Nonstationary Navier-Stokes Equations Research Articles

Analysis and Optimal Control of a System of Hemivariational Inequalities Arising in Non-Stationary Navier-Stokes Equation with Thermal Effects

Analysis and Optimal Control of a System of Hemivariational Inequalities Arising in Non-Stationary Navier-Stokes Equation with Thermal Effects

Convergence rates for the stationary and non-stationary Navier–Stokes equations over non-Lipschitz boundaries

In this paper, we consider the convergence rates for the 2D stationary and non-stationary Navier–Stokes Equations over highly oscillating periodic bumpy John domains with C2 regularity in some neighborhood of the boundary point (0,0). For the stationary case, using the variational equation satisfied by the solution and the correctors for the bumpy John domains obtained by Higaki and Zhuge [Arch. Ration. Mech. Anal. 247(4), 66 (2023)] after correcting the values on the inflow/outflow boundaries ({0} ∪ {1}) × (0, 1), we can obtain an O(ɛ3/2) approximation in L2 for the velocity and an O(ɛ3/2) convergence rates in L2 approximated by the so called Navier’s wall laws, which generalized the results obtained by Jäger and Mikelić [J. Differ. Equations 170(1), 96–122 (2001)]. Moreover, for the non-stationary case, using the energy method, we can obtain an O(ɛ3/2 + exp(−Ct)) convergence rate for the velocity in Lx2.

A mathematical analysis to the approximate weak solution of the Smagorinsky Model for different flow regimes

This study delves into the numerical approximation of non-stationary Navier-Stokes equations within turbulent regimes, employing the Smagorinsky Model (SM). By treating the model as inherently discrete, we implement a semi-implicit time discretization using the Euler method. This approach includes comprehensive stability analyses, applicable to a spectrum of flow regimes, and an exploration of the asymptotic energy balance dynamics during fluid movements. The primary contribution of this study is found in its methodical approach to the numerical approximation of non-stationary Navier-Stokes equations within turbulent regimes using the Smagorinsky Model (SM). The adoption of a semi-implicit time discretization with the Euler method, coupled with a meticulous analysis of energy balance, establishes a robust foundation adaptable to diverse flow conditions.

Regularity of the Pressure Function for Weak Solutions of the Nonstationary Navier–Stokes Equations

We study the nonstationary system of Navier–Stokes equations for an incompressible fluid. Based on a regularized problem that takes into account the relaxation of the velocity field into a solenoidal field, the existence of a pressure function almost everywhere in the domain under consideration for solutions in the Hopf class is substantiated. Using the proposed regularization, we prove the existence of more regular weak solutions of the original problem without smallness restrictions on the original data. A uniqueness theorem is proven in the two-dimensional case

Regularity of the Pressure Function for Weak Solutions of the Nonstationary Navier–Stokes Equations

Regularity of the Pressure Function for Weak Solutions of the Nonstationary Navier–Stokes Equations

Stability of stationary solutions to the Navier–Stokes equations in the Besov space

AbstractWe consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space for . It is clarified that if w is small in for and , then for every small initial disturbance with and (), there exists a unique solution of the nonstationary Navier–Stokes equations on (0, ∞) with such that and as , for , , and small .

A Meshless Algorithm for Modeling the Gas-Dynamic Interaction between High-Inertia Particles and a Shock Layer

This paper is devoted to numerical modeling of a supersonic flow around a blunt body by a viscous gas with an admixture of relatively large high-inertia particles that, after reflection from the surface, may go beyond the shock layer and change the flow structure dramatically. To calculate the gas-dynamic interaction of moving particles with the shock layer, it is important to take into account the large difference in scales of the flow around the particles and around the body. To make the computations effective, we use a meshless method to solve non-stationary Navier–Stokes equations. The algorithm is based on the approximation of partial derivatives by the least squares method on a set of nodes distributed in the calculation area. Each moving particle is surrounded by a cloud of calculation nodes belonging to its domain and moving with it in space. The algorithm has been tested on the problem of the motion of a single particle and a pair of particles in a supersonic flow around a sphere.

BICOMPACT SCHEMES FOR COMPRESSIBLE NAVIER–STOKES EQUATIONS

For the first time, bicompact schemes are generalized to non-stationary Navier–Stokes equations for a compressible heat-conducting fluid. The proposed schemes have an approximation of the fourth order in space and the second order in time, are absolutely stable (in the frozen-coefficients sense), conservative, and efficient. One of the new schemes is tested on several two-dimensional problems. It is shown that when the mesh is refined, the scheme converges with an increased third order. A comparison is made with the WENO5-MR scheme. The superiority of the chosen bicompact scheme in resolving vortices and shock waves, as well as their interaction, is demonstrated.

