Giorgio Martalò

Bio: Giorgio Martalò is an academic researcher from University of Parma. The author has contributed to research in topics: Shock wave & Mach number. The author has an hindex of 6, co-authored 17 publications receiving 98 citations. Previous affiliations of Giorgio Martalò include University of Bordeaux & University of Milan.

Slip Boundary Conditions for the Compressible Navier-Stokes Equations

Abstract: The slip boundary conditions for the compressible Navier–Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman–Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar–Gross–Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.

Shock Wave Structure of Multi-Temperature Euler Equations from Kinetic Theory for a Binary Mixture

Abstract: A multi-temperature hydrodynamic limit of kinetic equations is employed for the analysis of the steady shock problem in a binary mixture. Numerical results for varying parameters indicate possible occurrence of either smooth profiles or of weak solutions with one or two discontinuities.

Multi-temperature Euler hydrodynamics for a reacting gas from a kinetic approach to rarefied mixtures with resonant collisions

Abstract: Starting from a Boltzmann kinetic model for a gas mixture with bimolecular chemical reaction, hydrodynamic equations at Euler level are deduced by a consistent hydrodynamic limit in the presence of resonance, namely when the fast process driving evolution is constituted by elastic scattering between particles of the same species. The structure of the resulting multi-temperature and multi-velocity fluid-dynamic description is briefly commented on, and some results in closed analytical form are given for the special case of Maxwellian collision kernel.

Multi-temperature Hydrodynamic Limit from Kinetic Theory in a Mixture of Rarefied Gases

Abstract: Starting from the Boltzmann kinetic equations for a mixture of gas molecules whose internal structure is described by a discrete set of internal energy levels, hydrodynamic equations at Euler level are deduced by a consistent hydrodynamic limit in the presence of a two-scale collision process. The fast process driving evolution is constituted by mechanical encounters between particles of the same species, whereas inter-species scattering proceeds at the macroscopic scale. The resulting multi-temperature and multi-velocity fluid-dynamic equations are briefly commented on, and some results in closed analytical form are given for special simplified situations like Maxwellian collision kernels, or mono-atomic hard sphere gases.

Multi-temperature fluid-dynamic model equations from kinetic theory in a reactive gas: The steady shock problem

Abstract: Starting from a simple kinetic model for a chemical reaction, multi-temperature reactive Euler equations are derived for physical regimes in which evolution is driven by elastic collisions within the same species and mechanical relaxation is faster than the thermal one. The achieved hydrodynamic equations, where all inhomogeneous exchange rates take analytical closed form for simple collision models, are then used for the analysis of the steady shock problem. Results indicate that smooth shock profiles occurring for slightly supersonic flows bifurcate to weak solutions (jump discontinuity followed by a smooth tail) for increasing Mach number.

Kinetic Theory and Fluid Dynamics

Abstract: In this series of talks, I will discuss the fluid-dynamic-type equations that is derived from the Boltzmann equation as its the asymptotic behavior for small mean free path. The study of the relation of the two systems describing the behavior of a gas, the fluid-dynamic system and the Boltzmann system, has a long history and many works have been done. The Hilbert expansion and the Chapman–Enskog expansion are well-known among them. The behavior of a gas in the continuum limit, however, is not so simple as is widely discussed by superficial understanding of these solutions. The correct behavior has to be investigated by classifying the physical situations. The results are largely different depending on the situations. There is an important class of problems for which neither the Euler equations nor the Navier–Stokes give the correct answer. In these two expansions themselves, an initialor boundaryvalue problem is not taken into account. We will discuss the fluid-dynamic-type equations together with the boundary conditions that describe the behavior of the gas in the continuum limit by appropriately classifying the physical situations and taking the boundary condition into account. Here the result for the time-independent case is summarized. The time-dependent case will also be mentioned in the talk. The velocity distribution function approaches a Maxwellian fe, whose parameters depend on the position in the gas, in the continuum limit. The fluid-dynamictype equations that determine the macroscopic variables in the limit differ considerably depending on the character of the Maxwellian. The systems are classified by the size of |fe− fe0|/fe0, where fe0 is the stationary Maxwellian with the representative density and temperature in the gas. (1) |fe − fe0|/fe0 = O(Kn) (Kn : Knudsen number, i.e., Kn = `/L; ` : the reference mean free path. L : the reference length of the system) : S system (the incompressible Navier–Stokes set with the energy equation modified). (1a) |fe − fe0|/fe0 = o(Kn) : Linear system (the Stokes set). (2) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(Kn) (ξi : the molecular velocity) : SB system [the temperature T and density ρ in the continuum limit are determined together with the flow velocity vi of the first order of Kn amplified by 1/Kn (the ghost effect), and the thermal stress of the order of (Kn) must be retained in the equations (non-Navier–Stokes effect). The thermal creep[1] in the boundary condition must be taken into account. (3) |fe − fe0|/fe0 = O(1) with | ∫ ξifedξ|/ ∫ |ξi|fedξ = O(1) : E+VB system (the Euler and viscous boundary-layer sets). E system (Euler set) in the case where the boundary is an interface of the gas and its condensed phase. The fluid-dynamic systems are classified in terms of the macroscopic parameters that appear in the boundary condition. Let Tw and δTw be, respectively, the characteristic values of the temperature and its variation of the boundary. Then, the fluid-dynamic systems mentioned above are classified with the nondimensional temperature variation δTw/Tw and Reynolds number Re as shown in Fig. 1. In the region SB, the classical gas dynamics is inapplicable, that is, neither the Euler

