Numerical and Experimental Analysis of the Double-Diffusive Convection in a Linearly Stratified

1

Numerical and Experimental analysis of the double-diffusive convection in a linearly stratified medium

KarimChoubania,*, KaisCharfib, Safi Mohamed Jomâac, Mohammad Almeshaald andJamelOrfie

a Riyadh College of Technology, Mechanical Department, P.O.Box 42826 Riyadh 11551. KSA.

bFaculty of Sciences of Gafsa, Physical Department , University Sidi Ahmed Zarroug-2112 Gafsa- Tunisia.

c Ecole Nationale d’Ingénieurs de Tunis, Unité de Recherche Mécanique-Energétique, 1002 El Belvédère, BP 37- Tunisia.

d Alimam Mohammad Ibn saud Islamic University - Department of Mechanical Engineering,Kingdom of Saudi Arabia - Riyadh, 11432,

eMechanical Engineering Department College of Engineering, King Saud University, PO Box 800, Riyadh, 11421, KSA.

Correspondent author: Karim Choubani. E-mail address: ;

Abstract

In this paper, we numerically and experimentally studied the onset and the mechanisms of spatio-temporal evolution of the double convective diffusion in a saline medium linearly stratified and heated from below at a constant temperature.The numerical simulations were based on the solution of the Navier-Stockes, energy and concentration equations in stream-voracityformulation. The numerical method used is the Hermitian compact one. In the experimental studies, a transparent cavity with free surface was filled with linearly stratified solution. The solution was heated from below at a constant temperature. The convective motions were visualized by using shadowgraph to obtain more information on density exchange through the interfaces. Also, Particle Image Velocimetry (PIV) is used to measure the time evolution of velocity field.Flow visualizations by Shadowgraph and PIV allowed the comparison between experimental and numerical results. It was found that a multi-cellular flow oscillating in space and time mechanisms was observed in both experimental and numerical studies.

1. Introduction

The double-diffusive convection in a stratified-systems heated from below has many important applications, such as convection in oceans, atmosphere, lakes, solar ponds ( [1], [2],[3],[4],[5],[6],[7],[8] and[9]) and multi-component diffusion in crystal growth, coating, and casting processes ([10] and [11]). The phenomenon of double diffusion considered in this work is a horizontal fluid layer linearly stratified in density and heated from below. The theoretical modeling existing up-to-date needs considerable improvement. For example, for the situation in which the magnitude of the applied bottom heat is relatively low (solar heating), it is observed that single, warm, well-mixed layer forms near the bottom. This region is separated from the overlaying fluid by a boundary-layer ([12] and[13]).

The first elaborated models were one-dimensional; they were based on the input empirical correlations, and they offered little information on the flow structure in the system. The mechanism responsible for the heat and species transfer across the boundary layer, required for continuous growth of the mixed layer, is not well understood ([14]). Hence, models based on two-dimensional governing equations were used to explain the behaviour of a mixture of a fluid and a chemical species whose initial density distribution decreases linearly from the bottom to the upper surface, where the fluid layer is subjected to sudden heating at the bottom surface.

Ungan and Bergman [15], considered a similar problem. However, they imposed symmetrical condition and considered only thesystemwhich, numerically, restricts the physical stability of the problem. However, in unstableproblems and due to the nonlinearities, the symmetry can be broken. In the fluid dynamics area, one can find a good number of asymmetrical solutions, such as the problem under consideration. The use of symmetry condition is questionable. The transient evolution of the phenomenon depends on the initial perturbation imposed on the whole system.

