We investigate the multistability and bifurcation behavior in thermal vibrational convection through direct numerical simulations, focusing on the effects of varying initial conditions, vibrational Rayleigh number (${\text{Ra}}_{\mathrm{vib}}$), Strouhal number ($\text{St}$), and the aspect ratio ($\mathrm{\Gamma }$) at a fixed Prandtl number ($\text{Pr}=1$). In this study, the Nusselt and Reynolds numbers remain below the threshold for the fully turbulent regime, yet the system displays intricate nonlinear dynamics. By modulating the initial phase of external vibrations, we identify two distinct flow states: a periodic state with consistent large-scale circulation and periodic variations in the Nusselt number [$\text{Nu}\left(t\right)$] and the Reynolds number [$\text{Re}\left(t\right)$], and a chaotic state characterized by unstable flow structures and erratic $\text{Nu}\left(t\right)$ and $\text{Re}\left(t\right)$ fluctuations. The time-averaged Nusselt and Reynolds numbers are higher in the periodic state compared to the chaotic state. Fourier mode decomposition reveals that the periodic state is dominated by the single-roll ($M^{1,1}$) mode, while the chaotic state involves interplay among all modes. Initial velocity fields constructed from Fourier mode basis functions confirm that high-intensity $M^{1,1}$-dominated conditions yield the periodic state, while others produce the chaotic state. Parametric analysis highlights that bifurcations occur predominantly near $\text{St}=1$, where the vibration frequency matches the system's intrinsic timescale, fostering sensitivity to initial disturbances and multistability, and the bifurcation exhibits hysteresis across various ${\text{Ra}}_{\mathrm{vib}}$ and $\mathrm{\Gamma }$. This work elucidates the mechanisms driving flow state transitions in thermal vibrational convection, offering a novel framework for studying unsteady flow multistability with implications for heat transfer and flow control.

Predicting when a three-dimensional turbulent flow reaches its dissipation peak is essential for both theory and adaptive algorithms in simulations and experiments. Using direct numerical simulations (DNSs) of the Taylor-Green vortex (TGV) at resolutions of ${256}^3-{1024}^3$, we introduce and test a small-scale weighted diagnostic: the spectrum of ${|\nabla \times \mathit{\omega }|}^2$ (with $\mathit{\omega }=\nabla \times \mathbf{u}$), which, for incompressible flow, is equivalent to a $k^4$-weighted energy spectrum. We show that the peak wavenumber of this spectrum, $k_{\mathrm{peak}}{\left[\right|\nabla \times \mathit{\omega }|}^2]$, advances rapidly to intermediate-small scales and then levels off before the dissipation rate $\varepsilon \left(t\right)={\sum }_{k}2\nu k^{2}E\left(k\right)$ reaches its maximum. Across all resolutions, we observe robust temporal ordering $t_k<t_{\varepsilon }<t_{\mathrm{\Pi }}$, where $t_k$ marks the onset of the rapid rise of $k_{\mathrm{peak}}{\left[\right|\nabla \times \mathit{\omega }|}^2], t_{\varepsilon }$ is the time of the maximal $\varepsilon \left(t\right)$, and $t_{\mathrm{\Pi }}$ is when the cumulative flux $\left|\mathrm{\Pi }\right(K\left)\right|$ attains its largest peak scale. This early-warning signal correlates with the morphological transition to filament-dominated structures visible in $Q$-criterion isosurfaces and is consistent with integral-scale trends ($L_{\mathrm{int}},\lambda ,\eta$). The diagnostic is simple to compute from standard DNS data and highlights the incipient formation of high-curvature structures, where viscosity acts most strongly.

In this study, three-dimensional simulations of a reacting hydrogen jet in supersonic crossflow using a structure-preserving discontinuous Galerkin (DG) formulation are examined. The hydrogen jet, with a momentum flux ratio of five, is injected into a high enthalpy crossflow. The sensitivities of the solution to the grid element size and polynomial order are investigated to determine an accurate and computationally efficient approach to simulating high-speed airbreathing propulsion vehicles. The results demonstrate that $\mathrm{DG}(p=2$) solutions, which are nominally third-order accurate in smooth regions of the flow, show reasonable agreement with existing experimental results. The separation shock formation behind the jet is found to be heavily grid dependent and necessary for accurate simulations of the reacting jet in supersonic crossflow. It is determined that the highest considered cell resolution and polynomial order are required to capture the upstream separation shock and, consequently, the flame stabilization point. The mixing and combustion mode is also determined using the flame index and demonstrates the flow is heavily skewed toward a non-premixed diffusion mode, which is consistent with previously run simulations of this case using traditional finite volume schemes and sub-grid scale modeling approaches. Beyond this analysis, given the prevalent use of structured hexahedral meshes in the computation of realistic high-speed, chemically reacting flows, the novelty of this work lies in demonstrating accurate simulation of such a flow on a fully unstructured tetrahedral mesh. This highlights the potential of these methods to handle both complicated geometries and complex physics.

