Equipment Safety in Renewable Energies Exploitation

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1 European Association or the Development o Renewable Energies, Environment and Power Quality (EAEPQ International Conerence on Renewable Energies and Power Quality (ICREPQ 11 Las Palmas de Gran Canaria (Spain, 1th to 1th April, 11 Equipment Saety in Renewable Energies Exploitation B. Nsom 1 and K. Bouchlaghem 1, 1 Université de Bretagne Occidentale. LBMS EA. IU de Brest. BP 9169 Rue de Kergoat. 91 BRES Cedex (France Phone: , Fax: , blaise.nsom@univ-brest.r Ecole Polytechnique de Sousse. Boulevard Khalia Karoui-Sahloul SOUSSE (unisia Phone: , Fax: , bouchlaghemkarim@yahoo.r Abstract : his paper considers the dam-break problem in a horizontal smooth 1D channel, or the purpose o equipment saety in renewable energies exploitation. he luid is water and it can be described by a Newtonian model, provided that the inertial eects be neglected versus the viscous ones in the momentum balance. Assuming the shallow water approximation, a non dimensional equation is built rom the continuity and the Navier-Stokes equations in the limit o zero-inertia and it is solved analytically in two limits: short time and long time. hese solutions are then combined into a single, universal model. Limitations o the model are examined by comparison to a converged inite dierence numerical solution o the low equation. Key words Dam ailure, Finite dierence method, Flow regimes, Shallow water approximation, Similar solution. 1. Introduction Nowadays, ydraulics which represents 9% o the sources o renewable energies used or producing electricity in the world is in progress mostly in the developing countries. ydraulics is the only renewable energy which allows to make a stock o energy. Owing to the huge quantities o water retained behind dams, genuine energy reservoirs are available and ready to produce electricity at request. he bigger the dam is, the greater the pressure at its bottom is. Due to the system malunction or an act o war (e.g. Dnieproghes, Ukraine in 191, the dam can collapse. he water released downstream can destroy ields, goods, inrastructure and kill people and animals. Since Ritter s original work on dam-break low [1], many studies have been perormed ocusing on experiments, theory and numerical methods []. Dam-break low has become a classical hydraulic problem with such a large complexity that a higher degree o reproduction o real conditions raises new studies. Consider a dam obstructing a horizontal smooth channel, dry downstream and with a given quantity o luid upstream (with height h, contained between a ix plate and a dam. At initial time, the dam collapses and the luid is released downstream (positive wave, while a negative wave propagates upstream (negative wave. From dam-collapse date to time where negative wave reaches the ix plate, Ritter [1] gives the so-called inertial solution, stating that the wave ront advances with a constant speed o gh, while the negative wave moves back with constant speed gh. he luid is water and the low is described by the Navier Stokes and continuity equations, together with the non slip condition. Assuming the shallow water approximation, this system o equations leads to the Saint-Venant equations [], a one-dimensional hyperbolic system. he complete hydrodynamic equations describing this unsteady low in open channel were solved by Faure and Nahas [], using the method o characteristics. unt [], comparing one-dimensional turbulent low model down a slope with its viscous counterpart, concluded that the viscous low model gives the best description or debris lows. Indeed, these lows develop within a long domain, i.e. a domain o space that is much longer than it is wide, so short time behavior described by the previous studies are inappropriate to give a complete description o these natural lows. Natural lows generally erode their bed and transport sediments. For image analysis purpose, experimentalists generally slow down the low by using viscous complex mixtures o water with diverse additive. Nsom et al.[6] and Nsom [7] perormed an experimental study with glucose-syrup luids characterized with adjustable viscosity and density. unt [] built similarity solutions or such geological lows down a sloping 1D channel. Also, Schwarz [8] achieved a numerical study o viscous thin liquid ilms down an inclined plane. Solving ree surace lubrication equations, including the eects o RE&PQJ, Vol.1, No.9, May 11

