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Open channel hydraulics for engineers. Chapter 4 non uniform flow

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Linking up with Chapter 2, dealing with uniform flow in open channels, it may be noted that any change in the flow phenomenon (i.e. flow rate, velocity, flow depth, flow area, bed slope do not remain constant) causes the flow to be non-uniform. This chapter will discuss the effect of change in any one of the above quantities, including specific energy, critical depth and slope, and flow types. How to draw water surface profiles will also be introduced.

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Nội dung Text: Open channel hydraulics for engineers. Chapter 4 non uniform flow

  1.  OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Chapter NON-UNIFORM FLOW _________________________________________________________________________ 4.1. Introduction 4.2. Gradually-varied steady flow 4.3. Types of water surface profiles 4.4. Drawing water surface profiles _________________________________________________________________________ Summary Linking up with Chapter 2, dealing with uniform flow in open channels, it may be noted that any change in the flow phenomenon (i.e. flow rate, velocity, flow depth, flow area, bed slope do not remain constant) causes the flow to be non-uniform. This chapter will discuss the effect of change in any one of the above quantities, including specific energy, critical depth and slope, and flow types. How to draw water surface profiles will also be introduced. Key words Non-uniform; specific energy; critical; gradually-varied steady flow; water surface profiles _________________________________________________________________________ 4.1. INTRODUCTION 4.1.1. General In the previous Chapter 2, the flow was uniform under all circumstances under consideration. In many situations the flow in an open channel is not of uniform depth along the channel. In this chapter the flow conditions studied relate to steady, but non-uniform, flow. This type of flow is created by, among other things, the following major causes:  Changes in the channel cross-section.  Changes in the channel slope.  Certain obstructions, such as dams or gates, in the stream’s path.  Changes in the discharge – such as in a river, where tributaries enter the main stream. A non-uniform flow is characterized by a varied depth and a varied mean flow velocity: V h  0 or 0 (4-1) s s If the bottom slope and the energy line slope are not equal, the flow depth will vary along the channel, either increasing or decreasing in the flow direction. Physically, the difference between the component of weight and the shear forces in the direction of flow produces a change in the fluid momentum which requires a change in velocity and, from continuity considerations, a change in depth. Whether the depth increases or decreases depends on various parameters of the flow, with many types of surface profile configurations possible. Fig. 4.1. illustrates some typical longitudinal free-surface profiles. Upstream and downstream controls can induce various flow patterns. In some cases, a hydraulic jump might take place. A jump is a rapid-varied flow phenomenon; calculations were developed in Chapter 3. However, it is also a control section and it affects the free surface profiles upstream and downstream. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 70
  2. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- upstream control downstream control control sluice gate sharp-crested hydraulic jump weir rapid gradually rapid gradually rapid varied varied varied varied varied flow flow flow flow flow upstream control critical depth hc supercritical flow subcritical control downstream flow hydraulic control jump overflow (critical depth) rapid hc varied gradually flow varied flow rapid gradually rapid varied varied varied flow flow flow Fig. 4.1. Examples of non-uniform flow 4.1.2. Accelerated and Retarded flow An idealized section of a reach of a channel with accelerated and retarded flow conditions is shown in Fig. 4.2a and Fig. 4.2b, respectively. As flow accelerates, with the rate of flow constant, the depth h must decrease form point 1 to point 2, and a water surface profile as shown in Fig. 4.2a results. Retarded flow will produce water surface profiles as shown in Fig. 4.2b. Significant in each one of the above cases is the fact that now the water surface is a curved line and not longer parallel to the channel bottom and the energy line, as was the case for uniform flow. The following points are made in connection with the above observations. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 71
  3. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  The water surface, as will be shown later, can have a concave or a convex shape.  The energy line is not necessarily a straight line; however, it is assumed that the energy gradient is constant along the length of a reach and the energy line will be represented and considered to have a slope ie = HL/L.  As was done in the case of uniform flow, it is here also accepted that the depth of flow, h, is equal to the pressure head in the energy equation. Obviously, this applies only when the slope of the channel bottom is small. For very steep slopes, allowances for this discrepancy must be made.   energy-head line V 1 2 HL 2g water surface V 22 2g h1  p1 hydraulic grade line  h2  p2 i  z1 L z2   datum Fig. 4.2a. Accelerated flow   energy-head line V 12 HL 2g V 22 water surface 2g h1  p1  hydraulic grade line h2  p2 i  z1 L z2   datum Fig. 4.2b. Retarded flow ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 72
  4. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 4.1.3. Equation of non-uniform flow 1 2 ie he V2 2g water surface (V+dV) 2 2g h flow h+dh ib zb dl 1 2 Fig. 4.3. Non-uniform flow Consider a non-uniform flow in an open channel between section 1-1 and section 2-2, in which the water surface has a rising trend (i.e. the energy-head gradient is less than the bed slope) as shown in Fig. 4.3. Let V = velocity of water at section 1-1; h = depth of water at section 1-1; V+dV = velocity of water at section 2-2; h+dh = depth of water at section 2-2; ib = slope of channel bed; ie = slope of the energy grade line; dl = distance between section 1-1 and section 2-2; b = average width of the channel, Q = discharge through the channel, zb = change of bottom elevation between section 1-1 and section 2-2, and he = HL, change of energy grade line between section 1-1 and section 2-2. Since the depth of water at section 2-2 is larger than at section 1-1, the velocity of water at section 2-2 will be smaller than that at section 1-1. Applying Bernoulli’s equation at section 1-1 and section 2-2: V2 (V  dV) 2 zb  h   (h  dh)   he (4-2) 2g 2g  V  dV   i .dl  ib .dl   h    h  dh   e  2 V2 (4-3) 2g 2g  (dV)2 i b .dl  dh   i e .dl , V.dV neglecting (small of second order) (4-4) g 2g ib    ie dh V.dV or (dividing by dl) (4-5) dl g.dl ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 73
  5. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  ib   ie dh V.dV (4-6) dl g.dl We know that the quantity of water flowing per unit width is constant, therefore q = V.h = constant (4-7)  0 dq (4-8) dl 0 d(Vh) or (4-9) dl Differentiating the above equation (treating both V and h as variables),  0 V.dh h.dV (4-10) dl dl    dV V dh (4-11) dl h dl dV Substituting the above value of in Eq. (4-6), yields dl V 2 dh  ib    ie dh (4-12) dl gh dl dh  V 2  1    ib  ie (4-13) dl  gh  i i   b e2 dh (4-14) dl  V  1    gh  Notes: The above relation gives the slope of the water surface with respect to the bottom of the channel. Or in other words, it gives the variation of water depth with respect to the distance along the bottom of the channel. The value of dh/dl (i.e. zero, positive or negative) gives the following important information:  If dh/dl is equal to zero, it indicates that the slope of the water surface is equal to the bottom slope. Or in other words, the water surface is parallel to the channel bed.  If dh/dl is positive, it indicates that the water surface rises in the direction of flow. The profile of water, so obtained, is called backwater curve.  If dh/dl is negative, it indicates that the water surface falls in the direction of flow. The profile of water, so obtained, is called downward curve. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 74
  6. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Example 4.1: A rectangular channel, 20 m wide and having a bed slope of 0.006, is discharging water with a velocity of 1.5 m/s. The flow is regulated in such a way that the slope of the water energy gradient is 0.0008. Find the rate at which the depth of water will be changing at a point where the water is flowing 2 m deep. Solution: Given: width of the channel: b = 20 m bed slope: ib = 0.006 velocity of water: V = 1.5 m/s slope of energy line: ie = 0.0008 depth of water: h=2m dh Let be the rate of change of water depth. Using equation in (4-14): dl i i  b e2 dh = 0.0059 Ans. dl  V  1    gh  4.2. GRADUALLY-VARIED STEADY FLOW 4.2.1. Backwater calculation concept Gradually varied flow is a steady, non-uniform flow in which the depth variation in the direction of motion is gradual enough to consider the