Models of non-boussinesq lock-exchange flow

Transcription

1 Under consideration for publication in J. Fluid Mec. Models of non-boussinesq lock-excange flow R. ROTUNNO, J. B. KLEMP, G. H. BRYAN AND D. J. MURAKI 2 National Center for Atmosperic Researc, Boulder, CO 837, USA 2 Department of Matematics, Simon Fraser University, Burnaby, BC V5A S6, Canada (Received April 3, 2) Nearly all analytical models of lock-excange flow are based on te sallow-water approximation. Since te latter approximation fails at te leading edges of te mutually intruding fluids of lock-excange flow, solutions to te sallow-water equations can be obtained only troug te specification of front conditions. In te present paper, analytic solutions to te sallow-water equations for non-boussinesq lock-excange flow are given for front conditions deriving from free-boundary arguments. Analytic solutions are also derived for oter proposed front conditions conditions wic appear to te sallow-water system as forced boundary conditions. Bot solutions to te sallow-water equations are compared wit numerical solutions of te Navier-Stokes equations and a mixture of successes and failures is recorded. Te apparent success of some aspects of te forced solutions of te sallow-water equations, togeter wit te fact tat in a real fluid te density interface is a free boundary, sows te need for an improved teory of lock-excange flow taking into account nonydrostatic effects for density interfaces intersecting rigid boundaries. Key Words: Lock-excange flow, Gravity currents, Sallow-water equations.. Introduction Lock-excange flow results from te adjustment under gravity of two fluids of different densities initially separated by a vertical partition in a oriontal cannel (Fig. a). In addition to gravity and pressure-gradient forces, a model of lock-excange flow must reckon wit stress at te cannel walls, stress and diffusion between te two fluids and, in cases involving a liquid-gas interface, surface-tension effects. Given te matematical complexity attaced to tese processes, te more tractable two-layer sallow-water equations, in wic te aforementioned processes are eiter neglected or simply represented, ave been applied to lock-exange flow by Rottman & Simpson (983, RS), Keller & Cyou (99), Klemp, Rotunno & Skamarock (994, KRS), Sin, Daliel & Linden (24), Lowe, Rottman & Linden (25, LRL) among oters. Judging te relative merits of tese differing applications of te sallow-water equations against laboratory data is difficult owing to te influence of te aforementioned neglected effects. However te gap between sallow-water teory and laboratory experiments can be bridged in certain cases by using numerical integrations of less approximate fluid-flow equations as surrogates for laboratory data as in KRS and Birman, Martin & Meiburg (25). In tis article we Te National Center for Atmosperic Researc is sponsored by te National Science Foundation.

2 2 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki ~ H a) Lock-excange-flow scematic ~ v f ~ d f ligter fluid ~ H b) Sallow-water teory ~ v f ~ d f eavier fluid ligter fluid ~ ~ ~ u f f eavier fluid ~ ~ u f f Figure. Scematic of lock-excange flow based on a) laboratory data and b) typical solutions to te sallow-water equations. Te vertical line at x = indicates te lock center were te eavier and ligter fluids are initially separated; te oriontal cannel walls are separated in te vertical by a distance H. After te release of te lock te eavier (ligter) fluid flows to te rigt (left) at speed ũ f (ṽ f ) wit tickness f ( d f ) measured some distance beind te complicated flow at te leading edge. ~ x extend te Boussinesq (density ratio of ligter to eavier fluid r ), two-layer sallowwater teory put forward by KRS to te non-boussinesq case and ten evaluate it and anoter sallow-water teory against compatible (i.e. free-slip, no surface-tension, etc.) numerical integrations of te Navier-Stokes equations for lock-excange flow. As te sallow-water equations (SWE) are based on te ydrostatic approximation, tey are incapable of describing flow wit strong oriontal variation, suc as at te leading edges of te mutually intruding interfaces sown in Fig. a. In order to use te SWE in suc cases, one admits solution discontinuities and appeals to a more complete pysical teory for conditions tat apply across tem. RS proposed using a formula developed by Benjamin (968) [is Eq. (22)] to relate te rigt-going front speed ũ f to its dept f in a Boussinesq dam-break calculation using te two-layer SWE. Benjamin s formula, based on mass and momentum balance across a control volume moving wit a steadily propagating gravity current (e.g. Simpson & Britter 979), gives ũ f in terms of te eigt f of te lower fluid well beind te complex flow of te gravity-current ead; in solutions of te SWE, Benjamin s control volume containing te gravity-current ead is represented by a simple discontinuity as sown in Fig. b. In teir application of te two-layer SWE to te Boussinesq lock-excange problem, RS found tat te leftgoing interface (Fig. a), wic was left to evolve freely, immediately became multivalued (see Fig. 7c of RS). KRS resolved te latter problem by recogniing tat te left-going interface must also be represented by a discontinuity tat satisfies Benjamin s front condition (Fig. b); Fig. 7d of KRS gives te solution of te SWE for te lockexcange problem in te Boussinesq limit. A continuing source of discussion in te literature is tat te Benjamin front condition admits a special dissipation-free solution along wit a continuum of solutions aving