On the two-step BDF finite element methods for the incompressible Navier–Stokes problem under boundary conditions of friction type

This article concerns the numerical approximation of the two-dimensional nonstationary Navier–Stokes equations with slip boundary conditions of friction type. It studies the well-posedness and error analysis properties of the numerical scheme in L2-norm for all positive time using the two-step backward differentiation formula (BDF) in time and the finite element approximation in space. Error estimates are proved under feasible assumptions on the regularity of the solution and with the aid of different versions of discrete Grownwall lemmas. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Decay estimates on Besov and Triebel-Lizorkin spaces of the Stokes flows and the incompressible Navier-Stokes flows in half-spaces

Consider the non-stationary Navier-Stokes equation in the upper half space of Rn. We prove the decay estimates of the strong solution and its derivatives in the setting of Besov and Triebel–Lizorkin spaces. This is the first time that the decay estimates of solutions to the non-stationary Navier-Stokes equations on Besov and Triebel-Lizorkin spaces are established. Our results not only extend the known results from Hardy spaces to Besov and Triebel-Lizorkin spaces but also imply new estimates on Sobolev spaces and give better decay estimates on Hardy spaces.

Enhancement of Chiller Performance by Water Distribution on the Adiabatic Cooling Pad’s Mesh Surface

Evaporative cooling is widely recognized as an energy efficient and environmentally-friendly air conditioning solution, and it has drawn a lot of market interest in recent years. However, this technology is accompanied by several challenges. For instance, insufficient evaporation due to poor and non-homogenous water distribution of the pre-cooling pad significantly reduces the cooling performance. The aim of the study is to develop a technique for numerical simulation of the distribution of a droplet liquid (water) on the mesh surface of an adiabatic cooler to improve the performance of air conditioning equipment. Modern computer-aided design (CAD)/computational fluid dynamics (CFD) programs were used to solve the issue. For the mathematical modelling of the medium motion, non-stationary Navier–Stokes equations were used. Parameters such as heat, mass transfer, and the efficiency of liquid droplet spraying were determined. The current study presents CAD modelling, conducted in SolidWorks platform, of water distribution on the adiabatic cooling pad’s mesh surface for improving air conditioning equipment performance. This study provides the methodology for computer modeling and numerical calculation of the parameters of adiabatic cooling, such as modelling of water atomization process. The results show that the use of additional metal mesh intended as cooling pads increases the mass transfer coefficient by Sh ≈ 15–40%; heat transfer coefficient Nu increases by ≈20–40%; and the atomization efficiency increases by ≈30–40%. The installation of metal pad mesh allows for equalized uniformity of the water distribution. The results imply that there are more opportunities to optimize the parameters of adiabatic cooling, which should be evaluated in further research on the subject.

Existence of Weak Solution to the Nonstationary Navier-Stokes Equations Approximated by Pressure Stabilization Method

Existence of Weak Solution to the Nonstationary Navier-Stokes Equations Approximated by Pressure Stabilization Method

Exact Solutions to the Navier–Stokes Equations and the Equation of Continuity for Twisted Axially Symmetric Flows of a Viscous Incompressible Fluid

We study exact solutions to the nonstationary Navier–Stokes equations and the equation of continuity for a special class of twisted axially symmetric flows of a viscous Newtonian fluid. We find relations describing the streamlines of such flows in the stationary mode.

Partially regular weak solutions of the stationary Navier-Stokes equations in dimension 6

By using defect measures, we prove the existence of partially regular weak solutions to the stationary Navier-Stokes equations with external force f in L_{text {loc}}^q cap L^{3/2}, q>3 in general open subdomains of {mathbb {R}}^6. These weak solutions satisfy certain local energy estimates and we estimate the size of their singular sets in terms of Hausdorff measures. We also prove the defect measures vanish under a smallness condition, in contrast to the nonstationary Navier-Stokes equations in {mathbb {R}}^4 times [0,infty [.

Finite Element Analysis of the Dynamics of Power-Law Fluid around an Obstacle in a Channel

Control of fluid forces is an emerging area of research with numerous engineering applications. The uneven wake behind an obstacle causes undesirable structural oscillations, which can lead to fatigue or structural failure. Controlling the wake phenomena could directly benefit a wide range of engineering applications, including skyscrapers, naval risers, bridges, columns, and a few sections of airplanes. This study is concerned with the time dependent simulations in a channel in presence of an obstacle aiming to compute fluid forces. The underlying mathematical model is based on nonstationary Navier–Stokes equations coupled with the constitutive relations of power law fluids. Because the representative equations are complex, an effective computing strategy based on the finite element approach is used. To achieve higher accuracy, a hybrid computational grid at a very fine level is used. Theℙ2−ℙ1elements based on the shape functions of the second and first-order polynomials were used to approximate the solution. The discrete nonlinear system arising from this discretization is linearized by Newton’s method and then solved through a direct linear solver PARADISO. The code validation study is also performed for Newtonian fluids as a special case, and then the study is extended to compute drag and lift forces for other cases of viscosity as described by the power law index. When looking at the phase plot, it can be seen that for the Newtonian case n = 1, there is only one closed orbit after the steady state is reached, whereas forn=0.5, there are multiple periodic orbits. Moreover, the effects of shear rate on the drag-lift phase plot are also discussed.

