

ORIGINAL ARTICLE 

Year : 2018  Volume
: 3
 Issue : 1  Page : 1321 

Numerical and experimental analysis of the doublediffusive convection in a linearly stratified medium
Mohammed Almeshaal^{1}, Karim Choubani^{2}
^{1} Department of Mechanical Engineering, Al Imam Mohammad Ibn Saud Islamic University, Riyadh, Kingdom of Saudi Arabia ^{2} Department of Mechanical, Riyadh College of Technology, Riyadh, Kingdom of Saudi Arabia
Date of Submission  03Jul2018 
Date of Acceptance  10Sep2018 
Date of Web Publication  19Oct2018 
Correspondence Address: Dr. Mohammed Almeshaal Department of Mechanical Engineering, Al Imam Mohammad Ibn Saud Islamic University, Riyadh Kingdom of Saudi Arabia
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/ijas.ijas_7_18
Introduction: In this paper, we numerically and experimentally studied the onset and the mechanisms of spatiotemporal evolution of the double convective diffusion in a saline medium linearly stratified and heated from below at a constant temperature. Methodology: The numerical simulations were based on the solution of the Navier–Stokes,energy, and concentration equations in streamvorticity formulation. 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 using shadowgraph to obtain more information on density exchange through the interfaces. Furthermore, particle image velocimetry (PIV) is used to measure the time evolution of the velocity field. Results: Flow visualizations by Shadowgraph and PIV allowed the comparison between experimental and numerical results. Conclusion: It was found that a multicellular flow oscillating in space and time mechanisms was observed in both experimental and numerical studies.
Keywords: Double diffusion, flow visualization, numerical simulations, solar pond
How to cite this article: Almeshaal M, Choubani K. Numerical and experimental analysis of the doublediffusive convection in a linearly stratified medium. Imam J Appl Sci 2018;3:1321 
How to cite this URL: Almeshaal M, Choubani K. Numerical and experimental analysis of the doublediffusive convection in a linearly stratified medium. Imam J Appl Sci [serial online] 2018 [cited 2019 Feb 21];3:1321. Available from: http://www.eijas.org/text.asp?2018/3/1/13/243623 
Introduction   
The doublediffusive 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],[9]} and multicomponent diffusion in crystal growth, coating, and casting processes.^{[10],[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 uptodate 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, and wellmixed layer forms near the bottom. This region is separated from the overlaying fluid by a boundary layer.^{[12],[13]}
The first elaborated models were onedimensional (1D); 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 2D governing equations were used to explain the behavior 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 a symmetrical condition and considered only the system which, numerically, restricts the physical stability of the problem. However, in unstable problems 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, was obtained 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 wellmixed 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 (Ra_{T}/Ras) on the evolving temperature, concentration, and flow fields. Kazmierczak and Poulikakos,^{[16]} concluded that the geometric aspect ratio of the system has a little impact on the phenomenon but the increase of the stability parameter Ra_{T}/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],[18]} Details on principle, types, and characteristics of solar ponds can be found in the review by Kalogirou.^{[19]} The stability of saltgradient solar ponds was studied by Ben Mansour et al.^{[20]} using a 2D, transient, variable property diffusive model under the climatic conditions of Tunis city, Tunisia. It was found that the presence of heat extraction with its cooling effect tends to stabilize the pond. Giestas et al.^{[21]} presented a numerical study based on the NavierStokes equations to investigate effect of solar radiation absorption on the stability of solar pond. In another study, Giestas et al.^{[22]} analyzed effects of nonconstant diffusivities on the stability of the pond. Suarez et al.^{[17]} analyzed numerically the problem of 2Dcoupled transient double diffusionconvection for salt gradient solar ponds. They incorporated latent heat (evaporation) and sensible heat losses, solar shortwave, and longwave radiation fluxes at waterair 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 spatiotemporal evolutions of the doublediffusive convection 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 an SPGZ. Visualizations by Shadowgraph and PIV allowed comparisons with numerical simulations results.
Mathematical Formulation   
The system of interest is shown schematically in [Figure 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 thermocapillary effect and Dufour and Soret secondary effects, the equations governing these transport phenomena in unsteady state conditions are as follows:
Continuity equation:
Momentum equation:
Energy equation:
Salinity equation:
In accordance with the usual Boussinesq approximation, the fluid density and thermosphysical 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:
The boundary conditions are:
The initial conditions are:
Before attempting to solve the system of Equations 14, it is convenient to replace these equations in dimensionless form and to write them with respect to the Cartesian system of [Figure 1].
