Idrodinamica (a.a. 2011/2012)

1. Idrodinamica (a.a. 2011/2012) Moto uniforme negli alvei naturali Marco Toffolon con contributi da presentazioni di Guido Zolezzi Matilde Welber Gary Parker

2. Contatti Marco Toffolon email: marco.toffolon@ing.unitn.it Laboratorio Didattico di Modellistica Idrodinamica (2° piano, corridoio centrale) tel.: 0461 28 2480

3. Moto uniforme: introduzione

4. Introduzione Moto uniforme: cosa significa? perché è importante? uniforme: «uguale» ovunque quali condizioni devono essere soddisfatte? quali sono i limiti di questa definizione? utilizzo: scala delle portate (principalmente) nota la profondità  stima della portata (locale) o viceversa

5. Equazioni Equazioni 1D (De Saint Venant) Equazione di continuità (conservazione della massa) Equazione di bilancio della quantità di moto (2° principio della dinamica) (con sistema di riferimento inclinato)

6. Equazioni Termine d’attrito Moto uniforme Raggio idraulico Velocità d’attrito Bilancio di forze all’equilibrio Relazioni (generali) di moto uniforme

7. Chiusura Moto uniforme Problema della chiusura: definire un legame tra tensioni e velocità Coefficiente di Chézy (adimensionale) alveo rettangolare “largo”

8. Formule di resistenza Moto turbolento pienamente sviluppato (Re>105) Coefficiente di Chézy (adimensionale) “scabro” “liscio” Coefficiente di Chézy (dimensionale) Coefficiente di Gauckler-Strickler (dimensionale) Alveo rettangolare largo

9. Confronti con moto nei tubi Darcy-Weisbach Chézy Chiusura Stima coefficiente di resistenza diagramma di Moody Colebrook-White

10. Resistenza nei canali Colebrook-White (tubazioni) Canale a sezione circolare in regime liscio Canale a sezione circolare in regime scabro Canale di forma regolare in regime liscio fattore di forma per sezioni rettangolari larghe Canale di forma regolare in regime scabro Formula di Marchi (1961) per sezioni di forma regolare

11. Variabili Problema idraulico: stima della portata Problema semplificato (alveo rettangolare largo): 5 variabili Note 4 grandezze, la quinta può essere determinata In generale: • 5 «variabili»: • portata Q • pendenza if • livello idrico z • geometria della sezione (~larghezza) • scabrezza Ch(~diametro caratteristico dei sedimenti)

12. Problema di progetto • Determinare la profondità richiesta per il deflusso in moto uniforme della portata Q in un canale di larghezza b con pareti in cemento non perfettamente lisciate. • (Determinare il coefficiente m della scala di deflusso) • Dati: Q = 10 m3/s b = 4 m if = 0.001 scabrezza cemento  tabella coefficiente di Manning n = 1/ks [s m-1/3]

13. Portata negli alvei naturali

14. ALVEI NATURALI • Geometria irregolare • Variazione di scabrezza lungo il contorno • Come determinare la scala di deflusso e in generale le proprietà idrauliche f(Y) della sezione?

15. Adige, Trento, ponte S. Lorenzo, 24 gennaio 2002 ~ 100 m³/s B = 75 m Y = 2 m Corso di Idrodinamica – Anno 2009

16. Adige, Trento, ponte S. Lorenzo, 26 novembre 2002 ~ 1400 m³/s B = 100 m Y = 6 m Corso di Idrodinamica – Anno 2009

17. Adige, Trento, ponte S. Giorgio, 24 gennaio 2002 ~ 100 m³/s Corso di Idrodinamica – Anno 2009

18. Adige, Trento, ponte S. Giorgio, 26 novembre 2002 ~ 1400 m³/s Corso di Idrodinamica – Anno 2009

19. Adige a Trento, ponte S. Lorenzo Scala di deflusso scala idrometrica sezione rettangolare

20. Adige, Trento, Lungadige G. Leopardi Corso di Idrodinamica – Anno 2009

21. …ma quanto vale il coefficiente di scabrezza per l’Adige?? Adige, Trento Nord, zona depuratore Corso di Idrodinamica – Anno 2009

