tex seph can c emp een re x intak stron nderin am to of Qf flow imenta signin p. ement 2003�. For example, flood diversion schemes in the form of by- skii et al. �1980� reported a hydraulic model study of tangential intake for the Medeo dam in Kazakhstan. A comprehensive litera- D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. pass floodways or underground storage schemes have been de- signed and commissioned to protect urban centers in Hong Kong from flooding �Arega et al. 2008; Lee et al. 2008�. For hilly ter- rains the storm water can also be intercepted upstream and trans- ferred to an underground tunnel system via vortex drops. The hydraulic design of the storm-water flow intake structures is vital to the success of these diversion schemes. The flow in a vortex intake is typically three dimensional, and the intercepted flow is conveyed via a long vertical dropshaft into the tunnel system. Reviews of various types of vortex drop structures �scroll, spiral, tangential� have been given �e.g., Vischer and Hager 1995; Jain and Ettema 1987�. In general, vortex-flow intakes have been shown to give good hydraulic performance with respect to energy dissipation and air entrainment control �Jain and Ettema 1987�. ture review and hydraulic model study of vortex drops was car- ried out by Jain and Kennedy �1983�. By assuming energy conservation, the depth-discharge relationship for a tangential vortex intake was derived; there was fair agreement of the theory with experimental data �Jain 1984; Jain and Ettema 1987�. The optimal shape of the tangential inlet �dropshaft diameter, ap- proach flow channel width, inlet channel bottom and side slope angles� was developed from experiments. In addition, an empiri- cal relation for dropshaft diameter that satisfies a minimum air core area ratio of 25% is suggested �Jain and Ettema 1987�. A parametric study of energy dissipation of a tangential vortex in- take has also been reported �Toda and Inoue 1999�. More recently, the energy dissipation and air entrainment characteristics in a relatively short tangential vortex intake dropshaft was studied �Zhao et al. 2006�. Despite the previous advances made on the subject, several aspects of the hydraulics of tangential vortex intakes have not been addressed. First, a comprehensive account of the salient flow features corresponding to different designs has not been reported. For example, free surface profiles in the tangential inlet for dif- ferent discharges have not been measured. Second, the applicabil- ity of the one-dimensional �1D� depth–discharge relation of Jain �1984� remains to be clarified. Recent model studies revealed that under certain conditions, a significant hydraulic jump can occur in 1Research Assistant Professor, Dept. of Civil Engineering, The Univ. of Hong Kong, Pokfulam Rd., Hong Kong SAR, China. E-mail:
[email protected] 2Professor, Dept. of Civil Engineering, The Univ. of Hong Kong, Pokfulam Rd., Hong Kong SAR, China. E-mail:
[email protected] Note. Discussion open until August 1, 2009. Separate discussions must be submitted for individual papers. The manuscript for this paper was submitted for review and possible publication on February 25, 2008; approved on August 25, 2008. This paper is part of the Journal of Hy- draulic Engineering, Vol. 135, No. 3, March 1, 2009. ©ASCE, ISSN 0733-9429/2009/3-164–174/$25.00. 164 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 Hydraulics of Tangential Vor Daeyoung Yu1 and Jo Abstract: A tangential vortex intake is a compact structure that dropshaft. It has been studied in physical models and successfully a comprehensive account of the key flow characteristics has not b not available. In this study the hydraulics of tangential slot vorte flow in the tapering and downward sloping vortex inlet channel is some conditions, hydraulic instability and overflow can occur, re depends on the discharge at which flow control shifts from upstre theoretical design criterion for stable flow is developed in terms design, the flow in the tapering inlet evolves from supercritical different tangential vortex intake models are tested. The exper prediction. The present study provides a general guideline for de without unstable fluctuating flow associated with a hydraulic jum DOI: 10.1061/�ASCE�0733-9429�2009�135:3�164� CE Database subject headings: Vortices; Stormwater manag Introduction Urbanization and global climate change have resulted in more frequent occurrence of extreme rainfall events and consequent urban flooding. One effective and economic flood protection mea- sure in densely populated cities is to intercept the storm-water upstream of the urban area. Such urban flood control schemes are generally much less costly and more environmentally friendly than improvement works of existing drainage systems �DSD J. Hydraul. Eng. 2009 Intake for Urban Drainage H. W. Lee, F.ASCE2 onvey storm water efficiently as a swirling flow down a vortex loyed in urban drainage and hydroelectric plant applications, but ported and a theoretical design guideline of a tangential intake is es is investigated via extensive experiments. It is found that the gly dependent on the geometry of the inlet and dropshaft. Under g the design ineffective. It is shown that the hydraulic stability downstream �Qc�, as well as the free drainage discharge �Qf�. A and Qc as a function of the vortex inlet geometry. For a “stable” to subcritical flow smoothly as the discharge increases. Fifteen l observations are in excellent agreement with the theoretical