Comparison between 1D and 2D models to analyze the dam break

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Ignacio Valcárcel LencinaTRITA LWR Masters Thesisair trapping in the former or vertical accelerations onthe latter, and their range of application depends onthem.The validity of the shallow water approach was basedon comparisons of the model against problems with aknown analytical solution, which has been done bymany researchers over the past years.Concerning comparisons with in situ measurements,Quecedo applied the shalow water approach tosimulate breaking of a tailing dam in East Texas. Allthese comparisons do not concentrate on intervalswhere vertical accelerations are important, like theinitial instants following failure, or the interaction withobstacles like vertical dykes. In these cases, the errorscan be important.Quecedo used a more accurate approach to assess thevalidity of the shallow water approach in these casesfor which no known analytical solutions are available.Based on the results obtained, the authors concludedthat:(i) The shallow water approach predicts (i) faster floodwaves when propagating over dry beds, thus providingsafer predictions of flood wave arrival times, (ii)reasonable estimations of the evolution of water depthfar from the dam.(ii) If failure takes place onto a wet bed, the shallowwater results can be considered reasonable incomparison to those obtained using the Navier–Stokesapproach.(iii) If the free surface curvature is high, the shallowwater approach misspredicts the pressure. Thus, itshould not be used to calculate, for instance, stresseson structures.In spite of the shortcomings of the shallow waterapproach when applied to dam break problems, thecomputational effort required by other methods, suchas the Navier–Stokes Level-Set algorithm presentedherein, in actual problems makes the shallow waterapproach attractive for large computational domains.The Navier–Stokes approach would be used for theanalyses of small areas when knowledge of the threedimensionalstructure of the flow is needed.2.2 One-dimensional Shallow Water equations.Once we have seen that the shallow water equations(from now on “SWE”) are the most suitable todescribe the dam break problem, the next step is todescribe them for the one-dimensional approach.The SWE are frequently used for modelling bothoceanographic and atmospheric fluid flow. Models ofsuch systems lead to the prediction of areas eventuallyaffected by pollution, coastal erosion, and polar icecapmelting.As previously seen in [2.1], comprehensive modellingof the dam break using physical descriptions such asthe Navier-Stokes equations can often be problematic,due to the scale of the modeling domains. The SWE,of which there are a number of representations,provide an easier description of such phenomena.The SWE are obtained from the Navier-Stokesequations, assuming that vertical velocities andaccelerations are negligible and integrating over thewater depth.This 1D model investigates the settling of the dambreakwave over a variable bed as a function of time.A mathematical relation represents the shape of thebed so that is easy to change parameters.The 1D Saint-Venant’s-SWE are the following:∂z∂(zv)+ = 0∂t∂x2∂(zv)∂(zv ) ∂z∂zf+ + g ⋅ z ⋅ + g ⋅ z ⋅∂t∂x∂x∂x22∂ ( zv)∂ ( zv)−υc⋅ + g ⋅ z ⋅ Sf − E ⋅ = 022∂x∂xOr in the compact form:∂z+ ∇ ⋅ ( zv)= 0∂t∂(zv)T+ ∇ ⋅ ( zvv ) + g ⋅ z ⋅∇zS∂t−υ⋅ ∆(zv)+ g ⋅ z ⋅ Sf − E ⋅ ∆(zv)= 0cWhere z is the thickness of the water layer (m), v is thevelocity (ms -1 ), g the gravity constant (m s –2 ), Sf is thefriction term (explained later) and υ c the kinematicviscosity (m 2 s -1 ). The wind effect is not present inthe equations.The dispersion coefficient E has been set to 2 m 2 /s.This value takes into account the turbulence and theheterogeneity of the velocity on the vertical. It is amajor unknown in the case of dam break, but theresults that we will see later, show that ComsolMultiphysics has need of artificial diffusion forstabilizing its solution.The definition of the thickness of the water layer, z, isz s -z f , where z s and z f are the measures in figure 1.(1)(2)(3)(4)4
Comparison between 1D and 2D models to analyze the dam break waveFigure 1. Representative vertical section through the fluiddomain showing the bed of a lake and the water surface.2.3 Two-dimensional Shallow Water equations.Depth averaging of the free surface flow equationsunder the shallow water hypothesis leads to a commonversion of the two dimensional SWE, wich inconservative form is:∂z∂(zu)∂(zv)+ + = 0∂t∂x∂y∂(zu)∂(zu+∂t∂x2) ∂(zuv)+∂y(5)∂z∂zf (6)− f ⋅ zv + g ⋅ z ⋅ + g ⋅ z ⋅∂x∂x22∂ ( zu)∂ ( zu)+ g ⋅ z ⋅ Sfx− E ⋅ − E ⋅ = 022∂x∂y∂(zv)∂(zuv)∂(zv+ +∂t∂x∂y+ g ⋅ z ⋅ Sf2)∂z∂zf (7)+ f ⋅ zu + g ⋅ z ⋅ + g ⋅ z ⋅∂y∂yy22∂ ( zv)∂ ( zv)− E ⋅ − E ⋅ = 022∂x∂yThe equations above are valid under the followingassumptions:1. The fluid is well-mixed vertically with a hydrostaticpressure gradient.2. The density of the fluid is constant.3. We study water waves of long wave lengths.4. The viscosity term is negligible.Where z is the depth of water, zu and zv are unitdischarges along the co-ordinate directions. Inaddition, u and v are the velocities in the x and ydirections respectively, g is the acceleration due togravity and Sf x and Sf y are the friction terms in the xand y directions respectively. The turbulent dissipationterms (E) and Coriolis effect (f) are present in theequations, but not the wind effect, which is notsignificant in the usual valleys where the dam-breakoccurs.2.3.1 The friction term.Many two-dimensional depth-averaged models includeonly friction at the bottom. Specifically, models thatassume vertical channel side-walls and use free-slipboundary conditions do not account for the friction atthe walls. Neglecting this effect, open channel flowwould likely show a marked variation in water depthfrom the one measured experimentally.Evaluation of the friction slope that quantifies theenergy loss at the side-walls of the channel is required.It is common to use a uniform flow law, Manning orChezy formulae, to calculate this term. However, theselaws were developed for one-dimensional flow andmust be extended and properly incorporated to twodimensionalequations.Manning’s formula used in one-dimensional shallowwatermodels is expressed in the formSfn2A⋅ Q ⋅ Q= (8)2⋅ R4 / 3Where n is the Manning coefficient and R = A/P is thehydraulic radius, which depends on the wettedperimeter P. For a rectangular channel P = b + 2zand for an irregular basin P = b.A distinction must be made between an arbitrarychannel cross-section and a cross-section with verticalwalls (fig. 2). Most two-dimensional models assumevertical walls and free-slip wall boundary conditionsfor rectangular channels, while the equations used tocompute the friction term are based on irregular crosssections.To correct this inconsistency, the frictionslope equation has been modified here to reflect thevertical side-wall assumption. Basically, themodification ensures that the entire wetted perimeter(bottom width and side-walls) is accounted for.For an arbitrary cross-section, the wetted perimeter ina cell is equal to the bottom width b (fig. 2), so that thehydraulic radius adopts the form R = z, the waterdepth. In this case, the two-dimensional friction termsare written in the form (Brufau and García-Navarro2000):22n ⋅ u ⋅ u +Sf x 4/ 3zv2= (9)22n ⋅ v ⋅ u +Sf y 4 / 3zv2= (10)5
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Comparison between 1D and 2D models to analyze the dam break

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