Understanding the coupled dynamics of liquid-solid phase change and fluid flows is crucial in a wide range of geophysical and industrial applications. When freezing occurs in saline water, the newly formed ice is mushy, with a porous structure that traps the brine within the ice. In this work, which combines experiments and theoretical analyses, we investigate the long-term evolution of saline ice, comprehensively accounting for the coupled dynamics of multiscale fluid flow, heat and mass transfer, and phase change. We show that in a closed convective system the rapid formation of a mushy ice layer is followed by desalination (i.e., the expulsion of salt from the ice) processes that might lead to a slow asymptotic decrease of the ice thickness. Desalination of mushy ice reduces its porosity, which alters the dynamic thermal equilibrium and ice thickness by weakening buoyancy-driven convection within the mushy layer. In turn, changes in brine convection and ice thickness affect the further desalination of the ice. The long-term dynamics of the system can be accurately predicted by a one-dimensional model based on appropriate parametrizations of global heat and mass transfer properties. Furthermore, within the same theoretical model we explore the ice dynamics across a broader parameter space. Our findings advance the understanding of the coupled phase-change physics of saline solutions in the presence of convective fluid flows and provide a basis for explaining and predicting real-world phenomena such as the aging of sea ice.

We report experimental investigations of an unconfined liquid-solid fluidized bed at low to moderate Reynolds number. The fluidization velocity $U_f$ is measured when an upward flow is imposed, and the sedimentation velocity $U_{\mathrm{sed}}$ is measured when the flow is stopped. Only when the level of inlet flow fluctuation is sufficiently low will these two velocities coincide, as expected for an unbound system. A systematic comparison between $U_f$ and $U_{\mathrm{sed}}$ should therefore be made when attempting to determine fluidization laws. Comparisons with previous studies show that $U_f$ measurements gather on a single function of the ratio $\mathrm{\Phi }/{\mathrm{\Phi }}_{\mathrm{pack}}$ between the particle volume fraction $\mathrm{\Phi }$ and its packing value ${\mathrm{\Phi }}_{\mathrm{pack}}$, except for a factor $K$ that encompasses all effects other than concentration: Stokes number, Reynolds number, confinement, and inlet condition. Time-resolved measurements of the fluidized bed surface show two contributions: uncorrelated high-frequency fluctuations and low-frequency oscillations that remain correlated over minutes. The coherent oscillations correspond to the arrival of upward concentration waves at the bed surface. Their wavelength, which characterizes the length scale of large-scale heterogeneities, was known to diverge as the packing state is approached. Here, we found that it also increases with decreasing concentration, suggesting that the loss of global bed stability observed in previous work at high expansions may be related to the development of heterogeneous structures of a size comparable to that of the column.

This study experimentally examines the entry and penetration of a superhydrophobic sphere into a quiescent deep pool, with special emphasis placed on the primary and secondary pinch-off of the air cavity existing in its wake. By spanning previously underexplored density ratios and impact velocities, this work extends the known regimes of sphere water entry: notably, it demonstrates that lighter spheres, assisted by their wake cavities, can reverse the trajectories from descent to ascent, in contrast to the terminally descending behavior of heavier spheres reported in previous studies. A strong correlation between both primary and secondary pinch-off times with the impact Froude number and dimensionless density is revealed. A semiempirical scaling for cavity volume after primary pinch-off is proposed, offering predictive capability across a broader parameter space. A scalar force balance predicts a drastic decrease in buoyancy at the moment of pinch-off, which is reflected as an abrupt change in the deceleration of the sphere. This behavior is captured from time-resolved trajectory measurements obtained using two orthogonally placed high-speed cameras. These findings provide a comprehensive dataset comprising buoyancy transitions, force balances, and cavity morphologies.

