Fluid Mechanics at Interfaces 2

Case Studies and Instabilities

Roger Prudhomme, Stephane Vincent, Roger Prud'homme

• Publisher: Wiley-ISTE
• Year: 2022
• Print ISBN:
  9781786308177
• eBook ISBN:
  9781119902997
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Fluid Mechanics

Interfaces are present in most fluid mechanics problems. They not only denote phase separations and boundary conditions, but also thin flames and discontinuity waves. Fluid Mechanics at Interfaces 2 examines cases that involve one-dimensional or bi-dimensional manifolds, not only in gaseous and liquid physical states but also in subcritical fluids and in single- and multi-phase systems that may be pure or mixed.

Chapter 1 addresses certain aspects of turbulence in discrete mechanics, briefly describing the physical model associated with discrete primal and dual geometric topologies before focusing on channel flow simulations at turbulence-inducing Reynolds numbers. Chapter 2 centers on atomization in an accelerating domain. In one case, an initial Kelvin–Helmholtz instability generates an acceleration field, in turn creating a Rayleigh–Taylor instability which ultimately determines the size of the droplets formed. Chapter 3 explores numerical studies of pipes with sudden contraction using OpenFOAM, and focuses on modeling that will be useful for engines and automobiles.

Chapters 4 and 5 study the evaporation of droplets that are subject to high-frequency perturbations, a possible cause of instabilities in injection engines. The Heidmann model, which replaces the droplets in motion in a combustion chamber with a single continuously-fed droplet, is made more complex by considering the finite conduction heat transfer phenomenon. Finally, Chapter 6 is devoted to a study of the rotor blade surface of a Savonius wind turbine, considering both a non-stationary and a three-dimensional flow.

1
Turbulent Channel Flow to Re_τ = 590 in Discrete Mechanics

Discrete mechanics based on an acceleration-potential formulation presents itself as an alternative to Navier–Stokes equations; the acceleration on an edge is the sum of a direct contribution due to the difference of the scalar potential at the ends and of another induced by the circulation of the vector potential on a dual contour. The law of dynamics becomes a law of conservation of acceleration on the segment considered; acceleration is written as the sum of two components, one divergence-free and another curl-free. This equation of motion makes it possible to apprehend turbulent flows in direct simulation, taking into account the propagation of longitudinal and transverse waves. The first part of this chapter is a brief description of the physical model, closely associated with discrete primal and dual geometric structures.

The last section is devoted to simulating the turbulent channel flow at a turbulent Reynolds number of Re_τ = 590. It is demonstrated that discrete mechanics make it possible to accurately recover the mean velocity profiles of reference DNS and also to provide scale laws of the whole mean velocity profile from the wall to the center of the channel.

1.1. Introduction

Starting from the first drawings of Da Vinci (Lumley 1997), when one is interested in turbulence and its understanding, it can be chosen to adopt different complementary scientific approaches that have historically been experimental (Reynolds 1883), theoretical (von Kàrmàn 1948; Kolmogorov 1991) or numerical (Deardorff 1970). In the framework of this chapter, we choose the prism of the formulation of the models in fluid mechanics to propose a certain number of remarks and discussions related to turbulence and the phenomena it is sensitive to. If we rely here on the importance and meaning of models in the analysis and understanding of turbulence, this is particularly because new formulations of fluid mechanics equations have been published in recent years, more specifically those associated with discrete mechanics (DM) (Caltagirone 2019; Caltagirone and Vincent 2020).

Figure 1.1. Discrete geometric structure: a set of primitive planar facets S are associated with the segment Γ of unit vector t whose ends a and b are distant by a length d. Each facet is defined by a contour Γ* oriented according to the normal n such that n · t = 0; the dual surface ∆ connecting the centroids of the cells is also flat. For a color version of this figure, see www.iste.co.uk/prudhomme/fluid2.zip

From a general point of view, the notion of a discrete medium is directly derived from the principle of relativity of velocity and the weak equivalence principle: gravity accelerates all objects regardless of their mass or the materials from which they are made. All the contributions brought by the mechanical effects such as viscosity, compression or inertia, as well as all the source terms, are written as the sum of a free divergence term and a zero rotational term following the Helmholtz–Hodge decomposition. The motion equations of discrete mechanics reveal the role played by two quantities, namely the scalar and vector potentials, both associated with acceleration, a quantity considered as absolute. Even though the results obtained with this set of equations are generally the same as those of the continuous media provided by the Navier–Stokes equations, many formal differences exist. In particular, the density, as such, has disappeared from the momentum equation in favor of a scalar potential being the ratio between pressure and density. Many properties of the continuum are recovered intrinsically by the discrete mechanics formalism, in parti...

Table of contents

• Cover
• Table of Contents
• Title Page
• Copyright
• Preface
• 1 Turbulent Channel Flow to Reτ = 590 in Discrete Mechanics
• 2 Atomization in an Acceleration Field1
• 3 Numerical Simulation of Pipes with an Abrupt Contraction Using OpenFOAM
• 4 Vaporization of an Equivalent Pastille
• 5 Thermal Field of a Continuously-Fed Drop Subjected to HF Perturbations
• 6 Study of the Three-Dimensional and Non-Stationary Flow in a Rotor of the Savonius Wind Turbine
• List of Authors
• Index
• Summary of Volume 1
• End User License Agreement