Navier-Stokes solver using Green's functions II: spectral integration of channel flow and plane Couette flow

The Kleiser-Schumann algorithm has been widely used for the direct numerical simulation of turbulence in rectangular geometries. At the heart of the algorithm is the solution of linear systems which are tridiagonal except for one row. This note shows how to solve the Kleiser-Schumann problem using perfectly triangular matrices. An advantage is the ability to use functions in the LAPACK library. The method is used to simulate turbulence in channel flow at $Re=80,000$ (and $Re_{\tau}=2400$) using $10^{9}$ grid points. An assessment of the length of time necessary to eliminate transient effects in the initial state is included

High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures

The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl

Navier-Stokes solver using Green's functions I: channel flow and plane Couette flow

Numerical solvers of the incompressible Navier-Stokes equations have reproduced turbulence phenomena such as the law of the wall, the dependence of turbulence intensities on the Reynolds number, and experimentally observed properties of turbulence energy production. In this article, we begin a sequence of investigations whose eventual aim is to derive and implement numerical solvers that can reach higher Reynolds numbers than is currently possible. Every time step of a Navier-Stokes solver in effect solves a linear boundary value problem. The use of Green's functions leads to numerical solvers which are highly accurate in resolving the boundary layer, which is a source of delicate but exceedingly important physical effects at high Reynolds numbers. The use of Green's functions brings with it a need for careful quadrature rules and a reconsideration of time steppers. We derive and implement Green's function based solvers for the channel flow and plane Couette flow geometries. The solvers are validated by reproducing turbulent signals which are in good qualitative and quantitative agreement with experiment

An Optimal Control Approach to Bounding Transport Properties of Thermal Convection.

This dissertation explores and develops an optimal control approach to upper bounds on transport properties of fluid flows inspired by the physical phenomenon of buoyancy-driven Rayleigh-Bénard convection. This method is applied in the context of three different problems: the Lorenz equations, the Double Lorenz equations, and the Boussinesq approximation to the Navier-Stokes equations. Rather than restricting attention to flows that satisfy an equation of motion, we consider incompressible flows that satisfy suitable bulk integral constraints and boundary conditions. Bounds on transport are formulated in terms of optimal control problems where the flows are the "control" and a passive scalar tracer field is the "state". All three problems lead to non-convex optimization problems. Sharp upper bounds to the Lorenz equations are proven analytically, and it is shown that any sustained time-dependence of the control variable strictly lowers transport. For the Double Lorenz equations an upper bound is proven and saturated by steady optimizing flow fields and any time-periodic stirring protocol strictly lowers transport. In contrast to the Lorenz equations, however, the optimizing steady flow fields (solutions to the Euler-Lagrange equations for optimal transport) are not solutions to the original equations of motion. In the Boussinesq equation context the optimal control problem is rigorously formulated for steady flows, and analytic upper bounds to transport are deduced using the background method. A gradient ascent procedure for numerically solving the associated the Euler-Lagrange equations for optimal transport is developed, including optimality conditions for the domain size. The numerically computed optimizing flow fields consist of convection cells of decreasing aspect ratio as one allows for a stronger flow fields. Implications for natural convective transport in the motivating Rayleigh-Bénard problem are discussed.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133426/1/sandre_1.pd