EXPERIMENTAL INVESTIGATION ON LAMINAR HIGHLY CONCENTRATED FLOW MODELED BY A PLASTIC LAW Session 5

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1 EXPERIMENTAL INVESTIGATION ON LAMINAR HIGHLY CONCENTRATED FLOW MODELED BY A PLASTIC LAW Sessio 5 Sergio Brambilla, Ramo Pacheco, Fabrizio Sala ENEL.HYDRO S.p.a. Polo Idraulico Strutturale, Milao; Seior Hydraulic Egieer; Studet, Civil Egieerig, Politecico di Milao. SUMMARY A lamiar viscoplastic model Herschel-Bulkley for a steady uiform flow ad gradually varig flow of water ad clay at high cocetratio is preseted here. The model was aalysed experimetally at ENEL.HYDRO Polo Idraulico Strutturale laboratory of Mila by meas of a 15m log, 0.50m wide plexiglass walled laboratory flume with adjustable bottom slope. Rheological measuremets o samples of mud that were carried out promptly after the flume experimets were used to establish the variatio of its yield stress ad plastic viscosity with cocetratio of solid particles. A iterestig fidig was obtaied i the experimets preseted here: there is ot a cotiuous icrease i velocity with depth of flow as stated by Chezy s formula for the whole rage of cocetratio used. The lamiar viscoplastic model Herschel-Bulkey is able to reproduce the shape of the discharge vs. flow depth curve ad give some isight about the dyamical behaviour of the flow i coditio of steady flow. 1. INTRODUCTION It is well kow that mud ad debris flows ca cause devastatio i moutaious regios throughout the world where soils o steep slopes are saturated by heavy rais or sow melt. Much of the loss due to mud ad debris flow could be avoided if the path of probable flows could be predicted. Also, the desig of cotrol structures would be greatly improved by a ability to simulate such flows (Wright ad Kroe, 1987). May models have bee developed to determie the mud ad debris flows behavior, ad most of the them are based upo a costitutive equatio represetig the solid-water rheology. However, a theoretical model by itself is ot eough to describe ad explai the mud ad debris flow pheomea but it is ecessary to have experimetal ad field data to improve our kowledge of this kid of flows. The objective of this paper is to preset a theoretical ad experimetal ivestigatio based upo a Herschel-Bulkley model for the simulatio of lamiar ad viscoplastic free surface flows. We first examie the problem of a uiform steady mud flow. The model developed uses depth-averaged values for a uiform steady mud-fluid flow dow a iclied chael, i order to be able to explai some of the experimetal fidigs. Furthermore, from the equatios of the viscoplastic fluid a estimatio of the gradually ad spatially varied steady flow is also preseted. 2. EXPERIMENTAL SET-UP Experimets of steady, uiform flow ad gradually varied flow dow a iclied chael, of water- clay mixtures were coducted i a 15m log, 50cm wide chael. The chute agle was

1,93%. Clay particles with d 50 10µ m ad desity of 2650kg/m 3 were used. The grai size distributio is preseted i Figure 1.

2 1,93%. Clay particles with d 50 10µ m ad desity of 2650kg/m 3 were used. The grai size distributio is preseted i Figure clay silt fie sad coarse sad Weight percetage [%] Diameter [ µm] Figure 1. Grai size distributio of the atural fie mud suspesio. The chael bottom was i PVC. The mud flow was supplied by a pump located at the upper ed of the chael, it esured the steady circulatio. The mass flow rates were measured by a flow meter. The mixture clay-water was mixed i a power mixer for five hours to assure a uiform ad homogeeous mud mixture. The uiformity of the flow depth was cofirmed by markig the flow depth alog the chael side wall, which was clear ad allowed for direct visual observatio. The levels were recorded alog of the flume. Figure 2. Experimetal istallatio ENEL.HYDRO Hydraulic Laboratory of Mila-Italy.

3 3. RHEOLOGY The debris ad mud flows have differet characteristics depedig o the grai size distributio, the cocetratio of sedimet, ad the magitude of the flow. Previous rheological studies cocerig cocetrated mud suspesios have show that these materials are o-newtoia, viscoplastic fluids. The mai feature of these fluids is their yield stress which might be defied as the miimum shear stress to overcome for flow to take place. The yield stress of a mud or debris suspesio icreases expoetially with its solid cocetratio. Because of their high apparet viscosity, atural flows of cocetrated mud suspesios are lamiar. The rheological measuremets showed that water-clay mixtures are a visco-elastic fluid. The fluid respose to a applied shear rate it is deoted total stress. The shear depedecy of the mud viscosity ad yield stregth was visible i all the experimets. Due to shear depedecy of the rheological properties of mud a Herschel-Bulkley model have bee adopted sice it appears to be i keepig with the rheometrical measuremets i a very large, shear rate rage. This model expresses as follows: du du τ = τ c + k whe 0 ; (1) dy dy du τ τ c whe = 0. (2) dy where τ is the total shear stress magitude, du/dy the shear rate magitude ad τ c, k ad are fluid parameters. I this experimetal ivestigatio may geerally be take as 2/3 with water-clay mixtures with volume cocetratio Φ betwee 10 ad 19% experimetal results Herschel-Bulkley model Shear stress τ[pa] Φ =18,8% (21 C ) = 2/3 τ c = 19,013Pa k = 0,174Pa.s 2/ Shear rate du/dy[s -1 ] Figure 3. Flow curve of clay-water suspesio with 18.8% solid fractio.

