Analysis of some velocity-area methods for calculating open channel flow

Transcription

1 Hydrological Sciences - Journal - des Sciences Hydrologiques, 33,3, 6/988 Analysis of some velocity-area methods for calculating open channel flow INTRODUCTION JOÂO NUNO fflpôllto Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 096 Lisboa Codex, Portugal JOÂO MEMOSO LOUREERO Division ofhydrometry of the General Directorate of Water Resources and Reclamation (DGRAH), Portugal Abstract This article derives an expression for the calculation of the flow between two verticals in the cross section of a water course which, by considering a parameter (a), covers three of the calculation methods which are traditionally used. The analysis of the effects of this parameter enabled the authors to select one of the values which they recommend should be used. Analyse de quelques méthodes pour le jaugeage par exploration du champ des vitesses Résumé On a mis au point une expression pour le calcul du débit qui s'écoule entre deux verticales de mesure de vitesses dans la section d'un cours d'eau. Cette expression grace à la prise en compte d'un paramètre (a), permet la généralisation de trois méthodes traditionnellement utilisées. L'analyse de ce paramètre a permis de sélectionner une de ses valeurs, dont l'utilisation est recommandé par les auteurs. The amount of water which flows through a cross section of a water course (Fig. ) at a given moment is: Q = f V às () J s where V represents the velocity component normal to the element of area ds; V = V(x,y); and the integral covers the whole wetted area of the cross section S. The foregoing integral can be transcribed as follows:,b fd,b-.b Q=\\Vdydx=\Vdûx=\qdx (2) J 0 'o J o J o where V = V(x) represents the average normal velocity component over the vertical whose depth is d = d(x); and q = q(x), the flow over the same vertical Open for discussion until December

2 Joûo Nuno Hipôlito & Joâo Mimoso Loureiro 32 Fig. Cross section of a water course and definition of symbols. per unit surface width. The average velocity over a vertical can also be considered as a function of the wetted area which lies to the left of the vertical, limited by the bottom of the cross section and by the said vertical (Fig. 2): where: V = V(S) = V[S(x)] S(x) = V d{\) à\ or: às = d(x) àx \ 4 sp d (X) A H À Ê X SIX) jflçf Y ' Fig. 2 Cross section of a water course and definition of S(x). It can be seen that: F* V às = J Vd dx = Q (3) Velocity-area methods for liquid flow measurement in open channels are described in many text books and standards, as, for instance, in ISO

3 33 Analysis of some velocity-area methods (E) (ISO, 983) or in the WMO Manual on Stream Gauging (WMO, 980). Other methods for numerical integration can also be found in elementary text books on numerical analysis, as in Conte & de Boor (980). The following parts of the paper are designed to summarize in a single expression some of the gauging methods based on the average velocity V along the verticals, on their space intervals Ax. and on the expressions (2) and (3). STREAMGAUGMG Method (similar to that known as the mid-section method) Assuming that o ~ o(x^ varies!inesr! v ^^z a first order a r * T>r oxirr, ation^ between two consecutive verticals, designated i and / +, the flow between these two verticals, AQ\, will then be: + AQ) = j^' Ax. q àx = y (q. + q.j or Ax t AQ = IV. d. + V., d.,) (4) where Ax t = x i+ - x i represents the distance between the verticals, measured horizontally. Method 2 Assuming that V = V(x) and d = d(x) vary linearly between two consecutive verticals: Ag2 = J i+i y d & = M. '*,. V t d i * i V i (d i+ - d) d. (V.., - V.) + - (V.. - V.) (d.. - d) or Ax. r AQ] - -(V idt+ v M d t j + -<yid M + v i+l d: (5)

4 Joâo Nuno Hipôlito & Joâo Mimoso Loureiro 34 Method 3 (similar to that known as the mean-section method) Using the first_ integral of equation (3) for the calculation of the flow and assuming that V = V(S) varies linearly with 5 between the verticals: o *o -, i (fi S I where AS. = S i+ - S; represents the area between two verticals. assuming that d varies linearly between two verticals: Further Ax. AQ3 = (]/, + VM) (di + di+i) or àx. AQ3 = L ^ (V. d. + V., d.,) + - (V, d-, + V., d.) (6) Generalization regarding the methods described Examination of equations (4), (5) and (6) shows that the three methods can be expressed as a single equation: Ax. AQ, = «[a(f. d. + v M d M )+ ( - a) (V t d i+ + F. + rf.)] (7) where the parameter a weights the total of the products of the terms with subscripts that are equal and its complement weights the total of the products of the terms with interchanged subscripts. It can be seen that a = corresponds to method, a = 2/3 to method 2, and a = H to method 3. Equation (7) can be written in the following non-dimensional form: where: Q' = - [ce(l + d' V) + ( - a) (d' + V')} (8) Q, = d' = Ax. d. V. i i i d.jd. i + V i

5 35 Analysis of some velocity-area methods a' =.00 VALUES OF « MîO.SO T [ T r - ' I rd = 0.75,2.0-i d' = 0.25 VBLUES OFK VALUES OF OC O.S' 0.6"o».«f.Oo i 2-0 -i - I I to «o in V d = o.oo VBLUES 0F«VALUES OF <x o.5o.00 Fig. 3 o.o -i i i r i 0 p r " ï * ï p p r ' r *! w o w b o ô t o û i o v v Analysis of the influence of a on the evaluation of Q'. V = K. + /F,. Taking limits when Ax. goes to zero: lim d ' = A*.= 0 lim V' = A*.=0 and consequently, from equation (8): lim Q ' = A*.=o

