arXiv subjects

Tamer A. Zaki

Publications and source records attributed to Tamer A. Zaki.

At least 19 recordsLinked to original sources

arXiv2026

Latent-space variational data assimilation in two-dimensional turbulence

Starting from limited measurements of a turbulent flow, data assimilation (DA) attempts to estimate all the spatio-temporal scales of motion. Success is dependent on whether the system is observable from the measurements, or how much of the initial turbulent field is encoded in the available measurements. Adjoint-variational DA minimises the discrepancy between the true and estimated measurements by optimising the initial velocity or vorticity field (the `state space'). Here we propose to instead optimise in a lower-dimensional latent space which is learned by implicit rank minimising autoencoders. Assimilating in latent space, rather than state space, redefines the observability of the measurements and identifies the physically meaningful perturbation directions which matter most for accurate prediction of the flow evolution. When observing coarse-grained measurements of two-dimensional Kolmogorov flow at moderate Reynolds numbers, the proposed latent-space DA approach estimates the full turbulent state with a relative error improvement of two orders of magnitude over the standard state-space DA approach. The small scales of the estimated turbulent field are predicted more faithfully with latent-space DA, greatly reducing erroneous small-scale velocities typically introduced by state-space DA. Furthermore, latent-space DA is demonstrated to be robust to noisy measurements at the range of Reynolds numbers considered. These findings demonstrate that the observability of the system from available data can be greatly improved when turbulent measurements are assimilated in the right space, or coordinates.

physics.flu-dyn↗

arXiv2026

Neural ensemble Kalman filter: Data assimilation for compressible flows with shocks

Data assimilation (DA) for compressible flows with shocks is challenging because many classical DA methods generate spurious oscillations and nonphysical features near uncertain shocks. We focus here on the ensemble Kalman filter (EnKF). We show that the poor performance of the EnKF may be attributed to the bimodal forecast distribution that can arise in the vicinity of an uncertain shock location; this violates the assumptions underpinning the EnKF, which assume a forecast which is close to Gaussian. To address this issue we introduce the new neural EnKF. The basic idea is to systematically embed neural function approximations within ensemble DA by mapping the forecast ensemble of shocked flows to the parameter space (weights and biases) of a deep neural network (NN) and to subsequently perform DA in that space. The nonlinear mapping encodes sharp and smooth flow features in an ensemble of NN parameters. Neural EnKF updates are therefore well-behaved only if the NN parameters vary smoothly within the neural representation of the forecast ensemble. We show that such a smooth variation of network parameters can be enforced via physics-informed transfer learning, and demonstrate that in so-doing the neural EnKF avoids the spurious oscillations and nonphysical features that plague the EnKF. The applicability of the neural EnKF is demonstrated through a series of systematic numerical experiments with the inviscid Burgers' equation, the Sod shock tube, and a two-dimensional blast wave.

physics.flu-dyn↗

arXiv2026

The Level Set Ensemble Kalman Filter: Sequential Data Assimilation for Flows With Shocks

The ensemble Kalman filter is commonly used to perform sequential data assimilation in science and engineering because of its robust performance and scalability when the state space dimension is high. However, the standard EnKF generates spurious oscillations when applied to compressible flows with shocks; the cause is the uncertainty, across the ensemble, in the shock location. This paper develops the \emph{level set EnKF}, which overcomes this challenge by employing a nonlinear mapping from the original state space into a latent space in which the analysis step is undertaken. The latent representation comprises a level set function that encodes the discontinuity location and smooth state extensions, defined over the full physical domain, that represent the solution on either side of the discontinuity. We first formulate the level set EnKF as a specific instance of a more general approach of performing data assimilation in a latent space. We then demonstrate the feasibility of the level set EnKF for several one-dimensional compressible flows and a two-dimensional blast wave.

