One-dimensional two-equation two-fluid model stability

The one-dimensional incompressible two-fluid model based on physical closures is reduced to simpler two-equation models for horizontal or near-horizontal stratified and vertical bubbly flows. For stratified flow with small density ratio the model may be simplified further to obtain the one-dimensional shallow water theory equations. Characteristic, dispersion, and nonlinear analyses are performed to demonstrate that the models are linearly well-posed and nonlinearly bounded. Linear stability of the two-equation model for horizontal or near-horizontal stratified flow shows that the model is made well-posed by the hydrostatic force for a range of relative velocity surrounding the homogeneous condition. As the relative velocity increases, the model becomes unstable yet well-posed once the kinematic shallow water theory instability occurs, i.e., the viscous Kelvin-Helmholtz instability. However, unlike shallow water theory, the two-equation two-fluid model becomes ill-posed when the relative velocity reaches the dynamic inviscid Kelvin-Helmholtz instability, unless higher order modeling is incorporated, i.e., surface tension. Aside from these well-known results a new analytic expression for the kinematic instability is obtained because of the simplified mathematics of the two-equation model. Linear stability of the two-equation model for bubbly flow including the virtual mass and interfacial pressure forces shows that it is well-posed for low void fractions. It is now demonstrated that it becomes unconditionally well-posed when a bubble collision force is incorporated. It is also demonstrated that the two-equation model for bubbly flow becomes kinematically unstable when the shallow water theory stability condition is reached even though the bubbly flow model is more complicated than shallow water theory. Finally, the nonlinear evolution of the kinematic instability is simulated numerically for a case of Stokes bubbly flow. It is shown that the linearly unstable conditions result in a nonlinearly bounded limit cycle. The bounding mechanism is the viscous stress. Well-posedness and boundedness are of practical significance for nuclear reactor safety code verification because they enable convergence of the numerical two-fluid model. © 2013 by Begell House Inc.