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Solution of the St Venant Equations / Shallow-Water equations of open channel flow

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Title: Solution of the St Venant Equations / Shallow-Water equations of open channel flow


1
Solution of the St Venant Equations /
Shallow-Water equations of open channel flow
  • Dr Andrew Sleigh
  • School of Civil Engineering
  • University of Leeds, UK
  • www.efm.leeds.ac.uk/CIVE/UChile

2
Shock Capturing Methods
  • Ability to examine extreme flows
  • Changes between sub / super critical
  • Other techniques have trouble with trans-critical
  • Steep wave front
  • Front speed
  • Complex Wave interactions
  • Alternative shock fitting
  • Good, but not as flexible

3
More recent
  • Developed from work on Euler equations in the
    aero-space where shock capturing is very
    important (and funding available)
  • 1990s onwards
  • Euler equations / Numerical schemes
  • Roe, Osher, van Leer, LeVeque, Harten, Toro
  • Shallow water equations
  • Toro, Garcia-Navarro, Alcrudo, Zhao

4
Books
  • E.F. Toro. Riemann Solvers and Numerical Methods
    for Fluid Dynamics. Springer Verlag (2nd Ed.)
    1999.
  • E.F. Toro. Shock-Capturing Methods for
    Free-Surface Flows. Wiley (2001)
  • E.F. Toro. Riemann Problems and the WAF Method
    for Solving Two-Dimensional Shallow Water
    Equations. Philosophical Transactions of the
    Royal Society of London. A33843-68 1992.

5
Dam break problem
  • The dam break problem can be solved
  • It is in fact integral to the solution technique

6
Conservative Equations
  • As earlier, but use U for vector


7
I1 and I2
  • Trapezoidal channel
  • Base width B, Side slope SL Y/Z
  • Rectangular, SL 0
  • Source term


8
Rectangular Prismatic
  • Easily extendible to 2-d
  • Integrate over control volume V

9
2-dimensions
  • In 2-d have extra term
  • friction

10
Normal Form
  • Consider the control volume V, border O

11
Rotation matrix
  • H(U) can be expressed

12
Finite volume formulation
  • Consider the homogeneous form
  • i.e. without source terms
  • And the rectangular control volume in x-t space

13
Finite Volume Formulation
  • The volume is bounded by
  • xi1/2 and xi-1/2 tn1 and tn
  • The integral becomes

14
Finite Volume Formulation
  • We define the integral averages
  • And the finite volume formulation becomes

15
Finite Volume Formulation
  • Up to now there has been no approximation
  • The solution now depends on how we interpret the
    integral averages
  • In particular the inter-cell fluxes
  • Fi1/2 and F1-1/2

16
Finite Volume in 2-D
  • The 2-d integral equation is
  • H(u) is a function normal to the volume

17
Finite Volume in 2-D
  • Using the integral average
  • Where V is the volume (area) of the volume
  • then

18
Finite Volume in 2-D
  • If the nodes and sides are labelled as
  • Where Fns1 is normal flux for side 1 etc.

19
FV 2-D Rectangular Grid
  • For this grid
  • Solution reduces to

20
Flux Calculation
  • We need now to define the flux
  • Many flux definitions could be used to that
    satisfy the FV formulation
  • We will use Godunov flux (or Upwind flux)
  • Uses information from the wave structure of
    equations.

21
Godunov method
  • Assume piecewise linear data states
  • Means that the flux calc is solution of local
    Riemann problem

22
Riemann Problem
  • The Riemann problem is a initial value problem
    defined by
  • Solve this to get the flux (at xi1/2)

23
FV solution
  • We have now defined the integral averages of the
    FV formulation
  • The solution is fully defined
  • First order in space and time

24
Dam Break Problem
  • The Riemann problem we have defined is a
    generalisation of the Dam Break Problem

25
Dam Break Solution
  • Evolution of solution
  • Wave structure

26
Exact Solution
  • Toro (1992) demonstrated an exact solution
  • Considering all possible wave structures a single
    non-linear algebraic equation gives solution.

