Wojcech Bartnk, Andrzej Strużyńsk THE INFLENCE OF THE HYDRALIC PARAMETERS ON THE BEGINNING OF BED LOAD TRANSPORT IN MONTAIN RIVERS OBTAINED BY MEANS OF THE NISA PROGRAM Agrcultural nversty of Cracow Department of Hydraulc Engneerng The flow characterstcs n mountan rvers and streams, defnng the begnnng of bed load transport n nonunform materal, requre further knowledge of the hydraulc parameters. The most accurate results are obtaned by means of the Gordon [92] method whch descrbes the shear velocty on the bass of velocty dstrbuton. The NISA program has been used to formulate the profle velocty dstrbuton. The numercal smulaton demonstrated qualtatve agreement wth the measurement of crtcal shear stresses τ and the dmensonless parameter f under natural condtons. NOTATION C v - specfc heat of water E - constant f - shear stresses for d fracton or body force J c - crtcal slope h c - crtcal depth of flow K - thermal conductvty n j (s) - outward unt vector normal to the boundary p - pressure q - total heat flux q j - heat flux q s - heat source s - angle of straght lne accordng to = s (log y) + b or boundary parameter SF - shape factor T - temperature t - boundary parameter u - velocty u k - velocty vector - mean velocty n x-drecton * - shear velocty c - crtcal velocty for d fracton est- estmated velocty y - dstance from the wall Y - depth of water w - weghtng functon κ - von Karman constant
λ - second vscosty parameter µ - dynamc vscosty (coeffcent) ν - knematc vscosty of water ρ - densty of water σ j - boundary tracton τ j` - stress tensor resultng from nteracton of fluctuaton τ - shear stress τ w - wall shear stress Γ - functon of boundary condton accordng to Drchlet or Neumann ϕ - shape functon for pressure Φ - dsspaton term Ψ - shape functon for velocty Superscrpts: - - mean quantty - fluctuaton quantty + - dmensonless wall functon quantty KEY WORDS turbulent flow, velocty dstrbuton, bed load transport, ncpent moton, NISA program INTRODCTION Although new research pertanng to flow characterstcs n mountan rvers and streams, defnng the begnnng of bed load transport n non-heterogeneous materals requres further knowledge of hydraulc parameters, the drect measurement of shear stresses n natural condtons s qute complcated. Therefore, these have been determned on the bass of the ndrect method from the calculaton of local shear velocty from the velocty dstrbuton. The turbulent velocty dstrbuton over a rough bad has been defned by Prandtl and Karman. Ther unversal equaton has been modfed by Semons and Sentürk [1977] : * 30 y = 5. 75log k The measurement of the bottom level by means of Prandtl and Karman unversal equaton s taken nto consderaton. Accurate results are obtaned on the bass of the Wlcock [1994] method. One may use the k value obtaned from k = 2,85 d m.. In some lterature another method may be found to descrbe the frcton velocty - namely Gordon and McMahon [1992]: * = s (2) 5. 75 (1)
