Energy-conserving formulation of the two-fluid model for incompressible two-phase flow in channels and pipes

Energy‐consistent formulation of the pressure‐free two‐fluid model

“…Since the PFTFM is directly derived from the original TFM and describes the same physics, we expect that it conserves the same energy as the original TFM. This energy was presented by Buist et al 28 and is given by $$$$ e\left(\mathbf{q}\right)={\rho}_U{g}_n{\tilde{H}}_U+{\rho}_L{g}_n{\tilde{H}}_L+\frac{1}{2}\frac{q_3^2}{q_1}+\frac{1}{2}\frac{q_4^2}{q_2}. $$$$ Here $$$ {\tilde{H}}_U={\tilde{H}}_U\left({A}_U\left({q}_1,{\rho}_U\right)\right) $$$ and $$$ {\tilde{H}}_L={\tilde{H}}_L\left({A}_L\left({q}_2,{\rho}_L\right)\right) $$$ are general geometric terms representing the centers of mass of the upper and lower fluids, respectively (see Appendix A).…”

“…The cross‐sectionally averaged equations can be written in terms of the conservative variables $$$ \mathbf{q}\left(s,t\right) $$$: 28,34 $$$$ \frac{\partial \mathbf{q}}{\partial t}+\frac{\partial \mathbf{f}\left(\mathbf{q}\right)}{\partial s}+\mathbf{d}\left(\mathbf{q}\right)\frac{\partial p}{\partial s}=\mathbf{c}\left(\mathbf{q}\right), $$$$ with $$$$ {\mathbf{q}}^T=\left[{q}_1\kern1.50em {q}_2\kern1.50em {q}_3\kern1.50em {q}_4\right]=\left[\begin{array}{cccc}{\rho}_U{A}_U& \kern1em {\rho}_L{A}_L& \kern1em {\rho}_U{u}_U{A}_U& \kern1em {\rho}_L{u}_L{A}_L\end{array}\right]. $$$$ The cross‐sections are multiplied by the upper and lower fluid densities, $$$ {\rho}_U $$$ and $$$ {\rho}_L $$$, in order to obtain a mass per unit length, and the velocities are multiplied by the cross‐sections and densities to obtain a momentum per unit length.…”

“…As stated above, an expression for the free parameter $$$ \dot{K} $$$ can be determined based on the demand for energy conservation. To this end, we will first give an outline of the steps required to prove that the mechanical energy is a secondary conserved quantity of the PFTFM, based on our proof of energy conservation for the regular TFM 28 . In general, the process involves defining an expression for the mechanical energy, and (based on this expression) manipulating the governing equations of the model to obtain a local energy conservation equation, from which global energy conservation trivially follows.…”

“…In this work, the demand for energy conservation is used to find an appropriate expression for the rate of change of the volumetric flow rate. Since it is known that the TFM solution conserves energy, 28 replicating this property in the PFTFM naturally yields a PFTFM solution consistent with that of the TFM.…”

“…Due to the manner in which the PFTFM is derived from the TFM, it is natural to apply mass‐ and momentum conserving discretizations of the TFM to the PFTFM (unlike for two‐equation models which are not formulated as mass and momentum conservation equations per fluid). In the present paper, we will show that, starting from our recently presented staggered grid energy‐conserving discretization of the TFM, 28 an energy‐conserving semi‐discrete form of the PFTFM can be derived. This includes an expression for the rate of change of volumetric flow rate, as a function of the current solution throughout the domain.…”