River Mechanics CH3 Uniform Flow Redefining Uniform Flow
- Slides: 54
River Mechanics CH-3 Uniform Flow
Redefining Uniform Flow • Uniform flow if flow depth (h, Dh) as well as U, Q, roughness, and Sf remain invariable in different cross sections – Streamlines are rectilinear and parallel – Vertical pressure distribution is hydrostatic – Flow depth under uniform flow called normal flow depth – Se = S f = S w
Redefining Uniform Flow • Uniform flow is rare in natural and artificial channels – Only possible in very long prismatic channels far distance from an upstream or downstream boundary conditions
Continuity Equation • From last time… • Let us also assume the flow is steady…
Equation of Motion • Start with prismatic channel… – Friction force acting on the wetted perimeter: – Longitudinal component of gravity force: – What do we know about these forces in uniform flow?
Equation of Motion • We can then obtain the expression… • In hydrodynamics, we usually define:
Equation of Motion • Sometimes you will see the use of a friction coefficient…
Equation of Motion • You can also write the Darcy-Weisbach equation as…
Different Friction (Resistance) Coefficients • Darcy-Weisbach (f) – data generally based on circular cross-sections and standard roughness values • Chezy Coefficient (C) – useful as long as the flow is turbulent • Manning’s (n) • Coefficient for Mobile Bed – estimation for an immobile bed is difficult and even more so for mobile bed
Friction Coefficient, f • Usually for pipes we use the Moody diagram or the relation of Colebrook-White for turbulent flow • For circular cross-sections, you can use the experiments performed on pipes, with the following modification:
Friction Coefficient, f • Colebrook-White (turbulent flow) – written for channels as… – See Table 3. 1 for equivalent roughness (ks) for artificial channels – For granular beds, usually use ks = d 50 – Suggestion by researchers to modify the hydraulic radius by a factor, f: • Rectangular (B = 2 h) section • Large Trapezoidal section • Triangular (Equilateral) section f = 0. 95 f = 0. 80 f = 1. 25
Friction Coefficient, f • For most channels, ks is large and flow is turbulent, so Re is large: – Rough, turbulent flow: What does this imply? – Justification for use of Chezy equation, where C is only a function of the relative roughness (ks /Rh)
Friction Coefficient, f • For turbulent, rough flow…
Friction Coefficient, f • If rough channels of large widths (Rh = h), f can be obtained from measurements of point velocities: – Graf derives this expression assuming logarithmic distribution (see data on next slide):
Friction Coefficient, f • Obtained from experimental measurements at two depths (z’=0. 2 h and z’=0. 8 h):
Chezy Coefficient, C • Only valid for turbulent, rough flow • Estimated using empirical methods based on the hydraulic radius (m, s): – Bazin formula – established with data from small artificial channels:
Chezy Coefficient, C – Kutter formula – established with data from artificial channels and larger rivers: – Forchheimer:
Chezy Coefficient, C – Manning’s Equation:
Manning’s Equation • Only valid for turbulent, rough flow – Actually Manning’s assumes a coefficient that stays constant for a given roughness – Chezy coefficient changes depending on the relative roughness (Rh) – Typically, n = 0. 012 -0. 15 for natural and artificial channels
Discharge Calculations • Based on Manning’s equation… – Sometimes you will see the use of a term called the conveyance, K(h) – measure of the capacity for the channel to transport water:
Normal Depth • Solve Manning’s equation for h = hn… – Note that the normal depth can only exist on slopes that are decreasing (Sf>0)
Composite Sections • Can solve for case where different parts of the cross-section have different roughness or bed slope: – Apply formula of discharge for each subsection
Exercise 3. A - Graf A trapezoidal channel with bottom width of b = 5 m and side slopes of m = 3 is to built of mediumquality concrete to convey a discharge of Q = 80 m 3/s. The channel slope is Sf = 0. 1%. Flow is uniform at a temperature of 10 o. C. (a) Calculate the flow depth using both the Manning’s coefficient and the friction factor. (b) Verify whether the flow is laminar/turbulent and subcritical/supercritical.
Bed Forms – Mobile Bed • Mobile bed – channel composed on noncohesive solid particles which are displaceable due to the action of flow – Bed deformations depending on flow: • Fr < 1 – Subcritical and two potential regimes: – No Transport and Flat Bedform: Velocity does not exceed the critical velocity for that sediment – Transport and Mini-dune or Dune: Growing lengths l
Bed Forms – Mobile Bed – Bed deformations depending on flow: • Fr = 1 – Critical: – Transport and Flat: Dunes which are already long are washed out and the bed appears to be flat (state of transition) • Fr>1 – Supercritical: – Transport and Anti-dunes: Dunes that travel in the upstream direction, water surface becomes wavy (impacted by dune)
Bed Forms – Mobile Bed • Geometry of dunes idealized as triangular by Graf
Bed Forms – Mobile Beds • Why are we concerned with bed forms? – They increase the resistance to flow… – Researchers have used superposition to analyze for the effects of roughness due to particles (t’) and roughness due to bed form (t’’):
Friction Coefficient – Mobile Beds • Two types of methods: – Direct Calculation: Determine the overall or entire f – Separate Calculation: Determine f’ using prior formulas and then f’’ using other formulas
Friction Coefficient – Mobile Beds • Direct Calculation (see textbooks for all formulas): – Sugio (1972): • • KT = 54 (mini-dunes) KT = 80 (dunes) KT = 110 (upper regime) KT = 43 (rivers with meanders) – Grishanin (1990):
Friction Coefficient – Mobile Beds • Separate Calculation (see textbooks for all formulas): – Einstein-Barbarossa: American Rivers (0. 19<d 35[mm]<4. 3 and 1. 49 x 10 -4<Sf< 1. 72 x 10 -3)
Friction Coefficient – Mobile Beds – Alam-Kennedy: Artificial: 0. 04<d 50<0. 54 (mm) Natural: 0. 08<d 50<0. 45 (mm)
Discharge – Mobile Bed • Two velocities of concern with non-cohesive, mobile beds: – Velocity of Erosion (Critical Velocity) – permissible maximum velocity (UE or UCr) – Velocity of Sedimentation – permissible minimum velocity (UD) UD < UCr
Discharge – Mobile Bed • UD – minimum velocity necessary to transport the flow containing solid particles in suspension – Recommended Range: 0. 25 < UD [m/s] < 0. 9
Discharge – Mobile Bed • UCr – expressed in terms of the velocity or the critical shear stress, to, Cr – Note that the Hjulstrom diagram uses velocity next to the bed by assuming ub = 0. 4 U – Neill’s Relation:
Discharge – Mobile Bed • UCr – also common to use dimensionless shear stress: – Shields developed a relation between the dimensionless shear stress and the friction/particle Reynolds number:
Shields-Yalin Diagram
Example 3. B A river has a variable discharge in the range of 10 <Q [m 3/s] < 1000. At one particular crosssection, the width of the bed is 90 m and the banks have a slope of 1: 1. Use ss = 2. 65, d 50 = 0. 32 mm, d 35 = 0. 29 mm, and d 90 = 0. 48 mm. The water temperature is 14 o. C. The bed slope is Sf = 0. 0005. (a) Determine the stage-discharge curve assuming turbulent, rough flow. (b) At what depth will erosion and deposition begin to occur?
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