transverse pressure distribution as being hydrostatic. This allows the flow to be treated as one-dimensional with no transverse pressure gradients other than those due to gravity. For subcritical flows the flow situation is controlled by the downstream flow conditions. A downstream hydraulic structure (e.g. bridge piers, gate) will increase the upstream depth and create a “backwater” effect. This concept has been introduced shortly in section 4.1.3. The term “backwater calculation” refers more generally to the calculation of the longitudinal free-surface profile for both subcritical and supercritical flows. The backwater calculation is developed assuming:  a non-uniform flow  a steady flow  that the flow is gradually varied, and  that, at a given section, the flow resistance is the same as for a uniform flow with the same depth and discharge, regardless of trends of the depth. 4.2.2. Equation of gradually-varied flow In addition to the basic gradually-varied flow assumption, we further assume that the flow occurs in a prismatic channel, or one that is approximately so, and that the slope of the energy grade line can be evaluated from uniform flow formulas with uniform flow resistance coefficients, using the local depth as though the flow were locally uniform. Referring to Fig. 4.4., the total energy head at any cross-section is ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 75
  7. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- H  z  h  V2 (4-15) 2g in which z = channel bed elevation; h = water depth,  = kinetic-energy correction coefficient as introduced in Chapter 2, and V = mean flow velocity.   slope of energy grade line, ie V 2 2g dH h H bed bed slope ib z  dx   datum Fig 4.4. Definition sketch for gradually-varied flow If this expression for H is differentiated with respect to x, the coordinate in the flow direction, the following equation is obtained: with E  h   V2  i e  i b  dH dE (4-16) dx dx 2g in which ie is defined as the slope of the energy grade line; ib is the bed slope (= - dz/dx); and E is the specific-energy head (i.e. the energy head with respect to the bottom). Solving for dE/dx gives the first form of the equation of gradually varied flow:  ib  ie dE (4-17) dx It appears from this equation that the specific-energy head can either increase or decrease in the downstream direction, depending on the relative magnitudes of the bed slope and the slope of the energy grade line. Yen (1973) showed that, in the general case, ie is not the same as the friction slope if (= 0/R, this equation will be introduced again in Chapter 7) or the energy dissipation gradient. Netherless, we have no better way of evaluating this slope than applying uniform-flow formulas such as those of Manning or Chezy. It is incorrect, however, to mix the friction slope, which clearly comes from a momentum analysis, with terms involving , the kinetic-energy correction (Martin and Wiggert, 1975). Note: The bed slope ie and the friction slope if are defined as: z H  o ie   = sin  tan  and i f    x x  R respectively, where H is the mean total energy-head, z is the bed elevation,  is the channel slope and o is the bottom shear stress. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 76
  8. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- The second form of the equation of gradually-varied flow can be derived if it is recognized    1  Fr 2 , provided that the dE dE dh dE that and that, applying equation (4-11), dx dh dx dh Froude number is properly defined. Then, equation (4-17) becomes: dh i b  i e  (4-18) dx 1  Fr 2 The definition of the Froude number in equation (4-18) depends on the channel geometry. For a compound channel, it should be the compound-channel Froude-number, while for a regular, prismatic channel, in which d/dh is negligible, it assumes the conventional energy definition given by Q2B/gA3. The ratio dh/dx in Eq. (4-18) represents the slope or the tangent to the water surface at any point along the channel. This relationship therefore indicates whether at any point along the channel the water surface is rising (backwater condition) or dropping (drawdown condition). Immediately the following deductions can be made:  When  0 , the slope of the water surface is dropping in the downstream dh dx direction and the depth decreases downstream.  When  0 , the slope of water surface is parallel to the channel bottom and dh dx uniform flow exists. This can be readily seen from Eq. (4-18) since, for this condition, ib = ie must equal zero.  When  0 , the slope of water surface rises in the downstream direction and the dh dx depth h increases downstream.  