Resistance Due to Vegetation

Transcription

1 Resistance Due to egetation by Dr. Craig Fiscenic May 000 Complexity alue as a Design Tool Cost Low Moerate ig Low Moerate ig Low Moerate ig OERIEW Drag is generate wen a flui moves troug vegetation. Te rag creates velocity graients an eies tat cause momentum losses. Tese losses are significant for a wie range of flow conitions, an existing tecniques for te preiction of resistance o not take tese into account, leaing to unerpreictions of resistance. Concepts of rag were employe to formulate two new resistance relations for cases wen ense vegetation is present in te flooway: n = K n R /3 C A g for unsubmerge vegetation, an: /6 K n n = R (RS Eqn7 ) / for fully submerge vegetation. alues for CA as a function of vegetation type an flow conition are presente in an accompanying tecnical note. PYSICAL PROCESSES Te primary resistance to te flow of water in unrestricte open cannel flows can be g attribute to te effect of viscous sear at te bounaries of te cannel. Tis realization le Prantl (904) to formulate a logaritmic velocity relation for open cannel flows. is propose logaritmic istribution as been sown to be a goo representation of te actual velocity istribution in cannels an canals tat o not ave obstructions in te water column (Cow 959). Te assumptions Prantl use to erive te logaritmic velocity istribution are not vali for vegetate flooways. In aition to te viscous sear at te bounaries of te cannel, te vegetative elements generate form rag losses. Te rag inuce by te vegetative elements impees te flow an causes a eficit in te local velocity. Fiscenic (996) foun tat te net effect of tis rag on te velocity profile was consistent for all of te ata present in te literature, regarless of te vegetation type or flow meium. Figure presents velocity istributions for tree flow conitions: unobstructe, submerge vegetation, an unsubmerge vegetation. Te eviation from te logaritmic profile for te two cases wit vegetation impeing te flow is clear. Figure sows velocity plots for several vegetation types an flow conitions. Te eigts an velocities are normalize for comparative purposes. Te profiles isplay similar caracteristics; retare velocities an a near-zero velocity graient witin te lower USAE Waterways Experiment Station, 3909 alls Ferry R., icksburg, MS 3980 ERDC TN-EMRRP-SR-07

2 portion of te vegetation canopy, a zone near te top of te vegetation wit a very ig velocity graient, an an approximately logaritmic istribution above te vegetation. 3 Figure. elocity istribution for submerge an unsubmerge vegetation. elocity istribution represents vegetation conition to te left 3.5 y/ U/U Figure. Normalize velocity istributions for various types of vegetation an flow conitions. Te combine ata represent 7 analyses of grasses, 3 analyses of srubs, an 6 analyses of trees ERDC TN-EMRRP-SR-07

3 Because vegetative rag can ave a profoun effect on te velocity an, tus, te water surface elevation, any expression of te flow conitions in a vegetate cannel must inclue rag. Te general relation for rag is: momentum loss uner most flow conitions. F ρ = C A () were F =rag force (ML/T ) ρ=flui ensity (M/L 3 ) C =an empirical, imensionless rag coefficient A=area of te obstruction normal to te flow (L ) =approac velocity of te flui (L/T) Fenzl an Davis (964) foun tat te relative influence of te vegetation rag an te soil sear for unsubmerge vegetation is a function of te eigt above te be z relative to te eigt of te vegetation an two eigts z an z between z = 0 an tat are a function of te seiment graation an te vegetation properties (Figure 3). Tey etermine tat: ) For a range of small epts of flow ( z < z < ), soil rougness is te preominant source of yraulic resistance. Uner tese conitions, resistance ecreases wit increasing ept of flow. ) At some greater flow ept ( z < z < ), resistance becomes essentially inepenent of small canges in soil rougness an increases wit increasing ept of flow. 3) For intermeiate epts ( z < z < z ), resistance is influence by bot soil an vegetative rougness. Fenzl an Davis foun tat te ratio of resistance ue to soil rougness to total resistance ecrease from a value of 0.5 at a ept of flow of 0. ft, to 0.07 at a ept of 0. ft. Tus, te significance of te soil properties witin te vegetate portion of te flooway is significant only for very sallow flows, an vegetative rag is te ominant source of Figure 3. Parametric escription for Fenzl an Davis (964) relations. RESISTANCE FOR UNSUBMERGED EGETATION A relation for unsubmerge vegetation can be formulate from te principle of conservation of linear momentum. Following a erivation similar to tat for te e Saint enant Equation, te sum of te external forces in a control volume (C) is equate to te rate of cange of linear momentum: ERDC TN-EMRRP-SR-07 3 Z Z Σ F = ma = m( ) () t Consiering only te x-component of te linear momentum, te rigt sie of Equation can be expane to form Equation 3 as follows: M ρ A = ( ρa) x+ x -( ρ qsu s ) x t t x Te external forces inclue gravity (F g ), pressure (F p ), rag (F ), an friction (F f ), for wic te x-component wic can be escribe as: F g = ρ ga xs o (4)

