Open channel hydraulics for engineers. Chapter 7 unsteady flow

This chapter introduces issues concerning unsteady flow, i.e. flow situations in which hydraulic conditions change with time. Many flow phenomena of great importance to the engineer are unsteady in character, and cannot be reduced to steady flow by changing theviewpoint of the observer. The equations of motion are formulated and the method of characteristics is introduced as main part of this chapter. The concept of positive and negative waves and formation of surges are described. Finally, some solutions to unsteady flow equations are introduced in their mathematical concepts.......
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Open channel hydraulics for engineers. Chapter 7 unsteady flow OPEN CHANNEL HYDRAULICS FOR ENGINEERS-----------------------------------------------------------------------------------------------------------------------------------Chapter UNSTEADY FLOW_________________________________________________________________________ 7.1. Introduction 7.2. The equations of motion 7.3. Solutions to the unsteady-flow equations 7.4. Positive and negative waves; Surge formation_________________________________________________________________________SummaryThis chapter introduces issues concerning unsteady flow, i.e. flow situations in whichhydraulic conditions change with time. Many flow phenomena of great importance to theengineer are unsteady in character, and cannot be reduced to steady flow by changing theviewpoint of the observer. The equations of motion are formulated and the method ofcharacteristics is introduced as main part of this chapter. The concept of positive andnegative waves and formation of surges are described. Finally, some solutions to unsteadyflow equations are introduced in their mathematical concepts.Key wordsUnsteady flow; method of characteristics; positive and negative waves; surge; numericalsolution._________________________________________________________________________7.1. INTRODUCTION In unsteady flow in an open channel, velocities and depths change with time at anyfixed spatial position. Open channel flow in a natural channel almost always is unsteady,although it often is analyzed in a quasi-steady state, e.g. for channel design or floodplainmapping. Unsteady flow in open channels by nature is non-uniform as well as unsteadybecause of the free surface. Mathematically, this means that the two dependent flowvariables (e.g. velocity and depth or discharge and depth) are functions of both distancealong the channel and time for one-dimensional applications. Problem formulation requirestwo partial differential equations representing the continuity and the momentum principlein the two unknown dependent variables. The full differential forms of the two governingequations are called the Saint-Venant equations or the dynamic wave equations. Onlythrough rather severe simplifications of the governing equations analytical solutions areavailable for unsteady flow. This situation has led to the extensive development ofappropriate numerical techniques for the solution of the governing equations.A complete theory of unsteady flow is therefore required, and will be developed in thischapter. The equations of motion are not soluble in the most general case, but we shall seethat explicit solutions are possible in certain cases which are physically very simple but arereal enough to be of engineering importance. For the less simple cases, approximations andnumerical methods can be developed which yield solutions of satisfactory accuracy.-----------------------------------------------------------------------------------------------------------------------------------Chapter 7: UNSTEADY FLOW 130OPEN CHANNEL HYDRAULICS FOR ENGINEERS-----------------------------------------------------------------------------------------------------------------------------------7.2. THE EQUATIONS OF MOTION7.2.1. Derivation of Saint-Venant equations Although the governing equations of conservation of mass and momentum can bederived in a number of ways, we apply a control volume of small but finite length, x, thatis reduced to zero length in the limit to obtain the final differential equation. Thederivations make use of the following assumptions (Yevjevich, 1975; Chaudhry, 1993): 1. The shallow-water approximations apply, so that vertical accelerations are negligible, resulting in a vertical pressure distribution that is hydrostatic; and the depth, h, is small compared to the wavelength, so that the wave celerity c = (gh)½. 2. The channel bottom slope is small, so that cos2 in the hydrostatic pressure force formulation is approximately unity, and sin  tan = io, the channel bed slope, where  is the angle of the channel bed relative to the horizontal. 3. The channel bed is stable, so that the bed elevations do not change with time. 4. The flow can be presented as one-dimensional with a) a horizontal water surface across any cross section such that transverse velocities are negligible, and b) an average boundary shear stress that can be applied to the whole cross-section. 5. The frictional bed resistance is the same in unsteady flow as in steady flow, so that the Manning or Chezy equations can be used to evaluate the mean boundary shear stress.Additional simplifying assumptions made subsequently may be true in only certaininstances. The momentum flux correction factor, , for example, will not be assumed to beunity at first, because it can be significant in river overbank flows.7.2.2. The equations of motion We proceed to obtain equations describing unsteady open channel flow. The termsused are defined in the usual way, and are illustrated in Fig.7.1. V2 2g h B b h h+h A H x P z Datum Fig.7.1. Defi ...