NUMERICAL SIMULATION OF 3D TURBULENT FLOW STRUCTURES AROUND AN ATTRACTING GROIN WITH LOCAL SCOUR

Transcription

1 NUMERICAL SIMULATION OF 3D TURBULENT FLOW STRUCTURES AROUND AN ATTRACTING GROIN WITH LOCAL SCOUR ICHIRO KIMURA Department of Cvl and Envronmental Engneerng, Matsue Natonal College of Technology 4-4 Nsh-kuma, Matsue , Japan TAISUKE ISHIGAKI Dsaster Preventon Research Insttute, Kyoto Unversty Fushm, Kyoto 6-835, Japan MASANORI HIROE West Japan Ralway Company 4-4, Shbata -chome Kta-ku, Osaka , Japan Predcton of flow structures around an attractng gron, of whch head s shfted toward downstream, s mportant because such gron has a dsadvantage n a submerged stuaton, namely, the flow over the gron attacks a bank and erodes t. In ths paper, 3D turbulent flow structures around an attractng gron wth scour hole are studed numercally. The basc equatons on generalzed curvlnear movable coordnate are used to consder the complex topography and free surface elevaton. A refned non-lnear k-ε model s adopted as a turbulence model n order to reproduce secondary currents of second knd and a vortex formatons. The computatonal condtons are same as those of the laboratory tests by Ishgak et al (003). The numercal results show that the fundamental turbulence structres wth a crcular flow, whch may be a man trgger of the local scour, can be reasonably predcted by the present model. Introducton In recent years, grons are set not only for the protecton of rverbank eroson but also for the envronmental preservaton n Japan. Grons have a functon to form a vared bed confguraton as well as a functon to deflect a flow and reduce velocty near the rverbank. The recent grons are usually set perpendcular to a rverbank, whch s called a deflectng gron. On the other hand, many old grons n Japan were set at angles to a rverbank. The gron whch head s shfted toward downstream s called attractng gron and s not commonly used n a submerged stuaton because a flow over the gron attacks a bank and erodes t. Ishgak et al (003) studed the flow structures around a old attractng gron bult about 400 years ago n Kyoto Prefecture and ponted out that the scour hole depends on the hydraulc condton and that there s a hydraulc condton for preventng severe bank eroson. They also referred to the possblty that the scour hole caused by the flow around an attractng gron functons as a botope. Those results Work partally supported by Grant n Ad for Scentfc Research, JSPS, No

2 ndcate that the attractng gron can be useful both for flood control and envronment under certan hydraulc condtons. Therefore, t s very mportant to predct the flow structures and bed confguraton around an attractng gron. A numercal smulaton wth RANS type modelng for turbulence has been recognzed as a powerful tool to predct detaled 3D flow structures around bluff bodes. Kmura and Hosoda (003) proposed a modfed non-lnear k-ε model through the consderaton of realzablty and appled t to the flows around a rectangular cylnder. Kmura et al (00, 003) modfed the same turbulence model to a generalzed curvlnear form and appled t to the flow around upstream / downstream nclned sub-merged grons. Those studes showed that the nonlnear k-ε model could predct not only tme-mean flow structures but also unsteady flow structures such as vortex sheddng from the tp of grons wth practcal accuracy. In ths study, characterstcs of 3D turbulent flow structures around an attractng submerged gron wth a scour hole are numercally nvestgated. A generalzed curvlnear movable coordnate s employed to smulate the free surface oscllaton and take nto account the complex topography. The computatons were performed under the condtons of the laboratory tests performed by Ishgak et al (003). The flow features around an attractng gron are dscussed through the comparson of computatonal and expermental results. Computatonal Model.. Governng Equatons The Reynolds averaged 3D flow equatons wth contravarant components of velocty vectors on a generalzed curvlnear movable coordnate system are used as governng equatons n ths study. The governng equatons are descrbed as V t + ε + t V g g = 0 [ V V W )] + V W + V W = F g p + [ v v ] + ν e ( () ρ k + t [ ε V W )] l D t ( + k W = g v v V + + l ε ν g k (3) σ k [ k V W )] ε l ε D t ( + ε W = Cε gl v v V Cε + + ν g ε (4) k k σ k where ξ I = generalzed curvlnear coordnate, t = tme, V = contravarant component of the velocty vector of flows, W = contravarant component of the velocty vector of grd moton, p = pressure, ν = molecular dynamc vscosty, ρ = densty of water, k = turbulent energy, ε = turbulent energy dsspaton rate, g and g = covarant and ()

