Unsteady Motion Research Articles

On space–time diversity in shock train self-excited oscillation mode during wide-range evolution in a scramjet isolator

The shock train self-excited oscillation can induce combustor instabilities and reduce engine margin. In a dual-mode scramjet, the shock train undergoes a complete evolution process, exhibiting structural changes closely tied to this inherent unsteadiness. This study aims to elucidate the space–time diversity in shock train self-excited oscillation mode and the underlying mechanisms during wide-range evolution. The experimental investigations were conducted at Ma = 1.95, capturing the complete evolution of the shock train. The results indicate the evolution can be categorized into three regimes based on structural characteristics. In regime I, the shock region gradually forms, followed by the occurrence of the mixing region in regime II. Regime III corresponds to inlet unstart. In regime II, isolator outlet pressure fluctuations exhibit higher frequency and lower amplitude compared to regime I, while the shock motion demonstrates lower frequency and higher amplitude. The shock train behaves in a large-scale, low-frequency (1.53 times the duct height, 10 Hz) unsteady motion in regime II, posing a potential threat to engine operation. Coherence and phase analysis reveal the disturbance source originates downstream. Proper orthogonal decomposition modal analysis shows two oscillation modes: low-frequency components correspond to shock motion, and high-frequency components correspond to pressure fluctuations across the entire pseudoshock. The propagating of downstream disturbance differs between the two regimes. In regime I, the shock train exhibits rigid-body motion synchronously. In regime II, the relative motion between each shock wave and the cumulative effect of pressure disturbance lead to frequency decay upstream, amplifying the shock train motion.

Accurate simulations of moving flexible objects with an improved immersed boundary-lattice Boltzmann method

Accurately capturing boundaries is crucial for simulating fluid–structure interaction (FSI) problems involving flexible objects undergoing large deformations. This paper presents a coupling of the immersed boundary-lattice Boltzmann method with a node-based partly smoothed point interpolation method (NPS-PIM) to enhance the accuracy of simulating moving flexible bodies in FSI problems. The proposed method integrates a multiple relaxation time scheme and employs a force correction technique to address boundary capturing inaccuracies. The effect of virtual fluid is accounted for through a Lagrangian point approximation, ensuring precise FSI force calculations for unsteady solid motions. NPS-PIM is utilized as the solid solver, constructing a moderately softened model stiffness by combining the finite element method (FEM) with the node-based smoothed PIM (NS-PIM). Simulations of flow fields near flexible objects with large deformations demonstrate that the proposed approach reduces numerical errors, improves computational efficiency compared to traditional FSI models using FEM and NS-PIM, and accurately captures the behavior of moving flexible bodies and detailed flow fields.

Transient Dynamics of a Porous Sphere in a Linear Fluid

This article describes the unsteady translational motion of a porous sphere with slip-surface in a quiescent viscous fluid. Apart from its radius a and density ρs , the particle is characterized by its permeability parameter γ , slip-length l and effective viscosity-ratio ϵ for interior flow. The Reynolds number for the system is assumed to be small leading to negligible convective contribution, though the transient inertia for both the liquid and the solid is comparable to viscous forces. The resulting linearized but unsteady flow-equations for both inside and outside the porous domain are solved in time-invariant frequency domain by satisfying appropriate boundary conditions. The analysis ultimately renders frequency-dependent hydrodynamic friction for the suspended body. The frictional coefficient is computed under both low and high frequency limit for different values of γ, l and ϵ so that the findings can be compared to known results with impermeable surface. Moreover, the parametric exploration shows that the sphere acts like a no-slip body even with non-zero l if aγ ≫ 1/√ϵ . Scaling arguments from a novel boundary layer theory for flow inside porous media near interfaces explain this rather unexpected observation. Also, computed fluid resistance is incorporated in equation of motion for the particle to determine its time-dependent velocity response to a force impulse. This transient response shows wide variability with ρs, γ, l and ϵ insinuating the significance of the presented study. Consequently, the paper concludes that slip and permeability should be viewed as crucial features of submicron particles if unexpected variability is to be explained in nano-scale phenomena like nano-fluidic heat conduction or viral transmission. Thus, the theory and findings in this paper will be immensely useful in modeling of nano-particle dynamics.

