Two-fluid Flow Research Articles

Numerical simulation of two-fluid flow of electrolyte solution with charged deforming interfaces

The authors’ volume-of-fluid based electrokinetic flow model for multiphase flow with fluid–fluid interfaces, recently extended by them to allow for interfacial charge, is refined and more comprehensively validated under dynamic conditions. Validation is performed using (i) an analytical solution for drop relaxation in the presence of ions and interfacial charge, and (ii) published experimental observations of the shape evolution of a critically charged water droplet in air, undergoing Rayleigh instability. Predictions based on two methods of representing the electrical force, in combination with two different numerical treatments of the interfacial charge, are compared. It is shown that the most accurate predictions, when there is interfacial charge, are obtained by using a “modified pressure” formulation (in which part of the Maxwell stress is incorporated into the pressure) in combination with a published treatment for solving the Poisson equation for the electrical potential that is similar to the Ghost Fluid Method; this is therefore the recommended model combination. The choice of method (modified pressure or standard formulation) is found to be unimportant when there is no interfacial charge. Finally, the recommended model is used to predict the steady state deformation of a hexadecane drop in an electrolyte solution during pressure-driven flow through a cylindrical microchannel, when both the channel wall and the drop interface carry electric charge. It is shown that, for low interfacial tension, electrokinetic forces can significantly enhance or reduce the drop deformation, compared with that of a drop with no interfacial charge, when the charges on the drop interface and channel wall are of the same or opposite sign, respectively.

Asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form

We construct families of approximate solutions to the initial value problem and provide complete mathematical proofs that they tend to satisfy the standard system of isothermal one pressure two-fluid flows in 1-D when the data are L1 in densities and L∞ in velocities. To this end, we use a method that reduces this system of PDEs to a family of systems of four ODEs in Banach spaces whose smooth solutions are these approximate solutions. This method is constructive: using standard numerical methods for ODEs one can easily and accurately compute these approximate solutions which, therefore, from the mathematical proof, can serve for comparison with numerical schemes. One observes agreement with previously known solutions from scientific computing (Evje and Flatten, 2003 [16]) We show that one recovers the solutions of these authors (exactly in one case, with a slight difference in another case). Then we propose an efficient numerical scheme for the original system of two-fluid flows and show it gives back exactly the same results as the theoretical solutions obtained above.

Approximation of the two-fluid flow problem for viscoelastic fluids using the level set method and pressure enriched finite element shape functions

The numerical simulation of complex flows has been a subject of intense research in the last years with important industrial applications in many fields. In this paper we present a finite element method to solve the two immiscible fluid flow problems using the level set method. When the interface between both fluids cuts an element, the discontinuity in the material properties leads to discontinuities in the gradients of the unknowns which cannot be captured using a standard finite element interpolation. The method presented in this work features a local enrichment for the pressure unknowns which allows one to capture pressure gradient discontinuities in fluids presenting different density values. The method is tested on two problems: the first example consists of a sloshing case that involves the interaction of a Giesekus and a Newtonian fluid. This example shows that the enriched pressure functions permit the exact resolution of the hydrostatic rest state. The second example is the classical jet buckling problem used to validate our method. To permit the use of equal interpolation between the variables, we use a variational multiscale formulation proposed recently by Castillo and Codina (2014) [21], that has shown very good stability properties, permitting also the resolution of the jet buckling flow problem in the the range of Weissenberg number 0 < We < 100, using the Oldroyd-B model without any sign of numerical instability. Additional features of the work are the inclusion of a discontinuity capturing technique for the constitutive equation and some comparisons between a monolithic resolution and a fractional step approach to solve the viscoelastic fluid flow problem from the point of view of computational requirements.

Using statistical learning to close two-fluid multiphase flow equations for a simple bubbly system

Direct numerical simulations of bubbly multiphase flows are used to find closure terms for a simple model of the average flow, using Neural Networks (NNs). The flow considered consists of several nearly spherical bubbles rising in a periodic domain where the initial vertical velocity and the average bubble density are homogeneous in two directions but non-uniform in one of the horizontal directions. After an initial transient motion the average void fraction and vertical velocity become approximately uniform. The NN is trained on a dataset from one simulation and then used to simulate the evolution of other initial conditions. Overall, the resulting model predicts the evolution of the various initial conditions reasonably well.

Global classical solution for a 3D viscous liquid-gas two-fluid flow model in a half-space

In this paper, we consider the three-dimensional viscous liquid-gas two phase flow model in a half-space $\Omega=\{x\in\mathbb{R}^{3}:x_{3}>0\}$ , where the initial vacuum is allowed. We prove the existence of the global classical solutions when the energy of the initial data is small enough.

