Extending a serial 3D two-phase CFD code to parallel execution over MPI by using the PETSc library for domain decomposition

To leverage the last two decades' transition in High-Performance Computing (HPC) towards clusters of compute nodes bound together with fast interconnects, a modern scalable CFD code must be able to efficiently distribute work amongst several nodes using the Message Passing Interface (MPI). MPI can enable very large simulations running on very large clusters, but it is necessary that the bulk of the CFD code be written with MPI in mind, an obstacle to parallelizing an existing serial code. In this work we present the results of extending an existing two-phase 3D Navier-Stokes solver, which was completely serial, to a parallel execution model using MPI. The 3D Navier-Stokes equations for two immiscible incompressible fluids are solved by the continuum surface force method, while the location of the interface is determined by the level-set method. We employ the Portable Extensible Toolkit for Scientific Computing (PETSc) for domain decomposition (DD) in a framework where only a fraction of the code needs to be altered. We study the strong and weak scaling of the resulting code. Cases are studied that are relevant to the fundamental understanding of oil/water separation in electrocoalescers.Comment: 8 pages, 6 figures, final version for to the CFD 2014 conferenc

A robust method for calculating interface curvature and normal vectors using an extracted local level set

The level-set method is a popular interface tracking method in two-phase flow simulations. An often-cited reason for using it is that the method naturally handles topological changes in the interface, e.g. merging drops, due to the implicit formulation. It is also said that the interface curvature and normal vectors are easily calculated. This last point is not, however, the case in the moments during a topological change, as several authors have already pointed out. Various methods have been employed to circumvent the problem. In this paper, we present a new such method which retains the implicit level-set representation of the surface and handles general interface configurations. It is demonstrated that the method extends easily to 3D. The method is validated on static interface configurations, and then applied to two-phase flow simulations where the method outperforms the standard method and the results agree well with experiments.Comment: 31 pages, 18 figure

Experimental and computational studies of water drops falling through model oil with surfactant and subjected to an electric field

The behaviour of a single sub-millimetre-size water drop falling through a viscous oil while subjected to an electric field is of fundamental importance to industrial applications such as crude oil electrocoalescers. Detailed studies, both experimental and computational, have been performed previously, but an often challenging issue has been the characterization of the fluids. As numerous authors have noted, it is very difficult to have a perfectly clean water-oil system even for very pure model oils, and the presence of trace chemicals may significantly alter the interface behaviour. In this work, we consider a well- characterized water-oil system where controlled amounts of a surface active agent (Span 80) have been added to the oil. This addition dominates any trace contaminants in the oil, such that the interface behaviour can also be well-characterized. We present the results of experiments and corresponding two-phase- flow simulations of a falling water drop covered in surfactant and subjected to a monopolar square voltage pulse. The results are compared and good agreement is found for surfactant concentrations below the critical micelle concentration.Comment: 7 pages, 9 figures, to be presented at the ICDL 2014 conferenc

Analysis of the two-fluid model and the drift-flux model for numerical calculation of two-phase flow

