Session ZC26: Flow Instability: General

Identifying Instabilities in Complex Fluid Dynamical Systems with Adaptive-Ensemble Bred Vectors

A new concept of bred vectors (BVs) has been developed to computationally study the spatial patterns and time evolution of the nonlinear instabilities in turbulent fluid dynamical systems by finding the error signatures and establishing their connections with the predictability characteristics of instabilities such as convective, baroclinic, thermal-acoustic modes.  A turbulent fluid dynamical system with high degrees of unsteadiness has a finite limit of predictability and a strong sensitive dependence on initial conditions. Initially, in small amplitude, dynamically significant instabilities will inevitably bring about fundamental changes in global flow behaviors in a finite time. Consequently, the inherent instabilities will deteriorate the prediction and potentially result in valueless information. This study will provide new insight on the identification of spatial and temporal signatures of instabilities and potenially assist in the development of flow control methods.     

Was Squire wrong - New 3D Oblique Spatially Evolving Instability Modes

In 1933 Squire showed that for time-evolving perturbations, 2D instabilities occur at the smallest Reynolds number – the critical Reynolds number. We prove that this is not necessarily so for spatially evolving 3D modes, for which we introduce both a complex streamwise and spanwise wavenumber giving rise to oblique 3D modes. Such modes admit in the $x-z$-plane neutral stability lines (NSL), which are oblique to the main flow direction, und perpendicular to it shows maximum growth. As such extending Squires idea by invoking symmetry methods in parameter space the key result is that oblique 3D instabilities at a Reynolds number below the critical 2D Reynolds number exist. Other than for temporally evolving modes, however, for spatially evolving modes the additional condition of group velocity (GV) v_g must be taken into account, which states that the GV has to propagate in the direction of the spatially increasing mode. For 3D instabilities the vector v_g of the GV must thus point into the unstable region, i.e. cross the above mentioned NSL in the $x-z$-plane. Details on the extension of Squire's theory based on Lie symmetries will be present and exemplified using plane Couette flow, which is known to not admit a time-evolving instability.   

Percolation and Creeping Flows

Dispersion of solid particles carried by a creeping flow is the outcome of the competition between stresses/pressure imparted by the fluid and the cohesion stresses/pressure of the solid particles.  We consider a two-dimensional creeping flow inside a rectangular cavity.  Close to the corner adjacent to a moving wall and a stationary wall, the pressure is negative and it diverges as the corner is approached. This negative pressure extends the solid particle and is responsible for its rupture.  Using the mean-field percolation model, we determine the distribution of fragments and their average size. We determine the percolation threshold line. Inside the non-percolating region, we determine a line beyond which the particles are completely dispersed.   

Particle Lag Effects in Shock-Driven Multiphase Instability with Solid Particles

The Shock-Driven Multiphase Instability (SDMI) occurs when a multiphase (particle-gas) medium is

instantaneously accelerated by the passage of a shock-wave. At the macro-scale, shock-driven multiphase mixing

(cloud-mixing) occurs when vorticity is deposited at an interface between two multiphase mixtures of differing

properties. In the limit of small, fast-reacting particles the interface evolves like the classic Richtmyer-Meshkov

Instability; however, as particle velocity equilibration times grow the vorticity deposition is reduced. Due to this, a

deeper understanding of the particle response time effects to better predict the level of mixing generated.

Experiments are performed in a shock tube facility to investigate shock-driven multiphase mixing for different

particle velocity equilibration times. A multiphase interface consisting of solid particles and carrier gas is

impulsively accelerated by a shock wave. The particle size distribution and concentration are characterized ex-situ

and in-situ utilizing Phase Doppler Anemometry and particle mass retention. The particle response time was studied

while maintaining the same effective Atwood by controlling the concentration, particle size, and material properties.

The multiphase interface development was captured through an ensemble of Mie-scattering images. These

measurements allow for the characterization of the vorticity deposition. Results are compared with theoretical for

the RMI and a SDMI vorticity deposition model is proposed.   

Using Automatic Differentiation to Search for Minimal Seeds in Channel Flow

A minimal seed is defined as the smallest amplitude perturbation that can cause a transition from a linearly-stable state to another, a simple concept but extremely difficult to find in high-dimensional dynamical systems such as sheared fluid flows. In this work we present a method for finding said minimal seeds efficiently in the setting of pressure-driven channel flow but our methodology is applicable to other settings. We use automatic differentiation to calculate partial derivatives of the energy growth at some target time with respect to initial perturbations to the laminar state permitting us to efficiently optimise over these to find that which maximises the energy growth. We then examine how the initial and final states and energy growth change with perturbation amplitude. In particular we are looking for a significant step in the energy growth accompanied by a qualitatively different final state indicating transition. We will start by discussing the simpler 2D problem and then move up to the 3D setting.   