3 Models of non-boussinesq lock-excange flow 3 dissipation. KRS pointed out tat te dissipation-free front condition in te SWE would imply a front speed tat is greater tan te speed at wic information can travel to it from te lock center (violating causality ) in bot te Boussinesq (r ) and cavity (r ) limits (see Fig. 3 of KRS). On te oter and some laboratory investigators promote te relevance of te dissipation-free front condition to experimental flows (e.g. Sin et al. 24), wile oter laboratory experiments (Simpson & Britter 979) and numerical experiments (Härtel, Meiburg & Necker 2; Bryan & Rotunno 28) find flows consistent wit a dissipative front condition for Boussinesq lock-excange flow. For non-boussinesq conditions (r < ), laboratory experiments (Keller & Cyou 99; Gröbelbauer, Fannelop & Britter 993; Lowe, Rottman & Linden 25) and numerical simulations (Birman, Martin & Meiburg 25; Étienne, Hopfinger & Saramito 25; Bonometti, Blacandar & Magnaudet 28) sow tat as r decreases from unity, te rigt-going front of relatively eavy fluid increases in speed and becomes more turbulent, wile te speed of te left-going front of relatively ligt fluid remains uncanged and becomes less turbulent. Tis latter feature led Keller & Cyou (99) to consider te left-going front as te realiation of te dissipation-free Benjamin front condition. Following Keller & Cyou (99), LRL used two-layer sallow-water teory to construct te solution between te dissipation-free left-going upper front and a rigt-going front obeying te (generally dissipative) Benjamin front condition across te range of r (Fig. 4b of LRL). Te causality problem raised by KRS of te impossibility of information flow in te SWE from lock center to te dissipation-free front was not addressed. In te present paper, we extend te KRS Boussinesq two-layer sallow-water teory to apply across te range of r. As in KRS, te present teory is based on te twolayer SWE and te application of te Benjamin front condition to te left- and rigtgoing fronts, respectively. As wit te KRS numerical integrations of te SWE in te Boussinesq limit, te present non-boussinesq solutions require te dissipative Benjamin front condition across te range < r for solutions tat obey causality. We ten verify tat tese numerical solutions are unique troug an independent, exact analysis using te metod of caracteristics. For comparison wit te present solutions, we ave also constructed analytical solutions following te LRL approac described above. Tese exact solutions offer a matematically firm explanation for te numerical-solution features suc as te apparent expansion fans and ones of constant state tat appear. Peraps more important is tat te exact analysis gives a clear picture of information flow troug te system as seen by te sallow-water teory. To evaluate te present and te LRL SWE solutions, we ave carried out two- and tree-dimensional numerical simulations using te Navier-Stokes (NS) equations. Te simulations are carried out under free-slip conditions at te cannel walls and witout surface-tension effects. One effect contained in te NS equations but not in te SWE is interfacial instability; ence viscous effects are unavoidable as tey effect te growt and ultimate disposition of unstable waves growing on te interface. As pointed out by (Benjamin 968, p ) te upper gravity current is completely stable as r wile it is unstable as r ; on te oter and te lower gravity current is unstable at any r. Consistent wit te foregoing arguments, te present numerical solutions indicate tat te caracter (laminar or turbulent) of te upper front is a function of bot r and te relative strengt of viscous effects troug te Reynolds number Re. For Re greater tan some tresold tat depends on r, we find better agreement between te NS solutions wit te present extension of te KRS teory tan wit te LRL teory; owever for Re less tan tat tresold, te numerical solutions indicate better agreement wit te LRL teory tan wit te present one.

4 4 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki Hence we are led to te conclusion tat for relatively smaller Re, te solution to te Navier-Stokes equations for te lock-excange problem is outside te SWE solution space for solutions respecting te causality condition. Our exact analysis of te LRL SWE solutions indicates tat all information flows from left-going dissipation-free front inward towards te lock center. From te point of view of te SWE, te left-going front must tus be viewed as an external agent; tat is, te matematical problem becomes a forced- (rater tan a free-) boundary problem. Numerical solutions of two-layer SWE for non-boussinesq lock-excange flow are described next in 2. Motivated by tese numerical solutions, wic are consistent wit information flow from te lock center outward, 3 describes an exact causal analytical solution to SWE using te metod of caracteristics. For reasons listed above, we also give in 3, analytical solutions of te SWE for prescribed frontal parameters at te leftgoing front of ligter fluid; tese solutions do not respect causality but may noneteless be useful descriptions of fluid flow features tat are beyond sallow-water teory. In 4 te SWE solutions to tose of te less-approximate NS equations and, in particular, examines te variation of te solutions wit bot density ratio r and Re. A summary and concluding remarks are given in Numerical Solution of te Two-Layer Sallow Water Equations Following RS, Keller & Cyou (99) and LRL, te two-layer SWE equations for flow in a oriontal cannel neglecting stress, diffusion and surface tension, can be written in terms of te lower-layer eigt and velocity; in nondimensional form tese are and were a = u ( )2 r( + ) ( ) 2 + r( ) t + u x + u x = (2.) u t + a u x + b =, (2) x and b = ( )3 ru 2 ( ) 3 + r( ) 2. (2.3a,b) In (2.)-(2.3), = /H and u = ũ/ g H, were is te dept, and ũ te velocity, of te lower layer; te independent variables are x = x/h and t = t g /H; te reduced acceleration due to gravity is defined by g = ( r)g. For future reference, note tat d = d/h = and, by continuity, v = ṽ/ g H = u/d, were v is te velocity and d is te dept of te upper layer. Te initial condition is u(x,) =, and (x,) = for x and (x,) = for x >. As mentioned in te Introduction, RS used a front condition to represent te gravity current at te leading edge of te rigt-going fluid in a Boussinesq sallow-water calculation, wile te left-going intrusion was left to evolve freely. As evidenced by te result of tat calculation (Fig. 7c of RS), (x,t) for te left-going interface immediately became multi-valued for t >. KRS demonstrated, troug an evaluation of te wavepropagation caracteristics at te leading edge of te disturbance propagating to te left into a reservoir of dept (were < wit = representing lock-excange flow), tat deeper eigts travel slower tan sallower eigts for >.5; ence multivalued solutions are to be expected, and te application of a front condition is required for Boussinesq lock-excange flow. For te present non-boussinesq case, te wavespeeds

5 Models of non-boussinesq lock-excange flow 5 associated wit system (2.)-(2) are given by c ± (u,) = 2 (u + a) ± 2 (u + a)2 4(au b) (2) [Eq. (3.3) of LRL]; setting u = in (2) gives c ( ) ( r) (2.5) for a disturbance propagating to te left into a reservoir of eigt. Following KRS one can deduce tat lower eigts travel faster tan iger eigts for > (+ r). In te Boussinesq limit, r = and KRS s result is recovered sowing tat lower eigts travel to te left faster tan iger ones for >.5. Te present analysis of Eq. (2.5) sows tat lower eigts travel faster tan iger eigts for any finite r and ence tere is te necessity for a front condition for lock-excange flow ( = ). We note in passing tat (2.5) illustrates just one of te many intricacies associated wit a moving contact line (Sikmuraaev 28, Capter 5). In te foregoing argument we are first considering te limit (lock excange) and ten r (cavity). However ad we first taken r, Eq. (2.5) would ten ave given te classical one-layer result c = implying tat tat te lower fluid takes no notice of te upper fluid, and terefore, of te upper bounding surface, even in te limit. Following Benjamin (968), application of mass and momentum conservation across te front of eac gravity current gives d f (2 d f )( d f ) v f = B(d f ), (2) + d f for te left-going front, and u f = r f (2 f )( f ) = + f r B( f). (2.7) for te rigt-going front. Wit te front conditions (2)-(2.7), numerical solutions of (2.)-(2) are computed across te range of r and sown in Fig. 2. Figures 2a, c and e sow snapsots at t = of te interface eigt (x) for r =.,.7 and, respectively, wile Figs. 2b, d and f sow te corresponding velocities u(x) and v(x) in te lower and upper layers, respectively. For te Boussinesq case (Figs. 2a, b), te solution as te required reflective symmetry [(x) = d( x), u(x) = v( x)]; te frontal parameters, f = d f =.3473 and u f = v f =.527 (cf. KRS s Fig. 7d). For te density ratio r =.7, Fig. 2c indicates tat (x) is no longer symmetric, altoug te front eigts are still equal to teir values in te r = case (Fig. 2a). Moreover te velocity distributions (Fig. 2d), indicate tat te lower-fluid front speed as increased, wile tat of te upper fluid as remained as it was in te r = case (Fig. 2b). Wit te density ratio reduced to r =, Fig. 2e indicates no cange in te tickness d f of te left-going front, wile tat of te rigt-going front f is reduced, wit respect to te r =. and.7 cases; te velocity distributions in Fig. 2f indicate a furter speed increase of te rigt-going front, but no cange in tat of te left-going front. In contrast wit te r =. and.7 cases, te case wit r = as bot and u independent of x for some distance beind te rigt-going front. To aid in te interpretation of te numerical solutions sown in Fig. 2, we examine te corresponding wavespeeds c ± (u,) given by (2). Figure 2b sows for te r =. case tat < c + u and, by symmetry, v c <. Wit te density ratio r =.7, Fig. 2d

