Advection Diffusion Model for Particles Deposition in Rayleigh-Bénard Turbulent Flows

International Conference Nuclear Energy for New Europe 2005 Bled, Slovenia, September 5-8, 2005

Paolo Oresta, Antonio Lippolis DIASS – Politecnico di Bari Viale del Turismo 8, 74100 Taranto, Italia [email protected], [email protected]

Roberto Verzicco DIMeG and CEMeC – Politecnico di Bari Via Re David, 200, 70125 Bari, Italia [email protected]

Alfredo Soldati DEM and CIFI – Università degli Studi di Udine Via delle Scienze, 208, 33100 Udine, Italia [email protected]

ABSTRACT

In this paper, Direct Numerical Simulation (DNS) and Lagrangian Particle Tracking are used to precisely investigate the turbulent thermally driven flow and particles dispersion in a closed, slender cylindrical domain. The numerical simulations are carried out for Rayleigh (Ra) and Prandtl numbers (Pr) equal to Ra = 2·108 and Pr = 0.7, considering three sets of particles with Stokes numbers, based on Kolmogorov scale, equal to Stk = 1.3, Stk = 0.65 and Stk = 0.13. This data are used to calculate a priori the drift velocity and the turbulent diffusion coefficient for the Advection Diffusion model. These quantities are function of the Stokes, Froude, Rayleigh and Prandtl numbers only. One dimensional, time dependent, Advection- Diffusion Equation (ADE) is presented to predict particles deposition in Rayleigh-Bénard flow in the cylindrical domain. This archetype configuration models flow and aerosol dynamics, produced in case of accident in the passive containment cooling system (PCCS) of a nuclear reactor. ADE results show a good agreement with DNS data for all the sets of particles investigated.

1 INTRODUCTION

The passive containment cooling system (PCCS) consists of an inner steel shell and an outer concrete shell [1]. During a severe nuclear accident, the radioactive core meltdown and the energy produced is responsible for the thermal gradient between the lower and upper zones of PCCS. Specifically, the energy released generates a strong convective flow, which carries passively radioactive aerosols in the confined containment. The steel shell provides to remove the heat from the containment, to have the steam temperature and pressure below the max

173.1 173.2 design value. During an accident, the water, located in a tank on the top of the containment, flows down on the outside of the steel shell to improve the heat removal away from the PCCS. In this way, the consequences of a severe core meltdown accident will be restricted inside the containment. Dispersion, deposition and re-entrainment in buoyancy-driven flow are of fundamental issue to evaluate nuclear reactor safety problems. Particles deposition plays a significant role in limiting the release of radioactivity to the environment; yet, there is still a lack of knowledge about the nuclear aerosol release mechanisms in the containment. The convective flow occurring in this containment system can be modelled as a confined Rayleigh-Bénard flow. The flow field produced in a closed domain with the lower plate maintained at a temperature higher than the upper plate, is dominated by Rayleigh-Bénard convection. We use a cylindrical domain with aspect ratio (diameter over cell height) 1/2, filled with air as archetype configuration to model the PCCS device. The temperature gradient kept between the plates and the natural inhomogeneity of the temperature field on the cylindrical domain concurs to produce different dynamics in terms of vortical structures. In our case, the flow field is turbulent, three-dimensional and time dependent. In such flow, three sets of inertial particles are randomly dispersed on the cylindrical domain. Particles clustering and dispersion are due to their interactions with turbulent eddies, characterized by a specific dimension and by a specific vortical time. Particles are sensitive to small size eddies and their simulated behaviour can be significantly modified if these vortical structures are not directly computed, but described with sub-grid turbulence model as RANS and LES. For this reason, the solution of the flow field with DNS approach is very important. The advantage of the DNS is the possibility of obtaining an accurate velocity field, without using turbulence models. This means that we can calculate also the high frequency velocity fluctuations - that has to be modelled in other kinds of numerical simulations like LES and RANS - and evaluate their effect on the particle behaviour. With DNS, the turbulence properties along particle trajectories are directly available; with this technique, the local mechanism driving particles transfer and particle preferential distribution can be investigated. The main problem of Direct Numerical Simulation is the computational cost, due to the very fine spatial and temporal resolution in comparison to the turbulent modelling. In such flow, Lagrangian Particle Tracking is used to have an accurate temporal and spatial history of particles deposition in the cylindrical domain. In several engineering applications, a model simpler than Direct Numerical Simulation and Lagrangian Particle Tracking is needed to describe the time history of particles deposition. For instance, in case of severe nuclear reactor accident, we are mainly interested in evaluating the time history of particle concentration averaged on the horizontal plate. Specifically, time dependent, one dimensional, Advection Diffusion model is presented in this paper to predict, with sufficient accuracy, particle deposition in the cylindrical domain dominated by the turbulent convective flow. In this work, Direct Numerical Simulation and Lagrangian Particle Tracking are used to model the drift velocity and turbulent diffusion coefficient. These quantities are calculated a priori because they are functions of dimensionless parameters only. Advection-Diffusion model was tested against numerical simulations database.

