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Demetriou, J.

Dimitriou, D.

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Conference Paper, Published VersionDimitriou, D.; Demetriou, J.Energy Loss in Submerged Hydraulic Jump Within InclinedChannelZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/109839Vorgeschlagene Zitierweise/Suggested citation:Dimitriou, D.; Demetriou, J. (2010): Energy Loss in Submerged Hydraulic Jump WithinInclined Channel. In: Sundar, V.; Srinivasan, K.; Murali, K.; Sudheer, K.P. (Hg.): ICHE 2010.Proceedings of the 9th International Conference on Hydro-Science & Engineering, August2-5, 2010, Chennai, India. Chennai: Indian Institute of Technology Madras.Standardnutzungsbedingungen/Terms of Use:Die Dokumente in HENRY stehen unter der Creative Commons Lizenz CC BY 4.0, sofern keine abweichendenNutzungsbedingungen getroffen wurden. Damit ist sowohl die kommerzielle Nutzung als auch das Teilen, dieWeiterbearbeitung und Speicherung erlaubt. Das Verwenden und das Bearbeiten stehen unter der Bedingung derNamensnennung. Im Einzelfall kann eine restriktivere Lizenz gelten; dann gelten abweichend von den obigenNutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Documents in HENRY are made available under the Creative Commons License CC BY 4.0, if no other license isapplicable. Under CC BY 4.0 commercial use and sharing, remixing, transforming, and building upon the materialof the work is permitted. In some cases a different, more restrictive license may apply; if applicable the terms ofthe restrictive license will be binding.     ENERGY LOSS  IN SUBMERGED HYDRAULIC JUMP  WITHIN INCLINED CHANNEL  Dimitriou D.1 and Demetriou J.2   Abstract: In this paper the mechanical energy and loss of energy in the submerged hydraulic jump within rectangular inclined (angle φ, oo 15φ3 ≤≤ ) open channels are presented, analyzed, discussed and compared to corresponding free jump quantities. The energy and loss of energy calculations are based on previous measurements of both jumps’ lengths and conjugate depths’ ratios. The loss of energy in the submerged jump is much smaller than in corresponding free jump, showing that the submerged jump is not an effective means to dissipate any excessive flow energy when compared to much larger free jump energy loss.  Keywords: Submerged Jumps. Energy Loss.   INTRODUCTION  The water flow under a sluice gate and subsequent hydraulic jump (when formed under suitable conditions) belong to some of the most important hydraulic phenomena which are appearing in practice within open water channels. Sometimes the hydraulic jump is free and some other times is submerged, depending on  the downstream flow conditions, i.e. on the free or obstructed downstream channel flow. In the last case an obstacle such as a solid body (for example another sluice gate or a weir) is raising the water level just after the sluice gate and the entire flow becomes submerged. The higher part of any hydraulic jump – under appropriate conditions – is permanently remaining at its place although the flow under it is steadily developing along the open channel.  Fig. 1 schematically shows the general flow case where the rectangular channel is inclined (angle φ) to the horizon – with a slope Jo=sinφ – and includes a sharp edged and very thin sluice gate (perpendicular to the channel floor) which has the same width as the channel, and a lower aperture α. The water discharge per unit width is q, the most important depths are, p1, p2, for the submerged jump, and d1, d2, for the free jump. Both jumps are schematically presented in Fig. 1 and show the difference of the two states of flow. The submerged jump has an inclined length L and the free jump a corresponding length Ld.For the submerged jump    1 Dr Mechanical Engineer, Davy Process Technology, London W2 6LE, UK,  Email: dimitrios.dimitriou@davyprotech.com 2 Associate Professor (RD), School of Civil Engineering, National Technical University of Athens, Greece, 12 Polykarpou St., N. Smyrni, Athens, 17123, Greece. Email : idimit@central.ntua.gr (corresponding author).  