Progress in Computational Fluid Dynamics, Vol. 12, Nos. 2/3, 2012
153
Advances in the Large-Eddy and Interface Simulation (leis) of interfacial multiphase flows in pipes D. Lakehal*, M. Labois and C. Narayanan ASCOMP GmbH, 8005 Zurich, Switzerland E-mail:
[email protected] E-mail:
[email protected] E-mail:
[email protected] *Corresponding author Abstract: Interface Tracking Techniques (ITM) are used for stratified slugflow in pipes. ITM is combined with the Very-Large Eddy Simulation (V-LES) approach to cope with turbulence. We first present 2D validation results, and then discuss new results of transition from stratified to slug flow. We will then present results of droplet entrainment in a stratified flow with lower water cut. While turbulence has long been addressed using RANS, we show that new approaches are mature enough to be employed as viable alternatives to statistical approaches, including the filter-based approach, also known as V-LES, which constitutes a blend between URANS and LES. Keywords: V-LES; very-large eddy simulation; LES; large eddy simulation; stratified flow; slug flow; flow transition; droplet entrainment. Reference to this paper should be made as follows: Lakehal, D., Labois, M., Thomas, S. and Narayanan, C. (2012) ‘Advances in the Large-Eddy and Interface Simulation (leis) of interfacial multiphase flows in channels and pipes’, Progress in Computational Fluid Dynamics, Vol. 12, No. 2/3, pp.153–163.
1 Introduction Gasliquid flow appear in various industrial processes and in the petroleum industry in particular, where mixture of gas with associated liquids (light and heavy components of oil, solid particles, hydrates, condensate and/or water) are produced and transported together. During their co-current flow in a pipe the sheared interfaces acquires various characteristic topologies or patterns (Badie, 2000; Hewitt, 2005). Understanding the transition from stratified flow to slug flow is important in hydrocarbon transportation systems, and as such it has constantly stimulated the research in this direction (Valluri et al., 2008). Slug flow is a commonly observed pattern in horizontal and upslope gas liquid flows. The regime is associated with large coherent disturbances, owing to intermittent appearance of aerated liquid masses that fill the pipe cross-section and travel downstream. This causes large pressure fluctuations and variations in flow rates, affecting process equipments. Earlier techniques for the prediction of slugs were based on various linear stability theories. The transition from stratified to slug flow regime has often been associated with the sudden growth of interfacial waves owing to the KelvinHelmholtz instability (Taitel and Dukler, 1976; Barnea and Taitel, 1993), without accounting for wall shear and interfacial stress. The long-wave assumption is still adopted
Copyright © 2012 Inderscience Enterprises Ltd.
in the formulation of 1D lump parameter models, because it lends itself to integral momentum balance. Although the criteria for linear instability based on these theoretical investigations show good agreement with the data as to slug formation, the basic assumptions are still questionable (e.g., Lin and Hanratty, 1986). Issa (2003) used the multi-fluid simulation approach for slugging phenomena in horizontal channels and developed a mechanistic approach to the prediction of hydrodynamic slug initiation, growth and subsequent development into continuous slug flow in pipelines. A systematic study of numerical simulation of slug flow in horizontal pipes using the two-fluid formulation was carried out by Frank (2003), who showed that the formation of slug flow strongly depends on the wall friction of the liquid phase. In simulations using inlet/ outlet boundary conditions, it was found that the formation of slug flow regimes strongly depends on the perturbation in the inlet boundary condition. He also showed that the length of the computational domain plays an important role in slug formation. Besides slug formation, the stratifying annular flow regime occurs in pipeline transport systems of natural gas and condensates, too. In this regime the liquid layer flowing on the wall is subject to droplet dispersion in the gas core. From the flow assurance standpoint, it is essential to understand the relationship between the continuous liquid