physics.flu-dyn↗

arXiv2026

Mitigating adjoint chaos in wall turbulence

Estimating past events in wall turbulence based solely on surface measurements and first principles is an ill-posed problem that is complicated by chaos. The sensitivity of a measurement to the earlier flow state is described by the adjoint Navier-Stokes equations, which are solved in reverse time starting from the measurement kernel at the sensing position and time. The resulting adjoint field is the spatio-temporal domain of dependence (DOD) of the sensor, which is a dual to the concept of the domain of influence (DOI) of an actuator in the linearized forward equations. In channel turbulence, the energy of each adjoint realization grows exponentially in backward time according to the Lyapunov exponent, even though the energy of the ensemble average should decay. We introduce a linear eddy-viscosity closure model in the ensemble-averaged adjoint equations, and directly compute the mean DOD and compare our prediction to the ensemble average. Furthermore, we demonstrate that the DOD of a wall-stress measurement and the DOI resulting from a wall-stress perturbation exhibit respective universal behaviors across Reynolds numbers. However, their spatio-temporal structures differ qualitatively, due to the time-asymmetry of the governing equations. The DOD field has a two-part structure: one component is associated with the Orr mechanism, characterized by rapid reorientation under mean shear, and the other is related to self-similar expanding streaky structures. These two components jointly define the sensitivity of the wall-stress measurement to past flow events.

physics.flu-dyn↗

arXiv2026

Feature-preserving Latent-EnKF for Data Assimilation of Flows with Shocks

The ensemble Kalman filter (EnKF) is widely adopted for sequential data assimilation, but fails for solutions with discontinuities, such as shocks in compressible flows. Uncertainty in shock location induces multimodal ensemble statistics that violate the Gaussian assumptions underlying the EnKF, producing large-scale spurious oscillations in the analysis state. We introduce a feature-preserving latent-EnKF that performs the ensemble update in a learned low-dimensional latent space, where shock and flow features admit a smooth manifold representation, thereby preserving sharp features during EnKF analysis. The updated latent state is mapped back to physical state through a shared decoder for all ensemble members. The algorithm eliminates the member-specific ordered training and positivity flooring used in prior approaches. Numerical experiments on a Sod shock tube and Mach 2 shock interaction with a 2D cylinder, using sparse and noisy observations, show accurate feature recovery of shocks and contact discontinuities without spurious oscillations.

physics.comp-ph↗

arXiv2026

Assimilation of wall-pressure measurements in direct numerical simulations of high-speed flow over a cone-flare geometry

Ensemble-variational (EnVar) assimilation of wall-pressure measurements in direct numerical simulations of Mach 6 flow over a cone-flare is performed. The experimental data include pressure spectra and intensities from seven wall-mounted PCB sensors positioned upstream, within, and downstream of the separation region induced by the compression corner. Assimilation of the first two sensors only, all upstream of separation, is insufficient to accurately predict the downstream flow. Assimilating all the sensor data is shown to be essential to correctly predict separation onset and the downstream wall-pressure data. Similar to the experiments, the assimilated flow features intense rope-like structures in the attached region. The simulations additionally predict a localized amplification of disturbances beneath the separation shock, where experimental data are not available. This amplification results from the interaction of the boundary-layer instability modes with the compression shock. The simulations also capture the sharp decrease in wall-pressure intensity across separation, and the amplification of low-frequency three-dimensional disturbances within the recirculation bubble. Additionally, the computations highlight the uncertainty in the post-separation predictions due to the low-frequency unsteadiness of the separation shock. Oscillations of the streamwise velocity modulate the boundary-layer thickness, which in turn introduces variability in disturbance amplification.

physics.flu-dyn↗

arXiv2025

Accelerating Hamiltonian Monte Carlo for Bayesian Inference in Neural Networks and Neural Operators

