Special Session 124: Recent Advances in Hydrodynamic Stability Analysis

A nonlinear Schr\{o}dinger equation for capillary waves on arbitrary depth with constant vorticity

Malek ABID Aix-Marseille Universit\`e France Co-Author(s):    Christian Kharif, Yang-Yih Chen and Hung-Chu Hsu

Abstract: A nonlinear Schr\{o}dinger equation for pure capillary waves propagating at the free surface of a vertically sheared current has been \red{used} to study the stability and bifurcation of capillary Stokes waves on arbitrary depth. \newline A linear stability analysis of weakly nonlinear capillary Stokes waves on arbitrary depth has shown that (i) the growth rate of modulational instability increases as the vorticity decreases whatever the dispersive parameter $kh$ where $k$ is the carrier wavenumber and $h$ the depth (ii) the growth rate is significantly amplified for shallow water depths and (iii) the instability bandwidth widens as the vorticity decreases. A particular attention has been paid to damping due to viscosity and forcing effects on modulational instability. In addition, a linear stability analysis to transverse perturbations in deep water has been carried out, demonstrating that the dominant modulational instability is two-dimensional whatever the vorticity. \newline Near the minimum of linear phase velocity in deep water, we have shown that generalized capillary solitary waves bifurcate from linear capillary Stokes waves when the vorticity is positive. \newline Moreover, we have shown that the envelope of pure capillary waves in deep water is unstable to transverse perturbations. Consequently, deep water generalized capillary solitary waves are expected to be unstable to transverse perturbations.

Symmetry Breaking in Chemical Systems: Engineering Complexity through Self-Organization and Marangoni Flows

Azam Gholami New York University, Abu Dhabi, UAE United Arab Emirates Co-Author(s):    Sangram Gore, Binaya Paudyal, Mohammed Ali, Nader Masmoudi, Albert Bae, Oliver Steinbock, Azam Gholami

Abstract: Far from equilibrium, chemical and biological systems can form complex patterns and waves through reaction-diffusion coupling. Fluid motion often tends to disrupt these self-organized concentration patterns. In this study, we investigate the influence of Marangoni-driven flows in a thin layer of fluid ascending the outer surfaces of hydrophilic obstacles on the spatio-temporal dynamics of chemical waves in the modified Belousov-Zhabotinsky reaction. Our observations reveal that circular waves originate nearly simultaneously at the obstacles and propagate outward. In a covered setup, where evaporation is minimal, the wavefronts maintain their circular shape. However, in an uncovered setup with significant evaporative cooling, the interplay between surface tension-driven Marangoni flows and gravity destabilizes the wavefronts, creating distinctive flower-like patterns around the obstacles. Our experiments further show that the number of petals formed increases linearly with the obstacle`s diameter, though a minimum diameter is required for these instabilities to appear. These findings demonstrate the potential to `engineer` specific wave patterns, offering a method to control and direct reaction dynamics. This capability is especially important for developing microfluidic devices requiring precise control over chemical wave propagation.

Ergodicity of randomly forced PDEs via controllability

Vahagn Nersesyan NYU Shanghai Peoples Rep of China Co-Author(s):

Abstract: The problem of ergodicity of randomly forced dissipative PDEs has attracted a lot of attention in the last twenty years. It is well understood that if all or sufficiently many Fourier modes of the PDE are directly perturbed by the noise, then the problem has a unique stationary measure which is exponentially stable in an appropriate metric. The case when the random perturbation acts directly only on few Fourier modes is much less understood and is the main subject of this talk. We will explain how the controllability properties of the underlying deterministic system can be used to study the ergodic behavior of the stochastic dynamics. The results will be illustrated through the examples of 2D Navier-Stokes and Ginzburg-Landau equations; however, the methods apply to a wide variety of systems as soon as they satisfy appropriate controllability conditions. This talk is partially based on joint works with Sergei Kuksin (Universite Paris Cite) and Armen Shirikyan (CY Cergy Paris Universite).

Staircase formation in fingering convection: a peculiar case of instability in the mean fields.