Shear velocty s calculated on the bass of the Prandtl and Karman equaton. The slope lne estmated accordng to the velocty dstrbuton n the semlogarthmc graph s used to calculate shear velocty (eq. 2). It may be utlsed when the velocty profle n the semlogarthmc scale becomes close to a straght lne [Bartnk,Stru yñsk 1996]. Ths allows to calculate shear stresses 2 τ = ρ * (3) The slope of the lne s grows together wth the slope of the water level and bottom roughness. Accordng to Wlcock, the velocty profle s n agreement wth the logarthmc dstrbuton wthn 2d 90 <y<y/5. For the calculated profles, the range s at least equal to the above. In order to calculate the profle velocty dstrbuton, the NISA program has been used. NISA PROGRAM Theoretcal bass of Nsa program. NISA s a specal program for computer aded engneerng (CAE) snce t covers a wde spectrum of flud flow analyss [EMRC 1995]. NISA/3D-FLID computes lamnar and turbulent flows of Newtonan and Non-Newtonan fluds. The program s nterfaced wth a DISPLAY, whch allows to create a graphcal tool for the presentaton of results,.e. plots of pressure, velocty, temperature and densty number. The equatons of contnuty, momentum and energy become ρ u t + u k u x ρ + ( ρu k ) = 0 t x p λ u = x + x x k k k k µ u u + x + j x j x j + ρf (4) (5) ρc v T t + u k T p u K T k x = k x + x x k j j ρq s + Φ + (6) The equatons (4) - (6) are vald for both lamnar and turbulent flow, but a very fne grd s requred to predct turbulent moton n the flow feld. To overcame the dffculty, Reynolds averagng technque [Tennekes 1972] s used to modfy the Naver - Stokes equatons for turbulent flow. The physcal equatons charactersng the flow feld are wrtten as :
u = u + u` p = p + p` ρ = ρ + ρ` T = T + T ` (7) where, the quanttes appearng n the governng equatons can be consdered to consst of mean and fluctuatng parts. The equatons are then averaged over a fnte nterval of tme. Mathematcally, the governng equatons are a set of ellptc, second-order partal dfferental equatons. Therefore, the type of boundary condtons by Drchlet or Neumann are n fact close to the real state. These are wrtten as u = u (s,t) on Γ u σ j =σ (s,t) n j (s) on Γ σ (8) T = T (s,t) on Γ T q = q a (s,t) +q c (s) +q r (s) on Γ q where q a, q c and q r are the appled, convectve and radatve heat flux, respectvely. In turbulent flow, hgh gradents of velocty exst near a sold wall (fg.1). To resolve ths phenomenon, the wall functons for turbulent flow s used : y Ym + m = *, m =, j (9) v and as * = τ w ρ s the shear velocty. Next, a non-dmensonal velocty + m s defned + m m =, m =, j (10) τ Fg.1.Turbulent magnary plane
Based on the dstance Y m j and the prevous nteracton soluton, * can be calculated as follows or + + + = Y for Y < 11. 6 (11) j j j ( ) + 1 + + j = EYj for Yj 11 6 κ ln. (12) Numercal problems are descrbed by usng Galerkn s method of Fnte Element Formulaton. The flow dscretzng s done by a number of smple shaped regons called elements. Wthn each element the dependent varable s approxmated by smple polynomal functons. The coeffcents of these polynomals are obtaned from nodal values of the dependent varable. Mathematcally, the velocty, pressure and temperature n an algnment can be wrtten as [Chung 1978] : T (, ) = ψ ( ) T ( ) ( ) T (, ) = θ T ( t) u x t u t p x, t = ϕ p t (13) T x t where the unknown u, p and T are column vectors of nodal ponts and Ψ, ϕ, θ are column vectors of shape functons. These are substtuted nto the governng equatons n the form of ( ψ ϕ θ ) f,,, u, p, T = R (14) where R s the resdual resultng from the use of the approxmatons of eq.13. The Galerkn form of the method of weghted resduals seeks to reduce