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

Comparison between 1D and 2D models to analyze the dam break waveTo conclude, the last assumption concerns about thebathymetry profiling. An unreal bathymetry has beengenerated for the 1D model by the function sin(x)/10,which have been developed only downstream of thedam, where it has the real influence.This function creates a group of small elevations of±10 cm downstream of the dam, in order to observethe big influence of the bathymetry profiling when thedam break wave sweep along the valley. In thedenominator of the function there is a 10, it has beenonly the result after some different tests to prove themost suitable profile for the simulations.3.2 Dependent variables.Our dependent variables for this 1D model are z andzv. Where z is the depth of water, and zv is the unitdischarge along the co-ordinate direction.3.3 Constants.In this model, we call ‘constants’ the variables that areindependent of the geometry. Constants are global,that is, they are the same for all geometries andsubdomains. In this 1D model the following constantshave been used:Kinematic viscosity: ν = µ / ρIt is expressed as the dynamic viscosity divided by thedensity of the fluid. Kinematic viscosity is a measureof the resistive flow of a fluid under the influence ofgravity. Kinematic viscosity (Greek symbol nu: ν c ) hasSI units (m 2 s -1 ). The kinematic viscosity of water at20ºC is 10 -6 m 2 /s, and that is the value we assume forour model. This viscosity term is negligible; it has noinfluence although is has been taken intoconsideration. In the 2D model, the viscosity term isneglected and this is not an imprecise assumption.Acceleration due to gravityAt the Earth's surface, denoted g, is approximately 9.8m/s 2 (meters per second squared).Eddy diffusion coefficientIn fluid dynamics, an ‘eddy’ is the swirling of a fluidand the reverse current created when the fluid flowspast an obstacle. The moving fluid creates a spacedevoid of downstream-flowing water on thedownstream side of the object. Fluid behind theobstacle flows into the void creating a swirl of fluid oneach edge of the obstacle, followed by a short reverseflow of fluid behind the obstacle flowing upstream,toward the back of the obstacle. This phenomenon ismost visible behind large emergent rocks in swiftflowingrivers.The ‘eddy diffusivity’ is the exchange coefficient forthe diffusion of a conservative property by eddies in aturbulent flow. Also known as eddy-diffusioncoefficient. In this model, this coefficient is relevant toget the stability in the simulations. Without thiscoefficient, the results have been not satisfactory at all.We need this artificial diffusion for stabilizing thesolution. The dispersion coefficient E has been set to2 m 2 /s.Manning's Roughness CoefficientRoughness coefficients represent the resistance toflood flows in channels and flood plains. Suggestedvalues for Manning's n, tabulated according to factorsthat affect roughness, are found in the literature.Roughness characteristics of natural channels are givenby Barnes (1967). Barnes presents photographs andcross sections of typical rivers and creeks and theirrespective n values. From these guides we fetch atypical value for flood plains, n = 0.025Figure 3. The Constants dialog box.3.4 Scalar expressions.The scalar expressions are used to define scalarexpression variables that are valid on all geometrylevels everywhere in the current geometry. Thefollowing expressions have been defined in the model:The bed profile z f has been defined in this way:z f = (0)*(x=52)This is the analytical expression that has been chosento define the bathymetry in the 1D model; it meansthat there is a flat bed profile the first 52 m (upstreamof the dam) and a ‘sinusoidal’ bed profile downstreamof the dam, that is, the last 48 m of the one hundredmetres-domain.The derivative of z f with respect to the variable x hasbeen introduced here for computational effortreasons. Computing this term apart, the computerdoes not need to do it in the simulation for every finiteelement of the mesh, it takes simply the value which issaved here as another expression. This term appears inthe equations as we saw in [2.2].The initial value of the water layer z 0 has been set inthe same way as the bed profile, that is: z 0 =(10)*(x=50). It is developed in detail in[3.7].The friction term seen in [2.3.1] has been introducedhere with a different expression but it is really thesame. All the expressions as possible have been setapart from the equations for own convenience andcomputational effort reasons.7