Effect of sweep angle on three-dimensional vortex dynamics over plunging wings

The effects of sweep angle and reduced frequency on the leading-edge vortex (LEV) structure over flapping swept wings in the Reynolds number (Re) range of O(104) are yet to be completely understood. With increasing interest in designing bio-inspired micro-air-vehicles, understanding LEV dynamics in such scenarios is imperative. This study investigates the effects of three different sweep angles (Λ=0°, 30°, and 60°) on LEV dynamics through high-fidelity improved delayed detached eddy simulation to analyze the underlying flow physics. Plunge ramp kinematics at two different reduced frequencies (k = 0.05 and 0.4) are studied to investigate the unsteady motion effects on LEV characteristics. The leading-edge suction parameter concept is applied to determine LEV initiation, and the results are verified against flow field visualization for swept-wing geometries. The force partitioning method is used to investigate the spanwise lift distribution resulting from the LEV. Distinct peaks in the lift coefficient occur for the high reduced frequency case due to the impulse-like plunging acceleration. This causes the LEV to detach from the leading edge more quickly and convect faster, significantly affecting the lift generated by the wing. As reduced frequency increases, the LEV breakdown mechanism switches from vortex bursting to LEV leg-induced instabilities. These results provide insights into the complex vortex structures surrounding swept wings at Re = 20 000, and the impact both sweep angle and reduced frequency have on the lift contribution of these flow features.

Role of chemical processes and porous media in thermal transport of Casson nanofluid flow: A study with Riga plates

A Riga plate is an electro-magnetic instrument working with anodes and magnets on plates. The Riga plate has numerous advantages in geophysical sciences, the field of engineering, sensors,and astrophysics.The present work investigated the transitory Electromagnetohydrodynamic radiative squeezing unsteady motion of Casson nanofluid through a parallel channel impacted by the Riga plate with endothermic/exothermic chemical processes and porous media. The PDEs transform into non-linear ODEs using the appropriate similarity transformations. The RKF-45 method and Shooting techniques are utilized to compute the solutions to these non-linear ODEs and related boundary conditions. The consequences of non-dimensional parameters on the corresponding profiles are illustrated graphically. And also studied some significant engineering coefficients. The primary findings of the present study are a rise in the thermal relaxation time parameter, temperature increases for the exothermic chemical reaction case, and opposite behavior in the endothermic chemical reaction case. Enhancement in activation energy parameter, the temperature drops for exothermic and increases for the endothermic case. The value of the modified Hartmann number rises, the velocity profile is also increased in the lower plate and the opposite behavior in the upper plate. Increase in Biot number, temperature decreases in the exothermic case and increases in the endothermic case.

A generalized mathematical model for the damped free motion of a liquid column in a vertical U-tube

A one-dimensional mathematical model is developed to analyze the unsteady characteristics of arbitrary damped free motion, denoted as Z(τ), in a rigid, constant-diameter vertical column of a U-tube filled with an incompressible Newtonian liquid, where τ represents time. This model utilizes a simplified unsteady momentum equation derived from the Navier–Stokes equations in a circular coordinate system. Moreover, it incorporates assumptions about the periodicity of arbitrary damped free oscillations and employs the Fourier series representation to characterize the damped free motion. The combination of assumptions made for periodicity, the simplified momentum equation, and the Fourier series representation makes the current mathematical model unique and novel compared to prevailing models in the literature. In this model, the governing partial differential equation contains two dependent variables: Z(τ) is the known variable, as one can measure from experiments, and the instantaneous velocity uz is the unknown variable. Fitting the experimental data into the Fourier series provides the Fourier coefficients associated with the specific experiment. The Laplace transform method is used to determine the analytical solution for uz corresponding to the known Z(τ). The analytical expressions for instantaneous flow characteristics of practical importance, including area-averaged velocity, wall shear stress, and acceleration/deceleration, are deduced from uz. The analytical solutions presented are valid for generalized unsteady motions, including underdamped oscillations with varying amplitudes and periods, underdamped oscillations with varying amplitudes and constant period, and overdamped motion that does not exhibit a single oscillation. The findings from the present model offer insights for formulating a new friction model.