2 METHODOLOGY

2.1 Flow Field

The thermal convection was developed in a closed cylindrical domain of diameter d, distance between the heated plates h, filled with a fluid characterized by kinematics viscosity ν, thermal diffusivity k, coefficient of thermal expansion α and thermal conductivity λ. The

Proceedings of the International Conference “Nuclear Energy for New Europe 2005” 173.3 aspect ratio Γ=d/h was set equal to 0.5 and the magnitude of the gravity vector is g. The lateral surface is adiabatic and the boundary conditions are no-slip. Hot Th and cold temperatures Tc are enforced respectively at the lower and upper plates as thermal boundary conditions; ∆=Th-Tc is the temperature difference kept between the horizontal plates. We can define the Rayleigh and the Prandtl numbers as follows:

g∆h3 Ra = ; Pr = k k

In our case, the fluid that fills the cylindrical domain is the air and, for this reason, we have set Prandtl number equal to 0.7. Different combinations between the Rayleigh and Prandtl numbers and the aspect ratio concur to produce the different dynamics in terms of vortical structures. Starting with the flow at rest and increasing Rayleigh number, there are three fundamental flow regimes and their onset is characterized by three critical Rayleigh numbers. At the onset of the convection the flow has a steady state and if the nonlinear terms exceed the viscous one the unsteady state of the flow is reached. The transition to chaotic and turbulent regimes is more gradual in comparison to the transition from the steady state to the time dependent regimes. In the turbulent regime the flow dynamics are not strongly dependent on the cell geometry. Critical Rayleigh numbers, in a cylindrical cell with Γ = 1/2 and Pr = 0.7, 4 5 5 results equal to Rac = 2.35·10 , Rau = 5.6·10 and Rat = 9·10 , respectively for the beginning of convection, unsteady and turbulent regimes [2], respectively. We carried out the numerical simulations with Rayleigh number set equal to Ra = 2·108. With Rayleigh number equal to Ra = 2·108 and Prandtl number Pr = 0.7, the flow is turbulent, three-dimensional and time dependent. For these set of values, despite of the axysimmetric domain, the flow is organized in a single roll that fills the whole domain and in two toroidal structures close to the plates [3]. The domain of Rayleigh-Bénard problem can be divided in two regions: the thermal boundary layers and the core region. Core region shows a Kolmogorov turbulence cascade mechanism. The energy is pumped into the system at large scale h and is transferred to the inertial range scales r and, finally, the energy is dissipated, because the viscosity at the Kolmogorov scales η. In the thermal boundary layer, there is the whole temperature gradient and the turbulence dynamics follow the Bolgiano theory. The Bolgiano scale represents the scale at which energy is injected in the system as “potential energy”. Buoyancy force converts this energy in kinetic energy. In the region close to the thermal boundary layer, the buoyancy force is the dominant effect in the system [3],[4]. For the present flow configuration (Pr=0.7, Γ=1/2), the thermal layer is smaller than the viscous one. The flow field is obtained solving the unsteady Navier- Stokes equations with the Boussinesq approximation, written in dimensionless form as follows:

