Chapter 3 Theoretical Discussion and Development of Model Equations 3.1 Introduction The need for studying the applicaility of the Boussinesq-type momentum equation with pre-assumed uniform centrifugal term (BTMU model) for modelling free surface flow prolems with non-hydrostatic pressure and/or nonuniform velocity distriutions has een discussed in the previous chapter. In susequent sections of this chapter, the nature and the hydraulic characteristics of free surface curved flow will e discussed. summary of the derivation of the governing equations of this study as well as a simplified version of these equations for weakly curved free surface flow situation will e presented firstinthischapter,followedyariefdiscussion of the development of a Boussinesqtype linear model equations ased on different simplifying assumptions. 3. General discussion 3..1 Characteristics of free surface curved flow Free surface flows are defined as flows where one or more of the oundaries is not physically constrained ut can adjust to conform to the flow conditions (Liggett, 1994, p393). freesurfaceissujecttoatmosphericpressure. Thefreesurfaceflow conditions are complicated y the fact that the position of the free surface is likely to change with respect to time and space and also y the fact that the depth of flow, the discharge, and the slope of the ottom flow oundary and of the free surface are interdependent. In unsteady free surface flow prolems, the location of the free surface changes continuously and its evolution has to e determined. In the case of steady free surface flows, the free surface is not known a priori and the ojective is to locate it correctly. The quantitative description of such prolems depends on successfully locating the position of the free surface. 47
48 Chapter 3. Theoretical Discussion and Development of Model Equations Free surface flows with streamlines possessing small radii of curvature, and discontinuities, are grouped together as rapidly varied flows (Jaeger, 1957, p71). Curvature of the solid flow oundaries may e one of the causes for the flows to deviate from a rectilinear path. The physical characteristics of these flows are asically fixed y the geometry of the solid flow oundary as well as y the state of the flows. The important feature of such flows is therefore the strong departure from hydrostatic pressure distriution induced y the strong vertical components of the centrifugal acceleration. The presence of a vertical component of the centrifugal acceleration also modifies the velocity distriution across the flow. Consequently, the flow prolems have a two-dimensional nature in the vertical plane. This suggests that the modelling of such flow prolems should take into account the streamline curvature effects. Since rapid variation in flow characteristics of free surface curved flows occurs in a relatively short length, the oundary friction which plays a primary role in a gradually varied flow is comparatively small and in most cases insignificant. few examples of curved flows in hydraulic engineering are: i) flow over a spillway; ii) dam-reak flow; iii) a hydraulic jump. curvilinear flow may e either a concave flow or convex flow depending on the position of the centre of curvature with respect to the flow oundaries. In concave flow where the centre of curvature is aove the flow oundary, the vertical component of the centrifugal acceleration and the gravitational acceleration act in the same direction. Consequently, the curve for the actual pressure distriution lies aove the curve for the hydrostatic pressure distriution. For the case of convex flow (the centre of curvature is elow the flow oundary negative curvature), the resulting pressure is less than the hydrostatic pressure. The deviation of the pressure from the hydrostatic distriution is strongly influenced y the degree of the curvature of the streamlines. For weakly curved flows with small curvature of streamlines such as gradually varied flow, the hydrostatic law of pressure distriution can e used to approximate the pressure distriution at a vertical section (Chow, 1959, p31). However, pressure correction factors should e included to take into account the effect of the centrifugal acceleration in the analysis of curved flow prolems with sustantial curvature of streamlines using the lowest-order one-dimensional equation such as the depth averaged Saint-Venant equations (see e.g., Yen, 1973). The velocity distriution at any section in free surface flow depends on the roughness of the channel flow oundary (sidewalls and ottom), the presence of free surface, shape of
Chapter 3. Theoretical Discussion and Development of Model Equations 49 thecross-sectionandthepresenceof ends. The roughness of the ottom flow oundary changes the velocity profile gradient in the region close to the ed. Similarly, the sidewalls of the channel section influence the velocity distriution pattern of the flow. The action of the sidewalls of the channel is to reduce the velocity very close to the surface due to the existence of a secondary motion which is superimposed on the main longitudinal motion of the flow (Gison, 1909, cited in Montes, 1998, p87). In contrast to the theoretical and experimental results of the location of maximum velocity for flow in awidechannel,flow in a narrow channel attains maximum velocity at a point eneath the free surface. The dip in maximum value of the velocity is well correlated with the aspect ratio of the channel defined y width to flow depth ratio. Rajaratnam and Muralidhar (1969) found that in smooth laoratory flumes where the aspect ratio exceeded 7 there was no perceptile influence of sidewalls, ut this decrease in surface velocity was very acute for narrow channels. lthough the flow in a straight and prismatic channel is three-dimensional due to the presence of spiral motion, the velocity components in the transverse direction are small and insignificant compared to the longitudinal velocity component. In addition to the effect of surface roughness, the normal acceleration due to the curvature of the streamline also modifies the distriution of the velocity at a vertical section. Experimental studies (Rajaratnam and Muralidhar, 1971; Rajaratnam et al., 1976; Ramamurthy et al., 1994) show that the variation of the horizontal velocity of the flow with depth is nonlinear for the case of free surface flow with considerale curvature of streamlines. The common practice for incorporating the effect of nonuniform velocity distriution in the development of flow governing equations ased on energy and momentum principles is y introducing thecoriolisandboussinesqcoefficients in the respective methods as a correction factor for the assumed mean velocity. When the flow is nearly steady and uniform, the contriutions of these two correction coefficients for such flow situation are relatively minor (Xia and Yen, 1994). The Boussinesq and Fawer assumptions for the distriution of the vertical flow velocity over the depth have een used widely to simplify the modelling of free surface curved flow prolems (Chaudhry, 1993, p163). However, any crude assumption for the distriution of the velocity of such flow may affect the final solutions of the flow equations.