6 6 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki a) c - v r =. b) u c + r =.7 c) d) u c + c- v e) x -5 5 r = f) c - v x c + u -5 5 Figure 2. Numerical solutions of te two-layer sallow water equations wit te front conditions (2)-(2.7) for r =.,.7 and. For eac r, te eigt of te interface is displayed in a), c) and e), and te layer velocities and wavespeeds (2) are displayed in b), d) and f). sows tat, as in te case wit r =., < c + u, but tat c + is closer in magnitude to u trougout te interval between te fronts. On te oter and for c, Fig. 2d for r =.7 sows tat v c for x and tat c for x, indicating tat no information can travel from te rigt to te left of x =. At a density ratio of r =, Fig. 2f sows tat c + > u, wile te distribution of c is qualitatively te same as for te r =.7 case. 3. Solutions of te SWE by te Metod of Caracteristics Te solution to te lock-excange problem sares wit te classic dam-break and (constant velocity) piston problems te property tat te governing equations, front condi-

7 Models of non-boussinesq lock-excange flow 7 tions, and te initial conditions are witout any implied space or time scales (Witam 974, p. 9). As a result, for t >, solutions must only depend on te similarity variable x/t, and te time evolution is simply a linear-in-time dilatation of te spatial structure. In tis section, te spatial profile, as a function of x/t, is determined by te metod of caracteristics. Te yperbolic equations (2.)-(2) can be written as ( u ) t [ u + b a ]( u ) x = (3.) were te wavespeeds c ± (2) are obtained as te eigenvalues of te matrix. Multiplication by te left-eigenvector (a c ±, ) gives te Riemann invariant relation (a c ± ) d dt du dt = (3) were te derivatives are along caracteristic trajectories (or rays ) defined by dx ± /dt = c ± (Witam 974, p. 6). As te Riemann invariant relation as no explicit dependence on x or t, it can be integrated as te ordinary differential du d = a c± (u,), (3.3) giving a relation between u and along rays x ± (t) (Courant & Friedrics 948, p. 44). In te present application we will be concerned wit two specific solutions of (3.3), tese are te (fastest) inbound rays tat are launced from eac of te two fronts. We define u + () using (3.3) wit te c + (u,) wavespeed and starting values given by te left-going frontal parameters (u L, L ). Likewise, we define u () wit te c (u,) wavespeed initialied wit te rigt-going frontal parameters (u R, R ). 3.. Free-boundary solutions Te locations of te left- and rigt-going fronts tat delimit te propagation of te lockexcange flow into regions of te quiescent fluid are determined ere by free-boundary arguments. First, conservation of mass dictates tat te fronts move wit te fluid speed; ence, on a ray diagram, te rigt-going front is te event line x/t = u R, and te left-going front is te event line x/t = v L. Second, te assumption tat te Benjamin relation applies provides anoter condition: ru R = B( R ) (2.7) for te rigt-going and v L = B(d L ) (2) for te left-going front. Determination of bot frontal variables, (u R, R ), or (v L,d L ), requires a tird condition. Tere are two possible scenarios for completing te front specification, and we refer to tese as te caracteristic Benjamin front and te time-like Benjamin front (Courant & Friedrics 948, p. 84; Jon 98, p. 28). Tese are defined below, were te rigt-going front is considered first. Te rigt-going front speed (u R ) must satisfy te inequality c R < u R c + R (3) to satisfy causality, tat is, forbidding te front to propagate faster tan te rigtside caracteristic speed (c + R ), and requiring tat te front ave influence on te trailing region of disturbed fluid. Te case of equality in (3) defines te caracteristic Benjamin front, were te event line coincides wit te rigt-going c + -ray. In tis case, te tird front condition is u R = c + R = c+ (u R, R ). (3.5) It follows from te caracteristic equation (2) tat (3.5) is satisfied only wen te

8 8 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki coefficient b(u R, R ) =, so tat from (2.3b), r u 2 R = ( R ) 3. (3) Simultaneous solution of (3) wit te Benjamin front condition (2.7) gives te unique solution r ur.5273 ; R.3473 (3.7) for te frontal parameters. It may be verified from (2) tat for frontal parameters (3.7), c R < u R, as required by (3). For te time-like Benjamin front, te frontal parameters are determined by a tird condition tat is dictated by an inbound ray (from te left). In tis case te front speed u R satisfies te strict inequality c R < u R < c + R (3) implying tat te c + -rays (from te left-going front) now propagate faster troug te disturbed fluid and overtake te rigt-going front. Terefore in tis case te rigt-going front parameters (u R, R ) derive from te solution of (3.3) for u + () for rays emanating from te left-going front togeter wit te condition (2.7). Considering now te left-going front, te causality inequality analogous to (3) is A left-going caracteristic Benjamin front is tus defined by c L v L < c + L. (3.9) v L = c L = c (u L, L ) (3.) and gives te tird determining condition. Following te same logic as for te rigt-going front, te condition (3.) leads to te relation v 2 L = ( d L ) 3 (3.) in analogy wit (3). For te left-going caracteristic front ten, combining (3.) wit te Benjamin front condition (2) gives te numerical solution for te frontal parameters v L.5273 ; d L.3473 (3) in analogy wit (3.7) except in tis case te result is independent of r. Te upper-fluid values (3) correspond to te lower-fluid values u L 86 ; L 527. (3.3) Tere is also te possibility for a left-going time-like Benjamin front. In tis case, te tird condition would be obtained by an inbound ray (from te rigt). Our calculations of c L using u L = u ( L ) from te solution of (3.3) sow, owever, tat tere are no solutions for wic te case of c L < v Lis pysically realied. Te analysis of c from te numerical SWE solutions sown in Figs. 2d and f is consistent wit tis result. We are now ready to construct solutions to te SWE by te metod of caracteristics and begin wit te Boussinesq case (r =.). Consistent wit Fig. 2a, we assume tat bot left- and rigt-moving fronts are of te caracteristic Benjamin type and tat terefore on a ray diagram te region of disturbed flow lies inside te cone c L x/t c+ R. Figure 3a sows te Riemann invariant u () tat emanates from te rigt-moving front [were (u,) = (u R, R ) given by (3.7)] togeter wit te Riemann invariant u + () tat emanates from te left-going front [were (u +,) = (u L, L ) given by (3.3)]. Tere is a unique crossing point were u + ( c ) = u ( c ) = u c, were u c 4 and c =.5. Tis crossing point tus defines two cones of influence on a ray diagram: te rigt-moving