Flerfasestrømning er av betydning i en lang rekke anvendelser, blant annet i olje- og gassindustrien, i den kjemiske og i prosessindustrien, inkludert i varmepumpende systemer, samt i sikkerhetsanalyse av kjernekraftverk. Denne avhandlingen analyserer modeller for tofasestrømning, og metoder for numerisk løsning av disse modellene. Den er derfor ett bidrag til å utvikle pålitelige ingeniørverktøy for flerfase-anvendelser. Slike verktøy trengs og forventes av ingeniører i industrien. Den tilnærmede Riemann-løseren framsatt av Roe har blitt studert. Roe-skjema for tre ulike modeller for tofasestrømning har blitt implementert i rammen av en standard numerisk algoritme for løsning av hyperbolske bevaringslover. Disse skjemaene har blitt analysert ved hjelp av referanse-regnestykker fra litteraturen, og ved sammenlikning med hverandre. Et Roe-skjema for den fire-liknings ettrykks tofluid-modellen har blitt implementert, og det har blitt vist at en andreordens utvidelse basert på bølge-dekomponering og fluksdifferanse-splitting virker godt, og gir forbedrede resultater sammenliknet med det førsteordens skjemaet. Det har blitt foreslått et Roe-skjema for en fem-liknings totrykks tofluid-modell med trykkrelaksering. Bruken av analoge numeriske metoder for fire-liknings- og fem-liknings-modellene gjorde det mulig med en direkte sammenlikning av en metode med og uten trykkrelaksering. Numeriske eksperiment demonstrerte at de to framgangsmåtene konvergerte til samme resultat, men at den fem-liknings trykkrelakserings-metoden var betydelig mer dissipativ, særlig for kontakt-diskontinuiteter. Videre viste beregninger at selv om fem-liknings-modellen har reelle egenverdier, så produserte den oscillasjoner for tilfeller der fire-liknings-modellen hadde komplekse egenverdier. Et Roe-skjema har blitt konstruert for driftfluks-modellen med generelle lukningslover. Roe-matrisen er helt analytisk for det tilfellet at man kan anvende Zuber-Findlay-slipp-loven som beskriver boblestrømning. Dermed er dette Roe-skjemaet mer effektivt enn tidligere fullt numeriske Roe-skjema for driftfluks-modellen. En isentropisk diskret-nivå-flerfasemodell har blitt presentert. En diskusjon av hvordan man kan ta hensyn til ulike interfase-trykkmodeller har blitt gitt. Med de passende modellene for interfase-trykk og -fart, var samsvaret svært godt mellom diskret-nivå-modellen og det fem-liknings Roe-skjemaet. Multi-steg- (MUSTA) metoden har som siktemål å komme nær oppstrøms-metodene i nøyaktighet, samtidig som den bevarer enkelheten til sentrerte skjema. Her har metoden blitt brukt på driftfluks-modellen. Når antallet steg økes, nærmer resultatene fra MUSTA-metoden seg det man får med Roe-metoden. De gode resultatene til MUSTA-metoden er avhengige av at man bruker et stort nok lokalt grid. Derfor er hovedfordelen med MUSTA-metoden at den er enkel, snarere enn at man sparer regnetid. En karakteristikk-basert metode for å spesifisere grensebetinglser for flerfase-modeller har blitt testet, og funnet å virke godt for transiente problem.This thesis analyses models for two-phase flows and methods for the numerical resolution of these models. It is therefore one contribution to the development of reliable design tools for multiphase applications. Such tools are needed and expected by engineers in a range of fields, including in the oil and gas industry. The approximate Riemann solver of Roe has been studied. Roe schemes for three different two-phase flow models have been implemented in the framework of a standard numerical algorithm for the solution of hyperbolic conservation laws. The schemes have been analysed by calculation of benchmark tests from the literature, and by comparison with each other. A Roe scheme for the four-equation one-pressure two-fluid model has been implemented, and a second-order extension based on wave decomposition and flux-difference splitting was shown to work well and to give improved results compared to the first-order scheme. The convergence properties of the scheme were tested on smooth and discontinuous solutions. A Roe scheme has been proposed for a five-equation two-pressure two-fluid model with pressure relaxation. The use of analogous numerical methods for the five-equation and four-equation models allowed for a direct comparison of a method with and without pressure relaxation. Numerical experiments demonstrated that the two approaches converged to the same results, but that the five-equation pressure-relaxation method was significantly more dissipative, particularly for contact discontinuities. Furthermore, even though the five-equation model with instantaneous pressure relaxation has real eigenvalues, the calculations showed that it produced oscillations for cases where the four-equation model had complex eigenvalues. A Roe scheme has been constructed for the drift-flux model with general closure laws. For the case of the Zuber-Findlay slip law describing bubbly flows, the Roe matrix is completely analytical. Hence the present Roe scheme is more efficient than previous fully numerical Roe schemes for the drift-flux model. An isentropic discrete-equation multiphase model has been presented. The incorporation of different interfacial-pressure models was discussed, and examples were given. With the adequate models for the interfacial pressure and velocity, the agreement was very good between the discrete-equation model and the five-equation Roe scheme. The flux-limiter centred (FLIC) scheme was tested for the four-equation two-fluid model. Only the first-order version (FORCE) of the scheme was found to work well, but it was rather diffusive. The purpose of the multi-stage (MUSTA) method is to come close to the accuracy of upwind schemes while retaining the simplicity of centred schemes. Here it has been applied to the drift-flux model. As the number of stages was increased, the results of the MUSTA scheme approached those of the Roe method. The good results of the MUSTA scheme were dependent on the use of a large-enough local grid. Hence, the main advantage of the MUSTA scheme is its simplicity. A multiphase characteristic-based boundary-condition method has been tested, and it was shown to be workable for transient problems.PhD i energi- og prosessteknikkPhD in Energy and Process Engineerin

Influence of surfactants on the electrohydrodynamic stretching of water drops in oil

In this paper we present experimental and numerical studies of the electrohydrodynamic stretching of a sub-millimetre-sized salt water drop, immersed in oil with added non-ionic surfactant, and subjected to a suddenly applied electric field of magnitude approaching 1 kV/mm. By varying the drop size, electric field strength and surfactant concentration we cover the whole range of electric capillary numbers ($Ca_E$) from 0 up to the limit of drop disintegration. The results are compared with the analytical result by Taylor (1964) which predicts the asymptotic deformation as a function of $Ca_E$. We find that the addition of surfactant damps the transient oscillations and that the drops may be stretched slightly beyond the stability limit found by Taylor. We proceed to study the damping of the oscillations, and show that increasing the surfactant concentration has a dual effect of first increasing the damping at low concentrations, and then increasing the asymptotic deformation at higher concentrations. We explain this by comparing the Marangoni forces and the interfacial tension as the drops deform. Finally, we have observed in the experiments a significant hysteresis effect when drops in oil with large concentration of surfactant are subjected to repeated deformations with increasing electric field strengths. This effect is not attributable to the flow nor the interfacial surfactant transport

