Cu06997 lecture 10_froude

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  • 1. CU06997 Fluid dynamicsFroude number (page 148)5.9 Critical depth meters (page 155 – 158)1
  • 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. 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. 2222 yBgQyH v𝐻 = y +𝑄22𝑔 ∙ 𝐵2∙1𝑦2ExampleB= 2 m, Q = 6 m3/syBH𝐻 = y + 0.45 ∙1𝑦22
  • 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. 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. Critical Depth and Critical VelocitycyH 23min Sub-critical flow Supercritical flow𝑦𝑐 =𝑄2𝑔 ∙ 𝐵23𝑉𝑐 = 𝑔 ∙ 𝑦𝑐23h = y in this graph
  • 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. 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. 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. 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. 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. 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. Critical bed slope channel /river4
  • 15. Hydraulic jump [watersprong]When supercritical flow [schietend] changes to subcriticalflow [stromend] a hydraulic jump will occur5
  • 16.   2122321vvvvvwaHydraulic jump, energy loss5