9 Models of non-boussinesq lock-excange flow 9 front influences te region c c x/t c + R wile te left-moving front influences te region c L x/t c+ c, were c c = c (u c, c ) and c + c = c + (u c, c ). Te overlapping ones of influence tus define a cone c c x/t c + c were = c and u = u c is te one of constant state required to satisfy bot Riemann invariant conditions. Referring to Fig. 4a, te upper panel sows tis one of constant state (tin line segment) in terms of (x/t) and corresponds to te cone c c x/t c + c in te ray diagram directly below. Te oter curves in Fig. 3a are te wavespeeds and te upper-layer velocities as determined by teir respective Riemann invariant velocities u ± (). Figure 3a indicates tat te region v L x/t c c is uninfluenced by te u () Riemann invariant and ence tat region must be an expansion fan. Te solution (x/t) can be deduced parametrically from x/t = c [u + ()] (3) were c [u + ] is defined by te values on te so-labelled curve in Fig. 3. Likewise te region c + c x/t u R is also an expansion fan wit te solution given parametrically by x/t = c + [u ()]. (3.5) Te solution (x/t) for tese two expansion fans is sown by te tick line segments in te upper panel of Fig. 4a. Finally, te rays are computed using dx ± /dt = c ± wit (3) or (3.5) and sown as tin lines in te lower panel of Fig. 4a. It may be verified tat te analytical solution for (x/t) sown in Fig. 4a is essentially identical to te SWE numerical solution sown in Fig. 2a. Following te same procedure as for te Boussinesq case, we next construct te solution for te non-boussinesq case r =.7. Figure 3b sows tat te r-dependence in te rigtgoing front speed (2.7) produces an upward sift in u () so tat te intersection wit te (uncanged) u + () sifts to c.397. Oterwise te logic of te solution construction is identical to te Boussinesq case, and te spatial profile is illustrated in Fig. 4b. In tis case wit r =.7, te one of constant state sifts towards te rigt-going front and tere is a narrowing of te rigt-side expansion fan. Again te analytical solution for (x/t) sown in te upper panel of Fig. 4b is again identical wit te SWE numerical solution sown in Fig. 2b, Wit furter decreases in r, te rigt-side expansion fan is eventually eliminated at a critical value of r were c + c = u R. At tis critical value of r cr (.582) te solution must undergo a cange of spatial caracter. For r < r cr, te intersection of te two Riemann invariant curves u + () and u () occurs at an -value tat would be less tan te value for te rigt-going caracteristic Benjamin front ( R ) from (3.7). Specifically, te c + c would exceed te propagation speed u R of te assumed caracteristic Benjamin front, wic suggests tat left-side rays are overtaking te rigt-going front. Te resolution is tat te construction sould now assume tat te rigt-going front parameters now satisfy te time-like condition (3). Figure 3c sows te u + () Riemann invariant curve extended to te value R 725 wic, wit u R 23, also satisfies te Benjamin front condition (2.7). In te absence of te rigt-side expansion fan, te rigt-going front conditions are now constant-state conditions. Figure 4c (lower panel) illustrates tat te cange to te time-like front is manifested by rays launced from te left-going front catcing up to te rigt-going front as inbound rays. All rays of te c type carry constant values of and u. We note tat for r < r cr te ray diagram resembles tat of te classic dam-break problem (Witam 974, p. 457). As wit te previous two cases te analytical solution for te eigt (x/t) in te upper panel of Fig. 4c matces te SWE numerical solution sown in Fig. 2c. A summary of te rigt-going frontal values of R and u R as a function of density ratio

10 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki a) r =. u +.5 u - =u R =c + R c + [u - ] c - [u - ] u c c c + c - c + [u + ] u - u + =u L v - c - [u + ] v v L =c - L R c L u + u c b) r =.7.5 u - =u R =c + c + R [u - ] c - [u - ] c + c c - c c + [u + ] u - u + =u L c - [u + ] v - v v L =c - L R c L c + R u - =u R c + [u + ] c) r =.5 c - [u + ] u + =u L -.5 v + v L =c - L R.3.5 L.7 Figure 3. Solution to te Riemann invariant equation (3.3) for u ± () for density ratio r = a).. b).7 and c). Starting values for te integrations of (3.3) are indicated by te points u = u R, = R and u + = u L, = L. In a) and b), (u c, c) denotes te crossing point were u + () = u () and te tick solid lines indicates te parts of te u ± () solution curves relevant for calculating te corresponding wavespeeds c ± [u + ], c ± [u ] and upper-layer velocities v ± (long-dased curves). In c) tere is no crossing point wic indicates a cange in caracter of te solution.

11 Models of non-boussinesq lock-excange flow t (c - L, L ) (c - c, c ) (c+ c, c ) a) r =. (c +, ) R R x/t x t (c - L, L ) b) r =.7 (c +, ) c c (c -, ) (c +, ) R R c c x/t x (c - L, L ) (c- c, R ) c) r = (c +, ) R R x/t t x Figure 4. Analytical construction of te solution to te sallow water equations for density ratio r = a)., b).7 and c). For eac case te eigt of te interface (upper panel) is plotted versus te combined coordinate x/t wit te corresponding ray diagram given directly below. In a) and b) te one of constant state were (x/t) = c (tin line segment) is located witin te cone c c < x/t < c + c indicated in te ray diagram, and te left and rigt-side expansion fans (tick line segments) are defined by te ray-diagram cones c L < x/t < c c and c + c < x/t < c + R, respectively. In c) te rigt-side expansion fan disappears.

