Modelling the Zero-Inertia, Horizontal Viscous Dam-Break Problem

Transcription

1 r WSEAS International Conference on APPLIED an TEORETICAL MECANICS, Spain, December 4-6, 7 8 Moelling the Zero-Inertia, orizontal Viscous Dam-Break Problem BLAISE NSOM, WILFRIED NDONG AND BLAISE RAVELO LIME/LBMS/PMB - Université e Bretagne Occientale IUT e Brest BP 969. Rue e Kergoat.. 9 BREST Ceex FRANCE blaise. nsom@univ-brest.fr Abstract: - Debris flows such as avalanches an lahars iffer from the classical am-break problem of hyraulics ue to the relative importance of viscous versus inertial forces in the momentum balance. An equation of motion escribing ebris flow in the limit of zero inertia is evelope an solve analytically in two limits: short time an long time. These solutions are then combine into a single, universal moel. Limitations of the moel are examine by comparison to a converge finite ifference numerical solution of the flow equation. Key-Wors: - Dam failure; Finite ifference metho; Flui height, Numerical moels; One imensional flow; Shallow water; Viscous flow, Wavefront. Introuction Consier a am obstructing a horizontal smooth channel, ry ownstream an with a given quantity of flui upstream (with heighth ), containe between a fixe plate an a am. At initial time, the am collapses an the flui is release ownstream (positive wave), while a negative wave propagates upstream (negative wave). From the amcollapse ate to the time when the negative wave reaches the fixe plate, Ritter [] gives the so-calle inertial solution, stating that the wavefront avances with a constant spee of gh, while the negative wave moves back with a constant spee of gh. Between these two extremities, the average spee U an the hyrograph are respectively given by ( x ) t ( gh x) U + gh = () gh () = t where the am is assume to be locate at x =. This configuration generally represents a flow generate by a am failure cause by an exceptional rainfall (e.g. Malpasset, France in 99) or by an act of war (e.g. Dnieproghes, Ukraine in 94). The flui is water an the flow is escribe by the Navier Stokes an continuity equations, together with the non slip conition. Assuming the shallow water approximation, this system of equations leas to the Barré e Saint-Venant equations [], a one-imensional hyperbolic system. The complete hyroynamic equations escribing this unsteay flow in an open channel were solve by Faure an Nahas [], using the metho of characteristics. unt [4], comparing a one-imensional turbulent flow moel own a slope with its viscous counterpart, conclue that the viscous flow moel gives the best escription for ebris flows. In this work, a -D moel is presente, aiming to provie practical laws useful to engineers. A priori knowlege of the spee of the floo wave is inee important because this will etermine the available time in which forecast an rescue measures nee to be effecte. Assuming the shallowwater approximation, equations of motion governing viscous am-break flow are built an put in non-imensional form, an the initial an bounary conitions are state. Then, an analytical solution is presente both for short time an long time behaviour. Finally, an explicit proceure was use here, which oes not take into account turbulence generate by ambreak wave, as the flow evelops over a ry smooth be []. Numerical results are shown an compare with the analytical ones in each regime.