50 Chapter 3. Theoretical Discussion and Development of Model Equations 3.. Streamline curvature parameters Free surface curved flow is characterised y the presence of curvature of streamlines. The two important parameters which define the geometry of such streamlines are the radius of curvature and the angle of inclination. The functional representation of a streamline for flow over curved oundaries can e written as z = z (x), (3.1) where z (x) is a streamline function at a height z aove a horizontal datum. For any single streamline passing through a location z aove the datum, knowing the form of z (x), its radius of curvature R and angle of inclination θ can e found. Thus, tan θ = d z (x), (3.) dx ³1+ d z (x) /dx 3/ R =. (3.3) d z (x) /dx t the free surface, it is easy to determine the radius of curvature and the streamline angle directly from the flow surface profile using equations (3.) and (3.3). For any position aove the flow oundary within the flow region, the streamline inclination and radius of curvature are evaluated ased on the values of the streamline function at different sections. However, the determination of the values of the streamline function at the interior points using the stream function equation requires knowledge of the velocity distriution of the flow over the depth. In flow surface profile computational prolems particularly over curved oundaries, the distriution of the velocity is not known in advance. This requires the assumption of either the distriution of the velocity or the streamline parameters for the modelling of such flows. Since at the free surface the streamline function is descried y the equation of the flow surface profile, sustituting z (x) =η(x) in equation (3.3) gives the following expression for the radius of curvature of the streamline at this level: 1 R = d η(x)/dx 1+(dη(x)/dx). (3.4) 3/ For flows with limited slopes of streamlines, 1+(dη(x)/dx) = 1, equation (3.4) reduces to the following approximate expression: 1 R = d η(x) dx. (3.5)
Chapter 3. Theoretical Discussion and Development of Model Equations 51 Theanalysisofcurvedflow prolems in the vertical plane needs further knowledge of the distriutions of the streamline curvature parameters over the depth of flow. The experimental study of flow over a circular crested weir (Ramamurthy et al., 1994) shows that the variation of the streamline inclination and curvature at a vertical section is linear over a wide range of depth except in a narrow segment elow the free surface. The assumption of a linear variation of these parameters fromtheedtothefreesurfaceis central to the existing theoretical models for free surface curved flow prolems such as the Boussinesq models. For instance, Boussinesq (1877) applied a linear variation of curvature of the streamlines with depth to model the flow prolem y the momentum principle. This supposition, along with the assumption of a linear variation of streamline angle, has een extensively utilised for developing flow models for such types of prolems (e.g., Matthew, 1963; Hager and Hutter, 1984a; Hager, 1985a; Bhallamudi, 1994; Dey, 1998a,). s noted y Montes (199), this linearity assumption restricts the applications of these lower-order models to quasi-uniform flow situations. Fawer (1937) employed exponential variation of the streamlines curvature to model similar flow prolems y the energy principle. Recently, Fenton (1996) applied a different approach ased on the centrifugal term, κ/ cos θ, to incorporate the effect of the curvature of the streamline y assuming a constant value of this term at a vertical section. Compared to all these methods, however, the Fawer s approach was not systematic or rigourous to model higher-order equations. ll these show that the two-dimensional nature of the flow (in the vertical plane) can e included implicitly in the one-dimensional models y assuming certain distriutions for the variation of curvature and/or slope of the streamline with depth. 3.3 Velocity distriution coefficients The energy correction coefficient, α, was originally introduced y Coriolis (1836). Based on momentum considerations, Boussinesq (1877) introduced the momentum correction coefficient, β, much in analogy to the energy correction coefficient with 1 β α. Using the definition of the rate of transport of momentum and kinetic energy y the flow through a channel cross-section, the Coriolis and Boussinesq coefficients are written as follows: α = R u3 d U 3, (3.6a)
5 Chapter 3. Theoretical Discussion and Development of Model Equations β = R u d U, (3.6) where: u = time-averaged point velocity, U = mean flow velocity. In these equations, the effects of secondary currents and turulence are not incorporated. Yen (1973) presented a general equations for α and β as tensors ased on the temporal average velocity over turulent fluctuation. The momentum and energy correction coefficients which reflect the extent of nonuniform distriution of velocity in a channel cross-section generally referred to as velocity distriution coefficients. These coefficients must e evaluated ased on known distriution of the flow velocity efore one can accurately use the lowest-order depth-averaged flow governing equations to analyse two-dimensional flow prolem in the vertical plane. In most practical cases, the velocity distriution coefficients have often een assumed to e unity and the flow equations solved in an approximate way. However, this could limit the application of such types of equations to flow situations where the vertical velocity component is insignificant. In curved free surface flow, the streamline curvature influences the distriution of the velocity over the depth of flow. This implies that the streamline curvature has a direct impact on the magnitudes of the energy and momentum correction coefficients. s discussed efore, the roughness of the channel also influences the distriution of the velocity. Li and Hager (1991) showed that for uniform free surface flow, the energy and momentum correction coefficients depend significantly on the friction coefficient only. ll these suggest that the magnitude of the energy and momentum correction coefficients depend oth on the roughness of the channel and curvature of the streamlines for the case of free surface flow with appreciale non-hydrostatic pressure distriution effects. The velocity distriution coefficients tend to vary from section to section for the case of nonuniform free surface flow. They also tend to vary with the flow depth and discharge. For parallel flow, the velocity distriution is strictly uniform across a channel section (neglecting the effect of oundary roughness) and the coefficients for the velocity distriution can e taken as unity. If the cross-section of a channel is regular and its
Chapter 3. Theoretical Discussion and Development of Model Equations 53 alignment is fairly straight, the effect of nonuniform velocity distriution on the magnitude of the velocity distriution coefficients are considered as small and often the values of these parameters are assumed to e unity. However, in most practical situations the values of the velocity distriution coefficients are greater than unity. Chow (1959, #.7) stated that for fairly straight prismatic channels, the momentum and energy correction coefficients vary approximately from 1.01 to 1.1, and 1.03 to 1.36 respectively and for channels with complex cross-section the coefficients for energy and momentum can easily e as great as 1.6 and 1. respectively, and can vary quite rapidly from section to section in case of irregular alignment. In this thesis, equations (3.6a) and (3.6) will e solved numerically using experimentally determined velocity profile data for flow over trapezoidal profile weirs to assess the variation of the velocity distriution coefficients along the length of the flow domain upstream of the axis of symmetry of these weirs. Some of the results will e used to examine the influence of the momentum correction coefficient on the solution of the numerical model. The numerical procedure for estimating α and β will e descried in the following susection. 3.3.1 Numerical estimation of the velocity distriution coefficients For the case of free surface flow in a prismatic rectangular channel, d = Bdz and = BH. Sustituting these values in equations (3.6a) and (3.6), and simplifying the resulting expressions yield α = R H 0 u3 dz U 3 H, (3.7) R H 0 β = u dz U H. (3.8) For discrete velocity data at a relatively small interval, the integral of the horizontal component of the velocity over the depth can e evaluated using the two-point formula or Trapezoidal rule. In general, for function, G(z), which has known values at different points in the domain the two-point formula (Press et al., 00, p135) can e written as Z z G1 + G G(z) dz = h + O(h 3 f 00 ), z 1 (3.9) where: G 1 and G = are the values of the function at z 1 and z respectively, h = (z z 1 ) is the step size.