1/ 2 Du ≈ Pr ’ → = −∇p + ∆ ÷ ∇2u + Q z , Dt « Ra ◊ (1)

DQ 1 = ∇2Q, Dt (Pr Ra)1/ 2

→ with z the unity vector pointing in the opposite direction respect to gravity, u the velocity vector, p the pressure and Q = (T-Tc)/(Th-Tc) the dimensionless temperature in which T is the

Proceedings of the International Conference “Nuclear Energy for New Europe 2005” 173.4 temperature in the cylindrical domain. Equations are written in dimensionless form using the 1/2 free-fall velocity Uf = [gαL(Th – Tc)] , the distance between hot and cold plates L = h and the time τf = L/Uf . The method used to solve the Navier-Stokes equations is Direct Numerical Simulation (DNS). The flow field is calculated with finite differences code in cylindrical coordinates [5]. Navier-Stokes equations are solved by a fractional step method. The nonlinear terms are treated explicitly, while the viscous terms are computed implicitly. The time advancement of the solution is obtained by a Runge-Kutta 3rd order and spatial derivates are discretized using Crank-Nicolson scheme. The mesh has 97×49×193 points along the azimuthal, radial and axial directions [3], to compute the smallest vortical structures defined by the Kolmogorov theory. We have stretched the grid points, near the boundary, in the radial and axial directions to capture the viscous boundary layer and the thermal one. Specifically, the grid clustering has, at the lower and upper plate, 10 points in the thermal layer. The time step size must be close to the characteristic time scale of the smallest vortical structures to well compute the flow field. Sometimes, this condition is weak to calculate an accurate solution of particles momentum equation. Specifically, the time step size must satisfy a stronger condition that depends on the characteristic particle time scale. We have developed this concept in the following paragraph.

2.2 Lagrangian particles tracking

In our case, we suppose a small volume fraction of the dispersed phase (diluted system). This assumption implies that only the flow effects on particles are important (one-way coupling) and we can neglect particles effects on the flow field and particle-particle interactions. We have modelled the particles as rigid spheres with density large compared to fluid density, and we consider the forces applied on their centres. Saffman force is neglected because its effects are restricted to the region near the wall and the aim of this work is to model the time evolution of the particle concentration averaged on horizontal plates of the cylindrical domain. Elastic reflection on the wall cylindrical domain was imposed as boundary condition for the particles. We have used, for the particle equations, a model with only inertia, weight and drag forces, because the other forces can be neglected [6,7]. We indicate with u, v, x and d, respectively the flow velocity, the particle velocity, the particle position, and the particle diameter, respectively. The adjacent dimensionless quantities are u, v, x and d. Particle momentum equation, based on the previous hypothesis, as follows:

dv dx m = f 3πµd(u − v) + m g; = v p dt p dt (2) 1/ 2 u − v d U f h ~ ≈ Ra ’ ~ f = 1+ 0.15Re0.687[8]; Re = = u~ − ~v d = ∆ ÷ u~ − ~v d d d ν ν « Pr ◊ Particle equation in dimensionless form is obtained using the same dimensionless quantities used for Navier-Stokes equations.

