Effect of Collision Force on Well-Posedness and Stability of the Two-Fluid Model for Vertical Bubbly Flows

2016 ◽  
Vol 184 (3) ◽  
pp. 353-362 ◽  
Author(s):  
Avinash Vaidheeswaran ◽  
William D. Fullmer ◽  
Martin Lopez de Bertodano
Keyword(s):  
Fluid Model ◽  
Bubbly Flows ◽  
Well Posedness ◽  
Two Fluid Model ◽  
Collision Force ◽  
Two Fluid

scholarly journals On a model for internal waves in rotating fluids

2018 ◽  
Vol 3 (2) ◽  
pp. 627-648 ◽  
Author(s):  
A. Durán
Keyword(s):  
Internal Waves ◽  
Fluid Model ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Rotating Fluid ◽  
Rotating Fluids ◽  
Mathematical Properties ◽  
Gravity Effects ◽  
Two Fluid Model ◽  
Two Fluid

AbstractIn this paper a rotating two-fluid model for the propagation of internal waves is introduced. The model can be derived from a rotating-fluid problem by including gravity effects or from a nonrotating one by adding rotational forces in the dispersion balance. The physical regime of validation is discussed and mathematical properties of the new system, concerning well-posedness, conservation laws and existence of solitary-wave solutions, are analyzed.

Interfacial Pressure Coefficient for Ellipsoids and Its Effect on the Two-Fluid Model Eigenvalues

2016 ◽  
Vol 138 (8) ◽  
Author(s):  
Avinash Vaidheeswaran ◽  
Martin Lopez de Bertodano
Keyword(s):  
Fluid Model ◽  
Pressure Coefficient ◽  
Well Posedness ◽  
Virtual Mass ◽  
Two Phase Flows ◽  
Two Phase ◽  
Pressure Coefficients ◽  
Interfacial Pressure ◽  
Two Fluid Model ◽  
Two Fluid

Analytical expressions for interfacial pressure coefficients are obtained based on the geometry of the bubbles occurring in two-phase flows. It is known that the shape of the bubbles affects the virtual mass and interfacial pressure coefficients, which in turn determines the cutoff void fraction for the well-posedness of two-fluid model (TFM). The coefficient used in the interfacial pressure difference correlation is derived assuming potential flow around a perfect sphere. In reality, the bubbles seen in two-phase flows get deformed, and hence, it is required to estimate the coefficients for nonspherical geometries. Oblate and prolate ellipsoids are considered, and their respective coefficients are determined. It is seen that the well-posedness limit of the TFM is determined by the combination of virtual mass and interfacial pressure coefficient used. The effect of flow separation on the coefficient values is also analyzed.

scholarly journals Comparison of well-posedness criteria of two-fluid models for numerical simulation of gas-liquid two-phase flows in vertical pipes

2021 ◽  
pp. 158-158
Author(s):  
Naghibi Falahati ◽  
V. Shokri ◽  
A. Majidian
Keyword(s):  
Fluid Model ◽  
Well Posedness ◽  
Fluid Models ◽  
Vertical Pipe ◽  
Numerical Diffusion ◽  
Solution Field ◽  
Two Fluid Model ◽  
Two Phases ◽  
Well Posed ◽  
Two Fluid

The purpose of the present study is to compare the well-posedness criteria of the free-pressure two-fluid model, single-pressure two-fluid model, and two-pressure two-fluid model in a vertical pipe. Two-fluid models were solved using the Conservative Shock Capturing Method. A water faucet case is used to compare two-fluid models. The free pressure two-fluid model can accurately predict discontinuities in the solution field if the problem's initial condition satisfies the Kelvin Helmholtz instability conditions. The single-pressure two-fluid model can accurately predict the behavior of flows in which the two phases are poorly coupled. The two-pressure two-fluid model is an unconditionally well-posed one; if in the free-pressure two-fluid model and single-pressure two-fluid model, the range of velocity difference of two phases exceeds certain limits, the models will be ill-posed. The two-pressure two-fluid model produces more numerical diffusion than the free-pressure two-fluid and single-pressure two-fluid models in the solution field. High numerical diffusion of two-pressure two-fluid models leads to failure to better comply with the problem's analytical solution. Results show that a single-pressure model is a powerful model for numerical modeling of gas-liquid two-fluid flows in the vertical pipe due to a broader range of well-posed than free-pressure models and less numerical diffusion than the two-pressure model.

