Flow Characteristics and Modelling of Head-discharge Relationships for Weirs

Transcription

1 Chapter 8 Flow Characteristics and Modelling of Head-discharge Relationships for Weirs 8.1 Introduction In Chapters 5 and 7, the formulations of the numerical models for the simulations of flow surface and bed pressure profiles of steady rapidly varied open channel flows were discussed. These models have been developed for the solutions of various types of Boussinesq equations for the purpose of examining the nature of their solutions. In this chapter, detailed descriptions of the free and submerged flow characteristics of a family of broad-crested weirs will be presented. The main differences between the flow behaviours of short- and broad-crested weirs will be briefly described first, followed by the discussion of the hydraulics of a broad-crested weir. Also, the nature of the weir flow problems and the procedure for the numerical simulation of these flow problems for establishing head-discharge relationships will be described. A brief discussion of the numerical results in comparison with experimental data will be presented at the end of this chapter. All discussions in this chapter will be focused on broad-crested weirs with finite upstream and downstream slopes. 8.2 Critical flow condition and flow control in open channel Critical flow theory Critical flow is an intermediate flow state between subcritical and supercritical flows in which the energy per unit weight of the flow is minimum for a given discharge or alternatively it is a flow state corresponding to maximum unit discharge for a constant head (Jaeger, 1949, cited in Hager, 1985b). This state of flow is characterised by flow possessing a velocity equal to the translational velocity of a small wave. The velocity of propagation of such a wave on still water in a non-rectangular channel (Montes, 1998, p35) reads as c = r ga αt, (8.1) 221

2 222 Chapter 8. Flow Characteristics and Modelling of Head-discharge where c is the velocity of the wave and T is the top width of the channel. Subcritical flow is known by the mean flow velocity less than the velocity of the wave whereas in supercritical flow the opposite feature is true. Thus, the changes in depth communicated by the passage of such waves depend on the relative velocity of the waves with respect to the local flow. In the vicinity of the critical section the flow possesses appreciable streamline curvatures and slopes which influence the magnitude of the critical depth of flow. A general expression, which accounts for the effects of the streamline curvature, for predicting the critical depth (Jaeger, 1957, p140) for flow in any cross-sectional shape of a channel is given by αq 2 T K p ga = α Fr 2 =1, (8.2) 3 K p where K p is the pressure correction coefficient. Assumption of constant values of α and K p at a section ( α/ H = K p / H =0) is used to develop this equation. In a number of cases of practical importance, however, the critical depth may be calculated using reasonable simplifying assumptions based on the condition of the flow problem considered. For instance, for transcritical flow over a broad-crested weir the deviations of the correction coefficients, K p and α, from unity are insignificant (α = K p = 1.0). For such type of weir, the resulting simplified equation can be used to estimate the critical depth of the flow. The critical flow condition described above allows determination of the head-discharge relationships for long-based weirs under free flow conditions Flow control sections in open channel At a critical control section, the relationship between the depth and the discharge is unique, independent of the channel roughness and other uncontrolled circumstances. Such a unique stage-discharge relationship offers a theoretical basis for the measurement of discharge in open channels. In practice the section of a long prismatic channel with mild slope serves as a controlfor establishing a relationship between head and discharge (Fenton, 2001). In this case the control is due to friction in the channel giving a unique relationship between the flow and the slope of the channel, the stage, channel geometry, and roughness. Montes (1998, p50) states that the unique features of a critical control section compared to other sections, which also exhibit a single valued stage-discharge relationship, is the fact that in a critical control section there is a passage from subcritical to supercritical flow so that the upstream region is isolated from small perturbations generated downstream. The precise location of such control sec-

3 Chapter 8. Flow Characteristics and Modelling of Head-discharge 223 tion is dependent on the slope of the channel, and it occurs in a channel reach where the slope changes from mild to steep. Artificial control structures such as a weir can also be used as a flow control device in the absence of a control section in open channels. Due to the existence of a critical section on the crest of the structures, the upstream flow becomes independent of the tailwater flow condition. This results in the establishment of a consistent relationship between the overflow head and the discharge. Similar to the channel control section, the discharge is computed from a single measurement of depth upstream of the control structure. The present study focused on a control established by an embankment type of flow control structure. 8.3 Classification of weirs and location of gauging stations General discussion and classification of weirs Aweirwithfinite crest width in the direction of the flow is referred to as long-based weir (Chadwick and Morfett, 1998, p403). The long-based weir may also be classified based on the value of the overflow head to crest length ratio as a broad-crested or short-crested weir. A broad-crested weir is an overflow structure with a horizontal crest above which the deviation from a hydrostatic pressure distribution because of vertical acceleration may be neglected. In other words, the streamlines are nearly straight and parallel. The criterion to obtain this situation as reported by Bos (1978, p15) is the length of the weir crest in the direction of flow L w, should be related to the total energy head over the weir crest H 1, as 0.08 H 1 /L w If H 1 /L w is less than 0.08, then the energy losses above the weir crest cannot be neglected, and undulations may occur on the crest. If H 1 /L w is less than or equal to 0.50, then only slight curvature of streamlines occurs above the crest and a hydrostatic pressure distribution may be assumed. Experimental studies of flow over a broad-crested weir indicate that the flow passes through the critical stateatsomesectiononthecrest,andthelocationofthissectionvariesappreciablywith head and weir proportions. The broad-crested weir is an intermediate case between a transition in which the flow is wholly curvilinear and in which boundary resistance predominates, and hence both accelerative and viscous effects must be considered in its analysis. However, the influ-

