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1 Models Encompassing Hydaulic Jumps in Radial Flows Ove a Hoizontal Plate A.J. ROBERTS and D.V. STRUNIN Depatment of Mathematics and Computing Univesity of Southen Queensland Toowoomba, Queensland 3 AUSTRALIA aobets@usq.edu.au; stunin@usq.edu.au Abstact: A cicula jump foms in the adial flow thickness when a downwads steam hits a hoizontal plate. Cente manifold theoy is used to igoously deive, fom the Navie-Stokes equations, a dynamic model fo slow hoizontal vaiations of flow thickness and velocity. An advantage of this appoach is the capacity to study non-stationay egimes and examine stability of the flow. Numeical solutions of the model epoduce expeimentally obseved eciculation unde the jump and quantitatively agee with expeiments of lamina flows. Key-Wods: Radial fluid flow, hydaulic jump, cente manifold. Intoduction A fluid jet impinging downwads onto a hoizontal suface geneates a flow with hydaulic jump, that is, a sudden elevation of a fee suface of fluid [] [3]. Fo cetain paametes of the flow the jump is stationay and adially symmetic. The phenomenon has been studied theoetically in many woks but thee is still lack of systematic appoach to the dynamics. One of the most simple models of the hydaulic jump is an inviscid flow exhibiting Rayleigh's shock []. Linea one-dimensional shocks ae consideed in ealy wok by Rayleigh [], whee, in paticula, ive boes wee analyzed as moving shocks. The basic assumptions of this model ae continuity of mass and momentum fluxes acoss the shock, but discontinuity of kinetic enegy flux. It was shown that in ode to meet a natual equiement that the kinetic enegy be lost in the shock, the fee suface behind the shock must be highe then befoe. Thee have been attempts to use this appoach togethe with potential theoy to model cicula hydaulic jumps [6] [7]. Howeve, this gave incoect pedictions fo the adius of the jump. It was found that the adius should essentially depend on the adius of the falling jet and, hence, on the adius of a nozzle whee the jet dischages fom. Expeiments have not confimed such a dependence. A moe elaboate model by Watson [6] suggested that pat of the flow is effectively viscous and pat effectively inviscid. Fo the viscid segment, a similaity velocity pofile was pescibed. This theoy also teated the hydaulic jump as the shock. The viscous pat was shown to play substantial ole in the dynamics. The model was simplified by Tani [8] who supposed the flow was totally viscid. The bounday laye equations wee aveaged ove the depth of the flow unde assumption of similaity velocity pofile. Boh et al. [7] developed this appoach futhe, and good pedictions fo the adius of the jump wee obtained. Thee ae two main shotcomings of these models they descibe only stationay states and they ae heuistic: as mentioned above, vetical velocity pofiles ae often appoximated by similaity distibutions. Fo example, Boh et al. [7] assumed a paabolic stuctue: u(;z) = μu(c C 2 2 ), = z=(), whee μu is the depthaveage velocity, is the adius, z is the vetical coodinate, is the flow thickness, and C, C 2 ae some constant coefficients. In this pape we use an appoach that ests on the solid basis of cente manifold theoy. Its detailed desciption can be found, fo instance, in [9]. Hee we biefly outline the main idea of this theoy. Conside a dynamical system _x = Ax f(x; y; a) ; _y = By g(x; y; a) ; ()

2 whee the ovedot denotes d=dt, a is a set of paametes, x and y ae geneally multidimensional vaiables, and f and g ae nonlinea functions. It is assumed that the eigenvalues of the m m matix A all have zeo eal pat, and the eigenvalues of the n n matix B all have stictly negative eal pat. Nea the oigin (x; y; a) =(;;) the linea dynamics dominate and the modes y ae diven exponentially quickly to due to the equations _y = By. These modes ae thus ignoed when consideing the linea long-tem evolution, and the dynamics ae appoximately descibed in tems of the neutal modes, x, which obey the equations _x = Ax. Then cente manifold theoy assets that this linea pictue is only modified by the nonlinea tems f and g: the modes y(t) go exponentially quickly to a manifold y = h(x; a), called the cente manifold; and theeafte the long-tem evolution of the system is descibed by the lowdimensional system _x = Ax f(x; h; a). Cente manifold theoy has been successfully applied to a numbe of poblems such as dispesion of contaminants in channels [], [], dynamics of thin films on inclined planes [2] and othes. 