g a tangential vortex intake that can convey the flow smoothly ; Drainage; Urban areas; Hydraulic jump; Drop structures; Shafts. A tangential slot vortex intake has a simple and compact shape, and produces a stable vortex flow in the dropshaft with sufficiently large air core. The concept of tangential vortex intake was probably first introduced by Jevdjevich and Levin �1953�. They compared five intake types through model experiments and found the tangential intake has the same effectiveness in minimiz- ing the air entrainment as the conventional spiral intake. Brooks and Blackmer �1962� carried out a comprehensive model study of a tangential vortex intake for the San Diego Ocean Outfall. Slis- .135:164-174. channel transition. If the channel is horizontal �S0=0�, the entire flow in the channel will be critical �negligible friction can be Junction l D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. the inlet channel; the water level in the tapering inlet can be higher than that in the approach flow channel, leading to undesir- able overflow situations and violations of energy conservation implicit in the 1D model �Lee et al. 2006�. Third, a general cri- terion for the design of tangential intakes does not appear to be available. Consequently, preliminary design often has to resort to experience gained from previous successful designs—e.g., the op- timal design of the Milwaukee vortex drop �Jain and Kennedy 1983� or extensive model studies. A more generic approach that would enable the systematic evaluation of alternative designs for a given problem is clearly of interest. The hydraulics of tangential vortex intakes is revisited in the present study. The flow behavior in the vortex intake is studied in a series of experiments. For each design, the changes in free surface profile in the entire intake with increase in discharge are measured. The shift of flow control from the upstream approach flow channel to the downstream junction is observed along with any flow instability. The flow characteristics of satisfactory de- signs are contrasted with those of inadequate designs. Based on the findings from the theoretical and experimental study, a general heuristic criterion for the design of tangential vortex intakes is presented. The measured depth–discharge relation and vortex air core area are compared with theoretical predictions. An example application of tangential intake design using the proposed theory is also presented. Hydraulics of Tangential Intake Depth–Discharge Relation and Control Shift Consider an approach flow in a rectangular channel of uniform cross-section. For a given geometry of a tangential inlet and a given discharge, the flow depth in the approach flow channel can be determined. For small discharges, the control is at the approach flow channel; the flow is critical at the transition from the channel to the steep tapering inlet �Fig. 1�—much like a mild-steep sloped Approach flow channel DropshaftTapering section B L D e θ P (a) z β yc Q high discharge low discharge y P Approach flow channel DropshaftTapering section y y (b) Fig. 1. Tangential vortex intake: �a� plan view; �b� side view JOU J. Hydraul. Eng. 2009 assumed for the short channel�. The approach flow channel depth–discharge relation is then determined by the critical depth equation y = yca =�3 �QB � 21 g �1� where y and B=channel depth and width, respectively; Q =discharge; yca=critical critical depth based on unit discharge q =Q /B; and g=acceleration due to gravity. The flow in the taper- ing inlet channel is supercritical. For large discharges, however, the control is at the junction between the downstream end of the tapering inlet and the drop- shaft; the flow at the junction is critical. The control at the junc- tion drowns the control at the upstream end of the tapering inlet; the flow in the approach and inlet channels are both subcritical. For the tapering inlet channel, the specific energy of the flow is E=V2 /2g+y cos �, where V and y=velocity parallel to and depth normal to bed, respectively, and �=bottom slope angle �Fig. 1�. At the junction, the critical depth is ycj =�3 �Q e �2 1g cos � �2� in which e=inlet width at the junction. The specific energy at the junction, Ej, is then given by Ej = 3 2� �Q cos �/e�2g �1/3 �3� Assuming negligible hydraulic losses, the specific energy in the approach flow channel is Ea = Ej − z �4� in which z=drop in elevation between the approach flow channel and the junction to dropshaft. Ea is related to the discharge and the flow depth, y, by Ea = y + �Q/B�2 2gy2 �5� Substituting Ea and Ej from Eqs. �3� and �4�, Eq. �5� can be written in dimensionless form as y * + 1 2y * 2 = E * �6� where y * =y /yca; E*=3 /2�B cos � /e� 2/3 −z * and z * =z /yca. For E * �3 /2, the flow control is in the approach flow channel. For E * �3 /2, the flow control shifts to the inlet junction �Jain 1984�. The shift in flow control can be viewed with respect to an in- crease in discharge. If we hypothesize a smooth transition from the upstream to the downstream control as Q increases, the shift of the flow control occurs at Q=Qc, when the flow is critical both at the upstream channel and at the junction. The control shift discharge Qc can be predicted by substituting Eqs. �3� and �5� in Eq. �4� and equating Q to the critical discharge given by Eq. �1� Qc = �ge�2z/3�3/2 �cos2/3 � − �e/B�2/3�3/2 �7� However, whether this shift of flow control occurs smoothly with- out flow instability, energy loss �and hence whether the above- mentioned 1D