4 4. UNIFORM LAMINAR FLOW Figure 4. Free surface flow o a iclied plae. Figure 4 presets a idealized sketch of free surface flows modeled as a yield visco plasticfluid. The free surface flows of fluids whose behaviour follows the model expressed by Herschel-Bulkley model are basically govered by: y ( ) y q = u = p h hs dy u 0 p h hs (3) + hs 2 1 U = q h hs = u p 1 (4) h where U [m/s] is the mea flow velocity, h [m] the flow depth, q [l/ms] the discharge by uit legth ad u p [m/s] the velocity of the plug flow. Figures 5a-5b show a tred i the values of the mea velocity agaist flow depth very differet to that of pure water. The icrease of depth is very little with a icrease i the flow velocity. This is better observed where the results obtaied are preseted. This behaviour is differet to that of the water flow typically represeted by the Chezy or Maig equatio i the case of turbulet flow or i the case of lamiar flow. This criteria could be useful for the desig of cotrol works for this kid of flows. Uiform depth h [m] experimetal results Herschel-Bulkley model Chézy equatio Φ =13,5% τc =10,177Pa k =0,117Pa.s =2/ Discharge by uit leght q[l/s/m]

5 Uiform depth h [m] experimetal results Herschel-Bulkley model Chézy equatio Φ =15,6% τc =13,318Pa k =0,143Pa.s =2/ Discharge by uit leght q[l/s/m] Figure 5a-5b. Stage-discharge relatioship at various solid fractio. 5. FREE SURFACE PROFILE IN STEADY FLOW We cosider here the case of a steady o-uiform flow i a ope chael. Applicatio of mometum theorem to the cotrol volume idicated i Figure 4. ad assumig gradually varied flow leads to: dh dx ( + 1) ( 2 1) 1 + q 1 τc + k ρgh ta θ ( h h p ) [( 1) h h p] ta + + = θ (5) 2 1 Fr h p () x = τc dh ρg cosθ ta θ dx (6) where τ c = yield stress, k = Herschel-Bukley fluid behaviour parameter, = 2 3, ρ = specific mass, u p = velocity of plug, h p (x) = depth of plug alog the chael, h p = depth of plug i uiform flow, θ = bed slope, Fr = Froude umber. The theoretical predictios usig equatio (5) were compared with experimetal results correspodig to rectagular chael flows ad showed a rather good global agreemet (see Figures 6a-6b). Uder the steady coditios, the depths of plug were recorded alog the flume. The measuremets have bee obtaied whe τ = τc ad the discharge was carried out from q steady to q ull. The mathematical model based upo the costitutive equatio Herschel Bulkley shows good capabilities to predict plug-flow of the water-clay mixtures i gradually varied flow.

6 0.075 flow to chael ed Flow depth h [m] Φ =12% =4,789Pa τc k =0,072Pa.s =2/3 water surface profile plug flow profile water profile (experimetal results) plug flow profile (experimetal results) x [m] flow to chael ed Flow depth h [m] Φ =10,4% =3,326Pa τc k =0,062Pa.s =2/3 water surface profile plug flow profile water profile (experimetal results) plug profile (experimetal results) x [m] Figure 6a-6b. Compariso betwee mathematical model ad experimetal results. 6. CONCLUSIONS A lamiar viscoplastic Herschel-Bulkley model for a steady uiform flow ad gradually varied flow of water ad clay at high cocetratio has bee verified here. The model shows good capabilities to predict flows of the studied water-clay mixtures. The model is able to reproduce the shape of the relatioship betwee mea velocity ad depth of flow for the whole rage of cocetratios. The mai source of error i ay flow predictio (i atural) comes from rheometrical measuremets eve whe these latter are doe carefully. I these coditios this problem appears as a critical poit for the predictio of free surface flow characteristics especially i fields where it is hard to carry out precise rheometrical experimets. 7. BIBLIOGRAPHY 1. V.G. Wright ad R.B. Kroe Laboratory ad umerical study of mud ad debris flow. AIRH Cogress, Lausae, 1987; 2. Z. Wa ad Z. Wag Hypercocetrated flow. Balkema, Rotterdam, 1994; 3. P. Coussot Mudflow rheology ad dyamics. Balkema, Rotterdam, 1997; 4. ENEL HYDRO S.p.a. Verifica teorico-sperimetale di u modello visco-plastico per lo studio delle correti ipercocetrate, rapporto fiale per l ao 1999.

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