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A new model for two-layer liquid-gas stratified flows in pipes with general cross sections

Sarswati Shah, Gerardo Hernández-Dueñas

Abstract

In this work, we derive a new model for immiscible two-layer gas-liquid stratified flows in pipes with general cross sections. The bottom layer is occupied by an incompressible fluid in liquid phase with hydrodynamics based on a hydrostatic pressure, following a shallow water approximation. The top layer is occupied by a compressible gas, following an ideal gas law leading to conservation of mass, momentum and energy. The two subsystems are linked through non-conservative products, representing momentum and energy exchanges between layers. The hyperbolic properties of the resulting model are analyzed, including the derivation of entropy inequalities, and the approximations of eigenvalues of the corresponding coefficient matrix. Numerical tests are included to demonstrate the merits of the model and the numerical approximations, including well-balancedness, Riemann problems, and perturbations and convergence toward steady states at rest. Besides simulations of water and air where the density difference between layers is significant, a case where such difference is not so pronounced (like gas and liquid hydrogen) is also shown.

A new model for two-layer liquid-gas stratified flows in pipes with general cross sections

Abstract

In this work, we derive a new model for immiscible two-layer gas-liquid stratified flows in pipes with general cross sections. The bottom layer is occupied by an incompressible fluid in liquid phase with hydrodynamics based on a hydrostatic pressure, following a shallow water approximation. The top layer is occupied by a compressible gas, following an ideal gas law leading to conservation of mass, momentum and energy. The two subsystems are linked through non-conservative products, representing momentum and energy exchanges between layers. The hyperbolic properties of the resulting model are analyzed, including the derivation of entropy inequalities, and the approximations of eigenvalues of the corresponding coefficient matrix. Numerical tests are included to demonstrate the merits of the model and the numerical approximations, including well-balancedness, Riemann problems, and perturbations and convergence toward steady states at rest. Besides simulations of water and air where the density difference between layers is significant, a case where such difference is not so pronounced (like gas and liquid hydrogen) is also shown.
Paper Structure (17 sections, 4 theorems, 66 equations, 8 figures)

This paper contains 17 sections, 4 theorems, 66 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Model formulation and its properties
  3. Basic equations
  4. Duct's geometry
  5. Pressure
  6. Saint-Venant averaging
  7. Streamline conditions
  8. Averaging process
  9. The model
  10. Energy conservation and entropy inequality
  11. Hyperbolicity
  12. Numerical Results
  13. Cylindrical pipe - well-balance test
  14. Riemann problem
  15. Perturbation (in water layer) and convergence to a steady state of rest
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let us consider system FEF1. Then the entropy $\eta$ and entropy flux $q$ given by where $I_1 = g \int_{ \mathcal{B}}^{ \mathcal{B}+h} (h + \mathcal{B} -z)\sigma dz$, define an entropy pair. That is, $\eta$ is a convex function of the conserved variables, and satisfy the following inequality

Figures (8)

  • Figure 1: Schematic. Left panel: 3D schematic of the duct's geometry. Right panel: pressure as a function of height.
  • Figure 2: Schematic representation of the eigenvalue locations. Left: case of 4 distinct real eigenvalues. They are located at the intersections of the two curves defined by the polynomials on both sides of \ref{['eq:4thDegreePoly']} (projected onto the horizontal axis). Right: case where one eigenvalue is a double zero, separating the regions of hyperbolicity.
  • Figure 3: Plot of eigenvalues versus $\epsilon$, maintaining other parameters fixed: $u_1 = 1,\; c_1 = 0.5, \;, u_2 = 1.2, \; c_2 = 0.9$. Approximations are also displayed ($\gamma_{k}^\pm, k = 1,2$), together with the bounds $\alpha_\text{min}, \; \alpha_\text{max}$.
  • Figure 4: Left: 3D view of the duct at time $t= 0.06$ with initial conditions, topography and channel's width given by \ref{['eq:IC_wellbalance']}, \ref{['bottombump']} and \ref{['cylinder']} respectively. Right: Profiles of topography and interface (top left), velocities (top right), gas density (bottom left) and gas energy density (bottom right) at $t=0.06$ using $200$ grid points.
  • Figure 5: Solution at $t=0.06$ using $200$ grid points. The initial conditions are given by \ref{['eq:IC_RP']}. Left panel: 3D view of the channel. Right panel: interface (top left), velocities (top right), gas density (bottom left) and gas energy density (bottom right) are shown.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof