Comparative Approaches of Calculation of the Back Water Curves in a Trapezoidal Channel with Weak Slope

Transcription

1 Proceeings of the Worl Congress on Engineering Vol WCE, July 6-8,, Lonon, U.K. Comparative Approaches of Calculation of the Back Water Curves in a Trapezoial Channel with Weak Slope Fourar Ali, Chiremsel Rachi, Abesseme Fouzi Abstract The theoretical evelopment having le to the ifferential equation governing the back water curves is base on the equation of Bernoulli applie between two crosssections of the stream ischarge running out in a channel of known geometrical profile. The graient of the pressure loss is evaluate, accoring to the authors, by calling upon the relations of the normal or uniform flow. The most preferre relations are the equations of the type Chézy an Manning-Strickler but selom that of Darcy- Weisbach. n this last relation, the coefficient of friction epens at the same time on the relative roughness of the walls of the channel consiere an the Reynols number characterizing the flow. The calculation of the graually varie flow rests on the knowlege of the nature of the geometrical involve slope. For this reason the classification of the back water curves is a carinal importance an must be controlle before even approaching the calculation itself epths of the graually varie flow. The classification of the back water curves consists, in the secon time, to locate the epth of the flow compare to the critical level epths an uniform if those ha suenly occurre for the same flow forwaring by the same channel consiere. The epth of the flow varies accoring to conitions' hyraulic an geometrical channel, it can be increase by the lover towars the ownstream or on the other han ecrease. The metho of classification of the back water curves is common to all the theoretical approaches intene for the calculation of the varie flow. nex Terms graually varie flow, back water curves, ifferential equation, equation of Bernoulli, metho of the variations of epth, metho of the sections, metho of Pavlovski.. NTRODUCTON The calculation of the back water curves for the flows graually varie on free face in the open channels makes it possible to euce the epth at any point from the channel. t consists in following the evolution of the watermark over Manuscript receive December, ; revise January 6,. Fourar Ali is Senior Lecturer in the Department of Hyraulics, nstitute of Civil Engineering, Hyraulics an Architecture, Batna University, Algeria s: afourar78@gmail.com. Chiremsel Rachi is Postgrauate Stuent in the Department of Hyraulics, nstitute of Civil Engineering, Hyraulics an Architecture, Batna University, Algeria e -mails: Ra4inf8@gmail.com. Abesseme Fouzi is PhD Stuent in the Department of Hyraulics, nstitute of Civil Engineering, Hyraulics an Architecture, Batna University, Algeria e -mails: fouzi.abesseme@hotmail.fr. the entire length of the channel consiere. A goo inspection of the watermark is regare as an essential conition in the choice for example of the site of proection of a structure, to envisage corrections of the rivers in orer to avoi a possible overflow in the event of rising. nee this hyraulic behavior eserves to give him a great interest. Several methos are propose for calculation of the movements. These last have the same aim; however their principle of calculation is ifferent. The principal obective of this stuy is to propose a comparative approach of calculation between three methos metho of the variations of epth, metho of the sections an metho of Pavlovski an to raw the interesting conclusions as for the reliability an to the flexibility of the methos use. n this stuy roughness in length of the channel is consiere constant, the channel is trapezoial, prismatic an with weak slope an the curve back water calculate is of type M. []. FLOW GRADUALLY VARES t is a permanent flow for which the flow remains constant in time. On the other han the changes of crosssection of stream ischarge, generally cause by breaks of slopes, return the flow varies. Moreover if the curve of the watermark is neglecte the flow is graually varies. The flow graually varies can exten on a very long istance that s to say that the epth, Dhx Dh, as well as the other parameters change only very slowly from one section to another movement of raising or lowering.. DFFERENTAL EUATON OF THE MOVEMENT GRADUALLY VARES Accoring to Bernoulli the variation of specific energy is the ifference between the slope of the channel an the slope of the line of energy. E J An knowing that specific energy E is relate to h an that h is relate to, we can write: SBN: SSN: Print; SSN: Online WCE

