A stability diagram can be made by computing the steady state of the two-fluid model for a range of liquid and gas velocities, and for each steady state computing whether the dispersion relation predicts complex angular frequencies. This is shown in Figure 4.4a and is similar to the map found in [137]. Alternatively, these steady states can be computed as a function of the given pressure gradient and hold-up - this is shown in Figure 4.4b.
The boundary between ill-posedness and well-posedness (shown in Figure 4.4 as the dividing line between the white and gray-shaded areas) is determined by the inviscid analysis which was described
4.7 Conclusion 51
(a) (b)
Figure 4.4: Two maps of the stability regions of the two-fluid model as described in section 3.5. The plots show whether steady states corresponding to the parameters on the axes are stable, unstable, or ill-posed. The dashed lines in (a) are lines of constant holdup. The dashed lines in (b) are lines of constant velocity.
in subsection 4.3.1. If complex eigenvalues result from the homogeneous equations via (4.14), this will lead to an unbounded growth rate (for small wavelength perturbations) and the system is thus ill-posed. This happens when the velocities, driven by the constant background pressure gradient, lie too far apart (according to (4.17)).
In the ‘well-posed stable’ area initial disturbances are damped. This is an area where it is safe to carry out simulations: we can have trust that our model will remain well-posed, as long as the initial disturbances are small enough.
There is an area where the dispersion relation based on the homogeneous equations, or equivalently the inviscid equations, yields real eigenvalues, but the dispersion relation based on the viscous equations yields complex eigenvalues. Here, the two-fluid model is well-posed, but the viscous two-fluid equations are unstable. This is called the ‘well-posed unstable’ area in Figure 4.4b. An initial disturbance will grow according to (4.38). Initial states in this area can often only be analyzed for limited amounts of time with the two-fluid model, since with time these states are likely to evolve into states which are ill-posed. On the other hand, these states might lead to the transition from wavy flow to slug flow.
4.7
Conclusion
If one wants to find a well-behaved solution to the two-fluid model equations, the equations must be well-posed. If the set is hyperbolic, with correct boundary conditions, the time-dependent problem is well-posed. The propagation without growth or damping of a waveform or the existence of a complete set characteristics with all real eigenvalues are sufficient conditions for proving hyperbolicity of a quasi-linear PDE system.
The considered two-fluid model is hyperbolic and thus well-posed when the velocity difference between the liquid and gas phase is not too large. This analysis concerns only the homogeneous part of the equations, which for the two-fluid model coincides with the inviscid equations. Linear stability analysis of the full (viscous) equations shows that the two-fluid model can be unstable and well-posed at the same time.
In our generation of training data for the neural network, we will need to take care that our two-fluid model is well-posed for the chosen parameter ranges, and stable so that there is no danger of ill-posedness arising after some time. For our two-fluid model is useless (produces unphysical results) when the model equations are ill-posed. This is a given with the current two-fluid model. Closure terms aimed at getting the two-fluid model to agree with high-fidelity simulations can do nothing to change the ill-posedness problem, since they are a source term in the model equations (4.8). They do however have an influence on the boundary between the well-posed stable and well-posed unstable regimes shown in Figure 4.4.
A similar linear stability analysis can be done for the incompressible Euler equations for 2D channel flow. There is a difference between the vertically bounded and vertically unbounded case, though they converge for short wavelengths. The inviscid 2D bounded dispersion relation and the inviscid two-fluid dispersion relation converge for long wavelengths. Since the inviscid dispersion relations do not depend on the closure terms, any kind of closure term will do nothing to change the discrepancy between the two-fluid model and the 2D analysis for low wavelengths. Put otherwise, the assumption L H is inherent to the employed two-fluid model, and cannot be remedied by appropriate closure terms.
With the analysis in this chapter, we have thus gained insight into the fundamental limitations of the two-fluid model, which we should not hope to improve upon with our closure terms learned from high-fidelity simulations.
Chapter 5
Viscous Validation
5.1
Introduction
In this chapter, we conduct viscous simulations with both our high fidelity (Gerris) and low fidelity (Rosa) models. We validate our models for the viscous case by comparison to the steady state analytical relations (3.80)–(3.83). By evaluating the convergence of the Gerris simulations to the analytical solution, we can make a well-founded choice for the resolution of the simulations. For Rosa we need to perform wavy unsteady simulations in order to make a good choice of spatial and temporal resolution. We compare Gerris and Rosa wavy unsteady simulations and analyze the results. Additionally, we discuss the complications in extracting the necessary neural network training data, in particular the stresses, from the high fidelity simulations.
We use Gerris to simulate 2D channel flow with periodic boundaries under a constant body force. For pipe flow the boundary conditions at top and bottom are no-slip. The constant body force is a combination of a constant background pressure gradient and a gravitational force arising due to inclination of the domain: BL= ∂p ∂s 0 + ρLg sin(φ), BG = ∂p ∂s 0 + ρGg sin(φ).
The body forces arising due to channel inclination are different for the liquid and the gas. However, we will limit our analysis here to cases driven by solely a background pressure gradient.
The physics behind Gerris is explained in chapter 2 and its numerical details are given in section 2.5. The 1D two-fluid model is described in chapter 3 and the numerical properties of the Rosa code are discussed in section 3.6.