Hidrológiai Közlöny 1955 (35. évfolyam)

11-12. szám - Kovács György: Az árhullámok levonulására jellemző hidrológiai mennyiségek meghatározása

1(12, Hidrológiai Közlöny 35. évf. 1955. 11—12. sz. Kovács Gy.: Árhullámok levonulása •t

1 .neHHH 3naMeHHÜ pacxoAa H rjiyŐHHhi, aaBHcaiuHx BO BpeMH npoABHweHHH naBOAKOBbix Bojui OT Mecra H BpeMenH. Hcnojib30BaHHbie B npouecce BbiBo/ia H BbiBe.ieiiHbie na űCHOBe CTaTHCTHMecKnx aaHHbix 3naqeHiiH noKa3aTejiefl p = 1 H C00TBGTCTBEHH0 r— 1,5 B nponecce^nx npaKTimecKoro iipuMeiienHíi MoryT H3MeH«Tbcsi. OanaKO, NEMI NPEIINOJIOJKHTE.IBHO MO>Ker OKA3ATB noMomb npii HCC^eaoBamiHx, CBH3aiiHbix c H3yieHiieM naBO^KOBbix BOJIM c rn;ipaB.iiiiqecKoii TO'IKII 3penHH, KOTopbie HccjieaoBaHníi cpe^H npaKTimecKHx 3a,aaq cnocoöcTByiOT iie.iy rn;ipo;iorHMecKHx np0rn030B, a TaK«<e pemenHio Bonpoca pacnjiacTbiBaiinji, 9i<cn.nyaTaUHH roptlbix BOflOXpaHHJIHIH, H T. n. Determination of the hydraulic characteristics of flood-wavcs G. Kovács The object of the study has been to elaborate a proper method, by wliioh — with aid of tho available hydrological and hydraulic data — all. characteristics of flood-waves, the celerity of travel as woll as the time and local variations in discharge and depth can be determined. The basis of the procedure proposed are the Hernouilli equation for non-permanertt flow and the oquation of continuity. In order to obtain mathematical solution, the riverbed has been substitutcd by an ideál ized channel the slopo of which equals the slope of watorsurfaco of permanent flow subsequent to l'loods. The depth of the idealized channel is to bo~ computed in tho cxamined sectioi^ from the level characterizing the cessation of flow. The flow being rectilinear, no losses occur apart from friction-losses. Whilo the riverbed is infinitely wide, hydraulic radius is equal to depth. The oxponent n in the Baohmetev formula expressing variation of modulos with Tlepths, is equal to 3. The assumption in question makes the introduction of tho specific discharge carried by a stripe of one meter width, and tho substitution of the section-aroa with depth possiblo. Tho two equations mentionod abovo eomprise four variables. In ordor to promoto tlie pössibilitv of the solution, the author endeavoured to ostablisb hydrologicál relationship between two variables, that their numbor may be roduced. Illustrating the stage-discharge relation in rectangular coordinates, the correspondont discharge and stage values during tho rising and the falling wave deseribo the well-known floodloop (stage-discharge loop) closelv fitting tho diseharge-curve for permanont flow. The rising limb starting from the diseharge-curve for permanent flow would again attain this curve monotonously incroasing, if the rising wave would not be followed by a falling wave, but permanent flow could dovolop. In the same manner, if permanent flow would precede and succeed the falling wa.ve, the falling limb would be characterizecl by a monotonous curve. The attenuation at the lower and upper end of the loop is duo to the superposition of waves rising and falling. Tlius it follows, that tho dischargo-depth relation for rising and falling waves could be characterized by a monotonous curve. Let us approach these eurves by parabolas liaving oxponents aocoidinglv cliosen. In rectangular coordinates, the origo of which is the lower intorsection of the parabola and the discharge-curvo, tlie ociuation of parabolas may bc written : for tho rising limb dm = a

2 (dg)P for the falling limb dm == a

± (öqV The data of four sections of the Danube and the Tisza Rivers (Nagymaros, Sztálinváros, Polgár and Tisza bő) show, that the oxponent of the parabola substituting the rising limb can be chosen with good approximation as unity P = l. and thus tho curve may bc substitutcd by a straight line. The adsantage of this approximation is, that tho integrál rosulting from tho basic equations can be solved in olosed form and numerical results may bo obtained with a computation stop by step. This alsó renders tho pössibilitv of determining tho limiting oonditioiis in case of other oxponents whon a graphical solution is tho only pössibilitv. Ilaving the integration performed and substituting the limiting oonditions, tho rising wave can bo oharactorized by the following exprossion : /i

3(w —1) + ——— l

n \Tnv) a(a — p) P(P — «) A 111 <u = xp • 1|! ^ 2 V 2 Aq (x — vt) where x and I indicate time and location v wave-celerity i.

2 surface-slope of permanent flow .subsequent to rising Aq totál discharge-incroaso during rising a = relation of the difference between Aq permanent discharge succeeding rise and the momentaneous discharge to totál dischargs-increase. t A

0, A

v A„, A.

t, a and p are constants depending on on tho celerity of travel of waves, on permanent discharge and its mean velocity subsequent to the wave. M • If the exponent is taken as unity, the waveceleritv will be constant and can be computed as a quotient of the differences of the discharge and depth proceding and succeeding rising. — ?i TÍZ., — tn I11 examining falling waves on basis of tho hydrological data of ttíts sections mentionod before, the? approximate valuo of the oxponent. r = 1,5 has been chosen. In replacement of tho non-uniform discharge a new variablo has been introduced í 1 — Q

1 V/r 1 l J wherc q non-uniform discharge m

1 and depth and discharge of lower permanent flow subsequent to wave Am and Aq totál variation in depth and discharge. The relation between variables 2 and t (timo) in the ORSC of x — 0, is expressed by tho differential equation dí da \ qI í ^ 1 (1 +z

2)

3 —( 1— — Gz — — Oz

3\ gm\ 3 q^O (1 +z

2)

3 — (1 +Gz

3Y whereas fixing time to a constant value, between variables x and z the following differential equation can