154
D. Lakehal et al.
film around the periphery and the thicker liquid pool at the bottom of the pipe (Brown et al., 2008). Until recently, droplet entrainment rates have long been determined with lump-parameter models based on heuristic correlations. In this mechanism, droplets are entrained from the film at the bottom of the pipe and are transported towards the upper part of the pipe where they deposit and form a liquid film, which drains downwards towards the bottom. Thus, there is a process of continuous renewal and drainage of the film in the upper part of the pipe. Replenishment of the film by droplet deposition is then equal, in fully developed flow, to the deposition of the droplets on the surface (Anderson and Russell, 1970). ITM have proven robust and accurate in predicting interfacial flows (Lakehal, 2008), revealing to some extent the occurrence events similar to KelvinHelmholtz driven instabilities leading to sealing in pipe flows. We extend the prospect here by addressing slug formation and subsequent droplet entrainment. Use is made of ITM to bypass the limitations of phase-average models, in particular as to inter-phase friction, combined with large scale, timedependent simulation to better capture turbulence anisotropy and/or non-homogeneity and its transient motion. This combination is arguably more interesting than using the two-fluid approach with k-e model; this is now known as LEIS (Lakehal, 2010), short for Large Eddy and Interface Simulation, and is implemented in the Computational MultiFluid Dynamics (CMFD) code TRANSAT User Manual (2010). LEIS could be used in conjunction with LES or V-LES, short for Very-Large Eddy Simulation. This concept is attractive since it combines the strength of ITM with the advantage of unsteady, large-scale prediction of turbulence, allowing capturing transients of interfaces and associated turbulence, while minimising modelling efforts of both turbulence and interface dynamics. We discuss here the results of two-phase flow regime transition to slug in a 37 m long pipe under turbulent flow conditions. The data were extracted from the WASP facility of Imperial College (Ujang, 2003; Ujang et al., 2006). Then we present first results of droplet entrainment and detachment in a low-water-holdup stratified pipe flow under turbulent, strongly sheared interfacial conditions.
2 Model description 2.1 Multiphysics code TransAT© TransAT© is a multi-physics, finite-volume code based on solving single and multi-fluid NavierStokes equations on structured multi-block meshes. The grid arrangement is collocated and can thus handle more easily Body-Fitted Coordinates (BFC) grids. The solver is pressure based (Projection Type), corrected using the Karki-Patankar technique for subsonic to supersonic compressible flows. High-order time marching and convection schemes can be employed; up to 3rd order Monotone and TVD-bounded schemes in space and 5th order RK in time. An algebraic
multigrid algorithm is employed for the pressure equation, involving relaxation, restriction and prolongation up to six levels, to achieve high rates of convergence and error estimation. Multiphase flows can be tackled using: ••
interface tracking techniques (Level Set, VOF with interface reconstruction, and Phase Field)
••
N-phase, phase-averaged mixture with Algebraic Slip
••
Lagrangian particle tracking (one-to-four way coupling).
Re-distancing of level set is ensured via 3rd order WENO scheme in Cartesian meshes and via fast marching BFC grids. Mass conservation is enforced using global and local mass and volume conserving schemes. Besides BFC meshing, TransAT also uses the Immersed Surfaces Technique (IST) to map complex geometries into a rectangular Cartesian grid (IST was used in the 3D problems treated here). Near wall regions are treated with Block Mesh Refinement (BMR), a sort of geometrical multi-grid approach in which refined grid blocks or manifolds are placed where adequate. This multigrid is different from the algebraic multigrid algorithm discussed above, in that each BMR block is separately solved and the solutions are passed from one block to the next as an initial flow field. The connectivity between blocks can be achieved in parallel up to 8-to-1 cell mapping. The combination IST/BMR saves up to 70% grid cells in selected 3D problems and allows for conjugate-heat transfer and rigid body motion.