27
Exact Solution
  • Consider the local Riemann problem
  • Wave structure

28
Possible Wave structures
  • Across left and right wave h, u change v is
    constant
  • Across shear wave v changes, h, u constant

29
Determine which wave
  • Which wave is present is determined by the change
    in data states thus
  • h gt hL left wave is a shock
  • h hL left wave is a rarefaction
  • h gt hR right wave is a shock
  • h hR right wave is a rarefaction

30
Solution Procedure
  • Construct this equation
  • And solve iteratively for h (h).
  • The functions may change each iteration

31
f(h)
  • The function f(h) is defined
  • And u

32
Iterative solution
  • The function is well behaved and solution by
    Newton-Raphson is fast
  • (2 or 3 iterations)
  • One problem if negative depth calculated!
  • This is a dry-bed problem.
  • Check with depth positivity condition

33
DryBed solution
  • Dry bed on one side of the Riemann problem
  • Dry bed evolves
  • Wave structure is different.

34
Dry-Bed Solution

  • Solutions are explicit
  • Need to identify which applies (simple to do)
  • Dry bed to right
  • Dry bed to left
  • Dry bed evolves h  0 and u  0
  • Fails depth positivity test

35
Shear wave
  • The solution for the shear wave is straight
    forward.
  • If vL gt 0 v vL
  • Else v vR
  • Can now calculate inter-cell flux from h, u
    and v
  • For any initial conditions

36
Complete Solution
  • The h, u and v are sufficient for the Flux
  • But can use solution further to develop exact
    solution at any time.
  • i.e. Can provide a set of benchmark solution
  • Useful for testing numerical solutions.
  • Choose some difficult problems and test your
    numerical code again exact solution

37
Complete Solution
38
Difficult Test Problems
  • Toro suggested 5 tests

Test No. hL (m) uL (m/s) hR (m) uR (m/s)
1 1.0 2.5 0.1 0.0
2 1.0 -5.0 1.0 5.0
3 1.0 0.0 0.0 0.0
4 0.0 0.0 1.0 0.0
5 0.1 -3.0 0.1 3.0
Test 1 Left critical rarefaction and right
shock Test 2 Two rarefactions and nearly dry
bed Test 3 Right dry bed problem Test 4 Left
dry bed problem Test 5 Evolution of a dry bed
39
Exact Solution
  • Consider the local Riemann problem
  • Wave structure

40
Returning to the Exact Solution
  • We will see some other Riemann solvers that use
    the wave speeds necessary for the exact solution.
  • Return to this to see where there come from

41
Possible Wave structures
  • Across left and right wave h, u change v is
    constant
  • Across shear wave v changes, h, u constant

42
Conditions across each wave
  • Left Rarefaction wave
  • Smooth change as move in x-direction
  • Bounded by two (backward) characteristics
  • Discontinuity at edges

43
Crossing the rarefaction
  • We cross on a forward characteristic
  • States are linked by
  • or

44
Solution inside the left rarefaction
  • The backward characteristic equation is
  • For any line in the direction of the rarefaction
  • Crossing this the following applies
  • Solving gives
  • On the t axis dx/dt 0

45
Right rarefaction
  • Bounded by forward characteristics
  • Cross it on a backward characteristic
  • In rarefaction
  • If Rarefaction crosses axis

46
Shock waves
  • Two constant data states are separated by a
    discontinuity or jump
  • Shock moving at speed Si
  • Using Conservative flux for left shock

47
Conditions across shock
  • Rankine-Hugoniot condition
  • Entropy condition
  • ?1,2 are equivalent to characteristics.
  • They tend towards being parallel at shock