3 3 contravatant component of metrc tensor, g = det(g ) and F = contravarant component of gravty acceleraton. ndcates a covarant dfferental, for nstance, A k k A = + A k g g g x x k p k k km m Γ, Γ m = = g + = m p (5) where Γ k s a Chrstoffel symbol... Turbulence Model To calculate a complex turbulent flow wth separaton and vortex sheddng, a nd-order non-lnear k-ε model by Kmura and Hosoda (003) s adopted as a turbulence model. The consttutve equatons of the model are descrbed as follows. s k k v' v' = Dt S kδ s g Dt [ Q + Q + 3Q3 ], D Cµ 3 ε t = ε (6) l β l Q = S glω + S gβlω (7) l k mβ l l k mβ l Q = S g ls S g ms gβ kδl g, Q3 = Ω g lω Ω g mω gβ kδl g 3 3 (8) S = g V + g V, Ω = g V g V (9) The model coeffcents are not constants but functons of the stran parameter S and the rotaton parameter Ω. In ths study, all the coeffcents are gven as functons wth one varable M for smplcty as follows (Kmura and Hosoda, 003). = 0.3 C µ mn 0.09, M, = M 0.35 f, = f, f M 3 = M (0) f = M + 0.0M, M = max[ S, Ω ], k β k β S = S gs gβ, Ω = Ω g ε Ω gβ () ε In above equatons, - 3 were adusted through the consderaton of the dstrbuton of turbulent ntenstes n a smple shear flow compared wth the prevous expermental results. C µ [Μ] was tuned to satsfy the realzablty n a smple shear flow and sngular ponts n both D and 3D flow felds..3. Outlne of Numercal Method The dfferental equatons governng the mean-veloctes and the turbulence feld are solved wth the fnte volume method on full-staggered grd system. The metrc tensors and the Crstoffel symbols are defned only at grd ponts to save computer memory and the values at other postons are nterpolated at each computatonal step. QUICK scheme s appled to the convecton terms and the central dfferencng s used for the dffuson terms n the momentum equatons. The hybrd central upwnd scheme s appled to the k and ε equatons for the computatonal stablty. Adams-

4 4 80mm 0 y 65º x 88mm sand 54mm Fgure. Attractng gron used n the laboratory test by Ishgak et al (003) (left:plan vew, rght:vertcal vew). Table Hydraulc parameters n the laboratlry test (Case A5) by Ishgak et al (003). H/h H (cm) Q (l/s) u /u c º 00mm h H Fgure. Bed confguraton n Case-A5 n the laboratory test by Ishgak et al (003). Bashforth scheme wth second-order accuracy n tme s used for tme ntegraton n each equaton. The basc equatons are dscretzed as fully explct forms and are solved successvely along the tme axs step by step. The pressure feld s solved usng teratve procedure at each tme step usng SOLA algorthm..4. Boundary and Intal Condtons Snce the present turbulence model s a hgh Reynolds number type, the wall functon approach s appled as the wall boundary condtons for k and ε. The wall frcton s evaluated by the log-law. At the downstream end of the computatonal doman, the longtudnal gradents of all varables are assumed to be zero. At the boundary nlet, the level of k s chosen to be (0.0U 0 ) (U 0 = averaged velocty). The value of ε at the nlet s determned from the value of k at the nlet by specfyng the rato D t /ν = 0. The free surface moton s solved by a smple relaton n equaton () snce the contravarant components of the velocty vector are used n the basc equatons. h = 3 g33 V t () where t = tme ncrement and h = surface elevaton durng t. To consder the rapd attenuaton of turbulent ntenstes n the depth-wse drecton near the free surface, the eddy vscosty s multpled by the followng dumpng functon (Hosoda, 990).