2.2 On the V-LES strategy for turbulent flow Very-Large Eddy Simulation (V-LES) (Labois and Lakehal, 2011) also known as filter-based approach (Johansen et al., 2004) is based on the concept of filtering a larger part of turbulent fluctuations as compared with V-LES. This directly implies the use of a more elaborate sub-scale statistical modelling strategy than a zero-equation model like in LES. The V-LES implemented in TransAT is based on the use of k-e model as a sub-filter model. The filter width is no longer related to the grid size (~Dx); instead it is made proportional to a characteristics length-scale (L), which should be larger than (Dx), but necessarily smaller than the macro length-scale of the flow. Increasing the filter width beyond the largest length scales will lead to predictions similar to the output of RANS models, whereas in the limit of a small filter-width (approaching the grid size) the model predictions should tend towards those of LES. If the filter width is smaller than the length scale of turbulence provided by the RANS model, then larger turbulent flow structures will be able to develop during the simulation, provided that the grid resolution and simulation parameters are adequately set (in particular regarding time stepping and the order and accuracy of the time marching schemes employed). For a more detailed presentation and a discussion on the values of the model constants, the reader can refer to
155
Advances in the Large-Eddy and Interface Simulation (leis) of interfacial multiphase flows in pipes Johansen et al. (2004) and Labois and Lakehal (2011). One of the key hypotheses in V-LES though is that the Kolmogorov equilibrium spectrum is supposed to apply to the sub-filter flow portion too.
onset on the instability, Thorpe observed the growth of the waves for approximately 0.52 s, before rolling up. At this time the amplitude of the waves was about 2amax = 6–8 mm (Fig. 1). The downward wave speed was also measured to 2.6 cm/s.
2.3 On interface tracking Interfacial flows refer to multi-phase flow problems that involve two or more immiscible fluids separated by sharp interfaces, which evolve in time. Typically, when the fluid on one side of the interface is a gas that exerts shear (tangential) stress upon the interface, the latter is referred to as a free surface. ITM’s are best suited for these flows, because the interface topology is accurately resolved. When the exact shape of the interfaces is not relevant one may resort to phase averaged approaches, where separate conservation equations are required for each phase (or as a mixture) with appropriate exchange forces. Both formulations can virtually deal with interfacial flows, but not with the same precision. ITM’s are the cornerstone algorithms of the single-fluid formalism that solves a set of conservation equations with variable material properties (Lakehal et al., 2002). The strategy is rather accurate as it minimises modelling assumptions and deals naturally with heat and mass transfer across interfaces.
3 Model validation: Kelvin-Helmholtz instabilities in Thorpe’s experiment 3.1 Problem description Thorpe’s (1969) experiment is a setup enabling to observe and characterise KelvinHelmholtz instabilities in an interfacial, stratified two-fluid flow. The complete details about the experiment and comparisons with CFD results can be found in Bartosiewicz et al. (2008). The experiment consists of a rectangular channel (Fig. 1) half-filled by water and half-filled by paraffin, with fluid material properties r1, r2 = 783, 1000 kg/m3, m1, m2 = 0.0017,0.001 Pa · s and surface tension s = 0.032 N/m. The channel is then gradually tilted at an angle of b = 4.1289° until surface instabilities develop and grow, before decaying. The authors of the experiment have changed the channel height (3 and 5 cm) and found no appreciable differences as to the main flow-instabilty parameters, including time for onset, critical wave length, wave number wave amplitude and wave speed (tonset, lc, kc, 2amax, c), respectively. Thorpe (1969) took ten pictures of the interface, separated by a time interval of 0.059 s. The first picture is taken at a time of onset of the instability has been observed. This time is tonset = 1.88 ± 0.07 s and includes half the time taken to tilt the channel (about 0.25 s). In his paper, Thorpe claims that this uncertainty might be even larger. The most unstable wave-number is estimated with the distance between two wave crests (Fig. 1) to be equal to lc = 2.5–4.5 cm (uncertainty raises from the different critical wavelengths observed under the same operating conditions). After the
Figure 1 Sketch of the Thorpe experiment
Very recently, the group of UCL Belgium (Schrooyen and Thiry, 2010) has reproduced the experiment of Thorpe at the laboratory scale, using more advanced flow acquisition material and control. This is now called the ‘The refurbished’ ‘Thorpe experiment’. Novel features of this experiment compared with the previous one include: ••
calculated acceleration ramp to minimize initial perturbations
••
high speed camera
••
PIV (2D and stereoscopic)
••
fluids fully characterised in-house (surface tensions, densities, viscosities).
