Analytical Solution for the Free Over-Fall Weir Flow Using Conformal Mapping and Potential Flow Theory

In this study an analytical approach is presented based on the potential flow theory and conformal mapping technique to solve the problem of an ideal and steady flow over a free over-fall weir. The results are arranged for rectangular sharp-crested over-fall weirs with different vertical aspect ratios (h/P, h is the water head above the weir and P is the weir height). To validate the results of this approach, the free over-fall weir d ischarge equation and the water free-surface profiles obtained with the potential flow theory have been calibrated with the experimental data. In conclusion, the information made available and the close correlation among the results of present analytical approach and the experimental data are adequate to warrant the recommendation of this method as a valuable supplement to existing methods employed in engineering design.


Introduction
Weirs are simp le and precise structures for flow measurement and control in open channels. Formerly, the weir equations were found mainly on the basis of experimental data. Based on the prevailing universal weir discharge equation, the discharge over the rectangular sharp-crested weirs could be revealed as [1] 1. 5 2 2 3 d q C gh   (1) where q is the discharge per unite length of the weir crest, h is the upstream water head, g is the gravitational acceleration and C d is the weir discharge coefficient. Formerly, the weir discharge coefficient was determined main ly on the basis of exp erimen tal data and emp irical eq uat ions . The flo w characteristics including the discharge coefficient, velocity and pressure distribution and water surface p rofiles over the weirs were the subject of many investigat ions [2][3][4][5][6][7][8][9]. The problem studied here is concerned with the 2-D, steady, irro tat ional and free -surface flo w. Th e flu id cou ld be assu med as in co mp ress ib le and inv is cid . Hen ce, th is nonlinear problem could be solved by various approaches. An alternat ive app roach is emp loy ing the potential flow theory to determine the flow pattern i.e. velocity potential and st ream fu nct ions t h roug h pass ing th e we irs . The determination of the flow pattern in the vicinity of a weir has man y ap p licat ions , su ch as t he determin at ion o f t he discharge coefficient, the calculat ion of the pressure force exerted over the weir and determination of the distortion of the velocity profile induced by flow contraction. Potential theory allows one to calcu late exp licitly the free streamlines past the weir. For the external flows above the bodies such as weirs, potential flow theory should work well, until the surface pressure gradient beco mes adverse. Having found velocity potential () and the equipotential lines or stream function along the flow lines fro m such analysis, the flow velocity and discharge coefficient would be co mputed by direct differentiation of potential function. From the mathematical v iewpoint, the direct calculation of free streamlined flows is restricted to t wo-dimensional, irrotational flows of incomp ressible fluids for which the solid boundaries are straight. As these strict limitations are seldom fu lly satisfied by real flu id fields, the practical value of the results is open to question and relates to the problem which is solved. The significance of the various restrictions can be shown only by co mparing the calculated results with those obtained fro m observations or real data. Afzalimehr and Bagheri [10] developed an equation for estimating the discharge coefficient of sharp-crested weirs based on potential flow theory. They compared the results with the experimental data and suggested an equation for the discharge coefficient for a large range of vertical aspect ratios.
Potential flo w theory leads to the analytic determination of flow characteristics based on a transformation between the physical plane and the complex potential plane using conformal mapping technique. Conformal mapping is a mathematical technique in which co mp licated geo metries can be transformed by a mapping function into simpler M apping and Potential Flow Theory geometries which still preserves both the angles and orientation of the orig inal geo metry [11]. Conformal mapping t ransformat ions are both mathemat ically interesting and practically useful [12][13][14][15][16][17]. Although amp le studies have been carried out to investigate the flow characteristics passing sharp-crested weirs but limited info rmation is availab le reported in the published literature on the free over-fall weir flow characteristics. Free over-fall weir includes a sharp-crested weir installed above a deep drop aimed to force falling streamlines to be in vertical orientation ( Figure 1). In this paper, the study focuses on modeling the 2-D flu id flo w over the free over-fall weirs using the conformal mapping technique. The Schwarz-Ch ristoffel transformation, a specific applicat ion of conformal mapping, was used to achieve the solution for flow over free over-fall weirs. The solution of potential flow for an over-fall weir will be an orthogonal flow net as shown in Figure 1. The application of computational tools to implement this conformal mapping transformation in order to co mpute the flow field, water surface profile and weir discharge equation is discussed. Finally, the results of analytical modeling were co mpared with the experimental data of the present study. These results will be supplemented with a quantitative error analysis in an attempt to explain any discrepancies among the models.

