Self-induced Unsteady Flow On Pooled Stepped Chutes

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510 Self-induced Unsteady Flow on Pooled Stepped Chutes Selbstinduzierte Abflussinstationaritäten bei Treppengerinnen mit eingetieften Stufen Jens Thorwarth, Jürgen Köngeter Abstract Pooled stepped chutes are used to increase the energy dissipation compared to stepped chutes and cascades with horizontal and flat steps. Despite that fact this geometry causes the formation of unsteady thus the inflow is constant. Based on hydraulic model test this paper presents data to calculate the parameters of the instable flow region (occurrence, wave heights and wave frequencies) from the geometric parameters (step height, depth of the pool) and discharge. Zusammenfassung Kaskaden oder treppenförmig ausgebildete Gerinne können mit zusätzlichen Endschwellen am Ende jeder Stufe versehen werden (→ eingetiefte Stufen). Diese Bauweise erhöht zwar Energieissipation, aber nach ausreichender Entwicklungslänge bilden sich instationäre Abflüsse in Form von schwallartigen Wellen aus. Diese Veröffentlichung beschreibt das Phänomen auf Grundlage hydraulischer Modellversuche. Es werden Aussagen zu Auftretensintervall, Wellenhöhen und -frequenzen sowie zur Entwicklungslänge gemacht. Damit kann mit diesem Beitrag überprüft werden, ob bei bestehenden oder projektierten Anlagen dieser Bauweise die Gefahr von Instabilitäten vorhanden ist. 1 Introduction During the last decades stepped spillways have become more popular to spill floodwater caused by their good energy dissipation and low cavitation risk. The hydraulic research community has focused on the investigation of the complex flow and provided first guidelines for the design of such hydraulic structures [1], [4]. By dint of endsills, so-called pooled steps, the macro-roughness and therewith the energy dissipation can be increased. This method is also used in steep mountain rivers to control the sediment transport. Furthermore, this geometry was frequently used to spill floodwater of dams built in the beginning of the 20th century (e. g. [5]). Anyway, in conjunction with pooled steps self-induced unsteady flow is reported [3], [5], which is focused by this article. 2 Physical Model and Measurement Technique The experiments are conducted using a 0.5 m wide and 12.0 m long channel with two different slopes of θ = 8.9° and θ = 14.6°. 22 horizontal steps with height of s = 0.05 m are equipped with weirs of a height between w = 0.01 m to 0.05 m (Δw = 0.01 m). The complete experimental setup including discharge measurement and definition sketches is described in [6]. 511 A video-based water level measurement procedure is used to measure time-depended flow depth and to analyse the flow patterns. This procedure can be used both in unaerated and aerated flow regions. An additional advantage is its ability to measure the water level of unsteady flow as well as the simultaneous detection of the water level of a nearly 1 m long section of the flume (depending from resolution and field of view). Basis for an accurate water level measurement are pictures with a high contrast between water body and the air above. To achieve this demand the inside flume is equipped with high illumination of 500 W halogen lamps and black light (UV-A, 365 nm) which causes a fluorescence of the fluorescein disodium salt added to the water. The camera is placed outside of the flume with an angle of the optical axes of 80° (camera looks upwards). This guarantees that only the water level which is in line with the flumes wall and not water packets behind are recorded. To compute and remove the image distortion caused by the non-rectangle view on the object and low focal length lenses a raster plate with black points on white background is recorded once before the measurement. Its raster points are automatically detected by segmentation of the image and morphological operations. The inverse distortion function is based on perspective projection and on the analytical plumb line method. The parameters of this function are computed by minimizing the accumulated leastsquares (using the Nelder-Mead Simplex Method) between its result (using the detected raster points as input) and a raster with constant grid spacing. Each image of the sequences taken on different position of the flume (Figure 1) is processed as follows: – Correction of non-uniform illumination by calculation and subtraction