Numerical Simulation of Free-fountains in a Homogeneous Fluid

The behaviour of plane fountains, resulting from the injection of dense fluid upwards into a large container of homogeneous fluid of lower density, is investigated. The transient behaviour of fountains with parabolic inlet velocity profile and Reynolds numbers, 50 ≤ Re ≤ 150, Prandtl numbers, Pr=7, 300 and 700, and Froude numbers, Fr = 0.25 to 10.0 are studied numerically. The fountain behaviour falls into three distinct regimes; steady and symmetric; unsteady and periodic flapping; unsteady and aperiodic. The analytical scaling of nondimensional fountain height, zm, with Fr and Re is zm ∼ Fr4/3−2γ/3Re−γ. The constant γ is found empirically for each of the regimes. The fountain height decreases with increase in Reynolds number in the steady region but increases with Reynolds number in the unsteady regimes. However, the fountain height increases with Froude number in all regimes. Numerical results and the analytical scaling show that zm is independent of Prandtl number in the range considered. The fountain exhibits periodic lateral oscillations, i.e., periodic flapping for intermediate Froude numbers ranging from 1.25 ≤ Fr ≤ 2.25

Characterisation of Low Reynolds Number Fountain behaviour

Experimental evidence for previously unreported fountain behaviour is presented. It has been found that the first unstable mode of a wall bounded three dimensional round fountain is a laminar flapping motion that can grow to a circling or multi-modal flapping motion. With increasing Froude and Reynolds numbers, fountain behaviour becomes more disorderly, exhibiting a laminar bobbing motion. The transition between steady behaviour, the initial flapping modes and the laminar bobbing flow can be approximately described by a function C =FrRe 2/3. The transition to turbulence occurs at Re > 120, independent of Froude number. For Fr > 20 and Re 120 these instabilities cause the fountain to intermittently breakdown into turbulent jet like flow. A regime map of the fountain behaviour for 0:7 < Fr < 55 and 15 < Re < 1100 is presented and the underlying mechanisms for the observed behaviour are proposed

Impinging plane fountains in a homogeneous fluid

The transient behaviour of plane fountains with a uniform inlet velocity, injected upwards into a quiescent\ud homogeneous fluid of lower density to impinge on a solid flat ceiling, is investigated. The Reynolds\ud number, the Froude number and the Prandtl number of these impinging fountains have the values in the\ud ranges of 50 6 Re 6 1000; 8 6 Fr 6 20 and 7 6 Pr 6 700, and the height of the solid ceiling away from\ud the fountain source is varied in the range of 10Xin 6 H 6 30Xin, where Xin is the half-width of the planar\ud fountain source slot. A scaling is found by dimensional analysis for the augmented spreading distance\ud (H þ Xd, where Xd is the spreading distance of the impinging fountain), which shows that\ud ðH þ XdÞ=Xin Fr43\ud 23\ud ðcþgþ2/ÞReðcþgÞPrgðH=XinÞ/, where the powers c; g and / can be determined empirically.\ud The direct numerical simulation results show that after the fountain impinges upwards on the ceiling\ud it spreads outwards along the ceiling until gravity forces it to fall. Two different scenarios are\ud identified. In the first scenario, a nearly constant measurable spreading distance is obtained at full\ud development. In the second scenario, however, the fountain floods the whole computational domain\ud and no spreading distance exists at full development. The numerical results further show that in the first\ud scenario the augmented spreading distance (H þ Xd) has the reduced scaling of ðH þ XdÞ=Xin \ud Fr2=3ðH=XinÞ1=2 for the plane impinging fountains with the parameter values in the ranges of\ud 50 6 Re < 125; 8 6 Fr 6 20 and 7 6 Pr 6 700

Behaviour of laminar plane fountains with a parabolic inlet velocity profile in a homogeneous fluid

The behaviour of plane laminar fountains with parabolic velocity inlet profile is studied using numerical simulation over the parametric range 0.25 ≤ Fr ≤ 10.0, 50 ≤ Re ≤ 150 and Pr = 7, 300, 700. The behaviour of the flow is most strongly affected by the Froude number and to a lesser extent by the Reynolds number, particularly for weak fountains at low Reynolds numbers. Behaviour is independent of the Prandtl number over the parametric range investigated. Three distinct regimes are observed: a steady symmetric pattern at low Froude numbers (Fr < 1.25), an unsteady periodic flapping characterised by lateral oscillations at intermediate Froude numbers (1.25 ≤ Fr ≤ 2.25) and an unsteady aperiodic flapping at higher Froude numbers. Based on dimensional analysis, the rise height of the fountain is shown to follow the correlation z(m)~Fr⁴/³⁻²/³⁽ᶜ⁺ᵈ⁾Re⁻⁽ᶜ⁺ᵈ⁾Pr⁻ᵈ. The constants c and d are determined from numerical results for each regime

