More Like this

1. Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier–Stokes equations

   https://doi.org/10.1017/prm.2018.33  

   Cheskidov, Alexey; Dai, Mimi (April 2019, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   Abstract Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional (3D) fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier–Stokes equations, is a mathematical analogue of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier–Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose intermittency is not extreme. 

   more » « less
2. On Bayesian data assimilation for PDEs with ill-posed forward problems

   https://doi.org/10.1088/1361-6420/ac7acd  

   Lanthaler, S; Mishra, S; Weber, F (July 2022, Inverse Problems)

   Abstract We study Bayesian data assimilation (filtering) for time-evolution Partial differential equations (PDEs), for which the underlying forward problem may be very unstable or ill-posed. Such PDEs, which include the Navier–Stokes equations of fluid dynamics, are characterized by a high sensitivity of solutions to perturbations of the initial data, a lack of rigorous global well-posedness results as well as possible non-convergence of numerical approximations. Under very mild and readily verifiable general hypotheses on the forward solution operator of such PDEs, we prove that the posterior measure expressing the solution of the Bayesian filtering problem is stable with respect to perturbations of the noisy measurements, and we provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier–Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become ill-posed, as well as the compactness of numerical approximants in a suitable metric on time-parametrized probability measures. 

   more » « less
3. Comparing Low-Mach and Fully-Compressible CFD Solvers for Phenomenological Modeling of Nanosecond Pulsed Plasma Discharges with and without Turbulence

   https://doi.org/10.2514/6.2022-0976  

   Taneja, Taaresh Sanjeev; Yang, Suo (January 2022, AIAA Scitech 2022 Forum)

   This work aims at comparing the accuracy and overall performance of a low-Mach CFD solver and a fully-compressible CFD solver for direct numerical simulation (DNS) of nonequilibrium plasma assisted ignition (PAI) using a phenomenological model described in Castela et al. [1]. The phenomenological model describes the impact of nanosecond pulsed plasma discharges by introducing source terms in the reacting flow equations, instead of solving the detailed plasma kinetics at every time step of the discharge. Ultra-fast gas heating and dissociation ofO2 are attributed to the electronic excitation ofN2 and the subsequent quenching to ground state. This process is highly exothermic, and is responsible for dissociation of O2 to form O radicals; both of which promote faster ignition. Another relatively slower process of gas heating associated with vibrational-to-translational relaxation is also accounted for, by solving an additional vibrational energy transport equation. A fully-compressible CFD solver for high Mach (M>0.2) reacting flows, developed by extending the default rhoCentralFoam solver in OpenFOAM, is used to perform DNS of PAI in a 2D domain representing a cross section of a pin-to-pin plasma discharge configuration. The same case is also simulated using a low-Mach, pressure-based CFD solver, built by extending the default reactingFoam solver. The lack of flow or wave dominated transport after the plasma-induced weak shock wave leaves the domain causes inaccurate computation of all the transport variables, with a rather small time step dictated by the CFL condition, with the fully-compressible solver. These issues are not encountered in the low-Mach solver. Finally, the low-Mach solver is used to perform DNS of PAI in lean, premixed, isotropic turbulent mixtures of CH4-air at two different Reynolds numbers of 44 and 395. Local convection of the radicals and vibrational energy from the discharge domain, and straining of the high temperature reaction zones resulted in slower ignition of the case with the higher Re. A cascade effect of temperature reduction in the more turbulent case also resulted in a five - six times smaller value of the vibrational to translational gas heating source term, which further inhibited ignition. Two pulses were sufficient for ignition of the Re = 44 case, whereas three pulses were required for the Re = 395 case; consistent with the results of Ref. [1]. 

   more » « less
4. Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

   https://doi.org/https://doi.org/10.1016/j.jde.2018.08.033  

   Larios, A; Pei, Y; Rebholz, L (February 2019, Journal of differential equations)

   The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization. 

   more » « less
5. Convex integration solution of two-dimensional hyperbolic Navier–Stokes equations ^*

   https://doi.org/10.1088/1361-6544/ad7f18  

   Wu, Jiahong; Yamazaki, Kazuo (October 2024, Nonlinearity)

   Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations. 

   more » « less