## Abstract

This analytical study deals with the spatio-temporal evolution of linear thermo-convective instabilities in a horizontal fluid layer heated from below (the Rayleigh--Benard system) and subject to a horizontal pressure gradient (Poiseuille flow). The novelty consists of a spatially inhomogeneous temperature, in the form of a two-dimensional bump imposed on the lower plate, while the upper plate is kept at a constant temperature. The inhomogeneous boundary temperature and the mean flow of the Rayleigh--Benard--Poiseuille system break the symmetries of the classical Rayleigh--Benard system. The instabilities of interest are therefore spatially localised packets of convection rolls. If a mode of this type is synchronized, it is called a global mode. Assuming that the characteristic scale of the spatial variation of the lower plate temperature is large compared to the wavelength of the rolls, global modes are sought in the form of Eigenmodes in the confined vertical direction, modulated by a two-dimensional WKBJ expansion in the slowly-varying horizontal directions. Such an expansion breaks down at points where the group velocity of the instability vanishes, i.e. at WKBJ turning points. In the neighbourhood of one such point, located at the top of the temperature bump, the boundedness of the solution imposes a selection criterion
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## Citation Formats

Martinand, D.
Analytical determination of 3-D global modes in Rayleigh-Benard-Poiseuille-type mixed convection flow; Determination analytique des modes globaux tridimensionnels en ecoulement de convection mixte du type Rayleigh-Benard-Poiseuille.
France: N. p.,
2003.
Web.

Martinand, D.
Analytical determination of 3-D global modes in Rayleigh-Benard-Poiseuille-type mixed convection flow; Determination analytique des modes globaux tridimensionnels en ecoulement de convection mixte du type Rayleigh-Benard-Poiseuille.
France.

Martinand, D.
2003.
"Analytical determination of 3-D global modes in Rayleigh-Benard-Poiseuille-type mixed convection flow; Determination analytique des modes globaux tridimensionnels en ecoulement de convection mixte du type Rayleigh-Benard-Poiseuille."
France.

@misc{etde_20802740,

title = {Analytical determination of 3-D global modes in Rayleigh-Benard-Poiseuille-type mixed convection flow; Determination analytique des modes globaux tridimensionnels en ecoulement de convection mixte du type Rayleigh-Benard-Poiseuille}

author = {Martinand, D}

abstractNote = {This analytical study deals with the spatio-temporal evolution of linear thermo-convective instabilities in a horizontal fluid layer heated from below (the Rayleigh--Benard system) and subject to a horizontal pressure gradient (Poiseuille flow). The novelty consists of a spatially inhomogeneous temperature, in the form of a two-dimensional bump imposed on the lower plate, while the upper plate is kept at a constant temperature. The inhomogeneous boundary temperature and the mean flow of the Rayleigh--Benard--Poiseuille system break the symmetries of the classical Rayleigh--Benard system. The instabilities of interest are therefore spatially localised packets of convection rolls. If a mode of this type is synchronized, it is called a global mode. Assuming that the characteristic scale of the spatial variation of the lower plate temperature is large compared to the wavelength of the rolls, global modes are sought in the form of Eigenmodes in the confined vertical direction, modulated by a two-dimensional WKBJ expansion in the slowly-varying horizontal directions. Such an expansion breaks down at points where the group velocity of the instability vanishes, i.e. at WKBJ turning points. In the neighbourhood of one such point, located at the top of the temperature bump, the boundedness of the solution imposes a selection criterion for the global modes which provides the growth rate (or equivalently the critical threshold), the frequency and the wave vector of the most amplified global mode. This study thus generalizes to two-dimensional cases the methods used and the results obtained for one-dimensional inhomogeneities. The analysis is first applied to a simplified governing equation obtained by an envelope formalism and the analytical results are compared with numerical solutions of the amplitude equation. The formalism is finally applied to the Rayleigh--Benard--Poiseuille system described by the Navier--Stokes equations with the Boussinesq approximation. (author)}

place = {France}

year = {2003}

month = {Jan}

}

title = {Analytical determination of 3-D global modes in Rayleigh-Benard-Poiseuille-type mixed convection flow; Determination analytique des modes globaux tridimensionnels en ecoulement de convection mixte du type Rayleigh-Benard-Poiseuille}

author = {Martinand, D}

abstractNote = {This analytical study deals with the spatio-temporal evolution of linear thermo-convective instabilities in a horizontal fluid layer heated from below (the Rayleigh--Benard system) and subject to a horizontal pressure gradient (Poiseuille flow). The novelty consists of a spatially inhomogeneous temperature, in the form of a two-dimensional bump imposed on the lower plate, while the upper plate is kept at a constant temperature. The inhomogeneous boundary temperature and the mean flow of the Rayleigh--Benard--Poiseuille system break the symmetries of the classical Rayleigh--Benard system. The instabilities of interest are therefore spatially localised packets of convection rolls. If a mode of this type is synchronized, it is called a global mode. Assuming that the characteristic scale of the spatial variation of the lower plate temperature is large compared to the wavelength of the rolls, global modes are sought in the form of Eigenmodes in the confined vertical direction, modulated by a two-dimensional WKBJ expansion in the slowly-varying horizontal directions. Such an expansion breaks down at points where the group velocity of the instability vanishes, i.e. at WKBJ turning points. In the neighbourhood of one such point, located at the top of the temperature bump, the boundedness of the solution imposes a selection criterion for the global modes which provides the growth rate (or equivalently the critical threshold), the frequency and the wave vector of the most amplified global mode. This study thus generalizes to two-dimensional cases the methods used and the results obtained for one-dimensional inhomogeneities. The analysis is first applied to a simplified governing equation obtained by an envelope formalism and the analytical results are compared with numerical solutions of the amplitude equation. The formalism is finally applied to the Rayleigh--Benard--Poiseuille system described by the Navier--Stokes equations with the Boussinesq approximation. (author)}

place = {France}

year = {2003}

month = {Jan}

}