Proceedings of ninth International Conference on Hydro-Science and Engineering (ICHE 2010), IIT Madras, Chennai, India. 2 - 5  August 2010 1085 Energy Loss In Submerged Hydraulic Jump Within Inclined Channel  the control volume is included between cross sections 1 and 2, where the local pressures are assumed to be hydrostatic. Cross section 1 is considered as coinciding with the contracted cross section of the free jump – at a distance x1.The submerged jump has a mean free surface profile starting from the outer face of the sluice gate, followed by a local fall along x1,and then it turns (along L)  towards the downstream horizontal.    Beyond x1, the lower limit of the submerged jump has a complicated form also ending at p2 depth, while between the upper free surface and the lower limit a roller is created  with its recirculating flow. The discharge q goes  across p1 and p2, while at cross section 1 the roller’s water is increasing the pressure on depth p1 and the entire pressure distribution (column t+p1)) is considered as hydrostatic, with a resultant force along the flow direction (x) ϕ⋅+⋅⋅ cos)p(tγ0.5 21 , where γ=specific water weight.             Fig. 1. Submerged and free jump geometry.  The one-dimensional continuity equation is 2211 VpVpq ⋅=⋅= , while the momentum equation along x-with negligible tractive force on the channel boundaries – is  )]p/1()p/1[(qJWcos]p)pt[(5.0 122o2221 −⋅⋅=⋅+⋅−+⋅ ρϕγ    (1)  The above equation has been solved in the past (Demetriou, 2006), for the theoretical ratio λt=p2/p1, after the experimental determination of the weight W among sluice gate, channel floor, cross section 2 and free surface. This theoretical ratio has also been successfully compared to corresponding (experimentally determined) ratio λe.  Fig. 1 also illustrates the mechanical energy (per unit water weight) H1 at section 1 (depth p1), and H2 at section 2 (depth p2), while ∆H = H1-H2 is the local loss of energy between conjugate depths p1 and p2, which is due both to tractive stresses (for φ>00) and internal friction. The present paper is dealing with H1, H2 and ∆H and the comparison of them  with corresponding quantities in the free jump, both in inclined rectangular channels. The most important parameters are λ=p2/p1, L/p2 (experimentally determined) and the Froude numbers Fp1(section 1)and Fp2(section 2), generally  with  Fp=q/g1/2·(corresponding depth)3/2          (2) 1086 Energy Loss In Submerged Hydraulic Jump Within Inclined Channel   where Fp1>1 and Fp2<1. Fp2=q/g1/2·p23/2 was mainly used here, while all Reynolds numbers had large enough values (turbulent flows).  For the free hydraulic jump the conjugate depths d1 and d2 (and their ratio δ=d2/d1), the length Ld/d2 and Froude number Fd2=q/g1/2·d23/2=Fd1·(d1/d2)3/2 are used, while any comparison with the submerged jump is meant with Fp2=Fd2=Fr2. The points which appear on the following figures came out as results of energy computations, based on the pertinent experimental data – and in combination with onedimensional energy expressions.    PREVIOUS EXPERIMENTAL RESULTS   2.1 For the submerged hydraulic jump within inclined rectangular channels, Retsinis et.al, (2005) and Demetriou et al (2006), have presented the following experimental equations:   a) For the jump length (with φ in degrees and Fp2=Fr2),  )]25.103.0(Fr)0018.00069.09053.2[(Fr)8.307.2(cospL'L 2222+⋅−++⋅⋅−⋅+−=⋅+⋅⋅=ϕϕϕϕϕ  (3)  The above length L is the distance between the sluice gate and cross section 2, while the distance x1 (between sluice gate and cross section 1) was measured as x1≅1.7·α. Since x1 is rather small – in comparison with L, it is reasonable to consider the distance 1-2 as approximately equal to L.  b) For the conjugate depth’s ratio (with φ in degrees)  λ=p2/p1=1- {[(L΄)2-1]/B1}  where,                          (4) )e16.05.0105.0840.1(21 Fr]e224.0022.0776.1[Bϕϕϕϕ−⋅+⋅−⋅⋅+⋅+= −   The above equations are used here for φ=3o-6o-9o-12o-15o  ,  0.08 ≤ Fr 2≤ 0.15 for φ = 9o, and smaller Fr2 ranges for other angles φ.  2.2. For the free hydraulic jump, Demetriou (2005), has experimentally given the following equations:   a) For the jump length (with φ in degrees and Fd1=Fr1),   1087 Energy Loss In Submerged Hydraulic Jump Within Inclined Channel  Ld/d2= [7.69-0.094·Fr1-(6.27/Fr1)]·cosφ)2J35.3( 3.1o −⋅−        (5)  b) For the conjugate depths’ ratio, the following equation was verified,  δ=d2/d1=0.5·[(1+8·Fr12)1/2-1]·e3.5·Jo          (6)  The above equations hold for 0o≤ φ ≤ 16o, 2 ≤ Fr1 ≤ 19 for φ=0o, and smaller Fr1(≥2) ranges for other angles φ. Eq (6) gives exactly the same δ vs Fr1 lines as Chow’s (1959), graphical straight lines.   