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<br />2,10 <br /> <br />which is valid for laminar flow with a lrough surface. Equation 2.22 can be <br />substituted into Equation 2.16 for <br /> <br />!L a2u _ Jill.. L <br />1m a/ - 8rm h2 <br /> <br />(2.23) <br /> <br />Now all the terms have been defi ned in [quat ion 2.16 , Val ues of K can be <br />estimated using empirical relationships of friction coefficients f versus Re <br />fDr natura I grass or other surfaces, A procedure for est imat 'i ng K va lues is <br />presented in Appendix B. $ <br />Combining Equations 2.16, 2,17, and 2.22 results in a quadratic equation <br />where the ve'J ocity is so I ved as a funct i on of depth, vi scos ity, yi e I d stress, <br />s lope and roughness. The so I ut i on of these equat ions incorporate severa I <br />assumptions. The objective of the entire simulation effort is to predict the <br />mean flow depths and velocity for each grid at a specific time, Computing the <br />depth integrated velocity is not the goal of this mathematical model. The <br />equat ions represent i ng turbul ent and vi scous fl ow are genera I i zed, empi ri ca I <br />equat ions defi ni ng a genera I flow regi me, turbul ent or I ami nal". By permitting <br />these equat i Dns to interact inca I cuI at i ng a SD I ut i on to Equat ion 2.16, the <br />principal assumption regarding the derivation of these empirical equations are <br />being violated (that is that the flow is either laminar or turbulent). It is <br />through their interaction, however, using realistic estimates of the variables <br />that a trans i t i on range for flow between expressly turbul ent and express I y <br />laminar flow can be determined. Knowledge of the relationships of viscosity and <br />yi e I d stress with concentration combi ned wi th an accurate est imate of the <br />roughness results in rea list i c va I ues of the average fl OI~ depth and vel oci ty both <br />for channel fl ow and overl and flow. Further, l"esults of app lyi ng MUDFLOW to <br />predi ct area of i nundat ion, maximum depths, a,nd fi na I depths for very vi scous <br />fl ow in whi ch the fl ow often halts on the a 11uvi a I f<1Il have been excellent <br />justifying the assumptions discussed herein. The important assumptions are: <br /> <br />L The equations are being solved for a mean flow depth h and velocity V; <br />not a depth integrated velocity U; du/dy is approximated by V/h. <br /> <br />2, Empirical relationships describing turbulent and laminar flow extend into <br />a transitional flow regime that is quasi,turbulent. The solution of the <br />combi nEld equat ions is an adequate representat i on of the mean fl ow vari ab I es <br />in this quasi,turbulent regime. <br />