Session H04: Flow Instability: General (5:45pm - 6:30pm CST)

A wavepacket-based optimization method for resolvent analysis

We report on the development of a semi-analytic method to approximate resolvent (pseudospectral) mode shapes and amplification levels. The method assumes that mode shapes can be accurately approximated by a sum of suitably-defined wavepackets. The small number of parameters prescribing the shape of these wavepackets may be found by solving a low-dimensional optimization problem, thus eliminating the need to form and decompose discretized linear operators. We demonstrate the applicability and assess the performance of this method on the Squire operator in laminar Couette and plane Poiseuille flow. In particular, we show that the method can be applied to cases where leading resolvent modes are affected by boundary conditions and/or multiple critical layers. We additionally explore the capabilities of this method to compute suboptimal mode shapes and amplification levels, and discuss prospects for applying this method to more complex systems.   

Viscosity-Stratified Nonlinear Optimal Perturbation and its Evolution in a 3D Channel

We show that the early stages of nonlinear optimal perturbation growth in a viscosity-stratified flow are different from those in a constant-viscosity flow. We use a numerical technique of direct-adjoint looping and nonlinear optimisation to obtain the nonlinear optimal perturbation which maximises the time-intgrated perturbation kinetic energy of the system. The geometry of flow is a 3-dimensional channel. The two walls of the channel are kept at different temperatures and the viscosity varies as an exponential function of temperature. The Orr and modified lift-up mechanisms are in operation at low (linear) and high (nonlinear) perturbation amplitudes, respectively, at our subcritical Reynolds number of 500 and Prandtl number 7. The nonlinear optimal perturbation contains more energy on the hot (less-viscous) side, with a stronger initial lift-up. However, as the flow evolves, the important dynamics shifts to the cold (more-viscous) side, where wide high-speed streaks of low viscosity grow and persist. This strengthens the inflectional quality of the velocity profile. We show why the linear optimal misses this physics by describing the mechanism behind this energy growth. The Prandtl number does not qualitatively change the findings with the current viscosity model.   

Experimental study of capillary jets of viscoelastic liquids

The development of capillary instability on low-velocity jets manifest itself by surface deformations, whose evolution depends on the injection conditions, namely the nozzle geometry, the injection velocity and the liquid properties. When the liquid is viscoelastic, some of its properties depend on the other injection conditions. Also, under some conditions, the jet exhibit structures called ``beads-on-a-string'' that might be subject to coalescence phenomenon. Statistical analyses of experimental data are conducted to study the dynamics of free low-velocity viscoelastic liquid jets. The data set is made up of jet images taken at successive positions downstream from the nozzle exit. The images have been treated to detect the liquid/gas interfaces. The main features of these interfaces are captured by carrying two different Proper Orthogonal Decompositions (POD) within each ROI. The one considers each pixel value of the raster images, and the other is carried on the measured volume-based scale distributions. The variation of the injection conditions brings important results about their impacts on the onset of the capillary instability, on its evolution and also on the coalescence and degradation mechanisms.   

Centrifugal Instability of Coand\u{a} Flows

The first observations of stationary streamwise vortices within a centrifugally unstable laminar wall-jet flowing over a circular (Coand\u{a}) cylinder are presented. Flow visualization was used to analyze streamwise vortices and shear layer development, while PIV was used to determine the vortices' size and strength. Exponential growth of the shear layer was characterized by a linear dependence of the growth-rate parameter exponent on the Goertler number (G). Extrapolation of the growth-rate parameter to zero produced the first-ever experimental estimate of the critical Goertler number, namely 3.1, which compared remarkably well with stability theory (3.5). Increases in G resulted in a second time-dependent wavy instability whose median frequency increased linearly. Transition occurred at G$=$28, and was accompanied by multiple vortex wavenumbers and a large range of frequencies. Furthermore, transition corresponded to a minimum separation angle as a function of G for the following reasons. With increasing G, the relatively thick vortex-driven shear layer was more susceptible to separation and thus the separation angle, relative to the slot, decreased. However, following transition to turbulence, the high-momentum fluid near the wall that resulted from turbulent mixing, produced a subsequent increase in the separation angle.   

Viscous stabilisation of Rayleigh-Plateau modes on a cylindrical filament through radial oscillatory forcing

Faraday waves are standing wave patterns appearing on a liquid surface subject to an oscillatory body force and have been observed in base states of Cartesian, spherical or cylindrical coordinate systems (Benjamin et. al., Proc. Roy. Soc. Lond., 1954; Adou et. al., J. Fluid Mech., 2016; Patankar et. al., J. Fluid Mech. 2018). In these geometries, under inviscid irrotational approximation, small-amplitude standing waves are governed by the Mathieu equation. Taking viscosity into account produces a memory term leading to an integro-differential, damped, Mathieu equation. In Cartesian geometry such an equation was derived by Beyer et. al., Phys. Rev. E, 1995. Here, we derive an analogous integro-differential equation governing Faraday waves on a cylindrical filament of a viscous liquid utilising the toroidal-poloidal decomposition (Boronski et. al., J. Comput. Phys. 2007). We demonstrate perpetual stabilisation of an unstable axisymmetric Rayleigh$-$Plateau mode $(kR_0<1)$ perturbation in the viscous case by choosing the strength of forcing appropriately. Linearised theory is compared with Direct Numerical Simulations using the open-source code Gerris showing good agreement. We also show the possibility of generating droplets in a controlled manner by fragmenting the filament.   