5 5 plan vew vertcal vew Fgure 3. Plan and vertcal vews of numercal grd at y = 0cm (8-layers model). ( h y) ε s f exp s = B, ( = 0) 3 / B (3) ks where sub-s ndcates the value at the surface. ε at the surface s evaluated by the followng formula to calculate the secondary currents of nd knd (Sugyama, 995). 3/ 4 3/ Cµ 0 ks ε s =, ( Cµ 0 = 0.09) (4) 0.4 y s At the begnnng of the calculaton, U (= velocty n the longtudnal drecton (xdrecton)) = U 0 (=averaged bulk velocty), V (= velocty n the transverse drecton (ydrecton) = 0, k = k n and ε = ε n (k n and ε n are the values of k and ε at the nlet boundary) are specfed over the whole computatonal doman. 3 3D Computaton of Flows around an Attractng Gron 3.. Computatonal Condtons Computatons are performed under the condtons of the laboratory tests by Ishgak et al (003). The experment was performed n a 0 m long, 0.9 m wde and 0.3 m deep straght flume. The mddle part of the flume was made of movable bed flled wth fne sand of whch the mean dameter was 0.6mm. Fg. shows the plan and cross-sectonal vew of the attractng gron model. The gron model was set on the rght hand sde bank of the expermental flume. The experment was performed under 5 dfferent submerged and non-submerged hydraulc condtons. The computaton was performed under the condtons of Case A5 (submerged case). The hydraulc parameters n Case A5 were lsted n Table. Fg. shows the bed topography n Case A5 at hour after the begnnng of the experment. A scour hole s formed at the downstream area of the gron. The computaton was performed under the fxed bed condton wth the scour hole. Fg. 3 shows plan and vertcal vews of the computatonal grd. The grd n the x-y plane was made usng geometrc seres. The number of the grd n a horzontal plane s

6 6 Fgure 4. Cross-sectonal flow patterns at x = 75, 00, 50 cm (8-layers model). Fgure 5. Cross-sectonal flow patterns at x = 75, 00, 50 cm (6-layers model). 80 (x-drecton) 50 (y-drecton). The grd n the z-drecton was made by dvdng the local depth nto layers wth equal thckness. To consder effects of a grd sze, two grds wth dfferent layer number (8-layers and 6-layers) were tested. The vertcal vew of the grd (8-layers model) n the x-z plane at y = 0 cm s shown n Fg Flow Patterns n Cross Sectons Fg. 4 shows the cross-sectonal flow patterns at x = 75, 00 and 50cm n the computatonal result (8-layers grd). The tme of the result s at seconds after the ntal condtons. Note that the scale of the vertcal drecton s enlarged n these fgures. The characterstc flow of attractng gron, namely, a strong flow toward the rverbank can be seen at x = 75 and 00cm. The flow tends toward the downward drecton behnd the gron and goes nto the sour hole. The marked flows are lkely to be a man cause of formatons of the deep scour hole. The flow at x = 50cm forms two vortces n the clockwse drecton. Fg.5 are the computatonal results n cross-sectons at x = 75, 00 and 50cm wth 6-layers grd. The flow patterns are smlar to those n Fg.4. Therefore, the grd wth 8-layers n a vertcal drecton s lkely to be enough to reproduce fundamental flow structures. However, slght dfference can be seen between two results,.e., the flow separaton behnd the gron s only captured by the 6-layers grd Flow Patterns n Vertcal and Horzontal Sectons Fg.6 (a) and (b) are computatonal velocty vectors wth the 6-layers grd n vertcal longtudnal sectons at y = and y = 0cm, respectvely. These fgures ndcate that an upward flow s domnant near the sde wall (y=cm) at the downstream regon of the

