River hydrodynamicsInteraction with structure

Hosoda, Takashi

Kouchi, Y.

Puay, H. T.

Saif, A.

English

Hydraulic Engineering Repository

Conference Paper, Published VersionHosoda, Takashi; Saif, A.; Puay, H. T.; Kouchi, Y.Spatial water surface variations in open channel flowsdownstream of side disturbancesVerfügbar unter/Available at: https://hdl.handle.net/20.500.11970/99702Vorgeschlagene Zitierweise/Suggested citation:Hosoda, Takashi; Saif, A.; Puay, H. T.; Kouchi, Y. (2010): Spatial water surface variations inopen channel flows downstream of side disturbances. In: Dittrich, Andreas; Koll, Katinka;Aberle, Jochen; Geisenhainer, Peter (Hg.): River Flow 2010. Karlsruhe: Bundesanstalt fürWasserbau. S. 659-664.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. 1 INTRODUCTION  The spatial amplification of the sand bars generat-ed downstream of a point bar along a river bend is well known as the over-deepening phenomena of sand bars. Struiksma et al.(1985) formulated this phenomena mathematically, and derived the criti-cal conditions for the spatial amplification of sand bars. Blondeaux & Seminara (1985) derived the analytical solutions on the temporal change of point bars in a sinuous meandering channel. The resonance relation between sinuous channel and bars is included in the solutions. It is well known nowadays that the condition of the spatial amplifi-cation of bars by Struiksma et al. (1985) is coinci-dent with the resonance relation by Blondeaux & Seminara (1985). In this paper, we don’t deal with the spatial amplification of sand bars but the spa-tial variations of water surface in steady open channel flows downstream of an obstacle attached on the side wall of a flume.  It is firstly shown theoretically that using the linearized equations of 2-D shallow flows, the pe-riodic wavy patterns can exist for the supercritical flows condition (Froude number >1), but the am-plitude of periodic wavy patterns always atte-nuates downstream direction. The standing waves without attenuation can exist only for the case of zero friction factor. It is also shown that the atten-uation rates of periodic wavy patterns increase with the increase of Froude number.  Hosoda & Nishihama (2006) studied the re-sponse of water surface to a sinuous open channel and the flow behavior near the resonance condi-tion. Although there is no resonance condition in the case studied in this paper, it is pointed out that if the spatial variations are given on the uniform bottom, the flow can be amplified in the down-stream of side disturbances. Hydraulic experiments are carried out to verify the theoretical findings shown in this paper, changing the hydraulic conditions such as depth, bottom slope, etc. 2 THEORETICAL CONSIDERATIONS Referring to the coordinate system shown in Fig-ure 1, common plane 2-D shallow flow equations are given by Eqs.(1), (2) and (3). 