54 Chapter 3. Theoretical Discussion and Development of Model Equations For known velocity values at m nodal points, the integral part of equations (3.7) and (3.8) ecome Z Z u 3 dz = 1 u dz = 1 mx i=1 mx i=1 u 3 i + u 3 i+1 Zi, (3.10) u i + u i+1 Zi, (3.11) where: Z i = normal distance etween level i and i +1, u i and u i+1 = are the local time-mean horizontal velocity at level i and i +1respectively. In order to otain relatively accurate numerical estimate of the integration, the step size of the integration should e reasonaly small. Similarly, the average velocity of the flow is estimated using the following expressions: R U = ud = Z udz = 1 R H 0 udz H, (3.1) mx (u i + u i+1 ) Z i. (3.13) i=1 Inserting equations (3.10) and (3.13) in equation (3.7), and equations (3.11) and (3.13) in equation (3.8) and simplifying the resulting expressions give the following equations to evaluate the energy and momentum coefficients numerically: α = 4H P m i=1 u 3 i + u 3 i+1 Zi ( P m i=1 (u i + u i+1 ) Z i ) 3, (3.14) β = H P m i=1 u i + ui+1 Zi ( P m i=1 (u i + u i+1 ) Z i ). (3.15) In this work, the velocity distriutions for free flows over trapezoidal profile weirs at different sections were measured using the coustic Doppler Velocimeter (DV) for the purpose of predicting the velocity distriution coefficients at these sections numerically. Because of the technical limitation of the velocity measuring instrument (DV instrument), it is difficult to otain velocity readings very close to the ed and free surface. Prediction of the unrecorded velocities especially in these regions is necessary for estimating the velocity distriution coefficients at a section. The logarithmic velocity profile law will e applied here to fit the measured velocity data at different sections in the sucritical flow region upstream of the edge of the crest of the weir. Hence, the velocities near the ed and free surface can e easily extrapolated using this velocity dis-
Chapter 3. Theoretical Discussion and Development of Model Equations 55 triution model for turulent flow over smooth or rough oundaries. For this purpose, the logarithmic velocity profile equation can e written as u(z) =a 0 + a 1 ln(z), (3.16) where: u(z) = time average velocity at a distance of z from the ed, a 0 = u κ ln 9u ν for smooth ed, or ³ a 0 = u 30 κ ln k s for rough ed, u = the oundary shear velocity, k s = the Nikuradse equivalent sand roughness, κ = von Kármán s constant (κ =0.4), ν = the kinematic viscosity of the fluid, a 1 = u /κ. t any vertical section in the flow domain, the friction velocity depends on the oundary friction and hence, a 0 and a 1 attain constant values at a section. The two constants in the aove equation, equation (3.16), are determined from experimental data using the Chi-square technique for unweighted data. The Chi-square merit function (Press et al., 00, #15.1-15.) for the velocity distriution model is written as χ (a 0,a 1 )= mx ui a 0 a 1 ln(z i ), (3.17) i=1 where σ i is the standard deviation of the measured data at point i, and u i is the oserved velocity at a normal distance of z i aove the flow oundary. In most practical cases, the uncertainties related to a set of measurements are unknown in advance. The common procedure is to assume all measurements have the same standard deviation and assign an aritrary constant value of σ to all points. fter fitting for the model parameters y minimising the Chi-square merit function, the value of σ is recomputed using the following equation: mx σ (u i a 0 a 1 ln(z i )) =. (3.18) m i=1 Since our aim is to determine the parameters of the model, equation (3.17) is minimised to yield a 0 and a 1. t its minimum, the partial derivatives of the Chi-square merit function, equation (3.17), with respect to a 0,a 1 vanish. χ mx ui a 0 a 1 ln(z i ) = =0, (3.19a) a 0 i=1 σ i σ i
56 Chapter 3. Theoretical Discussion and Development of Model Equations χ mx ln(zi )(u i a 0 a 1 ln(z i )) = =0. (3.19) a 1 σ i=1 i If we define the following expressions for the sum of the different variales, we can write the solutions of the aove equations, equations (3.19a) and (3.19), in simple forms. S = S z = S zz = S u = mx i=1 1, σ i mx ln(z i ), i=1 mx i=1 σ i u i σ i, mx (ln(z i )), mx u i ln(z i ) S uz =. σ i=1 i Using the aove relations, the solutions of the two unknowns are i=1 σ i a 0 = S zzs u S z S uz SS zz (S z ), (3.0a) a 1 = S uzs S z S u SS zz (S z ). (3.0) Equations (3.0a) and (3.0) give solutions for the logarithmic velocity distriution model parameters a 0 and a 1. Basedontheestimatedmodelparameters, equation (3.16) can e used to predict the point velocities very close to the ed and the free surface in order to estimate the values of α and β numerically at the section. The results of the analysis will e presented in Chapter 4. 3.4 Flow oundary resistance The shear stress at the flow oundaries offers resistance to flow. Henderson (1966, p88) stated that the existence of a free surface and the wide variety of possile cross-sectional shapes, each with its own distriution of shear stress round the solid flow oundary make the oundary shear nonuniform. Because of the variation in resistance along the wetted perimeter and ecause of the shape of the channel cross-section, secondary currents are usually set up in turulent free surface flows. The effect of these circulating currents in the plane of the channel cross-section is to modify the velocity distriution at a vertical section and also its pattern influences the distriution of the shear stress along the
