Nonlinear Incompressible Navier-Stokes Equation Research Articles

Robust Controller versus Numerical Model Uncertainties for Stabilization of Navier-Stokes Equations

Abstract We consider the stabilization of incompressible fluid flow using linearized and spatially discretized models. In order to potentially work in applications, the designed controller must stabilize the discrete model with a robustness margin that covers linearization, discretization, and modeling errors. We expand on previous results that a linearization error in the infinite-dimensional model amounts to a coprime factor uncertainty and show that H∞-robust controllers can compensate this in the discrete approximation. In numerical experiments, we quantify the robustness margins and show that the H∞-robust controller, unlike the LQG-controller, is capable of stabilizing nonlinear incompressible Navier-Stokes equations with an inexact linearization.

Energy stable boundary conditions for the nonlinear incompressible Navier–Stokes equations

The nonlinear incompressible Navier-Stokes equations with boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the en ...

Black holes as incompressible fluids on the sphere

We consider finite deformations of the p+2-dimensional Schwarzschild geometry which obey the vacuum Einstein equation, preserve the mean curvature and induced conformal metric on a sphere a distance $\lambda$ from the horizon and are regular on the future horizon. We show perturbatively that in the limit $\lambda$ approaches 0 the deformations are given by solutions of the nonlinear incompressible Navier-Stokes equation on the p-sphere. This relation provides a link between global existence for p-dimensional incompressible Navier-Stokes fluids and a novel form of cosmic censorship in p+2-dimensional general relativity.

Convergence Analysis of a Residual Local Projection Finite Element Method for the Navier–Stokes Equations

This work presents and analyzes a new residual local projection stabilized finite element method (RELP) for the nonlinear incompressible Navier-Stokes equations. Stokes problems defined elementwise drive the construction of the residual-based terms which make the present method stable for the finite element pairs $\mathbb{P}_1/\mathbb{P}_l$, $l=0,1$. Numerical upwinding is incorporated through an extra control on the advective derivative and on the residual of the divergence equation. Well-posedness of the discrete problem as well as optimal error estimates in natural norms are proved under standard assumptions. Next, a divergence-free velocity field is provided by a simple postprocessing of the computed velocity and pressure using the lowest order Raviart-Thomas basis functions. This updated velocity is proved to converge optimally to the exact solution. Numerics assess the theoretical results and validate the RELP method.

Experimental and Numerical Study of Laminar Round Jet Flows Along a Wall

In the present study, both experimental and numerical techniques were employed to study three-dimensional laminar wall jet flows. The wall jet was created using a circular pipe of diameter 7×10−3 m and flows into an open water tank. The inlet Reynolds numbers based on the pipe diameter and jet exit velocity were 310 and 800. A particle image velocimetry (PIV) was used to conduct detailed measurements at various streamwise-transverse and streamwise-spanwise planes. The complete nonlinear incompressible Navier–Stokes equation was also solved using a collocated finite volume based in-house computational fluid dynamics (CFD) code. The computation was performed for three inlet Reynolds numbers, namely, 310, 420, and 800. From the PIV measurements and CFD results, velocity profiles and jet half-widths were extracted at selected downstream locations. It was observed that the numerical results are in reasonable agreement with the experimental data. The distributions of the velocities, jet spread rates, and vorticity were used to provide insight into the characteristics of three-dimensional laminar wall jet flows.

Numerical Investigation of Three-Dimensional Laminar Wall Jet of Newtonian and Non-Newtonian Fluids

Three-dimensional laminar wall jets of a Newtonian fluid and two shear-thinning non-Newtonian fluids were numerically investigated. The complete nonlinear incompressible Navier-Stokes equation was solved using a colocated finite volume based in-house computational fluid dynamics code. For each fluid, the computation was performed at three Reynolds numbers. The results showed that the streamwise velocity profiles for the Newtonian fluid became self-similar but the more shear-thinning fluid never achieved a self-similar condition. Significant differences were observed among the profiles for the various fluids in the inner region. Although the transverse and spanwise components of the velocity decreased substantially with increasing Reynolds number, the values for the non-Newtonian fluids were generally an order of magnitude larger than the corresponding values for the Newtonian fluid. Depending on the specific fluid and Reynolds number, the apparent viscosities were up to 4 orders of magnitude higher than the dynamic viscosity of water. Consequently, the spread of the jet in both the transverse and spanwise directions, decay of the maximum streamwise velocity, and the skin friction coefficient depend strongly on both Reynolds number and nature of the fluid. The results also show that the jet half-width in the transverse direction is significantly higher than in the spanwise direction.