Incompressible Navier Stokes System Research Articles

A variational flexible multibody formulation for partitioned fluid–structure interaction: Application to bat-inspired drones and unmanned air-vehicles

We present a three-dimensional (3D) partitioned aeroelastic formulation for a flexible multibody system interacting with incompressible turbulent fluid flow. While the incompressible Navier–Stokes system is discretized using a stabilized Petrov–Galerkin procedure, the multibody structural system consists of a generic interaction of multiple components such as rigid body, beams and flexible thin shells along with various types of joints and connections among them. A co-rotational framework is utilized for the category of small strain problems where the displacement of the body is decomposed into a rigid body rotation and a small strain component. This assumption simplifies the structural equations and allows for the incorporation of multiple bodies (rigid as well as flexible) in the system. The displacement and rotation constraints at the joints are imposed by a Lagrange multiplier method. The equilibrium conditions at the fluid–structure interface are satisfied by the transfer of tractions and structural displacements via the radial basis function approach, a scattered data interpolation technique, which is globally conservative. For the coupled stability in low structure-to-fluid mass ratio regimes, a nonlinear iterative force correction scheme is employed in the partitioned staggered predictor–corrector scheme. The convergence and generality of the radial basis function mapping are analyzed by carrying out systematic error analysis of the transfer of fluid traction across the non-matching fluid–structure interface where a third-order of convergence is observed. The proposed aeroelastic framework is then validated by considering a flow across a flexible pitching plate configuration with serration at the trailing edge. Finally, we demonstrate the flow across a flexible flapping wing of a bat modeling the bone fingers as beams and the flexible membrane as thin shells in the multibody system along with the joints.

Strong solution for Korteweg system in bmo−1(RN) with initial density in L∞

In this paper we investigate the question of the existence of strong solution in finite time for the Korteweg system for small initial data provided that the initial momentum ρ 0 u 0 belongs to bmo −1 T (R N) for T > 0 and the initial density ρ 0 is in L ∞ (R N) with N ≥ 1 and far away from the vacuum. This result extends the so called Koch-Tataru theorem for the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any ρ(t, ·) with t > 0. We also prove the existence of global strong solution for initial data (ρ 0 − 1, ρ 0 u 0) ∈ (B N 2 −1 2,∞ (R N) ∩ B N 2 2,∞ (R N)∩L ∞ (R N))×(B N 2 −1 2,∞ (R N)) N. This result allows in particular to extend the notion of Oseen solution (corresponding to particular solution of the incompressible Navier Stokes system in dimension N = 2) to the Korteweg system provided that the vorticity of the momentum ρ 0 u 0 is a Dirac mass αδ 0 with α sufficiently small. IHowever unlike the Navier Stokes equations the property of self similarity is not conserved for the Korteweg system since there is no invariance by scaling because the term of pressure.

A Multi-scale Limit of a Randomly Forced Rotating 3-D Compressible Fluid

We study a singular limit of a scaled compressible Navier–Stokes–Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say \(\varepsilon \ll 1\), whereas the Rossby number scales like \(\varepsilon ^m\) for \(m>1\) large, then we show that any family of weak martingale solution to the 3-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as \(\varepsilon \rightarrow 0\), to the 2-D incompressible Navier–Stokes system with a corresponding random forcing term.

Enhanced convergence rates and asymptotics for a dispersive Boussinesq-type system with large ill-prepared data

In this article we prove highly improved and flexible Strichartz-type estimates allowing us to generalize the asymptotics we obtained for a stratified and rotating incompressible Navier-Stokes system: for large (and less regular) initial data, we obtain global well-posedness, asymptotics (as the Rossby number $\epsilon$ goes to zero) and convergence rates as a power of the small parameter $\epsilon$. Our approach is lead by the special structure of the limit system: the 3D quasi-geostrophic system.

Global Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes system

In this paper, we shall prove the global existence of weak solutions to 3D inhomogeneous incompressible Navier-Stokes system (INS) with initial density in the bounded function space and having a positive lower bound and with initial velocity being sufficiently small in the critical Besov space, B˙2,112. This result corresponds to the Fujita-Kato solutions of the classical Navier-Stokes system. The same idea can be used to prove the global existence of weak solutions in the critical functional framework to (INS) with one component of the initial velocity being large and can also be applied to provide a lower bound for the lifespan of smooth enough solutions of (INS).

Simulating water-entry/exit problems using Eulerian–Lagrangian and fully-Eulerian fictitious domain methods within the open-source IBAMR library

In this paper we employ two implementations of the fictitious domain (FD) method to simulate water-entry and water-exit problems and demonstrate their ability to simulate practical marine engineering problems. In FD methods, the fluid momentum equation is extended within the solid domain using an additional body force that constrains the structure velocity to be that of a rigid body. Using this formulation, a single set of equations is solved over the entire computational domain. The constraint force is calculated in two distinct ways: one using an Eulerian–Lagrangian framework of the immersed boundary (IB) method and another using a fully-Eulerian approach of the Brinkman penalization (BP) method. Both FSI strategies use the same multiphase flow algorithm that solves the discrete incompressible Navier–Stokes system in conservative form. A consistent transport scheme is employed to advect mass and momentum in the domain, which ensures numerical stability of high density ratio multiphase flows involved in practical marine engineering applications. Example cases of a free falling wedge (straight and inclined) and cylinder are simulated, and the numerical results are compared against benchmark cases in literature.