the error R (resdual) to zero. Ths s done by achevng ortogonalty between the resdual and weghtng functons of the element whch s expressed as ( f w) dω ( R w) dω Ω E = 0 (15) NISA/3D-FLID contans a lbrary of element types whch can be used to solve a wde range of ssues. These element types can be dvded nto two man categores, namely two-dmensonal (or axsymmetrc) and three dmensonal. For two-dmensonal or axsymmetrc ssues, the followng elements may be used : a ) 4 node soparametrc quadrlateral, b ) 8 node soparametrc quadrlateral, c ) 3 node soparametrc trangle, d ) 6 node soparametrc trangle. Ω E
NISA/3D-FLID Element Lbrary Nsa/3-D consst of the followng executve commands for 2-D elements (NKTP=2) (Tab.1): Executve Commands Table1 Element type NKTP = 2 Analyss Types 1. FLID 2. HEAT 3. FLHT Degrees of Freedom 1. 2 per node :, V 2. 1 per node : T 3. 3 per node :, V, T NORDR - Quadrlateral : 4, 8 or 9 nodes Shape / number of nodes ( NORDR = 1, 2, 30 ) - Trangle : 3 or 6 nodes ( NORDR = 10, 11 ) Materal Propertes Newtonan Flud Natural Convecton Porous Meda 4 propertes : DEN, VISC, COND, SPEC 8 propertes : DEN, VISC, COND, SPEC,, GRAV,BETA, TEMB, ANGL 5 propertes : DEN, VISC, COND, SPEC, RIST Nodal Output - VELOCITY (, V ) - PRESSRE ( P ) - TEMPERATRE ( T ) - STREAM FNCTION ( S ) - Turbulent : Knetc energy ( K ), dsspaton rate ( E ), eddy vscosty ( EDD ) - Newtonan and Non-Newtonan flud : shear stress ( SXX, SYY, SXY ) - Vortcty ( VOR ) Calculaton of the velocty profle The followng executve commands were ON for the typcal flud analyss run n NISA/3D-FLID: analyss types: FLID degrees of freedom: 2 per mode:, V shape and No. of node: Quadrlateral:4, Trangle:3 materal propertes: DEN,VISC, nodal output: Velocty (,V) Turbulence
Fg.2. Mesh for sphercal grans. Fg.3. Mesh for ellpsodal grans. The velocty profles were constructed for unform grans. The roughness was defned by mesh for sphercal and for ellpsodal grans (fg.2,3). In ths numercal experment, the gran dameter for both cases was 0.08 m. The comparson of resultng for plot velocty dstrbuton shows greater nfluence of bed materal on sphercal grans than on ellpsodal grans. Accordng to ths phenomenon, the shear velocty for the known parameter s (fg.4,5) and the shear stresses (tab.3) were calculated. The numercal results of velocty dstrbuton, calculatons of frcton velocty and shear stresses were compared wth the experment results. The values of crtcal shear stresses for modelled grans are very close to Sheld s dagram [Kenedy 1995] for ncpent moton of unform gran (respectvely f = 0.056 and 0.046). Recently, Bartnk and Mchalk [1994] presented a modfed Sheld's dagram for the ncpent moton of nonunform sedment. The correct determnaton of crtcal shear stresses s also strctly connected wth the shape factor of grans n ths dagram. Ths was acheved by
[m] 1 s = 1.55 0.1 est 0.01 0.50 1.00 1.50 2.00 [m s -1 ] Fg.4. Plot of velocty dstrbuton for sphercal grans. [m] 1 0,1 s = 1.40 est 0,01 0,70 1,20 1,70 2,20 [m s -1 ] Fg.5. Plot of velocty dstrbuton for ellpsodal grans. usng the radoactve tracer method and allowed to perform the drect measurement of crtcal parameters. The values of crtcal shear stresses for ellpsodal grans are very smlar to measurements under natural condtons ( f = 0.036 and 0.026 respectvely).
