A circular jump forms in the radial flow thickness when a downwards stream hits a horizontal plate. Centre manifold theory is used to rigorously derive, from the Navier-Stokes equations, a dynamic model for slow horizontal variations of flow thickness and velocity. An advantage of this approach is the capacity to study non-stationary regimes and examine stability of the flow. Numerical solutions of the model reproduce experimentally observed recirculation under the jump and quantitatively agree with experiments of laminar flows

Roberts, Tony

Strunin, Dmitry

English

University of Southern Queensland ePrints

Models Encompassing Hydraulic Jumps inRadial Flows Over a Horizontal PlateA.J. ROBERTS and D.V. STRUNINDepartment of Mathematics and ComputingUniversity of Southern QueenslandToowoomba, Queensland 4350AUSTRALIAaroberts@usq.edu.au; strunin@usq.edu.auAbstract: A circular jump forms in the radial ow thickness when a downwards stream hits a horizon-tal plate. Centre manifold theory is used to rigorously derive, from the Navier-Stokes equations, a dy-namic model for slow horizontal variations of ow thickness and velocity. An advantage of this appro-ach is the capacity to study non-stationary regimes and examine stability of the ow. Numerical solu-tions of the model reproduce experimentally observed recirculation under the jump and quantitativelyagree with experiments of laminar ows.Key-Words: Radial uid ow, hydraulic jump, centre manifold.1 IntroductionA uid jet impinging downwards onto a horizontalsurface generates a ow with hydraulic jump, thatis, a sudden elevation of a free surface of uid [1]{[3]. For certain parameters of the ow the jumpis stationary and radially symmetric. The phe-nomenon has been studied theoretically in manyworks but there is still lack of systematic approachto the dynamics.One of the most simple models of the hydraulicjump is an inviscid ow exhibiting Rayleigh'sshock [4]. Linear one-dimensional shocks are con-sidered in early work by Rayleigh [5], where, inparticular, river bores were analyzed as movingshocks. The basic assumptions of this model arecontinuity of mass and momentum uxes acrossthe shock, but discontinuity of kinetic energy ux.It was shown that in order to meet a natural re-quirement that the kinetic energy be lost in theshock, the free surface behind the shock must behigher then before. There have been attempts touse this approach together with potential theoryto model circular hydraulic jumps [6]{[7]. How-ever, this gave incorrect predictions for the radiusof the jump. It was found that the radius shouldessentially depend on the radius of the falling jetand, hence, on the radius of a nozzle where the jetdischarges from. Experiments have not conrmedsuch a dependence. A more elaborate model byWatson [6] suggested that part of the ow is eec-tively viscous and part eectively inviscid. For theviscid segment, a similarity velocity prole wasprescribed. This theory also treated the hydraulicjump as the shock. The viscous part was shown toplay substantial role in the dynamics. The modelwas simplied by Tani [8] who supposed the owwas totally viscid. The boundary layer equationswere averaged over the depth of the ow underassumption of similarity velocity prole. Bohr etal. [7] developed this approach further, and goodpredictions for the radius of the jump were ob-tained.There are two main shortcomings of these mod-els | they describe only stationary states andthey are heuristic: as mentioned above, verti-cal velocity proles are often approximated bysimilarity distributions. For example, Bohr etal. [7] assumed a parabolic structure: u(r; z) =u(C1  C22),  = z=(r), where u is the depth-average velocity, r is the radius, z is the verticalcoordinate,  is the ow thickness, and C1, C2aresome constant coeÆcients.In this paper we use an approach that restson the solid basis of centre manifold theory. Itsdetailed description can be found, for instance,in [9]. Here we briey outline the main idea ofthis theory. Consider a dynamical system_x = Ax+ f(x; y; a) ; _y = By + g(x; y; a) ; (1)1where the overdot denotes d=dt, a is a set of pa-rameters, x and y are generally multidimensionalvariables, and f and g are nonlinear functions. Itis assumed that the eigenvalues of the mm ma-trix A all have zero real