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. Background information • Why should we model rivers? • It is difficult and expensive to get data • The flow changes from day to day • Most of the time they are no problem

3. They cause disruption

4. They are dangerous

5. They Cause Financial and Personal loss

6. They cause structural damage

7. Human interference does not help

8. They are not new

9. Preventative Measures • build higher flood banks • divert the water with a relief channel • store the water • a combination of these

10. Design Considerations • Appearance • Effects on both upstream and downstream • The cost • The flood return period • Data availability

11. Consider that … • Floods cannot be prevented • It is neither economic nor practical to design for exceptionalfloods

12. Flood routingis the process of calculating backwater curves in unsteady flow. The Elements of Flood Hydraulics

13. Why do we need to route floods? To know: • Extent of flooding • Effects hydraulic structures • e.g. bridge piers, culverts, weirs • Size of flood relief channels • If flood relief measures will work • Give flood warnings

14. For each returnperiod • Take the flood hydrograph • Route this flow through the system • Ask if your proposal will work • Repeat for every proposal and return period

15. Objectives of this course • Understand necessary computational components • See different form of equations of unsteady flow • Use appropriate solution techniques By the end will • have programmed a model capable of simulating passage of a flood wave on a simple river network • have programmed a model to simulate extreme open channel flows and tested this with a dam break • But Today – just steady flow – like HEC-RAS steady

16. Functions / Programs • We will develop programs • Matlab functions equations • (could be any program / language) • Graphical representation • 1-D and 2-D • Input data • Solution data • Steady / Time dependent • Put function together for complete model

17. 2-d : Layout of Network

18. Section / Solution

19. Profile / Solution

20. 3-d, gis?

21. Flood routing achieved using the St. Venant Equations

22. St Venant Assumptions of 1-D Flow • Flow is one-dimensional i.e. the velocity is uniform over the cross section and the water level across the section is horizontal. • The streamline curvature is small and vertical accelerations are negligible, hence pressure is hydrostatic. • The effects of boundary friction and turbulence can be accounted for through simple resistance laws analogous to those for steady flow. • The average channel bed slope is small so that the cosine of the angle it makes with the horizontal is approximately 1.

23. Dam Break: real and dangerous

24. Dam break: difficult to solve • Idealised case • Sharp gradients

25. Dam Break: Animation • By the end of the course will be able to do something like this.

26. Basics: Consider Steady Flow • Today’s class will cover: • Components of a computational model • How to represent a network • Fundamental (steady) equations • Section properties • Friction formulas • Conveyance • Steady solutions: • uniform flow, • backwater curve.

27. How to represent channel network • Sections • Reach – group of sections • Boundary conditions • Internal: join reaches • “External”: define inflow and outflow • Together define river system

28. Diagrammatic picture

29. Sections • Look downstream. Left bank, Right bank

30. Sections • Variable roughness, shape, across section

31. Sections: File Format Local coordinates: x along channel, y across, z vertical SECTION AV2296_11909 8 0 22.61 0.5 5 19.89 0.04 15 14.44 0.04 45 14.44 0.04 47.5 17 0.5 60 17 0.5 65 18.87 0.5 75 22.61 0.5

32. Section Properties • Depth (d or y) – the vertical distance from the lowest point of the channel section to the free surface. • Stage (z) – the vertical distance from the free surface to a datum • Area (A) – the cross-sectional area of flow, normal to the direction of flow • Wetted perimeter (P) – the length of the wetted surface measured normal to the direction of flow. • Surface width(B) – width of the channel section at the free surface • Hydraulic radius (R) –area to wetted perimeter ratio (A/P) • Hydraulic mean depth (Dm) –area to surface width ratio (A/B) • Hydraulic diameter (DH) = equivalent pipe diameter • (4×R = 4A/P = D for a circular pipe flowing full) • Centre of gravity coordinates (centroid)

33. Function for Section Properties • Any section defined by coordinates (in file) • Common sections:

34. Steady Flow Equations • Conservation of energy

35. Energy / Bernoulli Equation hydrostatic pressure distribution Bed slope small: tan θ ≈ sin θ ≈ θ in radians

36. Momentum Equation • When flow is not hydrostatic, steep, discontinuous etc. • Hydraulic Jump b =momentum correction factor

37. Velocity Distribution

38. Velocity Distribution on Bend Hitoshi Sugiyama. See animation. http://www.cc.utsunomiya-u.ac.jp/~sugiyama/avs4/avs4eng.html

39. Calculation of a and b Function: Calculate the coefficients α and β for a given section and vel dist.

40. Reynolds Numebr • Using R as length scale • Using DH as length scale • For a wide river R = depth, DH = 4×depth. Function: Calculate Re (ReR or ReDH) for a given fluid, section, depth and velocity.

41. Froude Number, Fr • Critical Depth Fr = 1 • Fr < 1 sub-critical • upstream levels affected by downstream controls • Fr > 1 super-critical • upstream levels not affected by downstream controls Function:Calculate Fr, for a given section and discharge. Also dcritical.

42. Uniform Flow • Equilibrium – Friction balances Gravity Function: Calculate bed shear stress, to for given section, depth and bed slope.

43. Chezy C • assuming rough turbulent flow • shear force is proportional to velocity squared • thus Functions:Calculate V or Q for a given section and dn, C and bed slope. Also: normal depth, dn from Q, C, So, C from Q and So, dn, So from C, Q, dn.

44. Friction Formulae • Darcy-Weisbach for pipe • Full pipe • So = L / hf • and

45. Alternative form for f • Some texts give the value f is 4 times larger than quoted here • To clarify some text use l such that: • BE CAREFUL WITH FRICTION FORMULAE Functions: Calculate f or λ for a given section, depth, slope and discharge. Calculate f from C and vice versa

46. Colebrook-White equation for f • Originally developed for pipes • ksis effective sand grain size in mm • Implicit • Requires iterative solution • Use Altsul equation to start iteration

47. ks values • Some typical values of ksare Function: Calculate f or λ from ReR depth, section and ks.

48. Manning’s n • Most commonly used expression for friction • n relates to C: • In terms of discharge Function: Calculate Q from n, C from n, for given section.

49. Manning’s n values • Some typical values for n • Friction estimate great source of error

50. Computations in uniform flow • Typical and common calculations • Discharge from a depth = normal flow • Depth for a discharge = normal depth • Require iterative solution even for rectangular channel Function: Calculate dn or flow for given section and n, C or f , So,Q or dn.