Impinging plane fountains in a homogeneous fluid

Abstract

The transient behaviour of plane fountains with a uniform inlet velocity, injected upwards into a quiescent homogeneous fluid of lower density to impinge on a solid flat ceiling, is investigated. The Reynolds number, the Froude number and the Prandtl number of these impinging fountains have the values in the ranges of 50 ⩽ Re ⩽ 1000 , 8 ⩽ Fr ⩽ 20 and 7 ⩽ Pr ⩽ 700 , and the height of the solid ceiling away from the fountain source is varied in the range of 10 X in ⩽ H ⩽ 30 X in , where X in is the half-width of the planar fountain source slot. A scaling is found by dimensional analysis for the augmented spreading distance ( H + X d , where X d is the spreading distance of the impinging fountain), which shows that ( H + X d ) / X in ∼ Fr 4 3 - 2 3 ( γ + η + 2 ϕ ) Re - ( γ + η ) Pr - η ( H / X in ) ϕ , where the powers γ , η and ϕ can be determined empirically. The direct numerical simulation results show that after the fountain impinges upwards on the ceiling it spreads outwards along the ceiling until gravity forces it to fall. Two different scenarios are identified. In the first scenario, a nearly constant measurable spreading distance is obtained at full development. In the second scenario, however, the fountain floods the whole computational domain and no spreading distance exists at full development. The numerical results further show that in the first scenario the augmented spreading distance ( H + X d ) has the reduced scaling of ( H + X d ) / X in ∼ Fr 2 / 3 ( H / X in ) 1 / 2 for the plane impinging fountains with the parameter values in the ranges of 50 ⩽ Re < 125 , 8 ⩽ Fr ⩽ 20 and 7 ⩽ Pr ⩽ 700 .