2 Cente manifold model Li and Robets [3] used the cente manifold technique to descibe thin fluid flows on cuved substates. In paticula, thei model is applicable to flows on plane sufaces, which inteest us hee. A simila appoach was ealie used by Robets [2]. Let us descibe the main points of the appoach. The fluid dynamics ae govened by the continuity equation and Navie-Stokes u=; u u= ρ p μ ρ 2 ug; (3) whee taditional notations ae used. Attached to (2) (3) ae bounday conditions expessing no slip on the bottom plate, the fee suface kinematic elation, a jump in nomal stess on the fee suface caused by suface tension, and zeo tangential stess on the suface. The dimensional equations (2) (3) ae non-dimensionalized using chaacteistic thickness of the film, H, as the efeence length; μh=ff, whee ff is the coefficient of suface tension, as the efeence time; ff=μ as the efeence velocity; and ff=h as the efeence pessue. The non-dimensional fom of (2) (3) is u=; u u = p 2 u Bg ; () whee g is a unit gavitational vecto, R = ffρh=μ 2 is the Reynolds numbe, and B = ρgh 2 =ff is the Bond numbe. To set the equations to a fom teatable by the cente manifold appoach Robets [2] pefomed the following ticks. Fist, the hoizontal ae supposed ο ο " (non-dimensional), whee " is a small paamete. Second, the gavitational foce is egaded as a petubing nonlinea" tem by intoducing small paamete fi such that B = fi 2. Thid, the tangential stess on the fee suface is modified using an atificial paamete fl so that at fl = the lateal shea mode of slowest decay actually becomes a neutal mode, but at fl = the tangential stess bounday condition is ecoveed. The idea is to seek a solution as powe seies in fl, " and fi and substitute fl =into final expessions in ode to model the oiginal physical poblem. Convegence of the seies at fl = is confimed by calculations [3]. Accoding to the cente manifold technique the hoizontal velocity components u and v, vetical velocity w and pessue p ae epesented as functions of amplitudes" of the neutal modes: these ae and, as measues of the velocity, the hoizontal depth-aveage velocity components μu and μv. Cente manifold theoy guaantees that the solution is epesentable in the fom such that u(t) =U(; μu; " # μu μv = G(; μu; μv) : (6) In (6) the dependence on the paametes ", fl and fi is implicit. By adjoining the tivial = a new dynamical system is obtained fo u,, p, ", fl and fi. Once the tems of the oiginal dynamical system involving paametes " and fi ae teated as small petubations, the theoy leads to a physically adequate solution fo slow hoizontal vaiations (small "), and a elatively weak gavitational foce (small fi). 2

3 Analyzing (), (), (6) using compute algeba yields the model @(μv) ; ß ß2 μu 2ß2 2 Bg @μv @y " :83 μu (μu x μv y ) @y :33 2 y 2 yy 2 :62 x 2 :83 xx μu :69 y x 2 :833 xy μv ; (8) ß ß2 μv 2ß2 2 Bg @μu @x :83 μv (μu x μv y ) 2 2 :93@2 2 @x :967h y :33 2 x 2 2 xx 2 :6 y 2 :69 y x 2 :833 xy μu ; (9) In this pape we confine ou attention to adially-symmetic flows. To tansfom (8) and (9) to adial geomety, pola coodinates x = cos ', y = sin ' should be used unde assumption that, μu and μv do not depend on '. The equations fo the adial flow ae pesented futhe below. Meanwhile, the chaacteistic dimensional scales adopted in [3] ae inconvenient fo the paticula poblem. The scales ae essentially based μv on the suface tension coefficient, ff, which has a negligible ole in the adial dynamics, at least fo the flows epoted in []. We choose diffeent chaacteistic scales which ae typical fo the poblem in question: adius Λ, thickness Λ, velocity u Λ, pessue p Λ = ρg Λ and time t Λ = Λ =u Λ. These values ae estimated fom the govening equations as follows. Assuming that in the Navie-Stokes equations the non-stationay