scholarly journals On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

2018 ◽  
Vol 57 ◽  
pp. 432-447 ◽  
Author(s):  
N. Panicker ◽  
A. Passalacqua ◽  
R.O. Fox
Keyword(s):  
Fluid Model ◽  
Bubbly Flows ◽  
Two Fluid Model ◽  
Two Fluid

A One Dimensional Two-Fluid Model for Bubbly Flows in Fixed Beds

2002 ◽  
Author(s):  
M.-L. Bordas ◽  
A. Cartellier ◽  
P. Se´chet
Keyword(s):  
Pressure Drop ◽  
Void Fraction ◽  
Bubble Size ◽  
Fluid Model ◽  
Bubbly Flows ◽  
Two Phase ◽  
The Mean ◽  
Two Fluid Model ◽  
Fixed Beds ◽  
Two Fluid

Pressure drop and gas void fraction are important parameters for the design of multiphase packed bed reactors which are widely used in petrochemical industry. Several experimental studies have been devoted to the hydrodynamics of two-phase cocurrent upflow or downflow through fixed beds, and various correlations of limited range of validity are available in the literature. However, there is not yet a clear agreement on the form of the momentum equations to be used in such systems. Early attempts devoted to the pressure drop estimate were based on an extension of the Lockhart-Martinelli approach (Sweeney 1967), Rao et al. 1983). More recently, Attou at al. (1999) proposed the first serious attempt to adapt the Eulerian two-fluid model to cocurrent bubbly flows through packed beds. From an analysis of their proposal, it happens that the basic mechanical equilibrium for the gas phase needs to be reconsidered. In this scope, we derived a new model on the basis of the so-called hybrid approach initially developed for bubbly flows in ducts in absence of shear-induced turbulence (Achard and Cartellier 2000). As a first application, we considered a mean unidirectional flow of a bubbly mixture through a porous medium composed of beads uniform in size. For steady and fully established flows, and assuming a flat void fraction (α) profile, the resulting momentum equations for each phase write: Liquidphase:−dpdz=ρLg+fLS−fLG1−α(1)Gasphase:−dpdz=ρGg+fLS+fLGα(2) where fLS is the resultant of the liquid shear stress exerted on beads surface and on exterior walls, and where the quantity fLG = α F* / Vp represents the interaction force density between the gas and the liquid (F* is the mean force on bubbles and Vp = 4πa3/3 denotes the bubble volume, a being the bubble radius). The main difference with the model derived by Attou et al. is the presence of the fLS term in the gas phase equation. Without this term, the relative velocity of bubbles would be controlled by the axial pressure gradient dP/dz even in non accelerating flows which is unphysical. On the opposite, in the present model (1–2) the relative movement of bubbles is simply due to buoyancy. The set of equations (1–2) provides a mean to exploit the experimental data to derive the required closures, namely the evolution of the friction fLS with the gas content and that of the momentum exchange between phases fLG. Notably, from (1) and (2), one gets fLG=α(1−α)(ρL−ρG)g(3) In order to establish reliable closures, available experimental data of the literature are currently revisited under this framework. For the friction term, which is the principal contribution to the pressure drop, the usual closure law for fLS as given by an Ergun equation adapted to two-phase flows is under analysis. For the interfacial momentum transfer, the objective is to evaluate an “apparent” drag coefficient defined as Cd = F*/[ρL Ur2 π a2 / 2] where the mean relative velocity Ur is defined as the difference between the mean gas and liquid velocities averaged over a volume. Indeed, paralleling an approach already exploited for bubbly flows in ducts (Riviere and Cartellier 1999), it happens that the mean void fraction can be derived from equations (1) and (2) assuming a flat void fraction profile: β(1−β)−α(1−α)=(4π/3)α(1−α)[gδ2VSLνc](aδ)2fd(4) where δ is the typical size of the pores and where fd = (π/2) Rep Cd is expected to be a function of the bubble size, the porosity ε and the void fraction. To extract fd or Cd from (4), a characteristic bubble size must be specified. As shown Fig.1, the bubble size is controlled by the bed geometry and evolves between 0.2 δ and 3 δ in the dilute limit (Bordas et al. (2001)). Analysis of the existing data will be presented based on these size estimates, and comparison will be performed of this “apparent” drag with values measured for isolated bubbles in fixed beds (Fig.2).

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