Kazmierczak and Poulikakos [16],investigated numerically the transient double diffusion phenomena in a stably stratified fluid layer (water and salt) heated from below. The complete form of the governing equations, subjected to the Boussinesq approximation, wasobtained and solved. A constant heat flux condition was imposed at the bottom wall. It is found that this kind of boundary condition generates a single well-mixed flow region, which grows with time. The growth of the bottom region and the increase of the local and average bottom wall temperature were investigated. Furthermore, authors analyzed the effect of the geometric aspect ratio (H/L) of the system as well as the effect of the ratio of the thermal to the solutal Rayleigh number (RaT/Ras) on the evolving temperature, concentration and flow fields. Kazmierczak and Poulikakos [16], concluded that the geometric aspect ratio of the system hasa little impact on the phenomenon but the increase of the stability parameter RaT/Ras from 3 to 10 initiates significant 2D effects on the temperature and the concentration fields.

On the other sides, several theoretical and experimental studies have been conducted on solar ponds and their potential applications in heating, power generation, and desalination. The main aim of these investigations was to understand the thermal behavior of these ponds as both solar energy collectors and heat stores under different operating conditions ([17] and [18]). Details on principle, types, and characteristics of solar ponds can be found in the review by Kalogirou [19]. The stability of salt-gradient solar ponds was studied by Ben Mansour et al. [20] using a 2-D, transient, variable property diffusive model under climatic conditions of Tunis city, Tunisia. It was found that the presence of heat extraction with its cooling effect tend to stabilize the pond.Giestas et al. [21] presented a numerical study based on the Navier-Stokes equations to investigate effect of solar radiation absorption on stability of solar pond. In another study, Giestas et al. [22] analyzed effects of non constant diffusivities on the stability of the pond. Suarez et al. [17] analyzed numerically the problem of 2-D coupled transient double diffusion convection for salt gradient solar ponds. They incorporated latent heat (evaporation) and sensible heat losses, solar short wave, and long wave radiation fluxes at water-air interface.

In order to obtain an improved understanding of dynamic processes in the Solar Pond Gradient Zone (SPGZ), numerical and experimental studies have been conducted.The goal of the present study is to investigate the onset and the mechanisms of spatio-temporal evolutions of the double- diffusiveconvection in a saline medium linearly stratified and heated from below at a constant temperature. These conditions simulate the behavior of the fluid flow with heat and mass transfer within a Solar Pond Gradient Zone (SPGZ). Visualizations by Shadowgraph and Particle Image Velocimetry (PIV) allowed comparisons with numerical simulations-results.

2. Mathematical Formulation

The system of interest is shown schematically in Fig.1. The density of a mixture of water and salt decreases linearly from the bottom to the top. The sidewalls of the system are impermeable to heat and concentration.

Initially, the fluid is isothermal and at rest. Suddenly, a constant temperature is applied at the bottom wall and starts warming up the system. Because of the scenario described above, heat, species, and fluid transports are initiated in the system.

By neglecting Dufour and Soretsecondary effects,the equationsgoverning these transport phenomenain unsteady state conditions are:

- Continuity equation:

(1)

- Momentum equation:

(2)

- Energy equation:

(3)

- Salinity equation:

(4)

In accordance with the usual Boussinesq approximation, the fluid density and thermos-physical properties are assumed to be constant everywhere except in the buoyancy force term of the momentum equation (Equation 2), where the density follows the linear state equation:

(5)

The boundary conditions are:

(6)

(7)

(8)

The initial conditions are:

(9)

Before attempting to solve the system of Equations 1-4, it is convenient to replace these equations in dimensionless form and to write them with respect to the Cartesian system of Fig.1.

The dimensionless parameters needed to perform this task are defined as:

(10)

To overcome the problem encountered by calculation of the pressure P and in order to reduce the number of equations, stream function and vorticity function are used to replace the primitive variables U, V and P. The two dimensionless functions and, are introduced as:

(11)

Note that by definition, the stream function satisfies the continuity equation (1). Furthermore, the two momentum equations can be combined to eliminate the pressure gradient and obtain a unique vorticity equation.

