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## Plumes in confined environment

Although a lot of research efforts have been devoted to the study of self-similar solutions of plumes rising in free space, the laboratory experiments to produce these plumes were carried out in confined spaces. The bounded wall creates the recirculating flow along sidewalls and the entrainment of underlying fluid. Despite the contributions of the past experimental studies, much remains to be learned about the interaction of the plume motion with its surrounding. The dynamics of a continuous buoyant release in a volume of limited extent are very different to that of a release in a free space. In a confined region the environment is increasingly modified as the flow continues and the behavior of the source is also affected by this modified environment.

### Estimation of the physical properties

The diffusion coefficient and the dynamic viscosity of a mixture can be calculated via the component diffusion coefficients and dynamic viscosities, and component massfractions. We give their estimations for the case of a gas mixture and a glycerolwater mixture. We repeat here that in the case of helium-air mixture, helium is denoted by species 1 and air is denoted by species 2. In the case of glycerol-water mixture, glycerol is denoted by species 1 and water is denoted by species 2.

#### Velocity-pressure coupling: fractional-step projection method

The velocity-pressure coupling is handled by a fractional-step projection method.

The projection method was first introduced by [Chorin, 1968] to solve the timedependent Navier-Stokes equations for an incompressible fluid. In each time level the method is composed of two sub-steps. In the first sub-step the pressure is treated explicitly or ignored and a provisional velocity field is calculated. In the second substep the pressure is corrected by projecting the provisional velocity onto the space of divergence-free vectors. The method of [Chorin, 1968] is called the non-incremental form since the pressure is ignored in the first sub-step and the new pressure is calculated in the second sub-step. [Kan, 1986] proposed a second-order incremental pressure-correction scheme in which the pressure is treated explicitly and a pressure increment is solved in the second step. We make use of the incremental formulation in our numerical method.

**Determination of the inflow boundary condition**

At the inflow boundary the velocity wi(r, t) and concentration profile Y1i(r, t) have to be specified. First of all, let us consider a laboratory experiment of fluid injection in an enclosure. Before reaching the inflow boundary, the injected mass has to go from a source of mass Sm through an injection tube as is depicted in Fig. 3.4. Only the mass flux at the source is controlled where an averaged velocity w and a mass fraction Y1 are specified. However, in the computation the injection tube is not modeled and we have to specify the velocity and the mass fraction profiles at the Inflow so that the total mass flux injected in the system is exactly what has been injected at the mass source.

**Implementation of the numerical method, code performance**

The numerical method described in this chapter has been implemented in a 2D sequential code written in FORTRAN language. Two versions of the code have been developed corresponding to the glycerol-water case where the divergence is zero and the helium-air case where the divergence is different from zero. The code is run on a Linux platform. Its performance in the two cases mentioned above is reported in Table 3.1. We can see that the CPU time for glycerol-water case is approximately ten times the CPU time for helium-water case. This is because the Schmidt number in glycerol-water case is very large and the multigrid solver requires more cycles to converge to the same residual (10−12) in this case (17 cycles for glycerol-water case compared to 3 cycles for helium-air case).

**Table of contents :**

**Résumé**

0.1 Introduction

0.2 Modélisation physique

0.3 Méthode Numérique

0.3.1 Discrétisation spatiale

0.3.2 Discrétisation temporel

0.3.3 Couplage vitesse-pression: méthode de projection

0.4 Détermination des conditions aux limites entrantes

0.4.1 Flux diffusif négligeable (cas du mélange eau-glycérol) . . . .

0.4.2 Flux diffusif important (cas du mélange air-hélium) . . . . .

0.5 Panaches démarrants laminaires à haut nombre de Schmidt: comparaison avec les données expérimentales

0.5.1 Paramètrage numérique et méthode d’analyse

0.5.2 Comparaison avec l’expérience de [Rogers and Morris, 2009] .

0.6 Jets flottants du mélange hélium-air ou de l’hélium dans une cavité semi-confinée remplie d’air

0.6.1 Simulations plan du panache

0.6.2 Simulations du panache axisymmetric

0.7 Conclusions

Introduction

**1 State of the art **

1.1 Buoyant convection from isolated sources

1.2 Vertical steady plumes in a uniform unconfined environment

1.2.1 Laminar plane plumes

1.2.2 Laminar round plumes

1.2.3 Transition from laminar to turbulent plumes

1.2.4 Turbulent plumes

1.3 Laminar starting plumes

1.4 Plumes in confined environment

1.4.1 Theoretical models

1.4.2 Validation of the models

1.5 Numerical study of plumes

1.6 Motivation

**2 Physical modeling **

2.1 Governing equations for a binary mixture fluid flow at constant pressure and temperature ( [Taylor and Krishna, 1993])

2.1.1 Mass transport equation

2.1.2 Momentum equation

2.1.3 Equation relating the density and the mass fraction

2.1.4 Low Mach approximation

2.2 Estimation of the physical properties

2.2.1 The case of a gas mixture

2.2.2 The case of a glycerol-water mixture

2.3 The complete set of governing equations in dimensionless form .

2.4 Dimensionless parameters

**3 Numerical methods **

3.1 Spatial discretization

3.2 Temporal discretization

3.3 Velocity-pressure coupling: fractional-step projection method

3.4 Determination of the inflow boundary condition

3.5 Solution of the algebraic equations

3.5.1 Alternating Direction Implicit (ADI) method

3.5.2 Multigrid method

3.6 Implementation of the numerical method, code performance

3.7 Conclusion

**4 Laminar starting forced plumes at high Schmidt numbers: validation with experimental data **

4.1 Introduction

4.2 Description of the experiment of [Rogers and Morris, 2009]

4.3 Numerical set up

4.3.1 Numerical set up

4.4 General plume characteristics

4.4.1 Plume outer shape

4.4.2 Plume ascent velocity

4.4.3 Plume internal structure

4.5 Parametric study for the numerical simulation

4.5.1 Influence of variable viscosity

4.5.2 Influence of the inlet velocity profile

4.5.3 Influence of the convection schemes

4.5.4 Influence of the spatial resolution

4.6 Comparison with the experiment of [Rogers and Morris, 2009] .

4.6.1 General plume shape

4.6.2 Ascent velocity

4.6.3 Morphology of the plume heads

4.7 Conclusions

**5 Buoyant jets of helium-air mixture or helium in a partially confined air-filled cavity **

5.1 Introduction

5.2 Plane plume simulations

5.2.1 Physical configuration and numerical set-up

5.2.2 Flow description

5.2.3 Transition to unsteadiness

5.2.4 Comparison with scaling laws and similarity theory of laminar,plane plumes

5.3 Axisymmetric plume simulation

5.3.1 Description of the test cases

5.3.2 Description of the flow

5.3.3 Comparison with the finite element results for cavity 3/5 .

5.3.4 Comparison with GAMELAN experiment and other numerical results

5.3.5 Comparison with the model of [Worster and Huppert, 1983] .

5.4 Conclusions

**Conclusion **

List of Figures

List of Tables

A Grid convergence study of helium-air round plume simulations

**Bibliography **