2 both gravity and surace tension, he states a scaling law or the prediction o inger-width. In this work, a 1-D model is presented, aiming to provide practical laws, useul to engineers. Assuming the shallowwater approximation, equations o motion governing viscous dam-break low are built and put in nondimensional orm and the initial and boundary conditions are stated. hen, an analytical solution is presented both or short time and long time behavior. Zoppou and Roberts [9] tested the perormance o explicit schemes used to solve the shallow water wave equations or simulating the dam-break problem. Comparing results rom these schemes with analytical solutions to the dam-break problem with inite-water depth and dry bed downstream o the dam, they ound that most o the numerical schemes produce reasonable results or subcritical lows. So an explicit procedure was used here, which does not take into account turbulence generated by dam-break wave, as the low develops over a dry smooth bed [1]. his numerical solution is compared with the expermental and analytical results obtained in [7].. Problem statement A. Equation o motion Let h denote the height o luid at negative time in a smooth horizontal rectangular channel, g the gravity, ρ and µ the luid density and viscosity, respectively. Using a cartesian system o coordinates with the origin at the dam site, x-axis lying on the channel-length and the z-axis in the increasing vertical direction (ig. 1. Wall Fluid h -L Fig 1: Coniguration o horizontal dam-break low at negative time he luid is assumed to low mainly in the direction o x- axis with height h at the given control section o the abscissa x, at time t. So, the vertical velocities are negligibly small, and thereore the pressure is hydrostatic, the pressure in the low is given by p= p + ρ g( hz (1 where p denotes the (constant pressure at the ree surace. he balance between the pressure gradient and the viscous orces is thus expressed by 1 p h u = g = ν ( ρ x x z where the horizontal derivatives have been neglected in comparison with the vertical derivatives on the right-hand side o equ. ( because the length o the current is very much greater than its thickness. At the base o the luid layer the non slip condition writes u x,, t = ( ( Dam Considering that the shear stress at the top o the current is very much less than its value within the current, it can be approximated as ( x, h, t = u z ( the solution o equs. ( - ( is g u x, z, t = 1 hz hz ( ( ( ν x A complete determination o the unknowns u and h requires the equation o continuity which can be written here as h + = h udz (6 t x Substituting ( into (6 we obtain ( h ρg h = t 1µ x (7 I l denotes the reservoir length, we can assume the ollowing set o non dimensional variables: (,, ( h, x ρgh =, t 1 (8 h h µ l where subscript denotes the wave-ront, the equation o motion (7 then becomes, in the non dimensional orm: ( = (9 Equ. (9 is similar to the equation o motion obtained by Schwarz [1] and Barthes-Biesel [1], describing the evolution o a thin liquid layer lowing down a horizontal plane when surace tension eects can be neglected. B. Initial and boundary conditions Using (1, the luid height at initial time is given by: ( b (, = 1 or 1 otherwise Furthermore, a complementary boundary condition should be imposed upstream, assuming that a short time or an asymptotic solution is sought. hese boundary conditions are suggested by experimental observation. For the short time case, it is written as: ( = L, = 1 l with L= (11 which means that only a given luid quantity in the upper part o the reservoir is released downstream the very ew moments ollowing the dam collapse. While or the long time case, it is written as: ( =L, = (1 which means that there is no low at the ixed wall; so at that site, the ree surace is horizontal.. Numerical solution A. Equation o motion h 19 RE&PQJ, Vol.1, No.9, May 11

3 he problem to solve numerically is the same which has been solved analytically in the previous section by equs. (9-(1. o build a numerical procedure, it is necessary to deine the channel total length l t. he non dimensional extreme (downwards abscissa is L lt h l e=. his point is so ar rom dam site, that the low is supposed to never reach it during a given experiment (1D assumption, with total duration τ. his assumption constitutes the ollowing complementary boundary condition: ( e, = (1 his problem is solved by a inite dierence method. For this, the unction (, is computed in the set [ L, Le ] [,τ ], itsel discretized in a inite number o identical small rectangles with sides and. he equation will be approximated at grid points located at the ollowing coordinates in the [ L, Le ] [,τ ] set: ( i, j = ( L+ i, j i L L e, + j, τ (1 [ ], [ ] Notice that the equation o motion (9 can be put in the orm: = + (1 An heuristic approach considers the product ( in the right-hand side o eq.