Unsteady Flow Behaviors and Vortex Dynamic Characteristics of a Marine Centrifugal Pump under the Swing Motion

Due to the effects of swing motion, the performances and internal flow characteristics of marine centrifugal pump undergo some unsteady variations in the marine environment. The hydraulic test system with six degree of freedom parallel motion platform is established to study the pump performance characteristics at the different heel angles of steady roll position and pitch position. The pump head gradually decreases as heel angle increases. The pump head has decreased by 7% to reach the minimum at the 15° heel angle of roll position. At the same heel angle, the head at the roll position is lower than that at the pitch position under the rated flow condition. The fluid in the impeller passage is subjected to the additional inertial force of roll motion or pitch motion under unsteady swing motion, inducing some flow bias phenomena in the velocity field. The unsteady development of flow velocity induces the intense vortex motion, and the shedding and dissipation of interblade vortices are affected. The periodic flow-induced pulsation characteristics obviously appear in the impeller passage. The pulsation periodicity and pressure amplitude are influenced due to the swing motion. The pitch motion induces the greater hydraulic excitation and fluid-induced vibration amplitude. In addition to the pressure pulsation at the low frequencies, the pulsation amplitude at 20 times the shaft frequency is evident under pitch motion.

Оперативное диспетчерское управление водораспределением в оросительных системах по расчетному приращению объемов

Goal: to improve the methodological approach to developing an algorithm for operational dis-patch control of water distribution based on the calculated increment of water volumes in irri-gation canals. Materials and methods: the dispatch control methodology is described, which is based on the typification of flow conditions between two partitioning structures in technical and functional directions, as well as on the development of systems for unifying channels and their modes using similarity theory methods, which are widely used in hydromechanics and hydraulics. The basis for the implementation of the similarity theory is the principle of using similarity numbers, that is, combinations of dimensionless parameters that determine the char-acteristics of the unsteady motion of a heavy, incompressible, viscous fluid in channels. Re-sults: the general tasks of the dispatcher were determined to re-regulate the hydraulic regime of sections of the irrigation system, without tying them to specific types of hydraulic struc-tures. The main provisions of the method of dispatch control of water distribution based on the calculated increment of volumes, determination of the time of re-regulation between the main and compensating processes of re-regulation are substantiated. Four types of situations are identified when solving dispatch problems of water distribution, which are determined by comparing the volumes and flow rates installed on the site, for each of which the inherent re-regulation order is determined. Conclusion: the article outlines a new methodological ap-proach that ensures the joint use of the method of similarity theory and a simulation experi-ment with the Saint-Venant equations for steady and unsteady water flow in the canals of ir-rigation systems, to solve problems of operational dispatch control of water distribution ac-cording to the calculated increment of volumes.

Flow memory effects on the nonlinear hydrodynamic load of an ROV in translational maneuvering

This paper describes experimental and numerical investigations of the flow memory effect on the nonlinear hydrodynamic load response of a work-class remotely operated vehicle (ROV) under surge, sway, and heave impulse motions. The hysteresis loops of the dynamic drag coefficients during impulse motion tests are analyzed by comparing them with those obtained in steady drag tests. The areas of the loops and differences between the maximum and minimum dynamic and steady coefficients are used to quantify the flow memory effect. A novel unsteady hydrodynamic model for ROVs in translational maneuvering is established. This model considers the flow memory effect in terms of the unsteady hydrodynamic load response obtained from impulse motion response tests. The results given by the unsteady model are found to be in good agreement with those from CFD simulations. The unsteady hydrodynamic characteristics (including the asymmetric forces, force magnitude, and phase delay), motion amplitude, and frequency effect on the hydrodynamic forces of the ROV are analyzed by comparing the unsteady model results with those obtained from the steady model, revealing the rules governing the flow memory effect.

Dissipation Effects in the Tea Leaf Paradox

The Tea Leaf Paradox (TLP) describes unsteady fluid motions which help entrain and deposit suspended particles at the center of rotation. Various applications depend on the TLP for particle separations—spanning orders of magnitude in length scales—making it an important problem in fluid mechanics. Despite papers describing the phenomenon, the efficacy of particle separation using the TLP remains unclear as to the relative importance of, for example, hydrostatics, particle-fluid density ratio, wall friction, liquid bath aspect ratio and the rotation speed. The dynamics involved are notably complex and require a careful tuning of each variable. In this study, we have investigated the role of the limit of the aggregation dynamics in rotational flows within 3D-printed vessels of various sizes in tandem with particle imaging to probe the dissipation effects on the particle motions. We have found that the liquid bath aspect ratio limits how much aggregation may occur for a particle-fluid density ratio greater than unity (e.g., ρp/ρf>1), where ρp is the density of the particle and ρf is the ambient fluid density.