We report a comprehensive set of direct numerical simulation benchmarks of bypass transition in the narrow sense with inlet freestream turbulent intensity levels $0.75\%$, $1.5\%$, $2.25\%$, $3.0\%$, and $6.0\%$, respectively. Detailed descriptions of length scales and the rate of viscous dissipation are provided. We address two key physical questions. First, how do the decay rates and length scales of freestream turbulence over a transitional and turbulent boundary layer compare to those in spatially developing isotropic turbulence without the wall? Second, what bypass mechanisms drive turbulent spot inception at the intermediate range of freestream turbulence intensity level? We find that the boundary-layer freestream turbulence decay and length scales evolve similarly to their spatially developing isotropic turbulence flow without the wall counterparts. We also present evidence of the coexistence of two turbulent spot inception mechanisms at the inlet FST level $2.25\%$: the long low-speed streak primary and secondary instabilities (only in lower inlet FST levels) and the self-amplifying process of oblique vortex filaments interacting with a Δ-shaped low-speed patch underneath (prevailing only in higher inlet FST levels).

While experiencing atmospheric turbulence on a commercial flight can be uncomfortable, it rarely compromises the stability of the aircraft. The situation is quite different for small air vehicles that operate in urban canyons, around mountainous terrains, and in the wakes of marine vessels, where they could encounter highly unsteady atmospheric conditions with relatively strong gusts. The spatiotemporal scales of such disturbances can be larger than the characteristic aerodynamic scales of the small vehicles, making the relative effect of disturbance significantly stronger than what a large commercial aircraft would experience. The gust ratio can exceed 1 in these extreme flight environments, making stable flight difficult, if not currently impossible. We refer to the study of aerodynamics for gust ratios over 1 as extreme aerodynamics and identify major challenges that require breakthroughs, particularly with data-driven approaches. Extreme aerodynamics present unique opportunities for innovative analysis techniques to study rich flow physics problems with strong nonlinearity, transient dynamics, and low-dimensional modeling over a large parameter space. Some of the approaches discussed herein should apply to a wider range of fluid dynamics problems with similar challenges.

We introduce a physics-informed Gaussian process regression (GPR) method for data assimilation and uncertainty quantification of particle tracking velocimetry data. Unlike traditional methods based on regression, our approach transparently incorporates statistical information and physics such as mass conservation, boundary conditions, and statistical symmetries directly into the regression model. Furthermore, GPR quantifies prediction uncertainty and provides physics-constrained estimates of the two-point velocity covariance, a quantity of primary interest in turbulent flows. The methodology is demonstrated using synthetic and experimental data from three canonical turbulent flows: homogeneous isotropic turbulence (HIT), turbulent channel flow (TCF), and the turbulent wake behind a square prism (SPW). In all cases, we make comparisons relative to the performance of the vortex in cell method, $\mathrm{VIC}+$. For HIT, the model leverages isotropy to learn the velocity correlation function from even very noisy and sparse data, achieves a factor of two improvement over $\mathrm{VIC}+$ in velocity prediction error, and accurately quantifies the prediction uncertainty. For TCF, we introduce a scalable approach to train a high-dimensional GP model that respects wall-bounded flow physics. GPR significantly outperforms $\mathrm{VIC}+$ in terms of accuracy, uncertainty estimation, and resolution in this case. In the SPW case, GPR demonstrates improved accuracy in velocity prediction and improved coherence of the vorticity field obtained from independent snapshots of tracers. Our approach lays the groundwork for extensions to time-resolved data, inclusion of acceleration measurements, and reduced-parameter models based on resolvent analysis.