d~v f 1 → d~x (1 QFr 2 )(~ ~) ; ~ (3) = − u − v − 2 z = v dt St f Fr dt

2 τ p ρ pd U f St f = ; τ p = ; Fr = (4) τ f 18µ0 gL

Proceedings of the International Conference “Nuclear Energy for New Europe 2005” 173.5 where µ is the fluid dynamic viscosity. The Stokes number is a measure of the particle response to the fluctuations in the velocity field. We have indicated with Stf the Stokes number estimated, using the time scale of the larger eddy that fills the whole cylindrical domain. The Froude number (Fr) is a measure of the ratio between inertia and weight forces of the particle. Froude number can be set at most equal to 0.45 to satisfy the Boussinesq approximation, because it is equal to (α∆)1/2. The particle equations are solved using Lagrangian Particle Tracking approach. Particles are released when the flow is turbulent. The onset of particles motion starts when the flow shows a fully developed turbulent field. Initially, the particles have the same flow velocity and are randomly dispersed in the cylindrical domain. Time solution is obtained integrating the particle velocity equation and the particle position equation. The flow velocities on the particles positions are calculated by interpolating velocities from the closest 8 grid points. This interpolation scheme has the same 2nd order accuracy in space of the flow field solution. The particle position equation is solved using Runge-Kutta 3rd order and the particle velocity equation is solved using 2nd order implicit scheme; in this way, the particle solution is synchronous with the flow field solution and has the same accuracy in time of the flow field. We have carried out three simulations setting Stokes number, based on the large time scale, to Stf = 0.01, Stf = 0.005 and Stf = 0.001. The time integration is taken equal to Stf/3 in agreement to Elghobashi [9]

3 ADVECTION DIFFUSION APPROACH

Three dimensional, time dependent Advection Diffusion equation for particles system is written as follows:

≈ ’ ∂Φ ∆ 1 ÷ + ∇⋅(VΦ) = ∇⋅∆ D p ∇Φ÷ (5) ∂t « U f L f ◊ where Φ is the instantaneous value of particle concentration, V is the three-dimensional Eulerian convective velocity. The diffusive term accounts the effects due to Brownian motions [10,11]. Total velocity VT of the particle is related to the convective velocity V, by

1 ΦVT = ΦV − D p ∇ ⋅ Φ (6) U f L f

This equation accounts for all the fluid effects on particles, but its solution needs the instantaneous flow velocity field. For this reason, we try to simplify the above equation, preserving the basic information about the deposition of particles in the cylindrical domain. We have based our model on the same strategy used to obtain the RANS equations. We write the instantaneous values of Φ and V as the sum of an average value and a fluctuating term:

Φ(x,t) = φ(x,t) +φ'(x,t) V(x,t) = v(x,t) + v'(x,t) (7)

The average is taken over a time-interval T, long enough to smooth out turbulent fluctuations, but short enough to account for variations due to changes in the averaged quantities.

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1 T φ(x,t) =< Φ(x,t) >= —Φ(x,t)dt; φ'(x,t) = Φ(x,t) −φ(x,t) (8) T 0

1 T v(x,t) =< V(x,t) >= — V(x,t)dt; v'(x,t) = V(x,t) − v(x,t) (9) T 0

Using the cited hypothesis, we obtain the following equation:

∂φ ∂φ' + + ∇ ⋅ (vφ)+ ∇ ⋅ (vφ')+ ∇ ⋅ (v'φ)+ ∇ ⋅ (v'φ') = ∂t ∂t (10) ≈ ’ ≈ ’ ∆ 1 ÷ ∆ 1 ÷ = ∇ ⋅∆ Dp ∇φ ÷ + ∇ ⋅∆ Dp ∇φ'÷ « U f Lf ◊ « U f Lf ◊

We observe that the previous equation can be written as follows:

≈ ’ ∂φ ∆ 1 ÷ + ∇ ⋅ (vφ)+ ∇ ⋅ (< v'φ'>) = ∇ ⋅∆ Dp ∇φ ÷ (11) ∂t « U f Lf ◊

The Fickian model is used, following Young & Leeming [12], to estimate the fluctuation terms. We define the turbulent diffusion Dturb and the dimensionless turbulent diffusion Dturb

1 ~ < v'φ'>= −Dturb ∇φ = −Dturb∇φ (12) U f Lf

With this assumption we have:

≈ ’ ≈ ’ ∂φ ∆ 1 ÷ ∆ 1 ÷ + ∇ ⋅ (vφ) = ∇ ⋅ ∆ D p ∇φ ÷ + ∇ ⋅ ∆ Dturb ∇φ ÷ (13) ∂t « U f L f ◊ « U f L f ◊

