Population dynamics in compressible flows

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.     

Kolmogorov-Kraichnan scaling in the inverse energy cascade of two-dimensional plasma turbulence

Turbulence in plasmas that are magnetically confined, such as tokamaks or linear devices, is two dimensional or at least quasi two dimensional due to the strong magnetic field, which leads to extreme elongation of the fluctuations, if any, in the direction parallel to the magnetic field. These plasmas are also compressible fluid flows obeying the compressible Navier-Stokes equations. This Letter presents the first comprehensive scaling of the structure functions of the density and velocity fields up to 10th order in the PISCES linear plasma device and up to 6th order in the Mega-Ampère Spherical Tokamak (MAST). In the two devices, it is found that the scaling of the turbulent fields is in good agreement with the prediction of the Kolmogorov-Kraichnan theory for two-dimensional turbulence in the energy cascade subrange.     

Stationarity of Lagrangian Velocity¶in Compressible Environments

We study the transport of a passive tracer particle by a random d-dimensional, Gaussian, compressible velocity field. It is well known, since the work of Lumley, see [13], and Port and Stone, see [20], that the observations of the velocity field from the moving particle, the so-called Lagrangian velocity process, are statistically stationary when the field itself is incompressible. In this paper we study the question of stationarity of Lagrangian observations in compressible environments. We show that, given sufficient temporal decorrelation of the velocity statistics, there exists a transformation of the original probability measure, under which the Lagrangian velocity process is time stationary. The transformed probability is equivalent to the original measure. As an application of this result we prove the law of large numbers for the particle trajectory. Received: 1 May 2001 / Accepted: 4 December 2001     

Gas flow characteristics of argon inductively coupled plasma and advections of plasma species under incompressible and compressible flows

In this work, incompressible and compressible flows of background gas are characterized in argon inductively coupled plasma by using a fluid model, and the respective influence of the two flows on the plasma properties is specified. In the incompressible flow, only the velocity variable is calculated, while in the compressible flow, both the velocity and density variables are calculated. The compressible flow is more realistic; nevertheless, a comparison of the two types of flow is convenient for people to investigate the respective role of velocity and density variables. The peripheral symmetric profile of metastable density near the chamber sidewall is broken in the incompressible flow. At the compressible flow, the electron density increases and the electron temperature decreases. Meanwhile, the metastable density peak shifts to the dielectric window from the discharge center, besides for the peripheral density profile distortion, similar to the incompressible flow.The velocity profile at incompressible flow is not altered when changing the inlet velocity, whereas clear peak shift of velocity profile from the inlet to the outlet at compressible flow is observed as increasing the gas flow rate. The shift of velocity peak is more obvious at low pressures for it is easy to compress the rarefied gas. The velocity profile variations at compressible flow show people the concrete residing processes of background molecule and plasma species in the chamber at different flow rates. Of more significance is it implied that in the usual linear method that people use to calculate the residence time, one important parameter in the gas flow dynamics, needs to be rectified. The spatial profile of pressure simulated exhibits obvious spatial gradient. This is helpful for experimentalists to understand their gas pressure measurements that are always taken at the chamber outlet. At the end, the work specification and limitations are listed.     

Weak Solutions with Decreasing Energy¶of Incompressible Euler Equations

Weak solution of the Euler equations is an L²-vector field u(x, t), satisfying certain integral relations, which express incompressibility and the momentum balance. Our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations, as both viscosity and compressibility tend to zero; thus, we believe that weak solutions describe turbulent flows with very high Reynolds numbers. Every physically meaningful weak solution should have kinetic energy decreasing in time. But the existence of such weak solutions have been unclear, and should be proven. In this work an example of weak solution with decreasing energy is constructed. To do this, we use generalized flows (GF), introduced by Y. Brenier. GF is a sort of a random walk in the flow domain, such that the mean kinetic energy of particles is finite, and the particle density is constant. We construct a GF such that fluid particles collide and stick; this sticking is a sink of energy. The GF which we have constructed is a GF with local interaction; this means that there are no external forces. The second important property is that the particle velocity depends only on its current position and time; thus we have some velocity field, and we prove that this field is a weak solution with decreasing energy of the Euler equations. The GF is constructed as a limit of multiphase flows (MF) with the mass exchange between phases.     