Fully decoupled, linear and unconditional stability implicit/explicit scheme for the natural convection problem

The aim of this paper is to develop an unconditionally stable Euler implicit/explicit scheme for the natural convection equations based on an introduced exponential scalar auxiliary variable. Our numerical scheme involves treating the linear terms in the implicit scheme and nonlinear terms in the explicit scheme. Then, the considered problem is decoupled into a series of linearized subproblems, and these algebraic systems with constant coefficient matrixes can be solved efficiently. Compared with the existing theoretical findings of the implicit/explicit scheme for the incompressible flow [He YN. The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or nonsmooth initial data. Math Comput. 2008;77(264):2097–2124.; He YN, Huang PZ, Feng XL. H2-stability of first order fully discrete schemes for the time-dependent Navier-Stokes equations. J Sci Comput. 2015;62:230–264; He YN, Li J. A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations. J Comput Appl Math. 2010;235(3):708–725; Jin JJ, Zhang T, Li J. H2 stability of the first order Galerkin method for the Boussinesq equations with smooth and nonsmooth initial data. Comput Math Appl. 2018;75(1):248–288; Su J, He YN. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete Continuous Dyn Syst Ser B. 2017;22(9):3421–3438], our analysis removes the restriction on time step , and obtains the unconditional stability and convergence theoretical findings. Some numerical results are also provided to verify the established theoretical analysis and show the performances of the considered numerical scheme.

The reduced-order method of continuous space-time finite element scheme for the non-stationary incompressible flows

In this paper, we mainly concern with the order reduction for the unknown solution coefficient vectors about the classical continuous space-time finite element (CCSTFE) method of the two-dimensional (2D) non-stationary incompressible Navier-Stokes equations. Therefore, we first build the CCSTFE scheme for the 2D non-stationary Navier-Stokes equations with respect to stream–vorticity functions, give the existence and stability together with convergence to the CCSTFE solutions, and rewrite the CCSTFE scheme as a matrix form. And then, we use a proper orthogonal decomposition (POD) technique to build a reduced-order extrapolated continuous space-time finite element (ROECSTFE) scheme in the matrix form including very few unknowns but possessing high enough accuracy, employ matrix analysis to demonstrate the existence and stability together with convergence of the ROECSTFE solutions such that the theoretical analysis becomes very concise and convenient, and summarize the flowchart for solving the ROECSTFE scheme. Finally, we take advantage of the numerical simulations for the back-step flow and the flow around airfoil problem to verify the correctness of theoretical results.

Optimal Design for Suppressing Time Fluctuation Part of Two-Dimensional Jet in Crossflow

This paper proposes a shape optimisation method for suppressing a time fluctuation part of the two-dimensional jet-in-cross flow. Previously, the author has formulated a shape optimisation problem for suppressing transient fluids using Snapshot proper orthogonal decomposition (POD). This study extends it to a two-dimensional jet-in-cross flow. For this purpose, the eigenvalues of Snapshot POD are integrated to a cost function, where the constraint functions are the nonstationary Navier–Stokes equation and the eigenvalue equation of Snapshot POD. An objective functional is described using Lagrange multipliers and the finite element method as the sum of the cost and constraint functions. To verify the proposed optimisation strategy, using the gradient method for domain deformation, all triangles over a mesh are deformed, which causes the cost function to decrease. Finally, the eigenvalues of Snapshot POD in the initial and optimal domains are compared.

Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

We consider the stationary and non-stationary Navier-Stokes equations in the whole plane $\mathbb{R}^2$ and in the exterior domain outside of the large circle. The solution $v$ is handled in the class with $\nabla v \in L^q$ for $q \ge 2$. Since we deal with the case $q \ge 2$, our class is larger in the sense of spatial decay at infinity than that of the finite Dirichlet integral, i.e., for $q=2$ where a number of results such as asymptotic behavior of solutions have been observed. For the stationary problem we shall show that $\omega(x)= o(|x|^{-(1/q + 1/q^2)})$ and $\nabla v(x) = o(|x|^{-(1/q+1/q^2)} \log |x|)$ as $|x| \to \infty$, where $\omega \equiv {\rm rot\,} v$. As an application, we prove the Liouville type theorem under the assumption that $\omega \in L^q(\mathbb{R}^2)$. For the non-stationary problem, a generalized $L^q$-energy identity is clarified. We also apply it to the uniqueness of the Cauchy problem and the Liouville type theorem for ancient solutions under the assumption that $\omega \in L^q(\mathbb{R}^2 \times I)$.

Time-periodic Poiseuille-type solution with minimally regular flow rate

The nonstationary Navier–Stokes equations are studied in the infinite cylinder Π = {x = (x', xn) ∈ Rn: x' ∈ σ ∈ R n – 1: – ∞ < xn < ∞, n = 2, 3} under the additional condition of the prescribed time-periodic flow-rate (flux) F(t). It is assumed that the flow-rate F belongs to the space L2(0, 2π), only. The time-periodic Poiseuille solution has the form u(x, t) = (0, ... , 0, U(x', t)), p(x,t) = –q(t)xn + p0(t), where (U(x', t), q(t)) is a solution of an inverse problem for the time-periodic heat equation with a specific over-determination condition. The existence and uniqueness of a solution to this problem is proved.