6 Joâo Nuno Hipôlito & Joâo Mimoso Loureiro 36 Consequently, regardless of the value of a, equations (7) and (8) tend towards equivalent results when the distance between the verticals is reduced to zero. Such a fact confirms the consistency of any of the said methods or of any other which is obtained from the values which can be attributed to a. ANALYSIS OF TEE INFLUENCE OF a Analysis of the non-dimensional expression for Q ' In order to analyse the influence of a on the values calculated for the flow between two verticals, values of Q ' defined by equation (8) were calculated for a range of values of a, d ' and V (Fig. 3). In order to keep the range of d' between 0 and, / was taken to designate the vertical which corresponds to the greater depth, d., and / + the other vertical of lesser depth. An analysis of Fig. 3 enables one to draw the following conclusions: (a) when either of the non-dimensional variables V or d' is equal to unity, the value of Q' is independent of the value of a; if V =, then Q' = l A(l + d') while if d' =, Q' = Vz(l + V); (b) if V is less than (0 S V < ), the higher values of a will then correspond to higher values of Q ' ; and (c) if V is greater than (V > ), the higher values of a will then correspond to lower values of Q '. One should note that the conditions of conclusion (b) will occur more frequently than the conditions of conclusions (c), because higher average velocities generally correspond to greater depths. Analysis of the results for a fictional cross section Let us suppose that there is a semicircular cross section with a radius R (Fig. 4) in which the distribution of the velocity is defined by: which forms an approximate representation of the distribution of the velocity under turbulent conditions and where V M represents the maximum value for V (at r = 0). The flow which passes through the cross section will be: Q = 98 nr y M In order to analyse the influence of the value of a on the calculation of the flow through the section, the following steps were taken:

7 37 Analysis of some velocity-area methods Fig. 4 Y Semicircular cross section and definition of symbols. (a) verticals were considered to be spaced apart equally, the first and the last at the banks; and (b) the average velocity at each vertical was considered as being equal to the arithmetical average of the point velocities at depths which are 0.2 and 0.8 of the depth of the vertical. The relative error e is defined by the following expression: Q - Q c Q where Q represents the calculated flow for a given value of a and for a given number of verticals. Table shows the values for the relative errors which were obtained for a equal to.00, 0.67 and 0.50 and for a range of the number of verticals considered in the calculation between 3 and 20. An examination of Table enables one to draw the following conclusions: (a) for any given number of verticals, the calculation showing the smallest error always corresponds to a value for a which is equal to.00; and (b) the error decreases faster with the increase in the number of verticals when the value of a is equal to.00. Analysis of the results for a real cross section Figure 5 shows the micro-computer output for the flow calculation at the hydrometric station in Coruche on the Sorraia River in the Tagus basin. In the gauging procedure the pertinent variables were measured at 9 verticals. Table 2 shows the results for the calculated flow for a equal to.00, 0.67 and 0.50 and for a number of verticals considered in the calculation between 3 and 9. The relative error is not shown in Table 2, as was done in Table, because in this case the true flow is not known. The number of verticals disregarded between each pair of verticals is shown in the last column of Table 2. For example, when the number of verticals considered in the calculation was four, the verticals 0, 6, 2 and 8, as defined in Fig. 5, were considered and in between each pair used, five verticals were disregarded.

8 Joào Nuno Hipôlito & Joâo Mimoso Loureiro 38 Table Values of the relative error in the flow calculation for the fictional cross section Number of verticals a =.00 a = 0.67 a = Table 2 Values of the calculated flow (m 3 s' ) in the real section (Coruche) Number of verticals.00 a = 0.67 a = 0.50 Number of verticals disregarded Since the flow through the section is not known, one may assume that the computed flow indicated when 9 verticals are considered is the best estimate for a given value of a. Under these circumstances, the analysis of Table 2 enables one to draw the following conclusions with regard to the number of verticals used between 3 and 0: (a) for a given number of verticals, the calculation which shows the smallest error always corresponds to a value of a equal to.00; and

9 39 Analysis of some velocity-area methods Fig. 5 Illustration of the calculation for measuring the flow in a real section (Coruche) using Method.

10 Joâo Nuno Hipôlito & Joâo Mimoso Loureiro 320 (b) the error decreases faster with the increase in the number of verticals when the value of a is equal to.00. CONCLUSION This paper set out to present a general equation for the calculation of the flow between two verticals in a cross section of a water course which, by considering a parameter (a), includes three of the calculation methods that are traditionally used. Assuming accurate values for the depths (rf.) and for the normal component of the average velocity along each vertical (K.) as well as for the distances between successive verticals (Ac.), the behaviour of the general equation was analysed in the light of the values that could be attributed to a. The analysis which was undertaken leads us to recommend the use of a value of unity for a (which corresponds to Method, generally known as the mid-section method). Thus, the estimate of the flow which passes between two verticals should be calculated by means of the following equation: Ac- AQ. = (V.d.+ V.,d.,) Acknowledgements The authors wish to thank Professor A. Quintela for his support, criticism and suggestions, and Mr Pereira and Mr Vinagre of the Hydrometry Team of the Tagus Valley for the measurement of depths and velocities at each vertical of the Coruche Station. REFERENCES Conte, S, D. & de Boor, C. (980) Elementary Numerical Analysis. An Algorithmic Approach. International Student Edition, McGraw-Hill. ISO (983) ISO Standards Handbook no. 6,56-8. ISO, Geneva, Switzerland. WMO (980) Manual on Stream Gauging. Operational Hydrology Report no. 3. WMO - No. 59, Geneva, Switzerland. Received 7 December 986; accepted 9 November 987.