Observations of Instability in Spatially Periodic Channels with Different Aspect Ratios

Laboratory experiments are conducted for channel flows with one wall textured by spanwise grooves placed periodically in the streamwise direction. Prior numerical simulations indicate that the pure flow becomes unstable at a critical Reynolds number of order 100; the resulting secondary flow is predicted to exhibit stable waves reminiscent of Tollmien-Schlichting modes. Here, we examine instability onset and the resulting secondary flow as a function of the channel aspect ratio. Characterizing textured channels in low aspect ratio regimes could suggest new ways to improve mixing and transport at low Reynolds numbers with small additional costs in system pumping power.   

Mesh-free hydrodynamic stability

We develop a high-order mesh-free hydrodynamic stability analysis tool for complex geometries using radial basis

function-based finite differences (RBF-FD). Polyharmonic spline RBFs with polynomial augmentations (PHS+poly)

are used to construct the discrete linearized Navier-Stokes and resolvent operators on scattered nodes. This scheme

enables accurate, stable, and computationally efficient discretizations of the large matrix problems arising in two-

dimensional hydrodynamic stability analysis. The study addresses the trade-off between computational efficiency

and accuracy and provides best practices. Furthermore, the practical treatment of boundary conditions, including

the pole singularity in cylindrical coordinates, is examined and discussed. The numerical framework is validated

across various hydrodynamic stability theoretical methods and flows. This includes conducting linear stability (LST),

resolvent (RA), and wavemaker (WM) analyses for the canonical cylinder flow at Reynolds numbers ranging from

47 to 180. Additionally, RA and WM analyses are performed for a laminar zero-pressure-gradient (ZPG) flat-plate

Blasius boundary layer at a Reynolds number of 0 ≤ Re ≤ 6 × 10⁵, as well as the turbulent mean transonic jet at

Mach number 0.9 and a Reynolds number of approximately 10⁶. The comparisons of these benchmark problems with

the literature demonstrate the broad applicability, accuracy, and robustness of the mesh-free framework. Lastly, the

pioneering application of RA-based WM analysis on the Blasius boundary layer and turbulent jet offers new insights

into modal and non-modal growth in these flows.   

Instability islands in the radially heated Taylor-Couette flow

A Taylor–Couette setup with radial heating is considered where a Boussinesq fluid is sheared in the annular region between two concentric, independently rotating cylinders maintained at different temperatures. Linear stability analysis is performed to determine the Taylor number for the onset of instability. Two radius ratios corresponding to wide and thin gaps with several rotation rate ratios are considered. An important finding of the current study is the discovery of unstable modes in the Rayleigh-stable regime. Furthermore, instability islands are observed for both wide and thin gaps which can separate from or merge into open neutral--stability curves. Alternatively, instability islands can also morph into open neutral stability curves as the rotation rate ratio is changed. Instability islands are observed to be sensitive to changes in control parameters and their appearance/disappearance is shown to induce discontinuous jumps in the critical Taylor number.   

Electro-hydrodynamic Stability Analysis of Bilayer Electrified Jet

A one-dimensional electrohydrodynamic model is developed for the electrified two-fluid jet's temporal instability analysis, considering electrospinning conditions where a high electric field is applied to fabricate nanofibers. The Leaky dielectric model is incorporated into the nonlinear governing equations for calculating electrical Maxwell stresses at jet interfaces in the current work. The stability of the compound jet is examined for the periodic axisymmetric disturbance to address the capillary breakup of the electrified jet. Complex dynamics of the fluid deformation is thoroughly analyzed for two distinct instability modes as illustrated by the external axial electric field according to its strength. A surface tension driven capillary mode with a short wavelength leads the critical instability for the weak electric field. For the high electric field, a conducting mode with a long wavelength is more dominant. Competition between the two modes is explained in terms of the flow rate, material properties and process parameters. In addition, the location of the interfacial boundary layer between two immiscible fluids is also the most influential factor regarding jet deformation. Study of the coelectrospinning jet stability behavior enables one to modify either the material properties or the processing conditions during co-electrospinning such that the axisymmetric instability is suppressed, leading to smooth nanofibers formation for various common to high-end applications.     

Temporal Growth Analysis for Prandtl Slope Flows: Modal or Non-modal?

In 1942, Ludwig Prandtl introduced a simple 1D model to study stably stratified flows over sloped terrain such as nocturnal mountain or valley winds. It assumes a constant background stratification as well as uniform surface cooling on an infinite slope and has served as a canonical problem to enhance our understanding of stably stratified flows over non-flat surfaces. Prior investigations into the stability of Prandtl slope flows have indicated that the results from linear modal analysis were able to accurately capture the growth dynamics of small initial perturbations to the 1D laminar Prandtl slope profiles. This suggests that, in contrast to shear-driven flows, the linearized Navier-Stokes operator derived from Prandtl's base flow profile exhibits a large degree of normality, thus inhibiting strong transient growth rates that can dominate the modal growth. We present results from non-modal analysis over a large range of slope angles for both anabatic and katabatic slope flows to gather quantitative evidence for this assertion, which will help explain which dynamics are most significant during the transition to instability and turbulence in such flows. In a broader context, our results also help understand whether, despite its simplicity, linear modal analysis could be an adequate tool to comprehend the instability dynamics of Prandtl slope flows and related flow problems under the right circumstances.