Navier–Stokes equations

About: Navier–Stokes equations is a research topic. Over the lifetime, 18180 publications have been published within this topic receiving 552555 citations. The topic is also known as: Navier-Stokes equations.

Numerical simulation of vortex-induced vibration of a square cylinder at a low Reynolds number

Abstract: Vortex-induced vibrations (VIV) of a square cylinder at a Reynolds number of 100 and a low mass ratio of 3 are studied numerically by solving the Navier-Stokes equations using the finite element method. The equation of motion of the square cylinder is solved to simulate the vibration and the Arbitrary Lagrangian Eulerian scheme is employed to model the interaction between the vibrating cylinder and the fluid flow. The numerical model is validated against the published results of flow past a stationary square cylinder and the results of VIV of a circular cylinder at low Reynolds numbers. The effect of flow approaching angle (α) on the response of the square cylinder is investigated. It is found that α affects not only the vibration amplitude but also the lock-in regime. Among the three values of α (α = 0°, 45°, and 22.5°) that are studied, the smallest vibration amplitude and the narrowest lock-in regime occur at α = 0°. It is discovered that the vibration locks in with the natural frequency in two regimes of reduced velocity for α = 22.5°. Single loop vibration trajectories are observed in the lock-in regime at α = 22.5° and 45°, which is distinctively different from VIV of a circular cylinder. As a result, the vibration frequency in the in-line direction is the same as that in the cross-flow direction.

Numerical and Experimental Flow Analysis of Moving Airfoils with Laminar Separation Bubbles

Abstract: Experimental measurements and unsteady Reynolds-averaged Navier-Stokes simulations of the low-Reynolds-number flow past an SD7003 airfoil with and without plunge motion at Re = 60 k are presented, where transition takes place across laminar separation bubbles. The experimental data consist of high-resolution, phase-locked particle image velocimetry measurements in a wind tunnel and a water tunnel. The numerical simulation approach includes transition prediction which is based on linear stability analysis applied to unsteady mean-flow data. The numerical results obtained for steady onflow are validated against particle image velocimetry data and published force measurements. Good agreement is obtained for specific turbulence models. Flows with plunge motion reveal strong effects of flow unsteadiness on transition and the resulting laminar separation bubbles which are well captured in the simulations.

On the logarithmic mean profile

Abstract: Elements of the first-principles-based theory of Wei et al. (J. Fluid Mech., vol. 522, 2005, p. 303), Fife et al. (Multiscale Model. Simul., vol. 4, 2005a, p. 936; J. Fluid Mech., vol. 532, 2005b, p. 165) and Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) are clarified and their veracity tested relative to the properties of the logarithmic mean velocity profile. While the approach employed broadly reveals the mathematical structure admitted by the time averaged NavierStokes equations, results are primarily provided for fully developed pressure driven flow in a two-dimensional channel. The theory demonstrates that the appropriately simplified mean differential statement of Newton's second law formally admits a hierarchy of scaling layers, each having a distinct characteristic length. The theory also specifies that these characteristic lengths asymptotically scale with distance from the wall over a well-defined range of wall-normal positions, y. Numerical simulation data are shown to support these analytical findings in every measure explored. The mean velocity profile is shown to exhibit logarithmic dependence (exact or approximate) when the solution to the mean equation of motion exhibits (exact or approximate) self-similarity from layer to layer within the hierarchy. The condition of pure self-similarity corresponds to a constant leading coefficient in the logarithmic mean velocity equation. The theory predicts and clarifies why logarithmic behaviour is better approximated as the Reynolds number gets large. An exact equation for the leading coefficient (von Karman coefficient κ) is tested against direct numerical simulation (DNS) data. Two methods for precisely estimating the leading coefficient over any selected range of y are presented. These methods reveal that the differences between the theory and simulation are essentially within the uncertainty level of the simulation. The von Krmn coefficient physically exists owing to an approximate self-similarity in the flux of turbulent force across an internal layer hierarchy. Mathematically, this self-similarity relates to the slope and curvature of the Reynolds stress profile, or equivalently the slope and curvature of the mean vorticity profile. The theory addresses how, why and under what conditions logarithmic dependence is approximated relative to the specific mechanisms contained within the mean statement of dynamics. © 2009 Copyright Cambridge University Press.

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Abstract: Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ℝ3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ e ≤ 1, |v (x, t)| ≤ C ∗ r −1+e |t|−e/2 for − T 0 ≤ t < 0 and 0 < C ∗ < ∞ allowed to be large. We prove that v is regular at time zero.