arXiv subjects

Gustaaf B. Jacobs

Publications and source records attributed to Gustaaf B. Jacobs.

18 recordsLinked to original sources

arXiv2025

Robust Spectral Solver for High-Fidelity Investigations of Aerospike Nozzle Flow Dynamics

A spectral element solver is developed for the high-fidelity simulation of the unsteady flow over an aerospike nozzle. The Navier-Stokes solver is a kinetic-energy-preserving, discontinuous Galerkin spectral element method (DGSEM) combined with a hybridization of an entropy viscosity (EV) and a finite-volume subcell element (FVSE) shock-capturing scheme. The diffusive FVSE method is locally called only at locations where the EV method cannot sufficiently smooth the sharp solution gradients that suddenly appear in the supersonic, vortex-dominated jet generated by the aerospike nozzle. Two-dimensional tests of a perfectly expanded and an underexpanded nozzle flow demonstrate that the method is high-order accurate and captures unsteady flow phenomena at supersonic and hypersonic conditions. A resolved three-dimensional simulation at a Reynolds number of 95,000 shows that the solver implicitly models turbulent dissipation at the subgrid scales. To the authors' knowledge, these simulations represent the first DGSEM computations of the resolved, unsteady flow over an aerospike nozzle.

physics.flu-dyn↗

arXiv2025

Data-driven nonlinear aerodynamics models with certifiably optimal boundedness properties

Obtaining predictive low-order models is a central challenge in fluid dynamics. Data-driven frameworks have been widely used to obtain low-order models of aerodynamic systems; yet, resulting models tend to yield predictions that grow unbounded with time. Recently introduced stability-promoting methods can facilitate the identification of bounded models, but tend to require extensive brute-force tuning even in the context of simple academic systems. Here, we show how recent theoretical advances in the long-term boundedness of dynamical systems can be integrated into data-driven modeling frameworks to ensure that resulting models will yield bounded predictions of incompressible flows. Specifically, we propose to solve a specific set of convex semi-definite programming problems to (i) certify whether a system admits a globally attracting bounded set for the chosen modeling parameters, and (ii) compute a model with the optimal (tightest) bound on this globally attracting set. We demonstrate the approach via integration within the sparse identification of nonlinear dynamics (SINDy) modeling framework. Application on two low-order benchmark problems establishes the merits of the approach. We then apply our approach to obtain a low-order (6-mode) model of unsteady separation over a NACA-65(1)-412 airfoil at Re = 20, 0000, a flow that has been notoriously difficult to model using data-driven methods. The resulting model is found to accurately predict the dynamics of unsteady separation, with model predictions remaining bounded indefinitely. We anticipate this work will benefit future efforts in modeling strongly nonlinear flows, especially in settings where physically-viable long-term forecasts are paramount.

physics.flu-dyn↗

arXiv2025

The Footprint of Laminar Separation on a Wall-Bounded Wing Section at Transitional Reynolds Numbers

When a chordwise Reynolds number (Re) falls below about $10^5$ the performance of wings and aerodynamic sections become sensitive to viscous phenomena, including boundary layer separation and possible reattachment. Here, detailed measurements of the flow inside the boundary layer on the suction surface are shown for an aspect ratio 3 wing with wall boundaries. The separation lines and recirculation zones are shown on the wing and on the wall junction as Re and angle of incidence, ($α$) are varied. There is good agreement on the lowest Re case which has also been computed in direct numerical simulation. Though the flow at midspan may sometimes be described as two-dimensional, at $α\leq 6^\circ$ it is unrepresentative of the remainder of the wing, and the influence of the wall is seen in strong spanwise flows aft of the separation line. The geometry of the NACA 65(1)-412 section, used here, promotes a substantial chord length for the development of the recirculating regions behind separation making it apt for their study. However, the phenomena themselves are likely to be found over a wide range of wings with moderate thickness at moderate $α$.