4 224 Chapter 8. Flow Characteristics and Modelling of Head-discharge ence of the latter effect on the discharge capacity of the weir is limited to small overflow head. Short-crested weirs are those overflow structures in which the streamlines of the flow above the weir crest have pronounced curvatures and slopes (see Figure 8.1). This characteroftheflow streamlines has a significant influence on the head-discharge relationships of the structures. As discussed by Bos (1978, p27), the main difference between a broad-crested weir and a short-crested weir is that nowhere above the short crest can the curvature of the streamlines be neglected; there is thus no hydrostatic pressure distribution anywhere over the crest of the weir. Depending on the magnitude of the ratio of the overflow head to the length of the crest of the weir, the same flow measuring structure can act as a broad-crested weir or a short-crested weir. H 1 /L w = 0.33 is the delineating value which separates these two types of flow control structures. The two-dimensional flow patterns over a short-crested weir require the application of a higher-order flow model for the complete description of the flow problem. Figure 8.1: Flow over a short-crested weir (q = 462.1cm 2 / s) Gauging stations for overflow head and tailwater depth The head measurement station for free flow conditions should be located sufficiently far upstream of the structure to avoid the influence of the curvature of the flow surface on the head measurement. It should also be close enough to minimise the energy loss between the head measurement station and the structure. Harrison (1967) suggested that for a streamlined broad-crested weir the position of the upstream gauging station should be at least 1.7 times the overflow head upstream from the face of the structure. Bos (1978, p58) generalised the location of this station for different weir profiles and

5 Chapter 8. Flow Characteristics and Modelling of Head-discharge 225 recommended the position of the gauging station at a distance equal to between two to four times the maximum overflow head from the structure. The accuracy of the discharge measurement depends solely on the precision of the reading of the head over the control structure. In this thesis, a procedure based on the established knowledge for the location of this station will be incorporated in the numerical model, which simulates transcritical flow over trapezoidal profile weirs, for the purpose of developing headdischarge relationships for these weirs. Detailed discussion of the procedure will be presented in Section 8.7. Observation of the flow surface profile on the downstream side of a broad-crested weir for free flow conditions reveals that the tailwater depth increases gradually with distance in the direction of the flow. From the theoretical point of view, the downstream measuring gauge should be located at a section free of surface drawdown effects. However, it is difficult to generalise the location of this section due to the complex nature of the submerged flow behaviours downstream of the weir. For the purpose of analysing flow behaviour related to submerged flow conditions, the tailwater depth can be measured at a well-prescribed point in which the influences of the curvature of the water surface and the tailgate effect on the tailwater depth measurement are insignificant. 8.4 Submergence ratio and modular limit The submergence ratio at a flow control structure may be defined as the ratio of the downstream flow depth to the upstream depth of flow above the crest of the structure. For low submergence ratios, critical flow occurs at some section on the crest of the structure, and the tailwater conditions have no effect on the upstream flow depth. This flow condition is referred to as modular flow. For such flow condition, the discharge is computed from a single measurement of depth upstream of the control structure. At very high submergence ratios, critical flow no longer exists at any section on the crest of the structure. This non-modular flow condition requires that two flow depths upstream overflow depth and tailwater depth above the crest be measured to approximate the discharge of the control structures. The modular limit is the value of the submergence ratiowhentheflow just begins to be affected by the downstream level, that is, when the flow control structure begins to be drowned (see e.g., Bos et al., 1984, p66). From the practical point of view, the modular limit is very important in defining the limit up to which a flow control structure can be calibrated for discharge measurement. In terms of

6 226 Chapter 8. Flow Characteristics and Modelling of Head-discharge flow depths above the crest, the modular limit is described as: M L = h l, h 1 (8.3) where: M L = modular limit, h l = modular limit tailwater depth with reference to the crest, h 1 = sill-referenced upstream flow depth. The roughness of the surface of the control structure and approach channel have significant influence on the values of the modular limit (Kindsvater, 1964). The effect is to decrease its value. However, the height of the weir has little influence on the modular limit of the weir. Bos (1985, #4.25) presented an analytical procedure for estimating the modular limit of a long-throated flume (also valid for an hydraulically similar broadcrested weir)based on the estimation of the total energy losses between the upstream and downstream gauging stations. In this study, the limiting tailwater depth for a given discharge was determined experimentally by gradually raising the tailwater and observing the tailwater level at which the upstream water level began to rise. The result of these experiments to define the modular limit of the trapezoidal profile weirs is shown in Figure 8.2. This figure indicates that the modular limit of the weirs decreases with increasing of the ratio of the crest referencedheadtothelengthoftheweircrest,h 1 /L w. This implies that the effect of the curvature of the streamline of the flow over the crest is to decrease the modular limit of the weir (see e.g., Bos, 1978, p86). For the case of flow over a broad-crested trapezoidal profile weir, the curvature of the streamline at the control section weakly affects the modular limit of the flow. As a result such type of control structure has a higher modular limit compared to a short-crested trapezoidal profile weir (see Figure 8.2). This fact suggests that flow over a broad-crested weir takes place with minimum loss of head as compared to a similar short-crested trapezoidal profile weir. For a long broad-crested weir (H 1 /L w 0), an upper limit value of M L =0.85 is reached (Kindsvater, 1964). 8.5 Description of weir flow Nature of the flow problem Theoretical analyses of flows over trapezoidal profile weirs are complicated by a combination of effects related to the approach channel condition, geometry of the control

