Ueber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen

Oberbeck, A.

doi:10.1002/andp.18792430606

Cited by 356 publications

(167 citation statements)

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“…Now, assuming smallness of deviations of the density and pressure from their BRS values and exploiting the equation of state, one obtains a simplified equation, which is already very close to the standard Navier-Stokes equation in the Boussinesq approximation. Introducing (generalized) potential temperature (see Hauf and Höller 1987), one obtains the well-known system of hydrodynamic equations in the Boussinesq approximation (Oberbeck 1879 andBoussinesq 1903). Close to the ground, where one can neglect the dependence of the density on height, the system reads:…”

Section: Simplified Hydrodynamic Equations and Reynolds Decompositionmentioning

confidence: 99%

Turbulent fluxes in stably stratified boundary layers

L’vov¹,

Procaccia²,

Rudenko³

2008

Phys. Scr.

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We present here an extended version of an invited talk we gave at the international conference 'Turbulent Mixing and Beyond'. The dynamical and statistical description of stably stratified turbulent boundary layers with the important example of the stable atmospheric boundary layer in mind is addressed. Traditional approaches to this problem, based on the profiles of mean quantities, velocity second-order correlations and dimensional estimates of the turbulent thermal flux, run into a well-known difficulty, predicting the suppression of turbulence at a small critical value of the Richardson number, in contradiction to observations. Phenomenological attempts to overcome this problem suffer from various theoretical inconsistencies. Here, we present an approach taking into full account all the second-order statistics, which allows us to respect the conservation of total mechanical energy. The analysis culminates in an analytic solution of the profiles of all mean quantities and all second-order correlations, removing the unphysical predictions of previous theories. We propose that the approach taken here is sufficient to describe the lower parts of the atmospheric boundary layer, as long as the Richardson number does not exceed an order of unity. For much higher Richardson numbers, the physics may change qualitatively, requiring careful consideration of the potential Kelvin-Helmoholtz waves and their interaction with the vortical turbulence.

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Section: Simplified Hydrodynamic Equations and Reynolds Decompositionmentioning

confidence: 99%

Turbulent fluxes in stably stratified boundary layers

L’vov¹,

Procaccia²,

Rudenko³

2008

Phys. Scr.

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“…The Boussinesq system of hydrodynamics equations [4], [18], arises from a zero order approximation to the coupling between the Navier-Stokes equations and the thermodynamic equation [17]. The presence of density gradients in a uid leads to the conversion of gravitational potential energy into motion through the action of buoyant forces.…”

Section: The Modelmentioning

confidence: 99%

Spatial and time localization of solutions of the Boussinesq system with nonlinear thermal diffusion

Galiano¹

2000

Nonlinear Analysis: Theory, Methods & Applications

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“…Its roots stem from the end of the 19 th century; [6]. Nevertheless, only in 1996 Rajagopal, Růžička, and Srinivasa succeeded in its justification from the point of view of Continuum Mechanics and precisely clarified the applicability for concrete configurations; [7].…”

Section: Introductionmentioning

confidence: 99%

“…It is worth noticing that this perturbation parameter was already chosen by Oberbeck; [6]. For most liquids α < 10 −4 /K and thus, ε is small for a large range of temperature differences.…”

Section: Introductionmentioning

confidence: 99%

A limit problem in natural convection

Kagei

Růžička

Thäter

2006

Nonlinear differ. equ. appl.

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Abstract. We discuss a model limit problem which arises as a first step in the mathematical justification of our Boussinesq-type approximation [4], which takes into account dissipative heating in natural convection. We treat a simplified highly non linear system depending on a (perturbation) parameter ε. The main difficulty is that for ε = 0 the velocity is not solenoidal. First we prove that our system has weak solutions for each fixed ε. Moreover, while the chosen perturbation parameter ε tends to zero we show, that we arrive at the usual incompressible case and the standard Boussinesq approximation.2000 Mathematics Subject Classification: 35K55, 35Q35, 76E19, 76E20, 80A1.

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Ueber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen

Cited by 356 publications

References 0 publications

Turbulent fluxes in stably stratified boundary layers

Turbulent fluxes in stably stratified boundary layers

Spatial and time localization of solutions of the Boussinesq system with nonlinear thermal diffusion

A limit problem in natural convection

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