The dimensionless parameters needed to perform this task are defined as:
To overcome the problem encountered by calculation of the pressure P and to reduce the number of equations, stream function Ψ and vorticity function Ω are used to replace the primitive variables U, V, and P. The two dimensionless functions Ψ and Ω, are introduced as:
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:
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 Hermitian method where the function and its first and second derivatives are considered as unknowns. The method is fourth order accurate, O(h ^{4}), for Ψ and secondorder accurate, O(h ^{2}), i, in Ω, T, and S.
The 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 tridiagonal 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 [Figure 2]. At each step of time:  Figure 2: Algorithm Resolution of unsteady numerical procedure in (Ψ,Ω,T,S) formulation: IROUT: Outer iteration number, IRINMAX: Max inner iteration number, IROUMAX: Max outer iteration number, IRINMAX: Max inner iteration number, △τ: Step time, (△X, △Y): Step space in horizontal and vertical position, (NX,NY): Grid nodes, TESTΨ: Convergence test of Ψ
Click here to view 
The convergence of the stream function equation is achieved using internal iterations. The convergence criterion are defined by the following relation:
For the vorticity, temperature and salinity functions, the convergence criteria are:
χ is equal to 10^{−6} for the stream function Ψ and 10^{−4} for the Ω, T and S. [Figure 2] shows the flow diagram, i.e., algorithm of the solution.
Numerical Results   
The numerical simulations were conducted for the configuration [Figure 1] using the following parameters:
 H = 2.10^{−2} m, L = 10^{−1} m,
 A salt solution with △C_{0}= 10% of mass,
 A difference temperature: △T_{0}=(40–20)= 20°C,
 The values of Ra_{T}, Pr, Le, N, and Bi were, respectively:
 Ra_{T}= 6.1x10^{6}, 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 101 × 21 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 [Figure 3]. It was found that:  Figure 3: Spatiotemporal evolution of (a) density and (b) streamlines (numerical results)
Click here to view 
 At the beginning (t = 20 s), small vortices appeared in the bottom of the cavity while density remained stable
 After some time (t = 30 s), the number of vortices increases in the vertical direction [Figure 3]b inducing the acceleration of the diffusion of concentration. At the same time, density started to oscillate over the bottom surface [Figure 3]a
 As the time further increased, the recirculation zone extends to the horizontal direction, and density oscillation starts to be dumped in the vertical direction.
[Figure 4] shows the dimensionless density distribution for three positions of y (upward:19/20, 1/2, and 1/20).  Figure 4: Isodensity distribution with different position of y (numerical results)
Click here to view 
As the time increases, the 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 wellmixed layer of a limited thickness.
Experimental Setup   
All experiments were carried out in a rectangular Plexiglas tank with free surface and with inner dimensions of 20.10^{−2} m × 2.10^{−2} m. The sides of the tank, except for a window of (10.10^{−2} m × 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 adjustablethermostat to provide heating at a constant temperature over the base. The instructiontemperature is obtained with a precision of ±0.2°C. A photograph of the entire apparatus is shown in [Figure 5]a, and a schematic sketch is depicted in [Figure 5]b.
The desired linear salinity gradients were set up using a modified twotank technique of Oster.^{[24]} The resulting concentration was varying from C_{b} (high concentration) at the bottom to zero at the surface. The temperature profile of the stratified solution was measured by copperconstantan 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 to depict possible 3D regimes. All the thermocouples were connected to an Agilent data logger. The measurements of density were made using a singlepoint conductivity probe of 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 planelightsheet, generated by expanding the beam of a 20 mW HeNe laser by means of a cylindrical lens. This sheet can move parallel to the walls of the cell to depict a probably 3D regimes. Images were recorded by a CCD camera of resolution 578 × 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.
Experimental Results   
Experimental conditions associated with the linearly saltstratified solution heated from below at a constant temperature are fixed by: A = 5, △T_{0}= 20°C, Salt solution = (10% of mass).
Shadowgraph visualization [Figure 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. Hence, 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 upward while manifesting net entrainment from the upper layer.^{[3]}
Observations of the entrainment process during the heating of a nonlinear salinity gradient from below, lead us to distinguish two behaviors:
 An oscillatory regime corresponding to the deformation of the interface
 A steady twolayer 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 [Figure 7]. These eddies are responsible for the convective mixing, and they induce a significant shearing on the edges of the interface.  Figure 7: Particle image velocimetry visualization of the vortices at the bottom of the stratified layers
Click here to view 
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.