12 2 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki r are sown in Fig. 5. Free-boundary solutions for r > r cr.582 ave a caracteristic Benjamin front, wile for r < r cr ave a time-like Benjamin front. As can be inferred from Fig. 4, tere are two critical features tat coincide at te value r cr : (a) as r r cr from above, c + c u R (rigt-going expansion fan disappears); and (b) as r r cr from below, c + R u R (rays emanating from left-going front become parallel to te rigt-going front). Te summary in Fig. 5 describes te unique causal solutions assuming a left-going caracteristic Benjamin front and continuity on v L x/t u R. To sow tat te present solutions are indeed unique we consider te ypotetical case of a left-going time-like Benjamin front. A left-going time-like Benjamin front togeter wit a rigt-going timelike Benjamin front gives rise to a contradiction because bot Riemann invariants u ± () cannot satisfy te same end conditions u ± ( R ) = u R and u ± ( L ) = u L since an ODE inequality would follow from c [u ] c + [u + ]. A left-going time-like Benjamin front togeter wit a rigt-going caracteristic Benjamin front would require c L < v L (rays emanating from rigt-going front overtaking te left-going front) wic is found to occur only for te unpysical parameter regime r > (r cr ) >. Tus, te caracteristic solutions as described ere are te unique nonlinear solutions tat are continuous on v L x/t u R. 3. Forced-boundary solutions In tis subsection we analye te case considered by LRL of a dissipation-free, left-going front caracteried by te particular front conditions v L = u L = /2 ; d L = L = /2 (3) satisfying te Benjamin relation (2). Te caracteristic speeds (2) associated wit tese conditions satisfy te inequality v L < c L < c+ L, and ence te rays from te leftgoing front are bot directed into te lock excange region implying tat te frontal motion is not influenced by te flow witin te lock region. Terefore, from te point of view of te SWE, (3) violates causality as defined above (for furter discussion, see 4b) and (3) must be considered a forced-boundary condition. Matematically, te latter acts similarly to an initial condition and is commonly referred to as a space-like curve (Courant & Friedrics 948, p4; Jon 98, p8) on a ray diagram. Te construction of solutions to te SWE by te metod of caracteristics presented above is canged only in tat conditions (3)-(3.3) are replaced by (3). For te range of values r < r cr.5532, te Riemann invariant analysis leads to te rigt-going front being of te time-like Benjamin type. As seen in te example sown for r = in Fig. 6a, te spatial profile as a constant-state generated by te propagation of te left-going dissipation-free front for /2 < x/t < c L. Te non-causal, ence forced, nature of tis solution is evident in tat bot te c + and c rays are directed into te lock-excange region. Oterwise, te spatial structure of te solution for c L < x/t < u R is essentially like te free-boundary solution (Fig. 4c), consisting of an expansion fan and a constant-state following te rigt-going front. At larger values of r cr < r < r cr2 953, te spatial profile of te solution again as a constant-state attaced to te dissipation-free front followed by te two expansion fans typified by te rigt-going caracteristic Benjamin front case (e.g. Fig. 4b). Tis is illustrated for te case of r =.7 sown in Fig. 6b. However, unlike te free-boundary solutions, tere is a second critical value of r cr2 953 were te central constantstate coincides wit te dissipation-free-front conditions. Tis occurs were te Riemann invariant satisfies u (/2) = /2. Te implication of tis for values of r approacing te Boussinesq case, r cr2 < r, is tat solutions would seem to require c rays

13 2. Models of non-boussinesq lock-excange flow 3 a) u R.5 b).3 R r Figure 5. Rigt-side-front a) speed u R, and b) eigt R, as a function of r. Te present free-boundary solution (solid line) as two segments separated by te dot at r = r cr =.582 indicating te density ratio dividing caracteristic (r > r cr) from time-like (r < r cr) Benjamin fronts. Solutions wit te forced-boundary condition u L = L = /2 and free-boundary conditions for te rigt-side front (dased line) ave r = r cr =.5532 indicated by te star; te square at r = r cr2 = 953 is te limiting density ratio beyond wic continuous solutions do not exist for tese assumed frontal conditions. Te LRL solutions are same as te foregoing for r <.5532 but differ for r >.5532 (indicated by te solid gray line). Te front-speeds from te present 2D and 3D numerical simulations are nearly identical and indicated by te + ; data from te 2D simulations of Bonometti et al. (28) are indicated by te circles. tat cross. Matematically, tis situation is typically resolved by te appearance of a sock, or ydraulic jump. However, we coose not to pursue te analysis furter, as our computations based on te Navier-Stokes equations ( 4) suggest tat te dissipation-free front is not realied at tese larger values of r. Te solution for [u R (r), R (r)] using te

14 4 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki t t t a) r =, (v L, L )=(-.5,.5) (v L, L ) (c - L, L ) b) r =.7, (v L, L )=(-.5,.5) (v L, L ) (c - c, R ) c) r =.7, (v L, L )=(-.5,.5), LRL (v L, L ) -.5 (c - L, L ) (c + c, c ) (c - c, c ) (c - L, L ) (c + c, R ) (c - c, R ).5 (c + R, R ) x/t x (c + R, R ) x/t x (u R, R ) Figure 6. Analytical construction of te solution to te sallow water equations for te forced boundary condition u L = L = /2 and free-boundary conditions on te rigt-side front for a) r = and b) r =.7; c) LRL solution for r =.7 wit forced-boundary conditions at bot left and rigt fronts. x/t x

15 Models of non-boussinesq lock-excange flow 5 forced-boundary condition (3) is plotted on Fig. 5 as compared to te free-boundary solution. It is noted tat te rigt-going frontal parameters are little affected by tis cange in te left-going front. Finally we consider te solution procedure described in LRL (teir 3). LRL solve te Riemann invariant equation (3.3) for u + () starting from te left-going condition (3) and look for a crossing wit te Benjamin relation (2.7) to arrive at te rigt-going frontal parameters (u R, R ). For r < r cr te LRL procedure is te same as te procedure tat leads to te flow sown in Fig. 6a. However for r > r cr we find tat c + c < u R implying tat rays emanating from te left-going front do not reac te rigt-going front; ence in our construction of te solution for tis case, te left-going rays from te rigt-going caracteristic Benjamin front are necessary to complete te solution troug te rigtside expansion fan (Fig. 6b). Altoug it is possible to construct te solution for r > r cr (Fig. 6c) following te LRL procedure, one would need some pysical basis external to te SWE for assigning te derived frontal parameters (u R, R ) wic, as for te leftgoing front, must be considered a forced-boundary condition. In te limit as r, te LRL procedure produces te solution u = = /2 for /2 < x/t < +/2 (LRL, teir Figs. -2) and versions of te latter may be found in te literature dating back to te 94s (Yi 965, pp ). Neverteless, as sown in Fig. 5, tere is little difference in u R (r) produced by te present, te modified LRL or te LRL solutions to te SWE; te major difference is in R (r) for r cr < r <.. 4. Numerical simulations To assess te solutions of te two-layer SWE presented in te previous sections, we proceed ere to more general equations of fluid motion. In te present work we follow Étienne et al. (25) wo give te equations of motion for a mixture of two incompressible fluids of different densities. Te fluid density is given by ρ = ρ Φ + ρ l ( Φ) or, nondimensionaliing by te density of te eavier fluid ρ, ρ = r + ( r)φ, were Φ is te eavier-fluid volume fraction and ρ l is te density of te ligter fluid. Using te same nondimensionaliation as used for te sallow-water equations in 2, te equations expressing conservation of mass of te mixture, mass of te denser fluid and momentum of te mixture are, respectively, and ρ Du i Dt = p + 2 x i Re u i ( r) = x i r + ( r)φ DΦ Dt + Φ u i = 2 Φ x i ReSc x 2 i [λ(φ)(e ij u k δ ij )] x j 3 x k DΦ Dt, (4.) (4) ρ r δ i3 (4.3) were Re = ρ UH/η and Sc = (η/ρ )/κ are respectively te Reynolds and Scmidt numbers, η is a constant reference value for te dynamic viscosity, κ is te diffusion coefficient (assumed constant) and U = ( r)gh. Étienne et al. (25) let te dynamic viscosity µ = ηλ(φ) to allow for eiter constant dynamic viscosity (λ = ) or constant kinematic viscosity [λ = r + ( r)φ]. Seeking solutions tat are as close as possible to te pysical situation described by te two-layer SWE, we will focus on te limiting case Sc, signifying ero cross-species diffusion. Wit te assumption of constant dynamic viscosity, Eqs. (4.)-(4.3) simplify

