Page 1 of 18 Fluvial Dynamics M. I. Bursik ublearns.buffalo.edu October 26, 2008
1. Fluvial Dynamics We want to understand a little of the basic physics of water flow and particle transport, as so much of the world around us is critically dependent on the flow of water and transport of sediment in streams Page 2 of 18
2. Mass Wasting and Particle Physics Landslide phenomena can be treated with particle physics Sliding block problem Ballistics Method of Infinite Slope - each control volume of material treated independently Page 3 of 18
3. Channel Flow Open channel flow the flow of water in streams or engineered structures in which the water has a free surface This is the most ubiquitous particle transport and landscape modification system on the Earth s surface Page 4 of 18
4. Speed of water in reaches Uniform flow easiest - not changing forces are zero Page 5 of 18 What does it seem like the speed of water in a channel will depend on, intuitively? Gravity Gradient certainly Some measure of resistance This it turns out is related to the area of the bed, and a measure of the properties, mostly roughness, of the bed materials
5. Hydraulic radius and roughness parameter Measure of contact area is the hydraulic radius, R Page 6 of 18 R = A/P (1) Where A is cross-sectional area and P is the wetted perimeter Many measures of roughness Most common are Manning roughness parameter, Chezy constant, Strickler constant and friction factor Manning roughness parameter most fully dependent on roughness alone (bed properties not channel geometry)
6. Friction and drag Imagine bottom of stream with loosened clasts Page 7 of 18 Volume of water moved/time, V = U A c
7. Stream power Mass moved/time, m = ρ w UA b, where A b = A c So since energy (work), E = mu 2 /2 Work/time = power, P = ρ w UA b U 2 /2 or P = ρ w A b U 3 /2 Remember this is the power that it takes to move the fluid out of the way of the obstacle Page 8 of 18
8. Shape Page 9 of 18 Object shape is important Determines how water moves out of the way In a way, it affects the definition of A b We need to differentiate between blunt objects and hydrodynamically smooth objects Blunt - water meets object head-on, water forcibly ripped away, turbulence Smooth - water is diverted around the object, the volume diverted is more proportional to total exposed contact area With smooth, there is a strong gradient of deceleration next to the object Gradient is to no-slip condition at object surface
9. Drag Coefficient Drag coefficient, C D, characterizes the overall effect of the shape of the bed particles in counteracting flow motion C D varies with the change from form drag (blunt) to skin friction (smooth) C D is dimensionless What does its value depend on? Page 10 of 18
10. Drag Coefficient Page 11 of 18 C D = F (U, A b, µ, ρ w ) So: P f = C D A b ρ w U 3 /2 (2)
11. Balance of gravity and friction forces Balance to get velocity for uniform flow That is: ma = F ma = F g F f F g F f = 0 F g = F f Page 12 of 18
12. Derivation of Manning s Equation Equation of motion with friction factor By convention, when we deal with a stream, we use the friction factor, f f, rather than the drag coefficient F g = F f Page 13 of 18 (since P f /U = F f ) What is F g? F g = P f /U F g = f f A b ρ w U 2 /2
F g = ρ w gv sin ϑ Page 14 of 18 Chezy constant Manning roughness parameter ρ w gv sin ϑ = f f A b ρ w U 2 /2 The result is the Manning equation for uniform flow: Or the Chezy equation: U = 1 n M R 2/3 S 1/2 0 (3) U = C(RS 0 ) 1/2 (4) Question: N M = 0.03 s/m 1/3, R = 100 m, S f = 0.001. So mean stream velocity, Ū =...?
13. Velocity Distribution Page 15 of 18 U from the Manning equation is the mean speed. How is this related to the speed at the surface, which is easy to measure? within the fluid: ρ w gz sin ϑ = µ eff du(z) dz u(z) = ρ w g sin ϑz 2 /(2µ eff ) parabolic velocity profile with depth for streamflow not quite correct because of turbulence one result though is that (the U from Manning s equation) So through a vertical in uniform flow : U max (surface) = 1.5U (5) U max (surface) = 1 n M 1.5h 2/3 S 1/2 0 (6)
14. Gradually- and Rapidly-varied Flow Page 16 of 18 Uniform flow doesn t occur everywhere Gradually varied flow occurs where stream depth and speed change smoothly Rapidly varied flow occurs where stream depth and speed change suddenly In gradually and rapidly varied flow, it is a rather complicated calculation to find speed need to look at equations of motion a bit, and perhaps use a numerical program One can map out the reaches of any stream, such as the Niagara River defined by stretches of Uniform Flow and separated by short stretches of Gradually Varied or Rapidly Varied Flow To analyze motion in these changing stretches, Manning s Equation and its assumptions do not hold
15. Equations of Motion Continuity equation conservation of mass Page 17 of 18 ρ w UA = Q (7) which is discharge, and is constant for no inflow or outflow (either by tributaries, through groundwater, or by evapotranspiration) Assumes no inflow or outflow (no tributaries or distributaries; losses or gains with groundwater; evaporation) Momentum equation expresses Newton s second law or Bernoulli s equation ρ w gh + ρ w gy + (1/2)ρ w U 2 = E (8) which is a constant called the Bernoulli integral or total mechanical energy Bernoulli s equation works within a streamtube - tube made up of mean stream lines, which are parallel to channel walls and the free water surface in the case of streams
16. Hydraulic Jump In a region of rapidly varied flow, the height above a datum (y in the Bernoulli equation) does not change significantly, thus E = h + U 2 /(2g) is a constant. Either h can be big, or U can be big, to get the same E What happens on a graph of h (y-axis) versus E (x-axis)? There are two solution branches, defined by the Froude number, F r = ratio of kinetic energy to potential energy: Page 18 of 18 F r 2 = U 2 2gh (9)