Comparison of aerodynamic and aeroacoustic characteristics of fixed-pitch and variable-pitch rotors

This research conducts a comparative analysis of the aerodynamic and aeroacoustic characteristics of fixed-pitch and variable-pitch controlled multirotors [i.e., revolution per minute (RPM) and collective pitch control]. The study encompasses single-, twin-, and quad-rotor configurations under hovering flight conditions. The unsteady blade motion is modeled as a function of RPM and pitch, fluctuating with frequency and amplitude. To comprehensively account for wake interaction effects, a free-wake vortex lattice method combined with acoustic analogy is utilized. The findings reveal that unsteady blade motion and wake interaction effects cause fluctuations in thrust and tip vortex trajectory. The thrust and tip vortex behavior exhibited greater instability in response to RPM fluctuations than to pitch fluctuations. Consequently, the axial unsteady loading noise was more pronounced under RPM fluctuations compared to pitch fluctuations. In both twin- and quad-rotor configurations, the wake interaction significantly influenced the characteristics of thrust and tip vortex behavior. In addition, spectral analysis demonstrated that the frequencies of unsteady blade motion and wake interaction determine the frequencies of thrust and tip vortex fluctuations as well as unsteady loading noise.

Pressure-Sensitive-Paint-Driven Hybrid Unsteady Compressible Flow Simulation

A hybrid unsteady compressible computational fluid dynamics simulation (CFD) driven by pressure-sensitive paint (PSP) data is proposed, and the concept is demonstrated in two dimensions. By imposing the wall pressure acquired with PSP as the Dirichlet boundary condition, an unsteady compressible Euler equation solver is marched together with integral boundary-layer equations in time. As a result, an entire flowfield that fits the PSP measurement is created in the CFD domain. This concept is tested by taking PSP data from a past article of a wind-tunnel test using the NASA CRM65 airfoil and the steady and unsteady capabilities of this data-driven hybrid simulation are demonstrated for transonic flows in two dimensions, and the convergence characteristics are also investigated. It is found that a steady shock position, which cannot be accurately predicted by the Euler equation solver with a boundary layer, can be rectified by the hybrid simulation. Moreover, unsteady shock motion due to buffeting can also be captured well by the hybrid simulation.

Global well-posedness for two-phase fluid motion in the Oberbeck–Boussinesq approximation

This paper focuses on the global well-posedness of the Oberbeck–Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial state of the drop is close to a ball BR with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of BR. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck–Boussinesq approximation in a fixed domain. From there, we establish time-weighted estimates of solutions for the shifted equation using maximal Lp–Lq regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang [J. Differ. Equations 322, 101–134 (2022)]. Using time decay estimates of the semigroup, we then obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval (0, T) was proven by Hao and Zhang [Commun. Pure Appl. Anal. 22(7), 2099–2131 (2023)]. After that, we demonstrate that the solutions can be extended beyond T by analyzing the properties of the roots of algebraic equations.

Unveiling Turbulent Flow Dynamics in Blind-Tee Pipelines: Enhancing Fluid Mixing in Subsea Pipeline Systems

Blind tees, as important junctions, are widely used in offshore oil and gas transportation systems to improve mixing flow conditions and measurement accuracies in curved pipes. Despite the significance of blind tees, their unsteady flow characteristics and mixing mechanisms in turbulent flow regimes are not clearly established. Therefore, in this study, Unsteady Reynolds-Averaged Navier–Stokes (URANS) simulations, coupled with Explicit Algebraic Reynolds Stress Model (EARSM), are employed to explore the complex turbulent flow characteristics within blind-tee pipes. Firstly, the statistical flow features are investigated based on the time-averaged results, and the swirl dissipation analysis reveals an intense dissipative process occurring within blind tees, surpassing conventional elbows in swirling intensity. Then, the instantaneous flow characteristics are investigated through time and frequency domain analysis, uncovering the oscillatory patterns and elucidating the mechanisms behind unsteady secondary flow motions. In a 2D-length blind tee, a nondimensional dominant frequency of oscillation (Stbt = 0.0361) is identified, highlighting the significant correlation between dominant frequencies inside and downstream of the plugged section, which emphasizes the critical role of the plugged structure in these unsteady motions. Finally, a power spectra analysis is conducted to explore the influence of blind-tee structures, indicating that the blind-tee length of lbt = 2D enhances the flow-mixing conditions by amplifying the oscillation intensities of secondary flow motions.

