Critical control in transcritical shallow-water flow over two obstacles

Transcription

1 Lougboroug University Institutional Repository Critical control in transcritical sallow-water flow over two obstacles Tis item was submitted to Lougboroug University's Institutional Repository by te/an autor. Citation: GRIMSHAW, R.H.J. and MALEEWONG, M., 25. Critical control in transcritical sallow-water flow over two obstacles. Journal of Fluid Mecanics, 78, pp Additional Information: Tis article was publised in te Journal of Fluid Mecanics [ c Cambridge University Press] and te definitive version is available at: ttp://dx.doi.org/.7/jfm Metadata Record: ttps://dspace.lboro.ac.uk/234/9523 Version: Accepted Publiser: c Cambridge University Press Rigts: Tis work is made available according to te conditions of te Creative Commons Attribution-NonCommercial-NoDerivatives 4. International (CC BY-NC-ND 4.) licence. Full details of tis licence are available at: ttps://creativecommons.org/licenses/by-nc-nd/4./ Please cite te publised version.

2 Critical control in transcritical sallow-water flow over two obstacles Roger H.J. Grimsaw and Montri Maleewong 2 Department of Matematical Sciences, Lougboroug University, Lougboroug LE 3TU, UK 2 Department of Matematics, Faculty of Science, Kasetsart University, Bangkok, 9, Tailand Abstract Te nonlinear sallow-water equations are often used to model flow over topograpy In tis paper we use tese equations bot analytically and numerically to study flow over two widely separated localised obstacles, and compare te outcome wit te corresponding flow over a single localised obstacle. Initially we assume uniform flow wit constant water dept, wic is ten perturbed by te obstacles. Te upstream flow can be caracterised as subcritical, supercritical, and transcritical respectively. We review te well-known teory for flow over a single localised obstacle, were in te transcritical regime te flow is caracterised by a local ydraulic flow over te obstacle, contained between an elevation sock propagating upstream and a depression sock propagating downstream. Classical sock closure conditions are used to determine tese socks. Ten we sow tat te same approac can be used to describe te flow over two widely spaced localised obstacles. Te flow development can be caracterized by two stages. Te first stage is te generation of upstream elevation sock and downstream depression sock from eac obstacle alone, isolated from te oter obstacle. Te second stage is te interaction of two socks between te two obstacles, followed by an adjustment to a ydraulic flow over bot obstacles, wit criticality being controlled by te iger of te two obstacles, and by te second obstacle wen tey ave equal eigts. Tis ydraulic flow is terminated by an elevation sock propagating upstream of te first obstacle and a depression sock propagating downstream of te second obstacle. A weakly nonlinear model for sufficiently small obstacles is developed to describe tis second stage. Te teoretical results are compared wit fully nonlinear simulations obtained using a well-balanced finite volume metod. Te analytical results agree quite well wit te nonlinear simulations for sufficiently small obstacles. Introduction. Background Sallow water flow of a omogeneous fluid over bottom topograpy is a fundamental problem in fluid mecanics and as been eavily studied from various points of view. A widely used approac wen te topograpy is a single localised obstacle is te application of ydraulic concepts wic lead to te classification of te flow in terms of te value of te upstream Froude number, defined as te ratio of te uniform upstream flow to te linear long wave speed. Te flow is ten described as supercritical, subrcritical or transcritical depending on weter te upstream Froude number is greater tan unity, less tan unity, or close to unity respectively, see for instance te monograp by Baines (995) for a compreensive account of ydraulic teory and te issues involved. In te supercritical case waves generated by te flow interaction wit te obstacle propagate downstream away from te obstacle, and te flow at te obstacle location is a locally steady elevation. In te subcritical case, waves

3 propagate upstream and downstream away from te obstacle, and te flow at te obstacle location is a locally steady depression. Wen wave dispersion is considered, steady lee waves are also formed downstream of te obstacle. Bot tese cases can be well understood, at least qualitatively, using linearised teory. However, linearized teory fails in te transcritical regime, wic is te main interest ere, and ten a nonlinear teory is needed to describe te locally steady ydraulic flow over te obstacle, wic as an upstream elevation and a downstream depression, eac terminated by upstream and downstream propagating undular bores. A popular model ere in te weakly nonlinear regime wen te obstacle as a small amplitude is te forced Korteweg-de Vries (KdV) equation, see Akylas (984), Cole (985), Grimsaw and Smyt (986), Lee et al. (989), Binder et al. (26), Grimsaw et al. (27) and te recent review by Grimsaw (2). Various aspects of te extension to finite amplitudes in te long wave regime can be found in El et al. (26), El et al. (28), and El et al. (29). Tus transcritical sallow water flow is quite well understood for a single localised obstacle, but tere ave been comparatively very few studies of te analogous case wen te tere are two widely separated localised obstacles. In te context of tis article te most relevant is te article by Pratt (984) were a combination of steady ydraulic teory, numerical simulations using te nonlinear sallow water equations and laboratory experiments are used to infer tat te formation of dispersive waves between te obstacles is needed to obtain a stable solution. More recently Dias and Vanden-Broeck (24) Ee et al. (2, 2) ave examined te possible presence of suc waves for steady flows, wile Grimsaw et al. (29) considered te related problem of unsteady flow over a wide ole.tus a new feature of interest wen considering two obstacles is tat te waves generated by eac obstacle may interact in te region between te two obstacles, and ten te question is ow tis interaction migt affect te long-time outcome. In tis paper we examine tis scenario using te nonlinear sallow water equations, so tat altoug finite-amplitude effects are included, wave dispersion is neglected and te generated waves are represented as sock waves. Our empasis is on te transcritical regime for two widely-spaced localised obstacles. Te nonlinear sallow water equations are solved numerically using a well-balanced finite volume metod, and te results are sown in section 3. Te simulations are supplemented by a combination of fully nonlinear ydraulic teory wit classical sock closure conditions, and a reduced model used in te weakly nonlinear regime, presented in section 2. We conclude in section 4..2 Formulation Te basic model is one-dimensional sallow-water flow past topograpy, in wic te flow is described by te total local dept H and te dept-averaged orizontal velocity U. Te upstream flow is a constant orizontal velocity V >, and te forcing is due to a localised topograpic obstacle f(x) so tat te bottom is at z = +f(x) were is te undisturbed dept at infinity. Hencefort, we use non-dimensional coordinates, based on a lengt scale, a velocity scale g and a time-scale of /g., in terms of wic te equation system is ζ t + (HU) x =, H = + ζ f, () U t + UU x + ζ x =. (2) In tese non-dimensional coordinates, te constant upstream flow is F = V/ g, te Froude number. Here te topograpy f(x) consists of two obstacles, eac symmetrical, and placed 2