7 7 (a) y = cm (b) y = 0cm Fgure 6. Flow patterns n vertcal sectons at y =, 0 cm (6-layers model). (a) mddle layer (b) bottom layer Fgure 7. Plan vews of flow patterns at mddle and bottom layers (6-layers model). gron, whle a downward flow s domnant at the secton of y = 0cm. The flow separaton s not generated behnd the gron n both sectons. Fg.7 (a) and (b) show the plan vew of computatonal flow patterns wth the 6- layer grd at the mddle layer and at the bottom layer, respectvely. The flow from the man channel toward the rverbank can be seen ust downstream the gron at the mddle layer. The flow toward the rverbank also can be seen over the gron at the bottom layer. The computatonal results demonstrate agan the rsk of attractng grons to erode the rverbank. The flow pattern near the bed at the downstream regon of the gron s much dsturbed because of the complex topography of the scour hole. Fg.8 shows the comparson of surface-velocty dstrbutons n the longtudnal drecton n the expermental and numercal results. In the expermental result, the velocty profle has an nflecton pont at the downstream area of the gron because the flow over the gron s decelerated near the head of the gron. The computaton could capture the profle wth an nflecton pont though the profle s damped more rapdly than the expermental result.

8 8 (a) Experment (b) Computaton Fgure 8. Comparson of stream-wse velocty dstrbutons at a surface. 4 Conclusons It s mportant that hydraulc engneers have to recognze the rsk of attractng gron under a submerged condton and predct flow structures n detal n plannng such gron. In ths paper, 3D turbulent flow structures around an attractng gron wth a scour hole were studed numercally. A modfed non-lnear k-ε model, whch was tuned consderng the realzablty, was adopted as a turbulence model. A generalzed curvlnear movable coordnate was used to consder the complcated bed topography and water surface elevaton. Computatons were performed under the condtons of the laboratory test by Ishgak et al (003). The computatonal results ndcated that the present model could capture the fundamental aspects of 3D flow features, such as a flow whch attacks a rverbank at the downstream regon of an attractng gron. References Gatsk, T.B. and Spezale, C.G.(993). On Explct Algebrac Stress Models for Complex Turbulent Flows. J. Flud Mech.,54, Hosoda, T., (990). Ph.D. Thess, Kyoto Unversty (n Japanese). Ishgak, T., Ueno, T. and Tanaka, N. (003). Tradtonal Counter Measures for Flood n Kyoto Dstrct () Expermental Study on Old Attractng Gron n Kameoka. Annuals of Dsas. Prev. Res. Inst., Kyoto Unv., 46(B) (n Japanese). Kmura, I. and Hosoda, T.(003). A Non-lnear k-ε Model wth Realzablty for Predcton of Flows around Bluff Bodes. Int. J. Numer. Meth. Fluds, 4, Kmura, I., Hosoda, T. and Onda, S. (00). Predcton of 3D Flow Structures Around Skewed Spur Dkes by Means of a Non-lnear k-ε Model." Rver Flow 00 (eds. D.Bousmar and Y. Zech), Balkema,, Kmura, I., Hosoda, T., Onda, S. and Tomnaga, A. (003). 3D Numercal Analyss of Unsteady Flow Structures around Inclned Spur Dkes by Means of a Non-Lnear k-ε Model. Proc. of the Int. Symp. on Shallow Flows, Delft, The Netherlands, Part III, 05-. Sugyama, H., Akyama, M. and Matsubara, T. (995). Numercal smulaton of compound open channel flow on turbulence wth a Reynolds stress model. Journal of Hydraulc, Coastal and Envronmental Engneerng, 55 / II-3: (n Japanese).