HYDRAULIC CONDITIONS OF WATER FLOW IN RIVER MOUTH

Transcription

1 Prof. Dr. Zygmunt Meyer West Pomeranian University of Technology, Szczecin YDRAULIC CONDITIONS OF WATER FLOW IN RIER MOUT Astract: The outlet distance of river is the area at groing road at the river cross section and small slope of free surface. Additionally the river mouth is under strong influence of sea surges and ind action. The rising up sea level produces ackater curve upstream the river and the ind loing opposite to the main flo direction produces ind shear stress at the free ater surface. Bath: ind and surges change the relation river flo-depth. The paper give theoretical consideration at the phenomenon and the practical engineering relations hich can e used for calculations. The presented research leads to the conclusion that the ind ackater curve in the case of loer Odra river can reach the distance up to 9 km upstream.. INTRODUCTION River mouth is an area here the classical description: depth-flo in river ed fails. At the river mouth, usually the river cross-section area increases and free ater tale slope diminishes. We oserve there the ackater curve effects and ecause of very small slope, at the free ater surface ind shear stress plays important role. The ind loing opposite to the flo direction can create ackard currents changing significantly the vertical distriution of ater velocity (tachoida). The depth averaged velocity in river is also slo don so it affects the sediment transport especially at the ottom. The change of sediment composition at the ottom implies further roughness changes (e.q. roughness coefficient y Maning). All those phenomenon can e oserved in the Loer Odra River [,,,9,,,,4,5]. The free ater tale slope on this distance reaches -5, the river idth aout couple of kilometers and the depth averaged velocity drops don to,5, m/s. Additionally at the Loer Odra River e oserve arotrophic aves generated y atmospheric pressure changes and see surges due to the ind hich also affect the flo in the river mouth [, ]. In the Loer Odra River there is also oserved salt ater edge penetrating upstream the river [5]. In this paper the presentation of the phenomenon appearing in Loer Odra River is limited and does not cover: sediment stream analysis, arotrophic ave propagation and salt ater edge penetration upstream. They are presented in previous papers [,, 5].

2 Fig.. The map of Odra River catchments [Atlas geograficzny Polski 7] Fig.. Odra River outlet [Coufal ]

3 Fig.. Wind ackater curve in Odra River mouth [Liront ] Fig 4. Sal ater edge penetrating upstream [Meyer ]. MATEMATICAL DESCRIPTION OF PROBLEM. Basis equation of motion For the analysis of the ater flo in the river mouth area to dimensional vertically plane model, as applied. The asic system of coordinates as chosen in this ay that y - axe is directed vertically upard and the x - axe is horizontal [4,5,6,7,8,9,] see Fig. 5. Fig. 5. Scheme of flo applied for analysis

4 The asic equation of motion ere chosen as follos [,6,7]: d dt x p X + ρ x x ( ) + ( ) xx y xy () d dt y p Y + ρ y x ( ) + ( ) yx y In the aove equations folloing symols are used: the ater depth, x and y the ater velocity components in direction x and y respectively, p atmospheric pressure, t time, ij turulent shear stress components, shear stress at the river ottom, shear stress at the free ater surface, ρ ater density, X, Y the unit mass forces. The continuity of flo is expressed as We differentiate: equations (); other. It gives yy div () and equation (); y x () and then sutract them from each d y dt x d x dt y xx x y + y xy x xy yy x y (4) In eq. (4) it as assumed that: the unit mass forces X and Y have a potential, ater density vertical changes can e neglected, and that the atmospheric pressure does not very. If e referee eq. (4) to the asymptotic case for steady flo, e have then: for x xy (5) y Assuming the oundary conditions shon in Fig. 5 e have: for y ; xy and for y ; xy (6) The vertical distriution of the component of turulent shear stress after solving eq.(5) takes form: xy y ( y) ( + ) (7) This equation in the further part of the paper ill e used for description of tachoida prolem. The second asic equation refers to energy changes in the floing stream. We denote, as the mean depth averaged velocity: 4