theory is valid�, or overflow �unacceptable design� RNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 165 .135:164-174. =vz��D� �9�y* D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. depends on the geometry of the intake structure. This point will be investigated by means of systematic experiments. It should also be noted that Qc increases with e and most importantly z for given � and B. Free Drainage Condition Our experimental observations show that under some conditions, the supercritical inlet flow does not flow smoothly into the verti- cal dropshaft even when the discharge is smaller than the control shift discharge. The vortex flow, after turning 360° in the drop- shaft, may reenter the inlet or disturb the parallel inflow jet, lead- ing to a local swell or creating a backflow or even a hydraulic jump in the tapering section �see the upcoming Figs. 2 and 5�. It is hence desirable to design a tangential vortex inlet to avoid this condition. As a first-order approximation, the limiting condition of “free drainage” can be given by �z = vz�t �8� D e θ Vx vt r Vt (a) Air core B B Dropshaft Vortex flow b Q β A A a Vz yc cosβ (Throat) (b) β ∆z vxvz yc / cosβ (c) Fig. 2. Flow at junction: �a� plan view; �b� side view; and �c� perspective view 166 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 J. Hydraul. Eng. 2009 Vt where �z=drop height of water surface in dropshaft after the first 360° turn; vz=vertical velocity component at junction; �t=time for the first 360° turn in dropshaft; D=dropshaft diameter; and Vt=tangential velocity component at the dropshaft wall �Fig. 2�. The flow from the tangential inlet swirls down the dropshaft as an annular jet. The swirling flow inside the dropshaft can be ap- proximated as a free irrotational vortex �e.g., Jain and Ettema 1987�, characterized by the circulation strength or vortex constant c c = rvt = �D/2�Vt �10� where r and vt=radial position and horizontal tangential velocity respectively. By assuming a constant average horizontal x veloc- ity vx at the inflow junction �vx=Vx�, and constant vertical veloc- ity in the annular jet outflow in the dropshaft �vz=Vz�, the circulation strength c can be obtained from an application of the moment of momentum theorem �see the Appendix� c = vx �D − e� 2 = �Q e g�1/3cos4/3 � �D − e�2 �11� where vx=horizontal component of flow at junction and c =circulation constant. The circulation imparted by the tangential inlet depends on Q, �, e, and D. Noting that vz /vx=tan �, and using Eqs. �10� and �11�, Eq. �9� becomes �z = Vx tan � �D 2c/D = tan � �D 1 − e/D �12� The free drainage condition is satisfied when �z is larger than the critical flow depth at the junction, ycj and can be written as �z� ycj cos � for Q� Qc �13� The free drainage condition can then be obtained in terms of the tangential inlet geometry: Q� Qf = �tan � �D1 − e/D� 3/2 �ge cos2 � �14� where Qf is defined as free drainage discharge. If Qf�Qc, it is expected that a hydraulic jump takes place in the tapering section due to the blocking effect at the junction; this leads to undesirable fluctuating flow condition. Thus, to maintain the smooth and stable flow condition in the tapering section, a heuristic criterion for the design of the tangential vortex intake can be adopted as Qf � Qc �15� The overall capacity �Qc� and the flow stability condition �Qc �Qf� are governed by the geometric details of the vortex intake. It should be noted that Qf increases with �, e, and most impor- tantly D. Vortex Air Core Area The air core area generally decreases with discharge. A key de- sign parameter is the ratio of the air core area to the dropshaft cross-sectional area, �=b2 /a2, where a, b=radius of dropshaft and air core, respectively �Fig. 2�. � must be sufficiently large to allow free passage of air and ensure the stable operation of the vortex drop. A typical criterion is ��0.25 for the design dis- charge �e.g., Jain and Ettema 1987�. .135:164-174. Table 1. Parameters of Tangential Intake Models /B e .74 0 .22 0 .20 0 .30 0 .30 0 .30 0 .30 0 .30 0 .75 0 .75 0 .75 0 .75 0 .75 0 .75 0 .75 0 .33 0 .35 0 le flow D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. Following Binnie and Hookings �1948�, it is assumed that a critical section is established at the throat; the velocity of stream- ing �Vz� is equal to that of a small wave in the fluid. Let Vz be the vertical velocity of a “long wave” of annular form moving in an axial direction on the rotating surface at the throat. If V� additional velocity due to the wave motion at points where the infinitely small surface elevation is , the continuity equation becomes ���a2 − b2� + 2�b ��Vz + V�� = ��a2 − b2�Vz �16� Keeping only first-order terms, and noting that V��Vz, �b, we have ��a2 − b2�V� + 2�b Vz = 0 �17� and V� = 2 b a2 − b2 Vz �18� The radial change in pressure in the free vortex is given by �using Eq. �10�� �p �r = � vt 2 r = � c2 r3 �19� where p=pressure and �=density of water. The increase in pres- sure at any radius due to the spinning flow between radius b and b− is then �p = p�r=b� − p�r=b− � = c2 2g� 1�b − �2 − 1b2� �20� where =�g. On equating the increase in pressure head to the decrease in velocity head, we obtain �keeping first-order terms� �p = c2 2g�2b b4 � = �Vz + V�� 2 − Vz 2 2g = 2V�Vz g �21� Combining Eqs. �18� and �21� results in No. D �mm� B �mm� e �mm� L �mm� z �mm� Qc �L/s� Qf �L/s� D 1 124.0 167.0 28.0 651.9 326.0 18.0 8.9 0 2 124.0 102.0 28.0 651.9 326.0 24.7 8.9 1 3 124.0 103.0 28.0 271.9 136.0 6.6 8.9 1 4 127.0 98.0 