2 r WSEAS International Conference on APPLIED an TEORETICAL MECANICS, Spain, December 4-6, Problem statement. - Equations of motion Let h enote the height of flui at negative time in a smooth horizontal rectangular channel, g the gravity, ρ an µ the flui ensity an viscosity, respectively. Using a cartesian system of coorinates with the origin at the am site, x-axis lying on the channellength an the z-axis in the increasing vertical irection (fig. ). Wall Flui h Dam -L X Fig : Configuration of horizontal am-break flow at negative time The flui is assume to flow mainly in the irection of x-axis with height h at the given control section of the abscissa x, at time t. So, the vertical velocities are negligibly small, an therefore the pressure is hyrostatic, the pressure in the flow is given by p= p + ρ g( h ) () z where p enotes the (constant) pressure at the free surface. The balance between the pressure graient an the viscous forces is thus expresse by p h u = g = ν ρ x x z (4) where the horizontal erivatives have been neglecte in comparison with the vertical erivatives on the right-han sie of (.4) because the length of the current is very much greater than its thickness. At the base of the flui layer the non slip conition writes u ( x,, t) = () Consiering that the shear stress at the top of the current is very much less than its value within the current, it can be approximate as u (6) z ( x, h, t) = the solution of (4), () an (6) is g u( x, z, t) = h z( h z) (7) ν x A complete etermination of the unknowns u an h requires the equation of continuity which can be written here as h + = h uz (8) t x Substituting (7) into (8) we obtain ( 4 h ρg h ) = t µ x (9) If l enotes the reservoir length, we can assume the following set of non imensional variables: x gh ( ', ', ', ') (,,, t) h h h x f ρ h x x f t h µ l = () where subscript f enotes the wave-front, the equation of motion (9) then becomes, in the non imensional form: ( h' 4) h ' ' = () x t Eq. is similar to the equation of motion obtaine by Schwarz [6] an Barthes-Biesel [7], escribing the evolution of a thin liqui layer flowing own a horizontal plane when surface tension effects can be neglecte.. - Initial an bounary conitions Using (), the flui height at initial time is given by: h ' ', ' x t= = for x ' () h ' ', ' x t= = otherwise () Furthermore, a complementary bounary conition shoul be impose upstream, assuming that a short time or an asymptotic solution is sought. These bounary conitions are suggeste by experimental observation. For the short time case, it is written as: '( x' = l', t' ) = h with l= ' h l (4) which means that only a given flui quantity in the upper part of the reservoir is release ownstream the very few moments following the am collapse. While for the long time case, it is written as: h () ( x' = l', t' ) ' ' = x

3 r WSEAS International Conference on APPLIED an TEORETICAL MECANICS, Spain, December 4-6, 7 8 which means that there is no flow at the fixe wall; so at that site, the free surface is horizontal. For convenience, in the rest of the paper, non imensional quantities will be enote by corresponing capital letters with no primes. - Analytical solution Dam-break flow belongs to the general class of gravity currents; so the solution epens on the time scale [8]. First of all, the inertial regime, characterize by a fixe height at the am-site hols immeiately after the am collapse []. Then, a solution ominate by viscous effects appears an tens to an asymptotic form. The solution sought here will give the analytical expression for a short time ( T << ) an a long time ( T >> ) viscous solutions, as well as the critical height (a fixe point at the am-site in the inertial regime).. - Short time solution Assume a solution of the form ( X, f'η [ ] = = X η (6) ( where the variable η is efine in the interval [,]. If such solution exists, the flui height at the am-site ( X=, is constant. Substituting (6) in () gives = ct (7) where the constant c is etermine from the solution of the following orinary ifferential equation: f 4( η) f( η) + cη = (8) η η f ( η ) = f ( η + ) = (9) A polynomial approximation of (7) in the form of a Taylor series at η = in the form ( ) [( ) ( ) ( ) ] η η +... f( η) = c 4 8 η + 4 () an substitution of (7) in () prouce c =.48 () which etermines the solution from which (constant) flui height at the am site is obtaine: η= =. () ( ) 664 For the long time solution, a conition expressing the mass conservation may be written as: ( X, X= () which means that the flow only retains the initial (non imensional) volume of the reservoir V =. L an not the etails of its initial geometry. We seek a similar solution of the form ( X, = (, ( η) η= X + ( T+ ) (4) where variable η is efine in the interval [,], while the functions (, (, an ( η) are etermine by irect substitution in the equation of motion () an in the equation of mass conservation (). By setting = A h ' ( η) η we obtain: X ( ) ( ) f T = B T+ D ( ), T = A[ ( + ] 4 B = A () (6) To etermine constants A an C, the following ifferential equation has to be solve: {[ ( )]} ( ) 8 4 η η + η + ( η) = η η with the bounary conitions ( η) = Finally, we get ( η) = ( ) η ( η) = η, =. 84 (7) η = (8) A, B=. 86 (9) To etermine the transition point from the short time regime to the long time regime, both solutions previously obtaine are equate an the equation of motion of the front wave is finally obtaine as X f =.86 T (). - Long time solution