Theoretical Considerations
In comp lex systems, a Sch warz-Ch ristoffel technique is a conformal transformation of a simp le polygon interior area onto the upper half-plane Im(z)>0, wh ich is also used in potential flow theory and some of its applications. Suppose a polygon in the w-plane with vertices at w 1 , w 2 ,…, w n and interior angles of  1 ,  2 ,…,  n . Where these points map onto points x 1 , x 2 ,…, x n on the real axis of the z-plane (Figure 2). Let G denote the interior of the polygon and D the interior of the upper half-plane. Then a transformat ion w = F(z) maps D onto G is given by the Schwarz-Christoffel transformation as where the constants A and B demonstrate the size and position of G and x 1 , x 2 , …, x n are the real part of co mplex variable z (= x+iy) in the z-p lane [11]. The co mp lex potential w = F(z) is a conformal t ransformat ion which maps the stream function and velocity potential curves in w -plane (physical plane) to a set of horizontal and vertical lines in z-plane (co mplex plane). Hence, the main purpose of the analysis is to survey the appropriate form of co mplex potential F(z), which converts the complicated flo w field to parallel sets of horizontal and vertical lines [18] Applying the potential flo w theorem and conformal mapping procedure to discover the image of a flow passing an over-fall weir, two points x k might be chosen arbitrarily in z-plane corresponding to the comp lex variables w k in w -plane as ( Figure 3): Substituting the chosen x k and  k (k = 1, 2) into Eq.(2), yields to the following integration for co mp lex potential F(z): ; ; x w Consequence to integrating Eq.(4) and applying the conditions of F(0) = 0, F(P) = iP, constants A and B are calculated, hence the complex potential F(z) could be written as: Since z = re i , F(z) = F(r,) could be exp ressed as: To illustrate the interest of the approach, the equipotential and streamlines for different configurations in order to analyze the effects of the boundaries on the flow field were calculated. Afterward, the study focuses on the configuration and hydrodynamic aspects of the flow through over-fall weir and compares the solution with experimental data. While F(z) = w = u+iv, u and v could be exp ressed as the real and imaginary parts of Eq.(7), respectively. Since F(z) just involves the geometries of the prob lem and the hydraulic aspects are not included, u and v are representatives of velocity potential and flo w stream functions (e.g. x and y respectively). By deriv ing the real and imag inary parts of Eq.(7) an arb itrary point on orthogonal lines in the co mplex plane, is mapped to corresponding point (u,v) in physical plane. Defining the streamlines passing a free over-fall weir with the height P, the horizontal lines in z-p lane are mapped to the physical plane of w using Eq. (7). As the derived equation is very complicated and could not straightforwardly be separated to achieve the real and imag inary parts, hence a numerical co mputation was applied using Mathemat ica software. Based on the above procedure the pair of (u,v) were achieved for part icular x and y values. Figures 4 and 5 show typical representatives of flow streamlines and velocity potential curves respectively. The weir height was considered to be 1.5 m.
Besides, d ifferent figu res were p lotted to illustrate the effects of boundaries on the flow field. The weir contours and the bottom consist of solid constraints in the motion. The flo w is fo rced by the g rav ity and v iscosity and surface tension effects are neg lected . Results ind icated that , in general in the close vicinity of the weir crest, the streamlines deviate significantly. As expected, at a larger distance from the weir (about x/P5), less distortion of the equipotential lines is observed and the water free-surface is nearly horizontal. Th is result was proved by different investigators [9].

Model Experimentation
To investigate the water free-surface profile and over-fall weir discharge equations, experimental study was performed in a rectangular glass open channel of 11 m length, 0.4 m width and 1.0 m height. Experimental setup consists of a circulat ing system including main channel, pu mp, upstream and downstream reservoirs, etc. The longitudinal slope of the channel was set to be zero. Free over-fall weirs were made of 3 mm thick steel sheets with sharp edged and fully aerated condition. Two weirs with P=5 and 10 cm were considered and placed at the mid -length of the upstream in let above the corner of a 0.4 m h igh vertical drop. Series of tests were conducted for different flow conditions and each corresponding inlet d ischarge (Q) was measured using an ultrasound flow-meter with a precision of 0.1 lit/s. The minimu m of eight discharges between 20 and 60 lit/s were considered. To evaluate the velocity profiles aimed to investigate the flow structure at the test section, the 3-D velocity profiles were measured using an acoustic doppler velocimeter (ADV). The depths of flow along and across the channel and over the crest of the weirs were measured using a point gauge mounted on a movable beam with a precision of 0.1 mm. Experiments were conducted for sub-critical, stable and free over-fall conditions.