26.0 145.6 101.9 4.6 11.4 1 5 127.0 98.0 26.0 236.6 44.9 1.0 2.3 1 6 127.0 98.0 26.0 236.6 94.2 3.2 6.3 1 7 127.0 98.0 26.0 236.6 163.7 9.3 11.3 1 8 127.0 98.0 26.0 457.6 89.0 2.7 2.4 1 9 73.5 98.0 17.0 232.5 66.4 0.9 1.2 0 10 73.5 98.0 17.0 232.5 113.6 2.3 2.4 0 11 73.5 98.0 17.0 279.5 195.9 5.9 3.4 0 12 73.5 98.0 17.0 385.5 192.8 5.1 2.5 0 13 73.5 98.0 17.0 385.5 110.4 2.0 1.2 0 14 73.5 98.0 17.0 385.5 270.9 9.7 3.5 0 15 73.5 98.0 17.0 660.5 189.4 4.4 1.2 0 Jain and Kennedy �1983� 1 Brooks and Blackmer �1962� 1 Note: Observed control shift flow is Qc�obs� and flow type refers to stab JOU J. Hydraul. Eng. 2009 Vz = c �2� �a 2 − b2� b4 �1/2 �22� The discharge is then given by Q = Vz��a2 − b2� = c �2 ��a2 − b2�3/2 b2 �23� Using Eq. �11�, the air core area ratio �=b2 /a2 can then be ob- tained as a function of the discharge and vortex inlet geometry ��1 − ��3 2�2 = 4� Q2eg�3D6 cos4 �� 1/3 1 �1 − e/D� = Fa �24� where Fa=dimensionless flow parameter. This model differs from the vortex air core model of Jain �1984� in two key features: �1� the local tangential velocity is given by the free vortex, rather than assumed constant over the throat cross-section and equal to the inflow horizontal velocity at the junction and �2� the circula- tion strength is related to the vortex inlet geometry and discharge. As the discharge Q increases, both Eq. �24� and experiments �see the following� show that the air core decreases in size. For a given minimum �, Eq. �24� implies a maximum value of the dimensionless flow parameter Fa; e.g., for �=0.25, Fa�1.84 is a limiting value. Experiments Experimental Parameters A series of tangential vortex intake model experiments are carried out. Fifteen different intake designs are systematically tested. A wide range of nondimensional geometric parameters is employed to study the flow characteristics in relation to different intake geometries. In contrast to previous investigations �Brooks and Blackmer 1962; Jain and Kennedy 1983�, D /B is varied in the range of 0.74–1.3 �Table 1�. The length, L, and drop height, z, the bottom slope angle, �, and the tapering angle, �, cover a wide range �L /B=1.22–6.58; z /D=0.31–3.54; �=10.7–35.1°; and � /D L /B z /D z /L � �deg� � �deg� Qf /Qc Qc�obs� /Qc Flow type .23 3.75 2.52 0.50 26.6 12.5 0.49 0.89 HJ .23 6.14 2.52 0.50 26.6 6.7 0.36 0.73 HJ .23 2.39 0.99 0.50 26.6 17.0 1.34 1.06 S .20 1.22 0.66 0.70 35.0 31.0 2.48 0.79 S .20 2.15 0.31 0.19 10.7 18.8 2.39 1.55 S .20 2.15 0.66 0.40 21.7 18.8 1.94 1.19 S .20 2.15 1.15 0.69 34.7 18.8 1.22 0.97 S .20 4.41 0.66 0.19 11.0 9.5 0.89 1.33 HJ .23 2.21 0.84 0.29 15.9 20.5 1.35 1.74 HJ .23 2.21 1.44 0.49 26.0 20.5 1.06 1.21 HJ .23 2.69 2.52 0.70 35.0 17.1 0.58 0.80 HJ .23 3.78 2.52 0.50 26.6 12.3 0.49 1.01 HJ .23 3.78 1.44 0.29 16.0 12.3 0.63 1.65 HJ .23 3.78 3.54 0.70 35.1 12.3 0.36 0.62 HJ .23 6.58 2.52 0.29 16.0 7.2 0.28 0.85 HJ .25 2.18 0.98 0.60 27.5 16.8 1.44 .25 2.23 0.56 0.34 18.6 16.5 1.86 transition �S� or hydraulic jump �HJ�. RNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 167 .135:164-174. D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. =6.7–31.0°�. The experimental parameters of the models are summarized in Table 1 together with the nondimensional param- eters used by Brooks and Blackmer �1962� and Jain and Kennedy �1983�. The computed control shift discharge, Qc according to Eq. �7�, and the free drainage discharge, Qf �Eq. �14��, of each intake model and the ratio Qf /Qc are also summarized. The Qf /Qc ratio covers a broad range �0.17–2.58�; from Eq. �15� if Qf /Qc�1, the flow in the tangential intake is expected to drain smoothly through the tapering section and dropshaft. Experimental Setup Each of the tested intake designs consists of an approach flow channel of rectangular cross section, a tapering inlet channel of rectangular cross section, a circular dropshaft, a stilling basin and collector channel, and an outlet tank with a downstream control. Fig. 3 shows a schematic diagram of the experimental setup. The entire model of an intake structure is housed in a specially de- signed 6-m-high, three-level frame structure that allows conve- nient testing of different vortex dropshaft lengths. A steady discharge through the intake system can be generated by a recir- culating flow system. The flow is fed from a 2.4 m�0.7 m �1.0 m water storage tank and delivered to the elevated inlet tank by a 10 hp AJAX centrifugal pump �KSB Ajax Pumps Pty Ltd., Tottenham, Australia�. The flow enters the approach flow channel via a smooth inlet; the flow passes through the intake structure into the dropshaft, and the outflow from the collector tunnel and outlet tank is returned to the storage reservoir. A vari- ety of intake designs can be readily replaced and efficiently tested. The intake models are made of zinc steel to facilitate quick fabrication; selective intake models are made of perspex to facili- tate flow visualization. The inflow discharge is measured by a calibrated ultrasonic flowmeter �Controlotron System 1010, Con- trolotron Corp., Hauppauge, N.Y.