3D Navier Research Articles

Long-time [formula omitted]-stability of BDF2 time stepping for 2D Navier–Stokes equations

In this paper we study the H1-stability for all positive time of the BDF2 scheme for the 2D Navier–Stokes equations. More precisely, we discretize in time using the backward differentiation formula (BDF2), and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme admits this stability.

Global regularity of 2D Navier–Stokes free boundary with small viscosity contrast

This paper studies the dynamics of two incompressible immiscible fluids in two dimensions modeled by the inhomogeneous Navier–Stokes equations. We prove that if initially the viscosity contrast is small then there is global-in-time regularity. This result has been proved recently in Paicu and Zhang [Comm. Math. Phys. 376 (2020)] for H^{5/2} Sobolev regularity of the interface. Here we provide a new approach which allows us to obtain preservation of the natural C^{1+\gamma} Hölder regularity of the interface for all 0<\gamma<1 . Our proof is direct and allows for low Sobolev regularity of the initial velocity without any extra technicalities. It uses new quantitative harmonic analysis bounds for C^{\gamma} norms of even singular integral operators on characteristic functions of C^{1+\gamma} domains [Gancedo and García-Juárez, J. Funct. Anal. 283 (2022)].

Well-posedness and dynamics of 2D Navier–Stokes equations with moving boundary

We investigate the longtime dynamical behavior of 2D Navier–Stokes equations with a moving boundary. We obtain the well-posedness and dissipation through the penalty method. Then, we derive the regularity by applying a new penalty. Finally, we show that the induced dynamical system has pullback exponential attractors.

Stability of contact lines in fluids: 2D Navier–Stokes flow

In this paper we study the dynamics of an incompressible viscous fluid evolving in an open-top container in two dimensions. The fluid mechanics are dictated by the Navier–Stokes equations. The upper boundary of the fluid is free and evolves within the container. The fluid is acted upon by a uniform gravitational field, and capillary forces are accounted for along the free boundary. The triple-phase interfaces where the fluid, air above the vessel, and solid vessel wall come in contact are called contact points, and the angles formed at the contact point are called contact angles. The model that we consider integrates boundary conditions that allow for full motion of the contact points and angles. Equilibrium configurations consist of quiescent fluid within a domain whose upper boundary is given as the graph of a function minimizing a gravity-capillary energy functional, subject to a fixed mass constraint. The equilibrium contact angles can take on any values between 0 and \pi depending on the choice of capillary parameters. The main thrust of the paper is the development of a scheme of a priori estimates that show that solutions emanating from data sufficiently close to the equilibrium exist globally in time and decay to equilibrium at an exponential rate.

On the dynamics of vortices in viscous 2D flows

We study the 2D Navier–Stokes solution starting from an initial vorticity mildly concentrated near N distinct points in the plane. We prove quantitative estimates on the propagation of concentration near a system of interacting point vortices introduced by Helmholtz and Kirchhoff. Our work extends the previous results in the literature in three ways: The initial vorticity is concentrated in a weak (Wasserstein) sense, it is merely Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} integrable for some p>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>2$$\\end{document}, and the estimates we derive are uniform with respect to the viscosity.

Computational fluid dynamics (CFD) modeling of actual eroded wind turbine blades

Abstract. Leading edge erosion (LEE) is one of the most critical degradation mechanisms that occur with wind turbine blades (WTBs), generally starting from the tip section of the blade. A detailed understanding of the LEE process and the impact on aerodynamic performance due to the damaged leading edge (LE) is required to select the most appropriate leading edge protection (LEP) system and optimize blade maintenance. Providing accurate modeling tools is therefore essential. This paper presents a two-part study investigating computational fluid dynamics (CFD) modeling approaches for different orders of magnitudes in erosion damage. The first part details the flow transition modeling for eroded surfaces with roughness on the order of 0.1–0.2 mm, while the second part focuses on a novel study modeling high-resolution scanned LE surfaces from an actual blade with LEE damage on the order of 10–20 mm (approx. 1 % chord); 2D and 3D surface-resolved Reynolds-averaged Navier–Stokes (RANS) CFD models have been applied to investigate wind turbine blade sections in the Reynolds number (Re) range of 3–6 million. From the first part, the calibrated CFD model for modeling flow transition accounting for roughness shows good agreement of the aerodynamic forces for airfoils with leading-edge roughness heights on the order of 140–200 µm while showing poor agreement for smaller roughness heights on the order of 100 µm. Results from the second part of the study indicate that up to a 3.3 % reduction in annual energy production (AEP) can be expected when the LE shape is degraded by 0.8 % of the chord, based on the NREL5MW turbine. The results also suggest that under fully turbulent conditions, the degree of eroded LE shapes studied in this work show the minimal effect on the aerodynamic performances, which results in a negligible difference to AEP.

A geometric trapping approach to global regularity for 2D Navier–Stokes on manifolds

A geometric trapping approach to global regularity for 2D Navier–Stokes on manifolds

Entropy estimates for uniform attractors of 2D Navier–Stokes equations with weakly normal measures

This paper aims at the long-time behavior of non-autonomous 2D Navier–Stokes equations with a class of external forces which are [Formula: see text]-valued measures in time. We first establish the well-posedness of solutions as well as the existence of a strong uniform attractor, and then pay the main attention on the estimation of [Formula: see text]-entropy for such uniform attractor in the standard energy phase space.