(8 as a coeicient o diusion [16-18]. Indeed, the ollowing equations are considered : V V =, V (, = (, (16 his numerical scheme is tested using the von Neumann method to provide a stability criterion which is necessary to ensure the convergence o our non-linear problem. B. Algorithms Using aylor s ormula, the derivative o the unknown unction can be given by: ( (, + (, =, A, n1 n A= (, (17 n n n! Also, aylor s ormula can be used to write the non linear term in eq.(9: ( ( p1 p ( ( B p ( (,, = p! B= p [ ( +, ] + [ (, ] [ (, ] Introducing eq.(17 and eq.(18 in eq.(9 gives (, + (, [ ( +, ] = + C C= ( [ (, ] [ (, ] + R ( n! (,, n n with R, (, = (, D n n ( n1 n D= ( n! ( (, n ( R, (, is the residual term which is neglected to solve the numerical problem. Notice that this term can be numerically approximated knowing the solution at the ormer time step. Now let i, j= ( i, j (1 where i and j are given by eq.(, then the inite dierence equation to solve, which uses a irst order time scheme and a centred second order spatial scheme, is written as ( i, j 1 + = i, j+ i+ 1, j + i1, j i, j ( [ ] [ ] [ ] Notice that, j+ 1 corresponds to upstream Neumann condition given by eq.(. It is derived rom eq.(1, say, j+ 1= 1, j+ 1. Also, i i max denotes the maximum value that subscript i can reach, i.e. i max is rounded o to the integer that is closest to L + e, then downstream Dirichlet condition given by eq.( yealds ( = imax, j+ 1 In order to have a stability criterion, the equation (16 is discretized ollowing the same numerical scheme, i.e. a irst order time scheme and second order centred spatial scheme. he numerical problem is written : ( ([ V ] [ ] [ ] i 1, j + Vi 1, j V i j E =, Vi j = Vi j+ E, + 1, + ( with the same boundary conditions as, i.e. : V, j+ 1 = V1, j+ 1 and V = i max, j+ 1. Giving i max ˆ i kp /( i max 1 Vp, q = Vk, qe π + ( k = with ˆk, q V the Fourier component corresponding to wave number k at time = q, deined by : i max ˆ 1 iπ kp /( imax + 1 Vk, q = Vp, qe (6 i max+ 1 p= the equation ( is rewritten ˆ ˆ 16 Vk, q 1 Vk, q 1 sin ( π k + = i max 1 + he stability criterion consists in considering that ( π 16 k i max+ 1 1 sin < 1 As ( k, k sin π < 1 i max+ 1, we obtain the ollowing stability criterion: ( 8 1 (8 We can notice that the numerical scheme described by equ. (, makes I + 1, j + 1 i I, j. he ront wave velocity, deined as d V = must then veriy dt V (9 A McCormack inite dierence scheme can improve the accuracy o the solution when Ve. In our case, the time step must be chosen small enough to veriy the condition deined by equ. ( RE&PQJ, Vol.1, No.9, May 11

4 horizontal.. Results A. Free surace proile he ree surace proile is presented in ig.. A large time ater dam collapse, it completely diers rom Ritter s solution, i.e. when the luid is water, computed using equs.(1-( which is concave. his shows that the convex shape o ree surace proile or viscous dam-break low is intrinsic to the equations o motion governing the problem. Furthermore, a complete description o the low should include surace tension, introducing a complementary term in the equation o motion, say = 1 ( B where B denotes the Bond number, deined as ρgl B= and σ the luid surace tension. σ Computation o equ. ( was carried out using the procedure described in previous section or assigned glucose syrup concentration in water. Fluid physical properties (density, viscosity and surace tension were taken in [19]. For similar low coniguration, results were quite identical to those obtained rom equ. (9, i.e. when surace tension is neglected. In act, surace tension would aect viscous dam-break low, only in ilm lubrication conditions [1]. B. Fluid height =.6 = =6 =1 = =.1 = Figure : ime variation o ree surace proile Fig. also shows that during a very short while ater dam collapse (short time solution, the low height remains constant at dam site, with d ( =,.68 (1 in excellent agreement with the analytical solution (equ.. he viscous solution is characterized by a decreasing o the luid height at dam site. At a given location inside the reservoir, time variation o the luid height is shown in ig. which indicates that the luid height collapses or stations close to dam site ollowed by a smoother decrease or all upstream stations. While at given downstream station, low height increases abruptly at irst stage, then