Chapter Four GRADUALLY VARIED FLOW (GVF)

1. Chapter FourGRADUALLY VARIED FLOW (GVF) • 4.1 Introduction • A steady non uniform flow in a prismatic channel with gradual changes in its water surface elevation is termed as gradually –varied flow (GVF). • The back water produced by a dam or a weir across a river and the draw down produced at a sudden drop in a channel are few typical examples of Gvf. WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

2. Gradually Varied Flow Fig.4.1 WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

3. Gradually Varied Flow • In GVF, the velocity varies along the channel and consequently the bed slope ,water surface slope, and energy slope will all differ from each other. • Regions of high curvature are excluded in the analysis of this flow. • The two basic assumptions involved in the analysis of GVF are: WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

4. Gradually Varied Flow • The pressure distribution at any section is assumed to be hydrostatic. • The resistance to flow at any depth is assumed to be given by corresponding uniform-flow equation,such as the Manning’s formula,with the condition that the slope term to be used in the equation is the energy slope and not the bed slope. Thus, if in a gradually varied flow the depth of low at any section is y, the energy slope Sf is given by WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

5. Gradually Varied Flow • Sf = n2v2/R4/3 --------------------(4.1) • Where R = hydraulic radius of the section at depth y. 4.2 DIFFERENTIAL EQUATION OF GVF • Consider the total energy H of gradually varied flow in a channel of small slope and  =1 as : • H = Z + E = Z + Y +V2/2g -----4.2 • Where E = specific energy, WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

6. 4.2 DIFFERENTIAL EQUATION OF GVF • A schematic sketch of a gradually –varied flow is shown in fig.4.1 Since the water surface in general ,varies in the longitudinal (x) direction, the depth of flow and total energy are functions of x. Differentiating Eq.(4.2) with respect to x WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

7. 4.2 DIFFERENTIAL EQUATION OF GVF WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

8. 4.2 DIFFERENTIAL EQUATION OF GVF WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

9. 4.2 DIFFERENTIAL EQUATION OF GVF WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

10. 4.2 DIFFERENTIAL EQUATION OF GVF • The second form of the equation of gradually varied flow can be derived if it is recognized that dE/dx = dE/dy.dy/dx and that from chapter three dE/dy =1-F2. • Provided that the Froude number is properly defined. • Then Equation 4.8a becomes WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

11. 4.2 DIFFERENTIAL EQUATION OF GVF WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

12. 4.2 DIFFERENTIAL EQUATION OF GVF • The definition of the Froude number in equation 4.8 b depends on the channel geometry in which Froude number is given by the formula • While for a regular prismatic channel in which • negligible it assumes the conventional energy definition given by . WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

13. Reading Assignment • For non uniform gradually varied flow; friction slope is not parallel to bottom channel slope, but is evaluated using manning’s the chez’s equation. • There is no general clear solution although particular solutions are available for prismatic channels. Numerical methods are normally used. WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

14. 4.3 Classification of flow profiles • The general profile equation is written as: WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

15. 4.3Classification of flow profiles WOU/KIOT ,Department of WRIE Prepared vy Gurmu Deressa

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