Multi-scale Analysis of Compressible Viscous and Rotating Fluids

We study a singular limit for the compressible Navier-Stokes system when the Mach and Rossby numbers are proportional to certain powers of a small parameter . If the Rossby number dominates the Mach number, the limit problem is represented by the 2-D incompressible Navier-Stokes system describing the horizontal motion of vertical averages of the velocity field. If they are of the same order then the limit problem turns out to be a linear, 2-D equation with a unique radially symmetric solution. The effect of the centrifugal force is taken into account.     

Perturbation theory for large Stokes number particles in random velocity fields

We derive a perturbative approach to study, in the large inertia limit, the dynamics of solid particles in a smooth, incompressible and finite-time correlated random velocity field. We carry on an expansion in powers of the inverse square root of the Stokes number, defined as the ratio of the relaxation time for the particle velocities and the correlation time of the velocity field. We describe in this limit the residual concentration fluctuations of the particle suspension, and determine the contribution to the collision velocity statistics produced by clustering. For both concentration fluctuations and collision velocities, we analyze the differences with the compressible one-dimensional case.     

Active-to-absorbing phase transition subjected to the velocity fluctuations in the frozen limit case

The directed bond percolation process is studied in the presence of compressible velocity fluctuations with long-range correlations. We discuss a construction of a field theoretic action and a way of obtaining its large scale properties using the perturbative renormalization group. The most interesting results for the frozen velocity limit are given.     

Perturbational blowup solutions to the compressible 1-dimensional Euler equations

We construct non-radially symmetry solutions for the compressible 1-dimensional adiabatic Euler equations in this Letter. In detail, we perturb the linear velocity with a drifting term:

(1)     

Discrete Velocity Fields with Explicitly Computable Lagrangian Law

We introduce a class of random velocity fields on the periodic lattice and in discrete time having a certain hidden Markov structure. The generalized Lagrangian velocity (the velocity field as viewed from the location of a single moving particle) has similar hidden Markov structure, and its law is found explicitly. Its rate of convergence to equilibrium is studied in small numerical examples and in rigorous results giving absolute and relative bounds on the size of the second–largest eigenvalue modulus. The effect of molecular diffusion on the rate of convergence is also investigated; in some cases it slows convergence to equilibrium. After repeating the velocity field periodically throughout the integer lattice, it is shown that, with the usual diffusive rescaling, the single–particle motion converges to Brownian motion in both compressible and incompressible cases. An exact formula for the effective diffusivity is given and numerical examples are shown.     

Blow-up Criteria of Classical Solutions of Three-Dimensional Compressible Magnetohydrodynamic Equations

In this paper we consider the isentropic compressible magnetohydrodynamic equations in three space dimensions, and establish a blow-up criterion of classical solutions, which depends on the gradient of the velocity and magnetic field.     

Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves

In this paper, a three-dimensional (3D) finite-difference lattice Boltzmann model for simulating compressible flows with shock waves is developed in the framework of the double-distribution-function approach. In the model, a density distribution function is adopted to model the flow field, while a total energy distribution function is adopted to model the temperature field. The discrete equilibrium density and total energy distribution functions are derived from the Hermite expansions of the continuous equilibrium distribution functions. The discrete velocity set is obtained by choosing the abscissae of a suitable Gauss–Hermite quadrature with sufficient accuracy. In order to capture the shock waves in compressible flows and improve the numerical accuracy and stability, an implicit–explicit finite-difference numerical technique based on the total variation diminishing flux limitation is introduced to solve the discrete kinetic equations. The model is tested by numerical simulations of some typical compressible flows with shock waves ranging from 1D to 3D. The numerical results are found to be in good agreement with the analytical solutions and/or other numerical results reported in the literature.     

Intermittency and universality in fully developed inviscid and weakly compressible turbulent flows

We perform high-resolution numerical simulations of homogenous and isotropic compressible turbulence, with an average 3D Mach number close to 0.3. We study the statistical properties of intermittency for velocity, density, and entropy. For the velocity field, which is the only quantity that can be compared to the isotropic incompressible case, we find no statistical differences in its behavior in the inertial range due either to the slight compressibility or to the different dissipative mechanism. For the density field, we find evidence of "frontlike" structures, although no shocks are produced by the simulation.     