Francesco Paparella New York University Abu Dhabi United Arab Emirates Co-Author(s):    Francesco Paparella (New York University Abu Dhabi)

Abstract: Fingering convection is a special case of buoyancy--driven convection that occurs when the most diffusing of two buoyancy--changing scalars is stratified in such a way as to stabilize the fluid, and the least diffusing scalar in the opposite direction, while the overall density decreases upward. In those cases, a well--understood linear instability of the motionless state produces convective motions dominated by small coherent structures, known as `salt fingers`, that travel vertically in the domains carrying most of the fluxes. Once the convection is established, if the two Rayleigh numbers fall in a certain portion of the parameter space, a further instability occurs, which turns the horizontally averaged density field from a profile characterized by a constant gradient into a staircase--like profile that alternates regions of high and low density gradients. This talk explores the possible explanations of the staircase--forming instability, examining in particular some simplified models that undergo a similar phenomenology. If time allows, the analogy between these models and some well-known techniques for non--linear image denoising will be illustrated.

Stability Analysis of Two-Dimensional Laminar Elliptic Cylinder Wakes Using Reduced-Order Galerkin Models

Immanuvel Paul Khalifa University of Science and Technology United Arab Emirates Co-Author(s):    Immanuvel Paul

Abstract: The stability of two-dimensional laminar wakes behind elliptic cylinders is studied using a reduced-order Galerkin model. This approach employs proper orthogonal decomposition (POD) to extract the dominant coherent structures from flow simulations. It constructs a reduced dynamical system by projecting the governing Navier-Stokes equations onto the subspace spanned by these modes. The resulting low-dimensional Galerkin system effectively captures the essential dynamics of the wake flow while significantly reducing the computational cost. A linear stability analysis of the reduced-order system is performed to examine the onset of wake instability and identify the critical Reynolds number where the transition to unsteady flow occurs. The study further explores the influence of cylinder aspect ratio on wake behavior and the development of wake vortex shedding patterns. The reduced model demonstrates its capability to predict wake instabilities accurately and offers insights into the flow`s sensitivity to perturbations. These findings contribute to a better understanding of laminar wake dynamics and provide a foundation for designing flow control strategies to mitigate instability in practical applications.

Towards a Variational Theory of Hydrodynamic Stability

Haithem Taha University of California, Irvine USA Co-Author(s):    Ahmed Roman

Abstract: The projection of Navier-Stokes on the space of divergence-free vector fields is associated with a minimization problem. We recently formulated such a minimization problem in what we call The Principle of Minimum Pressure Gradient (PMPG). The principle asserts that an incompressible flow evolves from one instant to another in order to minimize the L2 norm of the pressure gradient. We proved that Navier-Stokes equation is the necessary condition for minimizing the pressure gradient cost subject to the divergence-free condition on the local acceleration. In the discretized domain, this problem is a convex quadratic programming problem, which has a closed-form solution. The resulting necessary condition is a quadratic ODE in velocity, directly without the need to solve for pressure, avoiding the need to solve the Poisson equation in pressure at every instant of time. This approach is expected to provide significant savings in computations. Moreover, it should facilitate the mathematical analysis of incompressible flows by exploiting tools from nonlinear systems theory to analyze the resulting quadratic ODE. Based on this view, we provide a conjecture for a necessary condition for nonlinear hydrodynamic stability: If an equilibrium solution is stable, it must minimize the L2 norm of the convective (and viscous) acceleration among all equilibrium, divergence-free solutions. This conjecture applies successfully to the ideal flow over an airfoil.

Well-posedness of Free Boundary Inviscid Flow-Structure Interaction models

Amjad Tuffaha American University of Sharjah United Arab Emirates Co-Author(s):    Igor Kuykavica. Sarka Necasova

Abstract: We obtain the local existence and uniqueness of solutions for a system describing interaction of an incompressible inviscid fluid, modeled by the Euler equations, and an elastic plate, represented by the fourth-order hyperbolic PDE. We provide a~priori estimates for the existence with the optimal regularity $H^{r}$, for $r>2.5$, on the fluid initial data and construct a unique solution of the system for initial data $u_0\in H^{r}$ for $r\geq3$. We also address the compressible Euler equations in a domain with a free elastic boundary, evolving according to a weakly damped fourth order hyperbolic equation forced by the fluid pressure. We establish a~priori estimates on local-in-time solutions in low regularity Sobolev spaces, namely with velocity and density initial data %$v_{0}, R_{0}$ in~$H^{3}$. This is joint work with Igor Kukavica and Sarka Necasova.