The shock-boundary layer interaction (SBLI) over the convex wall of a supersonic turbine vane is studied with a focus on extreme separation bubble events and the interplay between the bubble, streaks, and streamwise vortices. The present analysis is performed on a dataset computed by a wall-resolved large-eddy simulation (LES) of a supersonic turbine at Mach number 2.0 and Reynolds number of 395 000. Building on findings related to near-wall streaks and vortices that drive the bubble breathing motion, here, we employ conditional analysis to study extreme bubble events. First, mean and turbulence statistics are computed to examine differences between expanding and contracting bubble states. Then, finite-time Lyapunov exponent calculations are used to understand the interactions between streaks and streamwise vortices along large and small separation bubble events. Additionally, a deforming control volume methodology is applied to analyze the mass imbalance within the bubble and to identify regions that are more susceptible to mass inflow and outflow. In events where the recirculation region is small, near-wall high-speed streaks penetrate the bubble and lead to higher tangential Reynolds stresses upstream of the incident shock compared with those with large recirculation. The streaks are accompanied by streamwise vortices that meander and induce intense fluid mixing, leading to higher wall-normal and spanwise Reynolds stresses, and consequently, higher turbulent kinetic energy. This turbulent activity also causes intense fluctuations in wall pressure and skin-friction coefficient along the separation region. In contrast, during events with bubble expansion, high-speed streaks are advected over the separation region, with streamwise vortices appearing only downstream of the shock, resulting in minimal fluid mixing inside the bubble. The analysis of mass flux along the bubble surface reveals that during its contraction phase, mass flux out of the bubble occurs predominantly upstream of the incident shock because of high-speed streaks dragged toward the wall by the streamwise vortices. In the expansion phase, pronounced mass flux into the bubble is observed downstream of the shock, near reattachment. This increased fluid injection is associated with the presence of streamwise vortices and low-momentum flow structures near reattachment, suggesting that fluid entrainment by vortices plays a key role in the mass flux into the bubble.

We study the dynamics of prolate microswimmers traversing clean liquid-liquid interfaces. Using large-scale lattice Boltzmann simulations, we observe that the swimmers can be either trapped or cross the interface, depending on their initial angle, swimming speed, and the interfacial tension between the two fluids. The simulation results are rationalized by considering a competition between interfacial (thermodynamic) and self-propulsion (hydrodynamic) forces. The swimmers get trapped at the interface due to a thermodynamic trapping force, similar to trapping of passive colloidal particles at interfaces, when the forces from the interfacial tension dominate over the swimming forces. The trapping behavior can be captured by calculating a critical capillary number, which involves balancing the interfacial and self-propulsion forces. This prediction agrees remarkably well with the numerical simulations. Finally, our results demonstrate that the torque resulting in a reorientation of the swimmers parallel to the interface has both hydro and thermodynamic components.

Experiments are conducted over smooth and rough walls to examine the effect of pressure gradient history on turbulence statistics of turbulent boundary layers. A NACA0012 aerofoil of a chord with 1.25 m mounted in the freestream is used to impose different pressure gradients onto the flow. Measurements are taken at a single point, one chord downstream of the trailing edge of the aerofoil, using a single hot-wire probe. The turbulence statistics and premultiplied energy spectra nondimensionalized by directly measured skin friction are presented at the point of measurement. Measurements are taken at a friction Reynolds number, ${\text{Re}}_{\tau }$, that varies between 3000 and 27 000 depending on the surface and freestream speed, which ensures large-scale separation. The mean streamwise turbulence aligns with previous findings that adverse pressure gradients increase the inner-scaled energy in the outer region, while a favorable pressure gradient suppresses the energy. The individual and combined effects of pressure gradient history and roughness on the premultiplied energy spectra are examined. Examination of the difference between the pressure gradient and corresponding zero-pressure gradient spectra shows that adverse pressure gradients have a greater effect on the energy of the rough wall. The peak energy scales with a weighted integral of the pressure gradient history normalized by local quantities. In the outer region, this trend is linear, with the position and wavelength at which it is most affected appearing to be approximately a constant.

The complexity of wall-bounded turbulent flows has given rise to a variety of models that capture the essence of this physical problem, yet many of these models are built on different modeling assumptions. A plausible model is Townsend's attached eddy model (AEM), which is predicated on the assumption that the eddies have a geometric scaling with the distance from the wall and a constant characteristic velocity. With a limited set of modeling assumptions, AEM has shown potential to capture many of the preponderant features of the turbulent boundary layer (TBL) such as the log layer and Reynolds stress distributions. Other models, such as the stochastic one-dimensional turbulence (ODT) model from A. R. Kerstein, J. Fluid Mech. 392, 277 (1999) are built on a completely different set of modeling assumptions, yet show similarly convincing agreement when modeling the TBL. In this work, we rewrite the ODT formulation as a Markov process and simplify some modeling assumptions, which allows us to recast the equations into a form analogous to AEM. By distilling and simplifying ODT, we highlight the implicit similarities in the modeling assumptions found in AEM, which underscores the importance of (i) the eddy-probability estimation and (ii) the functions associated with the displacement and velocity induced by the eddy events. Comparison of the stochastic ODT model with a structural AEM model helps to identify the key features in TBL modeling along with some of the shortcomings in each of these models.