7 Chapter 8. Flow Characteristics and Modelling of Head-discharge Submerged flow 0.6 Free flow M L Kindsvater (1964) Lw = 40 cm Lw = 10 cm H 1 /L w Figure 8.2: Modular limit for trapezoidal profile weirs structure, property of the fluid and flow patterns. Experimental investigations are also complicated by the occurrence of several significantly different flow patterns under free and submerged flow conditions. The characteristic difficulties associated with the flow phenomena of these weirs makes both the theoretical and experimental analyses more complicated and difficult. The flow patterns for trapezoidal profile weirs involve boundary-layer growth under conditions of acceleration and separation, and non-hydrostatic pressure and nonuniform velocity distributions due to the curvature of the streamlines over the crest of the weir. It is evident that a general analytical expression for the discharge of the weir is impossible due to the complicated nature of the flow. However, existing theoretical procedures can be used to predict the discharge under specific conditions that are related to the overflow head to crest length ratio. On the other hand, a numerical model based on a higher-order governing equation may be used to establish head-discharge relationships for practical solution of the flow problems. Examining the flow profile over a trapezoidal profile weir shows that the upstream far side flow surface profile approaches the normal flow depth asymptotically for approach channel having mild or steep bed slope. On the downstream side of the structure, for different tailwater depths below the modular limit tailwater depth the flow

8 228 Chapter 8. Flow Characteristics and Modelling of Head-discharge changes from one flow regime to another without affecting the upstream flow situation. These flow characteristics of a trapezoidal profile weir complicate the locations of the inflow and outflow sections of the computational domain of the numerical model. However, systematic specification of the boundary values for a particular flow condition may give better numerical solution of the problem and overcome the limitations associated with the existing theory. The numerical solution of such model must be validated with experimental data for assessing the drawbacks of the model Flow behaviour over short- and broad-crested weirs The actual flow behaviour over short- and broad-crested control structures is quite complex, involving a three-dimensional velocity pattern as well as viscous effects. The viscous effects, which are more pronounced for the case of long broad-crested control structure, modify the distribution of the velocity of the flowandcausealossofenergy. For such structures, a correction factor should be introduced to account for viscous effects. For flow over a control structure, the two significant parameters which describe the flow characteristics are the ratios of the overflow head to the height of the control structure, and to the crest length of the structure. The former ratio is a function of the Froude number of the incoming flow in the channel and it indicates the significance of the velocity head of the flow in the prediction of the total overflow head. At relatively higher Froude number, the approach velocity cannot be neglected in determining the upstream total overflow head. The ratio of the overflow head to the crest length of the structure measures the shortness or breadth of the structure. The value of this ratio directly determines the degree of the curvature of the streamline of the flow over the crest of the control structure for free flow conditions. Escande (1939, cited in Fritz and Hager, 1998) classifies various types of flows over cylindrical-crested weirs. This classification is equally applicable to other geometrical shape flow control structures such as trapezoidal profile weirs (see e.g., Kindsvater, 1964; Wu and Rajaratnam, 1996; Fritz and Hager, 1998). Four different types of flow may occur depending on the height of the tailwater depth. These are the: 1. Free overflow; 2. Plunging flow;

9 Chapter 8. Flow Characteristics and Modelling of Head-discharge Surface wave flow; and 4. Surface jet flow. Free overflow For free overflow conditions, the critical depth is located at the crest of the trapezoidal profile weir somewhat downstream of the upper edge of the crest. The position of the critical section varies along the weir crest depending on the magnitude of the discharge asshowninfigures8.3and8.4. For a supercritical flow state at the outflow section, a pure transcritical flow without any shock wave can be observed under free flow conditions. For larger tailwater depth, a hydraulic jump occurs with its toe located at or downstream of the toe of the control structure (see Figure 8.5). Fritz and Hager (1998) noted that the flow configuration of the jump at the downstream side of the trapezoidal profile weir is identical to the classical hydraulic jump. The position of the jump varies along the bed of the downstream channel and is entirely controlled by the level of the downstream tailwater depth. For free flow conditions, when the surface tension effects are negligible, the discharge over the weir depends on the head above the crest of the weir and is independent of the tailwater depth. The common broad-crested weir equation can be applied to estimate the discharge of the structure provided that the curvature of the streamlines of the flow over the crest is insignificant. Plunging flow The first detailed experimental analysis of submerged flow patterns downstream of the trapezoidal profile weirs was given by Fritz and Hager (1998). Different flow characteristics were observed for the downstream submerged flow cases. These observations are very similar to the results of the experimental studies of submerged flow patterns downstream of a sharp-crested weir (except breaking wave) by Wu and Rajaratnam (1996). For a given discharge as the tailwater depth increases, the submerged flow downstream of the trapezoidal profile weir passes through several regimes. Figure 8.7 illustrates the observed flow surface profiles for different flow regimes for flow over a trapezoidal profile weir. For the plunging jet flow, the flow over the weir plunges into the tailwater with a concentration of forward flow along the bottom and a backward flow (surface roller) at the surface (see Figure 8.6). The main flow diffuses as a plane submerged jet along the downstream face of the structure, and hits this face and bed of the channel. The