Validation of a two-fluid model on unsteady liquid–vapor water flows

This paper is devoted to the validation of a two-fluid two-phase flow model in some highly unsteady situations involving strong rarefaction waves and shocks in water-vapor flows. The two-fluid model and its associated numerical method that were introduced in a previous work are first recalled, and details on the computational scheme and the verification of interfacial mass transfer terms are provided. Consistency with experimental data is checked in three configurations. First, a comparison with the speed of sound in a two-phase mixture is detailed. Afterwards, numerical approximations obtained with the two-fluid approach are discussed and compared with some experimental data documented in the Simpson water-hammer experiment and the high depressurization with flashing associated with Canon experiment.

Effect of viscous cross coupling between two immiscible fluids on elastic wave propagation and attenuation in unsaturated porous media

A central issue in the theoretical treatment of a multiphase system is the proper mathematical description of momentum transfer across fluid-solid and fluid-fluid interfaces. Although recent studies have advanced our knowledge on modeling the coupling behavior between a porous framework and the fluids permeating it, the effect of viscous resistance caused by two-fluid flow on elastic wave behavior in unsaturated porous media still remains elusive. In the present study, the theoretical model developed for describing immiscible two-phase fluid flows in a deformable porous medium related to harmonic wave perturbation is generalized to account for viscous cross coupling due to relative velocity between two adjacent fluids. The corresponding dispersion relations whose coefficients feature all elasticity, inertial-drag, and viscous-drag parameters are then precisely formulated, in a physical context characterizing three compressional waves and one shear wave.To evaluate quantitatively this as-yet unknown effect, numerical calculations are conducted to solve the dispersion relations for Columbia fine sandy loam bearing an oil–water mixture as a function of water saturation and excitation frequency. Our results show that the phase speed and attenuation coefficient of the P3 wave which has the smallest speed is strongly sensitive to the presence of viscous cross coupling, as expected since this wave is attributed primarily to the out-of-phase motion of the two pore fluids. Viscous cross coupling also exerts an impact on the attenuation coefficient of the shear wave and the P1 wave whose speed is greatest, which exhibits two opposite trends at different ranges of low and high water contents. Relative differences in these wave attributes are principally independent of excitation frequency. A sensitivity analysis is carried out to assess how changes in viscous cross coupling affect these differences, revealing that some of them become more significant as viscous cross coupling is stronger.

Mixed convection of electrically conducting and viscous fluid in a vertical channel using Robin boundary conditions

An analytical and semi-analytical method is used to solve two-fluid magneto-hydrodynanamic flow and heat transfer in the presence of a constant electric field in a vertical channel using Robin boun...

Absolute and convective instabilities in double-diffusive two-fluid flow in a slippery channel

Spatio-temporal instability of miscible two-fluid symmetric flow in a horizontal slippery channel is considered. Both fluids have the same density but different viscosity. A smooth viscosity stratification is created by a thin mixed layer between the fluids due to the presence of two species/scalars, which are diffusing at different rates. Our study suggests the existence of a rapidly growing absolute unstable mode for higher viscosity ratio with a highly viscous fluid close to the slippery channel wall. This instability is less stronger in the case of the equivalent single component two-fluid flow. The viscosity stratified single component (SC) and double-diffusive (DD) slippery flows are absolutely unstable for a wide range of parameter values, when a highly viscous fluid is adjacent to the slippery wall and the mixed layer is close to the channel wall with slip. The instability can be either enhanced or suppressed by wall slip and this is dependent on the location of mixed layer, inertial effects, diffusivity and the log-mobility ratios of the faster and slower diffusing species. This suggests that one can achieve early transition to turbulence due to the absolute instability in a viscosity stratified channel flow by making the channel walls hydrophobic/rough/porous with small permeability, which can be modelled by the Navier-slip condition.

Numerical simulations of compressible flows using multi-fluid models

Numerical simulations of two-fluid flow models based on the full Navier–Stokes equations are presented. The models include six and seven partial differential equations, namely, six- and seven-equation models. The seven-equation model consists of a non-conservative equation for volume fraction evolution of one of the fluids and two sets of balance equations. Each set describes the motion of the corresponding fluid, which has its own pressure, velocity, and temperature. The closure is achieved by two stiffened gas equations of state. Instantaneous relaxation towards equilibrium is achieved by velocity and pressure relaxation terms. The six-equation model is deduced from the seven-equation model by assuming an infinite rate of velocity relaxation. In this model, a single velocity is used for both fluids. The numerical solutions are obtained by applying the Strang splitting technique. The numerical solutions are examined in a set of one, two, and three dimensions for both the six- and seven-equation models. The results indicate very good agreement with the experimental results. There is an insignificant difference between the results of the two models, but the six-equation model is much more economical compared to the seven-equation model.

