Three-dimensional Incompressible Navier-Stokes Equations Research Articles

Simulation of different flow regimes in a narrow-gap spherical Couette flow

The spherical Couette flow (SCF) system with the flow between two concentric rotating spheres is one of the most convenient example in exploring the laminar-turbulent transition. The transitions occurred in this flow phenomena has a great relevance with the geophysical motions and has applications in hydrodynamic problems. In the present work, we numerically study different flow instabilities in a narrow-gap SCF between two concentric spheres with only inner sphere rotating and the outer sphere is fixed. The time dependent three-dimensional incompressible Navier-Stokes equations (INSEs) along with a parallelized line Gauss-Seidel(LGS) solver has been used for the simulation of different flow instabilities in a narrow-gap clearance ratios(CR), β=(R2−R1)/R1=0.10. The flow instabilities have been investigated for a range of Reynolds numbers (Re)170≤Re≤12000. For this narrow-gap CR, first we obtain the 0-,1-,2-,3-, and 4-vortex flows at Re=1430,1440,1460,1750 and 1950 respectively by using the Stokes flow (SF) as an initial condition. Further increasing the Re using the same initial condition the flow becomes wavy, spiral wavy, then return back to supercritical basic flow and finally becomes turbulent at higher Re. The computed numerical results have a good agreement with the existing numerical and experimental results. Further more, we have noticed some new flow states which have not been observed in the previous numerical and experimental studies.

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

LES study on the impact of airway deformation on the airflow structures in the idealized mouth–throat model

To investigate the impacts of upper airway deformation on the airflow structures, the airflow fields in the trachea are simulated using three geometrical models considering three different levels of airway deformations. Structured grids are used to create the high-quality grids. Large eddy simulation with the Smagorinsky sub-grid model is adopted to solve the three-dimensional in-compressible Navier–Stokes equations using the solver pisoFoam in the open-source CFD software OpenFOAM. The numerical results demonstrate that the airway deformation influences the main airflow structures depending on the deformation level. Particularly, it slightly impacts on the laryngeal jet such as the profile and the strength of laryngeal jet. The strength of the laryngeal jet increases slightly for the heavy deformation. In contrast, it impacts on the recirculation zone, secondary vortices, and turbulent kinetic energy more obviously. The increasing airway deformation will produce stronger secondary flow, smaller recirculation zone, and weaker turbulent kinetic energy. The turbulence intensity distribution varies as well. The obviously impacted flow region is mainly within the region of one to six tracheal diameters downstream the glottis.

A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains

We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the sense that the finite dimensional function spaces form a discrete de Rham complex. Conservation of mass, kinetic energy and helicity in the absence of dissipative terms is proven at the discrete level. Proper dissipation rates of kinetic energy and helicity in the viscous case are also proven. Numerical tests supporting the method are provided.

The primitive equations approximation of the anisotropic horizontally viscous 3D Navier-Stokes equations

In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting ε>0 to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders O(1) and O(εα), respectively, with α>2, for which the limiting system is the primitive equations with only horizontal viscosity as ε tends to zero. In particular we show that for “well prepared” initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as ε tends to zero, and that the convergence rate is of order O(εβ2), where β=min⁡{α−2,2}. Note that this result is different from the case α=2 studied in Li and Titi (2019) [38], where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order O(ε).

Numerical investigation of droplet impact on a solid superhydrophobic surface

The impact of microscale water droplets onto a solid superhydrophobic surface is numerically investigated. The multiphase problems are modeled by the three-dimensional incompressible Navier–Stokes equations and the liquid–gas interface is captured by the level set method. The numerical model is verified with our experimental impact results via the comparison of spreading factor ξ, which is defined as the ratio of the wetted surface area to droplet initial diameter. The simulation results suggest that when the droplet impacts with constant impact velocity and diameter, the maximum spreading parameter increases with the ambient temperature. As Weber and Reynolds numbers increase, the impact turns into doughnut-breakup regime; the droplet breaks up into a toroidal shape and a cavity is formed at the center. The results indicate that the diameter of the central cavity grows linearly related to the non-dimensional time. Finally, a new droplet impact spread/splash model that is governed by Weber and Reynolds numbers is proposed for superhydrophobic surface based on our numerical and experimental results.