10 230 Chapter 8. Flow Characteristics and Modelling of Head-discharge 0.3 Bed profile Flow surface (m) H1/Lw = H1/Lw = H1/Lw = H1/Lw = H1/Lw = Critical depth Horizontal distance (m) Figure 8.3: Free flow over a short-crested trapezoidal profile weir 0.3 Bed profile H1/Lw = Flow surface (m) H1/Lw = H1/Lw = H1/Lw = H1/Lw = Critical depth Horizontal distance (m) Figure 8.4: Free flow over a broad-crested trapezoidal profile weir

11 Chapter 8. Flow Characteristics and Modelling of Head-discharge 231 Figure 8.5: Transcritical flow over a weir with hydraulic jump (q =462.1cm 2 / s) position of the starting point of the plunging flow varies along the downstream face of the structure depending on the tailwater depth. The concentration of eddies decreases with increasing of tailwater depth of this flow regime. For tailwater level nearly equal to the upper limit tailwater depth, the starting point of the plunging flow is just at the downstream edge of the crest of the structure. Shifting of the plunging flow starting position by a small distance upstream of the downstream end section of the crest changes the flow pattern into surface wave flow. Figure 8.8 shows the velocity distribution profile for the plunging flow regime. It is this flow pattern which causes maximum erosive velocities on the downstream face of the trapezoidal profile weirs (Kindsvater, 1964). Figure 8.6: Flow pattern for plunging flow regime (q = 462.1cm 2 /s)

12 232 Chapter 8. Flow Characteristics and Modelling of Head-discharge Flow surface (m) Bed profile Free overflow Plunging flow Surface wave flow Surface jet flow 1: x = m 2: x = m q = cm 2 /s Horizontal distance (m) Figure 8.7: Typical flow profiles for different flow regimes 0.8 Measured (1, x = m) Measured (2, x = m) 0.6 h s /H Horizontal velocity (cm/s) Figure 8.8: Velocity distribution in plunging flow regime (q = 222.4cm 2 / s)

13 Chapter 8. Flow Characteristics and Modelling of Head-discharge 233 Surface wave flow Increasing the tailwater level to a point above the upper limit depth for plunging and below the modular limit tailwater depth results in the surface wave flow which is characterisedbythepresenceofthefirst standing wave near the downstream end of the weir crest followed by waves of decreasing amplitude (see Figures 8.7 and 8.9). The forward flow for this flow regime is along the surface and backward flow along the bottom and lower face of the weir. This fact is clearly indicated in the velocity profile which is shown in Figure The position and amplitude of the standing waves depend on the level of the tailwater depth. At higher tailwater depth, the stationary waves near the end section of the crest have considerable curvature. Further downstream, however, the curvatures of the waves are insignificant. For tailwater depth less than the modular limit tailwater level, the flow transition over the control structure is from subcritical to supercritical and then to subcritical with minimum loss of energy. The interesting thing is that the latter transition is with the formation of undular surfaces which resemble in character the nearly two-dimensional (without the appearance of any cross waves) undular hydraulic jump. Fawer (1937, cited in Jaeger, 1957, p152) stated that the surface of the jump is undular if the supercritical sequent depth is greater than 67% of the critical depth of flow. The analysis of the experimental data of this study confirms Fawer s observation (see Table 8.1). When the trough of the first standing wave passes through a flow section near the axis of symmetry of the weir, the flow over the structure completely changes to submerged flow. Compared to the plunging flow regime, the erosive tendencies of this flow regime are a minimum. Table 8.1: Supercritical sequent depth to critical depth ratios for undular jump Discharge (cm 2 / s) Conjugate depth, h 1 (cm) Critical depth, H c (cm) h 1 /H c (%) Surface jet flow Inthecaseofsurfacejetflow in which the tailwater level is above the modular limit tailwater depth, the approach flow is completely submerged. The configuration of the

14 234 Chapter 8. Flow Characteristics and Modelling of Head-discharge Figure 8.9: Surface wave flow regime (q =462.1cm 2 / s) 0.8 Measured (1, x = m) Measured (2, x = m) 0.6 h s /H Horizontal velocity (cm/s) Figure 8.10: Velocity distribution in surface wave flow (q = 222.4cm 2 / s)

