Mixed Convection equations
Convection Thermique Mixte: une devinette...
(Lagrée P.-Y.) à jour 06/04
LMM-Univ PARIS 6, B 162,
4 place Jussieu, 75252 PARIS
The thermal mixed convection boundary- layer flow over a flat horizontal cooled plate is revisited. It is shown that this flow is very similar to what happens in a free convection hypersonic boundary layer and that the observed branching singular solutions may be reinterpreted in the framework of ''triple deck''. In fact this is very similar to what happens in an hydrolic jump (in water), so this problem is a kind of hydrolic jump in thermal flow. Two saillant structures emerge, one in double deck, if the buoyancy is very small, and an other in single deck, if the buoyancy is of order one: this is a reinterpretation of Steinrück results. A numerical simulation of the unsteady boundary layer in the case of the impulsively started and cooled plate is done, it leads to the separation of the boundary layer as predicted by triple deck. Incomming flow at temperature T0 and velocity U0
---> ---> ---> ---> ---> ---> ---> ---> ---> | ---> ---> --> | ---> --> -> | gravity ---> ====================================================== | v Flat Plate at Temperature Tw There is a competition between the forced convection and the buoyancy induced natural convection. (Tw-T0) L Re-1/2 J is the Richardson Parameter: J = g a --------------- U02 This parameter gauges the mixed convection effect. If the temperature of the wall is lesser than the temperature of the flow, a self induced separation appears. The system to solve: òu òv -- + -- = 0 òx òy òu òu òu òp ò2u -- + u -- + v -- = - -- + --- òt òx òy òx òy2
òp 0 = - -- + J T òy òT òT òT ò2T -- + u -- + v -- = --- òt òx òy òy2 u=v=0 T=1 on the wall u=1 T=0 far over the plate. òp/òx=0 at the output. The final steady state depends of the size of the computational domain. If it is enough large, a separated bulb may be created. Preceding results stopped just before the point a zero skin friction. Thanks to triple deck, it may be shown that this flow is not "parabolic": there is up stream influence (like in hypersonic flows). the reduced skin friction function of the domain size, results are compared with the calculation of Wickern 1991 (compiled by Steinrück and refered as ''marching''. The size of the domain is xout=5 10 20 50 and 125. The boundary layer separation is obtained (separation buble). the displacement thickness function of the domain size, from the nose to x=15. (The size of the domain is xout=5 10 20 50 and 125). The thermal hydrolic jump may be observed as the "jump" of the displacement thickness where is the separation buble.