physics.flu-dyn↗

arXiv2025

Lagrangian Liouville models of multiphase flows with randomly forced inertial particles

Eulerian-Lagrangian models of particle-laden (multiphase) flows describe fluid flow and particle dynamics in the Eulerian and Lagrangian frameworks respectively. Regardless of whether the flow is turbulent or laminar, the particle dynamics is stochastic because the suspended particles are subjected to random forces. We use a polynomial chaos expansion (PCE), rather than a postulated constitutive law, to capture structural and parametric uncertainties in the particles' forcing. The stochastic particle dynamics is described by a joint probability density function (PDF) of a particle's position and velocity and random coefficients in the PCE. We deploy the method of distributions (MoD) to derive a deterministic (Liouville-type) partial-differential equation for this PDF. We reformulate this PDF equation in a Lagrangian form, obtaining PDF flow maps and tracing events and their probability in the phase space. That is accomplished via a new high-order spectral scheme, which traces, marginalizes and computes moments of the high-dimensional joint PDF and comports with high-order carrier-phase solvers. Our approach has lower computational cost than either high-order Eulerian solvers or Monte Carlo methods, is not subjected to a CFL condition, does not suffer from Gibbs oscillations and does not require (order-reducing) filtering and regularization techniques. These features are demonstrated on several test cases.

physics.flu-dyn↗

arXiv2024

Liouville models of particle-laden flow

Langevin (stochastic differential) equations are routinely used to describe particle-laden flows. They predict Gaussian probability density functions (PDFs) of a particle's trajectory and velocity, even though experimentally observed dynamics might be highly non-Gaussian. Our Liouville approach overcomes this dichotomy by replacing the Wiener process in the Langevin models with a (small) set of random variables, whose distributions are tuned to match the observed statistics.

math-ph↗

arXiv2023

SPARSE-R: A point-cloud tracer with random forcing

A predictive, point-cloud tracer is presented that determines with a quantified uncertainty the Lagrangian motion of a group of point-particles within a finite region. The tracer assumes a random forcing within confidence intervals to account for the empiricism of data-driven force models and stochasticity related to the chaotic nature of the subcloud scale dynamics. It builds on the closed Subgrid Particle-Averaged Reynolds Stress-Equivalent (SPARSE) formulation presented in Domínguez-Vázquez~\textit{et al.} [\textit{Int. J. Multiph. Flow.} 161, 104375] that assumes a deterministic forcing. SPARSE describes the first two moments of particle clouds with moment equations in closed-form, with a theoretical third-order convergence rate with respect to the standard deviations of the cloud variables. The cloud model alleviates computational cost and enhances the convergence rate as compared to Monte Carlo (MC) based point-particle methods. The randomness in the forcing model leads to virtual stresses that correlate random forcing and field fluctuations. These stresses strain and rotate the random cloud as compared to a deterministically forced cloud and thus determine to what extent the random forcing propagates into the confidence intervals of the dispersed solution. In symmetric flows the magnitude of the virtual stress is zero. Tests in analytical carrier-fields and in a decaying homogeneous isotropic turbulence flow computed with a discontinuous Galerkin (DG) compressible DNS solver are performed to verify and validate the SPARSE method for randomly forced particles.

physics.flu-dyn↗

arXiv2023

Spike formation theory in 3D flow separation

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in three-dimensional flows with arbitrary time dependence. Based on the exact evolution of the largest principal curvature on near-wall material surfaces, our theory identifies fixed and moving separation. Our approach is effective over short time intervals and admits an instantaneous limit. As a byproduct, we derive explicit formulas for the evolution of the Weingarten map and the principal curvatures of any surface advected by general three-dimensional flows. The material backbone we identify acts first as a precursor and later as the centerpiece of Lagrangian flow separation. We discover previously undetected spiking points and curves where the separation backbones connect to the boundary and provide wall-based analytical formulas for their locations. We illustrate our results on several steady and unsteady flows.