15 Chapter 8. Flow Characteristics and Modelling of Head-discharge 235 flow profile depends on the degree of submergence of the flow. For lower submergence ratios, undular surfaces can be seen on the crest of the weir. At higher submergence ratios, the flow profile becomes almost horizontal as indicated in Figures 8.7 and The forward flow is again along the surface with bottom recirculation in the region near the downstream end of the structure (see Figure 8.12). In both the surface wave and surface jet flow cases, the flow remains as a jet at the surface in the downstream channel. For flow control structures such as weirs, the discharge over the structure up to the modular limit tailwater depth depends on the upstream overflow head. Previous experimental investigations (see e.g., Kindsvater, 1964) for flow over broad-crested trapezoidal profile weirs show that this depth corresponds to the surface wave flow regime of the downstream submerged flow condition. This in turn implies that the free flow discharge equation can be used to estimate the discharge over a broad-crested type of such weir not only in the pure transcritical flow condition but also in the downstream submerged flow condition up to the modular limit tailwater depth. If the tailwater depth is above the modular limit depth, the flow control structure will submerge totally and moreover, it no longer acts as a flow control structure. Because of submergence, the discharge capacity of the structure decreases markedly. This is the behaviour of the surface jet flow regime Free flow transition ranges For a definite range of tailwater levels, a given discharge produces either a plunging flow or a surface wave flow on the downstream side of the trapezoidal profile weirs. The transition from plunging flow to surface wave flow, and vice-versa, occurs within a well defined range of tailwater levels (Kindsvater, 1957). The upper limit of the transition range corresponds to the maximum tailwater depth in which the plunging flow remains as a stable plunging flow. A small increment of the tailwater depth above the upper limit value abruptly changes this flowpatterntosurfacewaveflow. The lower limit of the transition range is the minimum tailwater level in which the stable surface wave flow pattern converts abruptly to plunging flow for tailwater depth slightly below this minimum value. Between the upper and lower limits of the transition either flow may occur. This particular switching mechanism adds to the complexity of the submerged flow pattern downstream of the weir. Figures 8.13 and 8.14 show details of flow near

16 236 Chapter 8. Flow Characteristics and Modelling of Head-discharge Figure 8.11: Submerged flow over a weir with 97% submergence ratio 0.8 Measured (1, x = m) Measured (1, x = m) 0.6 h s /H Horizontal velocity (cm/s) Figure 8.12: Velocity distribution in surface jet flow regime

17 Chapter 8. Flow Characteristics and Modelling of Head-discharge 237 the lower and upper limit of the transition range. In practice these sequences of events would normally occur during arisingandfallingofflood stages. The slope of the downstream face of the weir and surface roughness influence the transition range limits. Surface roughness tends to lower the upper and lower limit values of the transition range (Kindsvater, 1964). Figure 8.13: Flow near the lower limit transition range (q = 512.2cm 2 / s) In this study, the upper and lower limits of the transition range for flow over trapezoidal profile weirs were determined experimentally by gradually raising and lowering the tailwater and observing the levels at which the transition took place. Figure 8.15 shows the results of the tests made to define the lower and upper transition stage using the mean curves of the experimental data. In this figure the transition submergence, ξ t = h tc /h 1 (h tc and h 1 = crest-referenced tailwater depth and overflow depth respectively) is shown versus the ratio of the overflow head to weir crest length, H 1 /L w. The transition range separates the upper and lower limit curve, starting at 0 < ξ t < 0.60 for H 1 /L w =0, decreasing to < ξ t < for H 1 /L w = As the overflow head to weir crest length ratio increases, the upper and lower limit curves of the transition range rise gradually particularly for H 1 /L w 0.20 (see Figure 8.15). This shows that the effect of the curvature of the streamlines of the flow over the crest of the weir is to decrease the transition range of the free flow. It clear from this figure that flows are always plunging for tailwater level below the weir crest but above the subcritical sequent depth of the free jump (i.e., ξ t < 0). It also indicates that surface wave flows always occur for transition submergence value greater than (i.e., ξ t > 0.874).

18 238 Chapter 8. Flow Characteristics and Modelling of Head-discharge Figure 8.14: Flow near the upper limit transition range Transition range 0.6 ξt Upper limit Lower limit Kindsvater (1964) H 1 /L w Figure 8.15: Free flow transition range for trapezoidal profile weir

19 Chapter 8. Flow Characteristics and Modelling of Head-discharge Nature of bed pressure distributions in different flow regimes The level of the tailwater depth affects the distribution of the pressure on the surface of the trapezoidal profile weir. Figure 8.16 illustrates the observed bed pressure profiles in different flow regimes (the case of rising tailwater depth). In the regions around the corners of the broad-crested trapezoidal profile weir, the streamline curvatures for the free flow regime are very sharp (see Figure 8.4). Consequently, the pressure distributions strongly deviate from hydrostatic in these regions. For tailwater depth below the upper limit transition depth, the submerged flow on the downstream side of the weir becomes plunging flow. In this flow regime, the bed pressure on the downstream face of the trapezoidal profile weir is higher than the corresponding pressure for free flow situation due to the increasing of the level of the tailwater on this face of the weir. Examining the flow profile of this flow regime for tailwater depth very close to the upper limit transition depth (see Figure 8.14) reveals that the pressure distribution is nearly hydrostatic. For lower tailwater depth in the plunging flow regime, the curvature of the streamline influences the bed pressure distributions on the downstream face of the structure. 0.2 Weir crest Bed pressure (m) Free flow Plunging flow q = cm 2 /s Surface wave flow Surface jet flow Horizontal distance (m) Figure 8.16: Typical bed pressure distributions for different flow regimes (L w = 400 mm) The surface wave flow regime is characterised by the presence of waves of decreasing amplitude in the direction of the flow as shown in Figure 8.7. This implies that the