4 y F p = -ρ g A x (5) x F F f ρ = - C A A x (6) = -ρ ga xs f (7) were Fg = external gravity force on te C So = be slope Fp = external pressure force on te C F = external rag force exerte by te vegetation on te C A = A i /A x = vegetation ensity per unit cannel lengt L - F f = external friction force ue to sear on te bounary S f = friction slope (i.e. te slope of te momentum grae line) Equating te slope term in Equation to te slope term in Manning's Equation, a relation for Manning's n is establise as: / /3 C A n = K n R () g Alternative erivations of Equation from energy or sear consierations are possible. Te form varies epening on te assumptions mae. Equation requires an estimate of te rag coefficient C an te corresponing vegetation area. Tese are iscusse in furter etail later in tis capter. Tis equation is applicable only wen te vegetation is unsubmerge because it oes not account for te momentum flux ownwar into te canopy tat occurs wit overtopping flows (Figure 4). Te following section aresses submerge vegetation cases. Collecting tese terms an rearranging, te left-an sie of Equation gives: C Σ A y F = A ρ g x S o - S f - - (8) g x Using Equations 3 an 8, assuming te seepage inflow an te bounary sear are negligible, an rearranging yiels Equation 9: A C g = S o - g t - g x - y x Figure 4. Example of flow troug unsubmerge vegetation wic is te unsteay, graually varie version of te e Saint enant Equation for linear momentum replacing te bounary sear term wit a rag term. Te corresponing steay, graually varie equation is: A C S o y = - - (0) g g x x an te steay, uniform equation is: A C g =S o () RESISTANCE FOR SUBMERGED EGETATION Equation oes not work well wen te vegetation is fully submerge because it oes not take into account complex flow conitions above te flow canopy tat can increase or ecrease overall resistance epening on te ept of flow an te Reynols number. owever, normalize velocity profiles for tis case are efinitive, an present an alternative approac to te erivation of a resistance relation. 4 ERDC TN-EMRRP-SR-07

5 Meteorologists an flui mecanicists involve wit win power generation, soil erosion control, crop management, an oter relate activities ave stuie te beavior of wins insie an irectly above forest canopies an crops. A moifie logaritmic law escribing turbulent velocity profiles as been propose by most investigators for situations wen stratification as only a minor influence, an can be summarize by u (z - ) Equation 3, below: = ln U * k zo For (z-)/z 0 (3) were u = local velocity U * = (τ/ρ) / is te friction velocity = zero-plane isplacement k = von Karman's imensionless sear layer constant z = eigt above te groun z o = surface rougness Figure 5 grapically presents te parameters in Equation 3. Te z o term is generally taken to represent a mean value across an uneven surface an is often eliminate from te numerator. Te isplacement tickness is important for tall rougness elements suc as long grasses, agricultural crops, forests, an tick riparian vegetation. In te absence of rougness elements or wen tey are sufficiently sort, te isplacement eigt is zero. Te parameters z o an are usually etermine from measure win profiles. E I 40 G 0 T (cm) 0 Z' = Z o Z o EGETATED SURFACE Z' = Z + o Z o BARE SOIL E I G T (m) 6 4 E I G T (m) ELOCITY (M/S) Figure 5. elocity isplacement meto ERDC TN-EMRRP-SR-07 5