16 6 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki to and u i x i =, (4) Dρ Dt = (4.5) ρ Du i Dt = p + 2 u i x i Re x 2 ρ j r δ i3. (4) Equations (4) (4) are te same as solved by Bonometti et al. (28) were it is noted (p. 45) tat tere is in effect a finite value of Sc O( 3 ) due to te limitations of finitedifferencing across te sarp cange in ρ at te fluid-fluid interface. Again, in conformity wit te SWE, we will assume stress-free conditions at te upper and lower boundaries. Hereinafter (4) (4) are referred to as te Navier-Stokes (NS) equations. As noted in te Introduction and in previous work, te interface separating te eavierfrom te ligter-fluid flows is generally unstable; ence one expects a transition to turbulence beyond some tresold value of Re, and terefore, turbulent stress between te two fluids. To avoid turbulent stresses, one migt restrict attention to lower-re (laminar) cases; owever for a low-re flow tere would ten be viscous stress between te two fluids. Hence stress between te fluids is a generally unavoidable difference between te NS and te SWE solutions for lock-excange flow. In te present paper we will present solutions ranging from turbulent to laminar flow. Altoug te lock (Fig. ) is in principle two-dimensional, turbulent motion is fundamentally tree-dimensional and terefore we will explore solutions to (4)-(4) for variations in (r, Re) in bot two and tree dimensions. Details on te numerical-solution tecnique, grid resolution, solution verification, etc. are given in te Appendix. 4.. Results and comparison wit te SWE solutions To facilitate comparison of te SWE solutions ( 3) wit te NS solutions, it is convenient to plot te latter as a function of x/t at a time long enoug for te establisment of a statistically steady-state solution. Plotted in tis way, long-wave features of te NS solutions stand out more clearly, and sorter-wave features, suc as te leading-edge gravity currents are compressed, in analogue to te way tey are represented in te SWE. Figures 7a, c and d sow te density field ρ(x/t,) from two-dimensional simulations of te cases r =.,.7 and, respectively, wit Re = 4, wile Figs. 7b, d and e sow te y-averaged density field ρ(x/t,) from tree-dimensional simulations for te same cases (all at t = 6). Beginning wit te Boussinesq case r =, Fig. 7a indicates flow instability along te interface between te advancing fronts; owever, witout te ability to produce a turbulent cascade to smaller scales, te flow is dominated by large-scale billows. In tree dimensions, Fig. 7b sows tat te two-dimensional instability is able to break down into tree-dimensional turbulence wic diffuses te interface. For te non-boussinesq case r =.7, te two- and tree-dimensional simulations in Figs. 7c and d, respectively, also indicate turbulent flow along te interface wit a suggestion of a reduced level of turbulence for te left-going front. However for r =, te two- and tree-dimensional simulations in Figs. 7e and f, respectively, indicate laminar flow for te left-going front and turbulent flow for te rigt-going front. Tis simulated disappearance of turbulence from te left-going front wit decreasing r as been found in te laboratory and numerical studies reviewed in. Overlaid on te tree-dimensional numerical solutions are te present and te LRL solutions to te SWE. Some general points of comparison follow. Bot te present and te LRL solutions of te SWE agree wit te NS solutions in tat

17 a) r =., 2D Models of non-boussinesq lock-excange flow 7 b) r =., 3D avg c) r =.7, 2D d) r =.7, 3D avg e) r =, 2D f) r =, 3D avg - x/t - x/t Figure 7. Navier-Stokes simulations of lock-excange flow in two dimensions for density ratios r = a)., c).7 and e) and in tree dimensions for r = b)., d).7 and f). Sown from te tree-dimensional simulations is te y-averaged density field. Te tree contour intervals displayed in all plots are.,.5 and.9, wit te middle value empasied. Overlaid on te tree-dimensional solutions are te present solutions to te SWE (solid gray line) and tose proposed by LRL (dased gray line). Te tree oriontal gray lines in b), d) and f) mark =.3473,.5, and 527.

18 8 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki a) r =., 3D avg.5 b) r =.7, 3D avg.5 c) r =, 3D avg.5-2 x rel -4 Figure 8. Across-cannel-averaged density contours.,.5 and.9 plotted as a function of te distance relative to te left-going front x rel at t= 6 from tree-dimensional Navier-Stokes simulations of lock-excange flow for r = a)., b).7 and c) wic correspond, respectively, to Figs. 7b, 7d and 7f. Overlaid is te interface as computed from Benjamin s potential-flow solution. Te dotted lined denotes L = 527 for te left-going front according to te present teory. te speed of te left-going front is independent of r wile tat of te rigt-going front is inversely proportional to r [te NS u R (r) data points, plotted in Fig. 5a, are generally slower tan te SWE solutions]; te NS solutions all ave v L.5 as in te LRL SWE solution as compared wit v L =.527 in te present SWE solution. Te present SWE solution agrees wit te NS solutions in tat te interface generally slants from te upper left to te lower rigt in all cases wile te LRL solution approaces a level interface as r. In bot SWE solutions te rigt-going front tins wit decreasing r in agreement wit te NS solution. We note tat te only one of constant state tat clearly emerges in te NS solutions is te one attaced to te left-going front for r = in agreement wit te LRL solution; oterwise te NS solutions exibit an interface tat slopes from upper left to lower rigt approximately linearly in te variable x/t. Denoting te eigt of te middle density contour by ĥ and letting η = x/t, tis linear dependence can be expressed as ĥ(η) = ĥl (ĥl ĥr) η η L η R η L. (4.7) Altoug it is difficult to identify unambiguously te parameters in (4.7) from te NS solutions, it seems clear tat, for tese solutions wit Re = 4, ĥl decreases wit r, reacing te asymptotic limit ĥl = /2 between r =.7 and r =.