RESULTS. ANALYSIS AND DISCUSSION  For the submerged jump the mechanical energy H1 is (Fig. 1), H1 = L·Jo+p1·cosφ+(q2/p12·2·g), or, with the use of Fr2(= Fp2), λ=p2/p1, and after division by p1,   H1/p1= λ · (L/p2)·Jo+cosφ+0.5·λ3·Fr22      (7)  Also, H2=p2·cosφ+(q2/p22·2·g), or   H2/p1 = λ·[ cosφ+0.5·Fr22]        (8)  The local loss of mechanical energy between p1 and p2 is   ∆H/p1= (H1/p1)-(H2/p1) = λ·(L/p2)·Jo+(λ-1)·cosφ+0.5·Fr22·λ·(λ2-1)    (9)  or, in terms of ∆H/H1,   ∆H/H1=(∆H/p1)/(H1/p1)        (10)  Corresponding energies and loss of energies for the non submerged (=free) jump are given by similar equations, although – instead of p1, p2, λ, L and Fp2(=Fr2) – corresponding quantities d1, d2,  δ=d2/d1, Ld and Fd2(=Fr2) are used, while any comparison is meant for the same Froude number.  For the submerged jump Fig. 2 presents H1/p1 vs Fr2 at φ=3o, where the corresponding (solid) curve is freely extrapolated (dashed-short lines): H1/p1 is increasing with Fr2, for example from 6.9 (at Fr2=0.17) to 7.3 (at Fr2=0.20), i.e. the percentage increase is (7.3-6.9)·100/6.9 ≅ 8%.       1088 Energy Loss In Submerged Hydraulic Jump Within Inclined Channel           Fig. 2. H1/p1 vs Fr2  for φ=3o.  The same trends present H1/p1 vs Fr2 for φ = 6o-9o-12o-15o in Figs 3,4,5, with  the only difference (not visible in Fig.2) that after a certain Fr2 value corresponding curves are bending down, while the respective curve for φ=15opresents only a descending branch-in the present field of measurements. However the general H1/p1 level is considerably rising from φ= 3o to φ=15o.   Fig. 3. H1/p1 vs Fr2  for φ=6o.  Fig. 6 presents all previous curves (submerged jumps – solid lines) and all corresponding H1/d1 vs Fr2 (free jumps – dashed lines). Both families of lines are very systematic. For Fr2=const. H1/p1 for submerged jumps are increasing when angle φ is increasing, while to H1/d1=const. correspond larger Fr2 values when φ is decreasing. It is also clear that H1/d1 are generally larger than H1/p1 for any pair of lines with φ=const., i.e. the dimensionless energy at cross section 1 for submerged jump, is lower than the dimensionless energy at corresponding cross section for free (= non submerged) jump (at d1) – in the present field of measurements.As a simple example for φ=9ο and Fr2=0.13 H1/p1≅10 for submerged jump, while H1/d1≅17 for free jump, i.e. there is a percentage energy increase in free jump of (17-10)·100/10=70%.However, although the present measurements had a rather narrow Fr2 range, it may be predicted that any pair of corresponding lines – for the same angle φ – do not meet between them but they are incompatible curves. 1089 Energy Loss In Submerged Hydraulic Jump Within Inclined Channel  .                       Fig. 4. H1/p1 vs Fr2  for φ=9o.                  Fig. 5. H1/p1 vs Fr2  for φ=12o and φ=15o.                                                                                 Fig. 6. H1/p1 and H1/d1 vs Fr2  for submerged and free jump, 3o ≤ φ ≤ 15o.  Fig. 7 shows the dimensionless energy loss ∆H/p1 vs Fr2 at angle φ=3o for submerged jumps. When Fr2 is increasing ∆H/p1 are also increasing : As an example, for Fr2=0.17 ∆H/p1 is 0.15, while for Fr2 = 0.20 ∆Η/p1 becomes 1, i.e. the percentage increase is now much larger, (1-0.15)·100/0.15≅565%.  1090 Energy Loss In Submerged Hydraulic Jump Within Inclined Channel   Fig. 7. ∆H/p1 vs Fr2  for φ=3o.  Next Figs.8,9,10 and 11 present ∆H/p1vsFr2 for φ=6o-9o-12o-15o,which also show the same trends as in Fig.7,but the general level of the dimensionless losses is strongly increasing with  angle φ.          Fig. 8. ∆H/p1 vs Fr2 for φ=6o.    Fig. 9. ∆H/p1 vs Fr2 for φ=9o.                