When   , which requires that 1 – Fr2 = 0 or Fr = 1, the slope of the water dh dx surface must theoretically be vertical. This flow occurs when the flow changes from subcritical to supercritical, or vice versa, as indicated by the value of the Froude number. The formulas derived do not actually apply any longer due to the assumptions made. A vertical water surface also does not occur in reality; however, a very noticeable change in the water surface takes place. This is especially so when the flow changes from below hc to above hc. In such instance a phenomenon known as the hydraulic jump occurs. 4.3. TYPES OF WATER SURFACE PROFILES 4.3.1. Classification of flow profiles From the foregoing, it is evident that the relationship expressed in Eq. (4-18) provides a considerable amount of information as to the shape of the water surface profile in an open channel. Investigation of this formula yields the following results: ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 77
  9. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 1. The relationship between the slope of the channel bottom and the slope of the energy grade line determines whether the numerator of the equation is positive or negative. 2. The denominator of the equation is positive if Fr < 1.0 and vice versa. In other words, if the flow is subcritical (Fr smaller than 1) the denominator is positive, and if the flow is supercritical (Fr greater than 1) the denominator is negative. The conditions at which flow in an open channel can take place and the possible relationships between the observed depth ho, the normal depth at which flow is uniform hn, and the critical depth hc are illustrated in Fig. 4.5. It is evident from this figure that there are three zones of channel depths at which flow can be observed: Zone 1, with ho greater than hn and hc (i.e. ho > hn > hc) Zone 2, with ho between hn and hc (i.e. hn > ho > hc) Zone 3, with ho less than hn and hc (i.e. hn > hc > ho) ho hn hn ho hn hc hc hc ho ho > hn > hc hn > ho > hc hn > hc > ho Fig.4.5. Three zones of channel depths The relative bottom slope defines whether uniform flow is subcritical or supercritical. Determine the associated Froude-number Fre. Fre2    e e Ve2 R V2 gh e h e gR e where R is the hydraulic radius of the open channel flow. Subcript e denotes the equilibrium flow. The bottom/wall shear stress is defined as:  o  cf  Ve2   gR ei f  Ve2  f  b i i (the friction slope if = the bed slope ib ) gR e cf cf  Fre2   R e ib h e cf We have: Ae = Be.he  h e   A e Pe .R e , where Pe is the equilibrium wetted perimeter. Be Be ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 78
  10. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  Fre2   Be i b Pe c f In case of turbulent flow:   1.  R e Be For two-dimensional flow: 1 he Pe So, a proper approximation for Fre is: Fre2  ib cf  If ib < cf, we have a mild slope (M – type) The “uniform flow” is subcritical: Fre2 < 1, he > hc.  If ib > cf, we have a steep slope (S – type) The “uniform flow” is supercritical: Fre2 > 1, he < hc.  If ib = cf, we have a critical slope (C – type) Fre2 = 1, he = hc 1 Note: It can easily be derived that c f   n 2 gR 3 , where C is Chezy coefficient and n g 2 C is Manning’s. Two conditional channel bottom conditions or slopes exist. These do not really constitute open channel flow, but gravity flow can take place along them. They are as follows:  If ib < 0, we have an adverse slope (A – type)  If ib = 0, we have a horizontal slope (H – type) It should be noticed that hn = he. Note: The actual flow depends on the boundary condition, i.e. “mild”, “steep”, etc. does not tell us anything about the actual flow. 4.3.2. Sketching flow profiles In theory, for each of the five slope descriptions above there are three zones in which flow can be observed. It follows then that a total of 15 theoretical water surface profiles are possible, presented in Table 4.1. These profiles, together with illustrations of practical applications, are shown in Fig. 4.6. While this figure is for the most part self-explanatory, the following observations and explanations are presented for further clarification.  Mild slope (ib < cf). The M1 curve is generally very long and asymptotic to the horizontal and the line representing ho. The M2- and M3-curves end in a sudden drop through the line representing hc and a hydraulic jump, respectively.  Critical slope (ib = cf). Since hc = hn in this case, there is no zone 2, and only two water surface profiles exist, C1 and C3. The C2-curve coincides with the water surface that corresponds to uniform flow at critical depth. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 79