tem, inetia tems, pessue tem and viscous tem ae all of the same ode of magnitude one gets u Λ t Λ = u2 Λ Λ = u Λw Λ Λ = ρg Λ ρ Λ = νu Λ 2 Λ ; () whee ν = μ=ρ. The mass flux elation gives u Λ Λ Λ = Q; () (Q equals the total mass flux divided by 2ß), and the continuity elation gives u Λ Λ Solving () (2) esults in Q! =8 Λ = ν 3 ; u Λ = g = w Λ Λ : (2) Qνg 3 =8 : (3) Using scales (3) we deduce new non-dimensional fom of the equations (2) (3): u u= F p Re 2 u g F ; () whee the Foude numbe F and the new Reynolds numbe Re ae: F = u2 Λ g Λ ; Re = Λu Λ ν : (6) Substituting (3) into (6) we eveal that the Foude and Reynolds numbes ae epesented in tems of a single paamete which is the nondimensional total mass flux q: Re ==q 2 ; F = q; q= ν=8 : (7) Q 3=8 g=8 Let us show that the expession fo q indeed epesents the non-dimensional mass flux. Label tempoaily the dimensional quantities by tildes to distinguish them fom dimensionless quantities. 3

4 . η. 2 3 t η Figue : μ U2 =:. The flow evolves to stationay state. Steamlines of the eventual flow patten show eciculation unde the jump.. η t Figue 2: μ U2 =:8. The flow evolves to a settled oscillating egime.

5 j, mm η 2, mm Figue 3: The position of the stationay jump vesus the extenal thickness. Stas ae plotted using Fig. 3 of [], cicles coespond to ou model. By definition q = u, theefoe, q = (~= Λ )(~u=u Λ )(~= Λ ) = (~~u~)=( 2 Λ u Λ) = Q=( 2 Λ u Λ). Substituting hee (3) we eadily obtain the above fomula fo q. Compute algeba yields (the same esult follows fom (μ U); μ ß 2:67 q2 μ U 2 :66 q 2 μu :822 q : U μ U μ :77 μu 2 :83 μu 2 ρ ff :93 ( U) μ :8333 q 2 μu :66 q :83 q2! μu ; (9) whee μ U is the aveage adial velocity ( μ U 2 = μu 2 μv 2 ). See that the continuity equation is of the fist ode in and the momentum equation is of the second ode in. Hence, we need thee bounday conditions. We choose these to specify velocity and film thickness on the left end of spatial domain, that is at some elatively small fixed adius, and velocity on the ight end, that is at some fixed 2 >. 3 Numeical esults We solved equations (8) (9) using the dae2 solve developed by Robets []. To compae numeical esults with the expeiments we used dimensional paametes fom [] which wee the same in all the computations: = 6 mm, = : mm, 2 = 38 mm, and Q = 27=(2ß) ml/s. Once the total flux, coodinate and thickness on the left end ae stipulated, then the velocity on this end is obtained fom the continuity condition as U μ = Q=( ). Thus, the bounday conditions on the left end wee fixed, while the bounday condition specifying the velocity on the ight end, U2 μ,was fee to be vaied on ou choice fom one expeiment to anothe. Vaious magnitudes of U2 μ lead to vaious thicknesses on the ight end, 2. In some souces this thickness is called extenal height. In the laboatoy expeiments [] 2 was contolled by the height of a cicula im suounding the falling jet. Because the hydaulic jump is fomed fa away fom the im, the im does not act as immediate cause of the jump; it only contols the ight end bounday condition. Note that the jump is fomed even when thee is no im. The initial condition was chosen to epesent unifom thickness thoughout the flow and velocity deceasing like =. The latte choice povided a constant mass flux at the initial moment, which helped educe oscillations in ealy stages of the dynamics. We adopted (; ) ; μ U(; ) = q=( ) : Fo U2 μ > : we obseved that stationay flow is eventually fomed afte some tansitional evolution. An example of the flow dynamics and eventual steamline patten is shown in Fig.. In the jump aea a votex (eciculation aea) is situated, whee fluid paticles move along closed obits. This featue qualitatively agees with the laboatoy obsevations. Fo U2 μ < : unsteady behaviou of the flow is obtained (Fig. 2). Depending on the contolling bounday value of the velocity fom the ange μ U2 > : the flow settles upon a cetain thickness afte the jump, 2, and cetain jump coodinate j. The compaison of the numeical and expeimental esults fo the stationay flows ae pesented in Fig. 3. See that the points epesenting numeical solutions fom one line with the points epesenting expeiments.

