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Essentially nonoscillatory (ENO) and weighted essentially nonoscillatory (WENO) schemes are high order resolution schemes constructed for the hyperbolic conservation laws. Following their original ideas in this paper we extend these schemes to the one-dimensional bed-load sediment transport equations. The difficulties that arise in the numerical modelling come from the fact that a nonconservative product is present in the system. Our specific numerical approximations for the nonconservative product are based on two ideas. First is to include the influence of that term in the system upwinding and the second is to define the numerical approximation in such a way that the obtained scheme solves the system for the quiescent flow case exactly. As a consequence, the resulting schemes give excellent results, as it can be seen from the numerical tests we present. On the opposite, the numerical results obtained by applying the pointwise evaluation of nonconservative product on the same tests present unacceptably large numerical errors.

In this paper we extend the nonstaggered version of the central NT(Nessyahu-Tadmor) scheme to the balanced laws with geometrical source term. This extension is based on the source term evaluation that includes balancing between the flux gradient and the source term with an additional formulation that depends on the source term discretization. The main property of the scheme obtained by the proposed reformulation is preserving of the particular set of the steady-state solutions. We verify our scheme on two types of balanced laws with geometrical source term: shallow-water equations and nonhomogeneous Burger's equation. The presented results show good behaviour when compared with the analytical or numerical results obtained by using other numerical schemes in the considered problems. Furthermore, comparison with the numerical results obtained by the classical central NT scheme where the source term is pointwise evaluated shows that the proposed reformulation is essential.

Studija se bazira na numeričkim simulacijama propagacije poplavnog vala nastalog kao posljedica djelomičnog oštećenja ili potpunog rušenja brane Tribalj. Na temelju rezultata simulacija definirana su područja pod utjecajem poplavnih valova. Dodatno su iznesena svjetska iskustva iz problematike rušenja brana i matematičkog modeliranja takvih događaja. Računalni program korišten za simulacije validiran je na svjetski priznatim test-primjerima.

Grad Otočac smjestio se u zatvorenoj kraškoj dolini rijeke Gacke u priobalnoj regiji Lici. Dolina rijeke Gacke (450 m.n.m.) je masivom planine Velebit (1000-1700 m.n.m.) odvojena od Jadranskog mora. Tijekom povjesti grad Otočac višestruko je izložen poplavljivanju, te su prvi vodnogospodarski radovi u smislu prevencije i zaštite od poplava izvedeni još u 19 stoljeću. Početkom šezdesetih godina dvadesetog stoljeća sagrađen je Hidroenergetski sustav HE Senj kojim su vode iz Gacke doline i susjednog kraškog polja (Ličko polje), usmjeravaju prema hidroelektarni na samoj morskoj obali (Senj). Unutar sustava, sa osnovnom svrhom proizvodnje električne energije, zahvaćene su vode rijeka Like i Gacke, sa svojim većim i manjim pritocima. Okosnicu sustava čini akumulacija Krušćica (zapremine 142 milijuna m3 vode), tri hidrotehnička tunela (propusne moći od 49 do 60 m3/sek) i kompenzacijski bazen Gusić polje (zapremine 1, 5 milijuna m3 vode). Izgradnjom hidroenergetskog sustava izmijenili su se uvjeti propagacije poplavnih valova na pojedinim djelovima prirodnog hidrografskog sustava, te je došlo do promjene prirodnog vodnog (hidrološkog) režima rijeka Like i Gacke. U radu se analiziraju mogući scenariji poplavljivanja područja grada Otočca u ovisnosti o veličini poplavnih voda rijeka Like i Gacke, i u ovisnosti o mogućim upravljačkim manipulacijama na objektima HE Senj. Za provedbu dinamičke analize raznih (mogućih) scenarija poplavljivanja grada Otočca, kao i za iznalaženje optimalnog upravljanja s hidroenergetskim sustavom u vrijeme pojavljivanja velikih (poplavnih) voda, izrađen je matematički model propagacije poplavnih valova. Osim toga matematički je modelirano i tečenje malih voda u prirodnim koritima i artificijelnim dijelovima hidroenergetskog sustava, te su provedene ekološke analize vezane za biološke minimume u vodotocima. Te su analize provedene u svrhu rješavanja problema revitalizacije nekih područja, koja su izgradnjom hidroenergetskog sustava ostala bez vode.