Non-equilibrium condensation of wet steam flow within high-pressure thermo-compressor

In this study unsteady supersonic flow of wet steam through high-pressure thermo-compressor with non-equilibrium homogeneous condensation has been presented. Two-fluid multiphase flow formulations are applied for the simulation of unsteady, supersonic and compressible flow of wet steam. Non-equilibrium homogeneous condensation of we steam flow with considering suitable equations of state for both vapor and liquid is simulated via in-house CFD code. Results have shown that the proposed two-fluid multiphase approach can predict the non-equilibrium condensation of wet steam within high-pressure thermo-compressor. Quantitative validations of the calculated parameters with industrial data are presented. The validation of the in-house CFD code proves its usefulness for the modeling of steam condensing flows within high-pressure thermo-compressors which are very practical in desalination industry.

Analytical and Level Set Method-Based Numerical Study for Two-Phase Stratified Flow in a Pipe

A nondimensional analytical study for fully developed and three-dimensional numerical study of developing viscosity stratified flow is presented, at various values of viscosity ratio η, inlet area fraction of the less viscous fluid α 1,in , and Reynolds number. With increasing α 1,in , a change in the trend of axial variation in interface–from inlet to the fully developed region–is found at α 1,in = α 1,in,c . With increasing η, development length is found to asymptotically decrease for α 1,in = 0.2 and increase for α 1,in = 0.5 and 0.8. A physical model for flow development is also presented for single- and two-fluid flow. A favorable operating condition to reduce the cost of transportation of more viscous fluids by pipe is proposed.

Numerical simulation of particulate flow in interconnected porous media for central particle-heating receiver applications

The use of central solar particle-heating receivers (SPR) in concentrated solar power (CSP) systems offers numerous advantages over other central receiver concepts. The two main advantages are: (1) ordinary particulate minerals can allow collection temperatures approaching 1000°C compared with conventional molten salts which are limited to about 650°C, and (2) the low-cost high temperature particulate material can also be used as the storage medium in a highly cost effective thermal energy storage system. An innovative SPR design invented by researchers at King Saud University and Georgia Institute of Technology allows the particulate material to flow downward through a stationary porous structure where the concentrated solar energy is absorbed. The porous structure reduces the speed of the falling particulate material, i.e. increases its residence time within the receiver, thereby allowing a large temperature rise to be achieved in a single pass. The design increases absorption of the incident solar radiation (i.e. receiver efficiency) and reduces convective heat loss and particle loss (Al-Ansary et al., 2013). A numerical two-fluid solid–gas Eulerian–Eulerian flow model has been used to evaluate the effect of the porous structure on particulate flow through the receiver. The stationary porous structure was modeled using the packed bed concept in the FLUENT commercial CFD code. Two benchmarking experiments were conducted to assess the validity of the two-fluid flow and packed bed model for flow through the porous structure. This paper describes the two benchmarking experiments, and provides a comparison between the model predictions and the measured data. In the first experiment, particle flow rates through variable-width slits at the bottom of a constant head plenum were measured. In the second experiment, a porous layer was placed above the opening at the bottom of the constant head plenum to limit the particle discharge flow. Experiments with two different porous layer heights were conducted using two particle materials with different diameters and densities. The simulation results were in reasonable agreement with the test data.

Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows

In this work, we present a fast and parallel finite volume scheme on unstructured meshes applied to complex fluid flow. The mathematical model is based on a three-dimensional compressible low Mach two-phase flows model, combined with a linearised ‘artificial pressure’ law. This hyperbolic system of conservation laws allows an explicit scheme, improved by a block-based adaptive mesh refinement scheme. Following a previous one-dimensional work, the useful numerical density of entropy production is used as mesh refinement criterion. Moreover, the computational time is preserved using a local time-stepping method. Finally, we show through several test cases the efficiency of the present scheme on two- and three-dimensional dam-break problems over an obstacle.

Stability Analysis of a Gravity Driven Miscible Two-fluid Flow: Role of Wall Slip

A linear stability analysis of a gravity-driven miscible two-fluid flow with matched density and different viscosities, containing a thin mixed layer between the two fluids, down a slippery inclined plane is considered. Results reveal the stabilizing effects on the surface (S), the overlap (O1 and O2) and the TS (not shown) modes that exist for a configuration with more viscous fluid adjacent to the free surface under overlap conditions. There is a possibility of destabilizing the overlap mode (O1) when the mixed layer is very close to the slippery wall, indicating the dual role played by the wall slip. The stabilizing characteristics of slip can be favourably used to suppress the nonlinear breakdown that may happen due to the coexistence of the unstable modes in a flow over a substrate with no slip.