Laminar plane fountains impinging on a ceiling with an opposing heat flux

The behaviour of unsteady planar strong fountains, impinging on a ceiling with an opposing heat flux on the ceiling, is investigated numerically for 8:0 6 Fr 6 30:0 at Re ¼ 50 and Pr ¼ 7. The height of the ceiling is varied in 10 6 H=Xin 6 25, where Xin is the half-width of the fountain source, and the non-dimensional gradient of the temperature difference between the fountain source and the ceiling is varied for\ud 0:2 6 Dh=ðH=XinÞ 6 1:8. It is found that the fountain does not hit the ceiling, but instead stagnates and spreads at some distance from the ceiling due to the stratification of the fluids in the immediate vicinity of the ceiling. The scaling and direct numerical simulation results show that the augmented spreading distance Hd þ Xd has the scaling of Hd þ Xd XinFr2=9ðH=XinÞ1=2½Dh=ðH=XinÞ1=3 in the range studied, where Hd is the maximum fountain height, Xd is the spreading distance measured at Hd, respectively

Line fountain behavior at low-Reynolds number

In this paper, we present the line fountain behavior at\ud low-Reynolds numbers obtained by experiments. The experiments are conducted over the range of Reynolds number 2.1<=Re<=127 and Froude number 0.4<=Fr<=42. It is observed that the fountain behavior can be categorized broadly into four regimes: the steady; flapping; laminar mixing; and\ud jet-type mixing behavior, at full development. The critical Froude number for transition from a steady to unsteady flow varies with the Reynolds number. For Re>=60, the transition is independent of Re and is well described by the Fr~1.0 line. Over the range 10<Re<=50, the transition can be approximated by a constant FrRe^(2/3) line. For Re<=10, there is a higher dependency on the Reynolds number with a very sharp increase in the critical Froude number and it is hypothesized that the demarcation line follows Fr~Re^(-n), where n~2-4. The fountain exhibits flapping behavior in the range 13Re^(-2/3)<=Fr<=37Re^(-2/3). These observed fountain behaviors are mapped on to a Re-Fr plot. In addition,\ud the observed, non-dimensionalized fountain height zm is found to be in a reasonable agreement with previous results on laminar line fountains. In particular, the experiment results confirm the scaling zm~FrRe^(-1/2) in the steady regime which was obtained previously by scaling analysis

Height and stability of laminar plane fountains in a homogeneous fluid

The behaviour of plane fountains, resulting from the injection of a denser fluid upwards into a large body of a lighter homogeneous\ud fluid, is investigated numerically. The transient behaviour of fountains with a uniform inlet velocity, Reynolds number Re = 100, Prandtl\ud number Pr = 7, and Froude number 0.25 6 Fr 6 10.0 is studied numerically. In the present case, the density variation is as a result of\ud temperature difference between the fountain and the ambient fluids. Three distinct regimes are identified; steady and symmetric fountains\ud for 0.25 6 Fr 6 2.0, unsteady fountains with periodic lateral oscillation for 2.25 6 Fr 6 3.0, and unsteady fountains with aperiodic lateral\ud oscillations for Fr P4.0. It is found empirically that the non-dimensional fountain height, zm, scales differently with Froude number\ud in each of these regimes; in the steady and symmetric region zm Fr, in the unsteady and periodic lateral oscillation region zm Fr1:15 and\ud in the unsteady and aperiodic lateral oscillation region zm Fr4=3. The results are compared with previous numerical and experimental\ud results, where available and are consistent.\ud 2008 Elsevier Ltd. All rights reserved

Critical Froude number for transition from a steady to an unsteady fountain injected into a homogeneous fluid

In this paper, we present the critical Froude number for unsteadiness at full development of plane fountains. Numerical investigations are conducted over the range of Reynolds number, 6 <= Re <= 120. The critical Froude number for transition from a steady to unsteady flow varies with the Reynolds number. For Re >= 60, the transition is independent of Re and is nearly constant at a Froude number of Fr ~ 1.0. Over the range 6 < Re <= 50, there is a significant increase in the transition Froude number

Direct simulation of impinging plane fountains in a homogeneous fluid

In this paper we present the behaviour of plane fountains, injected into a homogeneous fluid of lower density, impinging on a ceiling. The transient behaviour of the impinging fountain with Reynolds number 100 ≤ Re ≤ 1000, Prandtl number Pr=7, and Froude number Fr = 4 and 5 is studied by direct numerical simulation using a fractional-step solution of the Navier–Stokes equations. When a vertical fountain impinges on a ceiling it spreads until gravity forces it to fall. The results show that the spreading distance is dependent on the source Froude number and independent of Reynolds number for range studied