The shallow water equations are accepted in many practical applications as the governing equations for the unsteady water flow in natural watercourses and in the case of flooding. For modelling a realistic problem it is necessary to use this nonhomogeneous system that includes source terms relevant to the varying bed topography. The first emphasis of this work is on using the numerical scheme in which the source term is appropriately discretized, so that the balance between the flux gradient and the source term is obtained. The first order scheme developed by Hubbard and Garcia-Navarro in [5] by an extension of the standard upwind finite volume numerical scheme based on the Roe’ s approximate solver, satisfies that balancing property, so we present it and use it in this work. The scheme is applied here to both one- and two-dimensional problems. Besides the problem of the correct source term discretisation an additional problem arises if the propagation over the dry bed occurs. We propose here a numerical technique that is necessary for appropriate treatment of this problem. This technique is crucial for the water flow computations over the natural watercourses. Simulation results corresponding to the test problems in the last section illustrate that the proposed numerical treatment successfully handles complex flow domains.

Studija sadrži hidrološki opis vodnog režima rijeke Kupe i V. Belice te opis izgrađenih 1D i 2D matematičkih modela strujanja. Nakon kalibracije provedene su simulacije strujanja na realnom koritu Kupe i V. Belice i okolnog poplavnog terena te na pet različitih varijanti regulacije korita Kupe i V. Belice. U završnom dijelu studije ocjenjene su učinkovitosti ispitivanih regulacijskih radova i dane smjernice za izradu idejnog rješenja zaštite naselja Kuželj od poplava.

When deciding whether to employ explicit or implicit approach in numerical approximation of open channel flow equations one has to bear in mind that explicit methods are generally easier to program, and require less computational effort per each time step than implicit methods. Explicit methods solve for one point at a time in certain time step - that is, one equation and one unknown. The main drawback of the explicit approach is that the unknown node must be within the zone of dependency of the nodes that are used to approximate the derivatives at the known time line – otherwise, the method will be unstable. Therefore, maximal time step is bound by Courant Friedrichs Lewy (CFL) condition. Implicit methods solve for the unknowns in a domain simultaneously, requiring solution of simultaneous algebraic equations. The advantage is that matrix of a linear system to be solved at each time level has its arguments from the new time level n+1 and thus the unknown nodes are always in the zone of influence. Generally, implicit methods also include data values from previous time level n but they come to the right hand side (RHS) of the linear system to be solved. Implicit schemes will yield convergence in a smaller number of time steps, since the time step is no longer constrained by a stability limit but on the other hand they require more computational effort consequent upon the need to solve block tridiagonal system of equations per each time step (providing finite difference approximation is employed). Use of implicit or explicit numerical method is also dependent upon the fact whether the objective of calculation is simply to reach steady state or analysis of transient solution is needed.

Hyperbolic systems with geometrical source terms arise in various applications. One such application is fluid flow with cross section area variations, which arise naturally in the study of fluid flow phenomena in ducts, pipes, shock tubes, and nozzles. In such applications the source term may have low regularity, which causes numerical difficulty particularly when steady state solutions are computed. The fix for this numerical difficulty is to approximate the source term using decomposition and upwinding, thus creating balance between numerical flux gradient and numerical source term. In this paper we introduce extension of the Q-schemes proposed in [1] to nozzle flow equations, i.e., Euler equations with geometrical source term.

1D model strujanja u otvorenim vodotocima korišten je za simulaciju strujanja vode u rijeci Rječini u predjelu od izvora do Tvornice papira na ulazu u urbano područje grada Rijeke. Pomoću izgrađenog modela provedene su nestacionarne simulacije propagacije vodnih valova duž rijeke Rječine.

