JSPS US-CANADA Seminar

A multi-dimensional extension of the Constantin-Lax-Majda model

Koji Ohkitani (Research Institute of Mathematical Sciences, Kyoto University, Japan)

We introduce a generalisation of the Constantin-Lax-Majda model to higher spatial dimensions, using higher-dimensional Hilbert transforms. Our model is explicitly solvable, and we provide examples of singularity formation with anisotropic scaling features. We also introduce suitable extensions incorporating advection effects. For those, we provide numerical simulations indicating that the transport terms can delay or suppress singularity formation.

This work is joint with Federico Pasqualott (Department of Mathematics, University of California, San Diego).

Self-similar solutions to the 3D Navier-Stokes equations

Koji Ohkitani (Research Institute of Mathematical Sciences, Kyoto University, Japan)

We present the forward self-similar profile for the 3D Navier-Stokes equations, representing the late stage of decaying Navier-Stokes flows. The existence of such a profile has been known, but its precise functional form has not been determined numerically, let alone mathematically. Here we determine the profile for the first time using numerical methods. This has been achieved by a combination of two things; a numerical method of solving the Navier-Stokes equations in the whole space and the explicit form of the linearised solution. Taking the initial data from the linearised solution, we solve the fully-nonlinear Navier-Stokes equations to observe its convergence to a steady solution in the dynamically scaled space. We have confirmed that the nonlinear correction is small, consistent with the previous perturbative analysis.

Developments in Dispersive Hydrodynamics

Mark Hoefer (University of Colorado Bouder)

Dispersive hydrodynamics encompass the collective, multiscale dynamics of interacting waves in dispersive nonlinear systems. Dispersive shock waves or DSWs are a canonical class of solutions. Scale separation leads to the mathematical framework of Whitham modulation theory, a long wavelength description of nonlinear dispersive wavetrains. From modulations of multiphase or multidimensional nonlinear wavetrains to stochastic DSWs, this talk will present a variety of developments and results in dispersive hydrodynamics utilizing modulation theory, numerical simulations, and experiment.

Neuroscience meets ocean engineering: How sound perception modeling can improve the performance of wave energy systems

Henrik Kalisch (University of Bergen, Norway)

In this work, we use an Echo State Network with a reservoir inspired by our current understanding of the auditory cortex in mammals in order to forecast time series of ocean waves. The prediction yields a phase-resolved estimate of the free-surface elevation at a fixed location which is essential in the operation of wave energy devices in order to maximize energy uptake. The network used here can predict these time series with fair accuracy up to a forecasting horizon of about two wave periods. The network is comparatively small and adapts naturally to changing conditions through continuous retraining during normal operation. This quick adjustment is achieved by updating only the output weights connecting the reservoir to the output neuron.

The L2 theory for compressible Euler equations and inviscid limit from Navier-Stokes equations

Geng Chen (University of Kansas, USA)

Compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general. It is well known that global BV solutions of system of hyperbolic conservation laws exist, when one considers small BV initial data.

In this talk, I will first discuss our recent L2 stability theory for compressible Euler equations using the method of relative entropy, with several collaborators. This theory is the second major stability theory for hyperbolic conservation laws, after the L1 theory in 1990s. As an application, we proved all BV solutions must satisfy the Bounded Variation Condition proposed earlier by Bressan and Lewicka as a sufficient condition for uniqueness, hence showed the uniqueness of BV solution. This is a joint work with Faile, Krupa and Vasseur.

Then I will introduce the recent result on the vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations. This is a famous open problem after Bressan-Bianchini’s seminal vanishing artificial viscosity limit result for BV solutions of hyperbolic conservation laws. This is a join work with Kang and Vasseur at UT Austin.