4 a distance L apart, wit respective maximum eigts (or depts) of ɛ,2. Our interest ere is wen ɛ,2 >, and te situation wen eiter or bot ɛ,2 < will be considered elsewere. We assume tat te separation distance L is muc greater tan te widt of te obstacles. Ten te main parameters are te Froude number F, and te maximum eigts ɛ,2. Tis system is to be solved wit te initial conditions H =, U = F, at t =. (3) Tis is equivalent to introducing te obstacles instantaneously at t = into a constant flow. Te solution will initially develop smootly, but being a nonlinear yperbolic system, we can expect te development of discontinuities in te derivatives of ζ, U. Te classical procedure is ten to introduce socks, given by S[ζ] + [HU] =, S[HU] + [HU H2 ] =. (4) Here S is te sock speed, and [ ] denotes te jump across te sock. In te absence of te bumps (f(x) = ), tese classical socks conserve mass and momentum. In te transcritical regime wen F, it will be useful also to consider a weakly nonlinear model for small-amplitude topograpy, given by ζ t ζ x ζζ x + f x 2 =, = F. (5) Here U = F + u and u ζ. Te reduced model (5) can be seen as a dispersionless forced KdV equation, see te afore-mentioned references. For convenience we present an alternative derivation in te Appendix. Te initial condition (3) is replaced by ζ =, at t =. (6) In tis weakly nonlinear limit, te sock conditions (4) reduce to Tis can also of course be directly deduced from (5). 2 Hydraulic flow 2. Steady solutions (S )[ζ] [ζ2 ] =. (7) Here we consider te ydraulic teory, and to begin wit we review te well-known teory (see Baines (995) for instance) for flow over a single obstacle obstacle. Ten we will sow ow tis can be extended to obtain analogous solutions for flow over two obstacles. Tus we seek steady solutions, so tat on omitting te time derivatives, equations (, 2) integrate to HU = ( + ζ f)u = Q, ζ + U 2 2 = B, (8) Here Q, B are positive integration constants, representing mass flux and energy flux respectively (strictly Q is volume flux, but we are assuming tat te fluid density as been scaled 3

5 to unity, and B is te Bernoulli constant, wile BQ is te energy flux). Eliminating H or U gives G 4/3 + 2 G = B + f, G = U 3/2 U = 2/3 Q 2/3 H/2 Q = Q. (9) /2 H3/2 wic determines te local Froude number G as a function of te obstacle eigt f. For noncritical flow, tis solution must connect smootly to U = F, ζ =, tat is G = F, at infinity, and so Q = F, B = F 2 /2. Noting tat ten te rigt-and side of te first expression in (9) as a minimum value of 3/2 ɛ m wen F =, it can be establised tat < ɛ m < + F 2 2/3 3F. () 2 2 Here ɛ m is te maximum obstacle eigt. Tis expression is plotted in figure at equality (note tat tis is te curve BAE in figure 2. in Baines (995)). It defines te subcritical regime F < F b < were F < G <, and te supercritical regime < F p < F were < G < F and a smoot steady ydraulic solution exists. In te subcritical regime a localised depression forms over te obstacle, and in te supercritical regime a localised elevation forms over te obstacle. For small (ɛ m ) /2 <<, recalling tat = F, we find tat p,b = ± (6ɛ m) /2 + ɛ m O(ɛ3/2 m ). () In te transcritical regime F b < F < F p () does not old and is replaced by ɛ m > + F 2 2/3 3F. (2) 2 2 Instead we seek a solution wic as upstream and downstream socks propagating away from te obstacle, and wic satisfies te critical flow condition at te top of te obstacle, tat is, wen f = ɛ m, G x. Tis condition implies tat 3Q 2/3 G =, = B + ɛ m at f = ɛ m. (3) 2 For a given ɛ m, tis relation defines B in terms of Q. At tis critical location U = U m = Q /3 and +ζ m ɛ m = Q 2/3. Te local Froude number varies over te range G < G < G + were ± denote te downstream and upstream values, It transpires tat in order for te socks to propagate away from te obstacle te flow is subcritical upstream were G < G <, ζ > ζ > ζ m, U < U < U m, and supercritical downstream < G < G +, ζ + < ζ < ζ m, U + > U > U m. Before proceeding we note tat te expressions (9) old bot upstream and downstream, yielding te relationsips and so U ± ( + ζ ± ) /2 = Q, U 2 ± 2 + Q U ± = wile G 4/3 ± 2 U 2 ± 2 + ζ ± = B, (4) Q 2 2( + ζ ± ) 2 + ζ ± + = B +, (5) + G 2/3 ± = B +. (6) Q2/3 For given Q, B, tese relations fix U ±, ζ ± completely. But we ave one relationsip (3) connecting B, Q, and so tere is just a single constant to determine. Tis is found using te classical sock closure described in te next subsection. 4