Singularities of certain finite energy solutions to the Navier-Stokes system

We continue and supplement studies from [G. Karch and X. Zheng, Discrete Contin. Dyn. Syst. 35 (2015), 3039-3057] on solutions to the three dimensional incompressible Navier-Stokes system which are regular outside a curve in \begin{document}$ \big(\gamma(t), t\big)\in \mathbb{R}^3\times [0, \infty) $\end{document} and singular on it. We revisit some of the existence results as well as some of the asymptotic estimates obtained in that work in order prove that those solutions belongs to the space \begin{document}$ C\big([0, \infty), L^2( \mathbb{R}^3)^3\big) $\end{document} .

Striated Regularity of 2-D Inhomogeneous Incompressible Navier–Stokes System with Variable Viscosity

In this paper, we investigate the global existence and uniqueness of strong solutions to 2D incompressible inhomogeneous Navier-Stokes equations with viscous coefficient depending on the density and with initial density being discontinuous across some smooth interface. Compared with the previous results for the inhomogeneous Navier-Stokes equations with constant viscosity, the main difficulty here lies in the fact that the $L^1$ in time Lipschitz estimate of the velocity field can not be obtained by energy method (see \cite{DM17,LZ1, LZ2} for instance). Motivated by the key idea of Chemin to solve 2-D vortex patch of ideal fluid (\cite{Chemin91, Chemin93}), namely, striated regularity can help to get the $L^\infty$ boundedness of the double Riesz transform, we derive the {\it a priori} $L^1$ in time Lipschitz estimate of the velocity field under the assumption that the viscous coefficient is close enough to a positive constant in the bounded function space. As an application, we shall prove the propagation of $H^3$ regularity of the interface between fluids with different densities.

Pointwise Decay in Space and in Time for Incompressible Viscous Flow Around a Rigid Body Moving with Constant Velocity

We present pointwise space–time decay estimates for the velocity part of solutions to the time-dependent Oseen system in 3D, with Dirichlet boundary conditions and vanishing velocity at infinity. In addition, similar estimates are derived for solutions to the time-dependent incompressible Navier–Stokes system with Oseen term, and for solutions to the stability problem associated with the stationary incompressible Navier–Stokes system with Oseen term.

A nonlinear estimate of the life span of solutions of the three dimensional Navier–Stokes equations

The purpose of this article is to establish bounds from below for the life span of regular solutions to the incompressible Navier-Stokes system, whichinvolve norms not only of the initial data, but also of nonlinear functions of the initial data. We provide examples showing that those bounds are significant improvements to the one provided by the classical fixed point argument. One of the important ingredients is the use of a scale-invariant energy estimate.

A variational approach to Navier–Stokes

We present a variational resolution of the incompressible Navier–Stokes system by means of stabilized weighted-inertia-dissipation-energy (WIDE) functionals. The minimization of these parameter-dependent functionals corresponds to an elliptic-in-time regularization of the system. By passing to the limit in the regularization parameter along subsequences of WIDE minimizers one recovers a classical Leray–Hopf weak solution.

Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

We consider a non-Newtonian fluid flow in a thin domain with thickness η e and an oscillating top boundary of period e. The flow is described by the 3D incompressible Navier-Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index p, with 9/5 ≤ p < +∞. We consider the limit when the thickness tends to zero and we prove that the three characteristic regimes for Newtonian fluids are still valid for non-Newtonian fluids, i.e. Stokes roughness (η e ≈ e), Reynolds roughness (η e e) and high-frequency roughness (η e e) regime. Moreover, we obtain different nonlinear Reynolds type equations in each case.

Uniform regularity for the incompressible Navier-Stokes system with variable density and Navier boundary conditions

We investigate the uniform regularity for the nonhomogeneous incompressible Navier-Stokes system with Navier boundary conditions and the inviscid limit to the Euler system. It is shown that there exists a unique strong solution of the Navier-Stokes equations in an interval of time that is uniform with respect to the viscosity parameter. The uniform estimate in conormal Sobolev spaces is established. Based on the uniform estimate, we show the convergence of the viscous solutions to the inviscid ones in L ∞ ( [ 0 , T ] × Ω ) L^\infty ([0,T]\times \Omega ) . This improves the result obtained by Ferreira et al. [SIAM J. Math. Anal. Vol. 45, No. 4, (2013), pp. 2576-2595], where L ∞ ( [ 0 , T ] , L 2 ( Ω ) ) L^\infty ([0,T],L^2(\Omega )) convergence was proved.

