A fast, decomposed pressure correction method for an intrusive stochastic multiphase flow solver

doi:10.1016/j.compfluid.2021.104930

Computers & Fluids

Volume 221, 15 May 2021, 104930

https://doi.org/10.1016/j.compfluid.2021.104930 Get rights and content

Highlights

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Application of decomposed pressure correction method to stochastic multiphase Navier–Stokes solver.
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Allows for use of various libraries to solve elliptic pressure Poisson equation in simulations of stochastic gas-liquid flows.
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Method offers a significant reduction in computational expense, which is further reduced as the mesh is refined and model order increased.

Abstract

Solution of the pressure Poisson equation is often the most expensive aspect of solving the incompressible form of Navier–Stokes. For a single phase deterministic model the pressure calculation is costly. Expanded to an intrusive stochastic multiphase framework, the simulation expense grows dramatically due to coupling between the stochastic pressure field and stochastic density. To address this issue in a deterministic framework, Dodd and Ferrante (“A fast pressure-correction method for incompressible two-fluid flows” Journal of Computational Physics, 273, 416–434, 2014) discuss a decomposed pressure correction method which utilizes an estimated pressure field and constant density to modify the standard pressure correction method. The resulting method is useful for improving computational cost for one-fluid formulations of multiphase flow calculations. In this paper, we extend the decomposed pressure correction method to intrusive uncertainty quantification of multiphase flows. The work improves upon the original formulation by modifying the estimated pressure field. The new method is assessed in terms of accuracy and reduction in computational cost with oscillating droplet, damped surface wave, and atomizing jet test cases where we find convergence of results with the proposed method to those of a traditional pressure correction method and analytic solutions, where appropriate.

Introduction

Methods of uncertainty quantification (UQ) can be placed into two categories: intrusive and non-intrusive. The non-intrusive category includes approaches such as Monte Carlo [1], collocation methods [2], and non-intrusive polynomial chaos (PC) [3]. The latter two essentially improve on a Monte Carlo by presenting a better way to select the input values so not as many simulations need to be run. In all cases a standard solver can be utilized, which is run many times with parameters selected from a distribution of inputs to compile a database of simulation results. This database is then used to calculate useful statistics of the system in question.

Unlike non-intrusive flow solvers, intrusive UQ methods require a change in the fundamental structure of the solver resulting from modified equations created by the inclusion of stochastic (random) variables. An intrusive solver is created with stochastic variables which store information as a function of added uncertainty dimensions. Stochastic variables developed as a function of uncertainty may take several forms, including PC [4] and Karhunen–Loeve expansions [5], [6]. Each of these methods offer their own advantages. The advantage of PC lies in the ability to utilize any number of uncertainty dimensions, the availability of a number of orthogonal basis function families, and the straightforward integration of continuous PC variables into many systems of differential equations. Assuming stable intrusive and non-intrusive UQ schemes, computational cost comparisons are based on the time it takes to generate a reliable source of statistical information on the system being modeled.

Considering UQ applications to multiphase flow dynamics, application of intrusive UQ methods to gas-liquid flows is a developing field. Le Maître et al. [7], [8] first developed the stochastic Navier–Stokes equations for single-phase incompressible flows utilizing a PC expansion. Since these works, several studies have implemented a PC-based approach to single-phase flows for a variety of test cases [9], [10], [11]. Previous work by Turnquist and Owkes [12] provided the first intrusive UQ method for gas-liquid multiphase flows named the multiUQ framework. The current work builds on this previous work by reducing the computational cost.

