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2D Modelling Finite Volume methods

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2D Modelling. Finite Volume methods. Flood Risk Concepts and ... Rotura de Balsa. Example 3. Llobregat river. DTM (Arc/info ASCII) Geometry. Geometry detail (1) ... – PowerPoint PPT presentation

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Title: 2D Modelling Finite Volume methods


1
2D ModellingFinite Volume methods
Flood Risk Concepts and Application In River
Basin Management
2
Obtaintion of equations from Navier Stokes
Equations
  • Navier Stokes Equations (3D)
  • Turbulent Flux Reynolds equations




3
Vertical integration of Reynolds equations



  • Leibnitz rule consideration of acting forces
    leads to

4
Saint Venant equations
  • Efective stresses




5
Simplification of Saint Venant equations(conserva
tive form)
  • Discard Coriolis forces, efective stresses and
    wind effects





6
In matrix form (conservative form)
  • Manning
  • Chezy

7
Saint Venant Equations in non conservative form




8
Non-lineal Hyperbolic system of equations theory
of Characteristics (privileged directions)
  • First family of Characteristics Characteristic
    cone

9
  • 2nd family of characteristics Characteristic
    Plane

10
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

11
Domain discretisation in 2D
  • Regular mesh (finite differences)
  • Non-structured irregular mesh (finite elements
    and finite volumes)

12
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
13
Wave propagations. The Riemann Problem
14
Strong Shock Wave
15
Rarefaction wave
16
The Riemann problem (hiperbolic 1D systems)
1D 2D
17
Solution of Dam Break in 1D (Particular case of
Riemann Problem)
18
Other 1D Riemann problems for SV equations
19
Finite Volumes in 2D
  • Integrating over a volume
  • Gauss Theorem
  • Volume averaging

20
  • In particular

21
Explicit schemes
  • Courant condition

22
Upwind schemes
  • Various possibilities of finite differences
  • Appliction to with
  • Centered scheme
  • Upwind scheme

23
2D Shallow water equations. Godunov Scheme (1st
order)
  • If H0

24
  • 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

25
Roe Approximate Riemann Solver
26
  • Entropy correction (avoid non physical solutions)
    (Hsrtrn and Hyman)

27
  • High resolution schemes

(m,n is the upwind wall of wall i,j)
28
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

29
Example 1. River Segre
30
Mesh
31
Water profile animation
32
Water depth
Discharge distribution
33
Velocity field
34
Example 2 Irrigation Reservoir Dam Break
35
Topography and mesh
36
Rotura de Balsa

37
Example 3. Llobregat river
DTM (Arc/info ASCII)
Geometry
38
Geometry detail (1)
39
Geometry detail (2)
40
Geometry detail (3)
41
Geometry detail (4)
42
Land use
43
Water Depth evolution
44
Example 3 Fluvia and Llierca confluence
N0.032 (brown), n0.065 (blue) n0.090 (green)
45
Flood in Fluvia and Llierca
46
Flood only in Llierca
47
Water depth
48
Velocity
49
Detail in Bridge
50
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