A flashing flow model for the rapid depressurization of CO2 in a pipe accounting for bubble nucleation and growth

Flashing flow is encountered in many industrial systems involving nozzles, valves and decompression of vessels and pipes. In the context of CO2 capture and storage (CCS), the design of safe and efficient CO2 transportation systems requires accurate flashing models, e.g., for safety analysis of pipe fractures and to predict the mass flow through relief valves. We propose a homogeneous flashing model (HFM) for flashing flow accounting for the underlying physical phenomena of the phase change: bubble nucleation, coalescence, break-up and growth. Homogeneous nucleation is modeled using classical nucleation theory and heterogeneous nucleation is approximated with constant rates of bubble creation and mass transfer from liquid to vapor. The flashing flow model is fitted for CO2 pipe depressurization data at various initial conditions. We find that the same, constant model parameters can be applied for the whole set of depressurization cases considered, as opposed to the conventional homogeneous relaxation model which typically is tuned on a case-by-case basis. For depressurization paths where the fluid state passes close to the critical point, we demonstrate that an accurate description of the flashing process along the length of the pipe can only be achieved when both homogeneous and heterogeneous nucleation are accounted for.publishedVersio

Calculation of the interface curvature and normal vector with the level-set method

acceptedVersio

Temperature response during rapid depressurization of CO2 in a pipe: Experiments and fluid-dynamics modelling

The temperature evolution of CO2 during full-bore pipe depressurization events is relevant for risk assessment, e.g., for CO2 transport in the context of CO2 capture and storage (CCS). We analyse and model the temperature evolution for four different initial temperatures between 0 = 4.6 ◦C and 0 = 40 ◦C at supercritical pressures. All the experiments showed an analogous temperature evolution, reaching similar minimum temperatures along the pipe. The warmer the initial temperature, the earlier dry-out and faster temperature recovery was observed. Also, the coldest experiment showed evidence of formation of more dry ice. We employ the homogeneous equilibrium model (HEM) with different heat-transfer correlations and a two-fluid model (TFM) with different slip models in order to understand the observed data. The results indicate that the heat transfer changes significantly with different thermodynamic states for the CO2. Also, the HEM and the TFM with a RELAP-type friction model performed reasonably well at the outlet and at the closed end of the pipe, but none of the tested models were able to fully describe the strong spatial and temporal gradients observed along the pipe during the depressurization.Temperature response during rapid depressurization of CO2 in a pipe: Experiments and fluid-dynamics modellingpublishedVersio

An exponential time-differencing method for monotonic relaxation systems

We present first and second-order accurate exponential time differencing methods for a special class of stiff ODEs, denoted as monotonic relaxation ODEs. Some desirable accuracy and robustness properties of our methods are established. In particular, we prove a strong form of stability denoted as monotonic asymptotic stability, guaranteeing that no overshoots of the equilibrium value are possible. This is motivated by the desire to avoid spurious unphysical values that could crash a large simulation. We present a simple numerical example, demonstrating the potential for increased accuracy and robustness compared to established Runge-Kutta and exponential methods. Through operator splitting, an application to granular-gas flow is provided.acceptedVersio

Experiments and modelling of choked flow of CO2 in orifices and nozzles

In order to accelerate the deployment of capture and storage (CCS), engineers need experimentally validated models, among other things, to predict the mass flow rate in process equipment and flow restrictions like valves, nozzles and orifices. There are few available, relevant data for choked flow in such geometries. To amend the situation, in this work, we report on six pipe-depressurization experiments from a pressure of 12 MPa and a temperature of 25 °C through three sizes (4.5, 9.0 and 12.7 mm) of orifices and nozzles. The results indicate that for the present cases, the choke point is at a non-equilibrium state. In order to predict quasi-steady choked flow in restrictions, the homogeneous equilibrium model (HEM) and the Henry–Fauske (HF) model are commonly used. The HEM often underpredicts the mass flow rate because it does not account for delayed phase transition. Here we develop a delayed HEM (D-HEM) where evaporation starts at the superheat limit described using classical nucleation theory. We then employ the HEM, D-HEM, and HF model in 1D CFD pipe simulations to describe the outflow of depressurization experiments and we also compare with experimental data for converging–diverging nozzles. In the CFD simulations, HF gave the best results, while HEM consistently underpredicted the mass flux. For the nozzle calculations, we found D-HEM to be the best model with a relative absolute error of 2.5 % for the predicted mass flux.Experiments and modelling of choked flow of CO2 in orifices and nozzlespublishedVersio