Abstract:
In this paper, we study semi-linearized compressible barotropic Navier-Stokes equations perturbed by noise in a 2-dimensional domain. We prove the existence and uniqueness of solutions in a class of potential flows.

Abstract:
Beltrami states for compressible barotropic flows are deduced by minimizing the total kinetic energy while keeping the total helicity constant. A Hamiltonian basis for these Beltrami states is also sketched.

Abstract:
We investigate diffusion in supersonic, turbulent, compressible flows. Supersonic turbulence can be characterized as network of interacting shocks. We consider flows with different rms Mach numbers and where energy necessary to maintain dynamical equilibrium is inserted at different spatial scales. We find that turbulent transport exhibits super-diffusive behavior due to induced bulk motions. In a comoving reference frame, however, diffusion behaves normal and can be described by mixing length theory extended into the supersonic regime.

Abstract:
for bounded and unbounded domains in r3, we establish the localization and the structure of the spectrum of normal vibrations described by systems of partial differential equations modelling small displacements of compressible stratified fluid in the homogeneous gravity field. we also compare the spectral properties of gravitational and rotational operators. our main result is the construction of weyl sequence for the essential spectrum, which is an explicit form of non-uniqueness of the solutions.

Abstract:
For a freely evolving granular fluid, the buildup of spatial correlations in density and flow field is described using fluctuating hydrodynamics. The theory for incompressible flows is extended to the general, compressible case, including longitudinal velocity and density fluctuations, and yields qualitatively different results for long range correlations. The structure factor of density fluctuations shows a maximum at finite wavenumber, shifting in time to smaller wavenumbers and corresponding to a growing correlation length. It agrees well with two-dimensional molecular dynamics simulations.

Abstract:
Organisms often grow, migrate and compete in liquid environments, as well as on solid surfaces. However, relatively little is known about what happens when competing species are mixed and compressed by fluid turbulence. In these lectures we review our recent work on population dynamics and population genetics in compressible velocity fields of one and two dimensions. We discuss why compressible turbulence is relevant for population dynamics in the ocean and we consider cases both where the velocity field is turbulent and when it is static. Furthermore, we investigate populations in terms of a continuos density field and when the populations are treated via discrete particles. In the last case we focus on the competition and fixation of one species compared to another

Abstract:
We study competition between two biological species advected by a compressible velocity field. Individuals are treated as discrete Lagrangian particles that reproduce or die in a density-dependent fashion. In the absence of a velocity field and fitness advantage, number fluctuations lead to a coarsening dynamics typical of the stochastic Fisher equation. We then study three examples of compressible advecting fields: a shell model of turbulence, a sinusoidal velocity field and a linear velocity sink. In all cases, advection leads to a striking drop in the fixation time, as well as a large reduction in the global carrying capacity. Despite localization on convergence zones, one species goes extinct much more rapidly than in well-mixed populations. For a weak harmonic potential, one finds a bimodal distribution of fixation times. The long-lived states in this case are demixed configurations with a single boundary, whose location depends on the fitness advantage.

Abstract:
The helical absolute equilibrium of a compressible adiabatic flow presents not only the polarization between the two purely helical modes of opposite chiralities but also that between the vortical and acoustic modes, deviating from the equipartition predicted by {\sc Kraichnan, R. H.} [1955 The Journal of the Acoustical Society of America {\bf 27}, 438--441.]. Due to the existence of the acoustic mode, even if all Fourier modes of one chiral sector in the sharpened Helmholtz decomposition [{\sc Moses, H. E.} 1971 SIAM ~(Soc. Ind. Appl. Math.) J. Appl. Math. {\bf 21}, 114--130] are thoroughly truncated, leaving the system with positive definite helicity and energy, negative temperature and the corresponding large-scale concentration of vortical modes are not allowed, unlike the incompressible case.

Abstract:
We establish various criteria, which are known in the incompressible case, for the validity of the inviscid limit for the compressible Navier-Stokes flows considered in a general domain $\Omega$ in $\mathbb{R}^n$ with or without a boundary. In the presence of a boundary, a generalized Navier boundary condition for velocity is assumed, which in particular by convention includes the classical no-slip boundary conditions. In this general setting we extend the Kato criteria and show the convergence to a solution which is dissipative "up to the boundary". In the case of smooth solutions, the convergence is obtained in the relative energy norm.

Abstract:
Landing gear noise prediction method is developed using Building-Cube Method (BCM). The BCM is a multiblock-structured Cartesian mesh flow solver, which aims to enable practical large-scale computation. The computational domain is composed of assemblage of various sizes of building blocks where small blocks are used to capture flow features in detail. Because of Cartesian-based mesh, easy and fast mesh generation for complicated geometries is achieved. The airframe noise is predicted through the coupling of incompressible Navier-Stokes flow solver and the aeroacoustic analogy-based Curle’s equation. In this paper, Curle’s equation in noncompact form is introduced to predict the acoustic sound from an object in flow. This approach is applied to JAXA Landing gear Evaluation Geometry model to investigate the influence of the detail components to flows and aerodynamic noises. The position of torque link and the wheel cap geometry are changed to discuss the influence. The present method showed good agreement with the preceding experimental result and proved that difference of the complicated components to far field noise was estimated. The result also shows that the torque link position highly affects the flow acceleration at the axle region between two wheels, which causes the change in SPL at observation point. 1. Introduction Great progress has been made in Computational Fluid Dynamics (CFD) in the past several decades, and nowadays it plays an important role in the design and analysis for aircraft development. The emerging problem for the commercial aircraft development is how to reduce the airframe noise from high lift device and landing gear. CFD has been widely used to predict the flow around the high lift device to reduce the aerodynamic noise, whereas the application of CFD to the landing gear is still limited [1–6]. The landing gear is constructed with many bluff components, and thus the noise source is not singular but many and thus complicated. To predict the noise from the real landing gear geometry precisely, all the components need to be modeled in the computation to treat all the interaction effect. Japan Aerospace Exploration Agency (JAXA) has developed a research landing gear model with detailed components based on the current two-wheel landing gear to understand the noise generation mechanism by experiments and numerical analysis. The model, Landing gear Evaluation Geometry model (LEG), includes all the components even with pins, tubes, and cavity as shown in Figure 1 [4–6]. To analyze the flow and acoustic field of the landing gear, the