5 υ x ( y)dy (8) And according to Saint-enant depth averaged stream energy procedure e have d x α dy dt x here: α is the Saint enant coefficient [7]. The integration in vertical direction of the unit mass force and pressure give: p X g ρ x x ( Rz) here: Rz denotes the free ater surface level referred to the datum level, g is the acceleration due to the gravity. The xx component of the turulent shear stress is taken according to the earlier orks [4] in the folloing form: xx κ ρ () x and it gives: () (9) x xx κ ρ x In the aove equations the value of κ denotes dimensionless parameter [4]. The last term In eq. () is integrated as: () xy y + dy ρ () Taking all the aove results i.e. egs. (8, 9,,,, ), the asic formula takes the final form: Rz + α κ x g + x x g (4) ρg The aove equation descries the steady non uniform flo in river mouth including folloing facts: varying river cross-section area and ind shear stress acting at the free ater surface. To salve the equation (4) it needs further to descrie the terms and. It needs to solve then the prolem of vertical distriution of velocity in river i.e. tachoida prolem.. Analysis of tachoida prolem ertical distriution of ater velocity in river in literature [6,7] is named tachoida Fig. 6. The asic equations descriing tachoida are: vertical distriution of turulent shear stress component xy (y) and the Boussinesq hypothesis descriing eddy viscosity coefficient influence [6, 7]. 5

6 Fig. 6. Flo factors in tachoida equation The distriution of the xy component of turulent shear stress as assumed according to the eq. (7), i.e. linear. This form of distriution has een confirmed in many researches [4, 5, 6, 7, 8, 9, ]. oever the prolem is still eing investigated. So e have ( y) dx xy( y) ρk( y) (5) dy here: K(y) is the eddy viscosity coefficient. If e put the relationship (5) to the eq. (7) it gives the asic formula for tachoida x ( y) y ( + ) dy + const ρk ( y) To solve the aove equation of tachoida it needs to descrie the eddy viscosity coefficient K(y). In literature, there are presented several models, among them the Prandtl distriution [6], Meyer [9] and that one assuming constant eddy viscosity coefficient. Prandtl model has never een applied for the case hen ind action occurs at the free ater surface. Sometimes exponential function is applied; as the system reaction for the unknon extortion [5, 6]. The detailed analysis of the eddy viscosity coefficient is not the aim of the present paper. To simplify calculations and to sho the ind influence the constant eddy viscosity coefficient as applied: ( y) const K (6) K (7) Additionally the oundary condition at the ottom as applied for x (8) y ; ( y) Including assumptions (7) and (8) e otain the solution of tachoida equation as follos: x ( y) y y + ρk ρk (9) That is the second order polynomial and for the case ith no ind shear stress it turns to the knon in literature tachoida of Bazin [6]. In the Fig. 7 examples of the curves given y eq. (9) are presented: 6

7 Fig. 7. Shape of tachoida curve for different The location of the point x max can e otained from folloing relations: y () + x max ρ K ( + ) () It is also possile to find the place y here velocity x. It gives: y () + Additionally, the depth averaged velocity according to the eq. (8) gives x( y) dy () 6 ρk The velocity at the free ater surface from eq. (9) comes to p ρk If the ind loing opposite to the main flo causes surface ack current its discharge can e estimated as: and the main flo After calculations it gives: ( y) dy (4) q x (5) y y ( y) dy q x (6) q ( + ) ρk + (7) 7

8 q (8) ρk ( + ) Further analysis of the flo needs description of the eddy viscosity coefficient K as a function of flo parameters. According to the literature [4, 5, 6, 7, 8, 9,, 6, 7] usually it is assumed that the coefficient is referred to the shear velocity or to the depth averaged velocity. So e have: K K κ u (9) here: u * - is the shear velocity and κ const is a coefficient. It is also very often used: K K κ () The parameter κ const is a coefficient hich has to e determined. Each of the aove description of the eddy viscosity coefficient leads to different tachoida. In the case of the uniform flo e have: + ρg I () here I is the river ottom slope. In the case of no ind e have: ρ u () The coefficients: κ and κ can e determined in the case and using Chezy formula. It gives: - for the case - for case κ g C () g κ (4) C In the aove equations the value C denotes velocity constant for Chezy equation. It is also possile to otain relations for the mean (depth averaged) flo velocity. We have: - for case κ + 9 κ + ρ ρ (5) - for case 6 κ ρ (6) To compare the results given in eqs. (5) and (6) it is convenient to introduce the ind shear effect, putting folloing parameter ξ, 8

9 So it comes to: - in the case ρgi ξ (7) ξ C I - ξ - C I f( ξ ) (8) ξ - in the case C I - ξ - ξ C I f( ξ ) (9) In the Tale, there are given the values of f (ξ); f (ξ) and also the ratio: ξ ξ - ξ f ( ) ( ξ ) η ξ (4) ξ ξ f( ξ ) The values of functions: f (ξ); f (ξ) for different ξ Tale. ξ f ( ξ ) f ( ξ ) η ξ ( ) From the point of vie of the practical application important matter is to sho ho each case: and, change the flo for the case hen: and so (4) (4) p and therefore ξ (4) + We put no ξ. 5 in the eqs. (8), (9) and it gives:, 5C I and, 5 C I (44) 9