−1/ 2 ∂φ 1 ≈ Ra ’ ~ + ∇ ⋅ (vφ) = ∆ ÷ ∇ 2φ + D ∇2φ (14) ∂t Sc « Pr ◊ turb

The equation (14) is three dimensional and time dependent. The deposition phenomena are developed mainly along the axial direction and we can average the equation (14) along the horizontal plates of the cylindrical domain, to obtain the following one dimensional form of the ADE equation:

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2 À∂φ ∂vdrift ∂φ ~ ∂ φ Œ = − φ − vdrift + Dturb Œ ∂t ∂z ∂z ∂z 2 Œ Œ Œ∂φ Ã = 0 z = 0 − lower plate; ∀t ∈[0;+∞[ (15) Œ ∂z Œ Œ Œφ = 0 z = 1− upper plate;∀t ∈[0;+∞[ ÕŒφ = 1+ ε t = 0 ∀z ∈ [0;1[; ∈[− 0.02;0.02]− random numer

In the equation (15) we have neglected the molecular diffusion term because we suppose that the main diffusion effects are accounted for by the turbulent one. The initial particles concentration for ADE model (eq. 15) was enforced using a random function, because Direct Numerical simulations are carried out starting from random particles positions. Random function has the magnitude of the same order of the particle concentration amplitude for DNS simulations. In equation (15) the drift velocity is approximated with the averaged particles velocity taken on the whole simulation time.

3.1 Model for drift velocity

Drift velocity is taken as particles velocity averaged on horizontal plates and on the simulation time. We have computed the averaged particles velocity on horizontal slice with height h/100. In Figure 1, it is shown the variation of averaged particles velocity in two points close to the lower and upper plates and, in the same figure, we can see the drift velocity in the two cited points. The time evolution of averaged particles velocity and drift velocity are presented in Figure 2 for each axial point.

Figure 1 Particles velocity close the upper and lower plates calculate with DNS and the averaged drift velocity used in the ADE model – Stokes number Stf=0.01.

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Figure 2 Particles velocity, averaged on horizontal slice, for three times instants (dashed lines). Solid line is time average of the last velocity. This figure is obtained for Stokes number Stf =0.01

We observe that the drift velocity depends linearly on the axial coordinate. We have tested this hypothesis for three DNS in which we have set Stokes number, based on large time scale, to Stf =0.01, Stf =0.005 and Stf =0.001. The second hypothesis to model the drift velocity is that it depends on the settling velocity. The dimensionless settling velocity vs et is defined as the particle velocity in a fluid at rest when it has acceleration equal to zero. The algebraic expression of settling velocity is obtained using the equation (3). The linear mathematical expression that accounts for these two hypotheses is written as follows:

St ~ ~ ~ f 2 vdrift = 1.3vset (z −1); vset = (1− QFr ) (16) Fr 2

Previous drift velocity equation shows a good agreement with DNS data for all the sets of particles investigated (Figure 3).

Figure 3 We present the fit between the averaged particles velocity on horizontal slice and on simulation time and the drift velocity for three Stokes numbers. - Stk =1.3, Stk =0.65 and Stk =0.13.

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The mathematical expression of drift velocity is completely predictive because it depends on the Stokes and Froude numbers only. The dimensionless temperature Q was set to 0.5. We have justified this choice in paragraph (4.1).

3.2 Model for turbulent dispersion

Turbulent diffusion coefficient is modelled using dimensional analysis. Basically, the quantity Dturb is proportional to the characteristic velocity U and to the characteristic length L.