In either intrusive or non-intrusive UQ methodology, much computational expense is devoted to solving the pressure Poisson equation. To numerically solve the incompressible Navier–Stokes equations, the standard pressure correction method (SPCM), first introduced by Chorin [13], is a commonly used approach. With the SPCM, time is discretized so that at every time step the convective, viscosity, and any source terms are evaluated and used to predict the velocity without the pressure term. Continuity (or mass conservation) is then used to enforce a divergence free condition at the next time step, while also creating an elliptic Poisson equation to solve for pressure. The approach makes it possible to solve the Navier–Stokes equations with imposed boundary conditions at reasonable computational expense. Further work has been done to expand the method, including improvements to order [14] and application to unstructured grids [15]. When using the SPCM in a multiphase scenario, the pressure Poisson equation becomes coupled to density, which adds computational cost and limits the possible algorithms used to solve. In an effort to counter this cost and following the work of Dong and Shen [16], Dodd and Ferrante [17] proposed a decomposed pressure correction method (DPCM) which would allow for using a fast Fourier transform (FFT) based solver. While numerical errors are added to the model, computational cost is reduced; certainly the trade off is worth consideration.

Given the computational cost improvements of a DPCM in the deterministic setting, it seemed reasonable to apply this methodology to the multiUQ framework [12]. Because of the coupled nature of non-linear terms in the stochastic Navier–Stokes equations due to the use of PC variables, the simulation expense grows at an exponential rate. This so-called curse of dimensionality increases the computational cost very rapidly for intrusive UQ. However, the same curse also affects non-intrusive methods. For example, the use of a Monte-Carlo [1] approach with two or more uncertain variables requires a way to compare the effect of one uncertain variable on another, compounding the number of simulations run to get convergent statistics. Due to this problem, understanding the interaction in uncertainty between multiple variables in a multiphase system is extremely expensive. Use of accurate and cost effective numerical techniques will bring these analyses within reach.

This narrative seeks to develop a more efficient pressure correction approach for stochastic multiphase flows by applying the DPCM to the multiUQ framework outlined in Turnquist and Owkes [12]. A mathematical development of the stochastic DPCM is introduced, followed by a derivation of the numerical methods. We then present test cases which illustrate the computational improvement over previously published methods and the error associated with the density decoupled approach. Finally, we close with a summary of the results and a discussion of where this work will fit in moving forward.

Section snippets

Mathematical development

Since the focus of this work is to develop an efficient pressure solver for stochastic multiphase flows, we begin with a development of the stochastic equations for fluid motion. Assuming the fluids are incompressible, this motion can be explained by the Navier–Stokes equations, where $\frac{\partial u}{\partial t} + u \cdot \nabla u = - η \nabla P + η \nabla \cdot [μ (\nabla u + \nabla^{T} u)] + η f_{σ} δ_{s}$ for velocity $u,$ time $t,$ specific volume $η = 1 / ρ$ (for density $ρ$ ), pressure $P,$ dynamic viscosity $μ,$ and surface tension force $f_{σ} = σ κ n,$ where $n$ is the interface normal vector, $σ$ is the

Numerical methodology

Computations are done on a two-dimensional rectangular domain with a structured Cartesian mesh. Scalar values such as pressure $P,$ level set $ψ,$ density $ρ,$ and viscosity $μ$ are held at the cell center. Subscripts on $P_{i, j}^{n}$ denote discrete spatial indexing in the $x$ and $y$ directions, respectively, while superscripts denote time discretization. Vector components of velocity $u,$ surface tension $f_{σ},$ and continuous normal vector $r$ are held at the cell walls. Second-order finite difference operators are

Test cases and computational assessment

Two test cases are used to evaluate the accuracy and efficiency of the proposed method, while a third is used to test the method on a more complicated scenario. First, an oscillating droplet case is used as there exists an analytic solution providing the oscillation period [22]. This case tests the ability of the surface tension force to drive flow. Second, a damped surface wave, which also has an analytic solution [23], is used to judge the accuracy of the interplay between viscosity and

Conclusions

We have presented a modified pressure correction method, extending the density decoupled approach of Dodd and Ferrante [17] to an intrusive stochastic multiphase solver. Deviations from the standard pressure correction method arise due to differences between the estimated pressure $\hat{P}$ and the actual pressure $P^{n + 1}$ . We propose a method to reduce these deviations by imposing a semi-Lagrangian extrapolation method for a better initial estimate of $\hat{P},$ then further improve $\hat{P}$ by iterating over the

CRediT authorship contribution statement

Brian Turnquist: Writing - original draft, Writing - review & editing, Methodology, Software, Conceptualization. Mark Owkes: Writing - review & editing, Supervision, Funding acquisition, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This material is based upon work supported by the National Science Foundation under grant nos. 1511325 and 1749779. Computational efforts were performed on the Hyalite High-Performance Computing System, operated and supported by University Information Technology Research Cyberinfrastructure at Montana State University.