physics.flu-dyn↗

arXiv2023

Analysis of an Explicit, High-Order Semi-Lagrangian Nodal Method

A discrete analysis of the phase and dissipation errors of an explicit, semi-Lagrangian spectral element method is performed. The semi-Lagrangian method advects the Lagrange interpolant according the Lagrangian form of the transport equations and uses a least-square fit to correct the update for interface constraints of neighbouring elements. By assuming a monomial representation instead of the Lagrange form, a discrete version of the algorithm on a single element is derived. The resulting algebraic system lends itself to both a Modified Equation analysis and an eigenvalue analysis. The Modified Equation analysis, which Taylor expands the stencil at a single space location and time instance, shows that the semi-Lagrangian method is consistent with the PDE form of the transport equation in the limit that the element size goes to zero. The leading order truncation term of the Modified Equation is of the order of the degree of the interpolant which is consistent with numerical tests reported in the literature. The dispersion relations show that the method is negligibly dispersive, as is common for semi-Lagrangian methods. An eigenvalue analysis shows that the semi-Lagrangian method with a nodal Chebyshev interpolant is stable for a Courant-Friedrichs-Lewy condition based on the minimum collocation node spacing within an element that is greater than unity.

math.NA↗

arXiv2022

Adjoint-based Particle Forcing Reconstruction and Uncertainty Quantification

The forcing of particles in turbulent environments influences dynamical properties pertinent to many fundamental applications involving particle-flow interactions. Current study explores the determination of forcing for one-way coupled passive particles, under the assumption that the ambient velocity fields are known. When measurements regarding particle locations are available but sparse, direct evaluation of the forcing is intractable. Nevertheless, the forcing for finite-size particles can be determined using adjoint-based data assimilation. This inverse problem is formulated with the framework of optimization, where the cost function is defined as the difference between the measured and predicted particle locations. The gradient of the cost function, with respect to the forcing can be calculated from the adjoint dynamics. When measurements are subject to Gaussian noise, samples within the probability distribution of the forcing can be drawn using Hamiltonian Monte Carlo. The algorithm is tested in the Arnold-Beltrami-Childress flow as well as the homogeneous isotropic turbulence. Results demonstrate that the forcing can only be determined accurately for particle Reynolds number between 1 and 5, where the majority of Reynolds number history along the particle trajectory falls in.

physics.flu-dyn↗

arXiv2022

Closed SPARSE -- a predictive particle cloud tracer

A closed and predictive particle cloud tracer method is presented. The tracer builds upon the Subgrid Particle Averaged Reynolds Stress Equivalent (SPARSE) formulation first introduced in [Davis et al., Proceedings of the Royal Society A, 473(2199), 2017] for the tracing of particle clouds. It was later extended to a Cloud-In-Cell (CIC) formulation in [Taverniers et al., Journal of Computational Physics, 390, 2019] using a Gaussian distribution of a cloud's influence over a mesh-based, velocity field solution. SPARSE corrects the cloud's trace to second order by combining a Taylor series expansion of the drag coefficient and Nusselt number correction factors around the mean relative velocity of a cloud of particles with a Reynolds decomposition of the particle equations to obtain a governing system for the first two statistical moments of the cloud's position, velocity and temperature. Here, we close the thus far unclosed SPARSE formulation by determining the velocity field in the vicinity of the mean cloud location using a truncated Taylor series velocity representation and by combining that with averaging. The resulting tracer is predictive. It enables the tracing of a cloud of particles through a single point and so reduces the required degrees of freedom in the accurate tracing of groups of particles. We demonstrate the accuracy and convergence of the method in several one-, two- and three-dimensional test cases.