10 From eq. (44) it can e concluded that the model represents flo ith igger resistance and so the mean velocity is smaller than in the case. The same conclusion concerns also the velocity at the free surface: - for the case ( ) ξ p C I (45) ξ - for the case and the maximal velocity: - for the case ( ) ξ p C I (46) ξ - for the case / ( ) ( ) x max C I ξ (47) ( ) ( ξ ) x max C I (48) ξ and the ack current flo at the surface and the main flo are as follos: - for the case ( ) ( ξ )( ξ ) q C I (49) 4 ξ - for the case 5 ( ) q ( ) C I ξ (5) 4 ( ) ( ξ )( ξ ) q C I (5) 4 ξ ( ) ( ξ ) q C I (5) 4 ξ From eqs. (45) to (5), it comes that in the model of turulence descried y case, the ind easier forms the ack current at the surface and the velocities are igger. In interesting case of flo can e considered if, so p and ξ ½ and e are looking for the river depth satisfying the same discharge in the case and. It reduces the prolem to eq. (44).

11 So e have: I 5 I 5, 5, 5 n n therefore (5) ( ), 6, 76, 8 In the aove equations the symol denotes the depth for the solution assuming model (case ) and - denotes the depth for the model (case ). To create the situation ith zero surface velocity and the same flo each case leads to different depth.. TE CONCEPT OF WIND BACKWATER CURE.. Equation of ind ackater curve The equation of the ind ackater curve comes from relation eq.(4). oever in this formula e have to introduce certain model of turulence to descrie the dependence. Including relations (5) and (6) it gives: ( ) f - for the case + ρg ρ + κ 4 + ρg 9 κ + ρ (54) - for the case + ρg + (55) C ρg The aove description means that no e have to introduce one of the formulae (54) or (55) instead of the last term in the equation (4). To sho the influence of ind in creating ackater curve it is easier taking the formula (55) i.e. the case. The relation eteen shear stress and velocity is then linear, ( ) f, (56) In engineering hydraulics it is often put critical depth kr as follo: αq kr (57) g Where also q const (58) Using the aove notations, the eq.(4) turns to the ordinary differential equation:

12 In the case We have: lim x kr d dx drz κ dx kr d dx q C ρg x eq.(59) turns to the formula relevant for the steady uniform flo. d dx And so the equation (59) gives: d and lim x dx (59) (6) q I (6) C ρq The symol denotes river depth for the steady uniform flo ith ind action. The equation (6) agrees ith earlier otained formula(8). We have: q I ξ (6) C Where no ξ is related to the case of steady and uniform flo: ρg ξ (6) I Sustituting formula (6) to the eq.(59) e otain finally folloing relation: d dx I kr kr d κ dx kr I kr (64) here: I ρg (65) I can e called ind slope. Eq.(64) is the asic description of the ind ackater curve. It differs from that one given in literature [,6,7] comparing the last term. It e put no: κ and I (66) and e otain the classical ell knon ackater curve equation d dx I (67) kr

13 If in the eq. (59) e put κ e tend to the case of ind affected ackater curve. It gives d dx I I (68) kr kr From the formal point of vie the aove formula is the second order differential equation. To solve it e need oundary conditions: x e have ( ) for (69) and d dx x (7) It means that the second derivative eq.(68) of depth, needs to assume as oundary condition first derivative of depth i.e. the free ater slope. So e can no descrie the ackater curve for the case hen river enters reservoir and the slope is equal to zero. The classical formula does not give the possiility to include such a case. To solve the classical equation it needs to give only the depth at the x. So x e have ( ) for (7) and therefore the slope at the x can e calculated as follo d dx x I ( ) kr ( ) (7) Note in the aove equation denotes the depth in river in steady and uniform flo and denotes depth of the piled ater at the x. ( ) Eq. (64) enales also to calculate the ackater curve for the case of flat (horizontal) ottom and no ind: I and (7) It gives then d kr d kr κ + (74) dx dx In practical calculations for rough estimation approximate formulae can e used. To otain the approximate solution folloing assumptions ere applied: - the piled part of the river depth is small ith comparation to the river depth, and