19 Models of non-boussinesq lock-excange flow 9 r =, 2D - x/t Figure 9. As in Fig. 7e, except for Re = 5 ; analytical solutions overlaid as in Fig. 7f. 4. Discussion In our judgement, te foregoing NS-SWE-solution comparison indicates only limited success for te SWE solutions. Tis comparison is complicated by te presence in te NS solutions of turbulent eddies along te unstable interface between te two fluid layers and te tickening of te intruding layers in te ead regions immediately beind te fronts. Bot of tese effects are absent in te idealied SWE solutions. Te one place were turbulence is suppressed in te NS solutions is in te region beind te left-moving front for r = and ere, te solution compares favorably wit te LRL SWE solution. However, even in tis case it must be recalled tat te condition (3) is externally forced in tat tere are no caracteristic curves tat reac te event line x/t =.5 from te lock region (see Fig. 6c). In te NS solution te evolving interface is obviously a free boundary as information must come from te lock region. One can reasonably infer tat local nonydrostatic effects must produce a propagation speed faster tan tat supported by te SWE for te conditions (3); owever, a more precise matematical model for tis effect is unknown to te autors. Accepting tat tere is an inerent tendency for te left-going front to satisfy te conditions (3) for small r, we investigate te conditions under wic tis occurs in te NS simulations. Te tendency ĥl(r) /2 wit decreasing r is also accompanied by a decrease in turbulence at te left-going front (Fig. 7). Tese results suggest a transition wit decreasing r from a NS solution more akin to te present SWE solution ( L = 527 at a dissipative Benjamin front) to one more akin to te LRL SWE solution ( L =.5 at a dissipation-free Benjamin front). To reinforce te point, we sow in Fig. 8 ρ(x,) from te tree-dimensional simulations (at t = 6) togeter wit te potential-flow solution found in (Benjamin 968, is 4.3). A comparison across te range of r sows tat, in

20 2 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki te absence of turbulence (Fig. 8c), te NS solutions closely approximate te Benjamin potential-flow solution; owever, as sown in Fig. 8a, in te Boussinesq case turbulence develops beind te ead producing a departure from te Benjamin potential-flow solution and a transition to a turbulent wake of reduced tickness. Somewat counterintuitively wit increasing viscosity te solution moves towards te potential-flow (inviscid) Benjamin solution. Indeed increasing te Reynolds number to Re = 5 for tis case of r = indicates instability and turbulence at te left-going front and a solution more akin to te present SWE solution (Fig. 9). Hence tere is a strong indication from te present NS solutions tat te caracter of te left-going front depends on bot Re and r. For r teory indicates te upper left-going current is unstable to disturbances of all wavelengts, but tat as r decreases from unity, te longest wavelengts are stabilied [Benjamin (968, pp ) and Fig. 5a of LRL]. Since viscous effects are strongest at te sorter wavelengts, and since longer wavelengts become stable for r <, it stands to reason tat instability is suppressed for combinations of smaller r and smaller Re; te evidence from Fig. 8c is tat viscous effects are large enoug to suppress turbulence but not large enoug to cause major departures from Benjamin s potential-flow solution for r = and Re = 4. Te rigt-going frontal parameter u R (r) (Fig. 5a) from any of te SWE solutions compares rater well wit te present NS solutions as well as tose produced in 2D simulations by Bonometti et al. (28) over a wider range of r. Tere is, owever, significant disagreement between te present model and te LRL model rigt-going frontal parameter R (r) for r > r cr were te latter produces R /2 as r. For te Boussinesq case we believe te present model is closer to te NS solution in tat Fig. 7b sows a significant overall tilt (from upper left to lower rigt) of (x/t) rater tan te level interface (x/t) = /2 predicted in te LRL SWE solution (see teir Figs. 2 wit r = ). 5. Summary and Conclusions Altoug te general sense of te circulation in lock-excange flow is easily deduced from te initial baroclinic distribution of density and pressure, more precise detail on te motion and nature of te evolving interface requires a fluid-flow model. In te present work we reviewed and advanced analytical models based on te sallow-water equations for non-boussinesq lock-excange flow. Tese analytical models were ten compared wit teir counterpart numerical solutions based on te Navier-Stokes equations. Nearly all of te existing analytical models of non-boussinesq lock-excange flow are based on te sallow-water approximation. Since te latter approximation fails near te leading edges of te mutually intruding flows (Fig. ), front conditions must be given at bot left- and rigt-going fronts in order to find solutions to te sallow-water equations. Tat a variety of suc solutions exist in te literature is due to te individual investigators coice of front conditions (e.g. KRS and LRL); furtermore permitting discontinuous solutions between te two fronts gives an even greater variety of solutions (e.g. KC; see Fig. 4a of LRL). In KRS te sallow-water equations were solved numerically for te Boussinesq lock-excange problem under te conditions tat te left- and rigt-going fronts satisfy te Benjamin front conditions (2) and (2.7), respectively, and tat te front is a free boundary influenced by te motion witin te lock region (i.e. it obeys causality ). Te present paper as extended te KRS numerical solutions to cover non- Boussinesq flows (Fig. 2). Using te metod of caracteristics, we ave found exact analytical solutions (Figs. 3-5) tat verify te numerical solutions of te sallow-water equations given ere and in