� with an accuracy of 1–2%. The longitudinal free surface profile along the centerline of both the approach flow channel and tapering inlet is measured in all mod- els using a point gauge. The transverse free surface profile of the Intake structure Inlet tank Dropshaft Stilling basin chamber Outlet tank Water storage reservoir Pump Downstream control Approach flow channel Collector tunnel Air vent Fig. 3. Experimental setup 168 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 J. Hydraul. Eng. 2009 tapering channel is measured for selected runs. The approach channel depth–discharge relation is determined using the water depth measured at one channel width upstream of the tapering section. For each design, the flow behavior in the vortex inlet is studied as the discharge increases. The control shift discharge �Qc�obs�� is determined from the measured depth–discharge curve. For selected cases, the air core area ratio is measured using a specially designed eight-leg ruler �Fig. 4�a��. The thickness of the vortex flow in the dropshaft is determined at the minimum air core level �Fig. 4�b��. The air core measurement device consists of eight tiny metal rods drilled into a 5-mm-thick transparent perspex disk of 15 mm diameter, at azimuthal intervals of 45°. On the rods are alternative markings of different colors at 5 mm in- tervals. The disk is screwed to a metal handle. In an experiment, the air core measurement device is lowered into the dropshaft and fixed at the throat of the air core with the approximate location. The markings on the eight legs, where the flow just cuts across, are recorded. The measurement is repeated a few times by sliding the device up and down. The air core area is then calculated by summing up the areas of the eight sectors of air core based on the eight measured flow thicknesses. The air core area measurement can be reproduced to within �5%. The diameter of each metal rod is 1.6 mm and the disturbance of the flow by the rods is negligible. Experimental Results and Analysis Flow Behavior in Tangential Intake There are two generic types of observed flow patterns: �1� inlet flow which drains smoothly and completely into the dropshaft and �2� flow with significant backup near the junction, caused by in- (a) Dropshaft Vortex flow surface Eight-leg ruler (b) Fig. 4. Air core area measurement: �a� eight-leg ruler; �b� air core area measurement at minimum air core level .135:164-174. D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. teraction of the inflow jet and the swirl in the dropshaft �Fig. 5�. Fig. 6 shows the observed characteristic flow features for different discharges of a model with Qf /Qc=0.49 �No. 1�. For small dis- charge �Fig. 6�a��, it can be seen that the flow is draining freely into the dropshaft; the water surface profile is smooth and flow is supercritical along the entire tapering section. As the discharge increases, however, a choking begins to occur at the junction. In the dropshaft, the vortex flow interacts with the inlet flow at the junction after 360° turning �Fig. 6�b��. The blockage of vortex flow in the dropshaft causes the supercritical flow in the tapering section to be transformed into a subcritical flow via a hydraulic jump before entering the dropshaft. As the hydraulic jump in- volves significant energy loss, the one-dimensional analysis is no longer valid. The free surface near the junction has a choppy appearance. In this case, there exist two controls in the intake: one at the approach flow channel and the other at the tapering section- dropshaft junction. The hydraulic jump in the tapering section can result in a fluctuating water level that is higher than that in the approach flow channel with the risk of undesirable overflow. With further increase in discharge the hydraulic jump becomes stronger and oscillates more vigorously, and moves upstream �Fig. 6�c��. Beyond a certain discharge, the entire approach flow channel flow becomes subcritical and the water surface is almost horizontal �Fig. 6�d��. Fig. 7�a� shows the measured free surface profiles for the dif- ferent discharges. The theoretical and measured depth–discharge relation, and the computed Froude number �F=V /�gy cos �� variation in the tapering inlet are shown in Figs. 7�b and c�, re- spectively. The free surface and F profiles show clearly: �1� for small discharges, a supercritical flow runs down the inlet and into the dropshaft, controlled by the critical flow at the upstream ap- proach flow channel; �2� the possibility of the existence of two controls both at the upstream channel and at the junction with the dropshaft; �3� for larger discharges, subcritical flow in the taper- ing channel occurs with critical flow at the junction. The measure- ment at this location is only indicative of the junction conditions as the most downstream depth measurement in the tapering inlet is taken at approximately 50 mm upstream of the dropshaft entry. From Fig. 7�b� the predicted control shift discharge �Qc� is somewhat higher than the observed value. More important, the fluctuating hydraulic jump occurs in the vortex inlet even for Q �Qc �Fig. 7�a and b�, Qc=18 L /s�. It can also be noted that when Q�Qc the depth changes more rapidly with discharge. Due to the flow backup and hydraulic jump, the measured depth is much greater than the theoretical prediction of the 1D model. In the theoretical prediction, Eq. �7�, the vertical drop z is taken as the Fig. 5. Backup flow in tapering and downward slope section of tan- gential intake JOU J. Hydraul. Eng. 2009 Fig. 6. Flow features in Model No. 1 �Qf =8.9 L /s, Qc=18.0 L /s, and Qf /Qc=0.49�: Q= �a� 4; �b� 13; �c� 15; and �d� 20 L /s RNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 169 .135:164-174. 