On Gagliardo–Nirenberg type inequalities in Fourier–Herz spaces

A variant of the Gagliardo–Nirenberg inequality in Hat–Sobolev spaces is proved, which improves certain classes of classical Sobolev embeddings. Some continuation criterion for the incompressible Navier–Stokes system is established as an application. A direct proof of the fractional Gagliardo–Nirenberg inequality in end-point Besov spaces is given and as a corollary, its counterpart in Fourier–Herz spaces is established.

Will mathematicians unleash the power of the Navier-Stokes equations?

Since 1934 when Leray proved the global existence of finite energy weak solutions to 3D incompressible Navier-Stokes equations, the regularity and uniqueness of Leray weak solutions has been core in the mathematical theory of incompressible fluid mechanics. This problem is also closely related to the Millionaires problem proposed by Clay Institute, namely, the global regularity or singularity to finite energy solutions of 3D incompressible Navier-Stokes system. The goal of this article is first to introduce the problem and then to survey some progresses on this problem.In Section 1 , following Fefferman, we first present the Millionaires problem concerning 3D incompressible Navier-Stokes equations.We also present the result of Leray on the global existence of finite energy weak solutions to the system. We conclude the introduction byremarking that the reason why we cannot solve the global regularity of 3D Navier-Stokes is that the system is energy super-critical. Whereas the energy lawis so far the only conservation law which we can find for 3D Navier-Stokes system.The goal of Section 2 is to present the main approaches toward the regularity of 3D Navier-Stokes system, namely, partial regularityof suitable weak solutions to 3D Navier-Stokes system, conditional regularity of Leray solutions, and well-posedness of 3D Navier-Stokes system with initial datain the critical spaces.Finally, in the last section, we comment about the possible singularity of 3D Navier-Stokes system.

The Transition Between the Navier–Stokes Equations to the Darcy Equation in a Thin Porous Medium

We consider a Newtonian flow in a thin porous medium $$\Omega _{\varepsilon }$$ of thickness $$\varepsilon $$ which is perforated by periodically distributed solid cylinders of size $$a_\varepsilon $$ . Generalizing (Anguiano and Suarez-Grau, ZAMP J Appl Math Phys 68:45, 2017), the fluid is described by the 3D incompressible Navier–Stokes system where the external force takes values in the space $$H^{-1}$$ , and the porous medium considered has one of the most commonly used distribution of cylinders: hexagonal distribution. By an adaptation of the unfolding method, three different Darcy’s laws are rigorously derived from this model depending on the magnitude $$a_{\varepsilon }$$ with respect to $$\varepsilon $$ .

Turbulence in vertically averaged convection

The vertically averaged velocity of the 3D Rayleigh–Bénard problem is analyzed and numerically simulated. This vertically averaged velocity satisfies a 2D incompressible Navier–Stokes system with a body force involving the 3D velocity. A time average of this force is estimated through time averages of the 3D velocity. Relations similar to those from 2D turbulence are then derived. Direct numerical simulation of the 3D Rayleigh–Bé nard is carried out to test how prominent the features of 2D turbulence are for this Navier–Stokes system.

Stability of Bifurcating Stationary Solutions of the Artificial Compressible System

The artificial compressible system gives a compressible approximation of the incompressible Navier–Stokes system. The latter system is obtained from the former one in the zero limit of the artificial Mach number $$\epsilon $$ which is a singular limit. The sets of stationary solutions of both systems coincide with each other. It is known that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion, then it is also stable as a solution of the artificial compressible one for sufficiently small $$\epsilon $$ . In general, the range of $$\epsilon $$ shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of $$\epsilon $$ can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.

On the stability of global solutions to the three-dimensional Navier-Stokes equations

We prove a weak stability result for the three-dimensional homogeneous incom-pressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence (u_{0,n})_{ n∈\in N} of initial data, bounded in some scaling invariant space, converges weakly to an initial data u0 which generates a global smooth solution, does u0,n generate a global smooth solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples u_{0,n} = nϕ0(n·) or u_{0,n}) = ϕ0(· − x_n) with |x_n| → ∞. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows

In geophysics, the shallow water model is a good approximation of the incompressible Navier-Stokes system with free surface and it is widely used for its mathematical structure and its computational efficiency. However, applications of this model are restricted by two approximations under which it was derived, namely the hydrostatic pressure and the vertical averaging. Each approximation has been addressed separately in the literature: the first one was overcome by taking into account the hydrodynamic pressure (e.g. the non-hydrostatic or the Green-Naghdi models); the second one by proposing a multilayer version of the shallow water model. In the present paper, a hierarchy of new models is derived with a layerwise approach incorporating non-hydrostatic effects to model the Euler equations. To assess these models, we use a rigorous derivation process based on a Galerkin-type approximation along the vertical axis of the velocity field and the pressure, it is also proven that all of them satisfy an energy equality. In addition, we analyse the linear dispersion relation of these models and prove that the latter relations converge to the dispersion relation for the Euler equations when the number of layers goes to infinity.