14 - the ratio of the critical depth to the ater depth is also small. This assumptions can e accepted in mammy practical cases at the river mouth. The asic equation for this case takes form d ( ) I I (75) dx After sustituting the dimensionless parameter ξ it gives x const exp I( + ξ ) (76) The constant value in eq.(76) is chosen in such a ay to satisfy oundary condition. x ( ) (77) and therefore hen x ( x) + [ ( ) ] exp I (78) and for the ind ackater curve x ( x) + [ ( ) ] exp I( + ξ ) (79) In the aove equations: and no ind and ith ind shear stress. If e consider flo in hich the discharge can derive the folloing relation among them: and so - for the case ith no ind 5 q I ns - for the case ith ind shear stress q n s are the depth in the river in uniform flo in the case of 5 ( ) I - ξ, q const e ξ (8) In the aove relation n denotes Maning roughness coefficient. Eq. (78) and (8) enale to compare the horizontal extent of the ackater curve in the case of no ind and ith ind shear stress. We have (8) (8) ( ) L ln I ( + ) s ξ in case ith ind shear stress and for the case ith no ind it gives L I ln ( ) s (8) (84) 4

15 In the aove equations folloing symols ere used: L - the horizontal extent of the ind ackater curve; L - the horizontal extent in the case ith no ind; s - the difference in the ater level ith ackater curve and in the case of uniform flo. Usually it is assumed: s.m (85) If e assume for loer Odra River. m; 5 I 5 ;. gives after calculations,,,. 6 m, and 67 km, L L 9 km ξ ; and ( ) 4. m it The calculations suggests that the ackater curve reaches in the river point Gozdoice. It can happen only in case of very strong ind. Conclusion comes that for the case ith ind shear stress the horizontal extent is smaller. The approximate formulae they are especially useful in calculations covering long distance, for example sediment transport changes. In calculations of sediment stream e need the mean velocity, depth and shear velocity along the river. They can e calculated from eq. (78) and (79) and the shear velocity from formulae (4), (5) and (75). We have u ρ g + (86) C ρ q (87) and so, the term hich appears in sediment transport calculations, takes form u g C + ρ (88). Wind oundary conditions The most important factor in calculations of the ind ackater curve is the ind shear stress. The second factor is the river ed slope and then depth of ater at the oundary x. If needs also to specify the flo parameters for uniform flo ith no ind. The most crucial is the relation descriing ind shear stress. In the literature the most common formula is [,,9,6,7]. κ ρ W (89) in this equation W denotes ind speed, ρ - density of ater, κ - dimensionless parameter. From field experiments for Loer Odra River it comes []. 5

16 6 6 κ (9) And no from equation (88) it can e concluded the ind speed W hich has significant influence on sediment transport. The evaluation gives that it depends on the term C W κ (9) g and this term should e of order ½. It allos to define the ind speed of significant importance for sediment stream C W κ (9) g W 5 8, hat In practical cases for Loer Odra River it means ind of the speed ( ) gives aout m/s. The scheme of the oundary conditions for the ind ackater curve are given in Fig. 8 Fig. 8. Boundary conditions for the ind ackater curve The characteristic feature of the ackater curve is the change of velocity profile. The surface velocity of ater is slo don y the ind loing opposite to the ater flo. But the ind effect diminish upstream. It is shon in Fig. 9. Fig. 9. The change of tachoida shape due to ind influence 6

17 In the extreme case of a very strong ind at the surface there can e created ack current. The ater at the upper layer is floing opposite to the main flo. It is shon in Fig.. Fig.. Tachoidas in river in case of ind ack current In Fig. there is shon the mechanism of formation of ind ack current at the surface layers. It can e seen that a circulation region is created in this area. It extends up to x x. And further upstream the normal ind ackater curve exists. This sort of flo has special meaning hen savages are discharged to the river. The ind is ale to push the savages upstream.. EALUATION EXAMPLE As the example of calculations of ind ackater curve in natural river Loer Odra River as chosen and distance from Szczecin to Gozdoice. Preliminary calculations presented in this paper shon that the ackater curve can cover the hole distance. In the calculations it as taken: ( ) 5, 5 m; results are given in Fig.. Q 55 m ; W m s ; s 5 I 5 and κ 6. The Fig. Wind ackater curve in Loer Odra River for Q 55 m [Liront ] s On Fig. these is shon the ind ackater curve for Loer Odra River in the case Q 5 m. s 7