Maximum entropy principle for rarefied polyatomic gases

Abstract: In this chapter, we prove, in the case of polyatomic rarefied gases, that the maximum entropy principle (MEP) gives the same closure of the system as that obtained in the phenomenological ET theory with 14 fields discussed in Chap. 5 The main idea is to consider a generalized distribution function depending not only on the velocity but also on an extra variable that connects with the internal degrees of freedom of a constituent molecule. On the basis of MEP, we again obtain the same binary hierarchy introduced in the previous chapter: the one is the usual momentum-type, F-series, and the other is the energy-type, G-series. The extra variable plays a role in the G-series. Thus we prove the perfect agreement between the ET theory and the molecular ET theory at least within 14-field theories. The agreement for any number of moments will be proved in Chap. 10

Multiscale Modeling of Gas Transport in Shale Matrix: An Integrated Study of Molecular Dynamics and Rigid-Pore-Network Model

Abstract: The physics of gas transport through shale systems remains ambiguous. Although several theoretical and experimental studies have been reported, most concentrate only on the permeability of shale kerogen. Shales, however, are composed of various proportions of organic matter and inorganic minerals (e.g., calcite and clay). Inorganic pores are larger than organic pores, thus affecting apparent permeability. To accurately predict the apparent permeability of shale, we couple molecular dynamics (MD) and a pore-network model (PNM) to develop a multiscale framework for gas flow through shales. First, we use nonequilibrium MD (NEMD) to study the pressure-driven flow behavior of methane (CH4) through organic, calcite, and clay [montmorillonite (MMT)] nanopores under reservoir conditions, from which, using the slip-corrected Poiseuille equation, we propose a mass-transport model accounting for the contributions of both the adsorbed-phase fluid and bulk fluid. Then, we incorporate these formulations into a shale PNM in which the influences of shale composition and bimodal pore-size distribution (PSD) are taken into account. We also develop an analytical model for the apparent permeability of shale matrix using the bundle-of-capillaries approach. In comparison with previous methods, our proposed models highlight the effect of relatively greater pore sizes in inorganic matrices. This work provides an efficient tool for better understanding gas transport through shale systems at both molecular and pore scales.

Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics

Abstract: The aim of this paper is to compare different kinetic approaches to a polyatomic rarefied gas: the kinetic approach via a continuous energy parameter $I$ and the mixture-like one, based on discrete internal energy. We prove that if we consider only $6$ moments for a non-polytropic gas the two approaches give the same symmetric hyperbolic differential system previously obtained by the phenomenological Extended Thermodynamics. Both meaning and role of dynamical pressure become more clear in the present analysis.

Discrete Boltzmann modeling of unsteady reactive flows with nonequilibrium effects.

Abstract: A multiple-relaxation-time discrete Boltzmann model (DBM) is developed for compressible thermal reactive flows. A unified Boltzmann equation set is solved for hydrodynamic and thermodynamic quantities as well as higher order kinetic moments. The collision, reaction, and force terms are uniformly calculated with a matrix inversion method, which is physically accurate, numerically efficient, and convenient for coding. Via the Chapman-Enskog analysis, the DBM is demonstrated to recover reactive Navier-Stokes (NS) equations in the hydrodynamic limit. Both specific heat ratio and Prandtl number are adjustable. Moreover, it provides quantification of hydrodynamic and thermodynamic nonequilibrium effects beyond the NS equations. The capability of the DBM is demonstrated through simulations of chemical reactions in the free falling process, sound wave, thermal Couette flow, and steady and unsteady detonation cases. Moreover, nonequilibrium effects on the predicted physical quantities in unsteady combustion are quantified via the DBM. It is demonstrated that nonequilibrium effects suppress detonation instability and dissipate small oscillations of fluid flows.