  11. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  Steep slope (ib > cf). All curves are relatively short. S1 is asymptotic to the horizontal, whereas S2 and S3 approach ho.  Horizontal slope and Adverse slope channels. In this case, hn is infinitely large and uniform flow cannot take place. Hence there are no H1- or A1-profiles. Table 4.1. Types of flow profiles in prismatic channels Designation Relation of ho Type of flow See Channel to hn and hc General type Fig. slope Zone Zone Zone Zone Zone Zone of curve 1 2 3 1 2 3 Mild M1 ho > hn > hc Backwater Subcritical 2 Fre < 1, M2 hn > ho > hc Drawdown Subcritical 4.6.a he > hc M3 hn > hc > ho Backwater Supercritical C1 ho > hc = hn Backwater Subcritical Critical C2 hc = ho = hn Parallel to Uniform 4.6.b 2 Fre = 1, channel bottom critical he = hc C3 hc = hn > ho Backwater Supercritical Steep S1 ho > hc > hn Backwater Subcritical 2 Fre > 1, S2 hc > ho > hn Drawdown Supercritical 4.6.c he < hc S3 hc > hn > ho Backwater Supercritical Horizontal None ho > hn > hc None None ib = 0 H2 hn > ho > hc Drawdown Subcritical 4.6.d H3 hn > hc > ho Backwater Supercritical Adverse None ho > (hn*) > hc None None 4.6.e ib < 0 A2 (hn*) > ho > hc Drawdown Subcritical A3 (hn*) > hc > ho Backwater Supercritical hn* in parentheses is assumed a possitive value. M1 M1 M1 horizontal NDL M2 M2 section of hn M2 enlargement hc M3 CDL M3 M3 Mild slope Fig.4.6.a. Mild slope (0 < ib < cf ) and examples of flow profiles CDL = critical-depth line; NDL = normal-depth line ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 80
  12. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- horizontal C1 C1 CDL = NDL hc = hn C3 critical slope C3 Fig.4.6.b. Critical slope (ib = cf > 0) and examples of flow profiles horizontal S1 S1 CDL S1 hc S2 hn section of S3 enlargement NDL steep slope S2 S2 S3 Fig.4.6.c. Steep slope (ib > cf > 0) and examples of flow profiles S3 horizontal H2 H2 CDL H3 hc H3 horizontal slope Fig.4.6.d. Horizontal slope (ib = 0) and examples of flow profiles A2 A2 hc CDL A3 A3 adverse slope Fig.4.6.e. Adverse slope (ib < 0) and examples of flow profiles ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 81
  13. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 4.3.3. Prismatic channel with a change in slope This channel is equivalent to a pair of connected prismatic channels of the same cross section but with different slope. Several typical flow patterns along a prismatic channel with a break or discontinuity in slope are shown in Fig. 4.7. M1 M1 S2 NDL S2 NDL M2 hc S3 CDL hn1 hn2 hn1 CDL hn2 hc steep (very long) NDL mild steeper milder (very long) reservoir M2 NDL NDL hn1 S2 H S2 hc hc CDL CDL mild hn2 steep NDL steep NDL reservoir reservoir M2 tailwater M2 NDL H NDL hn H hc hc hn NDL mild (very long) CDL mild (short) CDL free overfall Fig. 4.7. Flow profiles with a change in slope The profiles in Fig. 4.7 are self-explanatory. However, some special features should be mentioned:  The profile near or at the critical depth cannot be predicted precisely by the theory of gradually varied flow, since the flow is generally rapidly varied there.  In passing a critical line, the flow profile should, theoretically, have a vertical slope. Since the flow is usually rapidly varied when passing the critical line, the actual slope of the profile cannot be predicted precisely by the theory. For the same reason, the critical depth may not occur exactly above the break of the channel bottom and may be different from the depth shown in the figure. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 82
  14. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 4.3.4. Composite flow profiles with various controls Channels with a number of controls will have flow profiles that can be composed from the different types of flow profiles presented in the previous section. The ability to sketch the composite profiles is in many cases necessary for understanding the flow in the channel or for determining the discharge. In all cases it is necessary to identify firstly the controls operating in the channel, and then to trace the profiles upstream and downstream of these controls. Two simple cases are shown in Fig. 4.8; in the first case the slope is mild, in the second case steep. The curves for the mild-slope situation are self-explanatory, since they incorporate many of the features already discussed. For the steep-slope situation we have already seen that the critical flow must occur at the head of the slope – i.e. at the outflow from the reservoir; thereafter there must be an S2-curve tending towards the uniform-depth line. There must be an S1-curve behind the gate, and the transition from the S2- to