Treatment of Incompressibility and Boundary Conditions in 3-D Numerical Spectral Simulations of Plane Channel Flows

Kleiser, Leonhard; Schumann, U.

doi:10.1007/978-3-322-86146-7_17

Cited by 159 publications

(153 citation statements)

References 5 publications

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“…Both the Kleiser-Schumann [14]and Kim-Moin-Moser [13] methods begin by substituting the truncated Fourier expansion of the velocity field u. The various Fourier modes are coupled through the nonlinear term.…”

Section: The Kim-moin-moser Equationsmentioning

confidence: 99%

“…Instead, pressure is determined implicitly through the incompressibility constraint on the velocity field. One of the key algorithms for solving the Navier-Stokes equations in channel and plane Couette geometries is due to Kleiser and Schumann [14,1980]. Kleiser and Schumann introduced a numerical technique for enforcing the physically correct boundary conditions on pressure.…”

Section: Introductionmentioning

confidence: 99%

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Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow

Viswanath

Tobasco

2013

Journal of Computational Physics

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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 turbulence phenomena in good agreement with earlier simulations and experiment.

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Section: The Kim-moin-moser Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow

Viswanath

Tobasco

2013

Journal of Computational Physics

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“…As the simulation focuses on the linear and weakly nonlinear dynamics, the de-aliasing of the nonlinear term in the spatial expansion of the solution is not of crucial importance and is omitted to decrease the computational cost. Owing to the assumption of a divergence-free flow, the influence matrix method, introduced in Kleiser & Schumann (1980, 1984, is used to evaluate the pressure and the velocity field; the pressure gradient in the Navier-Stokes equation then being discretized by an implicit Euler scheme. Finally, the buoyancy term is also discretized in time by an implicit Euler scheme, since the energy equation is solved before the Navier-Stokes equations.…”

Section: Numericsmentioning

confidence: 99%

New wave generation

et al. 2010

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We present the results of a combined experimental and numerical study of the generation of internal waves using the novel internal wave generator design of . This mechanism, which involves a tunable source comprised of oscillating plates, has so far been used for a few fundamental studies of internal waves, but its full potential has yet to be realized. Our studies reveal that this approach is capable of producing a wide variety of two-dimensional wave fields, including plane waves, wave beams and discrete vertical modes in finite-depth stratifications. The effects of discretization by a finite number of plates, forcing amplitude and angle of propagation are investigated, and it is found that the method is remarkably efficient at generating a complete wave field despite forcing only one velocity component in a controllable manner. We furthermore find that the nature of the radiated wave field is well predicted using Fourier transforms of the spatial structure of the wave generator.

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“…Velocity and pressure are represented as Fourier expansions in the periodic streamwise and spanwise directions and as Chebyshev polynomials in the wall-normal direction. Channelflow uses the influence-matrix method of Kleiser & Schumann (1980) to integrate the Navier-Stokes equations forward in time. This method solves the Navier-Stokes equations at each time step via solutions of a sequence of onedimensional scalar Helmholtz equations forũ,ṽ,w andp for each wavenumber pair, with homogenous Dirichlet boundary conditions at the walls.…”

Section: Dns Resultsmentioning

confidence: 99%

Modelling for robust feedback control of fluid flows

et al. 2015

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This paper addresses the problem of designing low-order and linear robust feedback controllers that provide a priori guarantees with respect to stability and performance when applied to a fluid flow. This is challenging since whilst many flows are governed by a set of nonlinear, partial differential-algebraic equations (the Navier-Stokes equations), the majority of established control system design assumes models of much greater simplicity, in that they are firstly: linear, secondly: described by ordinary differential equations, and thirdly: finite-dimensional. With this in mind, we present a set of techniques that enables the disparity between such models and the underlying flow system to be quantified in a fashion that informs the subsequent design of feedback flow controllers, specifically those based on the H ∞ loop-shaping approach. Highlights include the application of a model refinement technique as a means of obtaining low-order models with an associated bound that quantifies the closed-loop degradation incurred by using such finite-dimensional approximations of the underlying flow. In addition, we demonstrate how the influence of the nonlinearity of the flow can be attenuated by a linear feedback controller that employs high loop gain over a select frequency range, and offer an explanation for this in terms of Landahl's theory of sheared turbulence. To illustrate the application of these techniques, a H ∞ loop-shaping controller is designed and applied to the problem of reducing perturbation wall-shear stress in plane channel flow. DNS results demonstrate robust attenuation of the perturbation shear-stresses across a wide range of Reynolds numbers with a single, linear controller.

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Treatment of Incompressibility and Boundary Conditions in 3-D Numerical Spectral Simulations of Plane Channel Flows

Cited by 159 publications

References 5 publications

Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow

Navier–Stokes solver using Green’s functions I: Channel flow and plane Couette flow

New wave generation

Modelling for robust feedback control of fluid flows

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