22. Resistenze negli alvei naturali

23. Resistenze negli alvei naturali VAL RIDANNA, ALTO ADIGE VAL PASSIRIA, ALTO ADIGE

24. + RESISTENZA DI FORMA Resistenze granulometria grossolana RESISTENZA DI GRANO granulometria fine RESISTENZA DI GRANO ESEMPI Sforzo al fondo maggiore

25. Granulometria ghiaia (gravel) sabbia (sand)

26. Granulometria Fiume Tevere monte valle

27. Granulometria Fiume Tagliamento superficiale (Wolman count) sotto lo strato di corazzamento (vagli) sabbia ghiaia

28. Riferimentibibliografici http://vtchl.uiuc.edu/people/parkerg/

29. RESISTANCE RELATIONS FOR HYDRAULICALLY ROUGH FLOW Keulegan (1938) formulation: where  = 0.4 denotes the dimensionless Karman constant and as = a roughness height characterizing the bumpiness of the bed [L]. notazione Manning-Strickler formulation: where r is a dimensionless constant between 8 and 9. Parker (1991) suggested a value of r of 8.1 for gravel-bed streams. Roughness height over a flat bed (no bedforms): where Ds90 denotes the surface sediment size such that 90 percent of the surface material is finer, and nk is a dimensionless number between 1.5 and 3. For example, Kamphuis (1974) evaluated nk as equal to 2.

30. COMPARISION OF KEULEGAN AND MANNING-STRICKLER RELATIONS r = 8.1 Note that Ch does not vary strongly with depth. It is often approximated as a constant in broad-brush calculations.

31. Ricostruzione di una relazione per il coefficiente di Gauckler-Strickler Formula empirica: Parker (1991) Kamphuis (1974) diametro passante 90% dei sedimenti Strickler (1923): Meyer-Peter & Müller (1948):

32. SKIN FRICTION AND FORM DRAG: THE CONCEPTS resistenza di forma «resistenza di grano» The drag force acting on a body can be decomposed into skin friction and form drag. The former is generated by the viscous shear stress acting tangentially to the body. The latter is generated by the normal stress (mostly pressure) acting on a body. The Newtonian constitutive relation for water is Here ij denotes the stress acting in the jth direction on a face normal to the ith direction, p denotes the pressure, ij denotes the Kronecker delta ( = 1 if i = j and 0 if i  j), ui = (u1, u2, u3) denotes the velocity vector and xi = (x1, x2, x3) denotes the position vector. The drag force Di on a body is given as where ji is evaluated at the surface of the body, ni denotes a local unit vector outward normal to the surface of the body, and dS denotes an infinitesimal element of surface area.

33. SKIN FRICTION AND FORM DRAG: THE CONCEPTS contd. The drag force Di can be decomposed into a component due to skin friction Dsi and a component due to form drag Dfi as follows: Drag due to skin friction consists of that part of the drag that pulls the surface of the body tangentially. Form drag consists of that part of the drag that pushes the body in normally. Only the former is thought to directly contribute to sediment transport. Now in the diagrams below let and denote the skin friction and form drag forces on the area element dS, denote a unit tangential vector to the surface in the x direction and denote a unit vector normal to the surface.

34. SKIN FRICTION AND FORM DRAG: THE CONCEPTS contd. Let D denote the drag force in the flow direction and nx denote the component of the unit outward normal vector to the surface in the flow direction. At sufficiently high Reynolds number, the drag on a streamlined body is mostly skin friction. The drag on a blunt body behind which flow separation occurs is mostly form drag. (The pressure in the separation bubble equilibrates with the low pressure at the point of separation.)

35. EINSTEIN DECOMPOSITION Einstein (1950); Einstein and Barbarossa (1952) When bedforms are not present, all of the drag on the bed is skin friction. This tangential drag force acts to pull the sediment along. When bedforms such as dunes are present, part of the drag is form drag associated with (most prominently) flow separation behind the dunes. Since this form drag is composed of stress that acts normal to the bed surface, it does not contribute directly to the motion of bed grains. As a result it is usually subtracted out in performing bedload calculations.