300 D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. difference in elevation between the approach flow channel and the midpoint of the junction �point P in Fig. 1�, with z= �L + l /2�tan �, where l=x length between junction and center section of dropshaft. In contrast, Figs. 8 and 9 show the corresponding flow obser- vations and measurements for an alternative intake design with Qf /Qc=1.34 �No. 3�. As the discharge increases, the flow changes smoothly without a distinct hydraulic jump for all discharges. The water level in the tapering section remains lower than that in the approach flow channel. For small discharges, the tapering inlet flow is supercritical with the control at the approach channel. The flow drains freely into the dropshaft; no apparent obstruction at the junction is observed �Figs. 8�a and b��. As the discharge in- −400 −200 0 200 400 600 −300 −200 −100 0 100 200 x (mm) y (m m ) 4 L/s 8 L/s 10 L/s 12 L/s 14 L/s 16 L/s 17 L/s 20 L/s (a) 0 10 20 30 0 100 200 300 400 Q (L/s) y (m m ) Q c measured predicted (b) −400 −200 0 200 400 600 0 1 2 3 4 x (mm) F r 4 L/s 8 L/s 10 L/s 12 L/s 14 L/s 16 L/s 17 L/s 20 L/s (c) Fig. 7. Observed flow characteristics in Model No. 1 �Qf =8.9 L /s, Qc=18.0 L /s, and Qf /Qc=0.49�: �a� free surface profile; �b� mea- sured and predicted depth-discharge relation �Eq. �6��; and �c� Froude number profile �F=V /�gy cos �� 170 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 J. Hydraul. Eng. 2009 creases toward Qc�=6.6 L /s�, the water level of vortex flow in the dropshaft rises; the water level after turning around 360° rises slightly above the invert level �Fig. 8�c��. However, unlike the substantial blockage for the previous case, the height of vortex flow above the invert level remains minimal until the entire taper- ing section flow becomes subcritical �Fig. 8�d��. As shown in Fig. 9�a�, the overall free surface profiles are Fig. 8. Flow features in Model No. 3 �Qf =8.9 L /s, Qc=6.6 L /s, and Qf /Qc=1.34�: Q= �a� 2; �b� 4; �c� 7; and �d� Q=9 L /s .135:164-174. 200 Stable flow Hydraulic Jump D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. smooth; and the water level in the intake rises gradually with Q. The depth–discharge relation and the F profiles are indicative of a gradual transition from a supercritical flow �with upstream con- trol� into a subcritical flow �with downstream control at junction� �Figs. 9�b and c��. Further, as the energy dissipation is minimal in this design, the predicted depth–discharge relation is in good agreement with the observations �Fig. 9�b��. Smooth flow is ob- served for all discharges both smaller and larger than Qc. The sharp contrast in flow behavior of the aforementioned two intakes is explained by the free drainage condition. When Qf /Qc�1, the interference by the swirling flow in the dropshaft to the inflow at the junction is minimal. The flow in the tapering section drains freely into the dropshaft. When Qf /Qc�1, how- −300 −200 −100 0 100 200 −100 −50 0 50 100 150 x (mm) y (m m ) 2 L/s 4 L/s 6 L/s 7 L/s 7.5 L/s 9 L/s 11 L/s (a) 0 5 10 0 50 100 150 200 250 Q (L/s) y (m m ) Q c measured predicted (b) −300 −200 −100 0 100 200 0 1 2 3 4 x (mm) F r 2 L/s 4 L/s 6 L/s 7 L/s 7.5 L/s 9 L/s 11 L/s (c) Fig. 9. Observed flow characteristics in Model No. 3 �Qf =8.9 L /s, Qc=6.6, and Qf /Qc=1.34�: �a� free surface profile; �b� measured and predicted depth-discharge relation �Eq. �6��; and �c� Froude number profile �F=V /�gy cos �� JOU J. Hydraul. Eng. 2009 ever, the blockage of swirling flow in the dropshaft is substantial leading to the backwater effect via a hydraulic jump. In Table 1 the predicted and observed control shift discharges �Qc� are compared. In general, the prediction is better for designs with Qf /Qc�1, with an average error of around �25%. The small difference of observed and predicted Qc is believed to re- flect the inherent limitation of using a 1D model to account for three-dimensional flow features in the tapering and downward sloping inlet section, including the junction. The sharp reduction of channel width results in a nonuniform transverse depth distri- bution; the water depth at the tapering side rises substantially higher than the opposite side �not shown�. As the shock wave propagates downstream, the transverse difference in depth gradu- ally decreases. This uneven transverse water surface profile is found for both smooth and hydraulic jump cases. In addition, the assumed hydrostatic pressure approximation is violated in the rapidly varied flow at both transitions �from the approach to ta- pering channel and at junction�. The observed flow feature of 15 tangential intake designs are shown in Table 1 and Fig. 10 with respect to Qc and Qf. The flow feature of an intake is categorized as stable flow when the flow drains freely into the dropshaft keeping the smooth water surface profile. If a hydraulic jump is observed in the tapering section, the flow is regarded as unstable and labeled hydraulic jump case. The observed flow is in excellent agreement with the prediction by Eq. �15�. The stable flows are found where Qf�Qc �above the 45° line in Fig. 10�; the hydraulic jump is observed where Qf�Qc. The results clearly demonstrate that Eq. �15� can be used to de- termine the appropriate design of a tangential vortex intake. Air Core Fig. 11�a� shows a typical measured air core, which is asymmet- ric. The thickness of vortex flow is the largest near the junction; as the vortex swirls in the dropshaft and its vertical velocity is accelerated, the flow thickness decreases. The nonuniform flow thickness results in the noncircular and eccentric air core, which is the case with both smooth flow and hydraulic jump cases. The air core area ratio decreases with discharge. When Q�Qc, the decrease rate appears to flatten off significantly—in accordance with the prediction of the proposed free vortex model �Fig. 11�b��. Fig. 12 shows the comparison of predicted and measured air core area ratios ���; the data of Jain and Kennedy �1983� and the prediction of Jain �1984� are also shown. Of the four designs, one 0 10 20 30 0 10 20 30 Q c (L/s) Q f (L /s ) Q f > Q c stable flow Q f < Q c hydraulic jump Fig. 10. Flow features in tapering inlet with respect to Qc and Qf RNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 171 .135:164-174. 