Numerical Simulation in Physics and Engineering: Trends and Applications

The aeroelasticity is the science which models, analyses and describes the coupled movements between a flow and a flexible structure. The different phenomena encountered can be classified using three (at least) adimensional numbers: the Strouhal number, the Reynolds number and the reduce frequency number (which despite its name, has no dimension). For sake of clarity, let us just mention in this abstract, that the reduce frequency is the ratio between the time necessary to a flow particle for flying over a flexible structure and the fundamental period of oscillation of this structure.

In the framework of quasi-static aeroelasticity, it is always assumed that the reduce frequency is smaller than the unity. It enables one to define the flow fields (velocity, pressure) from a static position of the structure. The effect of its position with respect to the flow leads to a modification of the stiffness (added aerodynamic stiffness). Furthermore, the relative flow velocity (difference between the flow velocity and the one of the structure) leads to introduce damping due to the flow and therefore modifies the static analysis of stability into the dynamic stability study (aerodynamic damping).

Recently, engineers have upgraded this approach by introducing the added mass concept. This is a mechanical effect due to the fact that the inertia of the structure should take into account the mass of flow which is involved in a movement. This is performed using an incompressible and inviscid model which gives a retroaction effect on the structure proportionally to its velocity. The two first parts of this text are devoted to a formulation of this three effects which are necessary in the dynamic modeling of a flexible (or not) structure immersed in a flow (air or water for instance). Examples in civil engineering and aerodynamics are given in order to illustrate the theoretical formulation. Few control aspects in a dynamic behavior of the coupled fluid-structure modeling are also discussed in a section of this text.

This chapter describes the phase-field approach to modeling interface dynamics, with particular emphasis on the mathematical formulation and computational aspects. We describe to approaches to derive a phase-field model. The first one can be understood as a regularization approach: We start with a sharp interface model which is later replaced by a diffuse interface. The second approach starts with a free energy functional and balance laws for mass, linear momentum and energy; the final governing equations are derived using the second law of thermodynamics and the Coleman-Noll procedure. We finish by illustrating how the phase-field method can be used to solve problems on complicated geometries using cartesian grids only. Some of the opportunities opened by the phase-field approach are illustrated with numerical simulations.

Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the internal energy equation, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either structured meshes or general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by fractional-step algorithms; these may be either based on semi-implicit pressure correction techniques or segregated in such a way that only explicit steps are involved (referred to hereafter as “explicit” variants). In order to ensure the positivity of the density, the internal energy and the pressure, the discrete convection operators for the mass and internal energy balance equations are carefully designed; they use an upwind technique with respect to the material velocity only. The construction of the fluxes thus does not need any Riemann or approximate Riemann solver, and yields easily implementable algorithms. The stability is obtained without restriction on the time step for the pressure correction scheme and under a CFL-like condition for explicit variants: preservation of the integral of the total energy over the computational domain, and positivity of the density and the internal energy. The semi-implicit first-order upwind scheme satisfies a local discrete entropy inequality. If a MUSCL-like scheme is used in order to limit the scheme diffusion, then a weaker property holds: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L^∞ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme, with the above-mentioned MUSCL-like scheme, the same result only holds if the ratio of the time to the space step tends to zero.

The transonic natural laminar flow wing will become an important feature of the next generation advanced civil transport aircraft, because it can greatly reduce the friction drag. In paper Tang et al. (Arch Computat Meth Eng 26:119–141, 2019) and Chen et al. (J Nanjing Univ Aeronaut Astron, 50(4):548–557, 2018), the problem of wave drag increase due to the expansion of laminar flow region in the optimization design of natural laminar airfoil is studied with Pareto game and EAs by using Shock Control Bump (SCB) and Trailing Edge Device (TED) respectively. In this paper, the numerical implementation of SCB and TED in the shape design optimization of natural laminar airfoils and the performance differences of the final optimal airfoil are compared and analyzed. The feasibility and potential of applying them to the optimization design of three-dimensional laminar wing are discussed.

In this paper, we will report our recent effort to apply the parareal algorithm to the time parallelization of an industrial code that simulates two phase flows in a reactor for safety studies. This software solves the six equation two-fluid model by considering a set of balance laws (mass, momentum and energy) for each phase, liquid and vapor, of the fluid. The discretization is based on a finite volume method on a staggered grid in space and on a multistep time scheme. Here, we apply a variant of the parareal algorithm on an oscillating manometer test case: the multistep variant allowing to deal with multistep time schemes in the coarse and/or fine propagators. Numerical results show that parareal methods offer the potential for an increased level of parallelism and is a good strategy to complement the current space domain decomposition implemented in the code.

A two-layer shallow water type model is proposed to describe bedload sediment transport for strong and weak interactions between the fluid and the sediment. The key point falls into the definition of the friction law between the two layers, which is a generalization of those introduced in Fernández-Nieto et al. (https://​doi.​org/​10.​1051/​m2an/​2016018). Moreover, we prove formally that the two-layer model converges to a Saint-Venant-Exner system (SVE) including gravitational effects when the ratio between the hydrodynamic and morphodynamic time scales is small. The SVE with gravitational effects is a degenerated nonlinear parabolic system, whose numerical approximation can be very expensive from a computational point of view, see for example T. Morales de Luna et al. (https://​doi.​org/​10.​1007/​s10915-010-9447-1). In this work, gravitational effects are introduced into the two-layer system without any parabolic term, so the proposed model may be a advantageous solution to solve bedload sediment transport problems.