~ ~ ~ Dturb = k1U D LD (17)

Turbulent diffusion is a measure of the flow velocity fluctuations effects on the particles. For this reason, we suppose that Dturb depends on the eddy velocity U and length L, for which we can observe the maximum interaction between the vortical structures and the particles. The quantities UD and LD are estimated using isotropy and homogeneous turbulence theory, because particle deposition in the cylindrical domain mainly depends on core dynamics. Specifically, the thermal layer effects are restricted close to the lower and upper plates. We have seen that Rayleigh-Bénard flow exhibits Kolmogorov turbulence cascade mechanism, in the core region, and Bolgiano turbulence dynamics in the thermal layer. In the isotropic and homogeneous turbulence, the relations between the velocity Uf , length h and time τf at the large scales are related to the Kolmogorov scales, η, Uk, τk, as follows:

1/ 2 η −3/ 4 U k −1/ 4 τ k −1/ 2 ≈ Ra ’ = Re f , = Re f , = Re f ; Re f = ∆ ÷ (18) h U f τ f « Pr ◊ where Ref is the Reynolds number evaluated at larger vortical scales. The analogue relations between the time τf and the time scales of the inertial range τr are:

2/ 3 1/ 3 τ f ≈ h ’ U f ≈ h ’ = ∆ ÷ , = ∆ ÷ (19) τ r « r ◊ U r « r ◊ where r is the characteristic scale of the inertial range. We use equations (18) and (19) to estimate the Stokes number for the inertial range scales (Str = τp/τr) as function of the Stokes number based on the Kolmogorov scales (Stk = τp/τk) and the unknown dimensionless eddy size r = r/h.

−1/ 4 1/ 4 τ p τ p τ k τ f ≈ Ra ’ ~ −2 / 3 τ p τ p τ f ≈ Ra ’ Str = = = Stk ∆ ÷ r , Stk = = = St f ∆ ÷ (20) τ r τ k τ f τ r « Pr ◊ τ k τ f τ k « Pr ◊

Stokes number is a measure of particle response to fluctuations in the velocity field. Specifically, particles with time scale on the same order as the fluid time scale - unity Stokes number - may be preferentially concentrated [13]. The size of the eddy, that interacts mainly with the particles and is in the inertial range if the Stokes number, based on Kolmogorov time scale, is greater than unity. For these particles, we can enforce the condition Str = 1 in the equation (20) to predict the eddy size for which we have the maximum interaction between vortical structures and particles.

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−3/8 ~ 3/ 2 ≈ Ra ’ r = Stk ∆ ÷ (21) « Pr ◊

With the previous dimensionless vortical size we can calculate, using the equations (17) and (19), the turbulent diffusion coefficient:

−1/ 2 ~ 2 ≈ Ra ’ Dturb = k1Stk ∆ ÷ Stk > 1 (22) « Pr ◊

Kolmogorov length scale is the smallest eddy size in the flow domain. For this reason, the effect of turbulence on particle path, is due to the Kolmogorov dissipative eddies, if the Stokes number, based on Kolmogorov scale, is less than or equal to unity. For these particles, we can enforce the conditions UD =Uk and LD =η (equation 17) obtaining the turbulent diffusion coefficient Dt urb as follows:

−1/ 2 ~ ≈ Ra ’ Dturb = k1∆ ÷ Stk ≤ 1 (23) « Pr ◊

4 RESULTS AND DISCUSSION

4.1 Description of flow field

Axial velocity field developed in the cylindrical domain with aspect ratio Γ=1/2, Rayleigh number equal to Ra=2·108 and Prandtl number set to Pr=0.7, is presented in Figure (4). In this figure, we can observe three instantaneous horizontal sections in which the axial flow velocity is plotted. Specifically, we have examined the section close to the lower plate (z=0.05), the section midway between the plates (z=0.5) and the section close to the upper plate (z=0.95). The values of the flow axial velocity was used to colour the stream tracers presented on the same figure. These stream tracers show that the flow field is organized in two kind of vortical structures. The first, indicated in Figure (4) as (A), fills the whole cylindrical domain, and the others are toroidal, indicated in figure as (B), close to the corner of lower and upper plates. The flow presented is very interesting, because it develops no- axisymmetric vortical structures in the axisymmetric cylindrical domain. Also temperature field exhibits very interesting behaviour. The Figure (5) shows the contours of instantaneous temperature field on a vertical section.