1 No. 3Air core area ratio = 44.3% D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. is a hydraulic jump case �No. 14, with Qf /Qc=0.36� and the other three are stable flow cases: No. 3, �Qf /Qc=1.34�, No. 6 �Qf /Qc =1.94�, and No. 7, �Qf /Qc=1.22�. The intake of Jain and Kennedy �1983� was reported as the stable case with Qf /Qc =1.44 �Table 1�. The measured air core areas for the different designs show some scatter, but a consistent trend. Despite the gross simplification of the complex problem �e.g., assumption of symmetrical air core�, Eq. �24� is well supported by the experi- mental data. The air core area ratio decreases gradually with the discharge; the measured air core is generally greater than pre- dicted. In general, our free vortex model predicts a larger air core than Jain’s model, which seems to be borne out by the data for ��0.4. For the three stable flow experiments, the measured � is greater than around 0.4. Smaller � are obtained for an experiment with unstable hydraulic jump, in which significant interaction of the inflow jet with swirling flow is expected �and key model assumptions break down�. For this unstable design, the data show that � can fall below the minimum requirement of 0.25 when the dimensionless flow parameter Fa exceeds about 1.2. There is scant data for small air core area ratios; more experiments are needed to discern the behavior of the air core for large discharges. It appears that a conservative design criterion can be based on a limiting dimensionless flow parameter of Fa =4�Q2e /g�3D6 cos4 ��1/3�1 / �1−e /D� 1. It can be shown that this minimum air core criterion translates to D�k�Q2 /g�1/5, where k depends on � and e /D �see later discussion�. −50 0 50 −60 −40 −20 0 20 40 60 x (mm) y (m m ) Flow → Air core (a) 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Q/Q c (obs) λ Measured data Prediction by free vortex theory (b) Fig. 11. Measured air core in dropshaft �No. 3, Qc=6.6 L /s�: �a� Q=11.3 L /s; �b� air core area ratio vs. normalized discharge 172 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 J. Hydraul. Eng. 2009 Design Guideline Basic Design Principles Based on this study, the hydraulic design of tangential vortex intakes can be based on several basic guidelines: 1. The design discharge is typically based on a hydrologic as- sessment of the site. For a given design discharge Qmax, the minimum dropshaft diameter can be sized from an empiri- cally based relation, D= �Q2 /g�1/5 �e.g., Vischer and Hager 1995�. A sufficiently small value of dimensionless junction width e /D is required to produce an effective swirling flow and air core of adequate size. On the other hand, e cannot be too small in order to satisfy other constraints. For a given bed slope angle �, Eq. �24� predicts that � decreases with e /D, but stays within a narrow range for �=10–40° �Fig. 13�. A minimum relative throat area of �=0.25 can be achieved with e /D�0.3. This is consistent with optimal designs de- veloped from model studies, with e /D=0.2–0.25 �Jain and Kennedy 1983; Jain 1984; Lee et al. 2006�. 2. This relation is also consistent with our free vortex model for the minimum air core area relation �Fig. 12 and Eq. �24��: 0 0.5 1 1.5 0 0.25 0.5 0.75 4(Q2 e /(g π3 D6 cos4β))1/3 /(1 − e/D) λ No. 6 No. 7 No. 14 J & K (1983) Jain (1984) (e/D = 0.25) Present study Fig. 12. Comparison of measured air core area ratio with theoretical prediction �Eq. �24�� 0 0.1 0.2 0.3 0.4 0.5 0 0.25 0.5 0.75 1 e/D λ β = 10° β = 20° β = 30° β = 40° Fig. 13. Variation of air core area ratio with relative junction width e /D at Q=�gD5 �based on Eq. �24�� .135:164-174. 4� Q2e �1/3 1 = 1 �25� periments. It is shown that a hydraulic jump with significant en-ergy loss may form in the tapering inlet channel, leading to D ow nl oa de d fro m a sc el ib ra ry .o rg b y Is ta nb ul U ni ve rs ite si on 0 6/ 19 /1 3. C op yr ig ht A SC E. F or p er so na l u se o nl y; al l r ig ht s r es er ve d. g�3D6 cos4 � �1 − e/D� D = k�Q2g � 1/5 = � 4 � �e/D�1/3 1 − e/D 1 cos4/3 � �3/5�Q2g � 1/5 �26� The dropshaft sizing parameter k depends on both � and e /D. Based on the discharge–dropshaft diameter relation of most existing vortex dropshaft �including scroll and tangen- tial type�, k=1–2 �Jain and Kennedy 1983�. Eq. �26� shows that for given e /D, k increases weakly with �. Over the range of �=10–40°, k=0.97–1.19 for e /D=0.2; and k =1.05–1.29 for e /D=0.25. For conservative design pur- poses, a value of k=1.2 can be adopted. 