# <span class=strong>Reception and Poster Session</span><br><br/><br/><b>Poster submissions welcome from all participants</b><br><br/><br/><a<br/><br/>href=/visitor-folder/contents/workshop.html#poster><b>Instructions</b></a>

Monday, February 22, 2010 - 4:40pm - 6:30pm

Lind 400

**Dynamical constraints for 3D Euler and possible blow-up**

Robert Kerr (University of Warwick)

Co-author Miguel D. Bustamante (UCD Dublin).

While characterization of singular behaviour of simulations of the Euler

equations using the localized spatio-temporal growth of the vorticity

modulus would be preferred, because numerical simulations are discrete,

this approach cannot reliably follow its position and value

at times and positions very close to the potential singularity.

With this paradigm in mind, the natural alternative is methods that

identify possible singular evolution in terms of global quantities

determined over suitably-defined areas and volumes of the dynamical system.

This poster presents methods based on the conservation of integrals

defined on symmetry planes of a commonly used class of initial

conditions. These are:

(i) strong tools designed to validate any numerical simulation,

irrespective of the conclusions on the potentially singular behaviour, and

(ii) sharp inequalities and exact results to estimate more precisely the

vorticity growth exponents in a blow-up scenario.**Axisymmetric Euler-α equations without**

swirl: Existence, uniqueness, and radon measure valued

solutions

Dongjuan Niu (Capital Normal University)

The global existence of weak solutions for the

three-dimensional

axisymmetric Euler-alpha (also known as Lagrangian-averaged

Euler-α) equations, without swirl, is established, whenever

the initial unfiltered velocity v_{0}satisfies

curlv_{0}/r is a

finite Randon measure with compact support. Furthermore, the

global existence and uniqueness, is also

established in this case provided that curlv_{0}/r belongs to

L^{p}_{c}(R^{3}) with p>3/2. It is worth mention that

no such results are known to be available, so far, for the

three-dimensional Euler equations of ideal incompressible

flows.**Spatial analyticity for the Navier-Stokes equations**

Animikh Biswas (University of North Carolina)

Joint work with C. Foias.

We augment the (3d) Navier-Stokes system with an equation

governing the space analyticity

radius of the solutions and identify a suitable domain where

this equation is well-posed. This

allows us to study the maximal analyticity radius of the

(regular) solutions. We also obtain certain estimates of the analyticity radius of the

solution on the entire regularity interval.**Alternate powers in radial dependencies for Serrin's swirling vortex solutions**

Mikhail Shvartsman (University of St. Thomas)

Joint work with Doug Dokken and Kurt Scholz.

In this work we consider a modification of the model developed by J. Serrin where velocity, in spherical coordinates, decreases in proportion to the reciprocal of the distance from the vortex line.

Serrin’s model has three distinct solutions, depending on the kinematic viscosity and the value of a “pressure” parameter. These are: down-draft core with radial outflow, downdraft core with a

compensating radial inflow, and updraft core with radial inflow (single-cell vortex). Recent studies, based on radar data of selected severe weather events show that the ratio of natural log of vorticity

and natural log of grid spacing have a linear relationship, suggesting a fractal-like phenomenon with a constant ratio. In two of the cases the ratio is close to 0.6. We have attempted Serrin’s approach seeking solution with the assumption that the velocity decreases in proportion to the reciprocal of the distance to the power 0.6. This ansatz leads to a substantially more complicated boundary-value problem for sixth order system of nonlinear differential equations. However, the spherical variable has not been eliminated from the right-hand side of the system That contradicts the original

assumption that the velocity would be represented by ratio of a function of the angle with the vortex line to the corresponding reciprocal of the distance to the power not equal to one. We discuss some

specific cases of the system and applications of a shooting method. Also dependence on the radial variable is studied as small variations in the solutions would indicate that our system

is essentially independent of the parameter.**Steady flow of the second grade fluid past an obstacle**

Ondřej Kreml (Charles University in Prague)