Chapter 3. Theoretical Discussion and Development of Model Equations 57 solid oundary of the flow. In most practical cases of flow simulations (except for flow in curved channels), the effects of secondary currents are negligile. lthough the distriution of shear stress is fairly uniform across the wetted oundaries, the assumption of uniform distriution of oundary shear stress is employed in the modelling of free surface flow prolems. Flow resistance in free surface flow is influenced y a numer of factors esides viscosity and flow oundary roughness that are commonly considered. The following factors, which have significant contriutions to flow resistance in open channel flow, are identified (Rouse, 1965): cross-sectional shape of the channel; non-uniformity of the channel in oth profile and plan; and flow unsteadiness. ny change in shape or size of the flow section that occurs with distance along the channel axis will necessarily e accompanied y a change in the wetted perimeter and the distriutions of velocity and shear. Similarly, a change in depth, either as a function of time or distance, produces a change in the velocity distriution. For the case of gradually varied flow, it remains customary to ignore such niceties on the assumption that the resistance at any section is approximated y the conditions of uniformity. The error involved in this assumption is negligile due to the weak curvature of the streamlines. However, the assumption of conditions ofuniformitydoesnot give satisfactory results for flows with appreciale curvatures of streamlines due to the rapid change of thesizeoftheflow section with distance. Flow resistance equation such as the Manning s equation neglects the effects of oundary non-uniformity, channel alignment and unsteadiness on the flow resistance. However, the changes in size or shape of the crosssection occurring along a channel can play a more dominant role in the flow resistance than the oundary roughness (Rouse, 1965). s mentioned efore, the velocity distriution in free surface curved flow is nonuniform due to the presence of centrifugal acceleration. The resistance term asically derived for the depth-averaged Saint-Venant flow equations is applicale for flow on an inclined straight channel with uniform flow velocity. Dressler and Yevjevich (1984) have shown that the curvature of the oundary influences the hydraulic resistance contriution of the ed and the wall. The result of the study indicates that the resistance effect on the flow
58 Chapter 3. Theoretical Discussion and Development of Model Equations is increased for flow over concave ed and decreased for flow over convex oundary. However, the modified resistance function for the Dressler equations, which includes the ed curvature effect, is too complicated for practical application especially for the modelling of flow prolems with considerale curvatures of streamlines. 3.5 Hydraulically smooth and rough surfaces Flow oundaries may e classified as hydraulically smooth and rough surfaces ased on the physical nature of the roughness elements. The height of the protrusions of the roughness elements relative to the thickness of the laminar su-layer determines the effect of the oundary roughness on the resistance of the flow. In flow profile simulation procedure, such surface characteristics have to e considered for reasonale estimation of the frictional resistance of the flow oundaries. For flow on flat plates or in pipes, Schlichting (1960, p51) gives the following expression to estimate the critical roughness height: where k c is the critical roughness height. k c = 5C z g ν U, (3.1) ccording to Schlichting (1960, p50), an hydraulically smooth surface is a flow oundary which has a roughness height less than the critical roughness height predicted y equation (3.1). For such flow oundary, the surface irregularities are so small that all roughness elements are entirely sumerged in the laminar su-layer. Consequently, the roughness of the surface has no effect upon the flow outside the laminar su-layer. For hydraulically rough surfaces, the roughness height of the protrusions of the flow oundary exceeds the limiting value of the critical roughness height. The roughness elements of such surface have sufficient height and angularity to extend their effects eyond the laminar su-layer. In the current study, curved flows over oth hydraulically smooth and rough surfaces are considered for the purpose of numerical simulations. In the course of flow profile computations, different resistance equations are used for estimating the frictional resistances of the flow oundaries depending on the oundary surface conditions. In the following section, the modelling of the hydraulic resistancetermsaswellasthepredictionsofthe slope and curvature of the flow surface in a quasi-uniform flow region will e discussed.