涡流管内三维强旋转流场数值模拟与分析

涡流管内可压缩气体的强旋转流动是涡流管能量分离的根本原因和驱动力,因而涡流管内流场研究是揭示涡流管能量分离物理机制的首要关键问题。由于涡流管内可压缩气体的三维强旋转湍流流动,实验测量中存在诸多问题,而CFD数值模拟技术对此具有很大的优势。文中以涡流管内部流场为研究对象,建立了涡流管计算域模型并进行网格划分,讨论了边界条件、湍流模型以及线性方程组求解策略等问题,对不同冷气流率下的涡流管内三维强旋流流场结构特性进行数值模拟,获得了不同冷气流率下的旋转运动、轴向运动、径向运动和循环流的分布特性。研究表明Realizableκ-ε湍流模型能够充分反映强旋流动特点,数值模拟结果与文献中实验值基本吻合。     

Population dynamics at high Reynolds number

We study the statistical properties of population dynamics evolving in a realistic two-dimensional compressible turbulent velocity field. We show that the interplay between turbulent dynamics and population growth and saturation leads to quasilocalization and a remarkable reduction in the carrying capacity. The statistical properties of the population density are investigated and quantified via multifractal scaling analysis. We also investigate numerically the singular limit of negligibly small growth rates and delocalization of population ridges triggered by uniform advection.     

Wave equation and dispersion relations for a compressible rotating fluid

A fundamental non-classical fourth-order partial differential equation to describe small amplitude linear oscillations in a rotating compressible fluid, is obtained. The dispersion relations for such a fluid, and the different regions of the group and phase velocity are analyzed.     

谐振管中混合工质的热声分离效应

本文在可压缩SIMPLE算法的基础上,基于先求解二维热声谐振腔内的速度场和温度场,然后利用求出的速度和温度值求解组分浓度场的思路,对混合工质的热声分离现象进行了数值研究。研究结果表明:在热声效应和热扩散效应共同作用下,声波能够将混合物中的两种气体分别泵向谐振腔的压力腹点和节点,使得混合气体在声波传播方向上逐渐分离。另外,通过考察5种不同管径下谐振管内径向浓度的变化规律,详细研究了浓度边界层在径向上的影响范围以及不同管径下的分离效率,并提出最佳管径应为16倍浓度渗透深度左右。     

Integral one-point statistical characteristics of vector fields in stochastic magnetohydrodynamic flows

We study integral statistical characteristics of a vector passive tracer (homogeneous at the initial time) in a velocity field that is assumed to be a Gaussian random field homogeneous in space and delta-correlated in time. Such statistical characteristics describe the dynamical system as a whole in the entire space, separating out the field generation processes, which allows us to not digress into details of the dynamics related to the advection of these quantities. The density field gradient (in the general case of a compressible fluid) and the magnetic field vector with its spatial derivatives (in an incompressible fluid) are such a tracer. We study the isotropization in time, helicity, and dissipation of these fields in the absence of molecular diffusion effects. We formulate a method of successive approximations for the variance of the density field and the mean magnetic field energy that allows the solutions valid in the entire time interval to be obtained in the first order in molecular diffusion coefficients.     

自由汇流旋涡Ekman抽吸演化机理

自由汇流旋涡形成过程中有抽吸现象发生, 是一个比较复杂的气液两相耦合过程, 其中所涉及的Ekman层耦合及演化机理具有重要的科研价值与实际意义. 针对上述问题, 提出了一种自由汇流旋涡Ekman抽吸演化机理建模与分析方法. 基于多相流体体积VOF模型与湍动能-耗散(k-ε)模型, 建立了面向汇流旋涡Ekman抽吸演化的两相动力学模型. 基于上述模型, 分析初始转动速度分量、排流量与Ekman抽吸过程的内在联系, 并揭示相关流场分布规律. 研究结果表明: 初始扰动不同, 汇流旋涡的吸气孔、抽气孔距离容器底面边界的高度保持不变; 初始扰动加强, 吸气阶段转速增加, Ekman边界层厚度及抽吸高度增加, 抽吸、贯穿阶段Ekman抽吸现象减弱; 初始扰动恒定, Ekman抽吸高度保持不变, 与排流量变化无关. 研究结果可为自由汇流旋涡形成机理方面的研究提供有益参考, 也可为冶金、化工领域的旋涡抑制控制提供技术支持.     

Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.