Provedena je prva faza analize strujanja dimnih plinova u elektrofilteru TE Plomin 1. Cilj studije je bio optimizirati strujanje kroz elektrofilter radi povećanja njegove učinkovitosti.

Claw2d 1.0 is a computer program that performs numerical solution of initial – boundary value problems for two-dimensional conservation laws. In the present version only 2d shallow water equations are included together with appropriate numerous boundary conditions that model lakes, see estuaries, coasts and walls, bridges, weirs, dams, etc. Numerical solution is based on the application of Q-schemes. The complete program’s code is object oriented and it is written in C++. User interface is developed using the Microsoft Developer Studio Visual C++ 6.0 and it enables the user to easily build a new project, add all the components of the model, define all the necessary data, perform all computations and examine graphically and numerically all the obtained results. Graphical representations are 2d and 3d and are implemented through OpenGL. The programs is and can be applied to problems of flow in natural watercourses, artificial channels or propagation of flood waves.

In this paper a solution of a flooding problem with the aid of a computer program for the open channel flow and flooding simulations is presented. The CFD (computational fluid dynamics) approach is based on the mathematical model for the one-dimensional non-stationary open channel flow extended with the floodplain compartment model. The developed computer program implements state of the art numerical schemes for the resulting initial-boundary value problem i.e. schemes specialized for hyperbolic conservation laws with significant source term. The computer simulations are based on the complex three-dimensional geodetic data of channel and surrounding terrain developed using standard GIS tools. Case study presented in this paper is the problem of flooding in the Gacka river basin (Lika, Croatia) which water supplies the hydroelectric power plant Senj. The flooding may be caused when the power plant is stopped or in the case of heavy rainfall. Analyzing simulations’ results, the construction of new bypass channel is proposed, as well as its best location and dimension. The bypass channel completely solves the flooding problem in the area.

The numerical procedure for an optimization of a ship hull from a hydrodynamic standpoint is presented in the paper. In the procedure, a potential flow solver coupled with the genetic algorithm has been used. As an objective function, the wave resistance has been chosen. The optimization procedure has been applied for the Series 60 (CB=0.60) hull taken as reference hull. Depending on imposed geometrical constraints, several variations of the ship forebody have been obtained. The results from the numerical predictions indicate that the obtained hull forms yield a reduction in wave resistance.

U radu su dani rezultati računalnih simulacija pucanja brane Valići i širenja poplavnog vala rijekom Rječinom, dobivenih primjenom state of the art numeričkih metoda za hiperboličke zakone ravnoteže.

This paper presents a CFD (computational fluid dynamics) approach applied in flood forecasting and prevention in the case of the lower part of the river Rjecina (Croatia, Europe).Rjecina is a torrential river, which in its lower part flows trough the city of Rijeka and into the Adriatic Sea. Key flood risk factors for the urban area surrounding the watercourse are possible sediment settlement, bridges that narrow the primary riverbed profiles, high tides etc. In order to design flood prevention measures acceptable in the urban environment a computer model was developed. This model is based on the 2D shallow water (i.e. St. Venant) equations. Since these equations are hyperbolic conservation laws with a significant source term, Q-schemes developed by Bermuddez, Dervieux et a1. were applied in the numerical computation. In this way the resulting computer application can simulate nonstationary river flow and flooding, dam breaks, supercritical flow, hydraulic jumps etc. First step in the application of this approach on the case of the lower part of the river Rjecina was digitaliztion of geodetic data and development of the numerical mesh. The model was then verified by comparison with measurements. In this paper some of the numerous simulation results are presented: concerning the possible sediment accumulation and concerning influence of the bridges. The main advantage of the developed computational model verified computational accuracy and the simplicity, with which hypothetical changes in the riverbed and surrounding urban area can be introduced. Obviously, the computer application can and will be applied to other flood prediction case studies.

