Nonlinear long-wave interfacial stability of two-layer gas-liquid flow

The flow of two immiscible viscous fluids in a thin inclined channel is considered, in either a cocurrent or countercurrent regime. Following the air-water case, which is found in a variety of engineering systems, we allow the upper fluid to be either compressible or incompressible. The disparity of the length scales and the density and viscosity ratios of the two fluids is exploited through a lubrication approximation of the conservation of mass and the Navier-Stokes equations. As a result of this long-wave theory, a coupled nonlinear system of partial differential equations is obtained that describes the evolution of the interfacial thickness and the leading-order pressure. This system includes the effects of viscosity stratification, inertia, shear, and capillarity, and reduces to the single-phase falling film Benney equation for sufficiently thin liquid films and constant gas density. The case of two incompressible fluids is investigated first. Since the experimental conditions for this effective system are unclear, we consider several ways to drive the flow: either by fixing the volumetric flow rate of the gas phase or by fixing the total pressure drop over a downstream length of the channel, or by fixing liquid flow rate and gas pressure drop. The forcing with prescribed pressure drop results in a single evolution equation whose dynamics depends nonlocally on the interfacial shape. From weakly nonlinear analysis in this case, we obtain the modified Kuramoto-Sivashinsky equation with an additional integral term, influencing the speed of propagation but not the shape of the interfacial wave. For the strongly nonlinear case, admissible criteria for Lax shocks, undercompressive shocks and rarefaction waves are investigated. Through a numerical verification we find that these criteria do not depend significantly on the inertial effects within the more dense layer. The choice of the local/nonlocal boundary conditions appears to play a role in the transient growth of undercompressive shocks, and may relate to the phenomena observed near the onset of flooding. We then perform a linear stability analysis when the gas phase is compressible. The base-state profile for the density is spatially dependent when a pressure drop over the length of the channel is prescribed. The case when zero pressure drop is prescribed is amenable to a normal-mode analysis. When the liquid film thickness is sufficiently thin, the stability matches that of the single-phase falling film case with the exception that the compressible quiescient gas is stabilizing. When the liquid film thickness is sufficently thick, the density mode within the thin gas layer is destabilizing. In the general case, over a finite domain, a general stability diagram of film thickness and pressure drop is found. For sufficently large countercurrent pressure drops, the interfacial mode becomes unstable, with the location of the largest deformation found near the liquid inlet

Instability and dripping of electrified liquid films flowing down inverted substrates

We consider the gravity-driven flow of a perfect dielectric, viscous, thin liquid film, wetting a flat substrate inclined at a nonzero angle to the horizontal. The dynamics of the thin film is influenced by an electric field which is set up parallel to the substrate surface—this nonlocal physical mechanism has a linearly stabilizing effect on the interfacial dynamics. Our particular interest is in fluid films that are hanging from the underside of the substrate; these films may drip depending on physical parameters, and we investigate whether a sufficiently strong electric field can suppress such nonlinear phenomena. For a non-electrified flow, it was observed by Brun et al. [Phys. Fluids 27, 084107 (2015)] that the thresholds of linear absolute instability and dripping are reasonably close. In the present study, we incorporate an electric field and analyze the absolute and convective instabilities of a hierarchy of reduced-order models to predict the dripping limit in parameter space. The spatial stability results for the reduced-order models are verified by performing an impulse-response analysis with direct numerical simulations (DNS) of the Navier–Stokes equations coupled to the appropriate electrical equations. Guided by the results of the linear theory, we perform DNS on extended domains with inflow and outflow conditions (mimicking an experimental setup) to investigate the dripping limit for both non-electrified and electrified liquid films. For the latter, we find that the absolute instability threshold provides an order-of-magnitude estimate for the electric-field strength required to suppress dripping; the linear theory may thus be used to determine the feasibility of dripping suppression given a set of geometrical, fluid, and electrical parameters

Evaporation in a Binary Liquid Falling Film

Thin liquid films are ubiquitous in both nature and industrial applications. We focus on the stability and dynamics of a fluid film of two miscible liquids falling along an inclined plane with one component is evaporating at the free surface. We utilize Navier-Stokes equation, heat equation and vapor-liquid jump conditions at the free surface. We non-dimensionalize the system and apply perturbation theory to obtain evolution equations for film and concentration. As the film travels, the film profile overlaps with the concentration profile, and three types of sharp transition will occur in the film profile: a. film peak speeds up and moves forward, b. film peak moves backward and then moves forward again, c. film will be slowed and two local peaks will form in the film

