Campuses:

<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 v0 satisfies
    curlv0/r is a
    finite Randon measure with compact support. Furthermore, the
    global existence and uniqueness, is also
    established in this case provided that curlv0/r belongs to
    Lpc(R3) 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
    (kuk)-1, (kvk)-1, or (k√style=text-decoration:overline>
    (uk,
    vk) )-1, where u_k and
    vk 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 vk)-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√ (uk,
    vk)
    )-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 107) 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>∼
    t1/3
    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.