Spacetime decay of mild solutions and conditional quantitative transfer of regularity of the incompressible Navier–Stokes Equations from ℝⁿ to bounded domains

Some analytical solutions of the 1D Navier Stokes equation are introduced in the literature. For 2D flow, the analytical attempts that can solve some of the flow problems sometimes fail to solve more difficult problems or problems of irregular shapes. Many attempts try to simplify the 2D NS equations to ordinary differential equations that are usually solved numerically. The difficulties that are associated with the numerical solution of the Navier Stokes equations are known to the specialists in this field. Some of the problems associated with the numerical solution are; the continuity constraint, pressure–velocity coupling and other problems associated with the mesh generation. This drives the generation of many schemes to simplify and stabilize the 2D Navier Stokes equations. The exact solution of the Navier Stokes equations is difficult and possible only for some cases, mostly when the convective terms vanish in a natural way. This paper is devoted to studying the possibility of finding a mathematical solution of the 2D Navier Stokes equations for both potential and laminar flows. The solutions are a series of functions that satisfy the Navier Stokes equations. The idea behind the solutions is that the complete solution of the 2D equations is a combination of the solutions of any two terms in the equations; diffusion and advection terms. The solution coefficients should be determined through the boundary conditions.

Part I: We consider the numerical solution of the Navier-Stokes equations governing the unsteady flow of a viscous incompressible fluid. The analysis of numerical approximations to smooth nonlinear problems reduces to the examination of related linearized problems. The well posedness of the linear Navier-Stokes equations and the stability of finite difference approximations are studied by making energy estimates for the initial boundary value problems. Flows with open boundaries (i.e., inflow and outflow) and with solid walls are considered. We analyse boundary conditions of several types involving the velocity components or a combination of the velocity components and the pressure. The properties of these different types of boundary conditions are compared with emphasis on the suppression of undesirable numerical boundary layers for high Reynolds number calculations. The formulation of the Navier-Stokes equations which uses an elliptic equation for the pressure in lieu of the divergence equation for the velocity is shown to be equivalent to the usual formulation if the boundary conditions are treated correctly. The stability of numerical methods which use this formulation is demonstrated. Part II: We consider the numerical solution of the stream function vorticity formulation of the two dimensional incompressible Navier-Stokes equations for unsteady flows on a domain with rigid walls. The no-slip boundary conditions on the velocity components at the rigid walls are prescribed. In the stream function vorticity formulation these become two boundary conditions on the stream function and there is no explicit boundary condition on the vorticity. The accuracy of the numerical approximations to the stream function and the vorticity is investigated.The common approach in calculations is to employ second order accurate finite difference approximations for all the space derivatives and the boundary conditions together with a time marching procedure involving iteration at each time step to satisfy the boundary conditions. With such schemes the vorticity may be only first order accurate. Higher order approximations to the no-slip boundary conditions have frequently been used to overcome this problem. A one dimensional initial boundary value problem containing the salient features is proposed and analysed. With the use of this model, the behaviour observed in calculations is explained.

The conventional no-slip boundary condition does not always hold in several fluid flow applications and must be replaced with appropriate slip conditions according to the wall and fluid properties. However, not only slip boundary conditions are still a subject of discussion among fluid dynamicists, but also their numerical treatment is far from being well-developed, particularly in the context of very high-order accurate methods. The complexity of these conditions significantly increases when the boundary is not aligned with the chosen coordinate system and, even more challenging, when the fluid slips along a curved boundary. The present work proposes a simple and efficient numerical treatment of general slip boundary conditions on arbitrary curved boundaries for three-dimensional fluid flow problems governed by the incompressible Navier–Stokes equations. In that regard, two critical challenges arise: (i) achieving very high-order of convergence with arbitrary curved boundaries for the classical no-slip boundary conditions and (ii) extending the developed numerical techniques to impose general slip boundary conditions. The conventional treatment of curved boundaries relies on generating curved meshes to eliminate the geometrical mismatch between the physical and computational boundaries and achieve high-order of convergence. However, such an approach requires sophisticated meshing algorithms, cumbersome quadrature rules on curved elements, and complex non-linear transformations. In contrast, the reconstruction for off-site data (ROD) method handles arbitrary curved boundaries approximated with linear piecewise elements while employing polynomial reconstructions with specific linear constraints to fulfil the prescribed boundary conditions. For that purpose, the general slip boundary conditions are reformulated on a local orthonormal basis to allow a straightforward application of the ROD method with scalar boundary conditions. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed solely on the polygonal mesh elements. Several benchmark test cases of fluid flow problems in non-trivial three-dimensional curved domains are addressed and confirm that the proposed method effectively achieves up to the eighth-order of convergence.