physics.flu-dyn↗

arXiv2021

Direct numerical simulation of cambered airfoil aerodynamics at Re = 20,000

A comprehensive and detailed overview of the flow topology over a cambered NACA 65(1)-412 airfoil at Re = 20,000 is presented for angles of attack ranging from 0° to 10° using high-order direct numerical simulations. It is shown that instabilities bifurcate the flow and cause it to change at a critical angle of attack from laminar separation without reattachment over a laminar separation bubble at the trailing edge to a bubble at the leading edge. The transition of the flow regimes is governed by the Karman vortex shedding of the pressure side boundary layer at the trailing edge, Kelvin-Helmholtz instabilities within the separated shear layer on the suction side, as well as three-dimensional instabilities of elliptic flow within the vortex cores and hyperbolic flow in the shear layer between subsequent Karman vortices. As the suction side shear layer transitions and reattaches, the interaction of the two and three-dimensional instabilities results in three-dimensional tubular structures and large-scale turbulent puffs. The formation and shifting of the laminar separation bubble defines the far-wake topology several chord-lengths behind the airfoil and is accompanied by a sudden increase of the lift force and decrease in the drag that underscores the sensitive nature of low-Reynolds number airfoil aerodynamics. Lift and drag polars are presented for direct numerical simulations, wind tunnel experiments, and simplified numerical procedures where incorrect prediction of the force coefficients is caused by the failure to correctly model the low-pressure region at the trailing edge that is caused by the time-dependent generation of the Karman vortices.

physics.flu-dyn↗

arXiv2020

Objective early identification of kinematic instabilities in shear flows

A kinematic approach for the identification of flow instabilities is proposed. By defining a flow instability in the Lagrangian frame as the increased folding of lines of fluid particles, subtle perturbations and unstable growth thereof are detected early based solely on the curvature change of material lines over finite time. The material line curvature is objective, parametrization independent, and can be applied to flows of general complexity without knowledge of the base flow. An analytic connection between the growth of Eulerian velocity modes perturbing a general shear flow and the induced flow map and Lagrangian curvature change is derived. The approach is verified to capture instabilities promptly in a temporally developing jet flow, an unstable separated shear flow over a cambered airfoil, and in the onset of a wake instability behind a circular cylinder.

physics.flu-dyn↗

arXiv2019

An explicit semi-Lagrangian, spectral method for solution of Lagrangian transport equations in Eulerian-Lagrangian formulations

An explicit high order semi-Lagrangian method is developed for solving Lagrangian transport equations in Eulerian-Lagrangian formulations. To ensure a semi-Lagrangian approximation that is consistent with an explicit Eulerian, discontinuous spectral element method (DSEM) discretization used for the Eulerian formulation, Lagrangian particles are seeded at Gauss quadrature collocation nodes within an element. The particles are integrated explicitly in time to obtain an advected polynomial solution at the advected Gauss quadrature locations. This approximation is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values and optional constraints for mass and energy preservation. An explicit time integration is used for the semi-Lagrangian approximation that is consistent with the grid based DSEM solver, which ensures that particles seeded at the Gauss quadrature points do not leave the element's bounds. The method is hence local and parallel and facilitates the solution of the Lagrangian formulation without the grid complexity, and parallelization challenges of a particle solver in particle-mesh methods. Numerical tests with one and two dimensional advection equation are carried out. The method converges exponentially. The use of mass and energy constraints can improve accuracy depending on the accuracy of the time integration.

math.NA↗

arXiv2019

Data-Driven Selection of Actuators for Optimal Control of Airfoil Separation

We present a systematic approach for determining the optimal actuator location for separation control from input-output response data, gathered from numerical simulations or physical experiments. The Eigensystem Realization Algorithm is used to extract state-space descriptions from the response data associated with a candidate set of actuator locations. These system realizations are then used to determine the actuator location among the set that can drive the system output to an arbitrary value with minimal control effort. The solution of the corresponding minimum energy optimal control problem is evaluated by computing the generalized output controllability Gramian. We use the method to analyze high-fidelity numerical simulation data of the lift and separation-angle responses to a pulse of localized body-force actuation from six distinct locations on the upper surface of a NACA 65(1)-412 airfoil. We find that the optimal location for controlling lift is different from the optimal location for controlling separation angle. In order to explain the physical mechanisms underlying these differences, we conduct controllability analyses of the flowfield by leveraging the dynamic mode decomposition with control algorithm. These modal analyses of flowfield response data reveal that excitation of coherent structures in the wake benefit lift control; whereas, excitation of coherent structures in the shear layer benefit separation-angle control.