6 The unsteady egimes at elatively lage values of 2 ( U2 μ < :) can be explained by too abupt change in the flow thickness and velocity in the jump aea. The lage 2 the lage the spatial deivatives enteing the govening equations. Recall that the accuacy of these equations is assued by the cente manifold theoy povided the spatial vaiations ae slow enough. This condition is violated fo extemely steep jumps. To extend the aea of applicability of the cente manifold model, it is necessay to take into account highe-ode tems in ". Nevetheless, fo elatively small extenal heights the model woks well, and stability of the solution is demonstated by Fig.. Conclusion A non-stationay adial model of thin fluid flow is developed to study the cicula hydaulic jump fomed by a steam of fluid hitting a hoizontal plate. Accuacy of the model, fo smooth flows, is assued by the cente manifold theoy. Stationay flow pattens ae obtained fo elatively small extenal heights. Fo the stationay egimes, the dependence of jump location against the extenal height is detemined, showing good ageement with available expeimental data. Stability of the egimes is demonstated. The model well epoduces a eciculation zone unde the hydaulic jump. Acknowledgement. We thank the ARC fo suppot of this eseach. Refeences [] T. Boh, C. Ellegaad, A.E. Hansen and Haaning, Hydaulic jumps, flow sepaation and wave beaking: an expeimental study, Physica B, Vol.228, 996, pp.. [2] T. Boh, C. Ellegaad, A.E. Hansen, K. Hansen, A. Haaning, V. Putkaadze and S. Watanabe, Sepaation and patten fomation in hydaulic jumps, Physica A,Vol.29, 998, pp. 7. [3] C. Ellegaad, A.E. Hansen, K. Hansen, A. Haaning, A. Macussen, T. Boh, J. Lundbeck Hansen and S. Watanabe, Ceating cones in kitchen sinks, Natue, Vol.392, 998, p [] V.T. Chow, Open-channel hydaulics, McGaw-Hill, 99. [] O.M. Rayleigh, On the theoy of long waves and boes, Poc. Roy. Soc., Vol.A9, 9, pp [6] E.J. Watson, The adial spead of a liquid jet ove a hoizontal plane, J. Fluid Mech., Vol.2, 96, pp [7] T. Boh, P. Dimon and V. Putkaadze, Shallow-wate appoach to the cicula hydaulic jump, J. Fluid Mech., Vol.2, 993, pp [8] I. Tani, Wate jump in the bounday laye, J. Phys. Soc. Japan, Vol., 99, pp [9] J. Ca, Applications of cente manifold theoy, Spinge-Velag, 98. [] G.N. Mece and A.J. Robets, A cente manifold desciption of contaminant dispesion in channels with vaying flow popeties, SIAM J. Appl. Math., Vol., 99, pp [] A.J. Robets and D.V. Stunin, Rigoous zonal model of contaminant dispesion in shea flows, In: Recent Advances in Applied and Theoetical Mathematics, Ed. N.E. Mastoakis (WoldSES Pess, Athens, Geece, 2) pp [2] A.J. Robets, Low-dimensional models of thin film fluid dynamics, Phys. Lett. A, Vol.22, 996, pp [3] Z. Li and A.J. Robets, The accuate and compehensive model of thin fluid flows with inetia on cuved substates, Univesity of Southen Queensland Woking Pape Seies, SC-MC-999, 999. [] A.J. Robets, staff/obetsa/. 6