Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure-preserving semi-implicit scheme
Ilya Peshkov, Michael Dumbser, Walter Boscheri, Evgeniy Romenski, Simone Chiocchetti, Matteo Ioriatti
备注：
The old title "Modeling solid-fluid transformations in non-Newtonian viscoplastic flows with a unified flow theory" has been changed to the title of the published version "Simulation of non-Newtonian viscoplastic flows with a unified first order hyperbolic model and a structure-preserving semi-implicit scheme"
We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids, also called yield-stress fluids. In contrast to the conventional approaches relying on the non-linear viscosity concept of the Navier-Stokes theory and representation of the solid state as an infinitely rigid non-deformable solid, the solid state in our theory is deformable and the fluid state is considered rather as a "melted" solid via a certain procedure of relaxation of tangential stresses similar to Maxwell's visco-elasticity theory. The model is formulated as a system of first-order hyperbolic partial differential equations with possibly stiff non-linear relaxation source terms. The computational strategy is based on a staggered semi-implicit scheme which can be applied in particular to low-Mach number flows as usually required for flows of non-Newtonian fluids. The applicability of the model and numerical scheme is demonstrated on a few standard benchmark test cases such as Couette, Hagen-Poiseuille, and lid-driven cavity flows. The numerical solution is compared with analytical or numerical solutions of the Navier-Stokes theory with the Herschel-Bulkley constitutive model for nonlinear viscosity.
Considerations on implementation of the stress-strain constitutive relations applied in Computational Fluid dynamics (CFD) simulation of cardiovascular flows have been addressed extensively in the literature. However, the matter is yet controversial. The author suggests that the choice of non-Newtonian models and the consideration of non-Newtonian assumption versus the Newtonian assumption is very application oriented and cannot be solely dependent on the vessel size. In the presented work, where a renal disease patient-specific geometry is used, the non-Newtonian effects manifest insignifican…
Direct numerical simulations of closely interacting infinite circular cylinders in a Bingham fluid are presented, and results compared to asymptotic solutions based on lubrication theory in the gap. Unlike for a Newtonian fluid, the macroscopic flow outside of the gap between the cylinders is shown to have a large effect on the pressure profile within the gap and the resulting lubrication force on the cylinders. The presented results indicate that the asymptotic lubrication solution can be used to predict the lubrication pressure only if the surrounding viscoplastic matrix is yielded by a macr…
Turbulent flows of viscoplastic fluids at high Reynolds numbers have been investigated recently with direct numerical simulations (DNS) but experimental results have been limited. For this reason, we carry out an experimental study of fully turbulent flows of a yield stress fluid in a rectangular aspect ratio channel with a high-resolution laser doppler velocimetry (LDA) setup. We employ aqueous Carbopol solutions, often considered to be a simple yield stress fluid. We formulate different concentrations to address the effect of the rheology of the fluid on the turbulence statistics at an appro…
We present the extension of an efficient and highly parallelisable framework for incompressible fluid flow simulations to viscoplastic fluids. The system is governed by incompressible conservation of mass, the Cauchy momentum equation and a generalised Newtonian constitutive law. In order to simulate a wide range of viscoplastic fluids, we employ the Herschel-Bulkley model for yield-stress fluids with nonlinear stress-strain dependency above the yield limit. We utilise Papanastasiou regularisation in our algorithm to deal with the singularity in apparent viscosity. The resulting system of part…
Numerical simulations of the flow in an extrusion damper are performed using a finite volume method. The damper is assumed to consist of a shaft, with or without a spherical bulge, oscillating axially in a containing cylinder filled with a viscoplastic material of Bingham type. The response of the damper to a forced sinusoidal displacement is studied. In the bulgeless case the configuration is the annular analogue of the well-known lid-driven cavity problem, but with a sinusoidal rather than constant lid velocity. Navier slip is applied to the shaft surface in order to bound the reaction force…
We present a methodology for simulating three-dimensional flow of incompressible viscoplastic fluids modelled by generalised Newtonian rheological equations. It is implemented in a highly efficient framework for massively parallelisable computations on block-structured grids. In this context, geometric features are handled by the embedded boundary approach, which requires specialised treatment only in cells intersecting or adjacent to the boundary. This constitutes the first published implementation of an embedded boundary algorithm for simulating flow of viscoplastic fluids on structured grid…
An elasto-viscoplastic thixotropic model for fresh concrete capturing flow-rest transition
University of California, Berkeley（加州大学伯克利分校）；Hong Kong University of Science and Technology（香港科技大学）；National University of Singapore（新加坡国立大学）；Ove Arup & Partners（奥雅纳工程顾问有限公司）
We demonstrate through numerical simulations and a mean field calculation that immiscible two-phase flow in a porous medium behaves effectively as a Bingham viscoplastic fluid. This leads to a generalized Darcy equation where the volumetric flow rate depends quadratically on an excess pressure difference in the range of flow rates where the capillary forces compete with the viscous forces. At higher rates, the flow is Newtonian.
