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

Viswanath, Divakar; Tobasco, Ian

doi:10.1016/j.jcp.2013.06.004

Cited by 4 publications

(17 citation statements)

References 37 publications

(89 reference statements)

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“…Each of the gradient ascent procedures described in §3 are of the forṁ 20) where x is the state vector, L is an "easily" invertible linear operator, N is a nonlinear operator, and f is a forcing function. We follow Viswanath & Tobasco (2013) and consider linear multi-step schemes as follows:…”

Section: Temporal Discretisationmentioning

confidence: 99%

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Wall-to-wall optimal transport in two dimensions

2020

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Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of velocity fields by a Péclet number Pe proportional to their root-mean-square rate-of-strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e., the Nusselt number Nu up to Pe ≈ 10 5 . The resulting transport exhibits a change of scaling from Nu − 1 ∼ Pe 2 for Pe < 10 in the linear regime to Nu ∼ Pe 0.54 for Pe > 10 3 . Optimal fields are observed to be approximately separable, i.e., products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound Pe 6/11 = Pe 0.54 as Pe → ∞ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh-Bénard convection are discussed.

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Section: Temporal Discretisationmentioning

confidence: 99%

“…The Kleiser-Schumann algorithm is a method for solving the modified Stokes problem (3.31), see Kleiser & Schumann (1980) ;Viswanath & Tobasco (2013). Since the modified Stokes problem is linear we solve wave-number by wave-number equations of the form (D 2 − β 2 n )u n = f n −∇p n (A 73) ∇ · u n = 0 (A 74)…”

Section: A3 Kleiser-schumann Algorithmmentioning

confidence: 99%

Wall-to-wall optimal transport in two dimensions

2020

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“…Use an appropriate (inverse) fast transform to evaluate the nonlinear terms on the clustered grid. 4. Solve the resulting linear problem.…”

Section: Introductionmentioning

confidence: 99%

“…In the current paper, we follow a different, more radical approach, which was also investigated by Viswanath [4]. The crucial step is to solve the BVP analytically using Green's function, thereby eliminating the need for numerical differentiation and linear solving altogether.…”

Section: Introductionmentioning

confidence: 99%

On Green’s function-based time stepping for semilinear initial–boundary value problems

van Veen

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tThe Fourier spectral approach is central to the analysis of semilinear initial-boundary value problems when considered with periodic boundary conditions. It aids theoretical arguments, such as renormalisation group analysis, as well as numerical simulation through standard pseudo-spectral methods. Fixed boundary conditions, in contrast, require the use of non-uniform grids, usually generated by bases of orthogonal polynomials. On such bases, numerical differentiation is ill-conditioned and can potentially lead to a catastrophic blowup of round-off error. In this paper, we apply ideas explored by Viswanath (2013) in the context of Navier-Stokes solvers to completely eliminate numerical differentiation and linear solving from the time-stepping algorithm in favour of numerical quadrature. We propose a concrete quadrature rule and test its stability, scalability and spectral accuracy on a Kuramoto-Sivashinsky template problem over a range of five orders of magnitude of the viscosity.

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“…The main application is to the integration of the Navier-Stokes equation in rectangular geometry and in the turbulent regime. In a companion paper [20], we present a computation of turbulent channel flow that reaches Re τ = 2380 using 10 9 grid points and M = 1024. Only 10 nodes of a small cluster are used in this computation.…”

Section: Introductionmentioning

confidence: 99%

Spectral integration of linear boundary value problems

Viswanath

2015

Journal of Computational and Applied Mathematics

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Spectral integration was deployed by Orszag and co-workers (1977Orszag and co-workers ( , 1980Orszag and co-workers ( , 1981 to obtain stable and efficient solvers for the incompressible Navier-Stokes equation in rectangular geometries. Two methods in current use for channel flow and plane Couette flow, namely, Kleiser-Schumann (1980) and Kim-MoinMoser (1977), rely on the same technique. In its current form, the technique of spectral integration, as applied to the Navier-Stokes equations, is dominated by rounding errors at higher Reynolds numbers which would otherwise be within reach. In this article, we derive a number of versions of spectral integration and explicate their properties, with a view to extending the Kleiser-Schumann and KimMoin-Moser algorithms to higher Reynolds numbers. More specifically, we show how spectral integration matrices that are banded, but bordered by dense rows, can be reduced to purely banded matrices. Key properties, such as the accuracy of spectral integration even when Green's functions are not resolved by the underlying grid, the accuracy of spectral integration in spite of ill-conditioning of underlying linear systems, and the accuracy of derivatives, are thoroughly explained.

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

Cited by 4 publications

References 37 publications

Wall-to-wall optimal transport in two dimensions

Wall-to-wall optimal transport in two dimensions

On Green’s function-based time stepping for semilinear initial–boundary value problems

Spectral integration of linear boundary value problems

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