A multiblock joint PDF finite-volume hybrid algorithm for the computation of turbulent flows in complex geometries

2011

J. Fluid Mech.

A probabilistic approach to model macroscopic behaviour of non-wetting-phase ganglia or blobs in multi-phase flow through porous media is proposed. The key idea is to consider a set of stochastic Markov processes that can mimic the microscopic multi-phase dynamics. These processes are characterized by equilibrium probability density functions (PDFs) and correlation times, which can be obtained from microscale simulation studies or experiments. A Lagrangian viewpoint is adopted, where stochastic particles represent infinitesimal fluid elements and evolve in the physical and probability space. Ganglion mobilization and trapping are modelled by a twostate jump process with transition probabilities given as functions of ganglion size. Coalescence and breakup of ganglia influence the ganglion size distribution, which is modelled by a Langevin type equation. The joint probability density function (JPDF) of the chosen stochastic variables is governed by a high-dimensional Chapman-Kolmogorov equation. This equation can be used to derive moment (e.g. saturation, mean mobility etc.) transport equations, which in general do not form a closed system. However, in some special cases, which arise in the limit of one time scale being smaller or larger than the others, a closed set of moment transport equations can be obtained. For slowly varying and quasi-uniform flows, the saturation transport equation appears in closed form with the mean mobility fully determined, if the equilibrium PDFs are known. Furthermore, it is shown how statistical parameters such as mobilization and trapping rates and equilibrium PDFs can be obtained from the birth-death type approach, in which ganglia breakup and coalescence are explicitly considered. A two-equation transport model (one equation for the total saturation and one for the trapped saturation) is obtained in the limit of very fast coalescence and breakup processes. This model is employed to mimic hysteresis in relative permeability-saturation curves; a well known phenomenon observed in the successive processes of imbibition and drainage. For the general case, the JPDFequation is solved using the stochastic particle method, which was proposed in our previous paper (Tyagi et al. J. Comput. Phys. 227, 2008, 6696-6714). Several one-and two-dimensional numerical simulation results are presented to show the influence of correlation times on the averaged macroscopic flow behaviour.

Section: Simulation Resultsmentioning

confidence: 99%

Probability density function approach for modelling multi-phase flow with ganglia in porous media

Tyagi

2011

J. Fluid Mech.

“…are not given here. Interested readers may refer to the numerical schemes those presented in Tyagi et al (2008), Jenny et al (2001), Rembold and Jenny (2006). The simulation test cases are constructed by idealizing the dynamics observed during the post-injection phase of CO 2 storage.…”

Section: Numerical Simulation Resultsmentioning

confidence: 99%

Probability Density Function Modeling of Multi-Phase Flow in Porous Media with Density-Driven Gravity Currents

Tyagi

2011

Transp Porous Med

A probability density function (PDF) based approach is employed to model multi-phase flow with interfacial mass transfer (dissolution) in porous media. The joint flow statistics is represented by a mass density function (MDF), which is transported in the physical and probability spaces via Fokker-Planck equation. This MDF-equation requires Lagrangian evolutions of the random flow variables; these evolutions are stochastic processes honoring the micro-scale flow physics. To demonstrate the concept, we consider an example of immiscible two-phase flow with the non-equilibrium dissolution of single component from one phase into the other-a model for solubility trapping during CO 2 storage in brine aquifer. Since CO 2 -rich brine is denser than pure brine, density-driven countercurrent flow is set up in the brine phase. The stochastic models mimicking the physics of countercurrent flow lead to a modeled MDF-equation, which is solved using our recently developed stochastic particle method for multi-phase flow (Tyagi et al. J Comput Phys 227:6696-6714, 2008). In addition, we derive Eulerian equations for stochastic moments (mean, variance, etc.) and show that unlike the MDF-equation the system of moment equations is not closed. In classical Darcy formulation, for example, the mean concentration equation is closed by neglecting variance. However, with several one-and two-dimensional simulations, it is demonstrated that the PDF and Darcy modeling approaches give significantly different results. While the PDF-approach properly accounts for the long correlation length scales and the concentration variance in density-driven countercurrent flow, the same phenomenon cannot be captured accurately with a standard Darcy model.

“…On the other hand, modeling molecular mixing remains a major challenge. In the following, we present a new approach, which is a combination of a hybrid joint PDF method [3,4,6,10], a transported progress variable, and the flamelet model. The probability density function (PDF) of a progress variable c can be described by a bimodal distribution.…”

Section: Background and Modeling Approachmentioning

confidence: 99%

“…An average of 20 particles per cell was used. All simulations were performed with our PDF simulation platform LEMSBK [10]. Peak values in the inflow velocity profile are 40 in the jet and 10 in the co-flow region.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A Lagrangian Joint PDF Approach for Turbulent Premixed Combustion

Rembold

Proc Appl Math and Mech

2006

A model for premixed turbulent combustion based on a joint velocity probability density function (PDF) method and a progress variable is presented. Compared with other methods employing progress variables, the advantage here is that turbulent mixing of the progress variable requires no modeling. Moreover, by applying scale separation, the Lagrangian framework allows to account for the embedded, quasi laminar flame structure in a very natural way. The numerical results presented here are based on a simple closure of the progress variable source term and it is demonstrated that the new modeling approach is robust and shows the correct qualitative behavior Background and Modeling ApproachTurbulent combustion is central for many engineering applications. Whereas the modeling approach based on mixture fraction and laminar flamelets proofed to be very successful for non-premixed combustion, currently there exists no general approach for premixed flames. In the Eulerian context a common model is the one by Bray, Moss and Libby (BML) [2]. They assume an infinitely thin flame and use a progress variable c ∈ {0, 1} to describe, whether the gas is burnt (c = 1) or unburnt (c = 0). The difficulty is to close the transport equation for the Favre average of c, i.e. to model the turbulent transport and mean source terms. Another modeling approach for premixed combustion is based on laminar flamelets and a level set equation to determine the flame position. For turbulent flames, however, it is not straight forward to achieve closure. The advantage of using joint probability density function (PDF) methods [9] is that reaction source term and turbulent convection appear in closed form. On the other hand, modeling molecular mixing remains a major challenge. In the following, we present a new approach, which is a combination of a hybrid joint PDF method [3,4,6, 10], a transported progress variable, and the flamelet model. The probability density function (PDF) of a progress variable c can be described by a bimodal distribution. Note that the use of such a PDF by the BML model for the temperature implies that the flame is infinitely thin. With such a presumed shape the PDF is specified by the Favre mean value of c, which can be obtained by solvingwhere the operators·, · and · denote Favre averaged, Reynolds averaged and Favre fluctuating quantities, respectively. Moreover, U is the fluid velocity, ρ the density and ω c the mean progress variable source term. A physical interpretation of Eq. (1) is given in [7]. By applying the BML model, the difficulties are the closures of the two terms on the right-hand side, i.e. of turbulent transport and production of c. In our model, each computational particle used by the joint PDF solution algorithm has a property c * ∈ {0, 1} that only indicates the arrival of an embedded quasi laminar flame at the fluid particle ( * denotes a particle property). The flame is therfore not assumed to be infinitely thin and its internal structure can be resolved. Herefore, the relative position of the particl...