The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Closed-Loop Turbulence Control: Progress and Challenges

Finite element methods for incompressible viscous flow

The objective of this paper is to introduce the essential ingredients of linear systems and control theory to the fluid mechanics community, to discuss the relevance of this theory to important open problems in the optimization, control, and forecasting of practical flow systems of engineering interest, and to outline some of the key ideas that have been put forward to make this connection tractable. Although many significant advances have already been made, many new challenges lie ahead before the full potential of this synthesis of disciplines can be realized.

A Linear Systems Approach to Flow Control

Fully resolved direct numerical simulations (DNSs) have been performed with a high-order spectral element method to study the flow of an incompressible viscous fluid in a smooth circular pipe of radius R and axial length 25R in the turbulent flow regime at four different friction Reynolds numbers Reτ = 180, 360, 550 and $1\text{,}000$. The new set of data is put into perspective with other simulation data sets, obtained in pipe, channel and boundary layer geometry. In particular, differences between different pipe DNS are highlighted. It turns out that the pressure is the variable which differs the most between pipes, channels and boundary layers, leading to significantly different mean and pressure fluctuations, potentially linked to a stronger wake region. In the buffer layer, the variation with Reynolds number of the inner peak of axial velocity fluctuation intensity is similar between channel and boundary layer flows, but lower for the pipe, while the inner peak of the pressure fluctuations show negligible differences between pipe and channel flows but is clearly lower than that for the boundary layer, which is the same behaviour as for the fluctuating wall shear stress. Finally, turbulent kinetic energy budgets are almost indistinguishable between the canonical flows close to the wall (up to y + ≈ 100), while substantial differences are observed in production and dissipation in the outer layer. A clear Reynolds number dependency is documented for the three flow configurations.

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Direct Numerical Simulation of Turbulent Pipe Flow at Moderately High Reynolds Numbers

SIMSON : A Pseudo-Spectral Solver for Incompressible Boundary Layer Flows

(Received 16 November 2004 and in revised form 18 July 2005) This work extends the estimator developed in Part 1 of this study to the problem of estimating a turbulent channel flow at Reτ = 100 based on a history of noisy measurements on the wall. The key advancement enabling this work is the development and implementation of an efficient technique to extract, from direct numerical simulations, the relevant statistics of an appropriately defined ‘external forcing’ term on the Navier–Stokes equation linearized about the mean turbulent flow profile. This forcing term is designed to account for the unmodelled (nonlinear) terms during the computation of the (linear) Kalman filter feedback gains in Fourier space. Upon inverse transform of the resulting feedback gains computed on an array of wavenumber pairs to physical space, we obtain, as in Part 1, effective and well-resolved feedback convolution kernels for the estimation problem. It is demonstrated that, by applying the feedback so determined, satisfactory correlation between the actual and estimated flow is obtained in the near-wall region. As anticipated, extended Kalman filters (with the nonlinearity of the actual system reintroduced into the estimator model after the feedback gains are determined) outperform standard (linear) Kalman filters on the full system.

State estimation in wall-bounded flow systems. Part 2. Turbulent flows

In applications involving the model-based control of transitional wall-bounded flow systems, it is often desired to estimate the interior flow state based on a history of noisy measurements from ...

/pdf/state-estimation-in-wall-bounded-flow-systems-part-1-4tdm60isws.pdf

State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows

This paper presents the application of feedback control to spatially developing boundary layers. It is the natural follow-up of Hogberg & Henningson (J. Fluid Mech. vol. 470, 2002, p. 151), whe ...

/pdf/linear-feedback-control-and-estimation-applied-to-6xul0qy8h4.pdf

Mattias Chevalier

Papers

SIMSON : A Pseudo-Spectral Solver for Incompressible Boundary Layer Flows

State estimation in wall-bounded flow systems. Part 2. Turbulent flows

State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows

Linear feedback control and estimation applied to instabilities in spatially developing boundary layers

Numerical simulations for high offset intake diffuser flows