6 2.2 Classical sock closure Outside te obstacle U = U ±, ζ = ζ ± are constants, downstream and upstream respectively, and are connected to te undisturbed values U = F, ζ = far downstream and upstream, using classical sock closure based on te sock conditions (4), see figure 2. Since te steady ydraulic flow over te obstacle conserves mass and energy, rater tan mass and momentum, tese are nontrivial conditions to apply. Furter, it transpires tat we cannot simultaneously impose upstream and downstream jumps wic connect directly to te uniform flow. Instead, we first impose an upstream jump as specified by Baines (995), see also El et al. (29). Tere is ten a downstream jump wic connects to a rarefaction wave, see figure 2. First we consider te upstream jump, wic connects ζ, U to, F wit S <. Te first relation in (4) gives and te second relation in (4) gives ζ (S U ) = U F, or S ζ = Q F, (7) ( + ζ )(U F )(S U ) = ζ ( + ζ 2 ). (8) Eliminating S, or U F yield te following expressions ( + ζ )(U F ) 2 = ζ 2 ( + ζ 2 ), (9) S = F [( + ζ )( + ζ 2 )]/2. (2) and ( + ζ )F ζ [( + ζ )( + ζ 2 )]/2 = Q. (2) Since we need S < it follows tat we must ave ζ > and U < Q < F. Te system of equations is now closed, as te combination of (5) and (2) determines ζ in terms of B, so tat finally all unknowns are obtained in terms of ɛ m from (3). Furter, te condition ζ > serve to define te transcritical regime (2) in terms of te Froude number F and ɛ m. Downstream, tis procedure also determines U + > F, ζ + <, but in general, tis cannot be resolved by a jump directly to te state F,. Instead we must insert a rigt-propagating rarefaction wave, see figure 2. Te rarefaction wave propagates downstream into te undisturbed state, F, and so is defined by te values U r, ζ r were U r 2( + ζ r ) /2 = F 2. (22) It is ten connected to te ydraulic downstream state U +, ζ + by a sock, using te jump conditions (4) to connect te two states troug a sock wit speed S + >. Tere are ten tree equations for te tree unknowns ζ r, U r, S + and te system is closed. In te weakly nonlinear regime, wen te forcing is sufficiently small (te appropriate small parameter is α ɛ m ), te rarefaction wave contribution can be neglected as it as te amplitude of order α 3 wile te sock intensity is O(α). In tis limit we can solve te system of equations by an expansion in α and find tat 3ζ ± = 2 (6ɛ m + β ± ) /2 + O(α 3 ), β ± = 3ζ 3 ± 2ζ ± , (23) 5

7 S ± = 3ζ ± 4 + ζ2 ± 32 + O(α3 ), G ± = + 3ζ ± 2 + γ ± + O(α 3 ), γ ± = 9ζ2 ± 8 ζ ± 2, (24) Q = + + ζ ± 3ζ2 ± 4 + O(α3 ). (25) Here β ± = O(α 3 ), γ ± = O(α 2 ) are small correction terms, wic if needed explicitly can be evaluated to leading order using te leading order solution for ζ ±. It is useful to note ere tat using (23) and (), te local Froude numbers G ± = ± (6ɛ m) /2 + O(α 2 ) = + p,b + O(α 2 ), (26) 2 and are independent of at te leading order in α. Also, since te transcritical regime is defined by s < < p it follows tat at te leading order in α, te local downstream and upstream Froude numbers G ± are outside tis transcritical regime, and ence te downstream and upstream flows are indeed fully supercritical and subcritical respectively. 2.3 Two obstacles Te same procedure can now be followed wen tere are two widely separated obstacles. Based on our numerical simulations reported in section 3, te solution evolves in two stages. In te first stage te teory described above can be applied to eac obstacle separately. Ten in te second stage wen te downstream propagating waves emitted by te first obstacle interact wit te upstream propagating waves emitted by te first obstacle, an interaction takes place and tere is an adjustment to a new configuration. Tere are several scenarios depending on te obstacles eigts ɛ,2 and te Froude number F. For instance if bot obstacles satisfy te condition () for subcritical or supercritical flow ten te obtained solutions for eac obstacle separately will again be obtained. On te oter and if bot obstacles satisfy te condition (2) for transcritical flow ten at te end of te first stage a downstream depression sock preceded by a rarefaction wave emitted by te first obstacle will meet an upstream elevation sock emitted by te second obstacle. Our numerical simulations sow tat tese generate a new sock between te obstacles. Te speed S int of tis sock can be found from (4) were te conservation of mass law implies tat S int (ζ 2 ζ + ) = ( + ζ 2 )U 2 ( + ζ + )U + + O(α 3 ) = Q 2 Q + O(α 3 ). (27) Here te O(α 3 ) error is due to te presence of te rarefaction wave. Since ζ 2 > > ζ + te sock moves in te positive or negative direction depending on weter Q 2 > (<)Q. Indeed using te expressions (24, 2) S int = 3 4 (ζ + + ζ 2 ) + O(α 2 ) = 4 ({6ɛ } /2 {6ɛ 2 } /2 ) + O(α 2 ), (28) and is independent of to tis order. Tus, tis sock will move towards te iger of te two obstacles, tat is, S int is positive or negative according as ɛ > ɛ 2 or ɛ < ɛ 2 respectively. Tis is followed by te interaction of tis sock wit eiter te second or first obstacle, followed eventually by an adjustment to a final localised steady state encompassing bot obstacles; tis is te second stage. Te final localised steady ydraulic state can now be determined as before, wit te criterion tat criticality occurs at te iger obstacle so tat te formulae in subsections 6

