UNIFORM FLOW CRITICAL FLOW GRADUALLY VARIED FLOW

Transcription

1 UNIFORM FLOW CRITICAL FLOW GRADUALLY VARIED FLOW

2 Derivation of uniform flow equation Dimensional analysis Computation of normal depth

3 UNIFORM FLOW 1. Uniform flow is the flow condition obtained from a balance between gravity and friction forces.. Uniform flow is often used as a design condition to determine channel dimensions 3. It requires the use of an empirical resistance coefficient 4. This coefficient has been subject of research since the 19th century 5. Design parameters include: channel slope, channel shape, soil conditions, topography and availability of land, risk and frequency analysis 6. The uniform flow depth is called NORMAL DEPTH

4 MOMENTUM AND ENERGY ANALYSIS FOR UNIFORM FLOW IN OPEN CHANNELS

5 MOMENTUM ANALYSIS FORCE BALANCE Gravity Force = Friction force AL sin 0 PL F p1 τ 0 P ΔL ΔL Wsenθ P = wetted perimeter τ 0 = mean boundary shear stress A = cross sectional area γ = specific weight of water Dividing by PΔL gives: Rsin S 0 0 / 0 W=γA ΔL θ F p A y Assuming small channel slope: S 0 = sinθ, where S 0 =channel bed slope, gives R

6 ENERGY ANALYSIS Energy Potential Energy + Pressure Energy + Kinetic Energy z y V g y z V g HGL EGL S 0 S f S w y z h f V g 1 Dr. Walter F. Silva y V y1 V1 z z1 1 h g g f 6

7 ENERGY ANALYSIS For uniform flow the bed slope is parallel to the water slope and to the energy slope as shown next Energy Slope S f dh dy dz d V ( ) dx dx dx dx g Recalling that for uniform flow the velocity is constant in the channel reach, dv /dx = 0, therefore: S f dh dx dy dx dz dx 0 Since the water depth is constant in uniform flow, dy/dx = 0, then: S f dh dz dx dx The energy slope is equal to: S f h l L where h l is the energy loss S 0

8 ENERGY ANALYSIS Expressing the slope in terms of the shear stress results in S h l 0 L 0 R F p1 τ 0 P ΔL ΔL Wsenθ This equation relates the shear stress with the channel slope, however, it does not includes the flow characteristics. W=γA ΔL θ F p A y There most be a relation between the shear stress and the flow velocity. This relationship is obtained from dimensional analysis

9 DIMENSIONAL ANALYSIS FOR UNIFORM FLOW

10 DIMENSIONAL ANALYSIS Consider that the shear stress is a function of Fluid density Fluid viscosity Acceleration of gravity Flow velocity Channel geometry (represented by the hydraulic radius) Channel roughness (represented by a parameter k) Other variables could also be included such as channel meandering and unsteady effects. The functional relation for the previous parameters is given by: f,, g, V, R, k 0

11 DIMENSIONAL ANALYSIS Applying Buckingham Π Theorem to this relation we get the following results V 0 Non-dimensional parameters namely: the Reynolds number (Re), the Froude number (Fr) and a relative roughness parameter. Not all of them are included in commonly used equations. f 1 Re Reynolds number and a roughness parameter are used for pipe flow. Others use the roughness parameter as the most important factor, assuming fully rough flow, where the viscous effects are not important There are procedures to include meandering and vegetation effects in the estimation of the roughness parameter, 1 Fr, k R

12 UNIFORM FLOW FORMULAS

13 UNIFORM FLOW FORMULAS There is an expression for the energy losses extensively used for pipe flow called the Darcy-Weisbach friction formula, given by: fv L h f gd where L is the pipe length, D is the pipe diameter and f is the Darcy-Weisbach friction factor. f is a function of the Reynolds number and the relative roughness. According to this equation the slope of the energy line is D h fv f L gd

14 UNIFORM FLOW FORMULAS For a direct use of this equation to an open channel, it is necessary to change the geometric characteristic. For a channel the convention is to replace the diameter by the hydraulic radius. For a pipe the hydraulic radius is: A D D R P 4D 4 D Therefore, for a channel: h f L 0 R fv g(4r) 0 V f 8

15 UNIFORM FLOW FORMULAS Also h f L 0 R fv g(4r) h f L C V 1 R S 0 C 1 = f 8g V C S0R This equation is called the Chezy s formula in honor to a French Engineer called Antoine Chezy who first proposed this formula in Several researchers tried to develop rational procedures for estimating the value of Chezy s constant, C.

16 UNIFORM FLOW FORMULAS Gauckler (1867) showed that C R Latter, Robert Manning, an Irish engineer, proposed in 1889 the following formula Nowadays the Manning s equation form is: S and should be called Gauckler-Manning formula. K n is 1.0 for the metric units system and 1.49 for English units system. 1 V C R 1 K n V R S n 3

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18 UNIFORM FLOW FORMULAS That Manning s equation has endured for more than a century as a uniform flow formula would seems to indicate that Manning s labors were not in vain, although the formula that bears his name probably would be surprising to him (Sturm)

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21 TOTAL ENERGY HEAD V H Z y g SPECIFIC ENERGY V E y g 1

22 E y E y V g Q ga y 1 and y are called alternate depths y 1 = supercritical flow, F r >1 y = subcritical flow, F r <1 y 1 y y 1 y 1 y y y c = critical depth, F r =1 (minimum specific energy)

23 Flow Regimes Froude Number Subcritical <1 Critical =1 Supercritical >1 V = Average flow velocity g = Acceleration of gravity D h = Hydraulic depth 3

24 Critical Depth is the flow depth corresponding to the minimum specific energy. Corresponds to F r =1 Fr 1 V gd V g( A / T ) Q A gd 4

25 D=y A=by 3 y c Q gb y b y E c 3 c 5

26 Water Depth (m) Pipe Diameter = 1.8 Discharge = Units System = SI Number of div = 40 Total Area =.54 Y Theta Area KE Spec. E SPECIFIC ENERGY CURVE FOR CIRCULAR CONDUIT Specific Energy (m)

27 Fundamental equations Computation methods

28 Definition: Steady, Non-Uniform flow in which the depth variation in the direction of motion is gradual enough that the pressure distribution can be considered hydrostatic Additional assumptions Small channel slope Energy losses could be estimated by using Manning s equation FUNDAMENTAL EQUATION Total Head z 1 + y 1 + V 1 g = z + y + V g + h f We are interested in the variation of y with x dy dx V 1 g y 1 HGL EGL z 1 S 0 z 1 S f S w y h f V g

29 Recalling that And dh dx = S f dz dx = S 0 (slope of the energy equation) (slope of the channel bottom) The derivative of the total head becomes: dy dx = S 0 S f 1 F F = Froude number

30 The normal depth together with the critical depth are used to classify the channel slope as mild, steep or critical. This classification is associated with the gradually flow profiles notation as M, S or C. Mild slope (M) y n > y c, in this case the uniform flow is subcritical Steep slope (S) y n < y c, in this case the uniform flow is supercritical Critical slope (C) y n = y c,, in this case the uniform flow is critical

31 Normal and critical depths divides the space above the channel bottom into three regions Zone 1 Zone NDL or CDL CDL or NDL Zone 3 There are 13 different types of surface profiles: 3 mild, 3 steep, critical, horizontal, adverse

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