Gustaaf B. Jacobs

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 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.

A data assimilation approach is coined that enables the discovery of forcing functions in Lagrangian, point-particle models from limited measurements of trajectory coordinates. Central to the proposed formulation of this inverse problem is the expression of the forcing function in terms of modal basis functions that are dependent on the relative velocity difference between a known carrier flow and the particle solution weighted with coefficients that are known within confidence intervals. The probability density function of the random forcing coefficients is inferred using a combination of the forward, particle model and its adjoint dynamics, which calculates the gradient of the cost function defined as the distance between the measured and predicted particle locations. To ensure convergence of the gradient-based optimization, multiple measurements may be required. If the measurements are noisy, samples of the forcing model within an assumed Gaussian distribution of the confidence interval of the measurement are computed using a Hamiltonian Monte Carlo method. The method is verified to correctly infer the forcing function of particles traced in the Arnold–Beltrami–Childress flow and a homogeneous isotropic turbulence. The confidence interval of the inferred forcing function with respect to a flow condition is improved if the particle is exposed more frequently to the flow condition. The forcing coefficients adapt the model to flow conditions that are outside of the limited range for which the point-particle models are typically known only empirically or within confidence intervals.

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. This strategy gives rise to an exact (deterministic, first-order, hyperbolic) Liouville equation that describes the evolution of a joint PDF in the augmented phase-space spanned by the random variables and the particle position and velocity. Analytical PDF solutions for canonical models of particle-laden flows serve to establish a relationship between the Langevin and Liouville approaches. Finally, our framework is used to derive a new analytical PDF model for fluidized homogeneous heating systems.

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.

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 centrepiece 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.

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.