8 2., 2.2 apply wit ɛ m = max[ɛ, ɛ 2 ], te same as if te combination of te two obstacles was a single obstacle. Indeed, te criticality determined at te first stage at te iger obstacle persists into te second stage, wile te flow at te lower obstacle adjusts in te second stage to be locally supercritical if tis lower obstacle is te second obstacle, or is locally subcritical if te lower obstacle is te first obstacle. Te final localised steady ydraulic state can now be determined as before, wit te criterion tat criticality occurs at te iger obstacle so tat te formulae in subsections 2., 2.2 apply wit ɛ m = max[ɛ, ɛ 2 ], te same as if te combination of te two obstacles was a single obstacle. Indeed, te criticality determined at te first stage at te iger obstacle persists into te second stage, wile te flow at te lower obstacle adjusts in te second stage to be locally subcritical if te lower obstacle is te first obstacle, or is locally supercritical if te lower obstacle is te second obstacle. Illustrative example taken from te numerical simulations are sown in figures 3 and 4 respectively. Note tat criticality is controlled by te iger obstacle wic as te same eigt in te two cases, and ence te same constant values of Q, B are generated in te region containing bot obstacles. Wen te obstacles ave equal eigts, ɛ = ɛ 2, ten also Q = Q 2 and te sock speed S int = (α 3 ), so tat te error term in (27) is needed to determine te sock speed. Tis error term is due to te neglected rarefaction wave, and wen tis as a negative mass flux as sketced in te scenario sown in figure 2, S int <. Te numerical solutions sow tat tis is indeed te case. Hence it is ten te second obstacle wic controls criticality. An example taken from our numerical simulations is sown in figure 5. In te region over bot obstacles combined tere is a steady state wit constant values of Q, B satisfying te relation (3). Te local Froude number G = at te crest of te second obstacle, were G passes smootly from subcritical G < to supercritical G >. Te flow is subcritical over te first obstacle, but G = at te crest of te first obstacle. At tis location tere is a discontinuity in te slope of G, and ence also in te slopes of U, H, but all quantities are continuous Tis can be deduced from (9, 3) were near te crest of eiter obstacle (G ) 2 3(ɛ m f) 2Q 2/3. (29) Tere are two possible solutions. We consider for simplicity te generic case wen ɛ m f δ(x ± L) 2, δ >. Ten at te second obstacle tere is a smoot solution for wic G C(x L), C = 3δ/2Q 2/3, but at te first obstacle te solution is G C x + L, wic is continuous but as a discontinuous slope. Tis can be regarded as a stationary contact discontinuity. Tis scenario is asymmetrical and so differs from tose considered by Pratt (984) wo examined only symmetrical configurations and sowed tese could not be stable. Furter e pointed out tat it is not possible to construct a steady stable solution using a stationary sock as tis would ten dissipate energy (see te last paragrap of is section and footnote on page 26). 2.4 Reduced model Before presenting te numerical results, it is useful to examine te same scenario presented above in subsections 2., 2.2, 2.3 using te reduced model, especially as ten te initial value problem can be solved, see Grimsaw and Smyt (986) and Grimsaw (2) for instance. Wit te initial condition tat ζ = at t =, equation (5) can be solved using 7

9 caracteristics, dx dt = 3ζ 2, dζ dt = f x 2, x = x, ζ =, at t =. (3) Te system (3) can be integrated to yield ζ 3ζ2 4 = 2 (f(x) f(x )), 3ζ = 2 { [f(x ) f(x)]} /2. (3) Here te upper sign is cosen until te caracteristic reaces a turning point were 2 = 3ζ and ten te lower sign is cosen. Wen = te upper (lower) sign is cosen on te left-and (rigt-and) side of te maximum point were f = ɛ m. Were caracteristics intersect, a sock forms wit speed S, given by (7) Ten wen 2 2 < 3ɛ,2, (32) tere is a critical x c for eac obstacle suc tat all caracteristics wit x < x c ave a turning point, propagate upstream and form an upstream sock. Oterwise all caracteristics wit x > x c ave no turning points, propagate downstream and form a downstream sock. Te critical point is defined by 3f(x c ) = 3ɛ, Ten, in te first stage, a steady solution will emerge over eac obstacle, terminated by upstream and downstream socks, determined by tat caracteristic emanating from x c and te corresponding steady solution is found using (3) ζ + 9ζ 2 = 6(ɛ m f(x)) 3ζ = 2 sign[x L]{6[ɛ m f(x)]} /2. (33) Te upstream (downstream) sock as a magnitude ζ were 3ζ = 2 ± {6ɛ m } /2, (34) respectively. Note tat ζ + >, ζ < so tat te upstream sock is elevation and te downstream sock is depression. Te speeds of tese socks are found from (7), tat is and S <, S + >. wile te local Froude number is 4S = 2 {6ɛ m } /2, (35) G = + 3ζ/2, and so G = {6ɛ m} /2. (36) 2 In te first stage, tis local steady solution olds only for eac obstacle separately. Wen tere are two obstacles te upstream elevation sock from te obstacle will meet te downstream depression sock from te obstacle. Tis generates a new sock, wit speed S int = 3 4 (ζ + + ζ 2 ) = 4 ({6ɛ } /2 {6ɛ 2 } /2 ), (37) wic is independent of, and is positive or negative according as ɛ > ɛ 2 or ɛ < ɛ 2 respectively. Tese results all agree wit te small amplitude limits of te corresponding expressions in te preceding subsections.. 8