0=∂∂+∂∂yvhxuh (1) hgxhgyuvxuu bxρτθ −=∂∂+∂∂+∂∂sin  (2) hyhgyvvxvubyρτ−=∂∂+∂∂+∂∂ (3) where ( )yx, : Cartesian coordinates, ( )vu, : ( )yx,  components of depth averaged velocity vectors, Spatial water surface variations in open channel flows downstream of side disturbances T. Hosoda, A. Saif & H. T. Puay Department of Urban Management, Graduate School of Engrg., Kyoto University, Kyoto, Japan Y. Kouchi Energia Economic and Technical Research Institute, Chugoku Electric Power Company, Higashi-Hiroshima, Japan ABSTRACT: This paper deals with the spatial variations of steady open channel flows downstream of an obstacle attached on the side wall of a flume. It is shown theoretically that using the linearized equations of 2-D shallow flows, periodic wavy patterns exist for supercritical flows (Froude number >1), but the amplitude of periodic wavy patterns always attenuates in the downstream direction. Standing waves can exist only for the case of zero friction factor. The attenuation rate increases with the increase of Froude number. These results are verified by hydraulic experiments carried out in this study. Keywords: Open channel flows, High velocity flow, Water surface profile, Instability River Flow 2010 - Dittrich, Koll, Aberle & Geisenhainer (eds) - © 2010 Bundesanstalt für Wasserbau ISBN 978-3-939230-00-7659h : depth, θ : bottom slope and ( )bybx ττ , : ( )yx, components of bottom shear stress vectors. For simplicity, the following formula with fric-tion factor, fc , is applied to evaluate bottom shear stress vectors. Figure 1 Coordinate system and explanation of symbols vvucuvuc fbyfbx 2222 , +=+= ρτρτ (4) Considering the deviation of depth and velocity from the uniform depth and uniform velocity, we can derive the common linearlized equations. The uniform depth and uniform velocity in the x  di-rection are given by the following equations: qUhUcgh f == 00200 ,sinθ  (5) where 0h : uniform depth, 0U : uniform velocity and q : unit width discharge. Using the following non-dimensional variables with prime, Eqs.(1), (2) and (3) can be trans-formed into the linearlized equations, Eqs.(6), (7) and (8). ')2/(,' yByLxx ==  '),'1(),'1( 000 vUvuUuhhh δδδ =+=+=  0'''''' =∂∂+∂∂+∂∂yvxhxu δβδδ  (6) )''2(''1''20hucxhFrxu f δδλδδ −−=∂∂+∂∂ (7) '''''20vcyhFrxv f δλδβδ −=∂∂+∂∂   (8) where the non-dimensional parameters, 0,, Frλβ  are defined as follows: 0000 ,,2/ ghUFrLhBL === λβ  where B : width of channel, L : wave length. From here, primes indicating non-dimensional variables are omitted for simplicity. Periodic standing wave solutions for small disturbances of depth and velocity components, vuh δδδ ,, can be written as: ( )xyah h ππδ 2cos2sin ⎟⎠⎞⎜⎝⎛=   (9) ( )uu xyau φππδ +⎟⎠⎞⎜⎝⎛= 2cos2sin     (10) ( )vv xyav φππδ +⎟⎠⎞⎜⎝⎛= 2cos2cos     (11) where hδ , uδ , vδ are small increment in h , u , v . Substituting Eqs. (9)-(10) into Eqs.(6)-(8) yields: 0cos)2/(sin2 =+ vvuu aa φπβφπ    (12a) 0sin)2/(2cos2 =+−− vvhuu aaa φπβπφπ    (12b) hfuufuuacaca)/(cos)/(2sin2λφλφπ+−=− (13a) uufhuuacaFraφλπφπsin)/(22)/1(cos2 20=−− (13b) vvfhvvacaFraφλπβφπcos)/()2/)(/(sin2 20−=+− (14a) vvfvv aca φλφπ sin)/(cos2 =−  (14b) It can be shown easily that Eqs.(12)-(14) have solutions with physical meaning only in the case of 0=fc . The solutions are given by Eqs.(15), (16) and (17). 20cos,0sinFraaa huuuu −== φφ   (15) 0cos,4sin20== vvhvv aFraa φβφ   (16) 161220β+=Fr   (17) Eq.(17) shows the relation between Froude number of flow and wave length of standing waves. Eq.(17) indicates that the standing waves exist under the condition of super critical flow. Introducing the spatial functions of amplitudes, )(),(),( xaxaxa vuh  in Eqs.(18), (19) and (20), the equations on )(),(),( xaxaxa vuh  can be derived as Eqs.