Stability of flow in a channel with longitudinal grooves

Abstract

The travelling wave instability in a channel with small-amplitude longitudinal grooves of arbitrary shape has been studied. The disturbance velocity field is always three-dimensional with disturbances which connect to the two-dimensional waves in the limit of zero groove amplitude playing the critical role. The presence of grooves destabilizes the flow if the groove wavenumber β is larger than β tran ≈ 4.22, but stabilizes the flow for smaller β. It has been found that β tran does not depend on the groove amplitude. The dependence of the critical Reynolds number on the groove amplitude and wavenumber has been determined. Special attention has been paid to the drag-reducing long-wavelength grooves, including the optimal grooves. It has been demonstrated that such grooves slightly increase the critical Reynolds number, i.e. such grooves do not cause an early breakdown into turbulence.

New instability mode in a grooved channel

“…Analysis of the transient growth demonstrated that optimal disturbances have the form of streamwise vortices (Szumbarski & Floryan 2006). Moradi & Floryan (2014) investigated the effects of longitudinal grooves (grooves parallel to the flow direction) and documented changes in the characteristics of the TS waves. They were able to demonstrate that short-wavelength grooves destabilize such waves while long-wavelength grooves stabilize them, when compared with their behaviour in a smooth channel.…”

“…The problem formulation has been described in detail by Moradi & Floryan (2014) and thus this presentation is limited to a short outline. Consider steady and fully developed flow through a smooth straight channel extending to ±∞ in the x-direction and driven by a constant pressure gradient along the x-direction.…”

“…Solution of (2.6) has to be determined numerically due to geometric complexities. A spectral discretization method based on the Fourier and Chebyshev expansions is used (Mohammadi & Floryan 2012;Moradi & Floryan 2014). The solution is assumed to be in the form of a Fourier expansion in the z-direction, i.e.…”

“…The modal functions are discretized using the Chebyshev expansions, and these expansions are truncated at Chebyshev polynomials of order N T . The Galerkin projection method is used to construct a system of linear algebraic equations, and the boundary conditions are enforced using the IBC concept combined with the tau method (Szumbarski & Floryan 1999;Floryan 2002;Moradi & Floryan 2014). The DT method (Cabal et al 2002) provides an alternative if grooves with very large amplitudes are of interest.…”

Spectrally-accurate algorithm for the analysis of flows in two-dimensional vibrating channels

“…The efficiency has been increased by an order of magnitude through the development of specialized solvers which account for the special structure of the coefficient matrix [38,39]. The method has been used to identify the laminar drag-reducing grooves [16][17][18][19][20] and to study the effects of various grooves on the flow stability [40][41][42][43][44][45][46]. This work is focused on the development of an efficient algorithm suitable for the analysis of changes in the pressure gradient required to drive a specified flow rate through a vibrating channel.…”