Dissipative Euler Flows Originating from Circular Vortex Filaments

The partial differential equations (PDE) governing the motions of incompressible ideal fluid in three dimensional (3D) space are among the most fundamental nonlinear PDEs in nature and have found a lot of important applications. Due to the presence of super-critical non-linearity, the fundamental question of global well-posedness still remains open and is generally viewed as one of the most outstanding open questions in mathematics. In this thesis, we investigate the potential finite-time singularity formation of the 3D Euler equations and simplified models by studying the self-similar spatial profiles in the potentially singular solutions. In the first part, we study the self-similar singularity of two 1D models, the CKY model and the HL model, which approximate the dynamics of the 3D axisymmtric Euler equations on the solid boundary of a cylindrical domain. The two models are both numerically observed to develop self-similar singularity. We prove the existence of a discrete family of self-similar profiles for the CKY model, using a combination of analysis and computer-aided verification. Then we employ a dynamic rescaling formulation to numerically study the evolution of the spatial profiles for the two 1D models, and demonstrate the stability of the self-similar singularity. We also study a singularity scenario for the HL model with multi-scale feature. In the second part, we study the self-similar singularity for the 3D axisymmetric Euler equations. We first prove the local existence of a family of analytic self-similar profiles using a modified Cauchy-Kowalevski majorization argument. Then we use the dynamic rescaling formulation to investigate two types of initial data with different leading order properties. The first initial data correspond to the singularity scenario reported by Luo and Hou. We demonstrate that the self-similar profiles enjoy certain stability, which confirms the finite-time singularity reported by Luo and Hou. For the second initial data, we show that the solutions develop singularity in a different manner from the first case, which is unknown previously. The spatial profiles in the solutions become singular themselves, which means that the solutions to the Euler equations develop singularity at multiple spatial scales. In the third part, we propose a family of 3D models for the 3D axisymmetric Euler and Navier-Stokes equations by modifying the amplitude of the convection terms. The family of models share several regularity results with the original Euler and Navier-Stokes equations, and we study the potential finite-time singularity of the models numerically. We show that for small convection, the solutions of the inviscid model develop self-similar singularity and the profiles behave like travelling waves. As we increase the amplitude of the velocity field, we find a critical value, after which the travelling wave self-similar singularity scenario disappears. Our numerical results reveal the potential stabilizing effect the convection terms.

7R47. Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics. - AJ Majda (Courant Inst of Math Sci, New York Univ, New York NY) and AL Bertozzi (Duke Univ, Durham NC). Cambridge UP, Cambridge, UK. 2002. 545 pp. Softcover. ISBN 0-521-63948-4. $40.00. (Hardcover ISBN 0-521-63057-6 $100.00). Reviewed by A Ogawa (Dept of Mech Eng, Col of Eng, Nihon Univ, T 963 Tamura-machi, Kooriyama-city, Japan).In this book, the theoretical and experimental analyses of the velocity fields with vorticity are applied to explain the physical phenomena of flow patterns of various types of vortex flows and of the flow structures in the boundary layer with high velocity gradient on a solid surface. Further, the vortex flow in nature is not always circular, but also elliptic rotational flows occur under stable and unstable conditions. Therefore, the fundamental mathematical and applied descriptions of vorticity are important factors in fluid mechanics in scientific and engineering applications. From these viewpoints, this textbook is reviewed as follows. In Chapter 1, Introduction to Vortex Dynamics for Incompressible Fluid Flows, the fundamental descriptions of the Euler and the Navier-Stokes equations, and the important quantities of velocity, vorticity, helicity, impulse, and moment of fluid impulse are defined. Chapter 2, entitled Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations, discusses the vorticity-stream formulations of 2D. Periodic flow, cat’s-eye flow, and also Beltrami flows are discussed with figures. The existence of a solution to either the Euler or the Navier-Stokes equations on the same time interval, given generally smooth initial velocity fields, is examined in Chapter 3, Energy Method for the Euler and the Navier-Stokes Equations. In Chapter 4, Particle-Trajectory Method for Existence and Uniqueness of Solution to the Euler Equation, the questions of existence, uniqueness, and continuation of solution to the Euler and the Navier-Stokes equations are discussed. Search for Singular Solutions to the 3D Euler Equations, is the title for Chapter 5, where an important unsolved research problem for incompressible flow and the current attempts to make progress on this problem are the main focus. Chapter 6, Computational Vortex Methods, deals with computational methods for simulations of the Euler and the Navier-Stokes equations with high Reynolds number condition for vortex dynamics, and includes a brief historical summary. In order to study the motion of slender tubes of vorticity at high Reynolds numbers, Chapter 7, Simplified Asymptotic Equations for Slender Vortex Filaments, describes formal, but concise, asymptotic expansions to the simplified asymptotic equations; they are seen to emerge with remarkable properties. In Chapter 8, Weak Solutions to the 2D Euler Equations with Initial Vorticity inL∞, a flow field having elliptic vorticity, like Thomson-Rankine combined vortex model, is examined. Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation is the title of Chapter 9, which describes a vortex sheet problem occurring in the vorticity layers. The last four chapters—Weak Solutions and Solution Sequences in Two Dimensions, 2D Euler Equation: Concentrations and Weak Solutions with Vortex Sheet Initial Data (with an example constructed by Greengard and Thomann), Reduced Hausdorf Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions, and Vlasov-Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions (with an analogy between vorticity and electron density)—address the mathematical theory connected with small scale structures and dynamics in high Reynolds number and inviscid flow. To sum up, this textbook for advanced undergraduate and beginning graduate students, is aimed at mathematicians and physicists to examine mathematical theories and techniques, and to explore their applications. It appears to be a little difficult to grasp the physical phenomena involved and to apply them directly to nature and engineering. Since there is a small gap between the actual fluid flows in nature and engineering and the contents in this textbook with the examples shown, it would be better if the authors had introduced and explained the examples from W Albring, Elementarvorga¨nge Fluider Wirbelbewegungen (Akademische-Verlag, Berlin, 1981), and L Lugt, Introduction to Vortex Theory (Vortex Flow Press, Inc, 1996). However, for graduate students, mathematicians, and physicists who can understand the contents of the books by GK Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, 1999) and by LD Landau and EM Lifshitz, Fluid Mechanics (Butterworth-Heinemann, Oxford, 1987), this peculiar textbook certainly contributes to understanding the fundamental concepts of fluid flow with vorticity, and to apply them to nature and engineering.

Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $C^{1,\alpha}$ initial velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations with boundary and $C^{1,\alpha}$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30,31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $C^{1,\alpha}$ initial velocity. We use a dynamic rescaling formulation and follow the general framework of analysis developed in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler or the 2D Boussinesq equations with $C^{1,\alpha}$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. In the previous version of this paper, we proved the blowup results for the 3D axisymmetric Euler equations with initial data $(u_0^{\theta})^2, u_0^r, u_0^z \in C^{1,\alpha}$. Though the velocity $u^r, u^z$ in the axisymmetric setting is $C^{1,\alpha}$, our interpretation that the velocity is $C^{1,\alpha}$ is not correct since the velocity in 3D also depends on $u^{\theta}$, which is not $C^{1,\alpha}$. This oversight can be fixed easily with a minor change in the construction of the approximate steady state and a minor modification to localize the approximate steady state.

In this thesis, new results excluding finite time singularities with localized assumptions/conditions are obtained for the 3D incompressible Euler equations. The 3D incompressible Euler equations are some of the most important nonlinear equations in mathematics. They govern the motion of ideal fluids. After hundreds of years of study, they are still far from being well-understood. In particular, a long-outstanding open problem asks whether finite time singularities would develop for smooth initial values. Much theoretical and numerical study on this problem has been carried out, but no conclusion can be drawn so far. In recent years, several numerical experiments have been carried out by various authors, with results indicating possible breakdowns of smooth solutions in finite time. In these numerical experiments, certain properties of the velocity and vorticity field are observed in near-singular flows. These properties violate the assumptions of existing theoretical theorems which exclude finite time singularities. Thus there is a gap between current theoretical and numerical results. To narrow this gap is the main purpose of the work presented in this thesis. In this thesis, a new framework of investigating flows carried by divergence-free velocity fields is developed. Using this new framework, new, localized sufficient conditions for the flow to remain smooth are obtained rigorously. These new results can deal with fast shrinking large vorticity regions and are applicable to recent numerical experiments. The application of the theorems in this thesis reveals new subtleties, and yields new understandings of the 3D incompressible Euler flow. This new framework is then further applied to a two-dimensional model equation, the 2D quasi-geostrophic equation, for which global existence is still unproved. Under certain assumptions, we obtain new non-blowup results for the 2D quasi-geostrophic equation. Finally, future plans of applying this new framework to some other PDEs as well as other possibilities of attacking the 3D Euler and 2D quasi-geostrophic singularity problems are discussed.