Hamiltonian Monte Carlo (HMC) is a powerful and accurate method to sample from the posterior distribution in Bayesian inference. However, HMC techniques are computationally demanding for Bayesian neural networks due to the high dimensionality of the network's parameter space and the non-convexity of their posterior distributions. Therefore, various approximation techniques, such as variational inference (VI) or stochastic gradient MCMC, are often employed to infer the posterior distribution of the network parameters. Such approximations introduce inaccuracies in the inferred distributions, resulting in unreliable uncertainty estimates. In this work, we propose a hybrid approach that combines inexpensive VI and accurate HMC methods to efficiently and accurately quantify uncertainties in neural networks and neural operators. The proposed approach leverages an initial VI training on the full network. We examine the influence of individual parameters on the prediction uncertainty, which shows that a large proportion of the parameters do not contribute substantially to uncertainty in the network predictions. This information is then used to significantly reduce the dimension of the parameter space, and HMC is performed only for the subset of network parameters that strongly influence prediction uncertainties. This yields a framework for accelerating the full batch HMC for posterior inference in neural networks. We demonstrate the efficiency and accuracy of the proposed framework on deep neural networks and operator networks, showing that inference can be performed for large networks with tens to hundreds of thousands of parameters. We show that this method can effectively learn surrogates for complex physical systems by modeling the operator that maps from upstream conditions to wall-pressure data on a cone in hypersonic flow.

stat.ML↗

arXiv2025

Lighthill's mechanism and vorticity cascade in the logarithmic layer of wall turbulence

We investigate Lighthill's proposed turbulent mechanism for near-wall concentration of spanwise vorticity by calculating mean flows conditioned on motion away from or toward the wall in an $Re_τ=1000$ database of plane-parallel channel flow. Our results corroborate Lighthill's proposal throughout the entire logarithmic layer, but extended by counterflows that help explain anti-correlation of vorticity transport by advection and by stretching/tilting. We present evidence also for Lighthill's hypothesis that the vorticity transport in the log-layer is a ``cascade process'' through a scale-hierarchy of eddies, with intense competition between transport outward from and inward to the wall. Townsend's model of attached eddies of hairpin-vortex type accounts for half of the vorticity cascade, whereas we identify necklace-type or ``shawl vortices'' that envelop turbulent sweeps as supplying the other half.

physics.flu-dyn↗

arXiv2025

Vorticity dynamics and drag for flows over a sphere and a prolate spheroid

The connection between the drag and vorticity dynamics for viscous flow over a bluff body is explored using the Josephson-Anderson (J-A) relation for classical fluids. The instantaneous rate of work on the fluid, associated with the drag force, is related to the vorticity flux across the streamlines of a background potential flow. The vorticity transport itself is examined by aid of the Huggins vorticity flux tensor. The analysis is performed for three flows: flow over a sphere at Reynolds numbers $Re=\{200,3700\}$ and flow over a prolate spheroid at $Re=3000$ and $20^{\circ}$ incidence. In these flows, the vorticity transport shifts the flow away and towards the ideal potential flow, with a net balance towards the former effect thus making an appreciably contribution to the drag. The J-A relation is first demonstrated for the flow over a sphere at $Re=200$. The drag is related to the viscous flux of azimuthal vorticity from the wall into the fluid and the advection of vorticity by the shear layer. In the wake, the azimuthal vorticity is advected towards the wake centerline and is annihilated by viscous effects, which contributes a reduction to drag. The analysis of the flow over a sphere at $Re=3700$ is reported for the impulsively started and stationary stages, with emphasis on the effects of unsteady separation and turbulent transport in the wake. The turbulent flux in the wake enhances the transport of mean azimuthal vorticity towards the wake centerline, and is the driver of the recovery of enthalpy downstream. The drag force for a prolate spheroid is mostly due to the transport of vorticity along the separated boundary layers. Primary and secondary separation contribute oppositely to the drag force, while the large-scale vortices only re-distribute vorticity. A mechanism for secondary separation is proposed based on the theory of vortex-induced separation.