3. The control shift discharge Qc can be evaluated from Eq. �7�. It is recommended that the free drainage discharge condition should be satisfied to avoid unstable flow and hydraulic jump conditions which may result in overflow when Q�Qc Qc� Qf �27� In addition it is advisable to limit the discharge not to exceed the free drainage value. Qmax� Qf �28� For a typical situation, B is determined from Qmax and a design unit discharge. L can be determined considering the site con- straints. The vertical drop z is the most important factor affecting the intake capacity as Qc�z1.5 �Eq. �7��. Based on our experi- ments and the literature, typical ranges of dimensionless width, drop, and length �B ,z ,L� are: B /D 0.8–1.2, z /D 0.5–1.5, z /L 0.35–0.7 �which is equivalent to �=19–35°�. As B and e are determined by other considerations, � is fixed once L �or �� is determined. Example Application Qmax and B are given as 14.0 m3 /s and 2.5 m, respectively. Site constraints dictate that L should not exceed 10 m. • Design 1: �1� From Eq. �26�, D=1.2�Qmax2 /g�1/5; the dropshaft diameter is determined as 2.3 m, which corresponds to B /D =1.25. �2� e can be determined 0.25D, i.e., 0.58 m. �3� � =26.6° and L=10 m are initially tried, and z=L tan �=5 m. �4� From Eq. �7�, the control shift discharge is Qc =26.7 m3 /s. Eq. �14� gives Qf =15.2 m3 /s. As Qf�Qc, this tangential intake design will generate hydraulic jump in the intake. Redesigning is required. • Design 2: �1� As Qc of the previous design is larger than Qf by approximately 170%, z is reduced to 3.5 m, keeping D, B, and e unchanged. This time Qc=18.2 m3 /s. �2� As Qf is dependent on � given D and e, L is reduced to 5 m to increase � to 35°. �3� Qf of this design becomes 21.2 m3 /s; now that Qf�Qc, a hydraulic jump will not occur in the intake; Qf is sufficiently larger than Qmax as well. This design satisfies all the basic criteria and can be expected to give satisfactory performance. Conclusions The hydraulics of a tangential vortex intake is studied theoreti- cally and experimentally. The contributions of the present work is twofold. First, the flow characteristics in the tapering inlet for increasing discharges are elucidated through comprehensive ex- JOU J. Hydraul. Eng. 2009 undesirable overflow situations. Second, a free drainage discharge condition is developed as a function of discharge and tangential inlet characteristics. Based on a theoretical analysis, a heuristic criterion in terms of the ratio of a critical flow control shift dis- charge and the free draining discharge is proposed to discern smooth and stable flow from unstable fluctuating flow in the in- take structure. Experiments involving 15 different tangential in- take models are carried out, covering a wide range of key dimensionless ratios of approach flow channel width, the length, bottom and side slope angle, and drop height of the tapering chan- nel reach, junction width and dropshaft diameter. The predicted flow characteristics for different vortex inlet designs are in excel- lent agreement with the proposed criterion. For stable inlet flows with minimal energy loss, the flow in the tapering inlet is smooth for all discharges; the depth–discharge relation can be well pre- dicted by a 1D model. The proposed vortex inlet flow stability criterion provides a simple and general guideline for the hydraulic design of a tangential vortex intake. Video clips of stable and unstable flow designs can be viewed from http://www.aoe- water.hku.hk/vortex. Acknowledgments This research was engendered by a series of model studies of vortex flow intakes for the Drainage Service Department �DSD� of Hong Kong Special Administrative Region, China, and in part supported by the Hong Kong Research Grants Council �RGC HKU7143/06E�. The assistance of Mr. Edward Lai in the experi- mental investigation is gratefully acknowledged. This paper was prepared during a research visit supported by the Alexander von Humboldt Foundation. Appendix. Vortex Circulation The tangential inlet inflow to the vortex dropshaft imparts a “spin” to the swirling flow. By assuming a free vortex in the dropshaft �Fig. 2�, the horizontal tangential velocity is given by c = rvt �29� where c=vortex constant and r and vt=radial coordinate and tan- gential velocity in the horizontal plane �with respect to center of dropshaft�. The moment of momentum theorem can be applied to a control volume with a vertical inflow section at the inlet junc- tion with the dropshaft, and a horizontal outflow section in the dropshaft below the invert of tapering inlet channel at or below the throat of air core. By neglecting wall friction, the sum of the angular momentum fluxes across the control surface �cs� is given by � cs �rà V��V · dA� = 0 �30� where r and V=radial coordinate and velocity vectors respec- tively, and dA=outward normal vector from the control surface with magnitude equal to the differential surface area dA. As V ·dA=0 at the solid wall, the angular momentum flux of the inflow and outflow must balance. At the inflow section, a critical flow section is established. If we assume axisymmetry and that the circulation is mainly contributed by the x component of the RNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2009 / 173 .135:164-174. inflow velocity �Vx, assumed constant over the cross section�, the angular momentum flux of the inflow can be estimated to first order by � 0 e �Vx yc cos � dy*�Vx�a − y*� = Vx2 yc cos � �ae − e22 � = QVx�a − e2� �31� where y*=horizontal distance normal to straight wall at the junc- tion; Q=qe; yc=ycj �Eq. �2��; and Vx= �q /yc�cos �. Similarly, the angular momentum flux of the outflow is given by � b a �rVt��Vz2�rdr� = cVz��a2 − b2� = Qc �32� Equating Eqs. �31� and �32� results in an expression for the vortex circulation constant c c = Vx�a − e2� = �Qe g� 1/3 cos4/3 ��D2 − e2� �33� The vortex circulation is seen to depend on the discharge and the tangential inlet geometric parameters �e ,D ,��. 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