Development of fuel cells has been boosted by global and regional environmental issues. Among others, the solid oxide fuel cell (SOFC) has been drawing much attention as a unit for distributed energy generation. Numerical analysis is used as a powerful tool in research and development of field cells, especially of the SOFC for which important and necessary thermal management information has scarcely been supplied experimentally. A central issue for achieving maximum benefit from the numerical analysis is how to properly model the complex phenomena occurring in the fuel cells. This chapter presents a numerical model for a tubular SOFC including a case with indirect internal reforming. In this model, the velocity field in the air and fuel passages and heat and mass transfer in and around a tubular cell are calculated with a two-dimensional cylindrical coordinate system adopting the axisymmetric assumption. Internal reforming and electrochemical reactions are both taken into account in this model. Electric potential field and electric current in the cell are also calculated simultaneously allowing their nonuniformity in the peripheral direction. A previously developed quasi-two-dimensional model was adopted to combine the assumed axisymmetry of velocity, heat and mass transfer fields and the peripheral nonuniformity of electric potential field and electric current. Details of those numerical procedures are described and examples of the calculated results are discussed. After presenting a re fundamental results, several strategies to reduce the maximum temperature gradient of the cell are examined for a case with indirect internal fuel reforming.

It is shown that the Euler equations of two-dimensional incompressible flow, with initial vorticity in L p , p > 1, possess weak solutions which may be obtained as a limit of vortex “blobs”; i.e., the vorticity is approximated by a finite sum of cores of prescribed shape which are advected according to the corresponding velocity field. If the vorticity is instead a finite measure of bounded support, such approximations lead to a measure-valued solution of the Euler equations in the sense of DiPerna and Majda [7]. The analysis is closely related to that of [7].

In this work, we explore the potential of Physics-Informed Neural Networks (PINNs) to process and combine data from supersonic wind tunnel experiments with computational methods. Specifically, we aim to infer velocity fields for a family of turbulent shockwave-boundary layer interactions (SBLI) at Ma = 2, using Background Oriented Schlieren (BOS) measurements. We train a neural network constrained by a set of Euler equations to predict velocity, pressure, and density fields from limited and noisy BOS data in an inviscid region of the flow field. We compare the results with PIV measurements to validate the predictions and analyze the effects of the BOS line-of-sight integration. The results suggest that, in this inviscid case, it is possible to reconstruct the velocity fields with an RMSE of < 5% even though the source data is corrupted by three-dimensional effects and measurement noise.

We study weak solutions of the two-dimensional (2D) filtered Euler equations whose vorticity is a finite Radon measure and velocity has locally finite kinetic energy, which is called the vortex sheet solution. The filtered Euler equations are a regularized model based on a spatial filtering to the Euler equations. The 2D filtered Euler equations have a unique global weak solution for measure valued initial vorticity, while the 2D Euler equations require initial vorticity to be in the vortex sheet class with a distinguished sign for the existence of global solutions. In this paper, we prove that vortex sheet solutions of the 2D filtered Euler equations converge to those of the 2D Euler equations in the limit of the filtering parameter provided that initial vortex sheet has a distinguished sign. We also show that a simple application of our proof yields the convergence of the vortex blob method for vortex sheet solutions. Moreover, we make it clear what kind of condition should be imposed on the spatial filter to show the convergence results and, according to the condition, these results are applicable to well-known regularized models like the Euler-$\alpha$ model and the vortex blob model.

The complete equations of average linear momentum balance for a single-phase fluid in an incompressible, homogeneous, porous medium are derived. The derivation begins with Euler's equation of motion /or a continuum and uses an integral transform recently developed by Slattery. For steady flow of compressible, Newtonian fluid, the usual equations of motion result. For transient flow, the space-time description of the pressure is determined in the lowest approximation by the telegrapher's equation. From the analysis a new phenomenological coefficient results which connects the viscous traction to the derivative of the linear momentum density. The magnitude of this coefficient determines the velocity of sound through the pore structure in this approximation to the pressure field. Introduction The modification of Darcy's law of momentum balance for steady, single-phase flow through porous media has been discussed for many years, The first modification was suggested by Forcheimer who added terms of higher order in the velocity. These can be expected to appear because the underlying microscopic equations of momentum balance are themselves nonlinear in the point velocity field. The Reynolds tensor pvv, which represents the convective flux of momentum density, appears in the momentum balance equation. Only in rectilinear flow (parallel stream lines) does the divergence of this tensor vanish. Since the steady flow stream lines in most porous media are not parallel, nonlinear dependence of the pressure gradient on the velocity should naturally appear. This nonlinearity has nothing to do with turbulence in the ordinary sense of random fluctuations in the pressure and velocity fields. It arises simply because the stream lines converge and diverge, even for steady flows. Klinkenberg demonstrated that the permeability coefficient in Darcy's law depends on the absolute pressure or, alternatively, on the density field. however, because he neglected inertial terms of the Forcheimer type, his correction coefficient could not be represented by a constant but tended toward a constant as the velocity decreased. Forcheimer's and Klinkenberg's modifications can be combined in a rigorous way to account for both inertia and slip during steady flow. This will be shown in a future paper. The transient change of pressure in porous media has been described by the diffusion equation. This form results from eliminating the velocity and density fields from a combination of the equations of motion in the form of Darcy's law, the continuity equation and an equation of state. Fatt suggested that the cause of deviations from the prediction of the diffusion equation for pressure transients lies not in the choice of Darcy's law as the equation of motion but on the existence of dead-end pores which might invalidate the averaged equation of continuity. On the other hand, Oroveanu and Pascal noted that the time derivative of the momentum density must be included in the equations of motion since this measures the local rate of change of momentum density. Their differential equation for pressure is the telegrapher's equation (neglecting gravity). However, their form of this equation predicts that the speed of pressure propagation through the pore structure is the same as that through the bulk fluid. M. K. Hubbert attempted a derivation of Darcy's law by volume averaging the Navier-Stokes equations. Since these equations represent momentum balance at a point within an open set of points containing the fluid itself, Hubbert's volume averaging cannot lead to terms involving transfer of momentum between the fluid and the walls of the pores.