20 240 Chapter 8. Flow Characteristics and Modelling of Head-discharge pressure distribution within this flow regime is non-hydrostatic. As the tailwater level approaches the modular limit tailwater depth value, the influence of the curvature of the flow surface on the pressure distribution increases with increasing of the curvature of the waves. Compared to the plunging flow bed pressure values, the bed pressures on the downstream face of the structure are relatively large due to the increasing of tailwater level and flow surface curvature (this is true for positive curvature effects). At a higher degree of submergence, the flow profile in the surface jet flow regime is nearly horizontal. For this condition, the pressure distribution within the flow region is purely hydrostatic. The shape of the bed pressure distribution curve along the length of the structure is similar to the inverted shape of the trapezoidal profile weir (see Figure 8.16). The bed pressure corresponding to this flow regime is the maximum of all the cases of the flow regimes related to the weir flow situations Hydraulics of broad-crested weirs Free flow discharge equation Because of the complex nature of the flow over weirs which is influenced by roughness, turbulence levels, geometry of the structure and several other parameters, it is difficult to develop a free flow discharge equation precisely. The discharge formulas, independent of the shape of the weir, relate the discharge to the upstream overflow depth via the discharge coefficient. Hence, the discharge coefficient represents the combined effects of all parameters that influence the free flow pattern. The accuracy of determining the discharge over the weir under free flow conditions depends on the reliability and validity of the discharge coefficient, and the sensitivity to the measurement of head over the flow control structure. For a broad-crested weir with rectangular control section, the head discharge relation that takes into account the approach velocity head is given by the following equation: Q = C d B p 2g where: C d = discharge coefficient, V 0 = approach velocity, B = width of weir perpendicular to the flow. µh 1 + αv 2 3/2 0, (8.4) 2g

21 Chapter 8. Flow Characteristics and Modelling of Head-discharge 241 In the above equation, equation (8.4), an energy head correction coefficient, α, is introduced to take into account non-uniformity of the approach velocity arising from the effects of the shape of the channel cross-section, boundary irregularities and curvature of the streamlines. For low overflow energy head with respect to weir height (H 1 /H w < 1/6), the magnitude of the approach velocity is very small. However, the non-uniformity of the approach velocity increases with increasing of the overflow energy head relative to the weir height. According to Bazin (1898, cited in Fritz and Hager, 1998), the energy head correction factor for large approach velocity is 5/3. French (1985, #8.3) based on the work of Bos (1978), provides the following criteria to analyse flow over a broad-crested weir: H 1 /L w < 0.08, flow over the crest is subcritical, and the weir cannot be used to determine the discharge H 1 /L w 0.33, the discharge equation used in this range will estimate the flow rate accurately H 1 /L w 1.50 to 1.80, theweirisnolongerbroadcrested but it should be classified as a short-crested weir H 1 /L w, the flow pattern over the weir crest is unstable and the nappe may separate completely from the crest. The weir characteristics approach a sharpcrested weir. Bos (1978, p28) suggests that for flow over a short-crested weir with rectangular control section, a head-discharge equation similar in structure to equation (8.4) can be used for discharge computation. The discharge coefficient takes into account the effect of the streamline curvature besides other factors. Such type of weir has a higher discharge coefficient compared to a broad-crested weir due to the substantial curvature of the streamline over the crest of the weir. Empirical discharge equation for submerged flow Equations of discharge for free flow have been derived on the basis of a simple energy analysis. The analysis was made possible because critical-flow control occurs on the crest of the control structure when the flow is under free flow conditions. For submerged overflow conditions, the flow passes over the structure in a subcritical state so that the discharge depends on both the upstream and downstream water levels. Several investi-