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The mixing process, interface thermodynamics, and early deformation of a cool liquid n-decane jet surrounded by a hotter moving gas initially composed of pure oxygen are analyzed at various ambient pressures and gas velocities. For this purpose, a two-phase, low-Mach-number flow solver for variable-density fluids is used. The interface is captured using a split Volume-of-Fluid method, generalized for the case where the liquid velocity is not divergence-free and both phases exchange mass across the interface. The importance of transcritical mixing effects over time for increasing pressures is shown. Initially, local deformation features appear that differ considerably from previous incompressible works. Then, the minimal surface-tension force is responsible for generating overlapping liquid layers in favor of the classical atomization into droplets. Thus, surface-area growth at transcritical conditions is mainly a consequence of gas-like deformations under shear rather than spray formation. Moreover, the interface can be easily perturbed in hotter regions submerged in the faster oxidizer stream under trigger events such as droplet or ligament impacts. The net mass exchange at high pressures limits the liquid-phase vaporization to small liquid structures.
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multiUQ is a novel tool that simulates gas-liquid multiphase flows and quantifies uncertainty in results due to variability about fluid properties and initial/boundary conditions. The benefit over a typical deterministic solver is that inexact information, such as variability in fluid properties or flow rates, can be included to determine the affect on simulation solutions. It is common to deploy non-intrusive methods which utilize many solutions from a deterministic solver to generate a distribution of possible results. Contrarily, multiUQ uses an intrusive uncertainty quantification method wherein variables of interest are functions of space, time, and additional uncertainty dimensions. The intrusive solver is run once, giving a distribution of solutions as an output, as well as desired statistics. We use polynomial chaos to create the stochastic variables, which represent a distribution of values at each grid point. The stochastic variables are substituted into the incompressible Navier-Stokes equations, which govern the stochastic fluid dynamics. A stochastic level set is used to capture the distribution of interfaces that are present in an uncertain multiphase flow. multiUQ is written in Fortran and uses a message passing interface (MPI) for parallel operation. Given the many applications of multiphase flows, including open flows, hydraulics, fuel injection systems, and atomizing jets, there is a massive potential benefit to calculating uncertainty information about these flows in a cost-effective manner.
Program Title: multiUQ
CPC Library link to program files: https://doi.org/10.17632/yp5pkd5x89.1
Developer's repository link: https://bitbucket.org/markowkes/multiuq
Licensing provisions: GPLv3
Programming language: Fortran 95/2003
Supplementary material: Wiki available at repository link.
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Solution method: This software utilizes an intrusive approach to uncertainty quantification of multiphase flows. Deterministic variables are expanded to include an added uncertainty domain by way of polynomial chaos. Stochastic Navier-Stokes govern the flow field in an incompressible one-fluid approach. Multiphase flows are created by coupling a stochastic level set and surface tension.
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2022, arXiv
A Volume-of-Fluid method for variable-density, two-phase flows at supercritical pressure
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View full text

A fast, decomposed pressure correction method for an intrusive stochastic multiphase flow solver

Highlights

Abstract

Introduction

Section snippets

Mathematical development

Numerical methodology

Test cases and computational assessment

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

J Comput Phys

Adv Water Resour

J Appl Math

SIAM J Sci Stat Comput

J Comput Phys

Phys Fluids

J Comput Phys

J Comput Phys

J Comput Phys

J Comput Phys

J Comput Phys

The Monte Carlo method

J Am Stat Assoc

A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations

44th AIAA Aerospace sciences meeting and exhibit. aerospace sciences meetings