In this article we investigate the two-dimensional incompressible rotating and stratified, just rotating, just stratified Euler equations, comparing with each other and with the normal Euler equations with the self-similar Ansatz. The motivation of our study is the following the presented rotating stratified fluid equations can be interpreted as a well-established starting point of various more complex and more realistic meteorologic, oceanographic or geographic models. We present analytic solutions for all four models for density, pressure and velocity fields, most of them are some kind of power-law type of functions. In general the presented solutions have a rich mathematical structure. Some solutions show nonphysical explosive properties others, however are physically acceptable and have finite numerical values with power law decays. For a better transparency we present some figs for the most complicated velocity and pressure fields. To our knowledge there are no such analytic results available in the literature till today. Our results may attract attention in various scientific fields.

SynopsisWhen a symmetric rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. These equations at the boundary contain a time derivative and thus are of a dynamic nature. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity and pressure fields of the fluid motion and the angular velocity of the rigid body.In this paper we formulate the physical problem for the case of rotation about an axis of symmetry as an abstract ordinary differential equation in two Banach spaces in which the velocity field is the only unknown. To achieve this, a method for the elimination of the pressure field, which also occurs in the boundary condition, is developed. Existence and uniqueness results for the abstract equation are derived with the aid of the theory of B-evolutions and the associated theory of fractional powers of a closed pair of operators.

Tracer particle path photographs of a rising isolated laminar thermal injected from a small source are presented. The presence of a velocity wake was clearly observed. The streaks of the tracers were sufficiently clear to deduce the velocity and vorticity fields. There is some indication that the velocity field approached a similarity state after an initial development distance. The vorticity distribution, calculated from the velocity field, was diffuse with no evidence of a distinctive core. Near the symmetry axis, the vorticity varied almost linearly with the radial coordinate and was independent of the axial coordinate. In the far field, the vorticity decreased monotonically.

Optimal regularity for the 2D Euler equations in the Yudovich class

Calculation of complex singular solutions to the 3D incompressible Euler equations

We discuss the non-linear growth of the induced-by-gravity kurtosis of the density and velocity fields, smoothed with a Gaussian filter. To do this, we take advantage of the exact perturbative technique [1,2] and the Zel'dovich approximation [3,4]. The kurtosis describes features such as sharpness or stretchiness of a distribution and the extent of its rare-event tail. In the case of the density field, our study extends previous works devoted to the calculation of the third moment, i.e. the skewness (see [5,61 and references therein). For the velocity field, due to its isotropy, all the odd moments are zero and consequently the study of the kurtosis is important because it represents the first moment where a departure from Gaussianity could be found. The time evolution equations for the matter density fluctuation 6(x, t) and the peculiar velocity field v(x, t) are the Poisson equation, the Euler equation and the continuity equation. We only consider Gaussian initial fluctuations. In order to compute the fourth order moment, we need to use the exact perturbative technique to solve approximately the previous equations up to third order solution. In the Zel'dovich approximation (ZA) the motion of particles from the initial comoving (Lagrangian) positions q is approximated by straight paths. The Eulerian position at time t is then given by tile uniformly accelerated motion x[q, t] = q + D(t) S(q) , where D(t) is the growth factor of linear density perturbations and S(q) is the displacement vector related to the primordial velocity field. An expression for the nth order perturbative corrections 5,,(X), when the density fluctuation field 6 is evolved according to ZA, is presented in [4]. First, we study the dependence of the induced-by-gravity excess kurtosis of the density field, smoothed with a Gaussian filter, namely the parameter $4 [(~4)--3(~ 2)2]/(6 2)3, on an initial (scale-free) power spectrum P(k) (x k ". More details can be found in [7]. According to the perturbative theory, the lowest order non-zero connected contribution to the induced-by-gravity kurtosis parameter $4 is