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EM 1110-2-1416 15 Oct 93 ilw ilw ilw ilw Pdt + P"dx + pvCiY + pw"'dz _ il (e ilw) _ il (e ilw) (4-3) dX ~ dY 2YdY il ilw ilp - 'dZ(Ezz-;jz) - "dz - pg - ~z = 0 b. Conservation of mass. The conservation of mass equation is: ilu + iJv + ~z = 0 (4-4) dX dY az where x,y.z = the Cartesian coordinate directions. U,V,w = velocity components in the x,y,z directions, respectively. t = time. g = the acceleration due 10 gravity, P = pressure. P = fluid density.1 e.a> eX)', etc. = the turbulent exchange coefficients which describe the diffusion of momentum in the direction of the first subscript 10 that of the second subscript. ~x' ;" ~z = terms representing the influence of boundary shear stresses. 4-4. Significance of Terms a. Accelerations. The terms in these equations represent forces (e.g., the pressure gradient ilp/ilx), local (temporal) accelerations (e.g., ilu/ilt), convective accelera- tions (e.g., uilu/ilx), and mass continuity. The momentum equations are derived by application of Newton's Second Law of Motion. The basic assumptions made are that the fluid is incompressible (constant density) and that the effects of turbulent momentum exchange can be simu- lated with an "eddy viscosity" (Boussinesq assumption). A rigorous derivation of these equations may be found in Rouse (1938) and French (1985). b. Forces. The forces in Equations 4-1 to 4-3 are those of gravity, pressure, boundary friction, and exchange of momentum due to turbulence. Some In general, density is a function of temperature, salinity, and pressure and is described with an additional "equation of state", see Sverdurp et al. (1942) and Wiegel (1964). 4-2 formulations of these equations may also include forces due to wind, ice, and the earth's rotation. For most riverine situations, wind and the earth's rotation (Coriolis effect) are not important; they may become important for bodies of water with length scales of tens of miles, and may become dominant for large bodies of water such as the Great Lakes. The continuity equation (4-4) repre- sents an accounting of water mass of constant density. Other formulations of these equations, such as used in estuaries, oceans, and lakes may include variable density. e 4-5. Use of Equations of Flow a. General. Equations 4-1 to 4-4 are applicable to all river and channel flow situations that satisfy the assumptions of constant density and a rigid (or at least slowly changing) boundary. The difficulty lies in solving the equations. The only reliable and routinely used engi- neering tool for solving the three-dimensional equations at this time (1991) is the physical model. Numerical models (computer programs), however, are routinely and successfully used for solving the two- and one- dimensional simplifications of the above equations. Three-dimensional numerical models are presently under development and undergoing field testing with some applications being reported. A major study of Chesa- peake Bay using a three-dimensional numerical model is reported by Kim et al. (1990) and Johnson et al. (1991). . . . . e b. Traditional approaches. "Traditional" approaches to river hydraulics studies separate continuity, or storage, routing HEC-l, (U.S. Army Corps of Engineers 19903) to determine the discharge, from the one-dimensional steady flow computations HEC-2, U.S. Army Corps of Engineers 1990b) used 10 determine water surface eleva- tions. Application of Equations 4-1 10 4-4 achieves the combined result of both routing and water surface eleva- tion computation in a single compotation. The "tradi- tional" techniques presented in Chapters 5 and 6 are based on simplifications of, or approximations 10, the equations presented above. There are many river analy- sis problems that can be satisfactorily evaluated with simplified methods. The focus of this chapter, however, is the analysis of more complex hydraulics problems in greater detail and resolution than is available with the traditional techniques. .: . . . 4-6. Two-Dimensional Flow Conditions a. General. For many rivers the width to depth ratin is 20 or more. In these cases, and for many common e