The settling efficiency, and stability with respect to settling, of a dilute suspension of infinite circular cylinders in a quiescent viscoplastic fluid is examined by means of direct numerical simulations with varying solid volume fraction, $ϕ$, and yield number, $Y$ . For $Y$ sufficiently large we find higher settling efficiency for increasing $ϕ$, similar to what is found in shear-thinning fluids and opposite to what is found in Newtonian fluids. The critical yield number at which the suspension is held stationary in the carrier fluid is found to increase monotonically with $ϕ$, while the t…
Two new control volume solvers multiFluidInterFoam and rheoMultiFluidInterFoam are presented for the simulation of Newtonian and non-Newtonian n-phase flows, respectively, fully accounting for interfacial tension and contact-angle effects for each phase. The multiFluidInterFoam solver modifies certain crucial aspects of the regular multiphaseInterFoam solver provided by OpenFOAM for Newtonian flows, improving its efficiency and robustness, but most importantly improving considerably its accuracy for surface tension driven flows. The new solver uses the volume-of-fluid (VOF) method based on the…
Implicit integration of the viscous term can significantly improve performance in computational fluid dynamics for highly viscous fluids such as lava. We show improvements over our previous proposal for semi-implicit viscous integration in Smoothed Particle Hydrodynamics, extending it to support a wider range of boundary models. Due to the resulting loss of matrix symmetry, a key advancement is a more robust version of the biconjugate gradient stabilized method to solve the linear systems, that is also better suited for parallelization in both shared-memory and distributed-memory systems. The …
We carry out an analytical study of laminar circular hydraulic jumps, in generalized-Newtonian fluids obeying the two-parametric power-law model of Ostwald-de Waele. Under the boundary-layer approximation we obtained exact expressions determining the flow, an implicit relation for the jump radius is derived. Corresponding results for Newtonian fluids can be retrieved as a limiting case for the flow behavior index n=1, predictions are made for fluids deviating from Newtonian behavior.
