Jae Ryong Kweon

Title: Jump discontinuities grazing corners and regularity of compressible Navier-Stokes flows

 

Abstract: In this talk I will discuss about a jump discontinuity grazing non-convex corners and regularities for compressible viscous Stokes flows. The jump discontinuity can be handled by constructing a vector field derived from the pressure jump on the interface curve. Piecewise regularity is derived in a suitable Sobolev space. I will also present some numerical simulations related to this issue.

 

References

[1] J. R. Kweon, R. B. Kellogg, Regularity of Solutions to the Navier-Stokes Equations for Compressible Flows on a Polygon. SIAM J. Math. Anal. 35 (2004) 1451-1485.
[2] J. R. Kweon. Corner singularity dynamics and regularity of compressible viscous Navier-Stokes flows. SIAM J. Math. Anal. 44 (2012) 3127-3161.
[3] J. R. Kweon. A jump discontinuity of compressible viscous flows grazing a non-convex corner. J. Math. Pures Appl. 100 (2013) 410 - 432.
[4] O-Sung Kwon and J. R. Kweon. Interior jump and regularity of compressible viscous Navier-Stokes flows through a cut. SIAM J. Math. 49 (2017) 1982 - 2008.
[5] J. R. Kweon and Minje Park. Interior jump and piecewise regularity for compressible Stokes equations in the T-shaped cavity domain. J. Differential Equations 267 (2019) 3693-3723.

 

 

 

 

Yasuhide Fukumoto

Title: Effect of compressibility in the reaction zone of a premixed flame and its implication to the Darrieus-Landau instability

Abstract: The Darrieus-Landau instability (DLI) is a linear instability of a planar premixed flame front which is identified as a density discontinuous surface. The effect of compressibility on the DLI is investigated by use of the method of $M^2$ expansions for small Mach numbers. We study the inner structure of the reaction layer, by applying the method of matched asymptotic expansions to an overall one-step irreversible chemical reaction expressed by the Arrhenius law. The temperature distribution is greatly influenced by the compressibility effect which originates from the material derivative of the pressure in the source term of the heat-conduction equation. This effect naturally embodies the volumetric heat loss, without having to include any artificial sink term, by decreasing the temperature, with the Mach number, on the burned side of the reaction zone, accompanied by the overshoot of the temperature in the midway of the reaction layer. An analysis of the burning-rate eigenvalue shows that the laminar flame speed sensitively drop down by the compressibility effect. The relevance of this result to the Mallard-Le Le Chatelier theory is pointed out. Slowing down of the laminar flame speed implies reduction of the growth rate of the DLI. The stabilizing effect of compressibility is illustrated from the viewpoint of vortex dynamics.

 

 

 

 

James Kelliher

Titel: 2D Euler equations with non-decaying vorticity and velocity

Abstract: All physical fluids are contained in some finite region, and so encounter some boundary, be it free, elastic, rigid, or moving. The interaction with this boundary is often the most difficult aspect of fluid flow to analyze. To avoid these difficulties and focus the analysis on the behavior away from the boundary, the fluid equations can be solved in the full space, thereby eliminating boundary terms. Decay of the data at infinity is usually assumed to keeps norms finite. Yet this is somewhat unsatisfactory as it implicitly defines a preferred origin near which the bulk of the fluid is concentrated.

An alternate approach is to assume that the fluid, as measured by its velocity and vorticity, is of the same order of magnitude throughout space. This approach was initiated by Ph. Serfati in 1995 for the 2D Euler equations in the plane. I will give an overview of this and subsequent work on the problem, first by Taniuchi 2004 and Taniuchi, Tashiro, and Yoneda 2010, and including later joint work of mine with David Ambrose, Milton Lopes Filho, and Helena Nussenzveig Lopes. To be physically relevant, it is also important to know that changes in initial data far away from a point should have a limited effect on solutions near the point for short time: I will discuss joint work with Elaine Cozzi in which we demonstrate this and also allow the velocity to grow at infinity.

 

 

 

 

Dmitry Kolomenskiy

Title: Computer Modelling of the Flight of a Bumblebee

 

Abstract: Miniaturization of various devices, being a continuing trend over recent decades, gave rise to a new, interdisciplinary take on insects as templates for bio-inspired small-scale designs. Contemporary research on insect flight integrates a biology approach to living organisms and a physics approach to complex systems. Flight as a mode of locomotion emerges from interaction of internal biological systems with the external environment. Although it is currently, or even generally, unrealistic to fully reduce this problem to first principles, it proves possible and useful to combine low-dimensional models of biological components with first-principle fluid dynamics models, for advancing our understanding of animal flight. In this talk, I will present flight experiments, numerical simulations and theoretical modelling of the flight of a bumblebee. The experiments have been conducted in free flight, and video recordings of hovering, forward flight and maneuvering have been acquired and analyzed to reconstruct the three-dimensional body motion and wing kinematics. The numerical simulations using FluSI, a pseudo-spectral Navier-Stokes solver with volume penalization, have allowed to assess the aerodynamic performance. The theoretical analysis has brought new insights in the bumblebee flight dynamics.