Comparison of unsteady MHD flow of second grade fluid by two fractional derivatives

A study is conducted on the unsteady motion of a free convective flow of second grade fluid, energy transfer, and Darcy’s law over an oscillatory smooth vertical plate. The study compares two different approaches in developing a fractional model: the Caputo-Fabrizio operator with nonsingular kernel and the constant proportional Caputo fractional operator with Fourier’s and Fick’s laws. By applying the Laplace method and transforming the provided set of equations into nondimensional form, we obtained semi-analytical results and presented these results through graphical analysis. The study examines how different flow parameters, including the fractional parameters, affect the velocity, mass, and heat profiles of the physical system. The results suggest that the physical model using constant proportional Caputo derivative leads to higher temperature, stronger concentration, and increased velocity compared to the Caputo-Fabrizio model. This highlights the importance of selecting an appropriate fractional model when studying complex physical systems. It is also depicted from the whole analysis that field variables with novel hybrid fractional derivative constant proportional Caputo exhibits a more effective and declining tendency than Caputo-Fabrizio.

Analysis of coherent structures over interacting barchan dunes based on tomographic particle image velocimetry

The influence of barchan dune interaction upon unsteady flow separation and wake dynamics around the fixed-bed downstream barchan dune (DBD) model are experimentally investigated at a Reynolds number of 2640 based on the tomographic particle image velocimetry (PIV) system. The time-averaged statistics including the mean velocities, recirculation area, vortex spatial topology, Reynolds stress, and turbulent kinetic energy were used to characterize the flow field and large-scale anisotropy. It was found that arch-shaped vortex “chains” with strong spanwise coherence shedding from isolated barchan crestline populate the whole wake region, while elongated rod-shaped vortex structures with strong streamwise coherence induced by the up-downwash flow around the DBD were found to fill the whole measurement range, which is closely related to “sheltering” effect on the incoming flow acting at DBD due to the presence of upstream barchan dune (UBD). Additionally, in order to study the complex dynamic features of these predominated vortex structure transformations, time-resolved planar particle image velocimetry was applied. This technique allows for providing complementary insights into the temporal behavior of the unsteady coherent flow structures populating the wake field in different experimental configurations. It was found that the basic unsteady flapping motion, vortex roll-up, and complex vortex interactions including vortex pairing, merging, and breaking up can all be analyzed by dividing into certain scales with ease in a combination wavelet and Lagrangian framework.

Unsteady hydrodynamics of a surface piercing and fully submerged body when entering a lock

Ship hydrodynamics in a constant waterway have been well studied. With a coordinate system fixed on the moving ship, the boundary value problem (BVP) is usually treated as a steady one. However, the hydrodynamics of a ship moving in confined waterways with abrupt changes in width or depth are complex due to their unsteady nature. The studies on such unsteady problems are insufficient. In the engineering practice, the hydrodynamic unsteadiness can be fully reflected by a scenario when a ship enters a lock. Prior studies have predominantly focused on predicting the ships' hydrodynamic forces without considering the unsteady terms on the free-surface boundary conditions. Obviously, such steady or quasi-steady methods overlooked the crucial unsteady phenomena on free water surface. To address this gap, the present study introduces a novel three-level difference scheme to discretize the free surface condition, preserving unsteady terms while maintaining temporal continuity of cells on the free surface. With the implementation of such fully unsteady BVP, we observed some interesting unsteady free surface motions, which were not well documented in the existing literature. To verify our new observations, as well as to validate the numerical method proposed in this study, two physical model tests were designed and conducted in a towing tank: a submerged ellipsoid enters a deep lock at relatively high speeds, and a box enters a shallow and narrow lock at very low speeds. The discussion is highlighted on the unsteady waves in front of the moving bodies, as well as the unsteady resistance induced by such waves.

