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Chapter 6

Chapter 6. Rapidly-varied Flow-1-Hydraulic Jump. 6.1 INTRODUCTION.

aaron-roberts
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Chapter 6

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  1. Chapter 6 Rapidly-varied Flow-1-Hydraulic Jump

  2. 6.1 INTRODUCTION Hydraulic jump is one subject which has extensively been studied in the field of hydraulic engineering. It is an intriguing and interesting phenomenon that has caught the imagination of many research workers since its first description by Leonardo da Vinci. The Italian engineer Bidone (1818) is credited with the first experimental investigation of this phenomenon.

  3. A hydraulic jump primarily serves as an energy dissipator to dissipate the excess energy of flowing water downstream of hydraulic structures, such as spillways and sluice gates. Some of the other uses are: (a) efficient operation of flow-measurement flumes,(b) mixing of chemicals, (c) to aid intense mixing and gas transfer in chemical processes, (d) in the desalination of sea water and (e) in the aeration of streams which arc polluted by bio-degradable wastes. A hydraulic jump occurs when a supercritical stream meets a subcritical stream of sufficient depth. The supercritical stream jumps up to meet its alternate depth. While doing so it generates considerable disturbances in the form of large-scale eddies and a reverse flow roller with the result that the jump falls short of its alternate depth.

  4. Figure 6.l is a schematic sketch of a typical hydraulic jump in a horizontal channel. Section 1, where the incoming supercritical stream undergoes an abrupt rise in the depth forming the commencement of the jump, is called the toe of the jump.

  5. The jump proper consists of a steep change in the water-surface elevation with a reverse flow roller on the major part. The roller entrains considerable quantity of air and the surface has white, frothy and choppy appearance. The jump, while essentially steady, will normally oscillate about a mean position in the longitudinal direction and the surface will be uneven. Section 2 which lies beyond the roller and with all essentially level water surface is called the end of the jump and the distance between sections 1 and 2 is the length of the jump, . The initial depth of the supercritica1stream is and is the final depth, after the jump, of the subcritical stream. As indicated earlier, wil1be smaller than the depth alternate to .

  6. The two depths and at the ends of the jump are called sequentdepths. Due to high turbulence and shear action of the roller, there is considerable loss of energy in the jump between sections 1 and 2. In view of the high energy loss, the nature of which is difficult to estimate, the energy equation cannot be applied to sections 1 and 2, to relate the various flow parameters. In such situations, the use of the momentum equation with suitable assumptions is advocated. In fact, the hydraulic jump is a typica1example where a judicious use of the momentum equation yields meaningful results.

  7. 6.2 THE MOMENTUM EQUATION FOR THE JUMP 6.2.1 General Equation for a Prismatic Channel The definition sketch of a hydraulic jump in a prismatic channel of arbitrary shape is presented in Fig. 6.2. The channel is inclined to the horizontal at an angle θ .Sections 1 and 2 refer to the beginning and end of the jump respectively.

  8. A control volume enclosing the jump as shown by dashed lines in the figure, is selected. The flow is considered to be steady. Applying the linear momentum equation in the longitudinal direction to the control volume, (6.1) where = pressure force at the control surface at section 1 = by assuming hydrostatic pressure distribution, where = depth of the centroid of the area below the water surface. = pressure force at the control surface at section 2 = if hydrostatic pressure distribution is assumed. (Note that if θ is small) = shear force on the control surface adjaent to the channel boundary.

  9. W sin θ = longitudinal component of the weight of water contained in control volume. = momentum flux in the longitudinal direction going out through the control surface = . = momentum flux in the longitudinal direction going in through the control surface = . The hydraulic jump is a rapidly-varied flow phenomenon and the length of the jump is relatively small compared to GVF profiles. The frictional force is usually neglected as it is of secondary importance. Alternatively, for smaller values of θ , (W sin- ) can be considered to be very small and hence is neglected. For a horizontal channel, θ = 0 and W sin θ = 0 .