This is a

joint work

with Pawel Konieczny. We study the steady flow of a second

grade fluid

past an obstacle in 3D. Arising equations can be split into a

transport

equation and an Oseen equation for the velocity. We use the

theory of

fundamental solutions to the Oseen equation in weighted

Lebesgue spaces

together with results of Herbert Koch to prove the existence of

a wake

region behind the obstacle, i.e. a region where the solution

decays

slower to the prescribed velocity at infinity.**Classical solutions of 2D rotating shallow water equations**

Chunjing Xie (University of Michigan)

We will discuss the classical solutions of two dimensional inviscid rotating shallow water equations with small initial data. The global existence and asymptotic behavior are obtained when the initial data has zero relative vorticity, where rotating shallow water system can be transformed to a symmetric quasilinear Klein-Gordon system. We also give the lower bound for the lifespan of classical solutions with general initial data. This is a joint work with Bin Cheng.**Effects of variable viscosity and viscous dissipation on the**

disappearance of criticality of a reactive third-grade fluid in a slab

Samuel Okoya (Obafemi Awolowo University)

We study the disappearance of criticality of a reactive fully

developed flow of an

incompressible, thermodynamically compatible fluid of grade

three with

viscous heating and heat generation between two horizontal flat

plates, where the

top is moving with uniform speed and the bottom plate is fixed

in the presence of

imposed pressure gradient. This is a natural continuation of

earlier work on

rectilinear shear flows. The governing coupled ordinary

differential equations

are transformed into dimensionless forms using an appropriate

transformation

and then solved numerically for thermal transition

(disappearance of criticality)

using Maple based shooting method. Attention is focused upon

the disappearance of

criticality of the solution set for various values of the

physical parameters and

the numerical computations are presented graphically

to show salient features of the solution set.**Spectral scaling of the two-dimensional Navier-Stokes-α and Leray-α model of turbulence**

Evelyn Lunasin (University of Arizona)

The Navier-Stokes-α model of turbulence is a

mollification of the Navier-Stokes equations in which the

vorticity is advected and stretched by a smoothed velocity

field. The smoothing is performed by filtering the velocity

field over spatial scales of size smaller than -α. The

statistical properties of the smoothed velocity field are

expected to match those of Navier-Stokes turbulence for scales

larger than α.

For wavenumbers k such that kα»1,

corresponding to spatial scales smaller than α, there

are three candidate power laws for the energy spectrum,

corresponding to three possible characteristic time scales in

the model equations. The three

possibilities depend on whether the

time scale of an eddy of size k^{-1}is determined by

(ku_{k})^{-1}, (kv_{k})^{-1}, or (k√style=text-decoration:overline>

(u_{k},

v_{k}) )^{-1}, where u_{_k}and

v_{k}are the

components of the filtered velocity field u and unfiltered

velocity field v, respectively, for wavenumber k.

Determining the actual

scaling requires resolved numerical simulations.

We measure the scaling of the energy spectra from

high-resolution simulations of the two-dimensional

Navier-Stokes-α model, in the limit as

α→∞. The energy spectrum of

the smoothed

velocity field scales as k^{-7}in the direct enstrophy

cascade regime, consistent with the dynamics dominated by the

time scale given by (k v_{k})^{-1}. We are

thus able to

deduce that the dynamics of the dominant cascading

conserved quantity, namely the enstrophy of the rough velocity

v, determines the power law for small scales.

For the two-dimensional Leray-α model, the time scale

given by (k√ (u_{k},

v_{k}) )^{-1}is understood to

characterize the dynamics of the conserved enstrophy. Indeed,

our numerical simulation of this model gives a k^{-5}

power

law in the enstrophy inertial subrange. This result supports

our claim regarding the characteristic time scale of the

two-dimensional NS-α model for wavenumbers kα»1.**Supercavitating flow in multiply connected domains**

Yuri Antipov (Louisiana State University)

Mathematical models of cavitation in fluids and quick and accurate numerical predictions are

essential at different stages in the design process and the evaluation of performance and cavitation

patterns. A question of particular interest is how to extend the traditional hodograph method used

for simply connected flows to more complicated cavitation models. These models are nonlinear and

include flows in multiply connected Riemann surfaces. We present two approaches for the

construction of a conformal map from a canonical parametric domain into an n-connected Riemann

surface of supercavitating flow. The first approach works when n is not greater than 3, and the

canonical domain is the exterior of n slits along the real axis. The map is given by quadratures and

expressed through the solutions of two Riemann-Hilbert problems on a symmetric hyperelliptic

