Roe‐type Riemann solver for gas–liquid flows using drift‐flux model with an approximate form of the Jacobian matrix

Abstract

This work presents an approximate Riemann solver to the transient isothermal drift-flux model. The set of equations constitutes a non-linear hyperbolic system of conservation laws in one space dimension. The elements of the Jacobian matrix A are expressed through exact analytical expressions. It is also proposed a simplified form of A considering the square of the gas to liquid sound velocity ratio much lower than one. This approximation aims to express the eigenvalues through simpler algebraic expressions. A numerical method based on the Gudunov's fluxes is proposed employing an upwind and a high order scheme. The Roe linearization is applied to the simplified form of A. The proposed solver is validated against three benchmark solutions and two experimental pipe flow data. Figure 4. Comparison of the numerical solution for the velocity with the reference presented in [17] for t = 0.8 s. (a) Upwind; (b) Lax Wendroff + van Leer limiter.

A High-Order Localized Artificial Diffusivity Scheme for Discontinuity Capturing on 1D Drift-Flux Models for Gas-Liquid Flows

“…where the expression of the speed of sound of the mixture c m is given in [45]. Numerical experiments led to the choice C ρ g = C ρ l = 0.1.…”

“…For an accurate numerical solution, it would be desirable to have an upwind resolution of all the waves inherent in the two-phase model, for example, building an approximate Riemann solver of Roe [43]. However, this is relatively complex, even for simpler drift-flux two-phase models [22,45]. For the driftflux two-phase models considered in this paper, since u g = u l , there is no tractable exact expression for the Jacobian matrix, as well as for the eigenvalues of the system of equations [50].…”

“…where the eigenvalue λ n max represents the maximum characteristic velocity of the system of equations at a given time level n. Although in drift-flux two-phase models the possibility of obtaining an analytical expression of the Jacobian matrix is quite difficult, it was possible to obtain approximate expressions for the eigenvalues of the isothermal drift-flux model [45]. Thus, the approximate characteristic velocities of the system of Eqs.…”

“…Thus, the approximate characteristic velocities of the system of Eqs. (2.1a)-(2.1c) can then be calculated following the formulas presented by Santim and Rosa [45].…”

“…However, the eigenvalues were evaluated numerically. Santim and Rosa [45] also proposed a Roe-type Riemann solver, presenting an approximate analytical form for both the Jacobian matrix and the eigenvalues of the system. It is worth mentioning that all the numerical schemes mentioned above are at best second-order accurate.…”