Uniqueness for active scalar equations in a Zygmund space

We consider a class of active scalar equations which includes, for example, the 2D Euler equations, the 2D Navier–Stokes equations, and various aggregation equations including the Keller–Segel model. For this class of equations, we establish uniqueness of solutions in the Zygmund space C ∗ 0 . This result improves upon that in (Trans. Amer. Math. Soc. 367 (2015) 3095–3118), where the authors show uniqueness of solutions in BMO. As a corollary of our methods, we establish the uniform in space vanishing viscosity limit of Hölder continuous solutions to the aggregation equation with Newtonian potential.

Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data

First-order convergence in time and space is proved for a fully discrete semi-implicit finite element method for the two-dimensional Navier–Stokes equations with L2 initial data in convex polygonal domains, without extra regularity assumptions or grid-ratio conditions. The proof utilises the smoothing properties of the Navier–Stokes equations in the analysis of the consistency errors, an appropriate duality argument, and the smallness of the numerical solution in the discrete L2(0, tm; H1) norm when tm is smaller than some constant. Numerical examples are provided to support the theoretical analysis.

Data assimilation with higher order finite element interpolants

AbstractThe efficiency of a nudging data assimilation algorithm using higher order finite element interpolating operators is studied. Numerical experiments are presented for the 2D Navier–Stokes equations in three cases: shear flow in an annulus, a forced flow in a disk with an off‐center cavity, and a forced flow in a box all satisfying Dirichlet boundary conditions. In all three cases, second order interpolation of coarse‐grain data is shown to outperform first order interpolation. Convergence of the nudged solution to that of a direct numerical reference solution is proved. The analysis points to a trade‐off in the estimates for higher order interpolating operators

Attractors of 2D Navier–Stokes system of equations in a locally periodic porous medium

This article deals with two-dimensional Navier–Stokes system of equations with rapidly oscillating term in the equations and boundary conditions. Studying the problem in a perforated domain, the authors set homogeneous Dirichlet condition on the outer boundary and the Fourier (Robin) condition on the boundary of the cavities. Under such assumptions it is proved that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier–Stokes system of equations with an additional potential and nontrivial right hand side in the domain without pores. For this aim, the approaches from the works of A.V. Babin, V.V. Chepyzhov, J.-L. Lions, R. Temam, M.I. Vishik concerning trajectory attractors of evolution equations and homogenization methods appeared at the end of the XX-th century are used. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) system of equations and prove the existence of trajectory attractors for this system. Lastly, we formulate the main theorem and prove it through axillary lemmas.

Dynamic Weight Strategy of Physics-Informed Neural Networks for the 2D Navier-Stokes Equations.

When PINNs solve the Navier–Stokes equations, the loss function has a gradient imbalance problem during training. It is one of the reasons why the efficiency of PINNs is limited. This paper proposes a novel method of adaptively adjusting the weights of loss terms, which can balance the gradients of each loss term during training. The weight is updated by the idea of the minmax algorithm. The neural network identifies which types of training data are harder to train and forces it to focus on those data before training the next step. Specifically, it adjusts the weight of the data that are difficult to train to maximize the objective function. On this basis, one can adjust the network parameters to minimize the objective function and do this alternately until the objective function converges. We demonstrate that the dynamic weights are monotonically non-decreasing and convergent during training. This method can not only accelerate the convergence of the loss, but also reduce the generalization error, and the computational efficiency outperformed other state-of-the-art PINNs algorithms. The validity of the method is verified by solving the forward and inverse problems of the Navier–Stokes equation.

Energy-Based Aerodynamic Analysis on the Blended-Wing-Body Aircraft with Boundary Layer Ingestion

This study is aimed at evaluating the energy saving benefit and its limitation of the boundary layer ingestion (BLI) applied to a transonic blended-wing-body (BWB) aircraft. The power balance method is adopted in the aerodynamic performance analysis of the aircraft. We further improve the one-dimensional analysis framework by adding the pressure volumetric work term in the power balance equation for the application in compressible flow with shock wave regions. A determining expression for the power-saving coefficient (PSC) of the BLI effect is also deduced, which is related to a BLI fraction, the airframe dissipation component ratios, and the specific thrust of the propulsor. This expression provides a way for the preliminary evaluation and comparison of BLI benefits applied in different aircraft configurations. The flow quantities of three different configurations are obtained by 3D Reynolds-averaged Navier–Stokes (RANS) numerical simulations for the calculation of aerodynamic dissipation components. The grid refinement and domain size study shows that the calculated force coefficients and total dissipation coefficient have a good convergence characteristic with the mesh refinement and are insensitive to the variation in domain size. The evaluated PSC of the BWB aircraft with a BLI propulsor is 5% using the previously developed determining expression. The maximum of PSC is 18.8%. The following comparison of dissipation components for the power-on and power-off cases shows that the benefit of BLI is slightly underestimated by the determining expression of PSC. The reason is that the variation in the boundary layer velocity profile at the trailing edge of the center body part is neglected, which is due to the installation of the propulsor. Furthermore, when the BLI propulsor is applied to a BWB aircraft, the variation in trailing edge vortex dissipation and volumetric pressure work term should be studied in detail.