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Figure 4 Instantaneous snapshot of flow field axial Figure 5 Vertical section of dimensionless temperature velocity in three horizontal planes at halfway between field. In this instantaneous snapshot we present the the plates and close to the lower and upper plate. The contours of dimensionless temperature Q: blue line stream tracers are plotted using the flow field axial Q[0;0.45], green line Q[0.45; 0.55] and red line velocity. Q [0.55; 1].

The contours are plotted in three ranges. The contours with blue lines are used to plot dimensionless temperature Q in the range [0, 0.45]; green lines are used to plot Q[0.45, 0.55], and red lines are used to plot Q[0.55, 1]. We can observe that, in the core region of cylindrical domain (Figure 10), the dimensionless temperature Q is close to 0.5. Specifically, the thermal layers effects are restricted close to the lower and upper plates, because they are very thin. This result implies that we can set Q=0.5 in drift and settling velocities equations (Equation 16) to estimate the averaged behaviour of particles deposition in the domain.

4.2 Phenomenology of particle dispersion

In this paragraph we have analysed particles interaction with the larger vortical structures to evaluate the macroscopic effect of the velocity flow field on particles transport and deposition. Specifically, we focused the analysis on the vertical section of cylindrical domain, because the flow effects on particles deposition are developed mainly along the axial direction. For this reason, we refer to the vertical slab that crosses the diameter of the cylindrical domain (Figure 6).

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Figure 6 Instantaneous slab of particles distribution. The vertical slab crosses the diameter of cylindrical domain. The particles are coloured using their axial velocity.

This instantaneous snapshot shows particles distribution for Stokes number, based on Kolmogorov scale, Stk = 1.3. We have used only the particles with Stokes number equal to 1.3, because the aim of this paragraph is to present qualitatively the interaction between the vortical structures and the particles. We have plotted the particles using circle with schematic size. They are coloured with their axial velocity. The Figure 6 shows that the particles go up on the right side of cylindrical domain and go down on the left one. This transport mechanism is in agreement with the vortical structures described in the previous paragraph. Particles follow the larger vortical structures that is responsible for their deposition. We observe that particles interact also with the toroidal vortices close to the lower and upper plates. Specifically, particles cluster close to the toroidal eddies. These vertical structures force back again the particles in the cylindrical domain.

4.3 Advection - Diffusion model

The equation to predict the time evolution of particles concentration field in Rayleigh-Bénard flow (Equations 15, 16, 22, 23) is function only of the Stokes, Froude, Rayleigh and Prandtl numbers only. In this way, ADE model is completely predictive because does not need the flow field information. The Figures (7), (8) and (9) show the DNS, dashed line, and the ADE, solid line, time concentration evolution for three Stokes numbers, based on Kolmogorov scale, Stk = 1.3, Stk = 0.65, Stk = 0.13.

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I I I

II II II

Figure 7 Particle concentration Figure 8 Particle concentration Figure 9 Particle concentration obtained using DNS (dotted line) obtained using DNS (dotted line) obtained using DNS (dotted line) and the one dimensional model and the one dimensional model and the one dimensional model (solid line) for Stokes number (solid line) for Stokes number (solid line) for Stokes number Stk =1.3. Stk =0.65. Stk =0.13.