the S1- curve must be via a hydraulic jump. Downstream of the sluice gate, the flow will tend to the uniform condition via an S2- or S3-curve; thence it proceeds over the fall at the end of the slope. In this case there is nothing that impels the flow to seek the critical condition. In Fig. 4.8 two profiles are drawn in dashed lines above the M3- and the S2-curve. These are loci of depths conjugated to the corresponding depths on the underlying real surface profiles, and are therefore known as “conjugate curves”. Obviously a hydraulic jump will occur where such a curve intersects the real (subcritical) surface profile downstream; the conjugate curve therefore provides a convenient means of determining the location of a hydraulic jump. M1 conjugate curve M2 hn reservoir hc M3 jump overfall mild slope conjugate curve S1 jump reservoir S2 S3 hc overfall hn steep slope Fig. 4.8. Examples of composite longitudinal profiles ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 83
  15. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- 4.4. DRAWING WATER SURFACE PROFILES 4.4.1. Direct-step method The computation of a flow profile by a step method consists of dividing the channel into short reaches and determining reach by reach the change in depth for a given length of a reach. In principle, the direct-step method could be applied to either Eq.(4-17) or Eq. )4- 18), but usually is associated with the former. Eq. (4-17) is put into finite-difference form by approximating the derivative dE/dx with a forward difference and by taking the mean value of the slope of the energy grade line over the step size x = (xi+1 – xi) in which the distance x and the subscript i increase in the downstream direction. The result is: E  Ei x i1  x i  i 1 (4-19) i b  ie where i e is the arithmetic mean slope of the energy grade line between section i and section i + 1, with the slope evaluated individually from Manning’s equation at each cross section. The variables Ei+1, Ei and i e on the right hand side of Eq. (4-19) all are functions of the depth h. The solution proceeds in a stepwise fashion in x by assuming values of depth h and therefore values of the specific-energy head, E. As Eq. (4-19) is written, x increases in the downstream direction. In general, upstream computations utilize Eq. (4-19) multiplied by (–1), so that the current value of the specific-energy head is subtracted from the assumed value in the upstream direction and x becomes (xi+1 – xi), which is negative. Therefore, if the equation is solved in upstream direction for an M2-profile, for example, the computed values of x should be negative for increasing values of h. Decreasing values of h should result also in negative values of x for an M1-profile. For an M3- profile, which is supercritical, increasing values of depth in the downstream direction correspond to decreasing values of the specific-energy head, and Eq. (4-19) indicates positive values of x, since i e > ib. Although the direct-step method is the easiest approach, it requires interpolation to find the final depth at the end of the profile in a channel of specified length. Some care must be taken in specifying starting depths and checking for depth limits in a computer program. In an M2-profile, for example, the starting depth should be taken slightly greater than the computed critical depth, if it is a control, because of the slight inaccuracy inherent in the numerical evaluation of critical depth. In addition, the M2-profile approaches the normal depth asymptotically in the upstream direction, so that some arbitrary stopping limit must be set, such as 99% of the normal depth. Example 4.2: A trapezoidal channel has a bottom width b of 8.0 m and a side slope ratio of 2:1. The Manning’s n of the channel is 0.025 m-1/3s, and the channel is laid on a slope of 0.001. If the channel ends in a free overfall, compute the water surface profile for a discharge of 30 m3/s. Solution: B Given: bottom width: b = 8.0 m side slope ratio: m:1 = 2:1 1 hn Manning’s n: n = 0.025 m-1/3s m=2 bed slope: ib = 0.001 b discharge: Q = 30 m3/s Compute the water surface profile. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 84
