2D Modelling Finite Volume methods

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2D ModellingFinite Volume methods
Flood Risk Concepts and Application In River
Basin Management
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Obtaintion of equations from Navier Stokes
Equations

• Navier Stokes Equations (3D)
• Turbulent Flux Reynolds equations

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Vertical integration of Reynolds equations

• Leibnitz rule consideration of acting forces
  leads to

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Saint Venant equations

• Efective stresses

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Simplification of Saint Venant equations(conserva
tive form)

• Discard Coriolis forces, efective stresses and
  wind effects

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In matrix form (conservative form)

• Manning
• Chezy

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Saint Venant Equations in non conservative form




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Non-lineal Hyperbolic system of equations theory
of Characteristics (privileged directions)

• First family of Characteristics Characteristic
  cone

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• 2nd family of characteristics Characteristic
  Plane

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2D numerical schemes

• Classic schemes (non conservative form of the
  equations)
• Characteristics
• Finite differences
• Implicit (ADI) (MIKE21, SOBEK)
• Explicit (Mac Cormack 2D)
• Finite elements (TELEMAC)
• High resolution schemes (conservative form of the
  equations) Shock capturing methods
• Finite volumes

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Domain discretisation in 2D

• Regular mesh (finite differences)
• Non-structured irregular mesh (finite elements
  and finite volumes)

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Finite differences ADI Schemes

• Alternating Direction Implicit (SOBEK, Mike-21)
• For coastal areas
• Regular rectangular meshes.
• Subcritical flow, smooth hydrographs
• Simplifications in flow transitions (artificial
  viscosity)

non-staggered grid
alternating directions
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Wave propagations. The Riemann Problem
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Strong Shock Wave
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Rarefaction wave
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The Riemann problem (hiperbolic 1D systems)
1D 2D
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Solution of Dam Break in 1D (Particular case of
Riemann Problem)
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Other 1D Riemann problems for SV equations
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Finite Volumes in 2D

• Integrating over a volume
• Gauss Theorem
• Volume averaging

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• In particular

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Explicit schemes

• Courant condition

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Upwind schemes

• Various possibilities of finite differences
• Appliction to with
• Centered scheme
• Upwind scheme

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2D Shallow water equations. Godunov Scheme (1st
order)

• If H0

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• At each wall 1D Riemann problem perpendiculat to
  the wall
• Determination of from solution of
  Riemman problems
• Approximate Riemann Solvers Approximate solution
  of the Riemann problem (HLL, Osher, Roe)
• Commonly used in 2D SV equations Roe Riemann
  solver

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Roe Approximate Riemann Solver
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• Entropy correction (avoid non physical solutions)
  (Hsrtrn and Hyman)

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• High resolution schemes

(m,n is the upwind wall of wall i,j)
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Practical aspects of 2D computations

• Build computation mesh
• Initial conditions
• Roughness coefficients
• Boundary conditions
• Inlet supercritical 3 conditions
• Inlet subcritical 2 conditions
• Outlet Subcritical 1 condition
• Outlet supercritical 0 conditions
• Compute

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Example 1. River Segre
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Mesh
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Water profile animation
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Water depth
Discharge distribution
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Velocity field
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Example 2 Irrigation Reservoir Dam Break
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Topography and mesh
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Rotura de Balsa

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Example 3. Llobregat river
DTM (Arc/info ASCII)
Geometry
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Geometry detail (1)
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Geometry detail (2)
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Geometry detail (3)
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Geometry detail (4)
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Land use
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Water Depth evolution
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Example 3 Fluvia and Llierca confluence
N0.032 (brown), n0.065 (blue) n0.090 (green)
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Flood in Fluvia and Llierca
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Flood only in Llierca
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Water depth
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Velocity
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Detail in Bridge
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