Fig. 10. ∆H/p1 vs Fr2  for φ=12o.    Fig. 11. ∆H/p1 vs Fr2 for φ=15o.    1091 Energy Loss In Submerged Hydraulic Jump Within Inclined Channel  Furthermore, Fig. 12 presents all previous ∆H/p1 vs Fr2 curves (solid lines) for submerged jumps and all ∆H/d1 vs Fr2 curves for free jumps (dashed lines). Both families of curves are very systematic. For Fr2=const. ∆Η/p1 for submerged jumps are largely increasing when angle φ is increasing. For example for Fr2=0.09  ∆H/p1≅0.6 (φ=9o), while for same Fr2 ∆H/p1≅ 3.7 (φ=12o), i.e. there is a percentage energy loss increase of (3.7-0.6)·100/0.6≅ 515% - when angle φ is increasing with a much smaller rate (~33%).  Finally, the most important result in Fig. 12 comes from a comparison between ∆H/p1 (submerged jump-solid lines) and ∆H/d1 (free jump-dashed lines) for the same Fr2 and angle φ: ∆H/d1 for any pair of φ = const. appears to be much larger than ∆H/p1, i.e. the dimensionless energy loss between conjugate jumps’ depths is considerably smaller for submerged jumps than for corresponding free jumps – in the present field of measurements. As a simple example for φ=9o and Fr2=0.13 ∆H/p1≅ 2.6 for submerged jump, while ∆Η/d1≅ 5.2 for free jump, i.e. there is a percentage energy loss increase in free jump of (5.2-2.6)·100/2.6=100%. This  fact looks rather reasonable if considering the flow nature of submerged jump-which is a rather calm flow phenomenon, and of the free jump which is far more violent: The submerged jump is not an effective means to dissipate excessive mechanical flow energy, although it presents some external similarities to the free jump.                            Fig. 12. ∆H/p1 and  ∆H/d1 vs Fr2 for submerged and free jump, 3o ≤ φ ≤ 15o. 1092 Energy Loss In Submerged Hydraulic Jump Within Inclined Channel  CONCLUSIONS  In this paper the mechanical energy and loss of energy in the submerged hydraulic jump within rectangular inclined (angle φ, with 3o ≤ φ ≤ 15o) open channels are presented, analyzed, discussed and compared to corresponding free jump quantities. The energy and loss of energy calculations are based on previous measurements by the second author  with other authors, concerning jump length and conjugate depths’ ratio, both for submerged and free jumps. Fig. 1 shows corresponding flow geometries, eq. (1) presents the one-dimensional momentum equation, while eq. 2 shows the main flow parameter, Froude number. Eqs. (3) to (6) give the results of the previous measurements, while eqs.(7) to (10) describe the corresponding energy  and loss of energy. Figs. 2 to 6 present the dimensionless energy at the beginning of submerged and free jumps, while Figs. 7 to 12 show the dimensionless energy loss of both hydraulic jumps: The loss of energy in submerged jumps is much smaller  than in free jumps and in this sense the submerged jump is not an effective means for the excessive flow energy dissipation – when compared to the free jump.   REFERENCES  Chow V.T. 1959. Open Channel Hydraulics, McGraw – Hill, p.427. Demetriou J. 2005. Unique length and Profile Equations for Hydraulic Jump in Sloping Channels, 17thCanadian Hydrotechnical Conference, Edmonton-Canada, August, p.p. 891-898. Demetriou J., Papathanasiadis T. 2006. A comparison Between Free – Submerged Hydraulic Jump in Horizontal Channel. 10th E.Y.E. Congress, October, Xanthi, Greece, 8 pages. Demetriou J., Retsinis G. 2006. Basic Flow Characteristics of Submerged Jumps Under Sluice Gates, IAHR, Int. Symposium on Hydraulic Structures, October, Ciudad Guayana, Venezuela, 8 pages. Demetriou J. 2006. Submerged – Inclined Hydraulic Jump. A Comparison Between Theory and  Measurement, 15th Congress APD – IAHR 2006, August, IIT Madras, Chennai, India, pp3-9. Retsinis G., Demetriou J. 2005. Discharge Measurements Through Sluice Gate in Submerged Open Channel Flow, 1st Int. Conference on E.P.S.M.S.O.,Athens, Greece, July, 6 pages.  1093

Energy Loss in Submerged Hydraulic Jump Within Inclined Channel

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Energy Loss in Submerged Hydraulic Jump Within Inclined Channel

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