Calibration of parameters of a mathematical model for hydraulic transient is an optimization problem, i.e. a problem of minimizing the difference between measurements and computations by changing values of the model parameters. In this paper the approach to this optimization task using genetic algorithm as well as excellent results of the application to the case study of a water supply pipeline in a hydroelectric power plant are presented.

In this paper a series of test including non-prismatic channels and supercritical flows, hydraulic jumps and smooth transitions is presented in order to investigate this problem a contribution to the research and comparison of one-dimensional (1d) and two-dimensional (2d) shallow water models is presented. For both mathematical models a state of the art numerical scheme, Q-scheme is applied and computer programs are developed. Results of measurements and of simulation of the two approaches on the propagation of water waves in the lower part of river Rjecina are compared. Differences between 1d and 2d shallow water modeling due to different mathematical and numerical modeling are analyzed.

In this paper a CFD (computational fluid mechanics) approach to the problem of hydroelectric power plants and flooding is presented. The applied method is mathematical modeling and computer simulation. The mathematical model is the open channel flow model, nonstationary, one-dimensional and extended with the floodplain compartment model. The computer simulation is based on the complex preprocessing of geodetic data of the river basin and numerical computation using state of the art numerical schemes. The developed computer program is applied on the case study, the problem of possible flooding in the river basin of Gacka and the town of Otocac. These flooding may be caused by stoppage of the work of the hydroelectric power plant Senj due to regular maintenance works, or errors in the manipulation with the objects in the hydro-energetic system of HE Senj. Also they can be caused by large flooding waves in the river Gacka in interaction with the work of HE Senj. Presented problem and results lead to the conclusion that the developed method is a complex and succesful tool for analysis and solution of flooding prevention.

Claw2d 1.0 is a computer program that performs numerical solution of initial – boundary value problems for two-dimensional conservation laws. In the present version only 2d shallow water equations are included together with appropriate numerous boundary conditions that model lakes, see estuaries, coasts and walls, bridges, weirs, dams, etc. Numerical solution is based on the application of Q-schemes. The complete program’s code is object oriented and it is written in C++. User interface is developed using the Microsoft Developer Studio Visual C++ 6.0 and it enables the user to easily build a new project, add all the components of the model, define all the necessary data, perform all computations and examine graphically and numerically all the obtained results. Graphical representations are 2d and 3d and are implemented through OpenGL. The programs is and can be applied to problems of flow in natural watercourses, artificial channels or propagation of flood waves.

Unificirani je pristup modeliranju strujanja u otvorenim i zatvorenim vodotokovima moguć i temelji se na jednadžbama za otvorene vodotokove. Ako se poprečni presjek cijevi modificira dodavanjem vrlo uskog i beskonačno visoke cjevčice iste jednadžbe mogu se primijeniti i na strujanje pod pritiskom u toj zatvorenoj cijevi. Dimenzije te fiktivne cjevčice moraju se prilagoditi tome da daju točnu brzinu zvuka. Značajna razlika između dimenzija cijevi i fiktivnog dodatka uzrokuje velike poteškoće u numeričkim simulacijama budući da se u trenutku kada strujanje sa slobodnom površinom postaje strujanje pod pritiskom prekid formira diskontinuitet u rješenju jednadžbi. Da bi se postiglo uspješno numeričko rješenje primijenjena numerička shema mora biti šok hvatajuća. U ovom su radu esencijalno neoscilirajuće sheme primijenjene i problem punjenja dovodnog tunela hidroelektrane Senj je riješen. Prikazani pristup daje stabilne numeričke rezultate i predstavlja izvrstan alat za simulacije punjenja tunela uopće.