4 r WSEAS International Conference on APPLIED an TEORETICAL MECANICS, Spain, December 4-6, Numerical solution To buil a numerical proceure, it is necessary to efine the total channel length l t. The non imensional extreme (ecreasing) abscissa is L e lt h l = () This point is locate so far from the am site that the flow is assume to never reach it uring a given experiment (D assumption) for a total uration τ. Then, the problem to solve numerically is escribe by the following equation of motion = T X 4 ( ) () associate with the following initial conitions: ( X,) = if X [ L,] () ( X,) = otherwise (4) an the bounary conitions: ( L, = T X () ( X e, = T (6) This problem is solve by a finite ifference metho. To o so, the function ( X, T ) is compute in the set [ L, Le] [,τ] an in turn iscretize into a finite number of ientical small rectangles with sies T an X. The equation will be approximate at the gri points locate at the following coorinates in set: the [ L, Le] [,τ] ( X i, T j ) = ( L+ i X, j [ L ] X L e, j [, τ ] i, + (7) T Notice that eq.() can be put in the form: = 4 + (8) T X X An heuristic approach [9-] consiers the prouct ( 4 ) in the right-han sie of eq.(8) as a coefficient of iffusion. Inee, the following equations are consiere : V V = 4 (9) T X an V ( X,) = ( X,) (4) The numerical scheme of the equations (9-4) is teste using the von Neumann metho to provie a stability criterion which is necessary to ensure the convergence of our non-linear problem. The erivative of the unknown function an the non linear term in equ. () were estimate using Taylor s formula. Then, the finite ifference equation to solve, which uses a first orer time scheme an a centre secon orer spatial scheme, is written as T 4 [( ) ( ) ] 4 i, j+ = i, j + i+, j + i, j ( ) T (46) i j ( ) (, )4 In orer to have a stability criterion, the equation (4) is iscretize following the same numerical scheme, i.e. a first orer time scheme an secon orer centre spatial scheme. The numerical problem is written as: V,, 4 T i j+ = Vi j+ V i+, j + V i, j V i, j ( ) ( ) (49) with the same bounary conitions as, i.e. : V = V an V = max, +., j+, j+ i By applying i max ˆ i kp /( i max ) V = V e π + () p, q k, q k= where V ˆk, q is the Fourier component corresponing to the wave number k at time T = q T, efine by : i max ˆ iπ kp /( i max+ ) Vk, q = Vp, qe () i max+ p= equation () is rewritten as: ˆ ˆ 6 T Vk, q Vk, q sin ( π k + = i max ) + () The stability criterion consists of consiering that: 6 sin ( kπ ) < XT i max+, k () Since sin ( kπ ) <, we obtain i max+ finally the following stability criterion j

5 r WSEAS International Conference on APPLIED an TEORETICAL MECANICS, Spain, December 4-6, 7 8 ( ) 8 X T (4) We can see that the numerical scheme escribe by the equation (46) gives + + if,. The front wave I, j I j X f velocity efine as Vf = must then t satisfy: Vf () T A McCormack finite ifference scheme can improve the accuracy of the solution when Ve. In our case, the time step must be T chosen small enough to satisfy the conition (). 4 Numerical results Notice that a complete escription of the flow shoul inclue surface tension, introucing a complementary term in the equation of motion, such as: = 4 T X X B X (6) where B enotes the Bon number, efine as ρ gl σ B= (7) an σ the flui surface tension. Computation of eq.(4) was carrie out using the proceure escribe in the previous section. The flui s physical properties (ensity, viscosity an surface tension) were taken in Weast et al. []. For a similar flow configuration, results were quite ientical to those obtaine from eq.(), i.e. when the surface tension is neglecte. In fact, the surface tension woul affect the viscous am-break flow only uner film lubrication conitions (e.g.: []). The results obtaine analytically an numerically are concorant. Moreover, they also agree with those obtaine experimentally by Nsom et al. [] an Nsom [4]. Notably, fig. shows that for a very short time after the am collapse (the short time solution), the flow height remains constant at the am-site with the value =. 664 an with a ecrease of the flui height in the long time regime. Upstream the flui height ecreases both in the short time an long time regimes, while ownstream at a given station the flui height abruptly increases, reaches a maximum an then ecreases. This maximum height an the corresponing time were accurately etermine. They are governe by the following equations: max ( ) ( ) ( ) X = α( X+ ) / ( X) ( ) ( X+ ) max = + T c ( αγ) (8) T (9) γ where α =. 84, γ =. 86 an T c =. 8. The time evolution of the wavefront was characterize. Three flow regimes coul be pointe out with concoring equations both analytically an numerically. An inertial regime vali immeiately after the am collapse corresponing to. 664 at the am-site an governe by T = 4 for T.4L ( ) T (6) meets Ritter s invisci solution []. Then, a viscous regime results with the following equation of motion: ( = T where.4l T.8L (6) This regime can be observe for a long perio for large values of L, which means that the asymptotic solution correspons to an infinite reservoir length. The short time viscous regime evolves into a long time viscous regime (fig.) T= T=. T=.4 T=.7 T=. T= T= X Fig.: Time variation of the free surface profile

