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Modelos Hidrodinâmicos

Modelos Hidrodinâmicos. Aula 4 Equations for 3D and 2D Hydrodynamic Models. Parameters and Boundary and Initial Conditions. Mass conservation. If P is the volumic mass , that has no Sources or Sinkes and has no diffusion because the net movement of molecules is the velocity ….

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Modelos Hidrodinâmicos

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  1. ModelosHidrodinâmicos Aula 4 Equations for 3D and 2D Hydrodynamic Models. Parameters and Boundary and Initial Conditions

  2. Mass conservation If P isthevolumicmass, thathas no SourcesorSinkesandhas no diffusionbecausethe net movementofmoleculesisthevelocity…. Ifincompressible:

  3. Momentum Conservation Se P for a quantidade de movimento por unidade de volume: Sourcesandsinks are Pressure forces (gravitationalis zero becausewe are interestedonlyon horizontal momentum)

  4. ShallowWaterEquations • HydrostaticPressure (vertical acelerationnegligeable). • If “z” isthe vertical axis, pointingupwards:

  5. Equações das águas pouco profundas

  6. Using the Leibnitz rule these equations can be integrated on vertical to obtain the equations of a 2D model.

  7. TheFinite Volume

  8. The 2D case TheAccumulation rate = flows in – flows out

  9. 1D Case TheAccumulation rate = flows in – flows out

  10. Momentum: 1D Case Ls Horizontal diffusion is negligible compared to vertical diffusion A Lb: wet perimeter

  11. The 1D Spatial Grid Qi-1 zi-1 Qi zi Qi+1

  12. Discretization A staggeredgridisconvenient. Temporal discretization can beexplicit, implicitou Crank-Nicholson

  13. Bottom shear stress α

  14. 2D Case

  15. Stability • Explicit (1D): • Implicit: Incondicionally stable • Explicit 2D:

  16. Boundary Conditions z0 Q1 z1 Q2 z2 • One can impose Free Surface levels and compute discharges or vice versa. • On sea side level is easier to know (tide) and on the land side river discharge use to be easier.

  17. Other boundary conditions • Bathymetry! • Surface shear stress, • Diffusive fluxes, • Advective fluxes.

  18. Initial conditions • Discharges/velocities, • Levels. • The good thing is that dissipative systems have low memory. Approximate initial conditions can be used. Usually zero velocity and horizontal free surface.

  19. Parameters • Friction coefficient, • Diffusion coefficient. • Surface friction coefficient if flux in not known (e.g. from a meteorological model).

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