U ovom su članku predstavljena dva računalna modela za propagacije poplavnih valova: modelom poplavnih terena prošireni jednodimenzionalni model strujanja u otvorenim vodotokovima i dvodimenzionalni računalni model plitkih voda. Kako su u oba ugrađene najsuvremenije numeričke sheme (ENO-sheme i Q-sheme), razvijeni računalni modeli omogućavaju uspješne simulacije stacionarnih i nestacionarnih strujanja, podkritičnih i nadkritičnih strujanja, hidrauličkih skokova, simulacije u složenim sustavima vodotokova s čvorištima, jezerima, preljevima, mostovima, utjecanjem u more, urbaniziranim i neurbaniziranim okolnim terenom, itd. Sve ove osobine modela vidljive su iz predstavljenih rezultata simulacija na slučajevima južnog kraka rijeke Gacke i donjeg toka Rječine.

U ovoj je studiji izložen proces razvoja matematičkog modela poplavljivanja donjeg toka Rječina i rezultati velikog broja simulacija dobivenih pomoću tog modela, a provedenih radi ispitivanja utjecaja različitih čimbenika na strujanja i poplavljivanja u tom vodotoku. Za modeliranje poplavljivanja donjeg toka Rječine korišten je nestacionarni dvodimenzionalni model plitkih voda sa ugrađenom numeričkom Q-shemom Bermudeza i Vasqueza. Nakon što je provedena digitalizacija geodetskih podataka donjeg toka Rječine, izvedena su i kalibracijska mjerenja te kalibracija parametara matematičkog modela poplavljivanja donjeg toka Rječine. Matematički model poplavljivanja donjeg toka Rječine primjenjen je na analizu utjecaja mora, mostova, nanosa i regulacijskih radova na poplavljivanje u donjem dijelu toka rijeke. Opisane su i mogućnosti daljnjeg korištenja modela za modeliranje drugih situacija i testiranje idejnih rješenja.

In this paper a contribution to the research and comparison of one-dimensional (1d) and two-dimensional (2d) shallow water models is presented. For both mathematical models a state of the art numerical scheme, Q-scheme is applied and computer programs are developed. Results of measurements and of simulation of the two approaches on the propagation of water waves in the lower part of river Rjecina are compared. Differences between 1d and 2d shallow water modeling due to different mathematical and numerical modeling are analyzed.

Osnovna svrha ovog kanala je da u slučaju prestanka rada jednog ili svih agregata HE Senj ili nailaska visokih voda rijeke Gacke, bez štetnih posljedica evakuira vodu preko Gornjeg Švičkog jezera u Donje Švičko jezero. Na taj način omogućava se nesmetana evakuacija nastalih viškova vode u smjeru ponornih zona, te se spriječava poplava u okolici čvorišta Šumečica (naselje Podgora), odnosno indirektno urbane jezgre grada Otočca. Za simulacije stacionarnih i nestacionarnih strujanja vode odteretnim kanalom tunel Švica - Gornje Švičko jezero korišten je računalni program za simulacije strujanja u otvorenim vodotokovima Claw1d 1.0. Taj je računalni program Windows aplikacija koju su razvili autori ovog idejnog projekta. Proračuni se stacionarnih i nestacionarnih strujanja u proizvoljnim sustavima otvorenih vodotokova u računalnom programu Claw1d 1.0 temelje na matematičkom modelu St. Venantovih jednadžbi proširenom modelom poplavnih terena i najsuvremenijim numeričkim shemama za hiperboličke zakon očuvanja. Računalni program Claw1d 1.0 već je provjeren i rabljen u prethodnim studijama.

In this paper a complete case study of the waterhammer in the hydroelectric power plant “Gojak”, Croatia is presented. The task consists not only in the mathematical modeling of the phenomenon but also in the development of a computer program, collection of all the relevant data on the pipeline and measurements of pressure during transient states. In the presented case all this is accomplished and comparison of measurement and computation shows that an excellent tool for waterhammer simulation in the pipeline in question is developed.

The mathematical model for the liquid-gas mixture flow in pipelines is an initial-boundary value problem for a nonlinear hyperbolic conservation law system. This hyperbolic conservation law system together with boundary conditions is numerically solved using the essentially non-oscillatory (ENO) schemes. The optimization problem is a boundary control problem, i.e. boundary conditions that cause pressure values in the pipeline as close as possible to the desired ones are to be found, considering given constraints. The applied optimization method is the genetic algorithm (GA) with two different variable-to-chromosome coding strategies: binary coding and real coding. The results of both GA strategies applied to two pipeline flow optimization problems are presented and compared.