36. RELATIONS FOR HYDRAULIC RESISTANCE IN RIVERS Sediment transport often creates bedforms such as dunes. These bedforms are accompanied by form drag, and so reduce the ability of the flow to transport sediment. Dunes in the Mississippi River, New Orleans, USA Image from LUMCON web page: http://weather.lumcon.edu/weatherdata/audubon/map.html Dunes on an exposed point bar in the meandering Fly River, Papua New Guinea

37. EINSTEIN DECOMPOSITION contd. Consider an equilibrium (normal) flow over a bed with mean streamwise slope if that is covered with bedforms. The flow has average depth Y and velocity U averaged over depth and the bedforms. The boundary shear stress averaged over the bedforms is given by the normal flow relation

38. EINSTEIN DECOMPOSITION contd. Now smooth out the bedforms, “glue” the sediment to the bed so it remains flat but offers the same microscopic roughness as the case with bedforms, and run a flow over it with the same mean velocity U and bed slope S. In the absence of the bedforms, the resistance is skin friction only. Due to the absence of bedforms the skin friction coefficient Cfs and the flow depth Hs should be less than the corresponding values with bedforms. Skin friction + form drag Skin friction only The difference between the two characterizes form drag.

39. EINSTEIN DECOMPOSITION contd. 0f = 0 - 0s = mean bed shear stress due to form drag of bedforms Cff = Cf – Cfs = friction coefficient associated with form drag Yf = Y – Ys = mean depth associated with form drag Skin friction + form drag Skin friction only The difference between the two characterizes form drag.

40. SKIN FRICTION Skin friction can be computed using the techniques developed in Chapter 5; where  = 0.4 and r = 8.1, or Skin friction + form drag Skin friction only The difference between the two characterizes form drag.

41. FORM DRAG OF DUNES: EINSTEIN AND BARBAROSSA (1952) One of the first relations developed to predict the form drag in rivers in which dunes predominate is that of Einstein and Barbarossa (1952). They obtained an empirical form for Cff as a function of qs, where (numero di Shields) denotes the Shields number due to skin friction and D35 is the grain size such that 35 percent of a bed surface sample is finer. Note that

42. FORM DRAG OF DUNES: ENGELUND AND HANSEN (1967) The total shear velocity u*, shear velocity due to skin friction u*s and shear velocity due to bedforms u*f, and the associated Shields numbers are defined as Engelund and Hansen (1967) determined the following empirical relation for lower-regime form drag due to dune resistance; or thus Note that bedforms are absent (skin friction only) when qs = q; bedforms are present when qs < q. The relation is designed to be used with the following skin friction predictor: Engelund and Hansen (1967) also present a form drag relation for upper-regime bedforms (antidunes).

43. FORM DRAG OF DUNES: ENGELUND AND HANSEN (1967) contd. No form drag Engelund-Hansen resistenza di forma resistenza di grano

44. FORM DRAG OF DUNES: WRIGHT AND PARKER (2004) The form drag predictor of Engelund and Hansen (1967) tends to work well for sand-bed streams at laboratory scale. It also works well at small to medium field scale, i.e. in streams in which dunes give way to upper-regime plane bed before bankfull flow is achieved. It works rather poorly for large, low-slope sand-bed rivers, in which dunes are usually never washed out even at or above bankfull flow. Wright and Parker (2004) have modified it to accurately cover the entire range. This relation is designed to be used with the skin friction predictor where strat is a correction for flow stratification which can be set equal to unity in the absence of other information (see original reference).

45. COMPARISON OF FORM DRAG PREDICTORS AGAINST FIELD DATA The Niobrara and Middle Loup are small sand-bed streams. The Rio Grande is a middle-sized sand-bed stream. The Red, Atchafalaya and Mississippi Rivers are large sand-bed streams. Engelund and Hansen (1967) Wright and Parker (2004)

46. Morfologia degli alvei naturali

48. Alatna river, Alaska • Configurazione planimetrica: • mono-pluricursale • struttura del campo di moto Nepal

49. FORME PLANIMETRICHE: meandri e braiding Tagliamento River, Italy Fly river, Papua

50. Meandri “siberiani” Waimakariri, NZ