# Cu06997 lecture 10_froude

Hogeschooldocent at Hogeschool Zeeland,Hz University of applied sciences.
Apr. 26, 2013
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### Cu06997 lecture 10_froude

• 1. CU06997 Fluid dynamics Froude number (page 148) 5.9 Critical depth meters (page 155 – 158) 1
• 2. Specific Energy V Channel bed as datum [m] Surface level [m] Total head H or Specific energy Es [m] y V2/2g Velocity head [m] y = Pressure head [m] = water depth [m] 𝐸𝑠 = 𝑦 + 𝑉2 2𝑔 𝑉 = Mean Fluid Velocity [m/s] y = p ρ∙g = Pressure Head / water depth [m] 1
• 3. Critical Depth V Reference /datum [m] Water depth y [m] y V2/2g Velocity head [m] y B g V yH 2 2  yBVQv  22 2 2 yBg Q yH v   H Suppose Q and B are given, what could by the value of H and y Total head H or Specific energy Es [m] 2 P1 P1
• 4. 22 2 2 yBg Q yH v   𝐻 = y + 𝑄2 2𝑔 ∙ 𝐵2 ∙ 1 𝑦2 Example B= 2 m, Q = 6 m3/s y B H 𝐻 = y + 0.45 ∙ 1 𝑦2 2
• 5. 0.00 1.00 2.00 3.00 4.00 5.00 6.00 0.300 0.590 0.880 1.170 1.460 1.750 2.040 2.330 2.620 2.910 3.200 3.490 3.780 4.070 H(totalhead)(m) y (water depth) (m) Sub-critical or Supercritical flow Stromend of schietend water Total head H=3/2*h Supercritical flow Schietend water Sub-critical flow Stromend water Example B= 2 m, Q = 6 m3/s 2 cyH 2 3 min 
• 6. 22 2 2 yBg Q yH v   𝐻 = y + 𝑄2 2𝑔 ∙ 𝐵2 ∙ 1 𝑦2 Differentiation [Differentiëren] dH/dy = 0 gives y B H 2 𝑦𝑐 = 𝑄2 𝑔 ∙ 𝐵2 3 Represents lowest point graph. Means point with the lowest H for a given Q and B
• 7. Critical Depth and Critical Velocity cyH 2 3 min  Sub-critical flow Supercritical flow 𝑦𝑐 = 𝑄2 𝑔 ∙ 𝐵2 3 𝑉𝑐 = 𝑔 ∙ 𝑦𝑐 2 3 h = y in this graph
• 8. Froude number 𝑦𝑐 = 𝑄2 𝑔 ∙ 𝐵2 3 𝑉𝑐 = 𝑔 ∙ 𝑦𝑐 2 𝐹𝑟 = 𝑉 𝑔𝑦𝑐 2 = 𝑉 𝑉𝑐 yc = critical depth [m] Q = discharge [m3/s] B = width [m] Vc = critical velocity [m/s] V = actual velocity [m/s] Fr = Froude number [-] Subcritical flow [stromend] Fr < 1 V < Vc Supercritical flow [schietend] Fr > 1 V > Vc 3
• 9. Froude number Fr>1 • Supercritical flow [schietend water] • Water velocity > wave velocity • Disturbances travel downstream • Upstream water levels are unaffected by downstream control Fr<1 • Subcritical flow [stromend water] • Water velocity < wave velocity • Disturbances travel upstream and downstream • Upstream water levels are affected by downstream control 3
• 10. Froude number<1 Subcritical [stromend] Consequences for strategy to calculate water levels What happens downstream affect the upstream water level So most of the time you start downstream and go upstream 3
• 11. Question 3de 50 m Ø300 PVC Ø500 beton Ø250 PVC Pump=20 l/s P4 P3 P2 GL +6.00 m Rain=66 l/s Waste=10 l/s Rain=225 l/s Waste=10 l/s +5,5 m Q=66 l/s v=0,93 m/s I=1:244 Q=291 l/s v=1,48 m/s I=1:166 Q=0 l/s v=0 m/s I=0 P1 In example m = 1,83
• 12. Froude number>1 Supercritical [schietend] Consequences for strategy to calculate water levels What happens downstream does not affect the upstream water level So most of the time you start upstream and go downstream 3
• 13. Critical bed slope channel /river Q and B (width channel) are given Step 1 Calculate yc Step 2 Calculate R and Vc Step 3 Calculate Sc using Chezy or Manning 𝑦𝑐 = 𝑄2 𝑔 ∙ 𝐵2 3 𝑉𝑐 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑐 𝑉𝑐 = 𝑅 2 3 ∙ 𝑆𝑐 1 2 𝑛 4
• 14. Critical bed slope channel /river 4
• 15. Hydraulic jump [watersprong] When supercritical flow [schietend] changes to subcritical flow [stromend] a hydraulic jump will occur 5
• 16.    21 2 2 3 21 vvv vv wa    Hydraulic jump, energy loss 5