Fluvial Hydraulics 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 = Sf = Sw

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 = d50 Suggestion by researchers to modify the hydraulic radius by a factor, f: Rectangular (B = 2h) section f = 0.95 Large Trapezoidal section f = 0.80 Triangular (Equilateral) section 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’=0.2h and z’=0.8h):

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 medium-quality concrete to convey a discharge of Q = 80 m3/s. The channel slope is Sf = 0.1%. Flow is uniform at a temperature of 10oC. (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 non-cohesive 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<d35[mm]<4.3 and 1.49 x 10-4<Sf< 1.72 x 10-3)

Friction Coefficient – Mobile Beds Alam-Kennedy: Artificial: 0.04<d50<0.54 (mm) Natural: 0.08<d50<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 < U < 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 = 0.4U 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 [m3/s] < 1000. At one particular cross-section, the width of the bed is 90 m and the banks have a slope of 1:1. Use ss = 2.65, d50 = 0.32 mm, d35 = 0.29 mm, and d90 = 0.48 mm. The water temperature is 14oC. 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?