part, and the eigenvaluesof the nnmatrix B all have strictly negative realpart. Near the origin (x; y; a) = (0; 0; 0) the lineardynamics dominate and the modes y are drivenexponentially quickly to 0 due to the equations_y = By. These modes are thus ignored when con-sidering the linear long-term evolution, and thedynamics are approximately described in termsof the neutral modes, x, which obey the equations_x = Ax. Then centre manifold theory asserts thatthis linear picture is only modied by the nonlin-ear terms f and g: the modes y(t) go exponen-tially quickly to a manifold y = h(x; a), calledthe centre manifold; and thereafter the long-termevolution of the system is described by the low-dimensional system _x = Ax + f(x; h; a). Centremanifold theory has been successfully applied toa number of problems such as dispersion of con-taminants in channels [10], [11], dynamics of thinlms on inclined planes [12] and others.2 Centre manifold modelLi and Roberts [13] used the centre manifoldtechnique to describe thin uid ows on curvedsubstrates. In particular, their model is appli-cable to ows on plane surfaces, which interestus here. A similar approach was earlier used byRoberts [12]. Let us describe the main points ofthe approach. The uid dynamics are governed bythe continuity equation and Navier-Stokes equa-tions:r  u = 0 ; (2)@u@t+ u  ru =  1rp+r2u+ g ; (3)where traditional notations are used. Attachedto (2){(3) are boundary conditions expressing noslip on the bottom plate, the free surface kine-matic relation, a jump in normal stress on thefree surface caused by surface tension, and zerotangential stress on the surface. The dimensionalequations (2){(3) are non-dimensionalized usingcharacteristic thickness of the lm, H, as the ref-erence length; H=, where  is the coeÆcientof surface tension, as the reference time; = asthe reference velocity; and =H as the referencepressure. The non-dimensional form of (2){(3) isr  u = 0 ; (4)R@u@t+ u  ru=  rp+r2u+Bg ; (5)where g is a unit gravitational vector, R =H=2is the Reynolds number, and B =gH2= is the Bond number.To set the equations to a form treatable bythe centre manifold approach Roberts [12] per-formed the following tricks. First, the horizontalgradients, @=@x and @=@y, are supposed small:@=@x  ", @=@y  " (non-dimensional), where " isa small parameter. Second, the gravitational forceis regarded as a perturbing \nonlinear" term byintroducing small parameter  such that B = 2.Third, the tangential stress on the free surface ismodied using an articial parameter  so that at = 0 the lateral shear mode of slowest decay ac-tually becomes a neutral mode, but at  = 1 thetangential stress boundary condition is recovered.The idea is to seek a solution as power series in ," and  and substitute  = 1 into nal expressionsin order to model the original physical problem.Convergence of the series at  = 1 is conrmed bycalculations [13]. According to the centre mani-fold technique the horizontal velocity componentsu and v, vertical velocity w and pressure p arerepresented as functions of \amplitudes" of theneutral modes: these are  and, as measures ofthe velocity, the horizontal depth-average veloc-ity components u and v. Centre manifold theoryguarantees that the solution is representable inthe formu(t) = U(; u; v) ;such that@@t"uv#= G(; u; v) :(6)In (6) the dependence on the parameters ",  and is implicit. By adjoining the trivial equations@"=@t = 0 ; @=@t = 0 ; @=@t = 0a new dynamical system is obtained for u, , p,",  and . Once the terms of the original dy-namical system involving parameters " and  aretreated as small perturbations, the theory leadsto a physically adequate solution for slow hori-zontal variations (small "), and a relatively weakgravitational force (small ).2Analyzing (4), (5), (6) using computer algebrayields the model [13]@@t  @(u)@x @(v)@y; (7)R@u@t  24u2+212Bgx R1:5041u@u@x+ 1:3464v@u@y+ 0:1577u@v@y+0:1483u(ux+ vy)+"4:0930@2u@x2+@2u@y2+ 3:0930@2v@x@y+4:8333x@u@x+y@u@y+1:9167x@v@y+1:9167hy@v@x+  0:50332y2 yy2+ 0:10612x2  0:5834xx!u+0:6094yx2  0:0833xyv; (8)R@v@t  24v2+212Bgy R1:3464u@v@x+ 1:5041v@v@y+ 0:1577v@u@x+0:1483v(ux+ vy)+"@2v@x2+ 4:0930@2v@y2+ 3:0930@2u@x@y+4:8333y@v@y+x@v@x+1:9167x@u@y+1:9167hy@u@x+  0:50332x2 xx2+ 0:10612y2  0:5834yy!v+0:6094yx2  0:0833xyu; (9)In this