(21a,b), (22a,b) and (23a,b). xθhz660( )xyxah h ππδ 2cos2sin)( ⎟⎠⎞⎜⎝⎛=   (18) ( )uu xyxau φππδ +⎟⎠⎞⎜⎝⎛= 2cos2sin)(   (19) ( )vv xyxav φππδ +⎟⎠⎞⎜⎝⎛= 2cos2cos)(   (20) 0cos2sin2cos=−+−vvhuuuuadxdaadxdaφπβφπφ (21a) 0sin22cos2sin=+−−−vvhuuuuaaadxdaφπβπφπφ (21b) hfuufhuuuuacacdxdaFradxdaλφλφπφ+−=+−cos21sin2cos20  (22a) uufhuuuuacaFradxdaφλπφπφsin221cos2sin20−=+−− (22b) vvfhvvvvacaFradxdaφλπβφπφcos2sin2cos20−=+− (23a) vvfvvvvacadxdaφλφπφsincos2sin=−− (23b)  We assume Eq.(24) as the mathematical expres-sion of )(xah , )(xau  and )(xav .  )exp()(),exp()(),exp()(xAxaxAxaxAxavvuuhhααα=== (24) Substituting Eq.(24) into Eqs. (21), (22) and (23), the basic relations on the spatial amplifica-tion/attenuation rate, α , and the phase lag be-tween depth variations and velocities can be de-rived as Eqs.(25), (26), (27) and (28). ,sin,cossinsincoscosuuuuuu QPQPφφφφ φφ ==  (25) vvvvvv QPQPφφφφ φφsinsincoscos sin,cos ==  (26) huhfhufuuAAFrAcAFrAcAP20220cos412πλαλαφ−⎟⎟⎠⎞⎜⎜⎝⎛ +−⎟⎟⎠⎞⎜⎜⎝⎛ += 222cos 42 uufuu AAcAQ πλαφ +⎟⎟⎠⎞⎜⎜⎝⎛ +=  ⎟⎟⎠⎞⎜⎜⎝⎛ +−⎟⎟⎠⎞⎜⎜⎝⎛ +−−=ufuhhfhuuAcAAFrAcAFrAPλαπλαπφ22122020sin uu QQ φφ sincos =  ⎟⎟⎠⎞⎜⎜⎝⎛ +−= vfvhv AcAAFrP λαπβφ 220cos  vhv AAFrP 220sin πβφ =  222sincos 4 vvfvvv AAcAQQ πλαφφ +⎟⎟⎠⎞⎜⎜⎝⎛ +==  22202204242πλαπλαλαα+⎟⎟⎠⎞⎜⎜⎝⎛ +−⎟⎟⎠⎞⎜⎜⎝⎛ +−⎟⎟⎠⎞⎜⎜⎝⎛ +fffcFrcFrc απλαλπλππ ++⎟⎟⎠⎞⎜⎜⎝⎛ +++2220422222fffccFrc (27) 0412222022 =+⎟⎟⎠⎞⎜⎜⎝⎛ +⎟⎟⎠⎞⎜⎜⎝⎛ +⎟⎠⎞⎜⎝⎛+πλαλαπβffccFr  2220204221πλαλαλαα+⎟⎟⎠⎞⎜⎜⎝⎛ +⎟⎟⎠⎞⎜⎜⎝⎛ ++⎟⎟⎠⎞⎜⎜⎝⎛ +−fffccFrcFr 661142422220220 −+⎟⎟⎠⎞⎜⎜⎝⎛ ++⎟⎟⎠⎞⎜⎜⎝⎛ +−⎟⎟⎠⎞⎜⎜⎝⎛ +−πλαπλαλααfffcFrcFrc (28) 0412222202=+⎟⎟⎠⎞⎜⎜⎝⎛ +⎟⎠⎞⎜⎝⎛−πλαπβfcFr  Substituting the following functional form, Eqs.(29) and (30) with λ/fc  into Eqs.(27) and (28), the coefficients in these equations can be de-termined as Eqs.(31) and (32). ⎟⎟⎠⎞⎜⎜⎝⎛+= λαfcaa 10  (29) ( ) ⎟⎟⎠⎞⎜⎜⎝⎛+−= λβ fcbFr 1202 116  (30) 1121,0202010 −+−==FrFraa  (31) 01 =b  (32) Eq.(31) indicates that the spatial disturbances always attenuate in the downstream direction of supercritical flows, and the attenuation rate de-creases with the increase of Froude number. Eq.(32) indicates that periodic disturbances can not exist in sub-critical flows. 3  HYDRAULIC EXPERIMENTS The laboratory tests were carried out to verify the results obtained by theoretical considerations. A schematic illustration of the experimental setup is shown in Figure 2. As shown in the figure, an obstacle with the shape of the function, )exp( 2BxAy −= , is at-tached at  350 cm from the upstream end at the left side wall of the flume. A plot of the obstacle shape is shown in Figure 3. The hydraulic experiments were conducted to examine the amplification or attenuation of water surface variations in the downstream of the ob-stacle. Water depth measurements were carried out using point-gauge instrument. 10 cases were performed under different hy-draulic conditions. 9 cases were carried under su-percritical flow condition and one case under sub-critical flow condition. The hydraulic variables for the laboratory tests are listed in Table 1.  Figure 4 shows photographs of flows in Run 1 and 4, while the contour maps of depth are shown in Figure 5.  The water surface variations along the both side walls are shown in Figure 6. It is pointed out that the amplitudes of depth variations attenuate in downstream direction for both cases as predicted theoretically in the former section, while we can not identify the magnitude of attenuation rates. It should be noted that since the depth distribution is anti-symmetric, it is necessary to consider the nonlinear effects in the theoretical analysis.  