of an 500 - Pixel average image. – Detection of the correct threshold by analysing the grey scale histogram. – Segmentation of gray-scale images (8 Bit, gray → 1 Bit, black & white) using the threshold calculated before. Now the pixels with value 0 (black) constitute to the background while nonzero pixels (white) belong to the water body. – Filtering of small white objects (usually water droplets, reflections on the glass walls). Computation of the white area boundaries, the most upper points of the boundaries represent the water surface. – Calculation of the real coordinates in m using the inverse distortion function from the pixel coordinate of the water surface From this data time series of the water level are extracted at specific points (e. g. over the weirs). This is the basis for the calculation of mean water levels, significant wave heights and wave frequencies. The error of flow depth measurement with this technique is lower 2 mm (overestimation of the true value) in the unaerated flow regions, mainly caused by surface tension. In the aerated flow regions there is a continuous transition from water to air. To value the error of measurement of the video based measurement procedure the characteristic mixture depth h90 with local air concentration of C = 0.9 is computed both from measurements with resistivity probes [6] and this video based technique (90. percentile of all values of the water level time series). The 512 comparison shows an average overestimation of the water depth measured by video of 12% (~ 10 mm - 15 mm), seeing that the waterbody under the detected height (by video) also contains air. Figure 1: Calibrated image sequence of self-induced wave at step 18-20 (camera turned clockwise by channel slope), white line represents the detected water level 513 3 Interval of Occurrence of Unsteady Flow The variation of both specific discharge q and the dimensionless weir height w/s shows two independent regions of self-induced transient flow (Figure 2) for each investigated channel slope θ. At low weir heights (region 1) up to 0.2 ≤ w/s ≤ 0.6 unsteady flow is observed between a discharge of 0.004 m³/(m s) ≤ q ≤ 0.015 m³/(m s) for θ = 8.9° (0.002 m³/(m s) ≤ q ≤ 0.0076 m³/(m s) and 0.2 ≤ w/s ≤ 0.4 for θ = 14.6°). At bigger weirs (0.8 ≤ w/s ≤ 1.0) this instable flow region is not existent anymore but another instable region (region 2) appears at 0.037 m³/(m s) ≤ q ≤ 0.064 m³/(m s) for θ = 8.9° (0.014 m³/(m s) ≤ q ≤ 0.027 m³/(m s) and 0.6 ≤ w/s ≤ 1.0 for θ = 14.6°). All instable regions are below the skimming flow regime, i. e. nappe flow and transition flow (for the description of the flow regimes see [2]). 4 Mechanism of Wave Development There is evidence that this phenomenon can be explained by the theory of resonance. The frequency of the waves (Figure 2) in the lower part of the studied stepped chute can already be detected as small fluctuations of the water surface (or waves) downstream of the hydraulic jumps at the first steps. The resonant frequency is depending on the pool geometry and the discharge and water depths respectively. In the instable region 1 the further amplification is caused by longitudinal seiches, e.g. the small waves are reflected at the inner sides of the basin and are growing from step to step by the principle of superposition. When the frequencies of the small fluctuations in the first basins agree with the resonant frequency the amplification reaches its maximum. This happens when the discharge equals the black dashed line shown within each region on the Figure 2. The instable region 2 is caused by a different mechanism. . On flat stepped spillways there is a smooth transition between nappe flow regime and skimming flow regime: The hydraulic jump is moving downwards gradually. By contrast this process happens abruptly on pooled steps (for w/s > 0.6). If the throw length of the jets is longer than approximately the half of the step length the jets can be lifted up by the small fluctuations formed in the hydraulic jumps above. This causes that the hydraulic jump is “blowed out” suddenly. The released water volume of the roller is accumulated in the downstream basins by the same process. 