paper we conne our attention toradially-symmetric ows. To transform (8) and(9) to radial geometry, polar coordinates x =r cos', y = r sin' should be used under assump-tion that , u and v do not depend on '. Theequations for the radial ow are presented furtherbelow.Meanwhile, the characteristic dimensionalscales adopted in [13] are inconvenient for the par-ticular problem. The scales are essentially basedon the surface tension coeÆcient, , which has anegligible role in the radial dynamics, at least forthe ows reported in [1]. We choose dierent char-acteristic scales which are typical for the problemin question: radius r, thickness , velocity u,pressure p= grand time t= r=u. Thesevalues are estimated from the governing equationsas follows. Assuming that in the Navier-Stokesequations the non-stationary term, inertia terms,pressure term and viscous term are all of the sameorder of magnitude one getsut=u2r=uw=gr=u2; (10)where  = =. The mass ux relation givesur= Q ; (11)(Q equals the total mass ux divided by 2), andthe continuity relation givesur=w: (12)Solving (10){(12) results inr= Q53g!1=8; u=Qg31=8: (13)Using scales (13) we deduce new non-dimensionalform of the equations (2){(3):r  u = 0 ; (14)@u@t+ u  ru =  1Frrp+1Rer2u+gFr; (15)where the Froude number Fr and the newReynolds number Re are:Fr =u2gr; Re =ru: (16)Substituting (13) into (16) we reveal that theFroude and Reynolds numbers are represented interms of a single parameter which is the non-dimensional total mass ux q:Re = 1=q2; F r = q ; q =5=8Q3=8g1=8: (17)Let us show that the expression for q indeed repre-sents the non-dimensional mass ux. Label tem-porarily the dimensional quantities by tildes todistinguish them from dimensionless quantities.30 1 2 3 4 50.40.60.811.200.050.1 r tη0.2 0.7885 1.37700.050.1 rηFigure 1:U2= 0:55. The ow evolves to stationary state. Streamlines of the eventual ow pattern showrecirculation under the jump.0 1 2 3 4 5 6 7 80.40.60.811.200.050.1 r tηFigure 2:U2= 0:48. The ow evolves to a settled oscillating regime.41 2 3 4 551015202530 r j , mmη2 , mmFigure 3: The position of the stationary jump ver-sus the external thickness. Stars are plotted usingFig. 3 of [1], circles correspond to our model.By denition q = ru, therefore, q =(~r=r)(~u=u)(~=r) = (~r~u~)=(r2u) = Q=(r2u).Substituting here (13) we readily obtain the aboveformula for q. Computer algebra yields (the sameresult follows from (7){(9)):@@t  1r@r(rU) ; (18)@U@t  2:4674 q2U2+1:4166 q2rrU   0:8225 q 1r 1:5041UUr  0:15771rU2  0:1483rU2+4:0930 q2@r1r@r(rU)+ 4:8333 q2rUr+ 0:1066 q22r2  0:5834 q2rr!U ; (19)whereU is the average radial velocity (U2=u2+ v2). See that the continuity equation is ofthe rst order in r and the momentum equationis of the second order in r. Hence, we need threeboundary conditions. We choose these to specifyvelocity and lm thickness on the left end of spa-tial domain, that is at some relatively small xedradius r1, and velocity on the right end, that is atsome xed r2> r1.3 Numerical resultsWe solved equations (18){(19) using the dae2solver developed by Roberts [14]. To comparenumerical results with the experiments we useddimensional parameters from [1] which were thesame in all the computations: r1= 6 mm, 1=1:5 mm, r2= 38 mm, and Q = 27=(2) ml/s.Once the total ux, coordinate and thickness onthe left end are stipulated, then the velocity onthis end is obtained from the continuity conditionasU1= Q=(1r1). Thus, the boundary condi-tions on the left end were xed, while the bound-ary condition specifying the velocity on the rightend,U2, was free to be varied on our choice fromone experiment to another. Various magnitudesofU2lead to various thicknesses on the right end,2. In some sources this thickness is called exter-nal height. In the laboratory experiments [1] 2was controlled by the height of a circular rim sur-rounding the falling jet. Because the hydraulicjump is formed far away from the rim, the rimdoes not act as immediate cause of the jump; itonly controls the right end boundary condition.Note that the jump is formed even when there isno rim. The initial condition was chosen to rep-resent uniform thickness throughout the ow andvelocity decreasing like 1=r. The latter choice pro-vided a constant mass ux at the initial moment,which helped reduce oscillations in