Chapter 3. Theoretical Discussion and Development of Model Equations 59 3.6 Predictions of flow resistance, slope and curvature of the flow surface 3.6.1 Flow resistance modelling In hydraulic engineering practice, the simulation of flow surface profile usually involves the use of Manning s equation as the appropriate means to compute the friction slope, which is included in the flow equations. The form of the Manning s equation that was originally estalished for steady uniform flow is assumed to provide a reasonale approximation for the friction slope in steady nonuniform flow. However, its application to unsteady nonuniform open channel flows has not een theoretically justified. Neglecting the effect of curvature of the flow oundary on flow resistance and also the contriution of the transverse velocity, the friction slope can e computed from the following equation: S f = n mq, (3.) R 4/3 h where: n m = Manning s roughness coefficient, R h = hydraulic radius. Equation (3.) applies only to channels with hydraulically rough oundaries. This restriction does not pose any prolem in engineering practice. However, in a laoratoryflume, usually made of plexiglass, the oundaries are hydraulically smooth and the Manning s equation is inappropriate. In such case, the Darcy-Weisach equation is used to estimate the flow resistance of the oundaries. The merican Society of Civil Engineers Task Committee (1963) reported that for open channel roughness similar to that encountered in pipes, the resistance equation similar to those of pipe flows are adequate for the estimation of the friction factor. The Darcy-Weisach equation for pipe flow resistance is read as where: f = friction factor, D = diameter of the pipe. S f = fu gd, (3.3) For open channel flow with smooth walls, the pipe diameter is replaced y 4R h. Using this fact in equation (3.3), S f = fq 8gR h. (3.4)
60 Chapter 3. Theoretical Discussion and Development of Model Equations Many different accurate formulas (e.g., Prandtl s, von Kármán s, and Colerook and White s equations) have een proposed in the literature to determine the friction factor. However, in these equations the friction factor appears on oth sides of the equations which need an iterative technique to solve these equations for f. The Haaland (1983) explicit accurate formula is used in this study to estimate the friction factor in equation (3.4). This explicit equation reads as à à k/d f = 1!! 10/9 log 3.4 10 + 6.90, (3.5) 3.70 where: k = roughness height, R e = 4R h U/ν is the Reynolds numer. In hydraulic practice, most free surface flow conditions are fully rough turulent and the friction factor depends on the equivalent relative roughness of the flow oundary only. However, for the experimental smooth flume, k = 0, and the friction factor is exclusively dependent on the Reynolds numer. R e f = 1 6.90 log 3.4 10. (3.6) R e s stated in the previous section, free surface curved flows on hydraulically smooth and rough flow oundaries are considered in the current study for validating a Boussinesqtype flow model. The development of this model will e descried in Section 3.7. Depending on the conditions of the solid oundaries of the flow prolems, either equation (3.) or (3.4) will e employed in the numerical models to estimate the friction slope at various nodal points in the computational domain. 3.6. Slope and curvature of the flow surface For the purpose of simulating curved flow prolems using the flow equations considered in this work, the inflow section of the computational domain is located in a quasiuniform flow region. Detailed description of the formulation of the oundary value prolem for the solutions of the model equations will e presented in Chapter 5. The quasi-uniform flow situation efore the inflow section facilitates the application of the gradually varied flow equation to compute the oundary values at this section accurately. Hence, the slope of the water surface, S H, is directly evaluated using the following equa-
Chapter 3. Theoretical Discussion and Development of Model Equations 61 tion: S H = dh dx = S 0 S f 1 βfr, (3.7) where Fr is the Froude numer. For the sucritical flow state upstream of a hump, the Froude numer squared is sufficiently small near the inflow section and can e neglected. Using this approximation, equation (3.7) can e written as dh dx = S 0 S f. (3.8) Differentiating equation (3.8) with respect to x gives κ H = d H dx = S0 f, (3.9) where κ H is the curvature of the flow surface. Now it is possile to otain an expression for Sf 0 in terms of the channel geometric properties. From the relation of discharge and conveyance factor, and Manning s law, S f = Q K (H), K = 1 5/3 n m P, /3 where: K = the conveyance factor, P = the wetted perimeter of the flow area. (3.30a) (3.30) and differentiating equation (3.30a) gives Sf 0 = Q dk dh K 3 dh dx. (3.31) Making use of equation (3.30), we can get an expression for the derivative of the conveyance factor: dk dh = 1 5 /3 d n m 3P /3 dh 5/3 dp. (3.3) 3P 5/3 dh Sustituting equations (3.8) and (3.3) in equation (3.31) and simplifying the resulting expression gives the following equationafterinsertinginequation(3.9): κ H = Q 5 /3 d n m K 3 3P /3 dh 5/3 dp (S 3P 5/3 0 S f ). (3.33) dh For the case of rectangular flow section, d/dh = B; dp/dh =;R h = /P and Q = qb. Making use of these facts, equation (3.33) ecomes κ H = q B ³ 5BR /3 3n m K 3 h 4R 5/3 h (S 0 S f ). (3.34)
6 Chapter 3. Theoretical Discussion and Development of Model Equations For hydraulically smooth oundaries such as plexiglass laoratory flumes, the Darcy- Weisach formula is used to compute the slope of the energy grade line or the friction slope. Following the same procedures as aove and approximating the hydraulic radius of the flow y the depth, the curvature of the flow surface for smooth oundaries ecomes κ H = 3fq 8gH 4 (S 0 S f ). (3.35) In this work, either equation (3.33) or (3.35) will e used depending on the flow oundary conditions to simulate free surface curved flow prolems using the Boussinesq-type energy and momentum equations. s mentioned in the previous section, only the evaluation of the friction slope term in equation(3.7)is affected y the condition of the flow oundary. 3.7 Development of the Boussinesq-type momentum equations 3.7.1 Summary of the derivation of the BTMU equations Fenton (1996) developed a Boussinesq-type momentum equation in a cartesian coordinate system ased on the assumptions which were stated in Chapter for curved flow in a prismatic channel with ed curvatures. Since the geometry of most hydraulic structures changes oth at the ottom and sidewalls oundaries, this equation will e redeveloped in more general form to include such geometric variations. In this section the method of the development of thisequationwillepresented. The one-dimensional dynamic equation (Fenton, 1996) which was developed ased on the consideration of momentum in a control volume was used to otain governing equations for flow with significant curvature of streamline. In this equation, the pressure gradient term should e expressed as a function of the centrifugal acceleration so that it can take into account the effect that the curvature of the streamline has on the pressure distriution. simple approximation ased on the distriution of the streamline curvature parameters is used here for this purpose. The simplified dynamic equation can e written as Q t + β Q + 1 Z p x ρ x d + gs f =0. (3.36) The possile variations of the channel width in the direction of the flow have een incorporated in this equation.