Stability of the Couette Flow for a 2D Boussinesq System Without Thermal Diffusivity

In this paper, we prove the stability of the Couette flow for a 2D Navier–Stokes Boussinesq system without thermal diffusivity for the initial perturbation in Gevrey- $$\frac{1}{s}$$ , ( $$1/3<s\leqq 1$$ ). The synergism of density mixing, vorticity mixing and velocity diffusion leads to the stability.

An immersed interface method for the 2D vorticity-velocity Navier-Stokes equations with multiple bodies

We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by nonconvex immersed bodies, multiply connected domains, and the calculation of force distributions on immersed surfaces. The immersed interface method is re-interpreted as a polynomial extrapolation of flow quantities and boundary conditions into the immersed solid bodies, reducing computational cost and enabling simulations with nonconvex bodies that could not be discretized with previous immersed interface methods. In the flow, the vorticity transport equation is discretized using a conservative finite difference scheme and explicit Runge-Kutta time integration. The velocity reconstruction problem is transformed to a scalar Poisson equation that is discretized with conservative finite differences, and solved using an FFT-accelerated iterative algorithm. The use of conservative differencing throughout leads to exact enforcement of a discrete Kelvin's theorem, allowing for simulations with multiply connected domains and outflow boundaries that have challenged other immersed interface vortex methods. We also explore novel methods for recovering time-dependent pressure distributions on immersed bodies within a vorticity-based method and present a novel control volume formulation for recovering aerodynamic moments from only the vorticity and velocity fields. The method achieves second order spatial accuracy and third order temporal accuracy, and is validated on a variety of 2D flows in internal and free-space domains.

Effect of slot‐die configurations on coating gap dependence of maximum and minimum wet thicknesses

AbstractThe dependence of the maximum and minimum wet thicknesses on the coating gap is derived for the slot‐die coating process, under different slot‐die configurations. Analytical expressions for the wet thickness and its derivative with respect to the coating gap are obtained using a simple flow model. The results indicate that, as expected, the minimum wet thickness increases linearly with the coating gap; however, the maximum wet thickness demonstrates a counterintuitive trend of decreasing as the coating gap increases, when a specific slot‐die configuration is assumed. Moreover, the results are also validated by numerically solving the complete two‐dimensional (2D) Navier–Stokes equation.

Strong $$L^2$$ convergence of time Euler schemes for stochastic 3D Brinkman–Forchheimer–Navier–Stokes equations

We prove that some time Euler schemes for the 3D Navier-Stokes equations modified by adding a Brinkman-Forchheimer term and a random perturbation converge in $L^2(\Omega)$. This extends previous results concerning the strong rate of convergence of some time discretization schemes for the 2D Navier Stokes equations. Unlike the 2D case, our proposed 3D model with the Brinkman-Forchheimer term allows for a strong rate of convergence of order almost 1/2, that is independent of the viscosity parameter.

Duffing–van der Pol nonlinear reduced-order model for explaining the phenomena and mechanism in pulsed jet flow separation control

A two-equation nonlinear reduced-order model is established to describe the interaction between external periodic excitation from a pulsed jet and unsteady flow separation. This model, which is based on a 2D Navier–Stokes equation, the Stuart vortex row model, and the Stuart–Landau model, can approximately describe the typical phenomena occurring in pulsed jet flow separation control. These phenomena include the frequency-dependent, threshold, and lock-on effects, which are related to the frequency, intensity, and phase of excitation. Two indices, namely, the maximum Lyapunov exponent and the entrainment degree, are introduced to identify the mechanism of flow control via external periodic excitation. These indices are used to reflect the order degree and momentum transfer of the flow field. A comparison of the results of the model and those of a numerical simulation in a curved diffuser shows that the nonlinear reduced-order model is effective for qualitatively featuring the behavior of pulsed jet flow separation control. Moreover, the mechanism of pulsed jet flow separation control is explained via the model. Three aspects (i.e., flow instability, flow field ordering, and momentum transfer) are assumed to function together, resulting in an efficient flow control performance.

The bleeps, the sweeps, and the creeps: Convergence rates for dynamic observer patterns via data assimilation for the 2D Navier–Stokes equations

We adapt a continuous data assimilation scheme, known as the Azouani–Olson–Titi (AOT) algorithm, to the case of moving observers for the 2D incompressible Navier–Stokes equations. We propose and test computationally several movement patterns (which we refer to as “the bleeps, the sweeps and the creeps”), as well as Lagrangian motion and combinations of these patterns, in comparison with static (i.e. non-moving) observers. In several cases, order-of-magnitude improvements in terms of the time-to-convergence are observed. We end with a discussion of possible applications to real-world data collection strategies that may lead to substantial improvements in predictive capabilities.