The measured flow-instabilty characteristics from UCL Belgium are shown in Table 1. The main differences with the original Thorpe experiment relate mainly to the values of the most unstable wave-number and time for onset of the instability. Table 1
Flow instability characteristics, experiments vs. CMFD
Flow-instabilty parameters
Thorpe’s exp. (1969)
UCL exp. (2010)
TransAT CMFD
lc [cm] kc [1/m] amax [mm] tonset [s]
2.5–4.5 139.6–251.3 6–8 1.88+/–0.007
3.32–4.45 141.2–189.2 6.1–7.3 2.09–2.19
4.1–4.7 133.7–153.2 6.05–7.23 2.1–2.2
3.2 Simulation setup and results The simulation of the test case has been very recently performed at ASCOMP, and first qualitative results are presented in Figure 2. Use was made of the TransAT CMFD code, using for this flow the Level Set technique to track the interface. High-order schemes were employed
156
D. Lakehal et al.
for space discretisation (Quick) and interface re-distancing (WENO). The time-stepping was adaptive and second-order implicit. The pressure solver was GMRES with SIP as pre-conditioner. The grid employed consisted of 1831 × 31 cells to cover a calculation domain of 1.8 m × 0.03 m. The liquidoil interface was initialised as a sine-wave with different modes with amplitude of 0.2 mm. The fluids properties were set as specified in the experiment, and the channel was then tilted at an angle of b = 4.1289°. Before discussing the comparison, it is perhaps useful to note that this problem is primarily dependent on the way initial conditions were set, in particular as to the interface. Several trials were performed before the right ingredients were found for a workable test-case. Although this piece of work lacks a detailed flow analysis as has been done previously, the results obtained compare very well with the experiment (Fig. 2). The first panels up to t = 2.29 s reflect the period of time during which the interface builds its inherent instability. The panels for t = 2.09 and 2.19 s are important ones, since they reflect the onset of the KH instability. TransAT simulations predict perfectly the trend, but with additional sub-modes that cannot be visualised in the experiment. Beyond the point of onset of instabilty, the prediction of the experimental instabilty characteristics (e.g., inclined steep, travelling waves) is overall excellent, in particular for the wave amplitude and time for onset, as shown in Table 1. Finally, while Thorpe observed that the wave ceases to grow at tdecay ≈ 2.4 s, the UCL experiments show actually that this occurs slightly later, at tdecay ≈ 2.59 s, which is exactly the outcome of the simulations by TransAT, a difference that can only attributed to the difference in fluid properties.
Figure 2 Comparison of interfacial model evolution and propagation (TransAT (b) vs. Experiment performed at UCL (a)) (see online version for colours)
4 Model validation: 2D slug flow formation in the Hawac experiment 4.1 Problem description The Horizontal Air/Water Channel (HAWAC) shown in Figure 3 was devoted to conduct co-current flow experiments. A special inlet device provides defined inlet boundary conditions by separate injection of water and air into the testsection. A blade separating the streams can be moved up and down to control the free inlet cross-section for each phase (Fig. 4), which influences the evolution of the flow regime. The cross-section of this channel is smaller than the one used earlier by Vallée et al. (2008), with dimensions 100 × 30 mm² (height × width). The test-section is about 8 m long, and therefore the length-to-height ratio L/h is 80, or L/Dh = 173 (Dh is the hydraulic diameter). The inlet device (Fig. 4) was designed for separate injection of air (through the upper part) and water (through the lower part) into the channel. As the inlet geometry introduces perturbations into the flow (bends, transition from pipes to rectangular cross-section), four wire-mesh filters were mounted in each part of the device. The filters were made of stainless steel wires with a diameter of 0.63 mm and have
a mesh size of 1.06 mm. The wire-mesh filters were used to provide homogenous velocities at the inlet, and to produce a pressure drop that attenuates the effect of the pressure surge created by slug flow on the fluid supply systems. Air and water streams merge at the trailing edge of a 500 mm long blade separating the phases downstream of the filter segment. The free inlet cross-section for each phase can be controlled by inclining the blade up and down. In this way, the perturbation caused by the first contact between gas and
Advances in the Large-Eddy and Interface Simulation (leis) of interfacial multiphase flows in pipes Figure 3 Schematic view of the horizontal channel with inlet device for a separate injection of water and air into the test-section
Figure 4 Schematic of the inlet device
157
Figure 6 shows the sealing mechanism over a larger portion of the channel, equivalent to Figure 7. Initial ripples (developed in the course of the simulation and not imposed) develop into larger waves, which when travelling at different speeds with the main flow come to merge together, creating a blocking effect leading to sealing or slugging. The slug travels at a speed dependent on the mixture velocity and density difference. The same process repeats itself with new flow events: early ripple-scale instabilities, to long waves, to slugs. The figure also shows other subtle details not visible in the phase-averaged simulations, including the detachment of some ligaments from the upcoming waves and far downstream from the slugs. Because of two-dimensional calculation, the sealing/slugging mechanism could not be reproduced in detail: slugging in a 2D channel tends to produce a higher water cut, since the waves cannot spread laterally. Be it as it may, the comparison with the former calculations and experiment is rather good. Figure 6 Interface contour predicted by TransAT (same conditions): flow zone at the slugging or sealing position (see online version for colours)
liquid can be either minimised or, if required, a perturbation can be introduced (e.g., hydraulic jump).