Chapter 3. Theoretical Discussion and Development of Model Equations 63 Defining a cartesian coordinate system (x, y, z), with x horizontally along the channel, y in the transverse direction perpendicular to the plane of the flow, and z vertically upwards as shown in Figure 3.1. The important term for descriing the two-dimensional nature of the flow in the aove equation is the pressure gradient term R p/ xd.the conventional approximation in hydraulic engineering is that the pressure is given y the equivalent static head of the flow aove each point. Z Free surface H(x) Z (x) Flow ed oundary X Figure 3.1: Definition sketch for free surface curved flow Here an attempt is made to go further than that approximation y allowing for centrifugal effects on the pressure of the flow. ssuming that the vertical pressure gradient is given y the usual gravitational component plus a centrifugal contriution, the expression for this gradient can e written as 1 p ρ z = g κv cos θ, where κ is the local streamline curvature, V is flow velocity tangent to the streamline, and θ is its angle of inclination to the horizontal. Note that in this method we are ignoring the curvature of the streamlines in the horizontal plane, which should e much less significant compared to the curvature in the vertical plane. This is essentially a two-dimensional theory, where the contriutions of the transverse velocity components are also neglected. In such case V = u + w where u and w are flow velocities in the
64 Chapter 3. Theoretical Discussion and Development of Model Equations x and z directions respectively, and in fact, V = u/ cos θ, so that the pressure gradient is given y 1 p u = g κ ρ z cos θ, where, in general, κ, u, andθ are all functions of position over the section. It is assumed that the centrifugal term can e expressed as a constant over the section (Fenton, 1996) such that the aove expression can e written as 1 p ρ z = g γ Q c, (3.37) where γ c is a coefficient such that Z κ u cos θ d γ c = = β Q κ cos θ U U = β κ cos θ, whichcanvaryfromsectiontosectionalongthe length of the channel. This approach is slightly different from the original Fenton s approach in the sense that it accounts for the effect of nonuniform velocity distriution. Integrating equation (3.37) in the vertical etween a general point at z and free surface at η, as we have asserted that the right hand side is constant over a section, p ρ = g + γ c Q (η z), (3.38) assuming that the free surface is horizontal in the transverse direction, such that the pressure is not a function of y. Now, differentiating equation (3.38) with respect to x 1 p g ρ x = Q η Q γ + γ c +(η z) c x x γ cq 3 x + γ cq Q. (3.39) x Integrating equation (3.39) over the cross-section of the channel with the sustitution of the first moment of area of the cross-section aout a transverse axis at the free surface level y d, where d is the depth of the centroid of the section elow the free surface yields: Z 1 ρ p x d = Q g η + γ c x + d Q γ c x γ Q c x + γ cq Q. x Sustituting this expression into the dynamic equation, equation (3.36), and after expanding and grouping similar terms: Q t + Q Q β + γ c d x Q β +γ c d x + g + γ c Q η x + Q d γ c x + gs f =0. (3.40)
Chapter 3. Theoretical Discussion and Development of Model Equations 65 The size of the cross-sectional area of a channel is influenced y the change in the free surface and ed elevations esides the variation of the channel width in the direction of the flow. For the purpose of simplifying the prolem, the mean ed slope across the channel is used to express the effect of the topographic level change. It can e shown that, η x = B x Z +(η Z ) B η x x = B x Z0 + H B x. (3.41a) The last term of equation (3.41a) is due to the variation of the geometry of the sidewalls of the channel. From depth of flow and free surface elevation relationships, η x = Z x + H x = Z0 + H x, (3.41) where: Z / x = the mean streamwise gradient of the ed and it should e evaluated across the section at any point, B = width of the channel at a section, η = mean flow surface elevation which is equal to the sum of the elevation of the ed and the depth of flow aove the ed. It is more convenient to use the free surface elevation, η, rather than the area of the flow cross-section,, in equation (3.40). Sustituting the expressions, equations (3.41a) and (3.41) into equation (3.40) gives the following equation: Q t + Q Q β + γ c d x + g Q B β + γ c d B + Q d γ c g x + Q β + γ c Z 0 HQ +γc d H x B x + gs f =0. (3.4) Nowwehavetoconsiderthetermsγ c and γ c / x. Fenton (1996) suggested a plausile approximation for the value of γ c as a weighted mean of contriutions from the ed and the surface: β H γ c = ω 1+Z 0 1 x + ω 0Z 00, (3.43a) γ c x = β 3 H ω 1+Z 0 1 x + ω 0Z 000 3, (3.43) where ω 0 and ω 1 are weighting factors. s descried in Fenton (1996), ω 0 represents the comined effect of the ed in determining the elevation of the surface and the associated dynamic pressures due to slow moving flow near the ottom of the flow oundary and might have a value slightly less than one, whereas ω 1 represents the contriution
