1
Turbulent Channel Flow to Ret = 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 Ret = 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 G of unit vector t whose ends a and b are distant by a length d. Each facet is defined by a contour G* 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 particular the conservation of mass, rotational or kinetic energy to cite a few. Trying to make the link between discrete or continuum mechanics formulations and analyzing or understanding of turbulence appears to us a relevant issue to be tackled.

Based on formulation differences, some features of discrete mechanics directly impact the understanding and modeling of turbulence. One of them relates to the value of the compressive viscosity for fluids, which is classically fixed by Stokes' law and is clearly questioned (Gad-el-Hak 1995; Rajagopal 2013). The treatment of the pressure in direct numerical simulation or the turbulent pressure in statistical turbulence modeling is directly related to the value of this compression viscosity, especially at the time scales of the small turbulent structures. Another aim of this chapter is to convince the reader that despite the fundamentally different nature of the discrete mechanics formulation, classical turbulent flow characteristics are nicely recovered and modeling issues can be revisited thanks to the DM approach.

This chapter is structured as follows. Section 1.2 provides a synthetic description of geometrical features of discrete mechanics together with a review of mathematical formulations, modeling issues and standard numerical aspects. The numerical simulation of the turbulent planar channel flow at a turbulent Reynolds number of Ret = 590 (Moser et al. 1999; Denaro et al. 2011) is then investigated in section 1.5. In particular, comparisons to DNS and LES results of the literature are discussed and fitting laws for the mean velocity profile from the wall to the center of the channel are obtained.

1.2. Discrete mechanics formulation

The derivation of the equation of motion is carried out on a segment G, shown in Figure 1.1, from the law of discrete dynamics where ? is the proper acceleration of the material medium and the second member represents the imposed accelerations. Integration between the a and b ends of the G segment gives:

The discrete motion equation is derived from the conservation equation of acceleration by expressing the deviations of potentials ø and ? as a function of velocity V. These "deviators" are obtained on the basis of the physical analysis of the storage-destocking processes of compression and shear energies; the first is written as the divergence of velocity, and the second as a dual curl of velocity. The physical modeling of these terms is developed in a book devoted to discrete mechanics (Caltagirone 2019) and a recent article on the subject (Caltagirone and Vincent 2020).

The vectorial equation of the movement and its upgrades are written as:

The quantities øo and ?o are the equilibrium potentials; they represent the storage of compression and shearing energies between the instant of initial mechanical equilibrium and that at the instant t. They are written:

The longitudinal and transverse celerities cl and ct are intrinsic quantities of the medium that can vary according to physical parameters. The terms and are, respectively, the deviators of the compression and shear effects. The second member is thus composed of two oscillators øo and ?o, which represent energies per unit mass, exchanging these with their respective deviators. The two terms in gradient and in dual rotation are orthogonal and cannot exchange energy directly; if an imbalance due to an external event occurs for one of these effects, then the acceleration is changed and the energy is consequently redistributed to the other term. The acceleration g represents gravity or any other source quantity and will also be written in the form of a Helmholtz-Hodge decomposition.

The physical parameters al and at are the attenuation factors of the compression and shear waves, respectively. They also depend only on the medium considered; for example a Newtonian fluid retains the shear stresses only for very weak relaxation time constants, of order of magnitude of 10-12 s and the factor at can be taken as zero. The updating of potentials at time t + dt is thus affected by these coefficients ranging between zero and unity. The velocity and possibly the displacement U are upgraded in turn. In the case where the density is not constant, it is also updated using the mass conservation in the form this quantity is only an explicit function of the divergence of velocity.

It should be noted that øo and ?o are energies per unit mass, each term reflecting the behavior of the medium or particle with respect to longitudinal and transverse waves. The term () represents the longitudinal wave where øo accumulates the compression energy contained in the deviator term or restores it over time according to cl. Similarly, () is the oscillator corresponding to the velocity transverse waves ct.

The time interval between two observations of the physical system in equilibrium is arbitrary, for example dt = 1020 s to obtain a stationary state, whereas dt = 10-20 s for the study in direct simulation of ? rays. The solution does not depend on this parameter if it is adapted to the mimicked physics; in the general case, the solution is of order 2 in space and time. Since all the terms are implicit on the variable velocity, or linearized as the terms of inertia, the system [1.2] is...