The Finite Element Method in Heat Transfer and Fluid Dynamics

International audienceThis work aims at modelling buoyant, laminar or turbulent flows, using a 2D Incompressible Smoothed Particle Hydrodynamics (ISPH) model with accurate wall boundary conditions. The buoyancy effects are modelled through the Boussinesq approximation coupled to a heat equation, which makes it possible to apply an incompressible algorithm to compute the pressure field from a Poisson equation. Based on our previous work (Leroy et al., 2014), we extend the unified semi-analytical wall boundary conditions to the present model. The latter is also combined to a Reynolds-Averaged Navier-Stokes approach to treat turbulent flows. The k − turbulence model is used, where buoyancy is modelled through an additional term in the k − equations like in mesh-based methods. We propose a unified framework to prescribe isothermal (Dirichlet) or imposed heat flux (Neumann) wall boundary conditions in ISPH. To illustrate this, a theoretical case is presented (laminar heated Poiseuille flow), where excellent agreement with the theoretical solution is obtained. Several benchmark cases are then proposed: a lock-exchange flow, two laminar and one turbulent flow in differentially heated cavities, and finally a turbulent heated Poiseuille flow. Comparisons are provided with a Finite-Volume (FV) approach using an open-source industrial code

Section: Time Discretisationmentioning

confidence: 99%

“…This approach has been widely used in computational fluid dynamics with mesh-based methods (see e.g. [2] regarding the finite elements method or [3] with the finite 456 A. LEROY ET AL. volume (FV) method) but is still rather new in the SPH publications.…”

Section: Introductionmentioning

confidence: 99%

Buoyancy modelling with incompressible SPH for laminar and turbulent flows

Leroy

Violeau

Ferrand³

et al. 2015

“…The nonlinear Navier-Stokes Equation (7) is solved by a hybrid solver based on a fixed-value iteration approach and a Newton-type solver. 37…”

Section: Fe Formulationmentioning

confidence: 99%

Shape morphing and topology optimization of fluid channels by explicit boundary tracking

Zhou

Lian

Sigmund

et al. 2018

Summary An integrated shape morphing and topology optimization approach based on the deformable simplicial complex methodology is developed to address Stokes and Navier‐Stokes flow problems. The optimized geometry is interpreted by a set of piecewise linear curves embedded in a well‐formed triangular mesh, resulting in a physically well‐defined interface between fluid and impermeable regions. The shape evolution is realized by deforming the curves while maintaining a high‐quality mesh through adaption of the mesh near the structural boundary, rather than performing global remeshing. Topological changes are allowed through hole merging or splitting of islands. The finite element discretization used provides smooth and stable optimized boundaries for simple energy dissipation objectives. However, for more advanced problems, boundary oscillations are observed due to conflicts between the objective function and the minimum length scale imposed by the meshing algorithm. A surface regularization scheme is introduced to circumvent this issue, which is specifically tailored for the deformable simplicial complex approach. In contrast to other filter‐based regularization techniques, the scheme does not introduce additional control variables, and at the same time, it is based on a rigorous sensitivity analysis. Several numerical examples are presented to demonstrate the applicability of the approach.

“…D is the Dirichlet-type boundary condition and N is the Neumann-type boundary condition. In the work, the generalized Newtonian model is employed, where the viscosity is a function of the second invariant of the deformation rate tensor [3,32] and, unlike Newtonian flows, the relationship between the sheer stress tensor and the deformation rate tensor is nonlinear, that is,…”

Section: Problem Statementmentioning

confidence: 99%

“…/, and H 1 . / are the standard notations with the usual meanings in the finite element literature [32][33][34]. The weighting and trial velocity function spaces V 0 h and V g h are…”

Section: Galerkin/least Squares Finite Element Formulationmentioning

confidence: 99%

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

Shiu

Hwang

Cai

2015

SummaryWe introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.