Numerical modelling of three-fluid flow using the level-set method

This paper presents a numerical model for simulation of three-fluid flow involving two different moving interfaces. These interfaces are captured using the level-set method via two different level-set functions. A combined formulation with only one set of conservation equations for the whole physical domain, consisting of three different immiscible fluids, is employed. Numerical solution is performed on a fixed mesh using the finite volume method. Surface tension effect is incorporated using the continuum surface force model. Validation of the present model is made against available results for stratified flow and rising bubble in a container with a free surface. Applications of the present model are demonstrated via a variety of three-fluid flow systems including (1) three-fluid stratified flow, (2) two-fluid stratified flow carrying the third fluid in the form of drops and (3) simultaneous rising and settling of two drops in a third stationary fluid.

MHD Two-Fluid Flow and Heat Transfer between Two Inclined Parallel Plates in a Rotating System.

Two-phase magnetohydrodynamic convective flow of electrically conducting fluid through an inclined channel is studied under the action of a constant transverse magnetic field in a rotating system. The fluids in the two phases are steady, incompressible, laminar, immiscible, and electrically conducting, having different densities, viscosities, and thermal and electrical conductivities. The transport properties of both the fluids are assumed constant. The bounding infinite inclined parallel plates are maintained at different constant temperatures, making an angle ϕ with the horizontal. Approximate solutions for velocity and temperature distributions are obtained by using a straightforward regular perturbation technique. An in-depth study has been done on the effects of rotation parameter, Hartmann number, inclination angle, the ratio of electrical conductivities, and viscosities of two fluids on the flow. It is observed that the effect of increasing rotation is to decrease the primary velocity. Further it is noticed that as the rotation increases, the secondary velocity increases for smaller rotation, while for larger rotation it decreases. It is also found that the temperature distribution decreases as the rotation increases.

Double-diffusive two-fluid flow in a slippery channel: A linear stability analysis

The effect of velocity slip at the walls on the linear stability characteristics of two-fluid three-layer channel flow (the equivalent core-annular configuration in case of pipe) is investigated in the presence of double diffusive (DD) phenomenon. The fluids are miscible and consist of two solute species having different rates of diffusion. The fluids are assumed to be of the same density, but varying viscosity, which depends on the concentration of the solute species. It is found that the flow stabilizes when the less viscous fluid is present in the region adjacent to the slippery channel walls in the single-component (SC) system but becomes unstable at low Reynolds numbers in the presence of DD effect. As the mixed region of the fluids moves towards the channel walls, a new unstable mode (DD mode), distinct from the Tollman Schlichting (TS) mode, arises at Reynolds numbers smaller than the critical Reynolds number for the TS mode. We also found that this mode becomes more prominent when the mixed layer overlaps with the critical layer. It is shown that the slip parameter has nonmonotonic effect on the stability characteristics in this system. Through energy budget analysis, the dual role of slip is explained. The effect of slip is influenced by the location of mixed layer, the log-mobility ratio of the faster diffusing scalar, diffusivity, and the ratio of diffusion coefficients of the two species. Increasing the value of the slip parameter delays the first occurrence of the DD-mode. It is possible to achieve stabilization or destabilization by controlling the various physical parameters in the flow system. In the present study, we suggest an effective and realistic way to control three-layer miscible channel flow with viscosity stratification.

The Improvement of the Wall Temperature Calculation Procedure for the Post-Crisis Region of the NPP Heated Channels

The reliable wall temperature prediction is important for the analysis of operation and safety of both nuclear reactors and direct-flow steam generators with high-temperature fluids. The onset of dangerous temperature excursion is possible in the post-crisis region of the heated channels when the inverted annular or the dispersed flow regime occurs. The wall temperature calculations for both flow regimes are represented. The data of Hammouda, Lapperiere, Chen, Miropolsky, Bennett and some other investigators were processed. The obtained average error and deviation are less than those which take place when special best estimate computer code KORSAR is used. Specified result was reached by the way of the engineering procedures correction and may be applied for the closure correlations improvement which is needed in two-fluid two-phase flow models. The conclusion is made that the correct choice of the integration step and the method of the critical quality determination produce effect on the adequate wall temperature prediction.

Moffatt eddies at an interface

It is shown that an infinite set of eddies can develop near the interface–wall intersection in a two-fluid flow. A striking feature is that the eddy occurrence depends on from what side of the interface the flow is driven. In air–water flows where the viscosity ratio is 0.018, the eddies develop if a driving source is located on (i) the air side for \({\alpha < 100.55^{\circ}}\), (ii) any side for \({100.55^{\circ} < \alpha < 146.55^{\circ}}\), and (iii) the water side for \({146.55^{\circ} < \alpha < 159.05^{\circ}}\), where \({\alpha}\) is the upper interface–wall angle.