Figure 7 shows the relation between wave length and Froude number for all cases. The solid line in Figure 7 is the linear theory given by Eq.(17). Experimental data are in good agreement with the theoretical curve based on linear analysis, although the nonlinear effect seems to be domi-nant.    20cm 30cm 5cm 800cm Run numberDischarge (cm3/s) Upstreamwater depth (cm)Upstream average velocity (cm/s)Froude number at upstream Bed slope Water Temperature(℃) Flow type 1 2 3 4 5 6 7 8 9 10 6,400 10,900 5,950 7,230 11,410 6,620 11,110 6,200 11,200 7,200 1.74 2.73 1.45 3.15 3.88 2.08 2.96 2.33 3.28 4.46 122.6 133.1 136.8 76.5 98.0 106.1 125.1 88.7 113.8 53.8 2.97 2.57 3.63 1.38 1.59 2.35 2.32 1.89 2.01 0.814 1/34 1/34 1/13 1/156 1/156 1/49 1/49 1/67 1/67 1/326 19.0 13.5 19.2 18.6 13.2 13.6 13.6 13.2 13.2 12.9 SupercriticalSupercriticalSupercriticalSupercriticalSupercriticalSupercriticalSupercriticalSupercriticalSupercriticalSub-critical Table 1 Hydraulic variables of experiments Figure 2 Schematic illustration of flume 0123450 5 10 15 20y(cm)x(cm)Figure 3 Shape function of an obstacle 662  Figure 4 Water surface variation downstream the obstacle (a) Run 1 Figure 4 Water surface variation downstream the obstacle (b) Run 4 x(m) y(m) x(m) y(m) (a) Run 1 (b) Run 4 (m) (m) Figure 5 Contour maps of depth 663 4 CONCLUSIONS This paper describes the spatial variations of flow depth in steady open channel flows downstream of an obstacle attached on the side wall of a flume. It is shown theoretically that using the linearized equations of 2-D shallow flows, periodic wavy patterns exist for supercritical flows (Froude number >1), but the amplitude of periodic wavy patterns always attenuates downstream direction. The attenuation rate increases with the increase of Froude number.  These results are verified by hydraulic experi-ments carried out in this study. Since the meas-ured depth distributions show very anti-symmetric feature, the further investigation including the non-linear effects is necessary to clarify the gen-eration mechanism of anti-symmetric depth distri-butions. Numerical simulation will be also applied to reproduce the flows observed in laboratory tests. REFERENCES Struiksma, N., Olesen, K.W., Flokstra, C. and de Vriend, H.J.; Bed deformation in curved alluvial channels, J. of Hydraulic Research, Vol.23, pp.57-79, 1985. Blondeaux, P. and Seminara, G.; A unified bar-bend theory of river meanders, J. of Fluid Mech., Vol.157, pp.449-470, 1985. Hosoda, T. and Nishihama, M.; Fundamental characteristics of high velocity flows in a sinuous meandering channel in the vicinity of resonance, River Flow 2006, pp.209-218, 2006. 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3h(m)x(m)right sideleft side00.010.020.030.040.050.060.070 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3h(m)x(m)right sideleft side(a) Run 1 (b) Run 4 Figure 6 Water surface variations along side wall 0123450 2 4 6 8 10 12 14 16Fr0βFigure 7 Relation between wave length and Froude number 664

Spatial water surface variations in open channel flows downstream of side disturbances

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Spatial water surface variations in open channel flows downstream of side disturbances

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