physics.flu-dyn↗

arXiv2019

The Kinematics of Lagrangian Flow Separation in External Aerodynamics

Kinematic aspects of flow separation in external aerodynamics are investigated in the Lagrangian frame. Specifically, the initial motion of upwelling fluid material from the wall is related to the long-term attracting manifolds in the flow field. While the short-time kinematics are governed by the formation of a material spike upstream of the zero-skin-friction point and ejection of particles in direction of the asymptotic separation line, the trajectories of the fluid tracers are guided by attracting ridges in the finite-time Lyapunov exponents once they leave the vicinity of the wall. The wall signature of this initial fluid upwelling event, the so-called \textit{spiking point} [Serra, M., Vetel, J., Haller, G., "Exact theory of material spike formation in flow separation", \textit{J. Fluid Mech.}, Vol. 845, 2018], is computed from the curvature of advected material lines and, for the first time, from high-order numerical derivatives of the wall-normal velocity obtained from direct numerical simulations of a circular cylinder and a cambered NACA 65(1)-412 airfoil. As the spline-based boundary parametrization of the airfoil profile induces oscillations, the principle spiking point can be recovered robustly through appropriate filtering. The short-term kinematics correlate strongly with the scaling lengths in the boundary layer.

physics.flu-dyn↗

arXiv2019

Method of Distributions for Systems with Stochastic Forcing

The method of distributions is developed for systems that are governed by hyperbolic conservation laws with stochastic forcing. The method yields a deterministic equation for the cumulative density distribution (CDF) of a system state, e.g., for flow velocity governed by an inviscid Burgers' equation with random source coefficients. This is achieved without recourse to any closure approximation. The CDF model is verified against MC simulations using spectral numerical approximations. It is shown that the CDF model accurately predicts the mean and standard deviation for Gaussian, normal and beta distributions of the random coefficients.

physics.comp-ph↗

arXiv2019

Multi-element SIAC filter for shock capturing applied to high-order discontinuous Galerkin spectral element methods

We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (B.W. Wissink, G.B. Jacobs, J.K. Ryan, W.S. Don, and E.T.A. van der Weide. Shock regularization with smoothness-increasing accuracy-conserving Dirac-delta polynomial kernels. Journal of Scientific Computing, 77:579--596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics (MHD) equations to several standard test problems with a variety of boundary conditions.

math.NA↗

arXiv2018

Uncertainty quantification in Eulerian-Lagrangian simulations of (point-)particle-laden flows with data-driven and empirical forcing models

An uncertainty quantification framework is developed for Eulerian-Lagrangian models of particle-laden flows, where the fluid is modeled through a system of partial differential equations in the Eulerian frame and inertial particles are traced as points in the Lagrangian frame. The source of uncertainty in such problems is the particle forcing, which is determined empirically or computationally with high-fidelity methods (data-driven). The framework relies on the averaging of the deterministic governing equations with the stochastic forcing and allows for an estimation of the first and second moment of the quantities of interest. Via comparison with Monte Carlo simulations, it is demonstrated that the moment equations accurately predict the uncertainty for problems whose Eulerian dynamics are either governed by the linear advection equation or the compressible Euler equations. In areas of singular particle interfaces and shock singularities significant uncertainty is generated. An investigation into the effect of the numerical methods shows that low-dissipative higher-order methods are necessary to capture numerical singularities (shock discontinuities, singular source terms, particle clustering) with low diffusion in the propagation of uncertainty.

physics.comp-ph↗