smoothly to a maximum value and inally decreases as shown in ig.... =-. =-. = Figure : ypical time variation o luid height at upstream stations =. =. =.7 =.9 =..1 1 Figure : ypical time variation o luid height at downstream stations C. Maximum heights o localize the maximum height at given down scenario described in Fig. shows that (, = ( While in the long time regime described by equ. (8, we have ' ( 1 + ( Ψ + Ψ' ( = λ α ( + 1 m So Ψ + λ Ψ'= ( which gives ( 1 λ = ( Introducing this solution in equ. (1, the maximum height at given downstream station is then ound as 1 1 max (6 ( ( ( 1 = ( α he corresponding graph (hyperbola is shown on Fig. max Figure : Variation o the maximum height at given station, vs the corresponding abscissa max Figure 6: Variation o the time o occurrence o maximum height or given abscissa and it occurs at time max such that ( 1 ( max ( 1 1 = + + c (7 γ γ α whose graph is shown on Fig RE&PQJ, Vol.1, No.9, May 11

5 D. Wave ront position ime evolution o the ront o the positive wave is presented in ig.7. It can be obtained either numerically (section or analytically using equ. (. his graph agrees with the experimental result obtained by Nsom [9] who ound the ollowing scaling law in this regime 1/ (8 in the long time regime, respectively. he results obtained using both methods are concordant U b Figure 9: ime variation o the velocity o the back wave U Figure 7: Evolution o the ront o the positive wave ront in short time viscous regime In the same low regime, the ront o the negative wave, obtained numerically or analytically using equ. (19 is shown on ig. 8. It can be observed that b ( decreases vs time with a slope itsel decreasing in the time, while or long time low regime, the graph o the equation o motion o the ront wave is shown on ig. 9. It can be obtained numerically (section or analytically using equ. (8 and this result agrees with the experimental result obtained by Nsom [9] who ound the ollowing scaling law in this regime 1/ (9 b Figure 8: Evolution o the ront o the positive wave ront in long time viscous regime In these experiments, the author perormed dam-break tests using well characterized water-glucose syrup solutions. Generally, in the literature, theoretical studies ocus on the long time solution also called asymptotic solution (e.g.: [7] and the scaling law obtained is o the orm o equ.(8. he originality o the present paper is to point out or the irst time numerically, the previous two viscous low regimes and to characterize them. E. Wave ront velocity he wave ront velocity is obtained rom the time derivation o (. It can be calculated analytically by a straightorward use o the corresponding equation o motion, obtained in section. While the numerical method consists in the ollowing centred second order scheme : ( w ( + ( U ( + = +Ο ( where U denotes the ront velocity and subscript w is used or b, s and l when reerring to the ront o back wave or positive wave in the short time regime and Figure 1: ime variation o the velocity o the positive wave in the short time regime U Figure 11: ime variation o the velocity o the positive wave in the long time regime hese graphs clearly show that or each wave, the velocity is time decreasing and tends to an asymptotic value which or the positive wave characterizes a 1D ilm lubrication. F. Comparison o the numerical results with the analytical ones Dam-break low belongs to the general class o gravity currents; so the solution depends on the time scale [11]. First o all, the inertial regime, characterized by a ixed height at the dam-site holds immediately ater the dam collapse [1]. hen, a solution dominated by viscous eects appears and tends to an asymptotic orm. he solution sought here will give the analytical expression or a short time ( << 1 and a long time ( >> 1 viscous solutions, as well as the dierent dynamic characteristics. Sedov [11] describes the method o investigating similar solutions o equ. (9 by means o a phase plane ormalism. In act, this equation o motion can be tackled by assuming a solution o the orm (, = Ω( Ψ( λ where φ( λ= (1 P( Let b ( denote the ront o the back wave and ( the ront o the positive wave. I c denotes the time where the back wave ront reaches the rear wall, the short time regime corresponds to a viscous solution such that c. While, or larger time, is everywhere less than 1. So, a solution should be sought such that ( b (, = 1 i c ( ( 1, i c 19 RE&PQJ, Vol.1, No.9, May 11

