On the accuracy of the calculation of transient growth in plane Poiseuille flow

Zhao, Shi; Duncan, Stephen R.

doi:10.1002/fld.3875

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…It is already discussed in the literature that there are two kinds of spurious eigenvalues that are obtained in generalized eigenvalue problems; namely, numerically spurious eigenvalues and physically spurious eigenvalues. 18,37 Numerically spurious eigenvalues would appear for the stability operator of linearized Navier-Stokes equations as its eigenspectrum contains an infinite number of eigenvalues. Thereby, with a finite number of grids, it is not possible to resolve the whole spectrum.…”

Section: Spurious Eigenvalues: Operator Matrix Correctionmentioning

confidence: 99%

A pseudospectral approach applicable for time integration of linearized N‐S operator that removes pole singularity and physically spurious eigenmodes

Nayak

Das

2019

Numerical Methods in Fluids

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Summary This paper addresses a modified singularity removal technique for the eigenvalue or optimal mode problems in pipe flow using a pseudospectral method. The current approach results in the linear stability operator to be devoid of any unstable physically spurious modes, and thus, it provides higher numerical stability during time‐based integration. The correctness of the numerical operator is established by calculating the known eigenvalues of pipe Poiseuille flow. Subsequently, the optimal modes are determined with Farrell's approach and compared with the existing literature. The usefulness of this approach is further demonstrated in the time‐based numerical integration of the linearized Navier‐Stokes operator for the adjoint method–based optimal mode determination. The numerical scheme is implemented with the radial velocity‐radial vorticity formulation. Even number of Chebyshev‐Lobatto grid points are distributed over the domain r∈[−1,1] omitting the centerline, which also efficiently provides higher resolution near the wall boundary. The boundary conditions are imposed with homogeneous wall boundary conditions, whereas the analytic nature of a proper set of base functions enforces correct centerline conditions. The resulting redundancy introduced in the process is eliminated with the proper usage of parity.

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Section: Spurious Eigenvalues: Operator Matrix Correctionmentioning

confidence: 99%

A pseudospectral approach applicable for time integration of linearized N‐S operator that removes pole singularity and physically spurious eigenmodes

Nayak

Das

2019

Numerical Methods in Fluids

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“…[22,23,26,37,109,110,115,116,154,155,164,179,206,213,222,223,267,275,385,487,488,498]. This class of problem is very difficult to handle mathematically due to the fact that the mathematical operators which arise in the instability analysis are non-symmetric and the resulting eigenfunctions are close to being linearly dependent.…”

Section: Poiseuille Flow Slip Boundary Conditionsmentioning

confidence: 99%

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Straughan

2015

Advances in Mechanics and Mathematics

113

111

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In this chapter we describe work of [366] and of [393] on thermal convection in a Darcy porous material in a vertical layer subject to different temperatures on the vertical walls.For the case of a single temperature the problem where the porous medium occupies a vertical layer subject to differential heating from one side to the other has attracted much attention. This problem has much application to insulation, such as in the area of building design, and is of importance in window double glazing. For the single temperature situation the first proof that a vertical porous slab of Darcy type which is held at fixed but different temperatures on the vertical walls is stable to perturbations from the equilibrium state is due to [156]. His analysis is based on the linear theory and analyses the two-dimensional problem.[409] further analysed this problem but in three-dimensions and he treated the completely nonlinear situation by employing energy-like integral methods. In particular, [409] produced a threshold for the Rayleigh number which guarantees global nonlinear stability, i.e. regardless of how large the initial perturbation may be. He also employed a generalized energy method to demonstrate that the result of [156] is true in the fully three-dimensional case although the initial data must be restricted. Articles dealing with other interesting aspects of convection in a vertical porous slab or pipe involving effects like double diffusion, and LTNE, include the papers by [32, 33, 36, 46, 211, 220, 221, 329, 330, 362] and [460]. The methods of energy stability techniques applied specifically to convection in a vertical porous slab are addressed in [144, 229] and [356], and these are discussed in detail in the book by [414, pp. 126-134].The analogous problem in LTNE to that of [156] was first addressed by [366]. Specifically [366] dealt with the [156] problem of thermal convection in a vertical porous medium, but he employed the theory of local thermal non-equilibrium, allowing for different fluid and solid temperatures. The interesting work of [366] demonstrates that the vertical configuration is always stable according to linear theory, even with the much more complicated LTNE theory. [393] continued the problem of [366] but they analysed the complete nonlinear three-dimensional situation.

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On the accuracy of the calculation of transient growth in plane Poiseuille flow

Cited by 2 publications

References 26 publications

A pseudospectral approach applicable for time integration of linearized N‐S operator that removes pole singularity and physically spurious eigenmodes

A pseudospectral approach applicable for time integration of linearized N‐S operator that removes pole singularity and physically spurious eigenmodes

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

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