4.2 Simulation setup and results HAWAC was simulated by TransAT using the level set technique to track the interface. The same space-time discretisation and interface re-distancing schemes as in the previous case were employed. Time-stepping was adaptive and second-order implicit. The grid employed consisted of 1831 × 31 cells covering only a 2D portion (in length) of the calculation domain. The interface was initialised at rest. The fluids properties were those of water and air. As to turbulence, use was made of the V-LES, with a filter scale of 10% the channel height, with the k-e model being employed to model the effect of sub-scale turbulence. The results shown in Figure 5 depict clearly the process evolving from an interface populated by ripples and wrinkles to sealing and slugging. The process is very precise, in that sealing occurs only after three waves have merged together in the direction of the flow. In all cases, the interface ripple-scale instabilities are predicted everywhere in the channel, including upstream the main perturbations, similar to the picture provided by the experiment (Fig. 7). Figure 5 Interface contour predicted by TransAT at JL = 1.0 m/s and JG = 5.0 m/s, with ∆t = 50 ms (depicted large part of the channel) (see online version for colours)
Figure 7 Measured sequences at JL = 1.0 m/s and JG = 5.0 m/s with ∆t = 50 ms (depicted part of the channel: 0–3.2 m from inlet)
5 Slug flow formation in the imperial college wasp facility 5.1 Problem description The experiments were performed at the Imperial College WASP facility with the test section mounted horizontally College (Ujang, 2003; Ujang et al., 2006). Gas and water were fed from two different entries perpendicular to the main pipe (Fig. 8). Slugs were monitored from close to the point where they were first initiated until they decayed or exited the pipe. Twin-wire holdup probes were used to monitor the liquid level at various locations along the pipe.
158
D. Lakehal et al.
Slugs were discriminated from large waves by measuring the velocity using cross correlation of the outputs of successive probes (the waves travel at a lesser velocity that that of the mixture and slugs travel at a velocity higher than that of the mixture). The length of the stainless steel test section is 37 m and its diameter is 77.92 mm, the pressure at the outlet is 1 atm, and the temperature is 25°C. The liquid water is introduced below a stratification plate at the bottom of the test-line and the gas is introduced above it. The superficial velocities of the two phases (air and water) are: UsL = 0.611 m/s and UsG = 4.64 m/s, respectively.
imposed filter width equal to 0.1 D (Labois and Lakehal, 2011). The inflow boundary conditions involve fixing the superficial gas and liquid velocities and the void fraction, as specified in the experiment. Specifically at the inflow, the following values for the turbulent flow conditions were set: gas superficial velocity UsG = 4.64 m/s; liquid superficial velocity UsL = 0.611 m/s; void fraction = 50%. Initial flow disturbances based on the wall shear Reynolds number were applied in the entire flow domain to sustain turbulence.