10 3 Numerical results 3. Numerical metod Te nonlinear sallow water equations (,2) can be written as U t + F x = G, (38) were U, F and G represent te density vector, flux vector, and source term respectively, [ ] [ ] [ ] H UH U =, F = UH HU 2 + H 2, G =. (39) /2 Hf x Te computational domain, < x < x L, is discretised by uniform cell size x. Te cell center is denoted by x i were x i /2 and x i+/2 refer te left and te rigt cell interface, respectively. In discretization form, equation (38) can be written as Ui n + F i+/2 n F i /2 n = Gi n (4) t x Superscript n refers to time step level. Te gradient of flux funtion is approximated by te difference of numerical fluxes at te left, Fi /2 n, and te rigt, F i+/2 n, of cell interfaces respectively. At te cell interface i + /2, U n+ i F n i+/2 = F(U n i+/2, U n i+/2+). (4) Numerical flux at te cell interface is a function of unknown variable on te left and te rigt limits, and [ ] [ ] H n Ui+/2 n i+/2 H = Hi+/2 n U, U n i+/2+ n i n2 i+/2+ = Hi+/2+ n U. i+ n2 Applying te ydrostatic reconstruction from Audusse et al. (24), H n i+/2 = max(, H i + f i f i+/2 ), and H n i+/2+ = max(, H i+ + f i+ f i+/2 ). Bottom slope is now included in te reconstruction of water dept. Te value of bottom eigt at te corresponding interface is approximated by upwind evaluation, f i+/2 = max(f i, f i+ ). To obtain a well-balanced sceme, te gradient of source term and flux difference must be balanced at steady state, Audusse et al. (24), so equation (4) can be rewritten as, U n+ i Ui n t wit modified numerical fluxes, + F n l (U n i, U n i+, f i, f i+ ) F n r (U n i, U n i, f i, f i ) x F n l (U n i, U n i+, f i, f i+ ) = F(U n i+/2, U n i+/2+) + 9 [ ], Hi n2 Hi+/2 n2 /2, =, (42)

11 [ ] Fr n (Ui n, Ui+, n f i, f i+ ) = F(Ui+/2, n Ui+/2+) n +, Hi+ n2 Hi+/2+ n2 /2. In tis work, we apply weigted average flux (WAF) proposed by Siviglia and Toro (29); Toro (992); Toro et al. (994) to obtain te approximation of F(Ui+/2 n, U i+/2+ n ). We also apply te minmod flux limiter based on te total variation diminising (TVD) proposed by Toro (992) in our numerical sceme to remove spurious oscillations wen simulating moving sock problem. In our simulations, we apply transmissive boundaries to allow waves to propagate outwards on bot boundaries. Te bottom elevation is assumed to be two Gaussian obstacles given by f(x) = ɛ exp ( (x x a ) 2 /w ) + ɛ 2 exp ( (x x b ) 2 /w ), were ɛ and ɛ 2 are te obstacle eigts, x a and x b = x a + L are te center locations of te first and te second obstacle respectively, and te widt of eac obstacle is w =. 3.2 Equal obstacle eigts 3.2. ɛ =., ɛ 2 =. Simulations for a subcritical case F =.5 are sown in figure 6. Initially, in te first stage (t = 5), steady depression waves are produced over eac obstacle, and small transient elevation waves travel upstream from eac obstacle. In te second stage t = 7 te transient wave from te second obstacle as passed over te first obstacle and proceeded upstream. In te final stage (t = 3) only te steady depression waves over eac obstacle are left. In tis case, te Froude number is outside te transcritical regime for bot obstacles, see () and figure,. Simulations for a transcritical flow case F = are sown in figure 7. In te first stage (t = 5) a transcritical flow is generated over eac obstacle separately, consisting of an elevation sock propagating upstream connected by a steady solution to a. a depression sock propagating downstream. Te depression sock from te first obstacle meets te elevation sock from te second obstacle at around t = 3 forming a single sock, wic ten propagates upstream. In te second stage (t = 4), tere is an adjustment in wic a locally steady subcritical depression wave forms over te first obstacle, wile a locally steady transcritical flow forms over te second obstacle. At te same time te elevation sock and depression sock outside bot obstacles continue to propagate in teir separate ways. As time increases (t = ), te flow over bot obstacles reaces a locally steady state wit criticality controlled by te second obstacle. Next, we examine a quantitative comparison between te nonlinear sallow water simulations te teoretical results from te reduced model presented in section 3.2. From te numerical simulations sown in figure 7 over te time range t = 4 to we find tat te respective sock magnitudes and speeds, ζ + =.2574, ζ =.267, S + =.88, S =.98. Wit ɛ m =. te local Froude numbers in equation (36) are G + =.3873, G =.627, wile te sock magnitudes from equation (34) are ζ + =.2582, ζ =.2582, and te sock speeds from (35) are S + =.937, S =.937. Tese values are in reasonable agreement wit te numerical determined values. Using te more exact formulas (23, 24) up to te O(α 2 ) terms leads to ζ + =.2468, ζ =.269 and S + =.87, S =.996, wic is an improvement. Note tat te effective small parameter ere is (6ɛ m ) /2 =.7746 and so is not small enoug for te reduced model to be completely accurate.

