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<br />e <br /> <br />i <br />, <br /> <br />e <br /> <br />t <br /> <br />i <br /> <br />e <br /> <br />Chapter 4 <br />Multidimensional Flow Analysis <br /> <br />4-1. Introduction <br /> <br />a. Definitions. Multidimensional flow analysis is the <br />description and/or predictinn of the detailed hydraulic <br />characteristics of a particular flow situation in more than <br />one dimension (direction). "Hydraulic characteristics" <br />refers to the following properties of the flow, discharge, <br />velocity, water surface elevation (depth), boundary shear <br />stress, rate of energy dissipation, and constituent or sedi- <br />ment transport rate. "Particular flow situation" refers 10 <br />the specifIC body of water, location therein, physical <br />setting, alternative design configurations, and flows <br />(steady or dynamic) 10 be studied. <br /> <br />b. Description. This type of analysis recognizes <br />velocity and depth variations in either two or three direc- <br />tions. For example, flow patterns in an estuary or at a <br />river confluence may exhibit significant velocities in both <br />the streamwise and transverse directions. A one-dimen- <br />sional flow model does not explicitly consider these <br />transverse effects. Horiwntal, depth-averaged, two- <br />dimensional flow models such as RMA-2 (King 1988, <br />Gee et al. 1990) are used in river hydraulics studies <br />mainly for two purposes: (1) 10 analyze two-dimensional <br />flow patterns in detail at some area of interest (such as at <br />bridge crossings, the confluence of two channels, flow <br />around islands, etc.) or (2) to analyze the flow behavior <br />on an unbounded alluvial fan or in a wide river valley. <br />Two- and three-dimensional models can be used for both <br />steady and unsteady flow conditions. Sediment transport <br />and water quality analyses can also be done with multi- <br />dimensional flow models such as TABS-2 (Thomas and <br />McAnally 1985). TABS-2 has primarily been used for <br />simulating the sedimentation processes in reservoirs, <br />estuaries, and complex river channels. <br /> <br />c. Techniques. The techniques discussed in this and <br />the following two chapters are strictly applicable only for <br />rigid boundary (bed and banks) situations. Techniques <br />that are used for movable boundary problems (Chapter 7) <br />are extensions of the techniques presented in Chapters 4 <br />through 6. In selecting an appropriate technique, or suite <br />of techniques, the engineer must identify the important <br />physical processes that need to be recognized in the <br />analysis. Resources and data necessary 10 manage and <br />perform the appropriate level of analysis need to be <br />identified early in the study plan (refer to Chapter 3). <br /> <br />EM 1110-2.1416 <br />15 Oct 93 <br /> <br />4-2. Umltatlons of One-Dimensional Analysis <br /> <br />Flow in a channel or river is quite often viewed as being <br />one-dimensional in the streamwise direction. This means <br />that the stage (water surface elevation), velocity, and <br />discharge vary only in the streamwise direction. Subdivi- <br />sion of cross sections, however, provides an approximate <br />method of accounting for transverse roughness and veloc- <br />ity distributions. This approach provides a simplified <br />mathematical description of the flow for water surface <br />elevation prediction (see Chapters 5 and 6). More <br />detailed analysis of flow velocities and directions requires <br />representation of the flow physics (conservation of mass <br />and momentum) in two and, sometimes, three dimen- <br />sions. The engineer should understand the capabilities, <br />limitations, and effort required to perform the various <br />levels of analysis described in this and the following <br />chapters. This information should be used to make an <br />informed decision regarding the technical approach <br />needed 10 meet the study objectives and 10 defme the <br />resources necessary 10 manage and perform the study. <br /> <br />4-3. Equations of Flow <br /> <br />The principles of mass and momentum conservation are <br />presented below in generalized three-dimensional form. <br />Simplifying assumptions allow the reduction of the equa- <br />tions to two dimensions and to one dimension. <br /> <br />a. Conservation of momentum. The conservation of <br />momentum equations in the x (horiwntal), y (horizontal), <br />and z (vertical) directions are respectively: <br /> <br />au au au au <br />Pdt + pudX + pvdY + pwdZ' <br />_ a (E au) _ a (E au) <br />dX "'dX dY X>'dY <br />_ a (E au) _ ap - 't = 0 <br />dZ XZCiZ dX x <br /> <br />(4-1) <br /> <br />pi + pll ~ + pv~ + pwi <br /> <br />_ a ( av) _ a (E,~) <br />dX Ey-'dX dY U dy <br />_ a (Ey av) _ ap _ 't = 0 <br />dZ 'dZ dY y <br /> <br />(4-2) <br /> <br />4-1 <br />