early stages ofthe dynamics. We adopted(r; 0)  1;U(r; 0) = q=(r1) :ForU2> 0:5 we observed that stationary owis eventually formed after some transitional evo-lution. An example of the ow dynamics andeventual streamline pattern is shown in Fig. 1.In the jump area a vortex (recirculation area) issituated, where uid particles move along closedorbits. This feature qualitatively agrees with thelaboratory observations. ForU2< 0:5 unsteadybehaviour of the ow is obtained (Fig. 2).Depending on the controlling boundary valueof the velocity from the rangeU2> 0:5 the owsettles upon a certain thickness after the jump, 2,and certain jump coordinate rj. The comparisonof the numerical and experimental results for thestationary ows are presented in Fig. 3. See thatthe points representing numerical solutions formone line with the points representing experiments.5The unsteady regimes at relatively large valuesof 2(U2< 0:5) can be explained by too abruptchange in the ow thickness and velocity in thejump area. The larger 2the larger the spatialderivatives entering the governing equations. Re-call that the accuracy of these equations is assuredby the centre manifold theory provided the spa-tial variations are slow enough. This condition isviolated for extremely steep jumps. To extend thearea of applicability of the centre manifold model,it is necessary to take into account higher-orderterms in ". Nevertheless, for relatively small ex-ternal heights the model works well, and stabilityof the solution is demonstrated by Fig. 1.4 ConclusionA non-stationary radial model of thin uid owis developed to study the circular hydraulic jumpformed by a stream of uid hitting a horizontalplate. Accuracy of the model, for smooth ows,is assured by the centre manifold theory. Station-ary ow patterns are obtained for relatively smallexternal heights. For the stationary regimes, thedependence of jump location against the exter-nal height is determined, showing good agreementwith available experimental data. Stability of theregimes is demonstrated. The model well repro-duces a recirculation zone under the hydraulicjump.Acknowledgement. We thank the ARC forsupport of this research.References[1] T. Bohr, C. Ellegaard, A.E. Hansen andHaaning, Hydraulic jumps, ow separationand wave breaking: an experimental study,Physica B, Vol.228, 1996, pp. 1{10.[2] T. Bohr, C. Ellegaard, A.E. Hansen, K.Hansen, A. Haaning, V. Putkaradze and S.Watanabe, Separation and pattern formationin hydraulic jumps, Physica A, Vol.249, 1998,pp. 111{117.[3] C. Ellegaard, A.E. Hansen, K. Hansen, A.Haaning, A. Marcussen, T. Bohr, J. Lund-beck Hansen and S. Watanabe, Creating cor-ners in kitchen sinks, Nature, Vol.392, 1998,p. 767.[4] V.T. Chow, Open-channel hydraulics,McGraw-Hill, 1959.[5] O.M. Rayleigh, On the theory of long wavesand bores, Proc. Roy. Soc., Vol.A90, 1914,pp. 324{328.[6] E.J. Watson, The radial spread of a liquidjet over a horizontal plane, J. Fluid Mech.,Vol.20, 1964, pp. 481{499.[7] T. Bohr, P. Dimon and V. Putkaradze,Shallow-water approach to the circular hy-draulic jump, J. Fluid Mech., Vol.254, 1993,pp. 635{648.[8] I. Tani, Water jump in the boundary layer, J.Phys. Soc. Japan, Vol.4, 1949, pp. 212{215.[9] J. Carr, Applications of centre manifold the-ory, Springer-Verlag, 1981.[10] G.N. Mercer and A.J. Roberts, A centremanifold description of contaminant disper-sion in channels with varying ow proper-ties, SIAM J. Appl. Math., Vol.50, 1990, pp.1547{1565.[11] A.J. Roberts and D.V. Strunin, Rigorouszonal model of contaminant dispersion inshear ows, In: Recent Advances in Appliedand Theoretical Mathematics, Ed. N.E. Mas-torakis (WorldSES Press, Athens, Greece,2000) pp. 64{70.[12] A.J. Roberts, Low-dimensional models ofthin lm uid dynamics, Phys. Lett. A,Vol.212, 1996, pp. 63{71.[13] Z. Li and A.J. Roberts, The accurate andcomprehensive model of thin uid ows withinertia on curved substrates, University ofSouthern Queensland Working Paper Series,SC-MC-9909, 1999.[14] A.J. Roberts, http://www.sci.usq.edu.au/sta/robertsa/.6

Models encompassing hydraulic jumps in radial flows over a horizontal plate

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Models encompassing hydraulic jumps in radial flows over a horizontal plate

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