6 r WSEAS International Conference on APPLIED an TEORETICAL MECANICS, Spain, December 4-6, 7 86 with the asymptotic equation ( T >> ) ( =.86T. 9 (6) The short time viscous regime evolves into a long time viscous regime (fig.) with the T >> asymptotic equation ( ) ( =.86T. 9 (6) Conclusion The horizontal viscous am-break flow was consiere. The channel was smooth an ry at the initial time. By applying the conservation of mass an momentum with the shallow water approximation, an equation of motion was erive an mae non imensional when the viscous forces were assume to be the ominant ones. Then, an analytical solution of the equation of motion was built. Immeiately after the am collapse, an inertial regime results. In this regime, the flow height at the am site has a (fix) characteristic value an the equation of motion of the front wave is a Ritter type. As the flow height insie the reservoir reaches this characteristic value, a short time viscous regime is establishe without any influence from the reservoir extening onto the front wave s equation of motion. Finally, a long time viscous results epening on the reservoir length. Also, a convex free surface profile with a self-similar form an a time variation of the flow height at both the upstream an ownstream stations were furnishe. Furthermore, the problem was consiere numerically. The previous equation of motion was of porous meium type an was approximate using an explicit finite ifference metho. The stability an convergence of the computations were insure using a criteria base on a heuristic approach. The very goo agreement between numerical an asymptotic solutions shows the consistency of the numerical scheme for both the short time an the asymptotic form of the long time solution which correspon respectively to the long an short reservoir length L.. References [] Ritter, A., Die fortpflanzung er wesser wellen. Ver Deutsch Ingenieure Zeitschrift, Vol.6, N, 89, pp (in German) [] e Saint-Venant, B., Théorie u mouvement non permanent es eaux, C. R. Aca. Sci., Vol.7, N, 87, pp.47-4 (in French) [] Faure, J. an Nahas, N., Etue numérique et expérimentale intumescences à forte courbure u front, La ouille Blanche, Vol.6, N, 96, pp (in French) [4] unt, B., Newtonian flui mechanics treatment of ebris flows an avalanches, J. yr. Div. (ASCE), Vol., N, 994, pp.-6 [] Shigematsu, T., Liu, P.L.F. an Oa, K., Numerical moeling of the initial stages of am-break waves, J. yr. Res. (IAR), Vol.4, N, 4, pp.8-9 [6] Schwarz, L.W., Viscous flows own an incline plane: Instability an finger formation, Phys. Fluis A, Vol., N, 989, pp [7] Barthes-Biesel, D., Rectification un film liquie sous l effet e la gravité et e la tension superficielle (in French) < ofs/informatique/georges.gonthier/pi98/film.h tml> (June, 998) [8] Simpson, J.E., Gravity Currents in the Environment an the Laboratory, John Wiley & Sons, New York, 987. [9] Smith, G.D., Numerical Solution of Partial Differential Equations, Oxfor University Press, New York, 969. [] Forsythe, G.E. an Wasow, W.R., Finite- Difference Methos for Partial Differential Equations, John Wiley an Sons, New York, 967. [] Richtmyer, R.D. an Morton, K.W., Difference Methos for Initial Value Problems, John Wiley an Sons, New York, 967. [] Weast, R.C., Astle,M.J. an Beyer, W.., es. anbook of chemistry an physics, CRC Boca Raton, Fla., 987. [] Nsom, B., Debiane, K. an Piau, J.M., Be slope effect in the am-break problem, J. yr. Res. (IAR), Vol.8, N 6,, pp [4] Nsom, B., orizontal Viscous Dam-Break Floo: Experiments an Theory, J. yr. Eng. (ASCE), Vol.8, N,, pp.4-46