66 Chapter 3. Theoretical Discussion and Development of Model Equations to the curvature from the surface and might have a value of aout 1/. This method approximates the curvatures of the streamline and the ed y the weakly curved flow expressions. Sustituting equations (3.43a) and (3.43) in equation (3.4) gives 1+Z 0 + Q t +β g 1+Z 0 +ω 1 β Q Z 0 +g 1+Z 0 β HQ 1+Z 0 + ω1 d H x + ω 00 Q Q 0 dz x Q B 1+Z β 0 + ω0 Z 00 d H B x H x + ω 1β Q d 3 H x + ω 0β Q dz 000 3 + Z 00 Z 0 (Sf + Z) 0 ω 1 β Q B d H H B x x 1+Z 0 H B + d ω 1 x + ω 0Z 00 =0. (3.44) x This equation is applicale to model unsteady free surface curved flow in a channel with continuous geometric changes in oth the ottom and sidewall oundaries. The terms of the cross-sectional properties for general channel cross-sections may ecome long and complicated, and it is necessary to simplify such expressions ased on reasonale assumptions that do not affect the overall quality of the numerical solution of the equation. Limiting the scope of the study to the case of unsteady curved flow in a rectangular channel with gradually varied width, then d = H/. Neglecting the product of the derivative terms, Q/ x H/ x, which is very small compared to other terms of the equation, equation (3.44) simplifies to 1+Z 0 1+Z 0 Q t +β HZ 00 + Q Q ω0 x + ω 1β Q H 3 H x 3 +ω 1 β Q Z 0 H x + 1+Z 0 g β Q B H x + g(s f + Z) 0 β HQ 1+Z 0 + H ω H B 1 x + ω 0Z 00 x +ω 0 β Q H Z 000 Z0 + Z00 H =0. (3.45) The continuity equation for no lateral inflow condition is given y t + Q =0. (3.46) x
Chapter 3. Theoretical Discussion and Development of Model Equations 67 For steady flow in a rectangular channel, equation (3.46) gives / t = B H/ t = Q/ x =0. For this specificcaseofflow, equation (3.45) reduces to ω 1 β Q H + 1+Z 0 β HQ d 3 H dx + ω 1β Q Z 0 d H 3 dx + 1+Z 0 g β Q B (g(sf + Z)) 0 + ω 0 β Q H Z 000 + Z00 Z0 H 1+Z 0 d + H ω H db 1 dx + ω 0Z 00 dx dh dx =0. (3.47) For the case of weakly curved free surface flow in the vertical plane, the curvature of the streamlines is limited so that the products of derivatives are of an order smaller in magnitude than the derivatives themselves and can e neglected. Using this approximation, equation (3.47) degenerates to ω 1 β Q H d 3 H dx + g β Q B dh 3 dx + ω 0β Q H 1+H Z 000 +g (S f + Z) 0 β HQ d H db ω 1 dx + ω 0Z 00 =0. (3.48) dx For the special case of unsteady flow in a constant width rectangular channel ( B/ x = 0), the discharge per unit width is q,thearea = BH, and the total discharge Q = Bq. Under this condition, equation (3.45) ecomes 1+Z 0 q 1+Z t +β 0 HZ 00 + q q ω0 H x + ω 1β q 3 H x 3 +ω 1 β q Z 0 H H x + 1+Z 0 gh β q H H x + gh (S f + Z) 0 Z +ω 0 βq 000 + Z00 Z0 =0, (3.49) H and for the same condition of channel geometry as aove ut for steady flow, equations (3.47) and (3.48) reduce to q d 3 H βω 1 dx + βω 1Z 0 q d H 3 H dx + 1+Z 0 gh β q dh H dx +gh 1+Z 0 (Z 0 Z + S f )+ω 0 βq 000 + Z0 Z00 =0, (3.50a) H q d 3 H βω 1 gh dx + β q dh Z 3 H dx + ω 0βq 000 + gh (Z 0 + S f )=0. (3.50) It is apparent from the aove procedure that equation (3.48) is a simplified version of equation (3.47) for flow situations that involve weak streamline curvature and slope. This implies that equation (3.47) incorporates relatively a higher degree of approximation for the effect of non-hydrostatic pressure distriution.
68 Chapter 3. Theoretical Discussion and Development of Model Equations In this thesis, equations (3.47) and (3.48) will e employed to simulate flow prolems that involve non-hydrostatic pressure and/or nonuniform velocity distriutions. The solutions of these two equations will e compared to examine the influence of the weakly curved flow approximation on the solution of equation (3.48). The numerical solution procedures for different test cases will e riefly descried in Chapter 5. Inserting the expression of γ c in equation (3.38) yields a general equation for the pressure distriution p ρ = Q H g + β (1 + Z 0) ω 1 x + ω 0Z 00 (η z). (3.51) Equation (3.51) clearly shows the nature of a higher-order approximation which is the normal hydrostatic pressure relation with a correction factor as a result of the vertical accelerations. If the curvatures of the free surface streamline and the ed are neglected i.e., H/ x = Z 00 =0, equation (3.51) reduces to the hydrostatic pressure equation. Since at a vertical section the curvatures of the ed and the streamline at the surface are constant, the excess pressure aove the hydrostatic varies linearly with the depth of flow. Replacing z y Z and then (η Z ) y H in equation (3.51) yields the following equation to predict the ed pressure profile: p ρ = gh + β q H H (1 + Z 0) ω 1 x + ω 0Z 00. (3.5) For the simplified flow equation, equation (3.48), the corresponding pressure equations caneotainedysetting1+z 0 = 1.0 in equations (3.51) and (3.5). In this work, the suggested values of the weighting parameters (ω 1 =0.50, ω 0 =0.95) will e used for the solutions of the equations. Nature of the flow equation The flow equation, equation (3.47), is a third-order nonlinear ordinary differential equation for which analytical solution is not availale except for very simplified cases. This equation requires three oundary conditions to e specified at the appropriate flow sections in order to otain a unique numerical solution of the equation for flow prolems where the effects of non-hydrostatic pressure distriutions are predominant. The oundary conditions may e specified at different locations in the solution domain depending on the nature of the flow prolems. The location of the external oundary conditions must e chosen carefully in order to avoid the influence of the streamline curvature on the evaluation of the oundary values.