New computer algorithms for surface intersection computation and for conformal projection of curves from a surface of revolution onto a plane are developed and implemented in a C++ program. The resulting CAD software is a turbine blade design tool aimed for engineers that work in the field of turbine blade optimization.

U ovoj je studiji izložen cjelokupni matematički model propagacije vodnih valova južnim krakom rijeke Gacke od ustave Šumečica do Donjeg Švičkog jezera. U pripremnoj fazi prikupljeni su podaci iz tehničke dokumentacije o hidroenergetskom sustavu HE Senj, podaci iz geodetskih mjerenja o južnom kraku rijeke Gacke te je u više navrata obiđen teren i snimljeno nekoliko video zapisa. Digitalizirane su državne topografske karte, podaci iz nivelmanskih knjižica, nacrti poprečnih i uzdužnih presjeka otvorenih vodotokova te je izrađen detaljan digitalni trodimenzionalni model otvorenih kanala “Tunel Lika-Gacka – čvorište Šumećica”, “Vivoza – brana Šumećica”, “Šumećica – Gornja Švica”, prirodnog korita južnog kraka Gacke od čvorišta Šumećica do Donjeg Švičkog jezera kao i čitavog okolnog zemljišta koji bi mogao biti zahvaćen poplavama. Iz tog su digitalnog modela terena izvedeni i pomoću splajn krivulja posebnih svojstava interpolirani svi geometrijski podaci o otvorenim vodotocima nužni za jednodimenzionalni matematički model, kao i stvorena složena nestrukturirana numerička mreža neophodna u dvodimenzionalnom matematičkom modelu. Postavljen je matematički model koji obuhvaća nestacionarno strujanje vode u otvorenim vodotocima, na mjestima račvanja kanala, kod preljeva, brana, mostova, utjecanja i istjecanja vode u akumulacije, jezera, more, strujanje vode pri poplavljavanju, itd. Jednodimenzionalni je model razvijen radi brzine simulacije, a dvodimenzionalni model radi detaljne simulacije poplavljivanja. Kao numeričke sheme odabrane su i primijenjene najnovije i za model plitkih voda najprimjerenije sheme: esencijalno neoscilirajuće sheme za jednodimenzionalni model i Q-sheme za dvodimenzionalni model, čime je osigurana sposobnost modela za simulaciju podkritičnog i nadkritičnog strujanja, hidrauličkih skokova, glatkih prijelaza, propagacije vodnih valova nastalih uslijed povećanog utoka u otvoreni vodotok, uslijed pucanja brana, uslijed zatvaranja ili otvaranja pojedinih zapornica ili zasuna, itd. Pri razvoju matematičkog modela dodatna je pažnja posvećena procjeni Mannigovog koeficijenta hrapavosti prema svojstvima korita otvorenih vodotokova i okolnog zemljišta i tablica eksperimentalno određenih vrijednosti poznatih iz literature. Model je utemeljen na znanstvenoj literaturi iz tog područja i analizi matematičkih modela komercijalnih računalnih programa (HEC-RAS 2.2, FLADWAV, MIKE 11, itd.). Provjerom na brojnim i klasičnim test-primjerima iz literature kao i usporedbom na istim problemima simulacije s komercijalnim računalnim programima utvrđena je visoka kakvoća razvijenog modela. Predstoji primjena razvijenog i provjerenog modela na simulaciju propagacije vodnih valova južnim krakom rijeke Gacke od ustave Šumečica do Donjeg Švičkog jezera radi nalaženja konačnog rješenja problema poplava na području Grada Otočca. Iz navedenog može se zaključiti da se izrađeni matematički model zasniva na najnovijim znanstvenim i stručnim znanjima te predstavlja značajan korak naprijed u korištenju matematičkih modela u zaštiti od štetnog djelovanja voda u Hrvatskoj i izvan nje.