Riemann surface. The second method is applicable for any n, and the parametric domain is the

exterior of n circles. To reconstruct the map, it is required to solve two Riemann-Hilbert problems

for piece-wise meromorphic, G-automorphic functions (G is a Schottky group). This method leads to

a series-form solution and does not need the solution to a Jacobi inversion problem (for the first

method, it cannot be bypassed).**Three dimensional stability of the Burgers vortex**

Yasunori Maekawa (Kobe University)

This is a joint work with Thierry Gallay. Burgers vortices are explicit

stationary solutions of the Navier-Stokes equations which are often used

to describe the vortex tubes observed in numerical simulations of

three-dimensional turbulence. In this model, the velocity field is a

two-dimensional perturbation of a linear straining flow with axial

symmetry. The only free parameter is the Reynolds number Re =

Γ/ν, where Γ is the total circulation of the vortex and

ν is the kinematic viscosity. We will show that the Burgers vortex is

asymptotically stable with respect to general three-dimensional

perturbations, for all values of the Reynolds number.**Numerical study of sectional curvature and Jacobi field**

in incompressible Euler equations

Koji Okitani (University of Sheffield)

We study some of the key quantities arising in the Arnold's theory (1966)

of the incompressible Euler equations both in two and three dimensions.

The sectional curvatures for the Taylor-Green vortex and ABC flows

initial conditions are calculated exactly in three dimensions.

We trace the time evolution of the Jacobi fields by direct numerical

simulations and, in particular, see how the sectional curvatures get more and

more negative in time. The spatial structure of the the Jacobi fields is

compared with the vorticity fields by visualizations.

The Jacobi fields are found to grow exponentially in time for the flows with

negative sectional curvatures.

In two dimensions, a family of initial data proposed by Arnold (1966)

is considered. The sectional curvature is observed to change its sign quickly

even if it starts from a positive value. The Jacobi field is shown to be

correlated with the passive scalar gradient in spatial structure.

On the basis of Rouchon's expression (1984) for the sectional curvature

(in physical space), the origin of negative curvature is investigated.

It is found that a 'potential' $\alpha_{\bm{\xi}}$ appearing in the

definition of covariant time derivative plays an important role,

in that a rapid growth in its gradient makes a major contribution to

the negative curvature.**Dissipative structures in the inviscid limit of 2D wall-bounded**

turbulence

Romain Nguyen van yen (École Normale Supérieure)

The behavior of solutions to the Navier-Stokes equations with no-slip

boundary conditions when the viscosity goes to zero

has been a long standing mathematical problem since its formulation by

Prandtl.

The main difficulty lies in the possible production of extreme velocity

gradients near boundaries.

We have undertaken a series of numerical experiments using a Fourier

mode expansion of the solution along with a volume penalization method

to impose the no-slip condition.

The results support a scenario in which the energy dissipation rate

remains strictly positive in the inviscid limit, due to a boundary layer

of tickness orders of magnitude smaller than the classical Re^{-1/2}

estimate.

When the initial condition is a Gaussian noise, a wavelet analysis of

the vorticity field after some time suggests that it has organized into

dissipative structures, that are recycled by detachment from the

boundary but visit the whole domain.

Some implications for modeling of boundary layer detachment phenomena

are briefly discussed.**Energy dissipation in the inviscid limit of a 2D dipole-wall**

collision

Romain Nguyen van yen (École Normale Supérieure)

The behavior of solutions to the Navier-Stokes equations with no-slip

boundary conditions when the viscosity goes to zero

has been a long standing mathematical problem since its formulation by

Prandtl.

The main difficulty lies in the possible production of extreme velocity

gradients near boundaries.

We have undertaken a series of numerical experiments using a Fourier

mode expansion of the solution along with a volume penalization method

to impose the no-slip condition.

The results support a scenario in which the energy dissipation rate

remains strictly positive in the inviscid limit, due to a boundary layer

of tickness orders of magnitude smaller than the classical Re^{-1/2}

estimate.

When the initial condition is a Gaussian noise, a wavelet analysis of

the vorticity field after some time suggests that it has organized into

dissipative structures, that are recycled by detachment from the

boundary but visit the whole domain.