2 Proceeings of the Worl Congress on Engineering Vol WCE, July 6-8,, Lonon, U.K. E E E h h h A h By equalizing the expressions, an we obtain: E B J 4 An finally: J B 5 n this expression, an are constant an A, B an J are relate to h. we consier that the pressure loss has the same value as in uniform moe for the same epth of water an the same flow, therefore accoring to the formula of Chézy we write: J C R A h The equation 5 is thus the ifferential equation x = f h of the watermark moving graually varies for a flow in permanent moe with appreciably parallel nets moving in block in uniform channel of weak slope. The general pace of the watermark is a function of the relative values of an c of the channel for the flow consiere. V. CURVES BACK WATER STANDARD M These curves answer the following inequalities: < c an h n > h c What correspons to a river flow. Three cases can occur: Branch M who correspons to the following conitions: h > h n > h c ; > J; F r < ; > The branch M is a back water curve of raising; it correspons to a graually elaye movement. Upstream this curve tens asymptotically towars the normal epth i.e. can be propagate upstream at an infinitely long istance. Downstream it tens asymptotically towars the horizontal. Branch M who correspons to the following conitions: h c < h < h n ; < J ; F r < ; < The branch M is a back water curve of lowering which correspons to a graually accelerate movement. Upstream this curve is connecte asymptotically to the normal epth for tening towars the ownstream while ecreasing perpenicularly towars the critical epth. Branch M who correspons to the following conitions: h < h c < h n ; < J; F r > ; > We can write: C R A J h gp B C Rh B C B B 6 The branch M is a back water curve of raising which correspons to a graually elaye movement, it leas to the proection close the critical epth making it possible to pass from the torrential moe to the river moe. V. METHODS OF CALCULATON The calculation an the exact construction of the forms of the free face require the integration of the ifferential equation: J B What comes own to saying that the flow, the slope of the be, its roughness is known. However one oes not know the imension of the free face. Consequently the variables are the coorinate an the corresponing epth h. The integration of the ifferential equation 5 conuit with an inefinite integral. t will thus be necessary to know the characteristics of the flow in a section of reference or control where there is a final relation between the flow an the epth of water. A.Metho of the variations of epth Δh is fixe The metho of the variations of epth or metho with irect steps is appropriate particularly well for the prismatic channels. t is use for a light ifference in epth. An to reuce the errors, it is avisable to tighten the variation the epths h an h This metho consists in seeking the value of the coorinate for a epth h very near to h using the equation: V V h h g g J h J h The coorinate is calculate an we will pass then to the following section. SBN: SSN: Print; SSN: Online WCE

3 Proceeings of the Worl Congress on Engineering Vol WCE, July 6-8,, Lonon, U.K. B. Metho of the sections Δ is fixe The metho implicit of the sections or with stanar step is use for the sections at short istances. t applies to the equation of the not uniform movement in the form: E E J t shoul be announce that calculations with this metho are long an complicate. The metho of the sections consists in etermining the epth of water h with the coorinate very near to h with a coorinate h By choosing a first value h' of section A' we calculate the values, J an A While carrying these values in the equation: J h J h h h g A A We obtain the value of h which will be probably ifferent from the selecte value h'. We start again by successive approximations until the value of h ata by this equation is equal to the last selecte value h', h" one will pass then to the following section. C. Metho of Pavlovski Posing: B P cin P cin : is calle kinetic parameter. The equation 5 can be written: J P cin 7 Accoring to the ébitance, the equation 7 becomes: K K P cin 8 The sizes of the secon member of the equation 8 present certain functions the epth h. this is why it is rational to write the equation 8 pennies the form: Pcin K K 9 Let us examine the expression of the kinetic parameter now P cin B K B K B Pcin Rh P Let us multiply the numerator an the enominator by C Then: BC K BC K K P cin gpa C R gp K K h BC gp Replacing the expression of the kinetic parameter eq. in the equation 9 we obtain: K K f h K K To carry out the integration of the equation, the sizes J an K / K must be expresse in the form of the explicit analytical functions of h. But this is not possible that for the bes of simple form. t thus proves that the function f h is so complex. t is impossible to solve it analytically. For the integration of the equation we suppose that the term J varies very little on the section subecte to integration. Pavlovski propose the following process: That is to say a certain section of the prismatic be with > within the limits of which we observe a not uniform flow of water. Let us inicate the epth of the current at the origin of the section by h the epth at the en by h an the length of the section by Δ. nex will inicate the following section in the irection of the flow. Accoring to an assumption which one alreay spoke, we aopt for section Δ: te c Let us introuce esignations: K, K, K K K K Let us amit that between the variables h an η there is a relation of the type: h = χ η where : h h 4 Or in the ifferential form: 5 Rewriting the equation by taking account of introuce esignations an the assumptions: 6 SBN: SSN: Print; SSN: Online WCE