 

 

 

 

Myoungjean Bae

Title: Detached shock past a blunt body

 

Abstract: In this talk, I will present a recent result on the existence of detached shocks past a blunt body $R^2$. The talk is based on a collaboration with Wei Xiang (City University of Hong Kong). Further details on the talk are as follows:

In $R^2$, a symmetric blunt body $W_b$ is fixed by smoothing out the tip of a symmetric wedge $W_0$ with the half-wedge angle $theta_win (0, frac{pi}{2})$. We first show that if a horizontal supersonic flow of uniform state moves toward $W_0$ with a Mach number $M_{infty}>1$ sufficiently large, %depending on $theta_w$,

then there exist two shock solutions, {emph{a weak shock solution and a strong shock solution}}, with the shocks being straight and attached to the tip of the wedge $W_0$. Such shock solutions are given by a shock polar analysis, and they satisfy entropy conditions. The main goal of this work is to construct a detached shock solution of the steady Euler system for inviscid compressible irrotational flow in $R^2setminus W_b$. In particular, we seek a shock solution with the far-field state being the strong shock solution obtained from the shock polar analysis. Furthermore, we prove that the detached shock forms a convex curve around the blunt body $W_b$ if the Mach number of the incoming supersonic flow is sufficiently large, and if the boundary of $W_b$ is convex.

If time allows, I will also discuss about related open problems and their difficulties.

 

 

 

 

Keh-Ming Shyue

Title: Singular solutions of dispersive systems

 

Abstract: Our goal in this talk is to discuss a dispersive regularization of the equations of barotropic uids. The governing equations are the Euler-Lagrange equations for a Lagrangian depending not only on the density, but also on the _rst material derivative of density. Such a regularization arises, in particular, in the modeling of waves in bubbly uids as well as in the theory of water waves.

We show that such terms are not always regularizing. The solution can develop shocks even in the presence of dispersive terms. In particular, we construct such a shock solution that connects a constant state to a periodic wave train. The shock speed coincides necessarily with the velocity of the corresponding wave train. The corresponding generalized Rankine-Hugoniot relations (jump relations) are derived. At such shocks the Rankine-Hugoniot relations to Whitham's equations (modulation equations) are also satisfied.

The numerical evidence of the existence of such shocks is demonstrated in the case of the Serre-Green-Naghdi equations and the Boussinesq system describing long water waves. To this aim, a robust high-order accurate numerical has been designed based on an appropriate operator splitting of the governing equations. In particular, it has been shown that such waves can dynamically be formed. Also, when such a wave is used for initial data, it is not destroyed by small perturbations. This proves a certain stability of these waves.

This is a joint work with French colleagues: S. Gavrilyuk, B. Nkonga, and Lev Truskinovsky.

 

 

 

 

Jiajie Chen

Title: Singularity formation for 2D Boussinesq and 3D Euler equations with boundary and some related 1D models

 

Abstract: In this talk, we will discuss recent results on stable self-similar singularity formation for the 2D Boussinesq and singularity formation for the 3D Euler equations in the presence of the boundary with $C^{1,alpha}$ initial data for the velocity field that has finite energy. The blowup mechanism is based on the Hou-Luo scenario of a potential 3D Euler singularity. We will also discuss some 1D models for the 3D Euler equations that develop stable self-similar singularity in finite time. For these models, the regularity of the initial data can be improved to $C_c^{infty}$. Some of the results are joint work with Thomas Hou and De Huang. 

 

 

   

 

Eiichi Sasaki

Title: Bimodal vortex solutions on a sphere

 

Abstract: Steady and time-periodic solutions of the two-dimensional Naveir--Stokes equations are studied on a sphere under steady forcings represented by a single spherical harmonic function. We investigate the bifurcation structure and find either simple steady or time-periodic bimodal solutions, consisting of only two pairs of positive and negative vortices, depending on the forcing profiles at very high Reynolds numbers.

 

 

 

 

Jinah Hwang

Title: 2-D RIEMANN PROBLEM FOR AN ISOTROPIC SYSTEM OF HYPERBOLIC CONSERVATION LAWS

 

Abstract: A Riemann problem is an initial value problem for conservation laws for which the initial condition is given as piecewise constant states. For a system of hyperbolic con- servation laws, most of the 2-D Riemann problem was studied under the assumption: only one planar wave is projected at each initial discontinuity. Since an n _ n system gener- ally can project n waves at each initial discontinuity, we consider 2-D Riemann problem for hyperbolic system without the restriction of one planar elementary wave at each initial discontinuity. We compute the analytic solutions and the numerical solutions. Numerical solutions con_rm the constructed analytic solutions and the result shows the rich behavior of interesting structure of the Riemann solution.