After the mentioned manipulations, the dimensionless governing equations and boundary and initial conditions become as:

(12)

(13)

(14)

(15)

;(16)

(17)

; (18)

(19)

3. Numerical simulation

The governing equations (12-19) were solved numerically. Variety of methods is available in literature and CFD software makes it possible to simulate a large range of industrial problems. In our case, the set of coupling equations (12-15) are solved using a difference finite scheme based on a compact Hermetian method where the function and its first and second derivativesare considered as unknowns. The method is fourth order accurate, forand second order accurate,in,and.

The A.D.I (Alternate Direction Implicit) technique is used to integrate the parabolic equations. Since the scheme is well documented in the literature, and has been widely used for natural convection and recirculation zones, there is no need for elaboration. The procedure has the advantage that the resulting tri-diagonal matrix instead of a matrix with five occupied diagonals can easily be solved by factorization algorithm.

Some difficulties were encountered in implementing the vortices boundary conditions but different approximations were tested and the Padé approximation was employed to overcome the numerical instability [23].

The set of equations is solved using an unsteady numerical procedure (Fig.2). At each step of time:

-The convergence of the stream function equation is achieved using internal iterations.The convergence criteria are defined by the following relation:

(20)

- For the vorticity, temperature and salinity functions, the convergence criteria is:

(21)

is equal to for the stream functionand 10-4for the, and. Fig.2 shows the flow-diagram, i.e., algorithm of the solution.

4. Numerical results

The numerical simulations were conducted for the configuration (Fig.1) using the following parameters:

-H=2.10-2 m, L=10-1 m,

-Asalt solution with C0=10% of mass,

-A difference temperature: T0=(40-20)=20 °C,

-The values of RaT, Pr, Le, N, and Bi were respectively:

RaT= 6.1x106, Pr = 4.68, Bi = 0.2, Le = 73, N = 7

Several grid sizes were tested to ensure that results are independent of the mesh size. A grid of 101x21 nodes was selected and used.

Results of the computations are presented in the form of contours plots of stream function and density distribution at different times. It was found that (Fig.3):

- At the beginning (t=20 s), small vortices appeared in the bottom of the cavity while density remained stable.

- After some time (t=30s), the number of vortices increases in the vertical direction(Fig.3b) inducing the acceleration of the diffusion of concentration. At the same time density started to oscillate over the bottom surface (Fig.3a).

- As the time further increased, the recirculation zone extends to the horizontal direction, and density oscillation starts to be dumped in the vertical direction.

Fig.4 shows the dimensionlessdensity distribution for three positions of y (upwards:19/20, 1/2, and 1/20).

As the time increases,appearance of the oscillatory movements near the bottom surface (y=1/20) starts.

One can notice that the heat and species are exchanged by molecular diffusion in the beginning of the process. Then, vortices appear and convective movement takes place inside the liquid and a double diffusion starts. The convective rolls are ejected from the bottom to upper surface but are quickly dumped in the vicinity of the hot plate. As the time increases, these rolls coalesce and give rise to a well-mixed layer of a limited thickness.

5. Experimental setup

All experiments were carried out in a rectangular Plexiglas tank with free surface and with inner dimensions of 20.10-2 m x 2.10-2m. The sides of the tank, except for a window of

(10.10-2m x 2.10-2 m), necessary for flow visualization, were insulated. The tank was fitted in the bottom with an Aluminum heat exchanger thermally insulated, through which circulates water coming from an adjustable-thermostat to provide heating at constant temperature over the base. The instruction-temperature is obtained with a precision of ± 0.2°C. A photograph of the entire apparatus is shown in Fig. 5a, and a schematic sketch is depicted in Fig. 5b.

The desired linear salinity gradients were set up using a modified two-tank technique of Oster [24].The resulting concentration was varying fromCb(high concentration) at the bottom to zero at the surface. The temperature profile of the stratified solution was measured by copper constantan thermocouples. The diameter of each thermocouple is about 0.2 mm. Temperature-measurements were obtained with a precision of ± 0.8%. Thermocouples were positioned at 4 mm intervals along the mid vertical axis, except for the lowest one which had been set at 2 mm above the bottom of the cell in order to depict possible three-dimensional regimes. All the thermocouples were connected to an Agilent data-logger. The measurements of density were made using a single-point conductivity probeof constant 0.954. Interface location and layer formation were monitored with a shadowgraph system. It uses the fact that the index of refraction of fluids is typically dependent on temperature.

