ā, defined as the phase‐interface surface fraction. This allows the change in the drop diameter to be quantified in terms of a probability density function. The source term in the equation describing the dynamics of the volumetric fraction of the dispersed phase α^D is related to the evaporation time scale τ_Γ. The performance of the new model is evaluated by performing a comparative analysis of the results obtained by simulating a polydispersed spray in a three‐dimensional duct configuration with the results of the Euler/Lagrange calculations performed in parallel. Prior to these calculations, some selected (solid) particle‐laden flow configurations were computationally examined with respect to the validation of the background, four‐equation, eddy‐viscosity‐based turbulence model. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical investigation of injection-induced electro-convection in a dielectric liquid between two eccentric cylinders

Numerical analysis of the 2D radial and azimuth electro-convection (EC) flow of dielectric liquid between two eccentric cylindrical electrodes driven by unipolar injection of ions is presented. The finite volume method is used to resolve the spatiotemporal distributions of the flow field, electric field, and charge density. The flow instability is studied in various scenarios where the radius ratio Γ = R_i/R_o ranges between 0.1 and 0.7 and the eccentricity η between 0.1 and 0.5. The bifurcation of the flow patterns depends on the electric Rayleigh number T, a ratio of the electric force to viscous force, and the two geometric parameters Γ and η. For an increasing T, the EC system develops from a weak steady convective state to chaos via different intermediate states experiencing pitchfork and Hopf bifurcations. The influence of Γ and η on the bifurcation behavior is also investigated. When Γ lies between 0.1 and 0.3, a novel periodic oscillation of the flow patterns has been observed.     

Subcooled forced convection film boiling over a vertical flat plate embedded in a fluid-saturated porous medium

The problem of subcooled forced convection film boiling on a vertical flat plate embedded in a porous medium was attacked exploiting similarity transformations on the governing equations and boundary conditions in both vapor and liquid layers. Similarity solutions were obtained to investigate the effects of the vapor super-heating and liquid subcooling. The heat transfer groupingNu _x /Ra _x ^1/2 is expressed in terms of a function of three parameters associated with the degree of liquid subcooling (Sub), the degree of vapor superheating (Sup) and the vapor buoyancy effect relative to the liquid forced convection effect (R). It is found that the level ofNu _x /Ra _x ^1/2 increases asSup orR decreases and asSub increases. Furthermore, asymptotic expressions were reduced considering the physical limiting conditions, namely, thin and thick vapor films.     

2D Burgers equation with large Reynolds number using POD/DEIM and calibration

Model order reduction of the two‐dimensional Burgers equation is investigated. The mathematical formulation of POD/discrete empirical interpolation method (DEIM)‐reduced order model (ROM) is derived based on the Galerkin projection and DEIM from the existing high fidelity‐implicit finite‐difference full model. For validation, we numerically compared the POD ROM, POD/DEIM, and the full model in two cases of Re = 100 and Re = 1000, respectively. We found that the POD/DEIM ROM leads to a speed‐up of CPU time by a factor of O(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of Re = 1000 has an accuracy with error O(10^?3) versus O(10^?4) in the case of Re = 100 when compared with the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small‐scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated ROM exhibit considerable agreement with the high fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.