6.6 Gradually Varied Flow

Transcription

1 6.6 Gradually Varied Flow Non-uniform flow is a flow for which the depth of flow is varied. This varied flow can be either Gradually varied flow (GVF) or Rapidly varied flow (RVF). uch situations occur when: - control structures are used in the channel or, - when any obstruction is found in the channel, - when a sharp change in the channel slope takes place. 55

2 Classification of Channel-Bed lopes The slope of the channel bed is very important in determining the characteristics of the flow. Let 0 : the slope of the channel bed, c : the critical slope or the slope of the channel that sustains a given discharge (Q) as uniform flow at the critical depth (y c ). y n is is the normal depth when the discharge Q flows as uniform flow on slope 0. 56

3 The slope of the channel bed can be classified as: 1) Critical lope C : the bottom slope of the channel is equal to the critical slope. 0 c or yn yc ) Mild lope M : the bottom slope of the channel is less than the critical slope. 0 c or yn yc 3) teep lope : the bottom slope of the channel is greater than the critical slope. 0 c or yn yc 4) Horizontal lope H : the bottom slope of the channel is equal to zero ) Adverse lope A : the bottom slope of the channel rises in the direction of the flow (slope is opposite to direction of flow). 0 negative 57

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5 Classification of Flow Profiles (water surface profiles) The surface curves of water are called flow profiles (or water surface profiles). The shape of water surface profiles is mainly determined by the slope of the channel bed o. For a given discharge, the normal depth y n and the critical depth y c may be calculated. Then the following steps are followed to classify the flow profiles: 1- A line parallel to the channel bottom with a height of y n is drawn and is designated as the normal depth line (N.D.L.) - A line parallel to the channel bottom with a height of y c is drawn and is designated as the critical depth line (C.D.L.) 3- The vertical space in a longitudinal section is divided into 3 zones using the two lines drawn in steps 1 & (see the next figure) 59

6 4- Depending upon the zone and the slope of the bed, the water profiles are classified into 13 types as follows: (a) Mild slope curves M 1, M, M 3. (b) teep slope curves 1,, 3. (c) Critical slope curves C 1, C, C 3. (d) Horizontal slope curves H, H 3. (e) Averse slope curves A, A 3. In all these curves, the letter indicates the slope type and the subscript indicates the zone. For example curve occurs in the zone of the steep slope. 60

7 Flow Profiles in Mild slope Flow Profiles in teep slope 61

8 Flow Profiles in Critical slope Flow Profiles in Horizontal slope Flow Profiles in Adverse slope 6

9 Dynamic Equation of Gradually Varied Flow Objective: get the relationship between the water surface slope and other characteristics of flow. The following assumptions are made in the derivation of the equation 1. The flow is stea.. The streamlines are practically parallel (true when the variation in depth along the direction of flow is very gradual). Thus the hydrostatic distribution of pressure is assumed over the section. 3. The loss of head at any section, due to friction, is equal to that in the corresponding uniform flow with the same depth and flow characteristics. (Manning s formula may be used to calculate the slope of the energy line) 4. The slope of the channel is small. 5. The channel is prismatic.. 6. The velocity distribution across the section is fixed. 7. The roughness coefficient is constant in the reach. 63

10 Consider the profile of a gradually varied flow in a small length of an open channel the channel as shown in the figure below. The total head (H) at any section is given by: H Z y V g Taking x-axis along the bed of the channel and differentiating the equation with respect to x: dh dz d V g 64

11 dh/ = the slope of the energy line ( ). f dz/ = the bed slope ( 0 ). Therefore, d V f 0 g Multiplying the velocity term by / and transposing, we get d V g 1 0 d f V g 0 f or 1 d V g 0 f This Equation is known as the namic equation of gradually varied flow. It gives the variation of depth (y) with respect to the distance along the bottom of the channel (x). 65

12 The namic equation can be expressed in terms of the discharge Q: 0 1 f Q T g A 3 The namic equation also can be expressed in terms of the specific energy E : de / 1 Q T g A 3 66

13 Depending upon the type of flow, / may take the values: (a) 0 The slope of the water surface is equal to the bottom slope. (the water surface is parallel to the channel bed) or the flow is uniform. (b) positive The slope of the water surface is less than the bottom slope ( 0 ). (The water surface rises in the direction of flow) or the profile obtained is called the backwater curve. (c) negative The slope of the water surface is greater than the bottom slope. (The water surface falls in direction of flow) or the profile obtained is called the draw-down curve. 67

14 Notice that the slope of water surface with respect to horizontal ( w ) is different from the slope of water surface with respect to the bottom of the channel (/). A relationship between the two slopes can be obtained: Consider a small length of the open channel. The line ab shows the free surface, The line ad is drawn parallel to the bottom at a slope of 0 with the horizontal. The line ac is horizontal. The water surface slope (w) is given by bc w sin ab cd bd ab Let q be the angle which the bottom makes with the horizontal. Thus cd cd 0 sinq ad ab 68

15 The slope of the water surface with respect to the channel bottom is given by bd ad bd ab w 0 This equation can be used to calculate the water surface slope with respect to horizontal. 0 w 69

16 Water Profile Computations (Gradually Varied Flow) Engineers often require to know the distance up to which a surface profile of a gradually varied flow will extend. To accomplish this we have to integrate the namic equation of gradually varied flow, so to obtain the values of y at different locations of x along the channel bed. The figure below gives a sketch of calculating the M1 curve over a given weir. 70

17 Direct tep Method One of the most important method used to compute the water profiles is the direct step method. In this method, the channel is divided into short intervals and the computation of surface profiles is carried out step by step from one section to another. For prismatic channels: Consider a short length of channel,, as shown in the figure. 71

18 Applying Bernoulli s equation between section 1 and, we write: V1 V y y g g 0 1 f or E E 0 1 f or E E 1 0 f where E 1 and E are the specific energies at section 1 and, respectively. This equation will be used to compute the water profile curves. 7

19 The following steps summarize the direct step method: 1. Calculate the specific energy at section where depth is known. For example at section 1-1, find E 1, where the depth is known (y 1 ). This section is usually a control section.. Assume an appropriate value of the depth y at the other end of the small reach. Note that: if the profile is a rising curve and, y y 1 y V y 1 1 n R if the profile is a falling curve. 3. Calculate the specific energy (E ) at section - for the assumed depth (y ). 4. Calculate the slope of the energy line ( f ) at sections 1-1 and - using Manning s formula 1 1 / 3 f 1 and And the average slope in reach is calculated V 1 n R / 3 f fm f 1 f 73

20 5. Compute the length of the curve between section 1-1 and - L 1, E E 1 0 fm or L 1, 0 E f E 1 1 f 6. Now, we know the depth at section -, assume the depth at the next section, say 3-3. Then repeat the procedure to find the length L,3. 7. Repeating the procedure, the total length of the curve may be obtained. Thus L L L... L 1,,3 n 1, n where (n-1) is the number of intervals into which the channel is divided. 74