4 The secon multiplier of the secon member of the éq.6 can be written: An by substitution in the éq.6 we obtain: Carrying out the integration of this last equality enters the limits of the section: Or: f η an η are lower than an f η an η are higher than. The expressions above can be transcribe efinitively in the following form: ] [ 7 Where: :,5 log à :,5log à The equation 7 makes it possible to calculate the back water curves in the prismatic bes of any form, if >. V. SOLUTONS b= [m], m =, n =.5[m -/ s], J f =.5, = [m /s], h n =.[m], h c =.9[m], J cr =.699, h =.[m]. Fig.. Plots the curve M using three methosa. b= 7[m], m =.5, n =.8[m -/ s], J f =., = 5[m /s], h n =.459[m], h c =.[m], J cr =.65, h =.5[m]. Fig.. Plots the curve M using three methosb. b= 5.[m], m =.5, n =.45[m -/ s], J f =.5, = 55[m /s], h n =.687[m], h c =.6[m], J cr =.6489, h =.6[m]. Fig.. Plots the curve M using three methosc. Proceeings of the Worl Congress on Engineering Vol WCE, July 6-8,, Lonon, U.K. SBN: SSN: Print; SSN: Online WCE

5 Proceeings of the Worl Congress on Engineering Vol WCE, July 6-8,, Lonon, U.K. b= 4[m], m =., n =.[m -/ s], J f =.75, = 85[m /s], h n =.94[m], h c =.49[m], J cr =.47, h =.49[m]. Fig. 4. Plots the curve M using three methos. b= 5.5[m], m =, n =.5[m -/ s], J f =.5, = [m /s], h n =.94[m], h c =.749[m], J cr =.8, h =.[m]. Fig. 5. Plots the curve M using two methosa. b= [m], m =.5, n =.[m -/ s], J f =., = 6[m /s], h n =.464[m], h c =.95[m], J cr =.46, h =.45[m]. V. NTERPRETATON The construction of the back water curves can be carrie out by various methos of which those which we use, in fact: metho of the sections, metho of Pavlovski an metho of the variations of epth. n all the cases this construction is carrie out section by section, while passing successively from one section to another upstream accoring to the current. The partition of a channel in sections carrie out accoring to the conitions of uniformity of the flow within the limits of each section of which the length can vary few hunres of meters until tens of kilometers. The most favorable partition is that which is one accoring to the uniformity of the slope of the watermark. The calculation an the layout of the free face, taking into account the bounary conitions which we prouce, make it possible, a priori, to confirm the assumption accoring to which all the methos use are asymptotic i.e. they give similar results. The watermark calculate by the metho by irect integration of Pavlovski iverges somewhat from that calculate by the metho of the variations of epth. All the calculate watermarks are more or less similar. Calculations by the metho of Pavlovski give a istance appreciably larger, i.e. The watermark converges more slowly towars the normal epth. The ifference of the results between the various branches M, M an M is ue to the asymptotic nature of the function Фη use in this metho. V. CONCLUSON The comparison of the results obtaine by the three methos Metho of the variations of epth, metho of the sections an the metho of Pavlovski makes it possible to reveal the avantages an the isavantages of each metho. n practice, it is a question of using the metho which is appropriate for the problem to be best solve. REFERENCES [] N.Kremenetski, D.Schterenliht, V.Alychev, L.Yakovleva. [] Hyraulics, MR-Moscow eition, p. 7-. [] Armano Lencastre. General hyraulics, Eyrolles P [4] M.Carlier, Hyraulics general an applie, p [5] Walter H Graf in collaboration with Mr. S. Altinakar, volume6, river Hyraulics, p Fig. 6. Plots the curve M using two methosb. SBN: SSN: Print; SSN: Online WCE