22 242 Chapter 8. Flow Characteristics and Modelling of Head-discharge gators proposed different empirical formulas to predict this non-modular discharge. Du Buat (1816) presented an equation for the computation of submerged discharge over a trapezoidal shaped weir. He considered the submerged flow as a flow consisting of free flow over the weir and flow through a submerged orifice under the tailwater referenced head. This simplifying assumption cannot be justified but gives a framework in which a constant can be attached to the equation. This equation has the form (Ellis, 1947, p79): Q s = C s B p h 1 h tc h h tc, (8.5) where: Q s = discharge passing under submerged condition, C s = coefficient of non-modular discharge. The value of C s in this equation must be determined by experiment. The accuracy of the prediction of the submerged discharge using equation (8.5) depends not only on the measurement of the upstream and downstream depths but also on the predetermined value of the submerged discharge coefficient. However, it is difficult to formulate a submerged flow equation independent of the tailwater depth. The most convenient alternative is an empirical solution based on experimental data analysis and free flow discharge equation (see e.g., Hager, 1994; Wu and Rajaratnam, 1996; Fritz and Hager, 1998). A simple functional relationship, which describes the effect of submergence on the discharge capacity of the weir, in terms of the free flow discharge, q free can be expressed as q sub = ϕ 0 q free, ϕ 0 = G µ htl h ll (8.6a), (8.6b) where: q sub = the submerged flow rate of the control structure, G = function for the submerged flow reduction factor, h tl = tailwater depth with reference to the modular limit depth (h t h l ), h t = the tailwater depth above the channel bed at the downstream gauging station, h ll = overflow depth with reference to the modular limit depth (H h l ), H = channel bed referenced upstream depth of flow at the corresponding gauging station, ϕ 0 = the submerged flow reduction factor.

23 Chapter 8. Flow Characteristics and Modelling of Head-discharge 243 It has to be noted that the ratio, h tl /h ll, isnotthesameasthesubmergence ratio of a flow control structure which was defined in Section 8.4. The above functional relationship can be determined from the plot of experimentally determined discharges (free and submerged) and submergence ratio with reference to the modular limit tailwater depth using data modelling techniques. Since the free flow discharge is constant over the full range of the conditions considered, the empirical solutions based on equation (8.6b) are adequate to predict the submerged discharge. This method will be applied in this study to establish an empirical relationship for the submerged discharge of the trapezoidal profile weirs. The results will be presented in Section 8.9. Submerged flow control structures are not recommended for the practical measurement of discharge. This is because of the following reasons (Bos, 1985, p66): the submerged flow reduction factor for a given control structure is not only a function of submergence ratio but also a function of the free overflow discharge. However, this discharge is to be measured. for a control structure with higher modular limit, the submerged flow reduction factor cannot be determined at the required accuracy. Any errors related to the measurement of both the upstream and downstream heads with respect to the crest of the weir directly influence the value of the submergence ratio. estimation of the non-modular flow rate of a structure requires measurement of both the upstream and downstream crest-referenced heads. From the practical point of view, however, measurement of two heads is time consuming and expensive. the submerged flow reduction factor for a given flow control structure is basically determined based on available experimental data. This requires the construction of different sizes of physical models to conduct the experiments in a laboratory for the complete range of discharges. In general, this process is relatively expensive. 8.6 Theoretical weir discharge coefficients The common head-discharge relationship for a flow control structure is formulated based on a number of idealised assumptions such as absence of energy losses between the gauging and control sections; uniform velocity distribution in both sections; and negligible streamline curvatures at the gauging and control sections. However, these simplifying assumptions are not often correct. A discharge coefficient must be intro-

24 244 Chapter 8. Flow Characteristics and Modelling of Head-discharge duced to take into account the effects of these assumptions in the estimation of discharge. This coefficient is theoretically determined by assuming hydrostatic pressure distribution at the crest and face of the flow control structure. However, the curve for the actual pressure distribution at these sections lies below the curveforthehydrostatic pressure distribution due to the effect of the negative curvature of the streamlines. It was shown by Matthew (1963) that the discharge coefficient for flow over a circular-crested weir is affected by both the streamlines curvatures and the absolute scale of the flow. Free flow over a weir is characterised by flow transition from subcritical to supercritical states. In the vicinity of this transition the streamlines of the flow have considerable curvature and slope. Owing to this fact, the application of the conventional method of analysis to such kind of flow problem results in underestimating the discharge capacity of the weir. Based on the BTMU equation, a general expression for weir discharge coefficients will be developed here. The theoretical discharge coefficient expression includes terms which account for the impact of the curvature of the streamlines. This implies that the effect of the non-hydrostatic pressure distribution is implicitly incorporated in the resulting discharge coefficient equation. Rewriting equation (8.4) for the head-discharge relationship under free flow conditions as q = C d p 2gH 3/2 1, (8.7) where: q = discharge per unit width, H 1 = upstream total energy head above the weir crest (h 1 + αv 2/2g). 0 Using the flow equation, equation (3.50a), the discharge capacity of the control structure as a function of the flow depth and other hydraulic parameters at a section can be written as q 2 = gh (H x + Zb ³ 0 ³ + S f) β ω 1 H 2ϕ xxx + ξ 0 H xx +2 ω 0 Z 000 b ω 1 b Z0 b + Z00 2 H + β Hx H 2, (8.8) where: H = flow depth above the bed, ϕ = 1+Z 02 b.