Recent developments on singular stochastic PDEs: uniqueness, non-uniqueness, and global theory

Kazuo Yamazaki (University of Nebraska, Lincoln, USA)

Partial differential equations forced by random noise are called stochastic partial differential equations (SPDEs). When the choice of random noise has rough spatial regularity, it can force the solution to become rough and consequently the product within the nonlinear term to become ill-defined according to Bony's estimates; such SPDE is called a singular SPDE. While this research area has seen significant developments due to the breakthrough works of Hairer (regularity structures) and Gubinelli, Imkeller, and Perkowski (paracontrolled distributions), many problems remain open, especially concerning global-in-time solution theory (unique or non-unique) and even local-in-time solution theory (unique or non-unique) in high dimensions. We discuss some recent developments in this direction.

When Roughness Matters – Engineered Microstructures for Cruise Efficiency and Drag Reduction

Isaac Choutapalli (Department of Mechanical Engineering, UTRGV)

Surface roughness is often seen as a performance penalty, but when engineered at the right scale and geometry, it can become a powerful passive flow-control tool. We explored two microstructured strategies aimed at AFRL-relevant missions: enhancing cruise efficiency of lifting surfaces and reducing skin-friction drag on non-lifting surfaces.

For lifting surfaces, we designed owl-inspired airfoils combining leading-edge tubercles, chordwise micro-barbules (100 µm high, 0.5 mm spacing), and a trailing-edge Gurney flap. Across Reynolds numbers from 69,000 to 138,000, these modifications consistently improved lift-to-drag ratio in the 2–5 degrees cruise AoA band, increased peak lift by up to ~25%, and preserved trim at cruise. At higher Re, they formed a distinct aerodynamic family with stronger lift carry-on toward stall, providing both efficiency and maneuver margin.

For non-lifting surfaces, we tested triangular porous texturing (TPT) on a zero-pressure-gradient flat plate using PIV. Our previous studies at lower Re showed solid micro-prisms achieved ~19% skin-friction reduction by slip-cutting high-speed streaks, while porous TPT sacrificed a small amount of drag reduction (~15%) to deliver +120% momentum flux via passive bleed - offering stall resistance capability. Experiments are underway to investigate these outcomes at higher Re. CFD simulations of the flat-plate flows are also underway to study how these geometries redistribute wall-normal momentum and shear, to enable informed tuning of microstructure parameters to target either drag minimization or flow separation control.

Together, the results show that “roughness,” when designed with purpose, can serve as an enabler of efficiency and control across a range of aerodynamic applications.

Multiple Phase Fluid Flow and Transport in Porous Media: Environmental Applications of Advection-Dispersion-Reaction Modeling for Water-Energy Nexus Research

Chu-Lin Cheng (School of Earth, Environmental and Marine Sciences, Department of Civil Engineering, UTRGV)

Modeling multiphase flow and transport in variably saturated porous media with hydraulic properties are crucial for many environmental and energy research (Water-Energy nexus), such as water supply, water treatment, pollutant removal, carbon sequestration and hydrogen storage. Scaling lab measurements from microscale to a mesoscale for numerical modeling has been a research focus for decades. Recent advances in nondestructive testing methods, such as XCT and neutron imaging, combined with inverse modeling and new upscaling techniques have improved our capability to measure hydraulic properties and model multiphase flow and transport. Examples in water and energy resources will be presented.

Computational Hemodynamic Modeling in Pediatric Patients with Congenital Heart Disease

Weuguang Yang (Deparment of Mechanical Engineering, UTRGV)

Congenital heart defects are a leading cause of infant mortality and present complex hemodynamic challenges. Computational modeling has become an essential tool for quantifying cardiovascular mechanics and increasingly complements experimental and clinical approaches. In this talk, I will present a computational framework for cardiovascular hemodynamic modeling through a series of applications, including patient-specific simulation of pulmonary circulation in pulmonary arterial hypertension, treatment planning for congenital heart disease, passive safety evaluation for in a novel impeller pump for single-ventricle physiology, and computational analysis of flow–surface interactions associated with slippery coatings on cardiac pacing leads. The presented studies leverage patient-specific geometries, multiscale 0D–3D coupling, and fluid structure interaction to interrogate clinically relevant hemodynamics.

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