physics.flu-dyn↗

arXiv2025

Lagrangian Finite-Time Fluctuation Relation in isotropic turbulence

The entropy generation rate in turbulence can be defined using the energy cascade rate as described in the scale-integrated Kolmogorov-Hill equation at a specified length scale. The fluctuation relation (FR) from non-equilibrium thermodynamics, which predicts exponential behaviour of the ratio of probability densities for positive and negative entropy production rates, was confirmed in prior work \citep{yao2023entropy}, but under certain limiting assumptions. We here examine the applicability of FR to isotropic turbulence under less stringent assumptions by analyzing entropy generation rates averaged over intervals ranging from one to several eddy turnover times. Based on time-resolved data at a Taylor-scale based Reynolds number $Re_λ= 433$, we find that the FR is valid in the sense that very close to exponential behaviour of probability ratios of positive and negative entropy generation (forward and inverse cascade of energy) is observed. Interestingly, finite-time averaging yields FR-consistent results only within a Lagrangian framework, along fluid trajectories using filtered convective velocities. In contrast, the FR does not hold with time-averaging at fixed (Eulerian) positions. Results provide evidence that the definition of entropy generation based on the scale-integrated Kolmogorov-Hill equation describes turbulent cascade processes that exhibit properties predicted by non-equilibrium thermodynamics.

physics.flu-dyn↗

arXiv2025

The origin of vorticity in viscous incompressible flows

In inviscid, incompressible flows, the evolution of vorticity is exactly equivalent to that of an infinitesimal material line-element, and hence vorticity can be traced forward or backward in time in a Lagrangian fashion. This elegant and powerful description is not possible in viscous flows due to the action of diffusion. Instead, a stochastic Lagrangian interpretation is required and was recently introduced, where the origin of vorticity at a point is traced back in time as an expectation over the contribution from stochastic trajectories. We herein introduce for the first time an Eulerian, adjoint-based approach to quantify the back-in-time origin of vorticity in viscous, incompressible flows. The adjoint variable encodes the advection, tilting and stretching of the earlier-in-time vorticity that ultimately leads to the target value. Precisely, the adjoint vorticity is the volume-density of the mean Lagrangian deformation of the earlier vorticity. The formulation can also account for the injection of vorticity into the domain at solid boundaries. We demonstrate the mathematical equivalence of the adjoint approach and the stochastic Lagrangian approach. We then provide an example from turbulent channel flow, where we analyze the origin of high-stress events and relate them to Lighthill's mechanism of stretching of near-wall vorticity.

physics.flu-dyn↗

arXiv2025

Variational data assimilation in wall turbulence: From outer observations to wall stress and pressure

Estimation of near-wall turbulence in channel flow from outer observations is investigated using adjoint-variational data assimilation. We first consider fully resolved velocity data, starting at a distance from the wall. By enforcing the estimated flow to exactly satisfy the Navier-Stokes equations, we seek a statistically stationary turbulent state that reproduces the instantaneous outer measurements. Such an estimated state provides full access to the unknown near-wall turbulence, including the wall shear stresses and pressure. When the first observation is within fifty viscous units from the wall, the correlation coefficient between the true and estimated state exceeds 95%. As the observations are further separated from the wall, at ninety viscous units, the accuracy of the assimilated wall stresses decreases to 40% at the wall. This trend is nearly independent of the Reynolds number. The Fourier spectrum of the estimation error is qualitatively consistent with the coherence spectrum between the outer and the inner state variables: observed long wavelength structures in the outer flow have deeper coherence into the unobserved near-wall region, and therefore the error is lowest at large scales. Nevertheless, the adjoint-variational approach provides a more rigorous quantification of the capacity to accurately predict the instantaneous near-wall turbulence from outer measurements. Lastly, we demonstrate the robustness of the estimation accuracy using filtered and sub-sampled outer observations.

physics.flu-dyn↗

arXiv2024

Multi evolutional deep neural networks (Multi-EDNN)

Evolutional deep neural networks (EDNN) solve partial differential equations (PDEs) by marching the network representation of the solution fields, using the governing equations. Use of a single network to solve coupled PDEs on large domains requires a large number of network parameters and incurs a significant computational cost. We introduce coupled EDNN (C-EDNN) to solve systems of PDEs by using independent networks for each state variable, which are only coupled through the governing equations. We also introduce distributed EDNN (D-EDNN) by spatially partitioning the global domain into several elements and assigning individual EDNNs to each element to solve the local evolution of the PDE. The networks then exchange the solution and fluxes at their interfaces, similar to flux-reconstruction methods, and ensure that the PDE dynamics are accurately preserved between neighboring elements. Together C-EDNN and D-EDNN form the general class of Multi-EDNN methods. We demonstrate these methods with aid of canonical problems including linear advection, the heat equation, and the compressible Navier-Stokes equations in Couette and Taylor-Green flows.