21 Models of non-boussinesq lock-excange flow 2 KRS. Tey also reinforce te finding tat only solutions wit frontal parameters implying dissipation at te fronts obey causality in te sallow-water equations. However it as been noted in bot laboratory and numerical experiments tat te left-going front becomes less dissipative, taking te form of a potential-flow solution found by Benjamin (968), as r, te ratio of ligter- to eavier-fluid density, decreases. Following LRL we ave found analytical solutions of te sallow-water equations by imposing te dissipationfree condition on te left-going front. As noted in KRS, te dissipation-free front moves at a speed greater te sallow-water-equation wave speed and ence, from te point of view of sallow-water teory, must be considered a forced-boundary condition. For r < r cr =.5532, our solutions assuming a left-going dissipation-free front ave rays emanating from te left-going front tat impinge on te rigt-going front and tus determine te rigt-going frontal parameters; tese solutions are te same as tose of LRL (Fig. 6a). For r > r cr we find tat rays emanating from te left-going front do not reac te rigt-going front and tat te solution must be completed wit a rigt-going expansion fan (Fig. 6b). Notwitstanding tat te left-going rays do not reac te rigtgoing front for r > r cr, LRL continue to look for te intersection of te solution to te Riemann invariant equation (3.3) wit te Benjamin front condition (2.7) to find te rigt-going frontal parameters; we ave constructed analytical solutions (Fig. 6c), wit te understanding tat tese must be regarded as solutions in wic bot te left- and rigt-going frontal parameters represent forced-boundary conditions. In an attempt to autenticate te various solutions to te sallow-water equations, we ave carried out bot two- and tree-dimensional numerical solutions of te Navier- Stokes equations for relatively large Reynolds number (Re = 4 ), very large Scmidt number (Sc >> ) and free-slip conditions at te upper and lower bounding surfaces. For Re = 4 and r =, te interface separating ligter and eavier fluid is generally turbulent; owever, as found in recent numerical studies, te left-going front becomes less turbulent wit decreasing r. In te present simulations wit Re = 4, te left-going front is essentially laminar at r = (Fig. 7e, f) and closely approximates te Benjamin potential-flow solution (Fig. 8c). A furter experiment keeping r = but wit a larger Re sows tat te left-moving front is again turbulent (Fig. 9) suggesting tere is a dependence on bot Re and r tat determines te caracter of te left-going front. Comparison of te present (free-boundary) wit te LRL (forced-boundary) solutions of te sallow-water equations wit teir counterpart numerical solution of te Navier- Stokes (NS) equations produced mixed results. Bot te free and forced solutions reproduced te NS-solution features of left-going frontal parameters (v L,d L ) independent of r, rigt-going front speed u R increasing wit r and rigt-going front eigt R decreasing wit r. Te present free-boundary teory produced v L =.527 wile te forced-boundary teory prescribed v L =.5 wic agrees closely wit te NS solutions. On te oter and, in te limit as r, te forced-boundary teory gives te level interface (x/t) =.5 between te left- and rigt-going fronts, wile te present free-boundary teory gives an interface tat is tilted from lower rigt to te upper left implying tat bot d L and L are less tan.5 in agreement wit te NS solutions. We noted tat te only place were te NS solutions produced a one of constant state is in association wit te left-moving front for relatively small r and Re. Bot free- and forced-boundary teories gives very similar predictions for u R (r) wic in turn compared well wit te NS solutions. In te generally nonydrostatic NS solutions, te evolving interface is of course a free boundary wose motion must be influenced by te flow in te lock region. We are unaware of an analytical teory taking account nonydrostatic effects and a density interface intersecting te rigid surfaces tat can be used to explain te evolution from

22 22 R. Rotunno, J. B. Klemp, G. H. Bryan and D. J. Muraki t = to te time wen te steadily propagating fronts are establised in lock-excange flow. In addition to explaining ow information flows from te lock region to te fronts, suc a teory may also sed ligt on wy te te upper-front speed v L.5 across te range of r wile te upper-front L clearly varies wit r in te NS solutions (Fig. 7). It may also explain wy NS front speeds are relatively insensitive to te interfacial dynamics bot two- and tree-dimensional simulations (wit very different versions of interfacial turbulence) give surprisingly similar predictions for te front speeds (Fig. 5a). D.J.M. is supported troug NSERC RGPIN Appendix A. Details of te Navier-Stokes solver Numerical integration of te Navier-Stokes (NS) equations (4) (4) requires solution of an elliptic equation to determine pressure p. Solution tecniques can be expensive in tree dimensions wit resolution ig enoug for adequately resolved direct numerical simulation (DNS), and can be difficult to implement effectively on modern distributedmemory computing systems. As an alternative, we replace te mass-continuity equation (4) wit a prognostic equation for pressure. Tis procedure eliminates te need to solve an elliptic equation, but introduces te need to account for acoustic waves. Te latter problem is addressed in te present study using te procedure developed by Klemp, Skamarock & Dudia (27). Our derivation of an appropriate pressure equation follows Corin (967); erein, we assume p is a function of ρ only, and we invoke an artificial speed of sound c s dp/dρ, ten using (4) (4.5) we find Dp Dt = u i ρc2 s. x i We set c s = g H to ensure tat acoustic waves propagate muc faster tan te flow of interest. Te time-integration metod and spatial discretiation follow Bryan and Rotunno (28, p. 548) except te subgrid turbulence parameteriation of KRS is replaced by explicit stress-divergence calculations [second term on rigt side of (4)]. Te domain extends from x = 9 to x = +9 for r =.99, from x = 9 to x = + for r =.7, and from x = 9 to x = +3.5 for r =. Te initial lock is located at x =. All simulations extend from y = to y = and = to =. Grid spacing is /32 in all directions. Following previously publised guidelines for consistency in DNS between resolution and Re (Moin & Maes 998, 2.), tis resolution is considered sufficient for our nominal setting Re = 4. Pressure at t = is determined using (4) and (4). Because u i (t = ) = everywere and ρ(t = ) is a function of x only, ten te elliptic equation tat applies at t = is 2 p x 2 ρ p ρ x x + 2 p 2 =, wic is solved using successive over-relaxation. To allow for development of tree-dimensional motion in 3D simulations, small-amplitude random oriontal-velocity perturbations are added to te initial state. Sligtly iger amplitude perturbations are inserted at x <. to crudely replicate laboratory experiments in wic turbulent motions are created by abrupt removal of a partition at t. Te solver is evaluated in two ways: comparison against a laboratory result, and comparison against previously publised numerical simulations. For te first evaluation, we

23 Models of non-boussinesq lock-excange flow 23 3D simulation D simulation, y avg experiment, sadowgrap image Figure. Direct numerical simulation of a Boussinesq lock-excange laboratory experiment at t = 7 sowing a rendition of te middle density surfaces (upper panel), an across-cannel average of te density field (middle panel) and a sadowgrap image from te laboratory experiment of Sin et al. (24). run a tree-dimensional simulation and compare against te Boussinesq lock-excange experiment of (Sin et al. 24, teir Fig. 2). Our numerical simulation uses r =.99 and Re = 4, similar to values for te experiment of Sin et al. (24). No-slip boundary conditions are used for tis simulation. Results at t = 7 are sown in Fig., werein te upper panel sows a view of te eigt of te middle density surface and te middle panel sows ρ(x, ) from te numerical simulation; te lower panel sows te sadowgrap image from Sin et al. (24). Te numerical simulation clearly captures te salient features of te experiment, suc as te propagation speed of te fronts, turbulent mixing along te interface, and a steeply sloped interface at x =. For comparison against previously publised results we simulate two-dimensional lockexcange flow across a large range of r following Bonometti et al. (28). For tese simulations we use te same settings as Bonometti et al. (28): a domain of 25 ; = /6; x = /64; and no-slip boundary conditions. used a different metod for nondimensionaliation tan we use erein; to allow direct comparison to teir results we