5 Dynamic Water Depths Figure 3 shows the water depths (mean water depths +/- half of the significant wave height HS) against discharge and different weir heights. The significant wave height is the mean of biggest third of the wave and was calculated by the Zero-Down-Crossing Method usually used to compute sea side parameters. It is obvious that the minimum water height (hmean – HS/2) in the periodic and unsteady discharge region occurs at the maximum significant wave height (cp. Figure 2 and Figure 3). This happens for q = 0.044 m³/(m s) (θ = 8.9°) and for q = 0.019 m³/(m s) (θ = 14.6°). Above this instable region there is still remaining a large wave height but these waves are non-periodic and mainly caused by splashing water, i. e. they do not fill the complete channel width. 514 The length of the wave development depends from the geometry (w/s) and the specific discharge q. For instance the significant wave height is 28 mm at step 5 for θ = 8.9° (47 mm at step 5 for θ = 14.6°) and increases to a mean of 100 mm at steps 20/21 for θ = 8.9° (63 mm for θ = 14.6° at steps 24/25). The data are valid for the discharges specified above and w/s = 0.8. 6 Conclusion The experiments show that pooled stepped chutes with different depths of pools cause the development of unsteady and periodic flow. In hydraulic model test waves with frequencies between 0.11 Hz and 0.90 Hz is observed, while the strongest instabilities occur with a frequency of ≈ 0.25 Hz. The formation of this unsteady flow can be explained by resonance: Small fluctuations caused by hydraulic jumps are amplified when they agree with the resonant frequency of the oscillating water body in the pools. The wave height can increase the mean water level up to a factor of 2. In conjunction with the hydrodynamic forces of the braking waves that can cause a hazardous underestimation in the dimensioning of such hydraulic structures. To estimate the occurrence of this self-induced unsteady flow in design the presented data can be scaled using the Froude's law, while it is not recommended to exceed a scaling factor of 1:10. Figure 2: Regions of unsteady flow, discharge 0 < q < 0.07 m³/(m s), dimensionless weir height 0 < w/s < 1.0, wave frequency 515 Figure 3: Mean water levels +/- significant wave height HS for different pool geometries of w/s = 0.6, w/s = 0.8, w/s = 1.0 and different channel slopes θ = 8.9° (step 21, x = 4949 mm) and θ = 14.6° (step 25, x = 6463.7 mm) Notation C hmean HS q s w w/s θ air concentration / void fraction [-] mean waterlevel [m] significant wave height [m] specific discharge [m³/(m s)] step height [m] weir height / depth of the pooled step [m] dimensionless weir height [-] channel slope [°] Literature [1] Boes, R. M.; Hager, W. H. (2003): Hydraulic Design of Stepped Spillways. In: Journal of Hydraulic Engineering, Vol. 129, No. 9, pp. 671-679. - ISSN 0733-9429 [2] Chanson, H. (2002): The Hydraulics of Stepped Chutes and Spillways. Lisse: Balkema. ISBN 90-5809-352-2 516 [3] Ganz, T. F. (2003): Entstehung und Entwicklung von Abflussinstabilitaeten bei Absturztreppen - Modellanalyse ueber den Einfluss von Gerinnegeometrie, Sohlrauheit und Suspensionsgehalt. Innsbruck: Leopold-Franzens-Universität (Dissertation). [4] Ohtsu, I.; Yasuda, Y.; Takahashi, M. (2004): Flow Characteristics of Skimming Flows in Stepped Channels. In: Journal of Hydraulic Engineering, Vol. 130, No. 9, pp. 860-869. ISSN 0733-9429; http:\\dx.doi.org\10.1061\(ASCE)0733-9429(2004)130:9(860) [5] Thorwarth, J.; Klein, P. (2005): Die getreppte Schussrinne der Sorpetalsperre Modellversuch zur Untersuchung von Abflussinstabilitäten. In: Energie und Wasserkraft zum 100. Todestag von Otto Intze: 35. IWASA, Internationales Wasserbau-Symposium Aachen 2005 / Ed. by J. KöngeterAachen: Shaker (Mitteilungen / Lehrstuhl und Institut für Wasserbau und Wasserwirtschaft, RWTH Aachen; 142), pp. 107-130. ISBN 3-8322-4286-4 [6] Thorwarth, J.; Köngeter, J. (2006): Physical Model Test on a Stepped Chute with Pooled Steps - Investigation of Flow Resistance and Flow Instabilities. In: Recent Developments on Hydraulic Structures: From Hybrid Modeling to Operation and Repairs ; Cuidad Guayana, Venezuela, October 2006 [International Symposium on Hydraulic Structures]; Ed. by Arturo Marcano [et al.] Caracas, Venezuela: Venezuelan Society of Hydraulic Engineering, pp. 477-486. - ISBN 980-12-2177-1 Authors’ Names and Affiliation Dipl-Ing. Jens Thorwarth Institute of Hydraulic Engineering and Water Resources Management Kreuzherrenstraße 7 52056 Aachen [email protected] Univ.-Prof. Jürgen Köngeter Institute of Hydraulic Engineering and Water Resources Management Mies-van-der-Rohe-Straße 1 52056 Aachen [email protected]