Natural convection in an air layer enclosed between two vertical plates with different temperatures

Heat convection and buoyancy effects in fluids

The effect of sloping boundaries on thermal convection is studied theoretically and in the laboratory in the context of a model in which fluid is contained in a differentially heated rectangular box of small aspect ratio (depth/length), inclined at an angle δ to the vertical. Like its two limiting cases, Benard convection and convection in the vertical slot, a basic state which exists for low Rayleigh numbers becomes unstable as this parameter is increased. The types of instability and indeed the manner in which the motions become turbulent depend crucially on δ. In our work with water the following general picture of the primary instabilities applies:
 For 90° > δ > 10° with the bottom plate hotter, the instabilities are stationary longitudinal convectively driven rolls with axes oriented up the slope. Near δ = 10° there is an upper and lower Rayleigh number cut off. If the Rayleigh number is too small diffusion damps the instabilities, but if it is too large they are damped by the development of a stable upslope temperature gradient in the mean flow.For 10° > δ > −10° (negative angles imply a hotter upper plate), transverse travelling waves oriented across the slope are the first instabilities of the mean flow. They obtain their kinetic energy via the working of the upslope buoyancy force.For - 10° > δ > −85° longitudinal modes are again observed. These are rather curious in that they may exist when the stratification is everywhere positive. The necessary energy for these modes comes out of the mean velocity field and out of the mean available potential energy.Agreement between the stability theory and the experiments is generally quite good over the whole range of δ, considering the approximations involved in finding a suitable basic flow solution.For Rayleigh numbers less than ∼ 106 turbulence is only possible for positive angles. For 85° > δ > 20° the development of unsteadiness involves the occurrence and the breaking of wavy longitudinal vortices in a manner reminiscent of the development of turbulence in cylindrical Couette flow.

Stability of the flow in a differentially heated inclined box

Correlation equations for free convection heat transfer in horizontal layers of air and water

Die Ergebnisse von Messungen des Warmedurchgangs durch ebene, waagerechte, von unten beheizte und oben gekuhlte Flussigkeilsschichten werden als Verhaltnis der scheinbaren Warmeleitzahl mit Einschlus des Warmetransports durch Konvektion zur Warmeleitzahl der ruhend gedachten Flussigkeit in Abhangigkeit vom Produkt aus der Grashof- und der Prandtlzahl dargestellt. Oberhalb eines kritischen Wertes dieses Produkles bei 1700 beginnt die Konvektion, die in dem untersuchten Bereich den Warmetransport bis zum sechsfachen Betrag des Transports durch Warmeleitung allein steigert. Optische Aufnahmen, die auf den Verschiedenheiten der Lichtbrechung in warmeren und kalteren Bereichen der Flussigkeit beruhen, liefern anschauliche Bilder des Verlaufs der Konvektionsbewegungen in der Flussigkeit. Die numerischen Ergebnisse sind in einer fur die praktische Benutzung leicht verwendbaren, universellen Kurve zusammengefast, die eine Nuseltzahl mit dem vorstehend erwahnten Produkt verknupft.

Wärmedurchgang in waagerechten Flüssigkeitsschichten

A description is given of a systematic investigation of the heat transfer and the convection phenomena in enclosed plane air layers in horizontal, vertical and oblique positions. Formulae for the heat transfer are given. Some interesting types of convection flow patterns are described and the influence they have upon the heat transfer is deduced from the measurements.

J. G. A. De Graaf

Papers

The relation between the heat transfer and the convection phenomena in enclosed plane air layers