math.NA↗

arXiv2023

Targeted computation of nonlocal closure operators via an adjoint-based macroscopic forcing method

Reynolds-averaged Navier--Stokes (RANS) closure must be sensitive to the flow physics, including nonlocality and anisotropy of the effective eddy viscosity. Recent approaches used forced direct numerical simulations to probe these effects, including the macroscopic forcing method (MFM) of Mani and Park ($\textit{Phys. Rev. Fluids}$ $\textbf{6}$, 054607 (2021)) and the Green's function approach of Hamba ($\textit{Phys. Fluids}$ $\textbf{17}$, 115102 (2005)). The resulting nonlocal and anisotropic eddy viscosities are exact and relate Reynolds stresses to mean velocity gradients at all locations. They can be used to inform RANS models of the sensitivity to the mean velocity gradient and the suitability of local and isotropic approximations. However, these brute-force approaches are expensive. They force the mean velocity gradient at each point in the averaged space and measure the Reynolds stress response, requiring a separate simulation for each mean velocity gradient location. Thus, computing the eddy viscosity requires as many simulations as degrees of freedom in the averaged space, which can be cost-prohibitive for problems with many degrees of freedom. In this work, we develop an adjoint-based MFM to obtain the eddy viscosity at a given Reynolds stress location using a single simulation. This approach recovers the Reynolds stress dependence at a location of interest, such as a separation point or near a wall, on the mean velocity gradient at all locations. We demonstrate using adjoint MFM to compute the eddy viscosity for a specified wall-normal location in an incompressible turbulent channel flow using one simulation. In contrast, a brute-force approach for the same problem requires $N=144$ simulations (the number of grid points in the non-averaged coordinate direction). We show that a local approximation for the eddy viscosity would have been inappropriate.

physics.flu-dyn↗

arXiv2023

Optimal two-dimensional roughness for transition delay in high-speed boundary layer

The influence of surface roughness on transition to turbulence in a Mach 4.5 boundary layer is studied using direct numerical simulations. Transition is initiated by the nonlinearly most dangerous inflow disturbance, which causes the earliest possible breakdown on a flat plate for the prescribed inflow energy and Mach number. This disturbance is primarily comprised of two normal second-mode instability waves and an oblique first mode. When localized roughness is introduced, its shape and location relative to the synchronization points of the inflow waves are confirmed to have a clear impact on the amplification of the second-mode instabilities. The change in modal amplification coincides with the change in the height of the near-wall region where the instability wave-speed is supersonic relative to the mean flow; the net effect of a protruding roughness is destabilizing when placed upstream of the synchronization point and stabilizing when placed downstream. Assessment of the effect of the roughness location is followed by an optimization of the roughness height, abruptness and width with the objective of achieving maximum transition delay. The optimization is performed using an ensemble-variational (EnVar) approach, while the location of the roughness is fixed upstream of the synchronization points of the two second-mode waves. The optimal roughness disrupts the phase of the near-wall pressure waves, suppresses the amplification of the primary instability waves, and mitigates the nonlinear interactions that lead to breakdown to turbulence. The outcome is a sustained non-turbulent flow throughout the computational domain.

physics.flu-dyn↗

arXiv2023

Spatio-temporal Signatures of Elasto-inertial Turbulence in Viscoelastic Planar Jets

