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

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Ignacio Valcárcel LencinaTRITA LWR Masters ThesisFigure 4. The Scalar Expressions dialog box.3.5 Geometry.To create the 1D geometry model, a line which is 100m in length has been drawn. This is the first step to doonce the dimensions and the dependant variables havebeen chosen. The figure below show the 1D geometrymodel:Figure 6. Bathymetry profile, z f , used in the model.Figure 7. The initial water surface profile and thebathymetry profileFigure 5. 1D geometry model3.6 Bathymetry profile.As it was written previously, the bed profile z f hasbeen defined in this way:z f = (0)*(x=52)In the figure below we can see the sinusoidal bedprofile only downstream of the dam, the zone ofinterest for us hydrodynamically speaking (notice thedifference in scale between the x and y direction). Thiskind of profile is theoretical, but the purpose here isshowing the influence of the topography of the bed onthe elevation of the water surface. To see thisinfluence, these small elevations of 10 cm have beenintroduced. A higher elevation profile has moreinfluence on the water surface, but it introduces muchmore instabilities in the model.3.7 Initial value.For a time-dependent analysis, initial conditions mustbe provided. An initial value for each dependentvariable is entered. We can also enter initial values forthe first time derivative of the dependent variables.These are used when solving time-dependentproblems containing second time derivatives. In thiscase there are no second time derivatives, then weenter zero in both edit fields.Only one value different from zero in the first editfield is entered, for the dependent variable z. Thisvalue is equal to z 0 because it was defined previously inthe scalar expressions [3.4]. At t=0 the value of theother dependent variable zv is equal to zero.The figure 7 above shows clearly the situation at t = 0,before the dam break occurs.The expression for the initial value is:z 0 = (10)*(x=50).8
Comparison between 1D and 2D models to analyze the dam break waveIn 1D, Comsol Multiphysics always uses regularrefinement, where it divides each element into twoelements of the same shape. The refinement of themesh has gone from 15 to 7680 elements, the finaldivision that has been chosen to define the model withan acceptable accuracy.Figure 8. The initial condition specification page in theSubdomain Settings dialog box.3.8 Meshing.A mesh is a partition of the geometric model intosmall units of simple shape. For 1D geometry thesoftware partitions the subdomains (intervals) intosmaller intervals (or mesh elements). The endpoints ofthe mesh elements are called ‘mesh vertices’.The value in the ‘Maximum element size’ edit fieldspecifies the maximum allowed element size, which bydefault is 1/15th of the size of the geometry.The value in the ‘Maximum element size scalingfactor’ edit field comes into play if we do not explicitlygive a maximum element size. In that case, thesoftware multiplies the default maximum element sizeby this factor; its default value is 1.The value in the ‘Element growth rate’ edit fielddetermines the maximum rate at which the elementsize can grow from a region with small elements to aregion with larger elements. The value must be greateror equal to one. The default value is 1.3, that is,element size can grow by 30% (approximately) fromone element to another.Table 2. Mesh parameters and statistics.ParameterMaximum element size 1Maximum element size scaling factor 1ValueElement growth rate 1.3X-direction scale factor 1.0Number of degrees of freedom 30722Number of elements 76803.9 Subdomain settings.The subdomain settings describe the physics on amodel’s main domain, which is divided intosubdomains. In this model, there is only onesubdomain.The PDE in its general form is used with a timedependent analysis. The PDE in the general form hasthe following aspect in the program:e2∂ u+ d∂t∂u⋅ + ∇ ⋅ Γ∂ta⋅a=2F(11)The next step is to introduce the equations seen in[2.2] adapting them to this general form, that is, the1D-SWE must be introduced in the proper formatthat Comsol Multiphysics requires.Figure 9. The flux vector edit fieldWhen trying to solve a problem that does notconverge, or having solved a problem where thesolution does not seem properly resolved, there isoften a need for re-solving the problem using a finermesh. In these situations it is sometimes easier torefine the existing mesh than to regenerate the meshfrom scratch using finer mesh settings.Figure 10. The source term edit field9
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