48
Shock analysis
  • Change frame of reference, add Si
  • Rankine-Hugoniot gives

49
Shock analysis
  • Mass flux conserved
  • From eqn 1
  • Using
  • also

50
Left Shock Equation
  • Equating gives
  • Also

51
Right Shock Equation
  • Similar analysis gives
  • Also

52
Complete equation
  • Equating the left and right equations for u
  • Which is the iterative of the function of Toro

53
Approximate Riemann Solvers
  • No need to use exact solution
  • Expensive
  • Iterative
  • When other equations, exact may not exist
  • Many solvers
  • Some more popular than others

54
Toro Two Rarefaction Solver
  • Assume two rarefactions
  • Take the left and right equations
  • Solving gives
  • For critical rarefaction use solution earlier

55
Toro Primitive Variable Solver
  • Writing the equations in primitive variables
  • Non conservative
  • Approximate A(W) by a constant matrix
  • Gives solution

56
Toro Two Shock Solver
  • Assuming the two waves are shocks
  • Use two rarefaction solver to give h0

57
Roes Solver
  • Originally developed for Euler equations
  • Approximate governing equations with
  • Where is obtained by Roe averaging

58
Roes Solver
  • Properties of matrix
  • Eigen values
  • Right eigen vectors
  • Wave strengths
  • Flux given by

59
HLL Solver
  • Harten, Lax, van Leer
  • Assume wave speed
  • Construct volume
  • Integrate round
  • Alternative gives flux

60
HLL Solver
  • What wave speeds to use?
  • Free to choose
  • One option
  • For dry bed (right)
  • Simple, but robust

61
Higher Order in Space
  • Construct Riemann problem using cells further
    away
  • Danger of oscillations
  • Piecewise reconstruction
  • Limiters

62
Limiters
  • Obtain a gradient for variable in cell i, ?i
  • Gradient obtained from Limiter functions
  • Provide gradients at cell face
  • Limiter ?i G(a,b)

63
Limiters
  • A general limiter
  • ß1 give MINMOD
  • ß2 give SUPERBEE
  • van Leer
  • Other SUPERBEE expression
  • s sign

64
Higher order in time
  • Needs to advance half time step
  • MUSCL-Hancock
  • Primitive variable
  • Limit variable
  • Evolve the cell face values 1/2Dt
  • Update as normal solving the Riemann problem
    using evolved WL, WR

65
Alternative time advance
  • Alternatively, evolve the cell values using
  • Solve as normal
  • Procedure is 2nd order in space and time

66
Boundary Conditions
  • Set flux on boundary
  • Directly
  • Ghost cell
  • Wall u, v 0. Ghost cell un1-un
  • Transmissive Ghost cell hn1 hn
  • un1 un

67
Wet / Dry Fonts
  • Wet / Dry fronts are difficult
  • Source of error
  • Source of instability
  • Common
  • near tidal boundaries
  • Flooding - inundation

68
Dry front speed
  • We examined earlier dry bed problem
  • Front is fast
  • Faster than characteristic.
  • Can cause problem with time-step / Courant

69
Solutions
  • The most popular way is to artificially wet bed
  • Give a small depth, zero velocity
  • Loose a bit of mass and/or momentum
  • Can drastically affect front speed
  • E.g. a 1.0 dambeak with 1cm gives 38 error
  • 1mm gives 25 error try it!

70
Conservation Errors
  • The conserved variable are h and hu
  • Often require u
  • Need to be very careful about divide by zero
  • Artificial dry-bed depths could cause this

71
Source Terms
  • Lumped in to one term and integrated
  • Attempts at upwinding source
  • Current time-step
  • Could use the half step value
  • E.g.

72
Main Problem is Slope Term
  • Flat still water over uneven bed starts to move.
  • Problem with discretisation of

73
Discretisation
  • Discretised momentum eqn
  • For flat, still water
  • Require

74
A solution
  • Assume a datum depth, measure down
  • Momentum eqn
  • Flat surface

and
75
Some example solutions
Weighted mesh gives more detail for same number
of cells
76
Dam Break - CADAM Channel with 90 bend
77
Secondary Shocks
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