12 Simulations for a supercritical flow case F =.5 are sown in Figure 8. Initially, in te first stage (t = 3), steady elevation waves are produced over eac obstacle, and small transient depression waves travel downstream from eac obstacle. At te beginning of te second stage (t = 7) te transient wave from te first obstacle is passing over te second obstacle and proceeded upstream. In te final stage (t = 4) only te steady elevation waves over eac obstacle are left. In tis case, te Froude number is outside te transcritical regime for bot obstacles, see () and figure,. It sould be noted tat in te reduced model te local Froude number (36) satisfies.627 < G <.3873 for ɛ m =.. Tis prediction is consistent wit te nonlinear simulations sown in figure 6 for subcritical flow, figure 7 for transcritical flow, and figure 8 for supercritical flow ɛ =.2, ɛ 2 =.2 Four simulations for F =.5,.,.5, 2. are sown in figures 9-2. Wen ɛ m =.2 transcritical flow occur in te range of.48 < F <.56, see () and figure. Te reduced model predicts transcritical flow wen.45 < F <.55, see (). Tus te flow is sligtly transcritical for F =.5,.5, respectively nearly subcritical or supercritical, wile it is transcritical for F =., and supercritical for F = 2.. In all cases we expect te reduced model to provide quite good intepretation. Te nearly subcritical case sown in figure 9 can be compared wit te subcritical case sown in figure 6 for ɛ = ɛ 2 =.. Altoug te first stage(t = 3, 6 is similar tere is now visible two small rarefaction waves propagating to te left, and in te second stage (t = 3, 8) a pronounced asymmetry develops wit a larger depression wave over te second obstacle. Tis is due to tis case being in te transcritical regime, and ence te second obstacle controls criticality. Te transcritical case sown in figure is qualitatively similar to tat in figure 7 for ɛ =., ɛ =.. From te numerical simulations sown in figure 7 over te time range t = 4 to 8 we find tat te respective sock magnitudes and speeds, ζ + =.36, ζ =.38, S + =.2535, S =.284. Wit ɛ m =.2 te local Froude numbers in equation (36) are G + =.5477, G =.4523, wile te sock magnitudes from equation (34) are ζ + =.365, ζ =.365, and te sock speeds from (35) are S + =.2739, S = Tese values are in reasonable agreement wit te numerical determined values. Using te more exact formulas (23, 24) up to te O(α 2 ) terms leads to ζ + =.3422, ζ =.3867 and S + =.263, S =.2853, wic is overall some improvement. But note ere tat te effective small parameter is (6ɛ m ) /2 =.954 and can ardly be considered small. Te nearly supercritical case sown in figure can be compared wit te supercritical case sown in figure 8 for ɛ = ɛ 2 =.. Altoug te first stage(t = 3) is rater similar tere is already an asymmetry in tat te elevation wave oter te second obstacle is already sligtly smaller tan tat over te first obstacle, indication tat te adjustment process to te second date is beginning. Tis adjustment continues at t = 3 and te final locally steady state is acieved at t = 66, 2, in wic tere is criticality controlled by te second obstacle, and a locally subcritical flow over te first obstacle. Te fully supercritical case is sown in figure 2 and can also be compared wit te supercritical case sown in figure 8 for ɛ = ɛ 2 =.. It is quite similar altoug te time ten to reac te second stage is muc sorter.

13 3.3 Unequal obstacle eigts 3.3. ɛ =., ɛ 2 =.2, and ɛ =., ɛ 2 =.2 A transcritical case (F = ) wen te second obstacle is larger is sown in figure 3 for quite small amplitudes. At te first stage (t = 5), eac obstacle generates elevation and depression socks tat can be described by te single obstacle teory. As time increases (t = 46) te depression sock from te first obstacle interacts wit te upstream elevation sock generated by te second obstacle. A new sock is formed, called an intermediate sock as described in te analysis of section 2. Since te second obstacle is larger, te intermediate sock travels upstream and pasts over te first obstacle, leaving a locally steady depression wave in a locally subcritical flow (t = ). Te speed of intermediate sock is greater tan te speed of te travelling elevation sock from te first obstacle. Tese two socks merge and finally form a new sock moving furter upstream (t = 8). Next, we compare quantitatively tese nonlinear simulations wit teoretical results of from section 2. For ɛ =., we find from te nonlinear simulations tat te upstream sock magnitude and speed are ζ =.822, and S =.65, wile te reduced model predicts tat ζ =.86 and S =.62, and using te more exact formulae (23, 24) leads to ζ =.828 and S =.69. Similarly, for te second obstacle wit ɛ 2 =.2, te downstream sock magnitude and speed from te simulations are ζ + =.34, and S + =.847 wile te reduced model predicts tat ζ + =.55 and S + =.866, and using te more exact formulae (23, 24) leads to ζ + =.32, and S + =.853. Tese comparisons sow very good agreement for tese small amplitude obstacles. Furter, te intermediate sock speed from te simulation is S int =.262, wile te teoretical expression (37) yields S int =.254. Also, note tat for te nonlinear simulations wen t = 8, te two upstream elevation socks merge to form a new one wit te new speed S =.867 wic is nearly te addition of S int and S (for ɛ =.). A case wit iger obstacle amplitudes, ɛ =., ɛ 2 =.2 is sown in figure 4. Te flow beaviour is quite similar to te smaller amplitude case. Here te intermediate sock speed from te simulation is S int =.286, but from equation (37), S int =.82. Te quite large difference is due to iger order nonlinear effects ɛ =.2, ɛ 2 =., and ɛ =.2, ɛ 2 =. A transcritical case (F = ) wen te first obstacle is larger is sown in figure 5 for quite small amplitudes. At te first stage (t = 5), eac obstacle generates elevation and depression socks tat can be described by te single obstacle teory. As time increases (t = 46), te downstream depression sock from te first obstacle interacts wit te upstream elevation sock generated by te second obstacle, and an intermediate sock is formed. Since te first obstacle is larger, it now controls criticality. Te intermediate sock travels downstream and passes over te second obstacle, leaving a locally steady elevation wave (t = 4) in a locally supercritical flow. Te speed of intermediate sock is greater tan te speed of te downstream travelling depression sock from te second obstacle. Tese two socks merge and form a new sock moving furter downstream (t = 8). Next, we compare quantitatively tese nonlinear simulations wit te teoretical results. For ɛ =.2, we find from te nonlinear simulations tat te upstream sock magnitude and speed are ζ =.7, and S =.88, wile te reduced model predicts tat ζ =.55 and S =.866, and using te more exact formulae (23, 24) leads to ζ = 2