  10. 6.2.2 Hydraulic Jump in a Rectangular Channel (a) Sequent Depth RGatio Consider a horizontal, frictionless, rectangular channel. Considering unit width of the channel, the momentum equation, Eq. (6.1), can be written in the form (6.2) Taking = =1.0 and not that by = discharge per unit width = = i.e., (6.3)

  11. On non-dimensionalising, where = Froude number of the approach flow = Solving for yields (6.4) This equation which relates the ratio of the sequent depths to the initial Froude number in a horizontal frictionless rectangular channel is known as the Belanger momentum equation. For high values of ,say > 8.0, Eq. (6.4) can be approximated for purposes of quick estimation of the sequent depth ratio as (6.4a)

  12. Equation (6.4) can also be expressed in terms of = =the subcritical Froude number on the downstream of the jump as (6.5) (b) Energy Loss The energy loss in the jump is obtained by the energy equation applied to sections l and 2 as (as the channel is horizontal, Fig. 6.1)

  13. Substituting for from Eq. (6.3) and simplifying (6.6) or (6.7) The relative energy loss But

  14. Substituting for from Eq. (6.4) and simplifying, Equation (6.8) gives the fraction of the intitial energy lost in the hydraulic jump. The variation of with is shown in Fig. 6.3 which highlights the enormous energy dissipating characteristic of the jump. At = 5, about 5o per cent of the initial energy in the supercritical stream is lost and at = 20, is about 86 per cent. Figure 6.3 also serves as a yard-stick for comparing the efficiencies of other types of jumps and energy-dissipating devices.

  15. Experimental studies by many research workers and specifically the comprehensive work of Bradley and peterka which covered a range of Froude numbers up to 20, have shown that Eqs (6.4) and (6.6) adequately represent the sequent-depth ratio and energy loss (respectively in a hydraulic jump formed on a horizontal floor.

  16. 6.3 CLASSIFICATION OF JUMPS As a result of extensive studies of Bradley and peterka the hydraulic jumps m horizontal rectangular channels are classified into five categories based on the Froude number of the supercritical flow, as follows: (i) Undular Jump: 1.0< 1.7 The water surface is undulating with a very small ripple on the surface. The sequent-depth ratio is very small and is practically zero. A typical undular jump is shown in Fig. 64(a).

  17. (ii) Weak jump: 1.7< 2.5 The surface roller makes its appearance at 1.7 and gradually increases in intensity towards the end of this range, i.e. 2.5. The energy dissipation is very small, is about 5 per cent at = 1.7 and 18 per cent at = 2.5. The water surface is smooth after the jump (Fig. 6.4(b)). (iii) Oscillating Jump: 2.5< 4.5 This category of jump is characterised by an instability of the high-velocity flow in the jump which oscillates in a random manner between the bed and the surface. These oscillations produce large surface waves that travel considerable distances downstream (Fig. 6.4 (c)). Special care is needed to suppress the waves in stilling basins having this kind of jump. Energy dissipation is moderate in this range; = 45 per cent at = 4.5.

  18. (iv)‘Steady’ Jump 4.5< 9.0 In this range of Froude numbers the jump is well-established, the roller and jump action is fully developed to cause appreciable energy loss (Fig. 6.4 (d)). The relative energy loss ranges from 45 per cent to 70 per cent in this class of jump. The ‘steady jump is least sensitive in terms of the toe-position to small fluctuations in the tailwater elevation.

  19. (v) Strong or Choppy Jump: > 9.0 In this class of jump the water surface is very rough and choppy. The water surface downstream of the jump is also rough and wavy (Fig. 6.4 (c)). The sequent-depth ratio is large and the energy dissipation is very efficient with values greater than 70 per cent. It is of course obvious that the above classification is based on a purely subjective consideration of certain gross physical characteristics. As such, the range of Froude numbers indicated must not be taken too rigidly. Local factors in stilling basin design can cause overlaps in the range of Froude numbers. Figure 6.5 (Plate 1) shows four typical hydraulic jumps in a rectangular laboratory flume.

  20. 6.4 CHARACTERISTICS OF JUMP IN A RECTANGULAR CHANNEL (a) The Length of the Jump The length of the jump is an important parameter affecting the size of a Milling basin in which the jump is used. There have been many definitions of the length of the jump resulting in some confusion in comparing various studies. It is now usual to take the length of the jump as the horizontal distance between the toe of the jump to a section where the water surface becomes essentially level after reaching the maximum depth ( Fig. 6.1). Because the water-surface profile is very flat towards the end of the jump, large personal errors are introduced in the determination of the length .