Modified one-way Navier-Stokes equations with PSE-like cost.

Spatial marching methods offer a low-cost alternative to global linear methods and direct numerical simulation for studying the linear and nonlinear stability of slowly varying flows, e.g., boundary layers and jets. The widely used parabolized stability equations (PSE) provide adequate accuracy in some cases, but tend to fail, due to regularization required to overcome the ill-posedness of the spatial march, for flows involving multiply instability mechanisms, transient growth, and acoustics. An alternative method called the One-Way Navier-Stokes equations (OWNS) uses a recursive filter to yield a formally well-posed spatial march that can accurately capture the entire downstream solution, but at one to two orders-of-magnitude higher computational cost than PSE. In this presentation, we introduce a modified OWNS method with cost approaching that of PSE. The speedup is achieved by modifying the OWNS recursion equations to achieve better cost scaling and by borrowing ideas from PSE while still avoiding its limitations. The accuracy and efficiency of the method are demonstrated using a hypersonic boundary layer and a turbulent jet..   

Periodic fluctuations of streamwise vortices in inertia-dominated intersecting flows

In the proximity of stagnation points, flow instabilities arise at relatively low Reynolds number, $Re\sim O~(10-100)$, resulting in the formation of vortices that can evolve time-periodic patterns. These flows are well studied where the stagnation point is downstream of a bluff body and the resulting vortices are in the spanwise direction (e.g. the von K\'{a}rm\'{a}n vortex street). However, they are less understood in intersecting flows, where the resulting vortices are stretched in the streamwise direction. In this work, quantitative flow velocimetry measurements and 3D numerical simulations, are used to characterize steady vortical flow fields at two intersecting rectangular channels with different degrees of confinement, determined by the channel aspect ratio (depth: width), $\alpha$. By tuning the parameters $\alpha$ and $Re$, we control the number of vortices in the steady flow field, their relative rotation direction and vortex core structure. These features will determine the onset parameters and dynamics of time-periodic fluctuations. The dimensionless description of our findings can be applied in various scales, for control over vortex-induced vibrations that harm structures such as bridges, buildings, wind turbines and pipes in terrestrial and marine environments.   

Investigation of tip-vortex instability in the near-wake region of a rotor

Most studies of stability of tip vortices generated from rotors have been focused in the far-field wake, where the vortices are usually modeled as interlaced, helical vortices. In the near wake, due to blade-vortex interactions and a reduction in the cross-sectional area of the slipstream, the tip-vortex configuration deviates from the circular, helical structures found in the far-field wake. These near-wake effects are well captured by the Landgrebe's empricial model for the vortex trajectories. The dependence of near-wake characteristics on the stability of tip vortices becomes significant especially in hovering rotors, where the vortices are convected away from the blade slowly when compared with rotors in forward or, axial flight. The dependence of near-wake characteristics also increases with decreasing thrust coefficients. A linear stability analysis is carried out for the Landgrebe's vortices to predict the growth rate and the instability modes, following closely the classical work of Gupta and Loewy (1974). The predictable quantities from the linear theory are functions of the wake age, with its values approaching the far-field values for very large wake ages. The theoretical results are then compared with the experimental observations conducted in our experimental setup.   

Resolvent Analysis Of Laminar Separated Flows Over Swept Wings

\sout{\textsc{WE STUDY LAMINAR SEPARATED FLOWS OVER SPANWISE-PERIODIC SWEPT WINGS AT HIGH ANGLES OF ATTACK USING RESOLVENT ANALYSIS. THREE-DIMENSIONAL POST-STALL FLOW PHYSICS OF SWEPT WINGS REMAINS LARGELY UNEXPLORED. WE USE RESOLVENT ANALYSIS TO GAIN INSIGHTS ON THE BEHAVIOR OF COHERENT STRUCTURES AND AMPLIFICATION MECHANISMS AS WE VARY THE SWEEP ANGLE, STROUHAL NUMBER, SPANWISE WAVENUMBER, AND ANGLE OF ATTACK. ADJOINT-BASED PARAMETRIC SENSITIVITY ANALYSIS IS LEVERAGED TO SPAN THE PARAMETER SPACE FOR DIFFERENT SPANWISE WAVENUMBERS AND STROUHAL NUMBERS. AT THE ZERO SPANWISE WAVENUMBER, THE RESOLVENT GAIN PEAK IS FOUND TO ALIGN WITH THE DOMINANT SHEDDING FREQUENCY. IN THIS CONFIGURATION, AN INCREASE IN SWEEP ANGLE IS SHOWN TO REDUCE THE RESOLVENT GAIN. AT THE NON-ZERO SPANWISE WAVENUMBERS, INCREASE IN SWEEP ANGLE SHIFTS DOMINANT GAIN TO HIGHER FREQUENCIES. FOR THE SWEEP ANGLE OF 45}}\sout{$^{\mathrm{O}}$\sout{\textsc{, 3D MODES ARE AMPLIFIED MORE THAN 2D MODES. THE PRESENT STUDY SERVES AS A FUNDAMENTAL BASIS TO REVEAL UNIQUE WAKE DYNAMICS SEEN FOR SWEPT WINGS.}}