Numerical simulation of a highly underexpanded carbon dioxide jet

The underexpanded jets are present in many processes such as rocket propulsion, mass spectrometry, fuel injection, as well as in the process called rapid expansion of supercritical solutions (RESS). In the RESS process a supercritical solution flows through a capillary nozzle until an expansion chamber where the strong changes in the thermodynamic properties of the solvent are used to encapsulate the solute in very fine particles. The research project was focused on the hydrodynamic modeling of an hypersonic carbon dioxide jet produced in the context of the RESS process. The mathematical modeling of the jet was developed using the set of the compressible Navier-Stokes equations along with the generalized Bender equation of state. This set of PDE was solved using an adaptive discontinuous Galerkin discretization for space and the exponential Rosenbrock-Euler method for the time integration. The numerical solver was implemented in C++ using several libraries such as deal.ii and Sacado-Trilinos

Numerical simulation methods for phase-transitional flow

The object of the present dissertation is a numerical study of multiphase flow of one fluid component. In particular, the research described in this thesis focuses on the development of numerical methods that are based on a diffuse-interface model (DIM). With this approach, the modeling problem posed by the presence of moving boundaries in the flow domain, namely the interfaces between different phases, can be solved in a way that preserves the characteristic physical features related to the interfaces, such as surface tension and phase transitions. The first, largest part of the dissertation describes how to apply the DIM formulation that has been adopted, commonly identified as Korteweg formulation, in numerical simulations, without altering the physical parameters of the fluid. The issues of stability and accuracy of the solution, which can be severely compromised by the elliptical and dispersive nature of the set of governing equations, are extensively discussed. Therefore, before discretizing the governing equations a transformation of variables is performed, which removes the most important dispersive terms and greatly increases the stability of the numerical method. The latter is tested on several benchmark two-phase flow problems and for various grid refinements, when a Van der Waals equation of state is used and the temperature is in the vicinity of the critical value. To study the behavior of the flow when the temperature and the velocity fields are coupled, not only isothermal but also non-isothermal simulations are performed and analyzed. This includes a phasetransitional flow where the initial temperature field is such that latent heat plays a major role. Next, the feasibility of a combination of the DIM formulation with Large Eddy Simulation (LES) for turbulent multiphase flow, which is typical in several industrial applications, is explored and tested on one of the isothermal flow simulations. First the various subgrid terms resulting from filtering the governing equations are assessed in an a priori analysis, and different models for the most important subgrid terms are evaluated. Subsequently, a real LES is performed with the best subgrid model based on this analysis and its results are compared with filtered results from a direct numerical simulation. The research carried out for DIM and DIM-LES simulations is intended as the first step towards the development of models for interface mass and heat transfer that can be applied in commercial flow solvers for turbulent phase-transitional flow on industrial problems. Therefore, this research represents an ideal bridge towards the last part of the dissertation, in which a CFD (Computational Fluid Dynamics) model is developed and tested for an industrial application of turbulent phase-transitional flow: the direct-contact condensation of superheated steam injected in water. This model is implemented in the commercial CFD software package ANSYS Fluent. The purpose of this work is twofold. On the one hand, a condensation model for the mass transfer rate at the steam–water interface, based on kinetic gas theory, is tested by comparison of the results with experiments conducted at the Department of Mechanical Engineering of TU/e within the scope of the same research project. By testing the phase change model, useful information can be obtained on the grid requirements and the turbulence model. On the other hand, comparison with experiments, also conducted at TU/e, can be made for the case of steam injected in a fully developed turbulent cross-flow of water in a square duct. To this purpose, results are shown for a three-dimensional simulation performed for the assigned geometry of the experimental setup and for one set of operating conditions used in the experiments. All simulations performed with Fluent are based on a Volume-of-Fluid (VOF) multiphase formulation and on the Reynolds-averaged Navier-Stokes (RANS) equations approach for turbulent flow. Both are typically adopted in the industrial two-phase flow CFD