Problem of computing streamlines of water flow inside the Francis turbine is solved using two-dimensional potential flow model and finite element method. The entire computational algorithm is implemented through developed numerical objects. The resulting software needs simple geometrical input, creates appropriate finite element mesh and computes streamlines efficiently and accurately, so it presents a good tool for turbine flow analysis.

Na temelju simulacija za postojeće stanje korita i mjerenja izvršena je kalibracija parametara modela. Usporedba rezultata simulacije i mjerenja pri kalibraciji modela pokazala je da je određivanje parametara modela na temelju poznavanja stanja korita i obimnih empirijskih podataka iz referenci ispravan i dobar pristup. Provedena je simulacija stacionarnih strujanja za postojeće stanje korita kao i za regulirano korito od Gornjeg Švičkog jezera do Donjeg Švičkog mosta. Razvijeni matematički model i softver omogućava pouzdanu simulaciju ne samo stacionarnih nego i nestacionarnih strujanja, ukjučujući podkritična i nadkritična strujanja, hidrauličke skokove, propagaciju vodnog vala po suhom koritu, poplavljivanje okolnih terena, utjecaj mostova, pregrada, preljeva, čvorišta i ostalih struktura na otvorenim vodotokovima. Snaga se razvijenog modela očituje u mogućnosti simulacija trenutnih i naglih podizanja pregrada, pucanja brana i simulacija propagacije udarnih valova (šok valova). Ove mogućnosti nemaju mnogi poznati komercijalni softveri. U studiji je razmotrena ideja prokopa odteretnog kanala koji bi povezivao kanal Šumečica - Gornja Švica s Gornjim Švičkim jezerom i izgradnje pregrade uzvodno od Gornjeg Švičkog jezera. Razmotrena je jedna moguća varijanta tog odteretnog kanala kada bi odteretni kanal imao poprečni presjek jednak porečnom presjeku kanala Šumečica - Gornja Švica. U studiji nije analizirana optimalna varijanta presjeka i položaja, što je s istim razvijenim softverom lako moguće napraviti nakon odluke o izgradnji tog kanala. Simulacije čiji su rezultati predočeni u ovoj znanstveno-stručnoj studiji temeljne su za predviđanje posljedica različitih scenarija propagacije vodnih valova u uvjetima sadašnjeg stanja južnog kraka rijeke Gacke te u uvjetima ispitanih regulacijskih radova i građevinskih zahvata. Postavljenim se matematičkim modelom i razvijenim softverom jednostavno i brzo mogu izvesti i druge simulacije prema zahtjevu investitora i priložiti u obliku dodatka ovoj znanstveno-stručnoj studiji.

The paper presents shape optimization method for Francis turbine spiral casing design. Two methods are applied: originally developed finite element potential flow model with conjugate gradient optimizer, as well as commercial FLUENT turbulent flow simulator with genetic algorithm as the search method. The objective functional is defined as the surface averaged value of the magnitude of the difference between calculated and desired velocities at the outlet, so as to give as homogenized spiral casing outflow as possible. The shape of the spiral casing is given by Bezier splines, the control points of which being optimizing parameters.

The shape optimization method in Computational Fluid Dynamics applications using Genetic Algorithm and commercial turbulent flow modeling software FLUENT is presented. Two test cases are solved: the optimum nozzle design and the optimum Francis turbine spiral casing design. The shape of the nozzle and turbine spiral casing are parameterized by cubic Bezier splines, the coordinates of control points being optimizing parameters. In the nozzle case, the energy losses, defined as the difference of total mechanical energy flux through inflow and outflow cross-sections, are minimized. The objective functional for spiral casing shape optimization is defined as the surface averaged value of the magnitude of the difference between calculated and desired velocities at the outlet, so as to give as homogenized spiral casing outflow as possible.