14 .65, S =.87. Similarly, for te second obstacle wit ɛ 2 =., te downstream sock magnitude and speed from te simulations are ζ + =.8, and S + =.67, wile te reduced model predicts tat ζ + =.86, and S + =.62, and using te more exact formulae (23, 24) leads to ζ =.82, S =.64. Tese comparisons sow very good agreement for small amplitude obstacles. Furter, te intermediate sock speed from te simulation is S int =.26, wile te teoretical expression (37) yields S int =.254. Also, note tat for te nonlinear simulations wen t = 4 8, te two downstream depression socks merge to form a new sock wit te new speed S + =.82, wic is nearly te addition of S int and S + (for ɛ =.). A case wit iger obstacle amplitudes, ɛ =.2, ɛ 2 =. is sown in figure 6. Te flow beaviour is similar to te smaller amplitude case. Here te intermediate sock speed from te simulation is S int =.28, but from equation (37), S int =.82. Again, te quite large difference is due to iger order nonlinear effects. 4 Summary Transcritical sallow-water flow over two localised and widely-spaced obstacles as been examined using te fully nonlinear sallow water equations (, 2) and wit a combination of numerical simulations and teoretical analysis based on ydraulic flow concepts. For a single obstacle, te solution is typically a locally steady ydraulic flow over te obstacle contained between an upstream elevation sock and a downstream depression sock. For te case of two obstacles tere are two stages. At te first stage, eac obstacle generates an upstream propagating elevation sock and a downstream propagating depression sock, eac well described by te single obstacle teory. Ten in te second stage te downstream propagating depression sock from te first obstacle interacts wit te upstream propagating elevation sock from te second obstacle to produce an intermediate sock, wic propagates towards te larger obstacle, or if te obstacles ave equal eigts, towards te second obstacle. Tere is an adjustment to a locally steady flow over bot obstacles were te iger obstacle obstacle controls criticality, or if te obstacles ave equal eigts, te second obstacle controls criticality Tis outcome agrees wit te analytical teory based on ydraulic flow concepts extended ere form a single obstacle to two obstacles. As is known, te case of flow over a single negative obstacle, or ole, is more complicated, as te sock waves are generated at te obstacle location, see Grimsaw and Smyt (986), Grimsaw et al. (27) and Grimsaw et al. (29). Hence we expect te case wen eiter or bot of te obstacles are oles could lead to different and more complicated scenarios, wic will be te subject of a future study. Furter te present study is restricted to non-dispersive waves and extensions to include even just weak dispersion using te forced KdV equation, or te fully nonlinear Su-Gardner equations, as done by El et al. (29) for a single obstacle, will certainly lead to rater different beaviour. In tat case, te socks are replaced by undular bores and te sock interactions described ere are replaced by te interactions of tese nonlinear wave trains. For instance some of te numerical simulations reported by Grimsaw et al. (29) using just te forced KdV equation indicate tat te interaction of tese nonlinear wave trains can produce very complicated beaviour. Tis also is a topic needing muc furter study. 3

15 Acknowledgements Tis work was supported by Tailand Researc Fund (TRF) under te grant no. RSA56838 to te second autor. References Akylas, T. R. (984). On te excitation of long nonlinear water waves by moving pressure distribution. J. Fluid Mec., 4: Audusse, E., Boucut, F., Bristeau, M.-O., Klein, R., and Pertame, B. (24). A fast and stable well-balanced sceme wit ydrostatic reconstruction for sallow water flows. SIAM J. Sci. Com., 25: Baines, P. (995). Topograpic effects in stratified flows. CUP. Binder, B., Dias, F., and Vanden-Broeck, J.-M. (26). Steady free-surface flow past an uneven cannel bottom. Teor. Comp. Fluid Dyn., 2: Cole, S. L. (985). Transient waves produced by flow past a bump. Wave Motion, 7: Dias, F. and Vanden-Broeck, J. M. (24). Trapped waves between submerged obstacles. J. Fluid Mec., 59:93 2. Ee, B. K., Grimsaw, R. H. J., Cow, K. W., and Zang, D.-H. (2). Steady transcritical flow over a ole: Parametric map of solutions of te forced extended Korteweg-de Vries equation. Pys. Fluids, 23:4662. Ee, B. K., Grimsaw, R. H. J., Zang, D.-H., and Cow, K. W. (2). Steady transcritical flow over an obstacle: Parametric map of solutions of te forced Korteweg-de Vries equation. Pys. Fluids, 22:5662. El, G., Grimsaw, R., and Smyt, N. (26). Unsteady undular bores in fully nonlinear sallow-water teory. Pys. Fluids, 8:2724. El, G., Grimsaw, R., and Smyt, N. (28). Asymptotic description of solitary wave trains in fully nonlinear sallow-water teory. Pysica D, 237: El, G., Grimsaw, R., and Smyt, N. (29). Transcritical sallow-water flow past topograpy: finite-amplitude teory. J. Fluid Mec., 64: Grimsaw, R. (2). Transcritical flow past an obstacle. ANZIAM J., 52: 25. Grimsaw, R. and Smyt, N. (986). Resonant flow of a stratified fluid over topograpy. J. Fluid Mec., 69: Grimsaw, R., Zang, D., and Cow, K. (27). Generation of solitary waves by trancritical flow over a step. J. Fluid Mec., 587: Grimsaw, R., Zang, D.-H., and Cow, K. W. (29). Transcritical flow over a ole. Stud. Appl. Mat., 22:

16 Lee, S.-J., Yates, G., and Wu, T.-Y. (989). Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mec., 99: Pratt, L. J. (984). On nonlinear flow wit multiple obstructions. J. Atmos. Sci., 4: Siviglia, A. and Toro, E. (29). WAF metod and splitting procedure for simulating ydroand termal-peaking waves in open-cannel flows. J. Hydraul. Eng., 35: Toro, E. (992). Riemann problems and te WAF metod for solving two-dimensional sallow water equations. Pilos. Trans. R. Soc. London Ser. A, 338: Toro, E., Spruce, M., and Speares, W. (994). Restoration of te contact surface in te HLL-Riemann solver. Sock wave, 4:

17 Appendix Te weakly nonlinear model (5) for small-amplitude topograpic forcing in te transcritical regime can be derived as follows. First, we introduce te Riemann variables so tat equations (, 2) become R = U + 2C, L = U 2C, C = H, (43) R t + (U + C)R x + f x =, L t + (U C)L x + f x =. (44) Ten we assume tat f α 2 were α, and tat ζ α, ζ t α 2, u = U F α, and = F α. Next, noting tat U + C = F + + O(α), we can find an approximation to te rigt-going Riemann invariant in te vicinity of te topograpy, R = F + 2 f 2 + O(α3 ), so tat u + ζ = ζ2 4 + f 2 + O(α3 ). (45) Here a transient propagating rapidly wit a speed F ++O(α) to te rigt is ignored. Ten we find tat for te left-going Riemann invariant, L = 2U (F + 2) + f 2 + O(α3 ) = F 2 2ζ + ζ f 2 + O(α3 ), U C = 3U 2 F f O(α3 ) = 3ζ 2 + O(α2 ). Tus finally te equation for L in (44) reduces to (5), wit an error of O(α 3 ). Similarly te mass sock condition in (4) reduces to (7) wit an error of O(α 3 ), wile te momentum sock condition as all terms of O(α 3 ). (46) 6

18 Figure : Plot of () at equality. Te intersection of te line ɛ m = constant wit te curve () defines F b,p respectively. Te region below te curve defines te subcritical and supercritical regimes, and te region above te curve is te transcritical regime. ζ ζ = ζ = ζ r ζ + ε m Figure 2: Scematic for closure using classical socks. 7

19 H G Q B x Figure 3: Hydraulic solution for te case F = and unequal obstacle eigts ɛ =., ɛ 2 =.2. In te steady region over bot obstacles Q =.8923 and B =.59, and G =.6584 at te crest of te first obstacle were te flow is locally subcritical. 8

20 H G Q B x Figure 4: Hydraulic solution for te case F = and unequal obstacle eigts ɛ =.2, ɛ 2 =.. In te steady region over bot obstacles Q =.8923 and B =.59, and G =.463 at te crest of te second obstacle were te flow is locally supercritical. 9

21 H G Q B x Figure 5: Hydraulic solution for te case F = and equal obstacle eigts ɛ = ɛ 2 =. In te steady region over bot obstacles Q =.9469 and B =.5464, and G = at te crest of te first obstacle, but G < in te vicinity of te first obstacle were te flow is locally subcritical. 2

22 .5.5 F =.5, t = 5, ε =., ε 2 = t = t = x Figure 6: Simulations for F =.5, ɛ =., ɛ 2 =.. 2

23 F =., ε =., ε 2 =. t = 5 t = 3 t = t = x Figure 7: Simulations for F =., ɛ =., ɛ 2 =.. 22

24 .5 F =.5, ε =., ε 2 = t = 3 t = 6 t = x Figure 8: Simulations for F =.5, ɛ =., ɛ 2 =.. 23

25 F =.5, ε =.2, ε 2 =.2 t = 3 t = 6 t = t = x Figure 9: Simulations for F =.5, ɛ =.2, ɛ 2 =.2. 24

26 F =., ε =.2, ε 2 =.2 t = 4 t = 2 t = t = x Figure : Simulations for F =., ɛ =.2, ɛ 2 =.2. 25

27 F =.5, ε =.2, ε 2 =.2 2 t = 2 t = 3 2 t = 66 2 t = x Figure : Simulations for F =.5, ɛ =.2, ɛ 2 =.2. 26

28 F = 2., ε =.2, ε 2 =.2 t = t = 2 t = t = x Figure 2: Simulations for F = 2., ɛ =.2, ɛ 2 =.2. 27

29 F =., ε =., ε 2 = t = t = t =.2.8 t = x Figure 3: Simulations for F =., ɛ =., ɛ 2 =.2. 28

30 F =., ε =., ε 2 =.2 t = 5 t = 8 t = t = x Figure 4: Simulations for F =., ɛ =., ɛ 2 =.2. 29

31 F =., ε =.2, ε 2 =..2.8 t = t = t = t = x Figure 5: Simulations for F =., ɛ =.2, ɛ 2 =.. 3

32 F =., ε =.2, ε 2 =. t = 5 t = 8 t = t = x Figure 6: Simulations for F =., ɛ =.2, and ɛ 2 =.. 3