Chapter 3. Theoretical Discussion and Development of Model Equations 69 This Boussinesq-type momentum equation is a one-dimensional flow equation which descries two-dimensional flow prolems where more vertical details are significant and essential. This implies that the effect of the vertical acceleration on the flow ehaviour is implicitly included in the equation. The first two terms of the equation the second and third derivatives of the flow surface profile determine the degree and the variation of the curvature of the streamline at the free surface respectively. In a quasi-uniform flow condition, the curvature of the flow surface is very small and the contriutions of these two terms are almost negligile. It is clear that for steady flow the forms of those streamlines, which are located very close to the flow oundary, are almost identical to the shape of the ed profile. The fifth term of the equation the second and third derivatives of the ed profile takes into account the effect of the curvature of the ed profile on the streamline ehaviour. This term makes the equation completely different from the lower-order Boussinesq-type momentum equations (see e.g., Rodenhuis, 1973). In a special case of curved flow in a prismatic channel where Z 000 = Z 00 the equation reduces to q d 3 H βω 1 dx 3 +gh + βω 1Z 0 q d H ³ H dx + 1+Z 0 gh β q dh H dx = B/ x =0, ³ 1+Z 0 (Z 0 + S f )=0. (3.53) Equation (3.53) implies that, in contrast to the Dressler equations derived for a curved oundary, this flow equation separately includes the effects of the curvature of the streamline and the flow oundaries. For the case of weakly curved free surface flow with negligile curvature of streamlines in a constant slope prismatic channel, the surface streamline and ed curvature terms vanish to zero. Under this flow condition, this Boussinesq-type equation, equation (3.47), reduces to the gradually varied flow equation, equation (3.54). This suggests that the numerical solution of the full governing equation is similar to the numerical solution of the gradually varied flow equation for such kind of flow situation. 1 βfr dh dx +(Z0 + S f )=0. (3.54) Inthecaseofflow with parallel streamlines, S f = Z 0 and d3 H/dx 3 = d H/dx = dh/dx =0. This implies that the uniform flow limit satisfies the flow equation and the flow depth does not change with space.
70 Chapter 3. Theoretical Discussion and Development of Model Equations pproximate analytical solution of the equation Fenton (1996) attempted to otain an approximate analytical solution for the steady version of the flow equation for the case of curved flow on a constant slope and constant width channel y limiting the scope of the study to the consideration of flow depth at depths close to the normal flow depth. Boussinesq also applied this principle for solving his equation analytically (see Jaeger, 1957, p15). Neglecting the square of the ed slope term and inserting Z 0 = S 0 in equation (3.53), the simplified flow equation for β =1.0 reads as ω 1 q d 3 H dx ω q d H 1S 3 0 H dx + gh q H dh dx + gh (S f S 0 )=0. (3.55) For the considered small perturations aout a normal depth of flow ³ H = H n + ĥ, theheadlossisapproximatedys f = S 0 (H n /H) m,where m =10/3 for Manning s Law. Then, linearising equation (3.55) aout this normal depth yields ω 1 q d 3 ĥ dx ω q d ĥ 1S 3 0 gh H n dx + n q d ĥ Hn dx ms 0gĥ =0, (3.56) where H n is the normal depth of flow, and ĥ is small perturations depth. Equation (3.56) is a third-order linear differential equation and its general solution is given y ĥ = C 1 e ε 1x/H n + C e ε x/h n + C 3 e ε 3x/H n, (3.57) where C 1,C and C 3 are constants which can e determined from the oundary conditions, and ε 1, ε and ε 3 are the roots of the characteristic equation corresponding to equation (3.56). Making use of an expansion of the ed slope and neglecting terms of the order of the square of the ed slope, the approximate expressions for the exponents are ε 1 = S 0 m 1 Fr + O S Fr 1 0, ε,3 = ± + S 0 ( Fr 1+ m/) + O S Fr Fr 1 0. (3.58) s discussed efore, this approximate analytical solution was used y the author of the equation to study the nature of the solution of the equation in su- and super-critical flow regions. Based on this investigation, he pointed out the prolem of the parasitic nature of one solution of the equation and suggested (ut did not use) the oundary value method for the numerical solution of the full equation. In this study, this method
Chapter 3. Theoretical Discussion and Development of Model Equations 71 has een applied to develop a numerical model for the solutions of the Boussinesq-type flow equations. 3.7. The Boussinesq-type momentum linear model equations The governing equations, equations (3.47) and (3.51), for flow over curved oundaries were developed ased on the assumption of a constant centrifugal term at a vertical section of the flow. This assumption gives a pressure distriution equation which descries a linear variation of non-hydrostatic pressure with the depth of flow. Even though the pressure equation accounts for the effect of the vertical acceleration, its application is limited to the prediction of pressure distriution for curved flow prolems with a single free surface. For a single free surface flow prolem with pronounced curvature of streamline, the accuracy of the prediction of the equation for pressure profile is questionale. Moreover, in the analysis of dual free surfaces flow prolems (for instance, flow over an inclined sharp-crested weir or flowinafreeoverfall)wherethepressureatthefreesurfacesofthe nappe is atmospheric, this equation predicts internal atmospheric pressure distriutions at any vertical sections. However, this prediction is inaccurate due to the converging nature of the streamlines in the flow region close to the rink section. The aove indicates the need for developing the Boussinesq-type linear model equations ased on different non-hydrostatic pressure correction for the purpose of comparison as well as assessing the overall quality of the solutions of the BTMU equations. This also creates a condition to examine the impact of the degree of the approximation for the effect of non-hydrostatic pressure distriution, which depends on the assumed distriution shape of the centrifugal term, on the numerical solutions of the Boussinesq-type equations for flow prolems with appreciale streamline curvatures. The following simplifying assumptions are made to develop the equations: the fluid is incompressile and non-viscous, the variation of the value of the centrifugal term, which is a function of the curvature and angle of streamlines, in the vertical direction is linear. cartesian coordinate system (x, y, z) which was defined in Figure 3.1 is also used to formulate the equations. For a curvilinear flow with vertical acceleration, a z,the intensity of the pressure p at z aove the datum is otained from the integration of the