25 Chapter 8. Flow Characteristics and Modelling of Head-discharge 245 For a trapezoidal-shaped flow control structure, the contribution of the bed curvature is zero, i.e. Zb 00 = Z000 b =0. Using this fact, equation (8.8) reduces to q 2 gh (H x + Zb 0 = + S f) β ω 1 (H. (8.9) 2ϕ xxx + ξ 0 H xx )+β Hx H 2 Equation (8.9) includes terms which reflect the effect of the curvature of the streamlines. However, for flow over a broad-crested weir with H 1 /L w 0.50, the flow surface over thecrestoftheweirhasalmostaconstant slope with negligible curvature (H xxx = H xx =0). For this kind of flow phenomenon, equation (8.9) becomes q 2 = gh3 (H x + Zb 0 + S f). (8.10) βh x Using equation (8.7) in equation (8.10) and further simplifying, one obtains à C d = 1 2H 3/2 1! µh 3 (H x + Zb 0 + S 1/2 f). (8.11) βh x Equation (8.11) relates the coefficient of discharge with the hydraulic parameters of the flow and the total head at the gauging stations. This equation is valid for flow over the weir with hydrostatic pressure distribution or insignificant curvature of streamline. In contrast to a broad-crested weir, the flow pattern over a short-crested weir is characterised by pronounced curvatures of streamlines. This behaviour influences the headdischarge relationship as well as the modular limit of the flow control structure. Inserting equation (8.7) into equation (8.9) and simplifying the resulting expression yields the following equation: Ã!Ã! 1 H (H x + Zb 0 C d = + S 1/2 f) 3/2 2H β ω 1 (H. (8.12) 1 2ϕ xxx + ξ 0 H xx )+β H x H 2 If the geometric characteristics of the surface streamline at a particular section (for instance, crest section near the axis of symmetry of the weir) over the crest of the weir is known in addition to the flow parameters, one can use equation (8.12) to estimate the coefficient of discharge for flow over a short-crested weir with H 1 /L w between 0.50 and The values of H xxx, H xx and H x can be determined numerically using experimental data for a given trapezoidal weir geometry to obtain the discharge coefficient, C d. Using measured values of discharge and overflow head, the experimental discharge coefficients were computed from equation (8.7). These values are compared with the theoretical discharge coefficients estimated from equation (8.12) for free flow conditions

26 246 Chapter 8. Flow Characteristics and Modelling of Head-discharge in Figure The figure also shows the mean trend curve for the experimental discharge coefficients and the corresponding equation. It can be seen from this figure that the agreement between the predicted and the experimental result is fairly good C d = (H 1 /L w ) (H 1 /L w ) (H 1 /L w ) C d Experimentally determined Predicted H 1 /L w Figure 8.17: Comparison of experimentally determined and predicted discharge coefficients 8.7 Model development for establishing head-discharge relationships Formulation of the boundary value problem As described before, a trapezoidal shaped weir over which water is flowing may be treated as a short- or broad-crested weir depending on the magnitude of the overflow head to weir crest length ratio. When free flow exists over the trapezoidal profile weirs, the common free flow discharge equation, equation (8.4), may be used to determine the quantity of water flowing over the weir with reasonable accuracy. However, this equation does not give satisfactory results for the case of flow over a short-crested weir due to the pronounced curvatures of streamlines over the crest of the weir. This indicates that a general model, which includes the effects of the curvature of the streamlines implicitly or explicitly, is essential for establishing discharge rating curves for such types of weirs. Therefore, the main objective of this part of the thesis is to examine the

27 Chapter 8. Flow Characteristics and Modelling of Head-discharge 247 feasibility of the BTMU equation for the development of head-discharge relationships for short- and broad-crested trapezoidal profile weirs. Hence, the research questions to be answered in this simulation study are: 1. As a one-dimensional model typically does not have the ability to define a coefficient of discharge, how can flow over a trapezoidal profile weir be accurately portrayed? 2. Does the numerical model give similar overflow heads to those determined experimentally? The solution procedure of the numerical model to estimate the crest-referenced head corresponding to the discharge over the trapezoidal profile weirs under free flow conditions using the BTMU equation requires the specification of three boundary values. It was mentioned in the preceding section that different flow regimes exist on the downstream side of this weir corresponding to different tailwater depths under the same free flow conditions. These flow situations complicate the precise measurement of the tailwater depth for free flow conditions. From the practical point of view, it is advantageous to measure one flow depth in the subcritical flow region for the purpose of calibrating the head-discharge relationships of the weirs. Because of these, a numerical procedure based on the specification of boundary values at the inflowsectiononlyisemployed here for the solution of the BTMU equation to provide head-discharge relationships for such types of weirs. This approachappliesasolutionprocedurebasedonthenewton- Raphson iterative scheme (similar to the method discussed in Chapter 5), which is entirely different from a solution procedure based on the conventional shooting method. The flow profile at the upstream far section is asymptotic to the normal depth of flow. Depending on the height of the flow control structure and the slope of the approaching channel, there are two possible flow surface profiles. If the height of the flow control structure is greater than or equal to the normal depth of the approach flow, the flow profile is M1 type or a backwater curve. Otherwise the profile is M2 or a drawdown curve. However, in most practical cases the flow control structure is such that the approach flow profile is a backwater curve. For this flow simulation problem, the unknowns are a set of flow depths along the length of the computational domain for a given discharge of the weir. Two additional equations that relate the flow parameters with the slope and curvature of the flow surface at the