Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity
The authors consider the system of one-dimensional drift-flux equations describing a two-phase flow and written as: \((\alpha_{g}\rho_{g})_{t}+(\alpha_{g}\rho_{g}u_{g})_{x}=0\), \((\alpha_{l}\rho_{l})_{t}+(\alpha_{l}\rho_{l}u_{l})_{x}=0\), \((\alpha_{g}\rho_{g}u_{g}+\alpha_{l}\rho_{l}u_{l})_{t}+(\alpha_{g}\rho_{g}u_{g}^{2}+\alpha_{l}\rho_{l}u_{l}^{2}+p)_{x}=-q\), where \(\alpha_{g}\), \(\alpha_{l}\in \lbrack 0,1]\) are the volume fractions of gas and liquid which satisfy \(\alpha_{g}+\alpha_{l}=1\), \(\rho_{g}\) the gas density, \(u_{g}\) the gas velocity, \(\rho_{l}\) the liquid density, \(u_{l}\) the liquid velocity, \(p\) the pressure for both gas and liquid, and \(q\) the external forces such as gravity and friction. Considering a 1D inviscid, compressible and isentropic liquid-gas two-phase flow on an inclined pipeline under gravity, which leads to special expressions for the external forces \(q\) and pressure \(p\), the authors simplify the above system as the following one written in a conservative form as: \(m_{t}+(m(v-\mu t))_{x}=0\), \(n_{t}+(n(v-\mu t))_{x}=0\), \(((m+n)v)_{t}+((m+n)v(v-\mu t)+(\gamma -1)(m+n)^{\gamma })_{x}=0\), where \(m\) and \(n\) are the gas and liquid masses defined as \(m=\alpha_{g}\rho_{g}\), \(n=\alpha_{l}\rho_{l}\), and \(v\) is the velocity \(v(x,t)=u(x,t)+\mu t\). Riemann initial conditions \((m,n,v)(x,t=0)=(m_{-},n_{-},u_{-})\), \(x<0\), \((m,n,v)(x,t=0)=(m_{+},n_{+},u_{+})\), \(x>0\), where \(m_{-},n_{-},m_{+},n_{+}>0\) are naturally deduced from physical considerations are imposed. The authors rewrite this last problem in a quasi-linear form for which they derive the associated characteristic equation: \((m+n)(\lambda -v+\mu t)((\lambda -v+\mu t)^{2}-(\gamma -1)\gamma (m+n)^{\gamma -1})=0\) and the three eigenvalues. They draw computations to determine and analyze the properties of the Riemann solutions to the last problem. The first main result proves that for any given Riemann initial conditions, there exists a unique solution to the last problem for fixed parameters \(\gamma\) and \(\mu\). Moving to the system: \(m_{t}+(m(v-\mu t))_{x}=0\), \(n_{t}+(n(v-\mu t))_{x}=0\), \(((m+n)v)_{t}+((m+n)v(v-\mu t))_{x}=0\), with the same initial conditions as above, the authors define the notion of generalized delta shock wave solution and they prove the existence of such solution if \(u_{+}<u_{-}\). They give its expression. In the last part of the paper, the authors describe the limits of Riemann solutions to the second system as \(\gamma \rightarrow 1+\).
scientific article; zbMATH DE number 7471746
scientific article