  16. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- First, the normal depth and the critical depth must be determined. Manning’s equation reads as: Q 1 2 1 .A.R 3 .i b 2 n with trapezoidal channel cross-section: A  y n (b  mh n ) h (b  mh n ) R  n A and hydraulic radius: P b  2h n 1  m 2 So, the Manning equation can be rewritten as: A [h n (b  mh n )] 2 5 5  A.      3 3 3 2 3 A Q.n A.R P  b  2h 1  m 2  2 2 1 P 3 3 ib 2  n   h n (8.0  2yn  3 30  0.025 5   23.72 m  hn = 1.754 m 8 8 3 3 or, m 8  2h 1  2  2 1 3 2 2 0.001  n  From the Froude formula: 1 Fr    2 V Q QB A 3 gD A 2 g A g.    B  QB   where B = b+ 2mh; D = A/B, the hydraulic depth. In case of critical flow:  h c  b  2h c   1 3   2 3 A g  2 Fr  1   c Ac 2 Q  b  2mh c  2 3 1 1 c 2 Bc 2 g  h c  8  2h c   3    hc = 1.03 m 2  30 5 8  2  2  h c  2 or 1 m 2 9.81 Due to hn > hc, this is a mild slope (ib = 0.001): we have an M2-profile that has a critical depth at the free overfall as boundary condition. The direct-step method, as applied to Example 4.2, can be solved in a spreadsheet (Microsoft Excel) as formatted in Table 4.2. The values of h are selected in the first column (1). The formulas for determining the specific-energy head E, column (5), and the slope of the energy grade line ie, column (6), for a given depth, are presented below. The arithmetic mean of ie (iebar = i e ) is computed in column (7), and the change in specific- energy head E, DelE, in the upstream direction is shown in column (8). Formulas applied in the spreadsheet: V2 Eh A  y(b  mh) 2g E  Del.E  E 2  E1 (nV)2 P  b  2h 1  m 2 ie  x  Del.x  Del.E R 3 4 (i b  i ebar ) R ; V Q Q (i  i ) x  Del.E /(ie1  i e2 ) P A iebar  e1 e2  (ie1  i e2 ) / 2 2 ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 85
  17. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- Then, the equation of gradually varied flow in finite difference form is solved for the E distance step x, as x  = - 0.028 m in the first step. i b  ie Table 4.2. Water surface profile computation by the direct-step method. h A R V E ie iebar Del.E Del.x Sum Del.x (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1.03 10.362 0.822 2.895 1.457 6.80E-03 0.00 1.04 10.483 0.829 2.862 1.457 6.58E-03 6.69E-03 1.62E-04 -0.028 -0.03 1.06 10.727 0.842 2.797 1.459 6.15E-03 6.36E-03 1.23E-03 -0.229 -0.26 1.08 10.973 0.855 2.734 1.461 5.75E-03 5.95E-03 2.35E-03 -0.476 -0.73 1.1 11.220 0.868 2.674 1.464 5.39E-03 5.57E-03 3.40E-03 -0.743 -1.48 1.12 11.469 0.882 2.616 1.469 5.06E-03 5.23E-03 4.36E-03 -1.032 -2.51 1.14 11.719 0.895 2.560 1.474 4.75E-03 4.90E-03 5.26E-03 -1.346 -3.85 1.16 11.971 0.908 2.506 1.480 4.47E-03 4.61E-03 6.09E-03 -1.687 -5.54 1.18 12.225 0.921 2.454 1.487 4.20E-03 4.33E-03 6.86E-03 -2.057 -7.60 1.2 12.480 0.934 2.404 1.495 3.96E-03 4.08E-03 7.58E-03 -2.460 -10.06 1.22 12.737 0.947 2.355 1.503 3.73E-03 3.84E-03 8.24E-03 -2.898 -12.96 1.24 12.995 0.959 2.309 1.512 3.52E-03 3.63E-03 8.87E-03 -3.377 -16.33 1.26 13.255 0.972 2.263 1.521 3.32E-03 3.42E-03 9.45E-03 -3.901 -20.23 1.28 13.517 0.985 2.219 1.531 3.14E-03 3.23E-03 9.99E-03 -4.474 -24.71 1.3 13.780 0.998 2.177 1.542 2.97E-03 3.06E-03 1.05E-02 -5.105 -29.81 1.32 14.045 1.010 2.136 1.553 2.81E-03 2.89E-03 1.10E-02 -5.800 -35.61 1.34 14.311 1.023 2.096 1.564 2.67E-03 2.74E-03 1.14E-02 -6.568 -42.18 1.36 14.579 1.035 2.058 1.576 2.53E-03 2.60E-03 1.18E-02 -7.419 -49.60 1.38 14.849 1.048 2.020 1.588 2.40E-03 2.46E-03 1.22E-02 -8.368 -57.97 1.4 15.120 1.060 1.984 1.601 2.28E-03 2.34E-03 1.26E-02 -9.430 -67.40 1.42 15.393 1.073 1.949 1.614 2.16E-03 2.22E-03 1.30E-02 -10.624 -78.02 1.44 15.667 1.085 1.915 1.627 2.06E-03 2.11E-03 1.33E-02 -11.975 -90.00 1.46 15.943 1.097 1.882 1.640 1.96E-03 2.01E-03 1.36E-02 -13.514 -103.51 1.48 16.221 1.110 1.849 1.654 1.86E-03 1.91E-03 1.39E-02 -15.279 -118.79 1.5 16.500 1.122 1.818 1.668 1.77E-03 1.82E-03 1.41E-02 -17.324 -136.11 1.52 16.781 1.134 1.788 1.683 1.69E-03 1.73E-03 1.44E-02 -19.715 -155.83 1.54 17.063 1.146 1.758 1.698 1.61E-03 1.65E-03 1.47E-02 -22.546 -178.37 1.56 17.347 1.158 1.729 1.712 1.54E-03 1.57E-03 1.49E-02 -25.945 -204.32 1.58 17.633 1.170 1.701 1.728 1.47E-03 1.50E-03 1.51E-02 -30.099 -234.42 1.6 17.920 1.182 1.674 1.743 1.40E-03 1.43E-03 1.53E-02 -35.283 -269.70 1.62 18.209 1.194 1.648 1.758 1.34E-03 1.37E-03 1.55E-02 -41.925 -311.63 1.64 18.499 1.206 1.622 1.774 1.28E-03 1.31E-03 1.57E-02 -50.732 -362.36 1.66 18.791 1.218 1.596 1.790 1.22E-03 1.25E-03 1.59E-02 -62.951 -425.31 1.68 19.085 1.230 1.572 1.806 1.17E-03 1.20E-03 1.60E-02 -81.023 -506.33 1.7 19.380 1.242 1.548 1.822 1.12E-03 1.15E-03 1.62E-02 -110.433 -616.77 1.72 19.677 1.254 1.525 1.838 1.07E-03 1.10E-03 1.63E-02 -166.658 -783.42 1.74 19.975 1.266 1.502 1.855 1.03E-03 1.05E-03 1.65E-02 -316.826 -1100.25 1.745 20.050 1.269 1.496 1.859 1.02E-03 1.02E-03 4.14E-03 -171.079 -1271.33 Note that at least three significant figures should be retained in E to avoid large round-off errors when the differences are small in comparison to the values of E. In the last column, the cumulative values of x are given, and these represent the distance from the starting point where the specified depth h is reached. After the first step, uniform increments in depth h, with h increasing in the upstream direction, are utilized. The values of h are stopped at the finite limit of 1.745 m, which is 99.5% of the normal depth. The length required to reach this point is 1271 m, which is the length required for this channel to be ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 86