The interplay between viscoelasticity and inertia in dilute polymer solutions at high deformation rates can result in inertio-elastic instabilities. The nonlinear evolution of these instabilities generates a state of turbulence with significantly different spatio-temporal features compared to Newtonian turbulence, termed elasto-inertial turbulence (EIT). We explore EIT by studying the dynamics of a submerged planar jet of a dilute aqueous polymer solution injected into a quiescent tank of water using a combination of schlieren imaging and laser Doppler velocimetry (LDV). We show how fluid elasticity has a nonmonotonic effect on the jet stability depending on its magnitude, creating two distinct regimes in which elastic effects can either destabilize or stabilize the jet. In agreement with linear stability analyses of viscoelastic jets, an inertio-elastic shear-layer instability emerges near the edge of the jet for small levels of elasticity, independent of bulk undulations in the fluid column. The growth of this disturbance mode destabilizes the flow, resulting in a turbulence transition at lower Reynolds numbers and closer to the nozzle compared to the conditions required for the transition to turbulence in a Newtonian jet. Increasing the fluid elasticity merges the shear-layer instability into a bulk instability of the jet column. In this regime, elastic tensile stresses generated in the shear layer act as an "elastic membrane" that partially stabilizes the flow, retarding the transition to turbulence to higher levels of inertia and greater distances from the nozzle. In the fully turbulent state far from the nozzle, planar viscoelastic jets exhibit unique spatio-temporal features associated with EIT. The time-averaged angle of jet spreading, an Eulerian measure of the degree of entrainment, and the centerline velocity of the jets both evolve self-similarly with distance from the nozzle. The autocovariance of the schlieren images in the fully turbulent region of the jets shows coherent structures that are elongated in the streamwise direction, consistent with the suppression of streamwise vortices by elastic stresses. These coherent structures give a higher spectral energy to small frequency modes in EIT characterized by LDV measurements of the velocity fluctuations at the jet centerline. Finally, our LDV measurements reveal a frequency spectrum characterized by a $-3$ power-law exponent, different from the well-known $-5/3$ power-law exponent characteristic of Newtonian turbulence.

physics.flu-dyn↗

arXiv2022

State estimation in minimal turbulent channel flow: A comparative study of 4DVar and PINN

The state of turbulent, minimal-channel flow is estimated from spatio-temporal sparse observations of the velocity, using both a physics-informed neural network (PINN) and adjoint-variational data assimilation (4DVar). The performance of PINN is assessed against the benchmark results from 4DVar. The PINN is efficient to implement, takes advantage of automatic differentiation to evaluate the governing equations, and does not require the development of an adjoint model. In addition, the flow evolution is expressed in terms of the network parameters which have a far smaller dimension than the predicted trajectory in state space or even just the initial condition of the flow. Provided adequate observations, network architecture and training, the PINN can yield satisfactory estimates of the the flow field, both for the missing velocity data and the entirely unobserved pressure field. However, accuracy depends on the network architecture, and the dependence is not known a priori. In comparison to 4DVar estimation which becomes progressively more accurate over the observation horizon, the PINN predictions are generally less accurate and maintain the same level of errors throughout the assimilation time window. Another notable distinction is the capacity to accurately forecast the flow evolution: while the 4DVar prediction depart from the true flow state gradually and according to the Lyapunov exponent, the PINN is entirely inaccurate immediately beyond the training time horizon unless re-trained. Most importantly, while 4DVar satisfies the discrete form of the governing equations point-wise to machine precision, in PINN the equations are only satisfied in an $L^2$ sense.

physics.flu-dyn↗

arXiv2022

Assimilation of wall-pressure measurements in high-speed flow over a cone

A nonlinear ensemble-variational (EnVar) data assimilation is performed in order to estimate the unknown flow field over a slender cone at Mach-6, from isolated wall-pressure measurements. The cost functional accounts for discrepancies in wall-pressure spectra and total intensity between the experiment and the prediction using direct numerical simulations (DNS), as well as our relative confidence in the measurements and the estimated state. We demonstrate the robustness of the predicted flow by direct propagation of posterior statistics. The approach provides a unique first look at the flow beyond the sensor data, and rigorously accounts for the role of nonlinearity unlike previous efforts that adopted ad-hoc inflow syntheses. Away from the wall, two- and three-dimensional assimilated states both show rope-like structures, qualitatively similar to independent schlieren visualizations. Despite this resemblance, and even though the planar second modes are the most unstable upstream, three-dimensional (3D) waves must be included in the assimilation in order to accurately reproduce the wall-pressure measurements recorded in the Ludwieg-Tube facility. The results highlight the importance of three-dimensionality of the field and of the base-state distortion on the instability waves in this experiment, and motivate future measurements that probe the 3D nature of the flow field.

physics.flu-dyn↗