5.2 Problem setup for CMFD
Figure 9 shows the measured liquid hold-up at different probe locations along the axis: 5.01, 5.69, 6.99 and 13.32 m. The signals display distinct large-wave structures developing along the pipe that could in fact be considered as slug-structures (Ujang, 2003). While a traditional slug is a structure blocking the cross section of the pipe completely, large-wave structures with a length scale larger than the pipe diameter can also be termed as ‘slugs’. The 3D simulations discussed next will help make the distinction between the different structures. Slugs or large-wave structures are captured around location x = 3 m and beyond (results not shown here). The periodicity of slug occurrence is clearly visible from these locations (x > 5 m).
Use was made here of the IST technique to mesh the pipe. The pipe CAD file was created using Rhinoceros software, and immersed into a Cartesian grid, as shown in Figure 8. The 2D simulations were performed in a pipe of length 17 m. The 3D simulations were performed in a shorter domain of 8 m, consisting of 715.000 cells, then in a longer one of 16 m, consisting of 1.200.000 cells. The simulation time for the 8 m pipe simulation was 10 days on a low-bandwidth Dell PC (2 nodes × 4 cores; Open MP parallel protocol) for 20 s real time, and 53 H on a high bandwidth 18 nodes IBM multicore computer (MPI parallel protocol). The LEIS approach was employed here, with a filter width of 0.1 D, combining the Level-Set technique for interface tracking and LES and URANS for superscale and subscale turbulence, respectively, segregated with an
5.3 Two-dimensional results
Figure 9 Measured liquid holdup for case UsL = 0.611 m/s and UsG = 4.64 m/s. Probe location from 5.01 to 13.319 m (see online version for colours)
Figure 8 Computational IST grid. The CAD file is immersed in a Cartesian grid. Air and water inflow are shown (see online version for colours)
Figure 10 depicts the calculated water holdup in 2D at different probe locations along the axis, from 5.65 to 14.4 m. While the signal is qualitatively similar to the measured one in terms of slug or large-wave structures intermittency, it is unclear whether slugs were indeed captured; various locations exhibit water holdup of about hL/D = 0.8–0.9. Be it as it may, large surface perturbations were already captured upstream close to the inflow, at x = 0.76 m, while the experiment there shows liquid holdup not exceeding hL/D = 0.2. For the locations shown in the figure, the experiment and CFD provide a similar picture as to wave frequency. It is also important to note that in some location immediately beyond full sealing (100% water holdup), e.g., at t = 12.5 s in panel 2, and at t = 37 s in Panels 2 and 3, the surface deformations show a sort of decay in their fluctuations. This can be explained by the
Advances in the Large-Eddy and Interface Simulation (leis) of interfacial multiphase flows in pipes Figure 10 Calculated liquid holdup for case UsL = 0.611 m/s and UsG = 4.64 m/s. 2D (L = 17 m). Probe location from 5.7 to 14.4 m
fact that when the channel/pipe has once totally filled up (HL/d = 1), the slug empties it, and a period of time is then required for the free surface to reestablishes before creating other instabilities.
5.4 Three-dimensional results The objective of the 3D simulations was to reproduce a series of slugs travelling in the pipe. To address the effect of pipe length on the results, the flow was reproduced in two pipes with different lengths: 8 m and 16 m, using the same flow conditions and grid resolution (proportional to the length of the pipe). Figure 11 shows the development structures at different probe locations along the axis, at 5.01, 5.695 and 6.995 m. These were obtained from 3D simulations in the short pipe (L = 8 m). The distinct patterns at different locations show the variations in the slug frequencies. Slug frequency decays as the location of the probe is moved further downstream. Slugs or large-wave structures are predicted rather at downstream locations close to the pipe end: x = 7 m, in contrast to the experiment and longer-pipe simulations, both indicating a shorter position for the early slugs. Further, in contrast to the 2D results discussed earlier, slugs or large disturbances of the surfaces are not predicted upstream close to the inflow, but rather downstream. These results are interpreted later on in terms of slug frequency, and compared to the longer pipe results. Turning now to the 16 m long pipe, the formation of the different types of slugs is well illustrated in Figure 12. The first panel exhibits a ‘large-scale slug’, which, in the experiments is often referred to as ‘operation slug’. This slug is formed upstream (x