In 1904, Prandtl introduced his famous boundary layer in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to 0. His Ansatz was that the solution of Navier Stokes equations can be described as a solution of Euler equations, plus a boundary layer corrector, plus a vanishing error term in $$L^\infty $$ in the inviscid limit. In this paper we prove that, for a class of smooth solutions of Navier Stokes equations, namely for shear layer profiles which are unstable for Rayleigh equations, this Ansatz is false if we consider solutions with Sobolev regularity, in strong contrast with the analytic case, pioneered by Sammartino and Caflisch (Commun Math Phys 192(2)433–461, 1998; Commun Math Phys 192(2)463–491, 1998). Meanwhile we address the classical problem of the nonlinear stability of shear layers near a boundary and prove that if a shear flow is spectrally unstable for Euler equations, then it is non linearly unstable for the Navier Stokes equations provided the viscosity is small enough.

Ninth International Conference on Numerical Methods in Fluid Dynamics

In an earlier paper related to recent results of Raugel and Sell for periodic boundary conditions, we considered the incompressible Navier-Stokes equations on 3-dimensional thin domains with zero (“no-slip”) boundary conditions and established global regularity results. We extend those results here by developing an attractor theory. We first show that under similar thinness restrictions trajectories of solutions approach each other in L 4 L^4 -norm exponentially. Next, for constant-in-time forcing data f 1 = f 1 ( x ) , f_1=f_1\left ( x\right ) , we suppose that f ( t ) → f 1 f\left ( t\right ) \rightarrow f_1 in L 2 L^2 as t → + ∞ , t\rightarrow +\infty , and show that if v v and w 1 w_1 solve the equations with forcing data f f and f 1 f_1 , respectively, then ‖ v ( t ) − w 1 ( t ) ‖ 4 → 0 \left \| v\left ( t\right ) -w_1\left ( t\right ) \right \| _4\rightarrow 0 as t → + ∞ . t\rightarrow +\infty . For similar thinness restrictions we show that the steady-flow equations with forcing data f 1 f_1 have a unique solution u s u_s . Under both thinness assumptions we then have that all solutions v ( t ) v\left ( t\right ) converge to u s u_s in L 4 L_4 as t → + ∞ t\rightarrow +\infty ; thus we have a one-point attractor for strong solutions. In fact, we have a one-point attractor for the Leray solutions as well. Moreover, under significantly more relaxed thinness assumptions we are able to show that Leray solutions nonetheless eventually become regular.

We study well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations, supplemented with no-slip velocity boundary conditions, a corresponding zero-normal condition for vorticity on the boundary, along with a natural vorticity boundary condition depending on a pressure functional. In the stationary case we prove existence and uniqueness of a suitable weak solution to the system under a small data condition. The topic of the paper is driven by recent developments of vorticity based numerical methods for the Navier-Stokes equations.

In this work, we focus on the Dynamical Low Rank (DLR) approximation of PDEs equations with random parameters. This can be interpreted as a reduced basis method, where the approximate solution is expanded in separable form over a set of few deterministic basis functions at each time, with the peculiarity that both the deterministic modes and the stochastic coefficients are computed on the fly and are free to adapt in time so as best describe the structure of the random solution. Our first goal is to generalize and reformulate in a variational setting the Dynamically Orthogonal (DO) method, proposed by Sapsis and Lermusiaux (2009) for the approximation of fluid dynamic problems with random initial conditions. The DO method is reinterpreted as a Galerkin projection of the governing equations onto the tangent space along the approximate trajectory to the manifold M_S , given by the collection of all functions which can be expressed as a sum of S linearly independent deterministic modes combined with S linearly independent stochastic modes. Depending on the parametrization of the tangent space, one obtains a set of nonlinear differential equations, suitable for numerical integration, for both the coefficients and the basis functions of the approximate solution. By formalizing the DLR variational principle for parabolic PDEs with random parameters we establish a precise link with similar techniques developed in different contexts such as the Multi-Configuration Time-Dependent Hartree method in quantum dynamics and the Dynamical Low-Rank approximation in the finite dimensional setting. By the use of curvature estimates for the approximation manifold M_S , we derive a theoretical bound for the approximation error of the S-terms DO solution by the corresponding S-terms best approximation at each time instant. The bound is applicable for full rank DLR approximate solutions on the largest time interval in which the best S-terms approximation is continuously differentiable in time. Secondly, we focus on parabolic equations, especially incompressible Navier-Stokes equations, with random Dirichlet boundary conditions and we propose a DLR technique which allows for the strong imposition of such boundary conditions. We show that the DLR variational principle can be set in the constrained manifold of all S rank random fields with a prescribed value on the boundary, expressed in low-rank format, with rank M smaller than S. We characterize the tangent space to the constrained manifold by means of the Dual Dynamically Orthogonal formulation, in which the stochastic modes are kept orthonormal and the deterministic modes satisfy suitable boundary conditions, consistent with the original problem. The same formulation is also used to conveniently include the incompressibility constraint when dealing with incompressible Navier-Stokes equations with random parameters. Finally, we extend the DLR approach for the approximation of wave equations with random parameters. We propose the Symplectic DO method, according to which the governing equation is rewritten in Hamiltonian form and the approximate solution is sought in the low dimensional manifold of all complex-valued random fields with fixed rank. Recast in the real setting, the approximate solution is expanded over a set of a few dynamical symplectic deterministic modes and satisfies the symplectic projection of the Hamiltonian system into the tangent space of the approximation manifold along the trajectory.