Decay estimates for three-dimensional Navier–Stokes equations with damping

In this paper, the optimal decay rates of solutions for three-dimensional incompressible Navier–Stokes equations with damping term uβ−1u are established. Bounds on the H.−s negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.

Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations

<p style='text-indent:20px;'>The stability and convergence of the Fourier pseudo-spectral method are analyzed for the three dimensional incompressible Navier-Stokes equation, coupled with a variety of time-stepping methods, of up to fourth order temporal accuracy. An aliasing error control technique is applied in the error estimate for the nonlinear convection term, while an a-priori assumption for the numerical solution at the previous time steps will also play an important role in the analysis. In addition, a few multi-step temporal discretization is applied to achieve higher order temporal accuracy, while the numerical stability is preserved. These semi-implicit numerical schemes use a combination of explicit Adams-Bashforth extrapolation for the nonlinear convection term, as well as the pressure gradient term, and implicit Adams-Moulton interpolation for the viscous diffusion term, up to the fourth order accuracy in time. Optimal rate convergence analysis and error estimates are established in details. It is proved that, the Fourier pseudo-spectral method coupled with the carefully designed time-discretization is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. Some numerical results are also presented to verify the established convergence rates of the proposed schemes.</p>

A comparison of different numerical schemes in spherical Couette flow simulation

We compare the performance of line Gauss–Seidel (LGS), point Gauss–Seidel (PGS), and alternating direction implicit (ADI) linear solvers used in the artificial compressibility method for the numerical solution of the three-dimensional incompressible Navier–Stokes equations. Spatial discretization is carried out using a fifth-order WENO scheme for the convective terms and a second-order central difference scheme for the viscous terms. A comparison is made by simulating the spherical Couette flow problem, with only the inner sphere rotating and the outer one fixed. OpenMP is used for numerical computation in parallel for the three schemes. First, we compare the numerical efficiency of the solvers by computing 0-vortex flow for a medium-gap σ=R2−R1/R1=0.25. Second, we make a residual comparison for a steady-state flow calculation based on the CFL number and artificial compressibility factor. Finally, we compare the three solvers for unsteady flow computations based on the artificial compressibility factor. The results show that the LGS solver is more reliable than the PGS and the ADI solvers. To show the accuracy of the LGS scheme, we compute different flow modes for an intermediate-gap clearance ratio σ=R2−R1/R1=0.14. The computed results have good agreement with the existing numerical results.

The Regularity Criteria and the A Priori Estimate on the 3D Incompressible Navier-Stokes Equations in Orthogonal Curvilinear Coordinate Systems

The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω 3 and one a priori estimate on the solution that H 3 u 3 L ∞ 0 , T ; L p ℝ 3 is bounded for 1 ≤ p ≤ ∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

A parallel non-nested two-level domain decomposition method for simulating blood flows in cerebral artery of stroke patient.

Numerical simulation of blood flows in patient-specific arteries can be useful for the understanding of vascular diseases, as well as for surgery planning. In this paper, we simulate blood flows in the full cerebral artery of stroke patients. To accurately resolve the flow in this rather complex geometry with stenosis is challenging and it is also important to obtain the results in a short amount of computing time so that the simulation can be used in pre- and/or post-surgery planning. For this purpose, we introduce a highly scalable, parallel non-nested two-level domain decomposition method for the three-dimensional unsteady incompressible Navier-Stokes equations with an impedance outlet boundary condition. The problem is discretized with a stabilized finite element method on unstructured meshes in space and a fully implicit method in time, and the large nonlinear systems are solved by a preconditioned parallel Newton-Krylov method with a two-level Schwarz method. The key component of the method is a non-nested coarse problem solved using a subset of processor cores and its solution is interpolated to the fine space using radial basis functions. To validate and verify the proposed algorithm and its highly parallel implementation, we consider a case with available clinical data and show that the computed result matches with the measured data. Further numerical experiments indicate that the proposed method works well for realistic geometry and parameters of a full size cerebral artery of an adult stroke patient on a supercomputers with thousands of processor cores.

Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces

Abstract In this paper, we prove a global well-posedness of the three-dimensional incompressible Navier-Stokes equation under initial data, which belongs to the Lei-Lin-Gevrey space Z a , σ − 1 $\begin{array}{} Z^{-1}_{a,\sigma} \end{array}$ (ℝ3) and if the norm of the initial data in the Lei-Lin space 𝓧−1 is controlled by the viscosity. Moreover, we will show that the norm of this global solution in the Lei-Lin-Gevrey space decays to zero as time approaches to infinity.

Influences of marshalling length on the flow structure of a maglev train

In this paper, transient numerical simulations of maglev trains of different marshalling lengths (2, 4, and 8-car group trains) were conducted in the open air and without wind. This was done by solving the three-dimensional incompressible Navier-Stokes equations using an SST K-ω double-equation IDDES turbulence model. The results were compared with the results of wind tunnel experiments to verify the feasibility of numerical simulation. The results show an increase in the marshalling lengths of the train affects the flow above and below the train. With the increase of the marshalling length, the position of the flow separation in the tail car is advanced. The turbulence generated by the average shear on the x, y-plane and the x, z-plane as a component of the turbulence of the wake region increases. The region that produces non-vorticial vorticity in the main vortex becomes narrow and moves towards tail car. The structural analysis of the wake indicates that the wake structure of the 8-car group train is quite different from the other two groups. Both the time-averaged slipstream and the gust analysis show that the maximum expected slipstream velocity at the track-side increases as the train marshalling length increases. At the platform height, the change in vertical position of the wake vortex structure of the 2, 4, and 8-car group trains caused the difference of the shear flow regions. This is why the maximum expected slipstream velocity generated by the 4-car group train at this position is largest. As the marshalling length of the train increases, the time-average drag and lift force coefficient of the tail car have a significant negative correlation.

Machine learning strategies for path-planning microswimmers in turbulent flows.

We develop an adversarial-reinforcement learning scheme for microswimmers in statistically homogeneous and isotropic turbulent fluid flows, in both two and three dimensions. We show that this scheme allows microswimmers to find nontrivial paths, which enable them to reach a target on average in less time than a naïve microswimmer, which tries, at any instant of time and at a given position in space, to swim in the direction of the target. We use pseudospectral direct numerical simulations of the two- and three-dimensional (incompressible) Navier-Stokes equations to obtain the turbulent flows. We then introduce passive microswimmers that try to swim along a given direction in these flows; the microswimmers do not affect the flow, but they are advected by it. Two nondimensional control parameters play important roles in our learning scheme: (a) the ratio V[over ̃]_{s} of the microswimmer's bare velocity V_{s} and the root-mean-square (rms) velocity u_{rms} of the turbulent fluid and (b) the product B[over ̃] of the microswimmer-response time B and the rms vorticity ω_{rms} of the fluid. We show that the average time required for the microswimmers to reach the target, by using our adversarial-reinforcement learning scheme, eventually reduces below the average time taken by microswimmers that follow the naïve strategy.

Numerical Analysis on Natural Convection Heat Transfer in a Single Circular Fin-Tube Heat Exchanger (Part 1): Numerical Method.

In this research, unsteady three-dimensional incompressible Navier–Stokes equations are solved to simulate experiments with the Boussinesq approximation and validate the proposed numerical model for the design of a circular fin-tube heat exchanger. Unsteady time marching is proposed for a time sweeping analysis of various Rayleigh numbers. The accuracy of the natural convection data of a single horizontal circular tube with the proposed numerical method can be guaranteed when the Rayleigh number based on the tube diameter exceeds 400, which is regarded as the limitation of numerical errors due to instability. Moreover, the effective limit for a circular fin-tube heat exchanger is reached when the Rayleigh number based on the fin gap size () is equal to or exceeds 100. This is because at low Rayleigh numbers, the air gap between the fins is isolated and rarely affected by natural convection of the outer air, where the fluid provides heat resistance. Thus, the fin acts favorably when exceeds 100.