6 It is customary to assume te von Kármán constant k = 0.4, to specify isplacement eigt as a function of te vegetation caracteristics an to solve for friction velocity an surface rougness eigt by fitting Equation 3 to measure ata. Surface rougness estimates ave been estimate for flow ata obtaine over ifferent agricultural crops an forests. A variance in te results approacing an orer of magnitue is common, even for flow over te same surface. Experimentalists frequently fail to obtain ata above te wake region of iniviual rougness elements (z > l.5); sometimes te ata are taken uring nonneutral conitions; an often upwin nonomogeneities istort te measure profiles. Table summarizes various investigators' estimates of z o an. In Table, s is te plant spacing, S is silouette area (te total area of te plant projecte normal to te flow), C f is te sear stan rag coefficient, an oter terms are as previously efine. Equation 3 as been foun to be a goo expression of te flui velocity profile for tat portion of te profile above about /3. owever, it oes not accurately represent te profile witin te lower portions of te vegetation canopy. For flow witin te vegetation canopy, ifferent profiles ave been propose by meteorologists using first- orer closure moels tat specify an ey iffusivity K an a rag coefficient C for constant foliage istribution. owever, none of tese expressions are consistent wit te observe zero velocity graient witin te lower alf of te vegetation canopy. Fiscenic (996) etermine tat te velocity profile witin te canopy coul better be approximate by: u cos( βξ ) = u cos( β ) 0.5 (4) Table. Formulae for Rougness Lengt Z o an Displacement Tickness Autor Z o = = Lettau (969) (0.5(s/S)) Massman (987).07(-)e**(-k(Cf/)**.5) *f(c LAI) Meroney (993) 0. < zo < < < 0.75 Otterman (98) Paescke (937) 0.5(-e**(-s/S)) /7.35 Seginer (974) (l/k)exp((-4k**3)/(lc a))**.33 -l/k Sellers (965).5 + ln ( in cm) Stanill (969) ERDC TN-EMRRP-SR-07

7 were C A u β = (5) K In Equations 4 an 5, u = local velocity u = velocity at te top of te uneflecte vegetation = uneflecte vegetation eigt z = eigt above te groun ξ = z/ C = a stan rag coefficient A = vegetation area base on ensity K = ey iffusion coefficient Fiscenic (996) investigate te relationsips between te lumpe rag-area term C A an te z o an terms in Equation 5. e foun tat tese parameters coul be approximate as functions of C A an tat, by replacing te K term in Equation 5, an expression for te mean velocity coul be reuce to a function of te vegetation eigt te flow ept y an te lumpe rag-area term C A (Equation 6):.5 =.6 e U y * 0 e 0.5 z (C A ) (C A ) z + z -.95 ln -( C A 0.3 e z Fiscenic s general equation is rater ifficult to work wit, but can be reuce to a simpler form wit some limiting assumptions. Assuming te C A value remains below unity an te flow ept is not more tan twice te eigt of te vegetation, estimates of witin 5 percent of te actual velocity can be obtaine by solving te integrals in Equation 6 to yiel: y -0.4 ) U * C A = y y ln By noting tat Manning's n value can also be expresse in terms of /U, te rigt-an sie (RS) of Equation 7 can be use to obtain a value of Manning's n for a particular flow ept y as follows: /6 K n n = R (RS Eqn 7 ) (. + C A ) g * 0.95 (8) (7) Equation 8 is applicable only in cases were te ept of te flow excees te eigt of te vegetation. Initial results of stuies by Fiscenic (996) suggest tat te practical application of Equation 8 is restricte to cases were te ept of flow is at least.y, toug aitional verification is neee. Because te flow ept must be specifie, te equation can be solve by iteration wen making normal ept computations or can be use to construct a N car for EC- analyses. Te vegetation eigt an a value for C A must be known in aition to te flow ept. Measurements of te vegetation eigt are relatively straigtforwar. Specifying appropriate values for C an A is te subject of an accompanying tecnical note. APPLICABILITY AND LIMITATIONS Tecniques escribe in tis tecnical note are applicable to stream restoration an assessment projects tat inclue floo conveyance as an objective, an were riparian vegetation is subject to flooing. Te range of applicability for te two resistance equations as not been fully explore. owever, te assumptions mae in teir erivation, early applications to fiel ata, an observations from laboratory stuies, suggest some preliminary limitations. Applicability is limite to steay uniform flow for te unsubmerge case (or at least quasi-steay, quasi-uniform flow). Properties tat likely efine te equation s applicability inclue: y ERDC TN-EMRRP-SR-07 7