Going beyond BEM with BEM: an insight into dynamic inflow effects on floating wind turbines

Abstract. Blade element momentum (BEM) theory is the backbone of many industry-standard wind turbine aerodynamic models. To be applied to a broader set of engineering problems, BEM models have been extended since their inception and now include several empirical corrections. These models have benefitted from decades of development and refinement and have been extensively used and validated, proving their adequacy in predicting aerodynamic forces of horizontal-axis wind turbine rotors in most scenarios. However, the analysis of floating offshore wind turbines (FOWTs) introduces new sets of challenges, especially if new-generation large and flexible machines are considered. In fact, due to the combined action of wind and waves and their interaction with the turbine structure and control system, these machines are subject to unsteady motion and thus unsteady inflow on the wind turbine's blades, which could put BEM models to the test. Consensus has not been reached on the accuracy limits of BEM in these conditions. This study contributes to the ongoing research on the topic by systematically comparing four different aerodynamic models, ranging from BEM to computational fluid dynamics, in an attempt to shed light on the unsteady aerodynamic phenomena that are at stake in FOWTs and whether BEM is able to model them appropriately. Simulations are performed on the UNAFLOW 1:75 scale rotor during imposed harmonic surge and pitch motion. Experimental results are available for these conditions and are used for baseline validation. The rotor is analyzed in both rated operating conditions and low wind speeds, where unsteady aerodynamic effects are expected to be more pronounced. Results show that BEM, despite its simplicity, can adequately model the aerodynamics of FOWTs in most conditions if augmented with a dynamic inflow model.

Lift tailoring on unsteady airfoils with leading-edge vortex shedding using an inverse aerodynamic approach

In this paper, we present a physics-informed approach to tailor the lift profile of an unsteady airfoil through the execution of an appropriate maneuver. In previous research, a low-order aerodynamic model based on the unsteady thin airfoil theory was developed for predicting the flowfield and loads on airfoils undergoing arbitrary motions. The theory was phenomenologically augmented using the concept of leading edge suction parameter (LESP) to incorporate the capability to predict intermittent leading edge vortex (LEV) shedding. The criticality of LESP was used to predict the onset and termination of LEV shedding and thus model the effect of LEVs on the flowfield and loads for a prescribed motion. In the current work, an inverse aerodynamic formulation is developed based on this framework for tackling the inverse problem: to obtain the motion kinematics required for generating a prescribed lift profile for an airfoil operating in the dynamic-stall regime. The LEV-modeling capability of the aerodynamic model enables the motion-design algorithm to take into account the effect of complex phenomena, such as dynamic stall and LEV shedding, which are not taken into account in previous research approaches. Several case studies are presented to demonstrate various scenarios such as lift tracking using pitching and heaving motions, lift cancellation during unsteady motion, and the generation of a given lift profile using two equivalent motions. The kinematic profiles generated by the inverse formulation are also simulated using a high-fidelity unsteady computational fluid dynamics solver to validate the predictions.

Experimental and Numerical Investigations of Mini-Tab Device for Active Flow Control

The next generation of aircraft is likely to exhibit a breadth of aeromechanic and aeroelastic behaviors that will be exacerbated by the turbulent flows generated in urban environments. Exploitation of novel flow control technologies will be critical in offering greater control authority to enable this new generation of aerial vehicles for faster, safer, and more efficient flight. This study experimentally investigates the efficacy of a mini-tab active flow control device using a novel dynamic test rig. Under steady conditions, the mini-tab response exhibited nonlinearity with deployment height. Under dynamic conditions, the lift and pitching moments displayed hysteresis around their quasi-steady counterparts at low reduced frequencies (k=0.05) but displayed significant deviations at middle to high reduced frequencies (k=0.15 to 0.31) due to the formation of a tab vortex that propagates downstream. It was demonstrated, for the first time experimentally, that mini-tab dynamic performance is insensitive to unsteady airfoil motions in pitch and plunge—representative of flutter onset. Finally, an existing mini-tab model was extended to capture vortex formation on the upper and lower airfoil surfaces and modified to incorporate mini-tab rate-dependence on vortex initiation. The model showed good agreement with experimental data in both periodic and transient scenarios.