The liquid was seeded with particles tracer. The illumination of the particles was provided by a plane-light-sheet, generated by expanding the beam of a 20 mW He-Ne laser by means of a cylindrical lens. This sheet can move parallel to the walls of the cell in order to depict a probably three dimensional regimes. Images were recorded by a CCD camera of resolution 578 x 512 pixels and 25 FPS. Images will be stored on a computer hard disk. Fluid velocities are determined by the displacements of the seeded particles at a given time separation, with an error of 10-7 m/s.

6. Experimental results

Experimental conditions associated with the linearly salt-stratified solution heated from below at a constant temperature are fixed by: A=5, T0=20°C, Salt solution = (10% of mass).

Shadowgraph visualization (Fig.6) shows that once the heating begins, a vertical oscillatory disturbance occurs in the bottom layer. This disturbance grows with time. This oscillatory motion seems to be the main mechanism responsible for the growth of the first mixed layer which strongly influences overall system performance. Indeed, when the mixed layer grows to a critical height, the interface behaves like a wall. The heat seems to be localized in the bottom layer. So, upper layer will be heated from the below at a constant temperature using a fictive base plate formed by a moving density interface. Thereby, the interface is so energetic to be shared by oscillatory motion, but it will be pushed up and the bottom layer rises. As the bottom layer grows, the same mechanism is reproduced in the upper edge of the interface; the later motion will be prompted. The motion appeared in the upper edge of the interface does not persist for a long time. This leaves the interface eroded essentially from the lower front. Since the convection in the lower layer is stronger than that in the upper layer, the differential turbulence levels across the interface cause its migration upwards while manifesting net entrainment from the upper layer [3].

Observations of the entrainment process during the heating of a non-linear salinity gradient from below, lead us to distinguish two behaviors:

i. An oscillatory regime corresponding to the deformation of the interface

ii. A steady two-layer regime where the interface is affected by an oscillatory motion in both lower and upper edges. In this regime, the density stratification acts as a barrier to motions that try to penetrate through it.

PIV visualization shows the appearance of eddies in the homogeneous layers (Fig.7). These eddies are responsible for the convective mixing and they induce a significant shearing on the edges of the interface.

It is of interest to mention that these observations based on the experimental tests come to confirm the above numerical results. Similar flow structures have been observed using both studies. For instance, numerical and experimental results showed that vortices appeared in the bottom then were travelled up, but not exceeding a fixed distance. During this process, density oscillated in space and time and then coalesces to form the first mixed layer. Time variations of the main variables of this problem obtained numerically have been confirmed by experiments.

7. Conclusion

This study presented numerical and experimental results of the double-diffusive convection in a linearly stratified medium of a salt solution heated from below at a constant temperature. Initially, the salt solution was at rest, isothermal, and its density decreased linearly with distance from the bottom wall.

Numerical results as well as experimental showed that vortices appeared in the bottom then were travelled up, but not exceeding a fixed distance. During this process, density oscillated in space and time and then coalesces to form the first mixed layer.

Experimentally we distinguish three steps of onset of the first mixed layer:

i. The onset, of aperiodic oscillations moving in space and time.

ii. A well mixed layer is born and an oscillation movement appears at the free surface later, the average wave-length is of the same magnitude of the stratified layer thickness. These oscillations become more structured with a given period and continue to propagate in the horizontal direction. During this phase, a well-mixed layer appears with an upper interface thin layer. At the same time a similar oscillation movement appears at the free surface and moves towards the bottom interface where it merges: a new interface was born.