Fish schooling behavior contains very interesting but complicated hydrodynamics. A simplified and classic fish schooling model was simulated and analyzed in the present study, by using two self-propelled flexible filaments in a tandem arrangment. The two filaments plung harmonically in the vertical direction and are free to move in the horizontal direction in an incompressible viscous fluid flow. The simulations were carried out by using an efficient immersed boundary projection method, where the additional momentum forcing is calculated in an implicit way and an approximate factorization procedure is applied to solve the incompressible Navier-Stokes equations. The coupling between the fluid flow and the flexible structures is realized through the additional momentum forcing. In the immersed boundary projection method, both the pressure and the momentum forcing are considered as the Langrangian multipliers to satisfy the continuity constraint and the no-slip condition at the immersed boundary, respectively. To ensure a long period simulation, a non-inertial frame, which is fixed to the leading edge of the upstream filament, was adopted in the Navier-stokes equations and the structure motion equation, by introducing a translational acceleration in the motion equations. In this way, it significantly reduced the computational cost by avoiding a large compuational domain. Two-dimensional simulations were performed in the present study. The Reynolds number based on the plunging motion was fixed at 200, while the different initial horizontal and vertical distances between the two filaments were considered in simulations. The stable states of the two tandem filaments and the mechanism of the vortex interactions were mainly concerned. The numerical results reveal that three stable states are formed with different horizontal distances and vertical distances between the two filaments, i.e., long-distance tandem state, short-distance tandem state and parallel state. For the long-distance tandem state, the upstream filament performs like the single-filament case and the motion of the downstream filament always tends to be convergent with different horizontal and vertical distances. The downstream filament was found to move through the upstream shedding vortices, and was barely affected by the offset in the vertical direction. For the short-distance tandem state, the shear layer of the upstream filament is merged with the shear layer of the downstream one before it is shed from the filament. This effect generates a strong vortex shedding, and consequently the propulsive force is enhanced with the cruising velocity increased as well. As a result, phase gaps of motions and drag forces between the upstream and downstream filaments arise. For the paralle state, when the initial horizontal distance of the two filamens is small while the vertical offset is large enough, the influence of the upstream vortex on the downstream filament is decreased. So the downstream filament moves forward and the parallel state is formed. In this case, the shear layers of two filaments shed synchronously and strong vortices are formed in the wake. Meanwhile the windward aera are about twice as compared with the tandem states, thus the cruising velocity is reduced. In summary, as a result of the interactions between upstream vortices and downstream vortices/structures, the forward speed of the parellel state is the slowest, and that of the short-distance state is the fastest, while that of the long-distance state is middle among the three stable states.