Blood flow, dam or ship construction and numerous other problems in biomedical and general engineering involve incompressible flows interacting with elastic structures. Such interactions heavily influence the deformation and stress states which, in turn, affect the engineering design process. Therefore, any reliable model of such physical processes must consider the coupling of fluids and solids. However, complexity increases for non-Newtonian fluid models, as used, e.g., for blood or polymer flows. In these fluids, subtle differences in the local shear rate can have a drastic impact on the fl…
In the above paper the author treats the boundary layer flow of an electrically conducting, non-Newtonian fluid along a vertical plate embedded in a Darcy-Brinkman-Forchheimer porous medium. The fluid is under the action of a constant transverse magnetic field and the radiation is taken into account in the energy equation. The boundary layer equations are transformed into ordinary ones and subsequently are solved numerically. However, there is a fundamental error in this paper which is presented below:
We propose a numerical scheme for simulation of transient flows of incompressible non-Newtonian fluids characterised by a non-monotone relation between the symmetric part of the velocity gradient (shear rate) and the Cauchy stress tensor (shear stress). The main difficulty in dealing with the governing equations for flows of such fluids is that the non-monotone constitutive relation allows several values of the stress to be associated with the same value of the symmetric part of the velocity gradient. This issue is handled via a reformulation of the governing equations. The equations are refor…
In this paper, the performance of two lattice Boltzmann method formulations for yield-stress (i.e. viscoplastic) fluids has been investigated. The first approach is based on the popular Papanastasiou regularisation of the fluid rheology in conjunction with explicit modification of the lattice Boltzmann relaxation rate. The second approach uses a locally-implicit formulation to simultaneously solve for the fluid stress and the underlying particle distribution functions. After investigating issues related to the lattice symmetry and non-hydrodynamic Burnett stresses, the two models were compared…
A natural framework for isogeometric fluid-structure interaction based on BEM-shell coupling
SISSA-International School for Advanced Studies（国际高等研究院）；Technische Universität Braunschweig（布伦瑞克工业大学）；University of Pavia（帕维亚大学）；Technische Universität München（慕尼黑工业大学）
We present an experimental and numerical study of immiscible two-phase flow in 3-dimensional (3D) porous media to find the relationship between the volumetric flow rate ($Q$) and the total pressure difference ($ΔP$) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold, making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium, introducing a threshold pressure $P_t$. In this regime, $Q$ depends quadratically on an excess press…
Viscoelastic fluids are non-Newtonian fluids that exhibit both "viscous" and "elastic" characteristics in virtue of mechanisms to store energy and produce entropy. Usually the energy storage properties of such fluids are modelled using the same concepts as in the classical theory of nonlinear solids. Recently new models for elastic solids have been successfully developed by appealing to implicit constitutive relations, and these new models offer a different perspective to the old topic of elastic response of materials. In particular, a sub-class of implicit constitutive relations, namely relat…
The present work focuses on the study of mixed convection of a purely viscous shear-thinning fluid in a horizontal annular eccentric duct. The inner and outer cylinders are heated with constant and uniform heat flux densities. The objective of this work is to study the effect of the variation of eccentricity, rheological behavior of the fluid as well as the thermodependency of the rheological parameters on the reorganization of the flow and thermal stratification caused by the buoyancy forces. At the entrance of the heating zone, the dynamic regime is assumed to be established and the temperat…
Most mathematics and engineering textbooks describe the process of "subtracting off" the steady state of a linear parabolic partial differential equation as a technique for obtaining a boundary-value problem with homogeneous boundary conditions that can be solved by separation of variables (i.e., eigenfunction expansions). While this method produces the correct solution for the start-up of the flow of, e.g., a Newtonian fluid between parallel plates, it can lead to erroneous solutions to the corresponding problem for a class of non-Newtonian fluids. We show that the reason for this is the non-…
The accuracy and stability of implicit CFD codes are frequently impaired by the decoupling between variables, which can ultimately lead to numerical divergence. Coupled solvers, which solve all the governing equations simultaneously, have the potential to fix this problem. In this work, we report the implementation of coupled solvers for transient and steady-state electrically-driven flow simulations in the finite-volumes framework. The numerical method, developed in OpenFOAM, is generic for Newtonian and viscoelastic fluids and is formulated for the Poisson-Nernst-Planck and Poisson-Boltzmann…
IB-Flows: an open-source multi-GPU immersed boundary code for fluid-structure interaction
Gran Sasso Science Institute (GSSI)（格兰萨索科学研究所）；INFN–Laboratori Nazionali del Gran Sasso（意大利国家核物理研究所-格兰萨索国家实验室）；Department of Physics and INFN, University of Rome, Tor Vergata（罗马托尔维加塔大学物理系及意大利国家核物理研究所）；NVIDIA Corporation（英伟达公司）；Physics of Fluids Group, University of Twente（特温特大学流体力学组）