Some implications for modeling of boundary layer detachment phenomena

are briefly discussed.**The spectrally-hyperviscous Navier-Stokes equations**

Joel Avrin (University of North Carolina)

We regularize the 3-D Navier-Stokes equations with hyperviscosity of

degree alpha, but applied only to the high wavenumbers past a cutoff m;

for now we are on a periodic box. Attractor estimates stay within the

Landau-Lifschitz degrees-of-freedom estimates even for very large m. An

inertial manifold exists for m large enough whenever alpha is at or

above 3/2. Galerkin-convergence and inviscid-limit results are optimized

for the high wavenumbers; the latter case is defined to mean that nu

goes to zero while the spectral hyperviscous term stays fixed.

Computational studies over many runs produce parameter choices that

facilitate close-to-parallel agreement (over a good-sized portion of the

inertial range) with the Kolmogorov energy-spectrum power law for high

(up to 10^{7}) Reynolds numbers.**Crossover in coarsenig rates in demixing binary viscous fluids**

Christian Seis (Rheinische Friedrich-Wilhelms-Universität Bonn)

We consider the demixing process of a binary mixture of two

liquids after a temperature quench. In viscous liquids, the

demixing is mediated by diffusion and convection. The typical

particle size src=/springer/greek/ell.gif> grows as a function of time t, a

phenomenon called coarsening. Simple scaling arguments based on

the assumption of statistical self-similarity of the domain

morphology suggest the coarsening rate: from src=/springer/greek/ell.gif>∼

t^{1/3}

for diffusion-mediated to src=/springer/greek/ell.gif>∼ t for flow-mediated.

In joint works with Yann Brenier, Felix Otto, and Dejan

Slepcev, we derive the crossover of both scaling regimes in

form of time-averaged upper bounds. The mathematical model is a

Cahn-Hilliard equation with convection term, where the fluid

velocity is determined by a Stokes equation. The analysis

follows closely a method proposed by Kohn and Otto, which is

based on the gradient flow structure of the evolution.**On fluid-rigid body interaction problem**

Zhouping Xin (Chinese University of Hong Kong)

We study the motion of a rigid body immersed in an incompressible ideal (or viscous) fluids. We first establish an existence of global (in time) existence of weak solution with natural far field condition for 2-dimensional Euler system. Then for viscous case, we prove that the corresponding generalized Stokes operator is the infinitesimal generator of an analytic semigroup on some appropriate spaces so that we can obtain local existence of strong solutions in such spaces.**High resolution simulations of the incompressible 3-D Euler equations: A lagrangian perspective**

Tobias Grafke (Ruhr-Universität Bochum)

Reacting to various new forms of lagrangian and local geometric

criteria for finite time blowup of the three-dimensional

incompressible Euler equations (J. Deng, T. Hou and X. Yu

(2005), P. Constantin (2001)) numerical simulations are carried

out. High resolution is achieved by utilizing an adaptive mesh

refinement technique. Lagrangian tracer particles are injected

into the flow to analyze the evolution of both vortex lines and

the inverse flowmap. Preliminary results regarding relevant

blowup criteria are presented.**On vorticity directions near singularities for**

the 3D Navier-Stokes flows

Hideyuki Miura (Osaka University)

This is a joint work with Yoshikazu Giga.

We give a geometric nonblow up criterion on the direction of the vorticity

for the 3D Navier-Stokes flow. We prove that under a restriction on

behavior in time (type I condition) the solution does not blow up if the

vorticity direction is uniformly continuous in the region where vorticity

is large.**A phase-field model and its numerical approximation**

for two-phase incompressible flows with different densities and viscosities

Xiaofeng Yang (University of South Carolina)

Modeling and numerical approximation of two-phase incompressible flows

with different densities and viscosities are considered using the diffusive

phase-field model. A physically consistent phase-field model that admits

an energy law is proposed, and several energy stable, efficient and accurate

time discretization schemes for the coupled nonlinear phase-field model

are constructed and analyzed. Ample numerical experiments are carried

out to validate the correctness of these schemes and their accuracy for

problems with large density and viscosity ratios.