In dense suspensions, both rigid particles and deformable red blood cells (RBCs) exhibit a tendency to migrate away from the walls and towards the centerline of the vessel in which they flow. Here we experimentally investigate the situation in which both particles and RBCs flow through bifurcating vessels, which is particularly relevant for targeted drug delivery. In particular, we focus on the effects of RBC concentration (hematocrit) and particle volume fraction. Via high-speed imaging and Lagrangian tracking, we locate and track tens of thousands of microspheres as they flow through microfluidic bifurcating channels of arteriole size, in physiological ranges of hematocrit and shear rate. Our findings show that the margination behavior of the spherical particles is quantitatively controlled by the RBC-to-particle volume ratio, i.e., the ratio between the hematocrit and the particle volume fraction. As this increases, interactions with RBCs induce significant particle velocity fluctuations and enhance their diffusivity. While the deformable RBCs populate the center of the vessel, particles marginate and form layers adjacent to the sidewalls of the parent branch, lining the lateral sides of the bifurcation and child branches. Despite this tendency, a particle-depleted layer, of thickness comparable to the particle radius, remains at the wall. This behavior is opposite to that when suspensions are devoid of RBCs, where particles concentrate in the center of the parent branch and at the medial side of the child branches.

We discuss the hydrodynamics of the radial expansion of an electric discharge (spark) channel at the early stage of the discharge process. Due to Joule heating, the strongly ionized gas in the channel expands and forms a cylindrical shock wave that, at the initial stage, controls its expansion. We show that such dynamics can be described approximately in terms of a self-similar solution to the hydrodynamics equations. We analyze the solutions in two regimes: for an ionizing shock front and for a nonionizing shock front. In the latter case, we find that the shock wave is followed by a sharp ionization wave that represents a sharp boundary between the strongly ionized gas (i.e., plasma) in the bulk of the channel and an outer shell region, where the gas is dense and relatively cold. We also analyze, qualitatively, the effects related to the heat transfer in the spark channel and discuss the influence of the heat production on the hydrodynamic profiles and the channel's expansion rate. We argue that the results of this paper can be used to assign the initial conditions for large-scale simulations of strong discharges in air gaps.

If a denser solution of a solute A lies above a less dense solution of a solute B in the gravity field, the Rayleigh-Taylor (RT) instability can trigger convective motions which enhance the mixing of the two solutions. Double-Diffusion (DD) and Diffusive-Layer-Convection (DLC) regimes can also develop in the case of an initially stable stratification, provided A and B diffuse at different rates. Here, we analyze numerically for bulk flows described by the Navier-Stokes equation coupled to evolution equations for the concentration of species A and B how the buoyancy-driven mixing induced by such instabilities can be affected by differential diffusion effects. We quantify the length of the mixing zone as well as the onset time and the growth rate of the instability as a function of the two parameters of the problem, namely, the buoyancy ratio $R$ and the ratio $\delta$ of diffusion coefficients. We show that, when nonmonotonic density profiles with a dynamic density difference $\mathrm{\Delta }{\rho }_m$ are triggered by differential diffusion, convection can be more intense and appear faster than in the single-solute case. We derive the scalings of the growth rate $\alpha$ of convective plumes and of the onset time of the instability as a function of $\mathrm{\Delta }{\rho }_0$ and $\mathrm{\Delta }{\rho }_m$ both in the RT regime affected by differential diffusion effects and in the pure DD or DLC regimes. These scalings allow to control how two miscible solutions can buoyantly mix by a careful choice of the two solutes involved.

Compound drops impacting on a solid surface play an important role in industrial applications, such as combustion, the food industry, and drug encapsulation. An intriguing phenomenon associated with this process is the occurrence of singular jets that are up to dozens of times faster than the impact velocity. These jets break into microdroplets, which can produce aerosols and affect the quality of printing technologies. The collapsing dynamics directly determine the cavity geometry, jet velocity, and entrapments. Here, we investigate experimentally and numerically the jetting process after a coaxial water-in-oil compound drop impacts onto a glass substrate with different releasing heights and volumetric ratios. After impact, the water core spreads and retracts, giving rise to a vertical jet initially made of oil. For certain values of the impacting velocity, high-speed and very thin jets are observed, the so-called singular jets. Depending on the volumetric ratio, one or two velocity peaks can be observed when varying the impact velocity, triggered by the contraction dynamics of a deep and cylindrical cavity. The self-similar time-evolution of the collapse for the first singularity regime follows a $1/2$ power law in time, which can be derived from bubble pinch-off. In contrast, the collapse at the second peak follows a $2/3$ power law, which can be accounted for by a balance between inertial and capillary forces.