18 Fig.. Wind ackater curve in Loer Odra River for Q 5 m [Liront ] s In Fig. these is shon the ater level changes at Gozdoice cross section due to ind influence Fig.. Water level changes at Gozdoice due to ind influence [Liront ] 5. CONCLUSIONS 5. The analysis of ind ackater curve formation is presented in this paper. 5.. The mechanism of formation of the ind ackater curve as estimated using the asic hydrodynamics equations of motion and the model of the vertical turulent momentum transfer. 5.. To models of the turulent momentum transfer ere considered one ased on the shear velocity and the other on the mean velocity. The mechanism of creation of the ind ackater curve ased on the models is the same. Different are the resulting depth and the surface velocity. In case of the model ith shear velocity it seems to e that the ater system presents stronger resistance against ind. So certain effects need stronger ind. The phenomenon needs further research. 5.4 From the analysis of the ind ackater curve it comes that in the case of very strong ind, loing opposite to the flo a ack current can e created. In this case ater at the upper layers flos opposite to the main flo. It has een confirmed in literature [Buchholz ]. 8

19 5.5. The presented model of the ackater curve needs further field experiments. Especially the ay of calculation of the ind shear stress, and the model of turulence in vertical momentum transfer. Biography [] Buchholz W. Wpły iatru na przepłyy ujściach rzek (Wind influence on river outlet flo) - Wyd. Inst. Morskiego Gdańsku 989. [] Coufal R. Bed Changes and Sediment Transport at River Mouth, Institute of ydroengeneering, Polish Academy of Science ydrotechnics No, Gdańsk 997. [] Liront D. Wpły prędkości iatru zmieniającej się zdłuż koryta dolnej Odtry na stany i przepłyy ody (Wind influence varing along river on depth and flo) - Rozpraa doktorska, Wydz. Bud. i Arch. Politechniki Szczecińskiej, Szczecin 999. [4] Meyer Z. ydrauliczne arunki dopłyu ody do ujęcia arunkach stratyfikacji gęstości (ydraulic condition of ater in flo to selective ithdraal) - Archium ydrotechniki 979, p [5] Meyer Z. ertical circulation in Reservoir Due to Selective Withdraal, Rozpray ydrotechniczne 98, p [6] Meyer Z. ertical Evaluation in Density Stratified Reservoir, Journal of the ydraulics Div. ASCE Y 7, 98, p [7] Meyer Z. Axisynnetric ertical Circulation in a Stratified Reservoir, Journal of ydraulics Research, 98, p. -5. [8] Meyer Z. Mechanics of Bottom Density Wedges, Archium ydrotechniki Nr /985. [9] Meyer Z. ertical Circulation in Density Stratified Reservoirs, Encyclopedia of Fluid Mechanics vol., Gulf Pulishing Co. uston, 986, p [] Meyer Z. Modified Logarythmic Tachoid Applied to Sediment Transport in River, Acta Geophysica, Institute of Geophysics Polish Academy of Sciences, W-a 9. [] Meyer Z., Coufal R. Wind Shear Stress Affected Sediment Transport at River Mouth, Proceedings of International Congress of ydro-science-engineering ICE 9 Mississippi USA 99. [] Meyer Z., Eertoski R. Influence of Atmospheric Pressure Changes on Channel Flo-Numerical Solution, 6-th German Polish Seminar on Coastal and Estuary Dynamics, Ueckermuende 997. [] Meyer Z., Eertoski R. Flo Friction Forces Effect on Baroscopic Wave Attenuation in River Mouth, Archives of ydroengineering and Environmental Mechanics, Gdańsk 998. [4] Meyer Z., Paceicz F. Wpły lokalnych zmian zierciadła ody na kształt krzyej spiętrzenia (Local river slope changes on ackater curve) - Konferencja Naukoa Wyrane Prolemy ydrotechniczne ujścia Odry do Bałtyku, Instytut Morski O/Szczecin, 984. [5] Meyer Z., Pluta M. Zastosoanie modelu przepłyu stratyfikoanego do analizy transportu rumoiska zagłęieniu dennym (Stratified flo model applied to to sediment transport in river cavern) - Konferencja pn. Zastosoania mechaniki udonictie lądoym i odnym pośięcona 7-leciu urodzin Profesora Piotra Wilde - Instytut Budonicta Wodnego PAN Gdańsk. [6] Prandtl L. Dynamika przepłyó (Dynamics of Flos) - PWN Warszaa 956. [7] Puzyreski R., Saicki J. Podstay mechaniki płynó i hydrauliki (Basies of Fluid Mechanics and ydraulics), PWN Warszaa