Supercritical and mixed regime flows

1. Supercritical and mixed regime flows

2. Background • Most open channel flows are above critical depth (subcritical flow with low velocities) • In steep channels flows may be below critical depth (supercritical flow with high velocities) • Energy considerations for an open channel govern whether the depth is above or below critical depth

3. Energy in open channels • For a given discharge, there are two possible depths for a given energy, (except at minimum energy -critical depth) • For a given discharge, there must be some minimum energy for this flow to exist (this minimum occurs at critical depth) Total energy = WS Elev + aV2/2g Total energy = WS Elev + [aQ2/2g]/A2 A = f(depth) = f(WS Elev)

4. Flow energy for a given discharge Critical Depth, yc Minimum energy

5. Subcritical and Supercritical Flow Definition is based on velocity: y > yc V < Vc Subcritical y = yc V = Vc Critical y < yc V > Vc Supercritical

6. Critical depth calculation • Energy Equation: E = Z + y + V2/2g = Z + y + (Q/A)2/2g • Energy is minimum at yc When E = Emin, dE/dy = 0 dE/dv = dZ/dy + dy/dy + (Q2/2g)2A-3dA/dy 0 = 0 + 1 + Q2/(gA3) dA/dy Since A = f(y), dA ≈ Tdy, dA/dy ≈ T • So at critical depth, Q2T/(gA3) = 1 or A3 /T = Q2/g

7. The Froude number At critical depth: Q2T/(gA3) = 1 (Q/A)2/(g A/T) = 1 The right hand side of the equation is a dimensionless ratio (the Froude number squared) Q/A = V (the mean velocity) A/T = D (the hydraulic depth)

8. Common form of the Froude number

9. Wave speed in an open channel cwave Dy Y = flow depth In a rectangular channel y = A/T

10. Velocity at yc = wave velocity

11. Speed of disturbance in open channel Wave movement in subcritical flow • Vavg < Vc • Wave can move upstream and downstream Wave movement in supercritical flow • Vavg > Vc • Wave can move only downstream

12. Flow Regime Types • Critical Flow Fr = 1 • Subcritical Flow Fr < 1 • Supercritical Flow Fr > 1

13. Expansion and contraction losses • Expansion and contraction losses used for supercritical flow should be lower than those used for subcritical flow • The standard coefficients tend to overestimate the losses in supercritical flow • If there is no contraction or expansion you may want to set the coefficients equal to zero • Typical values for gradual transitions in natural channels: • Contraction -- 0.05 • Expansion -- 0.10

14. “Controls” in open channel flow Location of a control • Critical depth at change in channel slope • Normal depth in a very long channel • Gate in channel How controls affect boundary conditions • Subcritical flow is subject to downstream BC • Supercritical flow is subject to upstream BC • Mixed flow regime requires both upstream and downstream BC

15. Upstream BC Downstream BC Internal BC Mixed Flow

16. Hydraulic jump • A hydraulic jump is a sudden dissipation of energy caused by a change from super-critical to sub-critical flow. • At the start of the jump, the flow height will begin to increase, and the velocity will slow creating an area of turbulence. • At the end of the jump, the flow height will level off again, and the fluid will continue flowing smoothly.

17. Hydraulic jumps • Hydraulic jumps dissipate a large amount of energy in open channel flows. This makes hydraulic jumps very useful in dam and spillway designs. • Often special features are needed to make jumps occur at desired locations such as at the bottom of a spillway. • Increasing surface roughness, adding a baffle wall, or sloping the basin floor can all help force a hydraulic jump.

18. Effect of the Froude number • The major parameter influencing a hydraulic jump is the Froude Number, Fr. • The best design range for the Froude number is 4.5 to 9.0. In this range, a well-balanced steady jump will occur with a large amount of energy dissipation. • A Froude number of 2.5 to 4.5 is the worst design range, as the jump in this range will create large waves that could cause structural damage.

19. At a hydraulic jump • Balance in pressure and momentum • S Forces = ma Well-defined jump in laboratory Weak jump in field

20. Controlled Hydraulic Jump

21. Flow near critical?

22. Roughness coefficients – steep streams Gravel bed streams

23. High Velocity, Near Critical Flow RAS profile plot

24. Critical Depth - Parabolic Method Water Surface Elevation 1 2 WScrit 3 Hmin Total Energy, H

25. HEC-RAS Cross Sections Right overbank Left overbank High Flow n1 n3 n4 n2 A1 P1 A2 P2 A3 P3 Ach Pch Klob = K1 + K2 Krob = K3 Kch Low Flow

26. Critical Depth - Slicing Method Water Surface Elevation Local minimum energy Top of bank or ineffective flow WScrit Hmin Total Energy H

27. RAS Output Table with Multiple Critical Depths

28. Critical Depth - Procedure • Program uses Parabolic method by default • This can be changed to the Slicing method • Program automatically switches to slicing method if any of the following occur: • Parabolic method fails to get answer within tolerance • Parabolic answer is at or near top of levee of ineffective flow area

29. Junctions in Supercritical Flow • Supercritical flows with changes in boundary alignments are generally complicated by standing waves. • In tranquil flow, backwater effects are propagated upstream, tending to equalize the flow depths in the main and side channels. • However, backwater cannot be propagated upstream in rapid flow, and flow depths in the main and side channels cannot generally be expected to be equal.

30. RAS junction analysis • Junctions for rapid flows and very small junction angles are designed assuming equal water-surface elevations in the side and main channels. • Model tests by the Corps on rapid-flow junctions have verified the use of the momentum equation for this purpose.  

31. Air entrainment in high velocity streams • For channels with high flow velocity the water surface may be slightly higher than the computed value due to air entrainment • For river flows this is usually not important • For chutes and concrete channels with Froude numbers greater than 1.6, this can be significant • The water surface with air entrainment is computed by RAS and can be displayed in the output tables • For 1.6<Fr<8.2, Da=0.906D(e0.061Fr) Da=depth with air entrainment D = computed depth Fr = Froude number e = 2.718282 (base of natural logs)

32. Hydraulic jump in a closed conduit Use cross section with lid

33. Cross section with lid • Grey area is lid • Lid must extend above HGL (what RAS considers as water surface)