Conclusion   
This study presented numerical and experimental results of the doublediffusive 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:
 The onset, of aperiodic oscillations moving in space and time
 A wellmixed layer is born, and an oscillation movement appears at the free surface later, the average wavelength 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 wellmixed layer appears with an upper interface thin layer. At the same time, a similar oscillation movement appears at the free surface and moves toward the bottom interface where it merges: a new interface was born
 This new interface becomes thinner and quasiplane, consequently, the stratification is nonlinear, and it was governed by the immigration of the single interface upward. This interface oscillated in time and space until the final homogenization of the whole system.
These results agreed with precedent ones but provided more details on the mechanism of double diffusion.
Nomenclature
Greek symbols
Subscripts
Exponents
Notations
Nondimensional numbers
Nondimensional quantities
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
References   
1.  Akrour D, Tribech M, Kalache D. A theoretical and numerical study of thermosolutal convection: Stability of a salinity gradient solar pond. Thermal Sci 2011;80:1567. 
2.  Anderson MD, Mcfadden BG, Coriell RS, Murray TB. Convective instabilities during the solidification of an ideal ternary alloy in a mushy layer. J Fluid Mech 2010;647:30933. 
3.  Choubani K, Zitouni S, Charfi K, Safi MJ, Akbarzadeh A. Experimental study of the salt gradient solar pond stability. Sol Energy 2010;84:2431. 
4.  Choubani K, Safi MJ. Experimental simulation of the salt gradient solar pond stability. J Renew Sustain Energy 2012;4:063143. 
5.  Gargett AE, Tanny J, Kelley DE, Fernando HJ, Ozsoy E. The diffusive regime of doublediffusive convection. Prog Oceanog 2003;56:46181. 
6.  Giestas MC, Milhazes JP, Pina H. Numerical modeling of solar ponds. Energy Procedia (Elsevier) 2014;57:241625. 
7.  Mishra M, Trevelyan PM, Almarcha C, De Wit A. Influence of double diffusive effects on miscible viscous fingering. Phys Rev Lett 2010;105:204501. 
8.  Scmid M, Busbridge M, Wüest A. Doublediffusive convection in Lake Kivu. Am Soc Limnol Oceanogr 2010;55:22538. 
9.  Thompson FA, Veronis G. Diffusivelydriven overturning of a stable density gradient. J Marine Res 2005;63:291313. 
10.  Chen CF, Turner JS. Crystallization in a doublediffusive system. J Geophys Res 1980;85:257393. 
11.  Ostrach S, Fluid mechanics of crystal growthThe Freeman Scholar Lecture, ASME, Transactions, Journal of Fluids Engineering 1983;105:520. 
12.  Meyer KA. A numerical model to describe the layer behaviour in saltgradient solar ponds. ASME J Solar Energy Engng 1983;25:6375. 
13.  Bergman TL, Incropera FP, Viskanta RA. Differential model for saltstratified, doublediffusive systems heated from below. Int J Heat Mass Transfer 1985;28:77988. 
14.  Incropera FP. Buoyancy effects in doublediffusive and mixed convection flows. Proc Int Heat Transfer Conf 1986;1:12130. 
15.  Ungan A, Bergman TL. A twodimensional numerical simulation of a linear stratified thermohaline system heated from below. Paper No. 87WA/HT19. ASME Winter Annual Meeting, Boston, MA; 1987. 
16.  Kazmierczak M, Poulikakos D. Transient double diffusion in a stably stratified fluid layer heated from below. Int J Heat Fluid Flow 1990;11:309. 
17.  Suarez F, Tyler SW, Childress AE. A fully coupled, transient doublediffusive convective model for saltgradient solar ponds. Int J Heat Mass Transfer 2010;53:171830. 
18.  Ben Mansour R, Nguyen CT, Galanis N. Numerical study of transient heat and mass transfer and stability in a saltgradient solar pond. Int J Therm Sci 2004;43:77990. 
19.  Kalogirou S. Seawater desalination using renewable energy sources. Prog Energy Combust Sci 2005;31:24281. 
20.  Ben Mansour R, Nguyen CT, Galanis N. Transient heat and mass transfer and longterm stability of a saltgradient solar pond. Mech Res Commun 2006;33:23349. 
21.  Giestas M, Pina H, Joyce A. The influence of radiation absorption on solar pond stability. Int J Heat Mass Transfer 1996;39:387385. 
22.  Giestas M, Pina H, Joyce A. The influence of nonconstant diffusivities on solar ponds stability. Int J Heat Mass Transfer 1997;40:437991. 
23.  Safi MJ, Loc TP. Development of thermal stratification in two dimensional cavity a numerical study. Int J Heat Mass Transfer 1994;37:201724. 
24.  Oster G. Density gradients. Sci Am 1965;213:706. 
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7]