In the paper a model of liquid-gas mixture flow in pipelines is presented. The result enables simulation of transient and cavitation phenomena so it provides a tool for prediction and prevention of pollution due to damages on pipelines as well as an environmentally improved pipeline design. Particularly an application to a pipeline Plase-Opatija in the coastal zone is presented in the paper. The applied methods are mathematical and numerical. The mathematical model consists of an initial-boundary value problem for a hyperbolic conservation law system, which is derived from the Kraneneburg's model for gas-liquid mixture flow in pipes. These hyperbolic conservation laws together with boundary conditions are then numerically solved using the essentially non-oscillatory schemes developed by Shu, Osher et al. The proposed cavitation model is tested on the Kranenburg's test example. The test shows excellent characteristics of the model, in fact it is superior to Kranenburg's numerical results. The model is applied to a real pipeline in order to predict cavitation phenomenon and improve pipeline change design.

In this paper the optimal strategy of filling the tunnel that conducts water to the hydroelectric power plant "Vinodol" is presented. The optimization problem consists in finding the sluice gate opening strategy that minimizes the time needed to fill the tunnel, taking into consideration two limitations: the pressure values must not exceed prescribed limits and no air-sockets should be created in the process. The tunnel filling dynamics is presented in detail. For the initial phase of the filling process open-channel theory is applied. Uniform and critical flow depths and conjugate hydraulic jump depths for all the tunnel sections and typically used discharges are calculated. The parameters for the model are estimated from the available data and measurements. A computer program that implements the model and enables the simulation of the filling process is developed. With the aid of this computer program a methodology for finding the optimal sluice gate opening heights is defined. This methodology gives the results that reduced the time needed to fill the tunnel from the existing 10 to only 5 hours, i.e. the improvement is very significant. Further smaller improvements can possibly be obtained after collecting new measurements that will give better model parameter estimation. Also in the sequel the authors will investigate an alternative mathematical model based on partial differential equations for the open channel flow with moving domain boundary.

In this paper a solution for valve closure optimization problem is presented. Nonstationary gas-liquid mixture flow in a pipeline is simulated using essentially nonoscillatory schemes for the appropriate conservation law model. The task of minimizing pressure oscillations generated by valve closure is solved using genetic algorithm. The obtained results show high accuracy and the approach is new and improved in comparison with the literature. Presented methods are applicable to any other similar pipeline optimization problem and have large application area in hydroelectric power plants, nuclear power plants, oil industry, ships, etc.

Predstavljen je matematički model nestacionarnog strujanja mješavine tekućine i plina u cjevovodu sastavljenim od cijevi, spremnika, ventila, zračnih komora, vodnih komora, crpki, turbina itd. Tako dobiveni inicijalno-rubni problemi za nelinearne hiperboličke sustave riješeni su numerički uz pomoć esencijano neoscilirajućih shema. Optimizacijski problemi za nestacionarna strujanja u cjevovodima riješeni su rabljenjem genetskog algoritma. Razvijen je računarski program za proračun i optimizaciju strujanja u proizvoljnim cjevovodima. Program je primijenjen na više problema proračuna i optimalnog upravljanja. Rezultati su uspoređeni s analitičkim rješenjima, rezultatima iz literature i izmjerenim podacima.

In the paper computer program “Egap” (ENO & GA Pipeline Optimizer) developed by the authors is presented. Program “Egap” implements the numerical model of hydraulic transient and cavitation phenomena based on the application of essentially nonoscillatory schemas on hyperbolic conservation laws, as well as optimization problem solution based on the application of the genetic algorithm. “Egap” can be used for pipelines in hydroelectric power plants, nuclear power plants, in oil industry, in ships, etc. It provides the capability to predict the consequences of manipulations with valves, pumps, turbines, etc. and to achieve optimal control in these operations with the goal of minimizing pressure oscillations or lengths of cavitation zones. “Egap” is developed using “Microsoft Visual C++ Developer Studio” so it is also characterized with a user-friendly Windows environment