Results of velocty dstrbuton and shear stresses analyss Table 2. shape of grans sphercal ellpsodal accordng to eq.(2) accordng to eq.(1) SF = 1 SF = 0.6 y y y unform grans nonunform grans [m.] [ms -1 ] [ms -1 ] 0.05 0.85 0.81 d [m] 0.08 0.08 d m [m] 0.08 0.08 0.07 0.90 1.10 SF 1.00 0.60 SF 1.00 0.60 0.09 1.22 1.29 s 1.55 1.40 k 0.23 0.14 0.13 1.46 1.48 *[ms -1 ] 0.27 0.24 *[ms -1 ] 0.22 0.18 0.19 1.69 1.64 τ [Nm -2 ] 72.5 59.2 τ [Nm -2 ] 46.40 33.08 0.28 1.94 1.87 f 0.056 0.046 f 0.036 0.026 Crtcal parameter of bed load moton under natural condton for the Raba rver Table 3. Year, number of run, rver 1987.1 Raba 1987.2 Raba 1988.3 Raba 1991.4 Raba d [m] 0.02 0.04 0.06 0.07 0.18 0.02 0.04 0.06 0.07 0.08 0.18 0.02 0.04 0.06 0.07 0.08 0.18 0.03 0.05 0.07 > 0.08 h [m] 1.07 1.07 1.10 1.18 1.15 1.26 1.34 1.12 1.20 1.32 1.31 1.43 1.21 0.95 0.90 0.90 1.35 0.57 0.81 0.71 0.76 I c 0.00315 0.00287 0.00298 0.00315 0.00333 0.00267 0.00279 0.00284 0.00315 0.00267 0.00302 0.00250 0.00230 0.00335 0.00375 0.00370 0.00301 0.00290 0.00290 0.00280 0.0028 τ c [Nm -2 ] 33.86 30.71 32.78 37.17 38.00 33.78 30.82 31.80 37.80 35.38 39.50 35.75 27.83 31.83 33.75 33.30 40.70 16.09 23.49 19.88 21.28 f 0.103 0.047 0.033 0.032 0.013 0.102 0.047 0.032 0.033 0.027 0.013 0.108 0.042 0.032 0.029 0.025 0.014 0.333 0.030 0.017 0.013 _ c [ms -1 ] 1.84 1.80 1.83 1.97 1.98 1.89 2.01 1.80 1.99 1.95 2.06 1.81 1.71 1.76 1.74 1.71 2.10 1.16 1.47 1.32 1.39
CONCLSIONS The paper presents a new methodology for calculatng crtcal condtons whch lead to the begnnng of bed load transport n mountan rvers and streams. Ths methodology dffers from the prevously employed methods based on measurements under natural condtons. The new methodology has the advantage of beng smple and flexble. Modfcatons can be easly made because more data becomes avalable. The tests also demonstrated a qualtatve agreement wth the measured crtcal shear stresses τ and the dmensonless parameter f for the natural condtons based on the Raba rver experments. Acknowledgements The materal presented s based on the NISA/DISPLAY program suppled by the INPOL COMPANY LTD of Cracow. REFERENCE Bartnk. W. 1992. Fluval Hydraulcs of Mountan Rver wth Moble Bed. Begnnng of Bed Load Moton., Zeszyty Naukowe Agrcultural nversty of Cracow, no. 171, n polsh. Bartnk W., Mchalk Al. 1994. Fluval Hydraulcs of Streams and Mountan Rvers wth Moble Bed., Hydraulcs Conference 1994, Buffalo, SA, Amercan Socety of Cvl Engneers, N.York, pp.767-771. Bartnk W., Srużyńsk A. 1996. Measurements of Basc Velocty Fluctuaton Characterstc n Rough Stream Bed., Zeszyty Naukowe Agrcultural nversty of Cracow., no. 16, pp.19-30, n polsh. Chung. T.J. 1978. Fnte Element Analyss n flud dynamcs., McGraw Hll. Gordon. N. D., McMahon A. 1992. Stream Hydrology., London. Jarrett. R. D. 1984. Hydraulcs of Hgh-Gradent Streams., Journal of the Hydraulcs Dvson., Amercan Socety of Cvl Engneers, vol. 110, no.11, pp.1519-1539. Kenedy J.F. 1995. The Albert Shelds Story., Journal of Hydraulc Engneerng., Vol.121, No.11, pp.766-771. Moore. M., Dplas. P. 1994. Effects of Partcle Shape on Bed Load Transport., Hydraulc Engneerng 94, pp.800-804. Semens. D. B., Sentürk. F. 1977. Sedment Transport Technology., Ford Collns. Tennekes H., Lumley J.L. 1972. A frst Course n Turbulence., The MIT Press, Cambrdge, MA. Wlcock. P. R., Barta. A. F., Shea. C. C. 1994. Estmatng Local Bed Shear Stress In Large Gravel-Bed Rvers., Hydraulc Engneerng 94, pp.834-838. Engneerng Mechancs Research Corporaton, 1995. NISA/3D-FLID (INCOMPRES- SIBLE) SERS MANAL., Centre for Engneerng and Computer Technology, Mchgan 48083 SA.