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. A338:43-68 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 Ω

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 • xi+1/2 and xi-1/2 tn+1 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 • Fi+1/2 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 xi+1/2)

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. PossibleWave 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* > hL left wave is a shock • h* ≤ hL left wave is a rarefaction • h* > 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. Dry–Bed 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 > 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 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. PossibleWave structures • Across left and right wave h, u change v is constant • Across shear wave v changes, h, u constant

42. t t Right bounding characteristic Left bounding characteristic hL, uL h*, u* x 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 • λ1,2 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