  18. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- considered hydraulically long, but that length varies, in general. The depth increments can be halved until the change in profile length becomes acceptably small. Alternatively, smaller increments in depth can be used in regions of rapidly changing depth, and larger increments may be appropriate in regions of very gradual depth-changes. A portion of the computed M2-profile is shown in Fig. 4.9. M2 water surface-profile computed by the direct-step method 2,4 1,8 depth (m) 1,2 0,6 0 -1200 -1000 -800 -600 -400 -200 0 distance upstream (m) Fig. 4.9. M2-curve drawn in example 4.2. Example 4.3: A trapezoidal channel with a bottom width of 5 m, a side slope of 1:1, and a Manning n of 0.013 m-1/3s carries a discharge of 50 m3/s at a bed slope of 0.0004. Compute by the direct-step method the backwater profile created by a dam that backs up the water to a depth of 6 m immediately in font of the dam. The upstream end of the profile is assumed at a depth equal to 1% greater than the normal depth. Solution: Given: bottom width: b = 5.0 m side slope ratio: m:1 = 1:1 Manning’s n: n = 0.013 m-1/3s bed slope: ib = 0.0004 discharge: Q = 50 m3/s water depth: h = 6.0 m (in front of dam) Compute the water surface profile. A M1 NDL Dam 6m CDL hn hc ib = 0.0004 Similar to Example 4.2: the normal depth and the critical depth are: [h n (b  mh n )] 5   3 Q.n hn = 2.87 m  b  2h 1  m 2  2 1 3 ib 2  n  ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 87
  19. OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------  h c  b  2h c   3    2  Q  b  2mh c  2 1 hc = 1.57 m g Because hn > hc the channel slope is mild. The profile lies in zone 1 and therefore it is an M1 curve. The range of depth is 6m at the downstream end and (101% x 2,87) = 2.90 m at the upstream end. Students should try to make a table computation, which is self- explanatory and draw an M1 curve as Fig. 4.10 below: M1 water surface-profile computed by the direct-step method 7 water depth (m) 6 in front of dam 5 4 3 2 1 0 -8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 distance upstream (m) Fig. 4.10. M1-curve drawn in example 4.3. 4.4.2. Direct numerical integration method The direct integration method is applicable to prismatic channels only. This method uses Eq. 4-18 as governing equation: dh i b  i e 1  Fr 2   dx  dh (4-20) dx 1  Fr 2 ib  ie In its integrated form, Eq. 4-20 becomes: 1  Fr 2  dx  x i1  x i   dh   g(h)dh x i1 h i 1 hi 1 (4-21) xi hi ib  ie hi The integrand on the right hand side of Eq. (4-21) is a function of h, g(h), which can be integrated numerically to obtain a solution for x, as shown as in Fig. 4.11. g(h) area = xi+1 - xi ho hi hi+1 hn h Fig. 4.11. Water surface-profile computation by direct numerical integration ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 88
  20. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- A variety of numerical integration techniques are available, such as the trapezoidal rule and Simpson’s rule, which are commonly used to find the cross-sectional area of a natural channel, for example. Simpson’s rule is of higher order in accuracy than the trapezoidal rule, which simply means that the same numerical accuracy can be achieved with fewer integration steps. Application of the trapezoidal rule to the right-hand side of Eq. (4-21) for a single step produces: g(h i 1 )  g(h i ) x i 1  x i   h i1  h i  (4-22) 2 To determine the full length of a flow profile, (xl – xo), multiple application of the trapezoidal rule results in   g(h o )  g(h l )  2 g(h i )  l 1  L  x l  x o  h  i 1  (4-23)  2      where L is the profile length and h = (hi+1 – hi) is the uniform depth-increment. Because the global truncation error in the multiple application of the trapezoidal rule is of order (h)2, halving the depth increment will reduce the error in the profile length by a factor ¼. By successively halving the depth interval, the relative change in the profile length can be calculated with the process continuing until the relative error is less than some acceptable value. ----------------------------------------------------------------------------------------------------------------------------------- Chapter 4: NON-UNIFORM FLOW 89
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