In this thesis numerical methods are developed for the simulation of turbulent flows governed by the incompressible Navier-Stokes equations. This is inspired by the need for accurate and efficient computations of the flow of air in windturbine wakes. The state-of-the-art in computing such flows is to use Large Eddy Simulation (LES) as a turbulence model. In LES the Navier-Stokes equations are filtered such that only the large, energy-containing scales of motion are simulated - the smaller scales are modeled. However, even with such a model, LES simulations remain expensive (not only in wind energy applications) and are typically ‘under-resolved’: the mesh is too coarse to resolve all important scales. Thus, an ongoing challenge is to construct numerical methods that are stable and accurate even on coarse meshes, and do not introduce false (‘artificial’) diffusion that can destroy the delicate features of turbulent flows. The approach taken is to construct high-order energy-conserving discretization methods. Such methods mimic an important property of the continuous incompressible Navier-Stokes equations, namely the conservation of kinetic energy in the limit of vanishing viscosity. The energy equation is, for incompressible flows, derived from the equations for conservation of mass and momentum. An energy-conserving discretization method is nonlinearly stable, independent of mesh, time step, or viscosity, and does not introduce artificial diffusion. The first part of this thesis addresses spatially energy-conserving discretization methods, in particular second and fourth order finite volume methods on staggered cartesian grids. Special attention is paid to the proper treatment of boundary conditions for high order methods. New boundary conditions are derived such that the boundary contributions to the discrete energy equation mimic the boundary contributions of the continuous equations. An important theoretical result is obtained: higher order energy-conserving finite volume discretizations are limited to second order global accuracy in the presence of boundaries. On properly chosen non-uniform grids, designed such that the maximum error is not at the boundary, fourth order accuracy can be recovered. The second part of this thesis addresses time integration of the incompressible Navier-Stokes equations with Runge-Kutta methods. Runge-Kutta methods are often applied to the spatially discretized incompressible Navier-Stokes equations, but order of accuracy proofs that address both velocity and pressure are missing. By viewing the spatially discretized Navier-Stokes equations as a system of differential-algebraic equations the order conditions for velocity and pressure are derived. Based on these conditions new explicit Runge-Kutta methods are derived, that have high-order accuracy for both velocity and pressure. These explicit methods are not strictly energy-conserving but can be efficient, depending when the time step is determined by accuracy instead of stability. However, for truly energy-conserving Runge-Kutta methods implicit methods need to be considered. High-order Runge-Kutta methods based on Gauss quadrature are proposed. In particular, the two-stage fourth order Gauss method is investigated and combined with the fourth order spatial discretization, resulting in a fourth order energy-conserving method in space and time, which is stable for any mesh and any time step. A disadvantage of the Gauss methods is that they are less suitable for integrating the diffusive terms, since they lack L-stability. Therefore, new additive Runge-Kutta methods are investigated: the diffusive terms are integrated with an L-stable Runge-Kutta method, and the convective terms with an energy-conserving Runge-Kutta method, both based on the same quadrature points. Unfortunately, their low stage order does not make them more efficient than the original Gauss methods. In practice, the second order Gauss method (implicit midpoint) is therefore the preferred time integration method. The third part of this thesis addresses actuator methods. Actuator methods are simplified models to represent the effect of a body (such as a wind turbine) on a flow field, without requiring the actual geometry of the body to be taken into account. Actuator forces introduce discontinuities in flow variables and should therefore be treated carefully. A new immersed interface method in finite volume formulation is proposed, which leads to a sharp, non-diffusive, representation of the actuator. This does not require the choice for a discrete Dirac function and regularization parameter. The ideas put forth in this thesis have been implemented in a new parallel 3D incompressible Navier-Stokes solver: ECNS (Energy-Conserving Navier-Stokes solver). The resulting method combines stability, no numerical viscosity and high-order accuracy. This makes it a valuable tool for simulating turbulent flow problems governed by the incompressible Navier-Stokes equations and suitable for the development and comparison of LES models. For the particular case of wind-turbine wake aerodynamics a number of simulations have been performed: flow over a wing as a model for a wind turbine blade, and flow through an array of actuator disks representing a wind farm.

The boundary condition switching method for solving stream function-vorticity equations is applied to solve the incompressible Navier-Stokes equations in primitive variable form. The key idea of this method is that on boundaries, where both Dirichlet and Neumann boundary conditions are known, the Neumann boundary condition is satisfied naturally while the Dirichlet boundary condition is used to substitute the unknown boundary condition of the other variable, for instance, vorticity in stream function-vorticity equations and pressure in Navier-Stokes equations. When solving the Navier-Stokes equations with primitive variables for incompressible laminar flows, the Dirichlet boundary condition is the velocity boundary condition by giving zero velocity at wall boundaries. The Neumann boundary condition are given by the continuity condition. In the methods solving Navier-Stokes equations with primitive variables, the velocity boundary conditions are implemented with the Neumann boundary condition satisfied by continuity equation. It is well known that these methods result in pressure oscillations using equal-order interpolation in finite element method (FEM) and using non-staggered grid in finite difference method (FDM). This difficulty could be removed by using the present method. Application of the method is tested by both FEM and FDM and the results show that the method does not result in any pressure oscillations. Thus, pressure solutions do not need to be filtered or smoothed.