A Liouville theorem for three-dimensional stationary nematic liquid crystal equations

In this paper, we prove a Liouville type theorem for three-dimensional stationary liquid crystal equationsunder the condition $u,~\\nabla~d~\\in~L^{9/2,~q}(\\mathbb{R}^3)$,where $u:~\\mathbb{R}^3~\\to~\\mathbb{R}^3$ denotes the velocity field and $d:~\\mathbb{R}^3~\\to~\\mathbb{S}^2$ is the moleculardirection. The proof of this theorem is inspired by Galdis ideas of using local energy estimates in provingthe Liouville theorem for three-dimensional incompressible Navier-Stokes equations in $L^{9/2}(\\mathbb{R}^3)$ and Schoens description on the harmonic maps. We generalize Galdis results to Lorentz spaces and nematic liquid crystal systems.

Ceiling effects on the aerodynamics of a flapping wing with advance ratio

The ceiling effect on the aerodynamics of flapping wings with an advance ratio is investigated by solving the three-dimensional incompressible Navier–Stokes equations. The aerodynamic forces and flow fields around the model wings flapping in a horizontal plane were simulated at various advance ratios, Reynolds numbers, as well as the distance between the wing and the ceiling. It is found that the ceiling could improve the aerodynamic forces at a low advance ratio and this improvement in aerodynamic forces decreases as the distance between the wings and ceiling increases, similar to the results under hovering condition. However, the flow fields show that the aerodynamic force enhancement is only caused by the increment in the relative velocity of the oncoming flow; the ceiling would no longer enlarge the angle of incidence of the oncoming flow at the range of advance ratios considered, which is different from that under hovering condition. As the advance ratio increases, the enhancement in aerodynamics from the ceiling effect decreases. This is mainly due to the degeneration of the ceiling effect at the outer part of the wing, where the effect of increasing velocity becomes rather small at a high advance ratio. The weakened “increasing velocity effect” is closely associated with the detachment of the leading-edge vortex at the outer part of the wing at a high advance ratio.

The viscosity vanishing limit and global well-posedness of the three-dimensional incompressible Navier–Stokes equations with smooth large initial data in spherical coordinates

This paper investigates the viscosity vanishing limit and the existence and uniqueness of the global strong solution on the three-dimensional incompressible Navier–Stokes equations without swirl in spherical coordinates. We establish the global existence and uniqueness of the smooth solution to the Cauchy problem for the three-dimensional incompressible Navier–Stokes equations for the any smooth large initial data without swirl in the sense of spherical coordinates. Also, by performing the viscosity vanishing limit for the global strong solution in time to the three-dimensional incompressible Navier–Stokes equations, we prove that there exists the unique and global strong solution to the Cauchy problem for the three-dimensional incompressible Euler equation without swirl in spherical coordinates with large initial data.

Existence of a weak solution to the fluid-structure interaction problem in 3D

We study a nonlinear fluid-structure interaction problem in which the fluid is described by the three-dimensional incompressible Navier-Stokes equations, and the elastic structure is modeled by the nonlinear plate equation which includes a generalization of Kirchhoff, von Kármán and Berger plate models. The fluid and the structure are fully coupled via kinematic and dynamic boundary conditions. The existence of a weak solution is obtained by designing a hybrid approximation scheme that successfully deals with the nonlinearities of the system. We combine time-discretization and operator splitting to create two sub-problems, one piece-wise stationary for the fluid and one in the Galerkin basis for the plate. To guarantee the convergence of approximate solutions to a weak solution, a sufficient condition is given on the number of time discretization sub-intervals in every step in a form of dependence with number of the Galerkin basis functions and nonlinearity order of the plate equation.

Chebyshev–Fourier collocation spectral method for the solution of swirling flow

Abstract The Chebyshev–Fourier collocation spectral method (CSM), in which the Chebyshev collocation method is applied in both radial and axial directions and the Fourier collocation method is applied in azimuthal direction, is developed to solve the Poisson equation with Neumann boundary condition in cylindrical coordinates. In order to avoid the axis at r = 0 , the Chebyshev–Gauss–Lobatto collocation point is adopted in radial direction and the number of points is set to be even. These direct solvers are developed to solve three-dimensional transient incompressible Navier-Stokes equations with projection method. The approach, in which the Neumann boundary condition is constructed for solving the Poisson equation based on the Chebyshev–Fourier CSM, is testified to be easily and efficiently implemented. Furthermore, the approach is applied for solving the swirling flows in an enclosed cylinder by the rotation of an end wall.