8 relative flow ept, Reynols number, soil particle size, an vegetation ensity. Work by Fenzl an Davis (964) suggests tat a minimum flow ept of approximately 0. is neee for te assumption of negligible sear to be accurate. Te actual minimum ept soul be a function of te soil particle size an vegetation ensity, but tis relation cannot be efine currently. At sufficiently great epts, rag soul be only a small component of resistance an flow can be assesse using te relation represente by Equation. Again, vegetation ensity woul play a role in efining tis ept, but a value of 0 soul be a conservative estimate. Equation 9 is suggeste for y>0: Equation 9 is base upon te relative rougness of sans, gravels, an cobbles. It is n = k n y /6 g 5.75log( 4y ) (9) not a continuous function wit Equation 8, but eiter soul yiel reasonable values of resistance for flow epts in te range of 0. egetation ensity is a critical eterminant in te egree of rag experience an can elp efine te applicability of te equations. A closely relate issue is te efinition of vegetation area for te rag relation. Te rag coefficients neee for te application of Equations an 8 also epen upon te efinition of vegetation area, as aresse in an accompanying tecnical note. ACKNOWLEDGEMENT Researc presente in tis tecnical note was evelope uner te U.S. Army Corps of Engineers Ecosystem Management an Restoration Researc Program. Tecnical reviews were provie by Messrs. E.A. (Tony) Dareau, Jr., an Jerry L. Miller of te Environmental Laboratory, an Drs. Stepen L. Maynor an Marian Rollings of te Coastal an yraulics Laboratory. POINT OF CONTACT Tis tecnical note was written by Dr. Craig Fiscenic, Environmental Laboratory, U.S. Army Engineer Researc an Development Center (ERDC). For aitional information, contact Dr. Craig Fiscenic, ( , fiscec@wes.army.mil), or te manager of te Ecosystem Management an Restoration Researc Program, Dr. Russell F. Teriot ( , terior@wes.army.mil). Tis tecnical note soul be cite as follows: Fiscenic, C.,(000). "Resistance Due to egetation," EMRRP Tecnical Notes Collection (ERDC TN-EMRRP-SR-07), U.S. Army Engineer Researc an Development Center, icksburg, MS. REFERENCES Cow,.T. (959). Open-cannel yraulics. McGraw-ill, New York. Fenzl, R.N., an Davis, J.R. (964). "yraulic resistance relationsips for surface flows in vegetate cannels." Transactions of te American Society of Agricultural Engineers 7, 46-5, 55. Fiscenic, J.C., (996). "elocity an resistance in ensely vegetate flooways," P.D. iss., Colorao State University, University Press, Fort Collins, CO. 8 ERDC TN-EMRRP-SR-07

9 Lettau,. (969). "Aeroynamic rougness parameter on te basis of rougness element escription," J. of Applie Meteorology 8, Massman, W. (987). "A comparative stuy of some matematical moels of te mean win structure an aeroynamic rag of plant canopies," Bounary-layer meteorology 40, Meroney, R.N. (993). "Win-tunnel moelling of ill an vegetation influence on win power availability," Literature Review, CSU Report CER9-93-RNM-, Colorao State University, Fort Collins, CO. Otterman, J. (98). "Plane wit protrusions as an atmosperic bounary," J. Geopysics Researc, 86, Paescke, W. (937). "Experimentelle Untersucungen z. Ravikeits un stabilitats problem in er boennaen Luftscic," Beitr. Pys. D. fr. Atmos, 4, Prantl, L. (904). "Über Flüssigkeitsbewegung bei ser kleiner Reibung," eranlung III Internationler Matematisce Kongress, eielberg 904. Seginer, I. (974). "Aeroynamic rougness of vegetate surfaces," Bounary Layer Meteorology 5, Sellers, P.J., Mintz, Y., Su, Y.C., an Dalcer, A. (986). "A simple biospere moel (SiB) for use witin general circulation moels," Journal of Atmosperic Sciences 43(6), Stanill, G. (969). "A simple instrument for te fiel measurement of turbulent iffusion flux," J. Appl. Meteorology, 8, 509. ERDC TN-EMRRP-SR-07 9