Water Surface Profile
In this section the analytical results obtained applying the potential flo w theory and conformal mapp ing technique, are presented. As mentioned before, with the advances in analytical methods, computational models of flow through hydraulic structures are increasingly being used in industry, but they still require validation fro m experiments. For this reason, the analytical results have been compared against experimental measurements of the present study. The laboratory measurements regard the water elevation above the over-fall weir in order to track the free-surface profile. Consequent to applying the proposed analytical model, the water free -surface profiles were calcu lated for particular over-fall weir geo metry by transforming the horizontal line passing from the upstream normal water depth (h) in complex p lane, into physical domain using Eq.(7). Co mparing the analytical results and experimental data indicated that, there were meaningful discrepancies between the analytical and experimental results of water free -surface. This is in part due to the simp lifying assumptions of the potential flow. Hence, the theoretical results were imp roved based on experimental data and trial and error procedure using a quadratic correlation as follows: where (y/h) act and (y/h) ana are the actual non-dimensional vertical coordinate and the dimensionless vertical coordinate of the water free-surface obtained based on analytical approach respectively. No doubt due to different combinations, many functions can be introduced for each trial and error turn but, one's judgment can achieve mo re accurate results.  Figure 6 shows the predicted non-dimensional flow profiles denoting y/h as a function of the non-dimensional distance x/P for subcritical approach flows respect to different h/P values. In Figure 6, the imp roved surface profiles of the analytical modeling are co mpared with the experimental profiles. As illustrated, the present imp roved analytical p rofiles agree very we ll with the model experimentation results. Results signified that, the nappe downstream the weir has a contraction preserving the continuum free -surface profile and property like a jet.
To evaluate the validity of the achieved analytical free-surface results compared with the experimental data, the error functions normalized root mean square error (NRMSE) and weighted quadratic deviation (WQD) expressed by Eqs. (12) and (13) were used. The coefficient of determination (R 2 ) is also calculated. M apping and Potential Flow Theory (12) (13) where f(x) and fˉ are the experimental values and the average of the experimental data respectively and F(x) is the estimated results. Contrary to R 2 , the values of NRM SE and WQD must be the smallest [19]. The reported NRMSE, WQD and R 2 values in this analysis are 0.17, 0.005 and 0.97, respectively, which show good agreement between theoretical and experimental results. Figure 7 shows the comparison between analytical normalized water free-surface profiles against the experimental data. In this figure, the majority of ±7% bounds are also shown. As almost all the data points lie with in ±7% tolerance, it means that the analytical model is in good agreement with the experimental results.

Discharge Coefficient
The complex potential of flow field includes two parts, (i) a real part, indicating the velocity potential ( ) and (ii) an imaginary part, wh ich specifies the stream function ().
According to Eq.(7) x = cte and y = cte are analogies to  and  functions, respectively. Based on numerical solution and applying nonlinear regression analysis the stream functions were expressed as follows: then: 0 k yU    (15) where U  and k 0 denote the ideal approaching flow velocity and correction factor taking into account the simp lifications and errors of analysis. The coefficient of determination for Eq. (14) is about 0.97. Note that Eq. (14) is valid at the vicinity of the weir crest, where was aimed for d ischarge calculation. Since, u=0 over the weir crest, comb ining Eqs. (14) and (15) (19) By assuming U  = q/(P+h), then y P would be determined based on a trial and error procedure. If the free jet flow theory is employed for flo w passing an over-fall weir, it tends to an appro ximate relation between U  and h as fo llo w: where k 1 is a correction factor. If Eqs. (19) and (20) were combined; the follo wing exp ression for estimating the flow discharge would be achieved.
It should be remarked that, the potential flow theory is valid for irrotational, ideal flow conditions, where the viscous effects are negligible. Assuming ideal flow condition for the flow passing a solid object is not valid in a thin layer surrounding the solid boundary. Likewise, surrounding an over-fall weir, concentrated vortices are developed and the potential flow assumptions can not be established. Hence, due to the importance of viscous and inertia effects as well as vortices consequences and flow separation, head loss arise through the flo w passage and consequently the flow discharge reduces compared with the ideal flow condition. The coefficient C 0 takes in to account these influences. The subject to h/P2.65. Figure 8 shows the variation of observed versus calculated 0 C . In this figure the majority of 5% bonds are also attached. As is illustrated analytical results well co mpared with the experimental ones. The corresponding NRMSE, W QD and R 2 values for the above relationship are 0.15, 0.003 and 0.98, respectively.

Conclusions
The overall goal of this study was to effectively model the flu id flow over free over-fall weirs using the conformal mapping technique based on the potential flo w theory. The solution for the inviscid flow over free over-fall weir has been mapped successfully using Schwarz Christoffel transformation. Consequent to employing the potential flow theory, equations for determining the water free-surface profile and the discharge coefficient of the free over-fall weirs were obtained. The analytical results were calibrated according to an experimental investigation. The predictions of the semi-analytical model agree well with the experimental data related to water surface profiles and discharge coefficient. The results reveal that the Schwarz-Christoffel is the most accurate transformation to solve such problems co mpared with the other ones. The results are validated for large rang of h/P (i.e. h/P2.65, which are more pract ical in design considerations).