Cu06997 lecture 10_froude

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Cu06997 lecture 10_froude

  1. 1. CU06997 Fluid dynamicsFroude number (page 148)5.9 Critical depth meters (page 155 – 158)1
  2. 2. Specific EnergyVChannel bed as datum [m]Surface level [m]Total head H or Specific energy Es [m]yV2/2g Velocity head [m]y = Pressure head [m]= water depth [m]𝐸𝑠 = 𝑦 +𝑉22𝑔𝑉 = Mean Fluid Velocity [m/s]y =pρ∙g= Pressure Head / water depth [m]1
  3. 3. Critical DepthVReference /datum [m]Water depth y [m]yV2/2g Velocity head [m]yBgVyH22 yBVQv 2222 yBgQyH vHSuppose Q and B are given, what could by the value of H and yTotal head H or Specific energy Es [m]2P1 P1
  4. 4. 2222 yBgQyH v𝐻 = y +𝑄22𝑔 ∙ 𝐵2∙1𝑦2ExampleB= 2 m, Q = 6 m3/syBH𝐻 = y + 0.45 ∙1𝑦22
  5. 5. 0.001.002.003.004.005.006.00 0.3000.5900.8801.1701.4601.7502.0402.3302.6202.9103.2003.4903.7804.070H(totalhead)(m)y (water depth) (m)Sub-critical or Supercritical flowStromend of schietend waterTotal headH=3/2*hSupercritical flowSchietend waterSub-critical flowStromend waterExampleB= 2 m, Q = 6 m3/s2cyH 23min 
  6. 6. 2222 yBgQyH v𝐻 = y +𝑄22𝑔 ∙ 𝐵2∙1𝑦2Differentiation [Differentiëren]dH/dy = 0 givesyBH2𝑦𝑐 =𝑄2𝑔 ∙ 𝐵23Represents lowest point graph.Means point with the lowestH for a given Q and B
  7. 7. Critical Depth and Critical VelocitycyH 23min Sub-critical flow Supercritical flow𝑦𝑐 =𝑄2𝑔 ∙ 𝐵23𝑉𝑐 = 𝑔 ∙ 𝑦𝑐23h = y in this graph
  8. 8. Froude number𝑦𝑐 =𝑄2𝑔 ∙ 𝐵23𝑉𝑐 = 𝑔 ∙ 𝑦𝑐2𝐹𝑟 =𝑉𝑔𝑦𝑐2=𝑉𝑉𝑐yc = 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 < VcSupercritical flow [schietend] Fr > 1 V > Vc3
  9. 9. Froude numberFr>1• Supercritical flow [schietend water]• Water velocity > wave velocity• Disturbances travel downstream• Upstream water levels are unaffected bydownstream controlFr<1• Subcritical flow [stromend water]• Water velocity < wave velocity• Disturbances travel upstream and downstream• Upstream water levels are affected bydownstream control3
  10. 10. Froude number<1 Subcritical[stromend]Consequences for strategy to calculate water levelsWhat happens downstream affect the upstream water levelSo most of the time you start downstream and go upstream3
  11. 11. Question 3de50 mØ300 PVCØ500 betonØ250 PVCPump=20 l/sP4 P3 P2GL +6.00 mRain=66 l/sWaste=10 l/sRain=225 l/sWaste=10 l/s+5,5 mQ=66 l/sv=0,93 m/sI=1:244Q=291 l/sv=1,48 m/sI=1:166Q=0 l/sv=0 m/sI=0P1In example m = 1,83
  12. 12. Froude number>1 Supercritical[schietend]Consequences for strategy to calculate water levelsWhat happens downstream does not affect the upstreamwater levelSo most of the time you start upstream and go downstream3
  13. 13. Critical bed slope channel /riverQ and B (width channel) are givenStep 1 Calculate ycStep 2 Calculate R and VcStep 3 Calculate Sc using Chezy or Manning𝑦𝑐 =𝑄2𝑔 ∙ 𝐵23𝑉𝑐 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑐𝑉𝑐 =𝑅23 ∙ 𝑆𝑐12𝑛4
  14. 14. Critical bed slope channel /river4
  15. 15. Hydraulic jump [watersprong]When supercritical flow [schietend] changes to subcriticalflow [stromend] a hydraulic jump will occur5
  16. 16.   2122321vvvvvwaHydraulic jump, energy loss5

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