Generating an underwater shock wave with well-controlled intensity and curvature radius is important for relevant experimental and numerical studies but still challenging. The present study employs the stiffened-gas equation of state (SG-EOS) for liquid water and proposes a method of theoretically designing a shock tube to transform a planar underwater shock wave into a cylindrical one with the preset intensity and curvature. Two fundamental relations, including the Chester-Chisnell-Whitham relation and the two-dimensional characteristic relation of shock dynamics, are formulated using SG-EOS. The design procedure of the shock tube profile is thus derived. The present study conducts typical numerical tests to examine the effectiveness of the proposed method. To evaluate the errors between numerical shocks and their theoretical cylindrical shapes, the normalized ${\mathcal{L}}^2$ and ${\mathcal{L}}^{\infty }$ distances are defined and found to be controlled within $O\left({10}^{-3}\right)$, or even $O\left({10}^{-4}\right)$ in some cases. And the relative errors of the postshock pressure and velocity magnitude can be controlled within 2%, or even 0.7% in some cases. Pressure profiles are also presented and found to continuously approach those of an ideal planar shock wave as the curvature radius increases.

Data-driven subgrid-scale (SGS) modeling in the large-eddy simulations (LESs) suffers from an inconsistency between the a priori tests and the a posteriori tests, which make training accurate SGS models a difficult task. We study the difference in filtered high-fidelity data and LESs to identify the numerical deviation between the two cases, which is a combined impact of commutation error, numerical errors, and error coupling. The impact of the numerical deviation is examined through two SGS model formulations: the eddy-viscosity and the complex nonlinear models. By incorporating numerical deviations into model training, we enhance consistency, stabilize simulations, and improve predictions of energy spectra in a posteriori tests. Our findings highlight that data-driven methods introduce significant nonlinearity and equation coupling, exacerbating inconsistencies compared to non-data-driven approaches. Finally, while the impact of the numerical deviation can be generalized, achieving accurate model predictions necessitates a physically grounded model form and an optimal filter width.

Gauss's principle of least constraint transforms a dynamics problem into a pure minimization problem, where the total magnitude of the constraint force is the cost function, minimized at each instant. A candidate motion whose evolution minimizes the Gaussian cost at each instant is guaranteed to satisfy Newton's equation. In other words, Newton's equation is the first-order necessary condition for minimizing the Gaussian cost, subject to the given kinematic constraints. The principle of minimum pressure gradient (PMPG) is to incompressible fluid mechanics what Gauss's principle is to particle mechanics. The PMPG asserts that an incompressible flow evolves from one instant to another by minimizing the $L^2$-norm of the pressure gradient force. A candidate flow field whose evolution minimizes the pressure gradient cost at each instant is guaranteed to satisfy the Navier-Stokes equation. Consequently, the PMPG transforms the incompressible fluid mechanics problem into a pure minimization framework, allowing one to determine the evolution of the flow field by solely focusing on minimizing the cost—without directly invoking the Navier-Stokes equation. In this paper, we show that the resulting minimization problem is a convex quadratic programming (QP) problem—one of the most computationally tractable classes in nonlinear optimization, which has a rich literature with many efficient algorithms. Moreover, leveraging tools from analytical mechanics and the Moore-Penrose theory of generalized inverses, we derive an analytical solution for this QP problem. As a result, we present an explicit formula for the projected dynamics of the spatially discretized Navier-Stokes equation on the space of divergence-free fields. The resulting ordinary differential equation (ODE) is ready for direct time integration, eliminating the need for solving the Poisson equation in pressure at each time step. It is typically an explicit nonlinear ODE with constant coefficients. This compact form is expected to be highly valuable for both simulation and theoretical studies, including stability analysis and flow control design. We demonstrate the framework on the benchmark problem of unsteady flow in a lid-driven cavity.