6 b i c b ( = ( 1 i c wo regimes can now be identiied, which correspond to two dierent physical mechanisms o reservoir emptying. he short time solution is such that ar downstream rom the dam, the luid seems to be at rest at a depth h, so that the reservoir s length l has no eect on this low regime, while or the long time solution, the low only retains the initial (non dimensional volume o the reservoir V = 1L and not the details o its initial geometry. o ind the short time solution, w estate that the inormation aects the luid contained between ( and b (, this suggests to take P( = ( b(, φ ( = b( ( Introducing equs. (1-(1 in the equation o motion (9, we get dψ ' λ Ψ + ' Ψ + P P d P = d d λ dλ dλ d ( b dλ For the long time solution, we state that, at = c, the ront o the negative wave reaches the rear wall ( b = 1, so or the long time solution, we can take ψ ( = 1 and P ( = + 1 (6 hen, introducing equ. ( in the equation o motion (9, we get ' dψ Ω ' λ Ψ + P Ψ P P d = d d (7 λ dλ Ω Ω dλ ( When computing the time and abscissa variation o the luid height in the short time viscous regime, we observed that the results rom the numerical method (ig. 1 were generally greater than those obtained rom the analytical method (ig. 1. Meanwhile, the results rom the two methods were concordant with a relative dierence less than 1%. he same remark holds or the long time viscous regime but the relative dierence being now less than %. o state on the accuracy o the methods used, we zoomed the dierent graphs obtained in the short time regime, on the dam site. We observed that the graphs observed rom the numerical method (ig. 1 intersected at a point which is more close to the one obtained experimentally, than in the analytical case (ig. 1. So, the numerical method seems more accurate than the analytical one.. Conclusion he low regimes o the horizontal viscous dam-break low are well known rom experimental studies. At initial time (when the dam collapses, the luid is released downstream (positive wave, while a negative wave propagates upstream. he low is inertial (Ritter s solution until the back wave reaches the ixed rear wall. hen, the viscous orces become higher than the inertial ones and a short time viscous regime takes place until. In this regime, the low height at dam site has a (ix characteristic value. As the relected wave overtakes the positive wave, the long time or asymptotic regime takes place. he present study considered the modelling o these two viscous low regimes. Applying the conservation o mass and momentum with the shallow water approximation, an equation o motion was derived and made non dimensional, when the viscous orces were assumed to be the dominant ones. It was o porous medium type and similar solutions built analytically. hen, the problem was considered numerically. he previous equation o motion was approximated using an explicit inite dierence method. he stability and convergence o the computations were insured using a criteria based on heuristic approach. he very good agreement between the numerical and the analytical solutions showed the consistence o the numerical scheme or both short time and long time solutions. he time evolution o the abscissa and velocities o the dierent ront waves were determined, as well as the dierent characteristic heights. Reerences: [1] Ritter, A.: Die ortplanzung der wesser wellen. Ver Deutsch Ingenieure Zeitschrit, Vol.6, N, pp. 97-9, 189 (in German [] Gill, M.A.: Dam-break problem. Encyclopedia o luid mechanics, 6, N.P. Cheremisino, eds., Gul, ouston (exas, 19-17, 1987 [] De Saint-Venant, B. : héorie du mouvement non permanent des eaux. C. R. Acad. Sci., vol. 7(, 17-1, 1871 (in French [] Faure, J. And Nahas, N. : Etude numérique et expérimentale d intumescences à orte courbure du ront.», La ouille Blanche, vol. 16(, 76-87, 1961 (in French [] unt, B.: Newtonian luid mechanics treatment o debris lows and avalanches. J. ydr. Div. (ASCE, vol. 1(1, 1-16, 199 [6] Nsom, B., Debiane, K. and Piau, J.M.: Bed slope eect in the dam-break problem. J. ydr. Res. (IAR, vol. 8(6, 9-6, [7] Nsom, B.: orizontal Viscous Dam-Break Flood: Experiments and heory. J. ydr. Eng. (ASCE, vol. 18(, -6, [8] Schwarz, L.W.: Viscous lows down an inclined plane: Instability and inger ormation. Phys. Fluids A, vol. 1(, -, 1989 [9] Zoppou, C. and Roberts, S.: Explicit Schemes or Dam-Break Simulations. J. ydr. Eng.(ASCE, vol. 19(1, 11-, [1] Shigematsu,., Liu, P.L.F. and Oda, K.: Numerical modeling o the initial stages o dam-break waves. J. ydr. Res. (IAR, vol. (, 18-19, [11] Simpson, J.E.: Gravity Currents in the Environment and the Laboratory, John Wiley & Sons, New York, RE&PQJ, Vol.1, No.9, May 11