The paper deals with theoretical analysis of non-stationary incompressible flow through a cascade of profiles. The initial-boundary value problem for the Navier–Stokes system is formulated in a domain representing the exterior to an infinite row of profiles, periodically spaced in one direction. Then the problem is reformulated in a bounded domain of the form of one space period and completed by the Dirichlet boundary condition on the inlet and the profile, a suitable natural boundary condition on the outlet and periodic boundary conditions on artificial cuts. We present a weak formulation and prove the existence of a weak solution. Copyright © 2006 John Wiley & Sons, Ltd.

The projection method to solve the incompressible Navier–Stokes equations was first studied by Chorin (Math Comput, 1969) in the framework of a finite difference method and Temam (Arch Ration Mech Anal, 1969) in the framework of a finite element method. Chorin showed convergence of approximation and its error estimates in problems with periodic boundary conditions assuming existence of a $$C^5$$ -solution, while Temam demonstrated an abstract argument to obtain a Leray–Hopf weak solution in problems on a bounded domain with the no-slip boundary condition. In the present paper, the authors extend Chorin’s result with full details to obtain convergent finite difference approximation of a Leray–Hopf weak solution to the incompressible Navier–Stokes equations on an arbitrary bounded Lipschitz domain of $${{\mathbb {R}}}^3$$ with the no-slip boundary condition and an external force. We prove unconditional solvability of our implicit scheme and strong $$L^2$$ -convergence (up to a subsequence) under the scaling condition $$ h^{3-\alpha }\le \tau $$ (no upper bound is necessary), where $$h,\tau $$ are space, time discretization parameters, respectively, and $$\alpha \in (0,2]$$ is any fixed constant. The results contain a compactness method based on a new interpolation inequality for step functions.

It may come as a surprise to most physicists and material scientists that up until recently, continuum hydrodynamics can not accurate model fluid(s) flow at the nanoscale. The problem lies not in the Navier Stokes (NS) equation, which expresses momentum balance and therefore must be valid, but in the boundary condition at the fluid-solid interface, required for the solution of the NS equation. Traditionally the no-slip boundary condition (NSBC) has always been used to solve hydrodynamic problems, which states that there can be no relative motion at the fluid-solid interface. The purposes of this article are to delineate the problem and to present its resolution through the application of Onsager’s principle of minimum energy dissipation, which underlies almost all the linear response phenomena in dissipative systems. Even though lacking in first principles support, NSBC was widely regarded as a cornerstone in continuum hydrodynamics, owing to its broad applicability in diverse fluid-flow problems. However, an important exception was pointed out some time ago in the so-called the moving contact line (MCL) problem in immiscible flows. Here the contact line is defined as the intersection of the two-phase immiscible fluid-fluid interface with the solid wall. In 1974, Dussan and Davis showed with rigor that under the assumptions of (1) fluid incompressibility, (2) rigid and flat solid wall, (3) impenetrable fluid-fluid interface, and (4) NSBC, there is a velocity discontinuity at the MCL, and the tangential force exerted by the fluids on the solid wall in the vicinity of the MCL is infinite. Subsequently, by employing molecular dynamic (MD) simulations, Koplik et al. (1988) and Thompson and Robbins (1988) have shown that near-complete slip occurs at the MCL, in direct contradiction to the NSBC. The failure of the NSBC and the lack of a viable alternative mean that the usual continuum hydrodynamics can not accurately model fluids flow on the micro/nanoscale. This has often been referred to as an example of the so-called “breakdown of continuum.”

We present the deep random vortex network (DRVN), a novel physics-informed framework for simulating and inferring the fluid dynamics governed by the incompressible Navier–Stokes equations. Unlike the existing physics-informed neural network (PINN), which embeds physical and geometry information through the residual of equations and boundary data, DRVN automatically embeds this information into neural networks through neural random vortex dynamics equivalent to the Navier–Stokes equation. Specifically, the neural random vortex dynamics motivates a Monte Carlo-based loss function for training neural networks, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVN can efficiently solve Navier–Stokes equations with non-differentiable initial conditions and fractional operators. Furthermore, DRVN naturally embeds the boundary conditions into the kernel function of the neural random vortex dynamics and, thus, does not need additional data to obtain boundary information. We conduct experiments on forward and inverse problems with incompressible Navier–Stokes equations. The proposed method achieves accurate results when simulating and when inferring Navier–Stokes equations. For situations that include singular initial conditions and agnostic boundary data, DRVN significantly outperforms the existing PINN method. Furthermore, compared with the conventional adjoint method when solving inverse problems, DRVN achieves a 2 orders of magnitude improvement for the training time with significantly precise estimates.