  21. Experimentally it is found that . The variation of with obtained by Bradley and Peterika is shown in Fig.6.6. This curve is usually recommended for general use. It is evident from Fig. 6.6 that while depends on for small values of the inlet Froude number, at higher values (i.e. > 5.0) the relative jump length is practically constant at a value of 6.1. Elevatorsi has shown that the data of reference l can be expressed as (6.9)

  22. (b) Pressure Distribution The pressures at the toe of the jump and at the end of the jump follow hydrostatic pressure distribution. However, inside the body of the jump, the strong curvatures of the streamlines cause the pressures to deviate from the hydrostatic distribution. Observations by Rajaratnam have shown that in the initial portions of the jump of pressures in the jump body will be less than the hydrostatic pressure. The deficit from the hydrostatic pressure increases with an increase in the initial Froude number .However, at the bottom of the channel and in a narrow region close to the bed, the pressures are essentially hydrostatic. Thus the pressure-head profile on the bed is the same as the mean water-surface profile. (c) Water Surface Profile A knowledge of the surface profile of the jump is useful in the efficient design of side walls and the floor of a stilling basin. Consider the coordinate system shown in Fig. 6.7. The coordinates of the profile are with the boundary condition that atx= 0 , y = 0 and at x = ,h = .In general, .

  23. and λ = x/X, where X = a length scale defined as the value of x at which h = 0.75( ). The variation of η with λ is given in Table 6.1. It may be noted that in the η - λ relationship the Froude number does not appear exp1icitly.In Eq. (6.10) X is the length sca1e and is given by (6.11)

  24. Equation (6.11) together with Table 6.l enables one to adequately predict the jump profile. Since the profile approaches at , asymptotically, the coordinates calculated from Table 6.1 may not exactly match the requirement of the end of the jump. For practical purposes it is suggested that the coordinates( , λ) be used to plot the profile up to λ 1.80 and then to smoothly finish the curve by joining the profile to the end of the jump at . (d) Velocity Profile When the supercritical stream at the toe enters the jump body, it undergoes shearing action at the top as well as at the solid boundaries. The top surface of the high-velocity now will have high relative velocities with respect to the fluid mass that overlays It. The intense shear at the surface generates a free shear layer which entrains the fluid from the overlying mass of fluid.

  25. The boundary shear at the bed causes a retardation of the velocity in a boundary layer. As a result of these actions the velocity distribution in a section at a distance x from the toe will be as shown in Fig. 6.8. It is seen that the velocity profile has two distinct portions-a forward flow in the lower main body and a negative velocity region at the top. In-the forward flow, the total volumetric rate of flow will be in excess of the discharge Q entering the jump at the toe. This is due to the flow entrainment at the shear layer. To maintain continuity, i.e. to account for the excess forward flow, a reverse now exists at the top. This situation results in the formation of the roller.

  26. The forward-velocity profile has zero velocity at the bed, maximum velocity at a distance and then gradually decreases to zero at a height above the bed. The region can be called the boundary layer part and the region the free-mixing zone. This velocity configuration indicates that the motion of the forward flow is similar to a wall jet except that the pressure gradient is adverse. The velocity profile and shear stress can be studied by following the methods of analysis similarto those used in the study of wall jets. The velocity u at a distance y from the bed in the boundary layer portion can be expressed by a velocity-defect law (6.12)

  27. where shear velocity and = maximum velocityat y = .In the free-mixing zone the velocity profile is found to be self-similar and can be expressed as (6.13) where= value of y at which .The maximum velocity at . It may be noted that the non-dimensionalised velocity profile is explicitly independent of and x . The scales of the above relationship are and which are given by (6.14) and (6.15)

  28. Both Eq. (6.14) and Eq. (6.15) are found to be independent of the initial Froude number . (e) Other Characteristics In addition to the characteristics mentioned above, information about shear stress and turbulent characteristics enhance one’s understanding of the jump phenomenon. It has been found that the initial boundary-layer thickness and relative roughness of the bed play a major role in these aspects. Useful information on these topics arc available in literature.

  29. EXAMPLE 6. 1 A spillway discharges a flood flow at a rate of 7.75 per metre width. At the downstream horizontal apron the depth of flow was found to be 0.50m. What tailwater depth is needed to form a hydraulic jump? If a jump is formed, find its (a) type, (b) length, (c) head loss, (d) energy loss as a percentage of the initial energy and (e) profile. Solution ,and sequent-depth: By Eq. (6.4) = required tailwater depth.