The fit between DNS data and ADE equation is obtained for k1=1 (eq. 15). In these figures we can see the good agreement between the DNS and the ADE approach for all the sets of particles investigated. Yet, we can distinguish two regions: the region close to the upper plate (I) and the region in the middle domain and close to the lower plate (II). In the three simulated cases, the region (I) occupies about 20 % of the cylindrical domain and the region (II) the remaining 80 %. For the three analysed cases, turbulent diffusion coefficient Dt urb and drift velocity vd rift are responsible, respectively, for the different slope of concentration profile near the upper plate and for the rate quickness of decrease of concentration. Specifically, the effect of turbulent coefficient on the concentration profile is played through concentration gradient. In the early instants, the axial concentration gradient is due to the perturbation of concentration field enforced as initial condition. The boundary condition at the upper plate keeps the concentration gradient, in the region (I), for the whole investigated time. The concentration time history, restricted to region (II), shows that the concentration value, after the early instants, is constant along the axial direction and its axial gradient is zero. This behaviour, common for the three kinds of particles, physically means that the turbulent diffusion moves the concentration along its gradient and, once the initial perturbation is eventually smoothed, the turbulent diffusion keep this stable configuration. These dynamics imply that, close to the lower plate, we can write the ADE equation, neglecting the axial derivative terms, as follows:

−1/ 4 À∂φ St ≈ Ra ’ Œ = −1.3 k (1− 0.5⋅ Fr 2 )∆ ÷ φ Œ ∂t Fr 2 « Pr ◊ Œ 1/ 4 Œ » St ≈ Ra ’− ÿ …−1.3 k (1−0.5⋅Fr2 )∆ ÷ Ÿ⋅t Ã … Fr2 « Pr ◊ Ÿ Œ Ω φ = e ⁄ (24) Œ Œ ÕŒφ = 1 t = 0

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The Figure (10) shows that the last equation is in good agreement with DNS results.

Figura 10 Time evolution of particles concentration close to the lower plate for the Stokes numbers equal to Stk = 1.3, Stk = 0.65 and Stk = 0.13. The dashed line indicate the DNS results and the solid line indicate the model prediction.

In this figure we have plotted the time evolution of the mean concentration value close to the lower plate and the solution of the equation (24) for particles with Stokes number, based on Kolmogorov scale, Stk = 1.3, Stk = 0.65, Stk = 0.13. We have set two Stokes number less than unity to validate the hypothesis for which the diffusion coefficient for particles with stokes number less than unity is the same. Specifically, this mean that, for all of this kind of “light particles”, the vortical structures with dissipative Kolmogorov size are responsible for the turbulent diffusion. In the case of Stokes number greater than unity, the turbulent diffusion depends on Stokes number because the vortical size, responsible for maximum particle dispersion, changes the size in the inertial range.

5 CONCLUSION

We have used the Advection-Diffusion approach (ADE) to predict the time history of the mean particles concentration value in the Rayleigh-Bénard flow. This model shows a good agreement against DNS simulations for three Stokes number, based on Kolmogorov scale, Stk = 1.3, Stk = 0.65, Stk = 0.13. In this paper, the drift velocity and the turbulent diffusion coefficient are computed using only the set-up values to define the flow field (Rayleigh number, Prandtl, Froude numbers) and to define the particles (Stokes number). This assumption is crucial and distinguishes our methodology from the recent ADE application to predict the particle concentration. For instance, Young and Leeming [12] proposed ADE model for particle dispersion in turbulent pipe flow computing the drift velocity with

Proceedings of the International Conference “Nuclear Energy for New Europe 2005” 173.15 momentum balance equation. Specifically, in our case, drift velocity changes linearly with the axial direction and is defined once the settling velocity is known. The eddies effect on the particles are accounted in the turbulent diffusion coefficient, that was estimated with dimensional analysis. We have used the characteristic turbulent velocity and characteristic length based on Kolmogorov theory to model the turbulent diffusion coefficient, because the core of Rayleigh-Bénard flow was dominated by Kolmogorov turbulence cascade mechanism. The presented ADE method is a useful tool to predict particles deposition in Rayleigh-Bénard flow in a slender closed cylindrical domain that models the radioactive aerosol deposition in the passive containment cooling system of a nuclear reactor.

ACKNOWLEDGEMENTS

One of the authors (P. O.) is grateful to DIASS (Taranto, Italy) for the financial support and to Marina Campolo and Andrea Giusti for the useful discussions. The support of CIFI and DEM (Udine, Italy) is gratefully acknowledged.

REFERENCES

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