  30. (a) Type: since = 7.0, a ‘steady’ jump will be formed (b) Since > 5.0, = 6.1 = length of the jump =6.1 4.71 =28.7 m (c) = head loss (d) (e) Profile : By Eq. (6.11) and

  31. Substituting these for the values of λ and η given in Table 6.l a relation between x and h is obtained. As suggested in Section 6.4 (c), the profile is calculated up to λ 1.80, i.e. up to x = 25.0 m and then is joined by a smooth curve to the end-of the jump at x = = 28.7 m. the change in the depth in this range would be 0.0433 3.16 = 0.14m. This being a flat curve, i.e., a change of 0.14 m in 3.70 m the procedure as above is adequate

  32. EXAMPLE 6.2 A rectangular channel carrying a supercritical stream is to be provided with a hydraulic-jump type of energy dissipator. If it is desired to have an energy loss of 5.0 m in the jump when the inlet Froude number is 8.5 determine the sequent depths. Solution , and By Eq. (6.4), By Eq. (6.7),

  33. ,and EXAMPLE 6.3 An overflow spillway (Fig.6.9) is 40.0 m high. At the design energy head of 2.5 m over the spillway find the sequent depths and energy loss in a hydraulic jump formed on a horizontal apron at the toe of the spillway. Neglect energy loss due to flow over the spillway face. (Assume = 0.738). Solution The discharge per metre width of the spillway is

  34. By the energy equation (Energy loss over the spillway is neglected) By trial-and –error By Eq.(6.4),

  35. Energy loss (Eq. 6.6) = Energy at section 1= 42.5m Percentage of initial energy loss

  36. 6.5 JUMPS IN NON-RECTANGULAR CHANNELS If the side walls of a channel are not vertical, e.g. in the case of a trapezoidal channel, the flow in a jump will involve lateral expansion of the stream in addition to increase in depth. The cross-sectional areas are not linear functions of the depth of flow. This aspect introduces not only computational difficulties in the calculation of the sequent-depth ratio but also structural changes in the jump. A brief introduction to this rather wide field of jumps in non-rectangular channels is given in this section.

  37. Consider a horizontal frictionless channel of any arbitrary shape, such as in Fig.6.2. For a hydraulic jump in this channel, the general momentum (Eq. (6.1)) with the assumption of = = 1.0 reduces to: (6.16) i.e. (6.16a) where A = area of cross-section any = depth of the centre of gravity of the area from the water surface. Rearranging Eq. (6.16)

  38. i.e. (6.17) i.e. (6.17a) The term is the specific force (Section 1.11 (c)).

  39. The specific force is a function of the depth of flow, channel geometry and discharge. A parabolic curve with two distinct limbs resembling the specific-energy curve is obtained for plots of vs y for a given in a given channel (Fig.6.10). The lower limb represents the supercritical flow and the upper limb the subcritical flow. An ordinate drawn at a given cuts the curve at two points A and B where the respective depths represent the sequent depths for the given discharge. The point C corresponding to the merger of these two depths is obviously the critical depth for the given flow . The specific-force diagram provides a convenient means of finding sequent depths for a given discharge in a given horizontal channel. If suitably non-dimensionalised, it can provide a quick graphical solution aid in cases involving a large number of calculations.

  40. For small and isolated calculations, Eq. (6.17a) is solved by a trial-and-error procedure to obtain the sequent depths. The energy loss due to a jump in a non-rectanglar horizontal channel is (6.18)

  41. EXAMPLE6.4 A trapezoidal channel is 2.0 m wide at the bottom and has side dope of 1.5 horizontal : 1 vertical. Construct the specific-force diagram for a discharge of 135 in this channel. For this discharge find the depth sequent to the supercritical depth of 0.5 m.

  42. Solution The channel cross-section is show as an inset Fig. 6.10. For a trapezoidal section

  43. Specific force Values of were computed using this equation for different y values, ranging from y = 0.1 m to y =6.0 m and is shown plotted in Fig. 6.10. From Fig. 6.10, the depth sequent to = 0.5 m is = 2.38 m (point B).

  44. (b) Sequent-depth Ratios Expressions for sequent-depth ratios in channels of regular shapes can be obtained by re-arranging the terms in Eq. (6.17) as: Noting that , and on re-arranging

  45. Substituting the expression for A, T and pertinent to the given geometry will lead to an equation relating the sequent-depth ratio to the inlet Froude number and other geometric parameters of the channel. In most non-rectangldar (6.19) channels Eq. (6.18) contains the sequent-depth ratio in such a form that it needs a trial-and-error procedure to evaluate it. Reference 7 gives useful information of hydraulic jumps in all shapes of channels. (c) Jumps in ExponentiaI Channels Exponential channels represent a class of geometric shapes with the area related to the depth as A = in which and a are characteristic constants. For example, values of 1.0, 15 and 2.0 for a represent rectangular, Parabolic and triangular channels respectively.

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