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1

Transient flow through a porous channel with ramped pressure gradient and velocity slip boundary condition

Jha Basant K., Yunus Zainab Sa'id

International Journal of Applied Mechanics and Engineering

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2022

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Vol. 27, no. 1

78--90

EN

A transient flow formation of an incompressible fluid through a horizontal porous channel assuming a ramped pressure gradient is considered with the velocity slip boundary conditions. The flow is a laminar flow caused by ramped pressure gradient along the flow direction. The equation governing the flow is modeled, and solved by the Laplace transformation technique to obtain a semi-analytical solution under slip boundary conditions. It was noted that the flow velocity increases as the slip parameter is increased.

2

Wpływ długości przewodu za zaworem kulowym na parametry uderzenia hydraulicznego w rurociągu z polietylenu o dużej gęstości

Kodura A.

Ochrona Środowiska

|

2018

|

Vol. 40, nr 4

15--20

PL

Przepływ nieustalony o właściwościach uderzenia hydraulicznego występuje stosunkowo rzadko w układach ciśnieniowych, tym niemniej jego skutki mogą wywoływać poważne problemy eksploatacyjne systemów wodociągowych. Zjawisko to jest złożone, a na jego przebieg wpływa wiele czynników. Jednym z nich jest oddziaływanie fali ciśnienia za zaworem wywołującym zjawisko uderzenia hydraulicznego. Wiąże się to z jednoczesnym wzbudzeniem fal ciśnienia o przeciwnych znakach w przewodzie przed i za zaworem. Przeprowadzone badania eksperymentalne potwierdziły znaczące oddziaływanie długości rurociągu poniżej zaworu kulowego na parametry uderzenia hydraulicznego. Dotyczy to szczególnie pierwszej fazy zjawiska. Stwierdzono wpływ długości tego rurociągu na opóźnienie wywołania zjawiska, rozumiane jako czas między rozpoczęciem zamykania zaworu a początkiem przyrostu ciśnienia. Długość przewodu wpływała także na czas przyrostu ciśnienia oraz wartości jego przyrostów. W przypadku tej samej długości przewodu przed i za zaworem, opóźnienia przyrostu ciśnienia były najmniejsze. Najdłuższe czasy opóźnienia zaobserwowano w skrajnych położeniach zaworu – blisko początku lub końca przewodu. Podczas uderzenia prostego wartości przyrostów różniły się nieznacznie od obliczonych ze wzoru Żukowskiego. W przypadku uderzeń nieprostych (złożonych), różnice między uzyskanymi przyrostami ciśnienia przy różnych długościach przewodu dochodziły do 40%.

EN

Transient flow with water hammer characteristics is not a common occurrence in pressure systems. However, its effects can lead to serious problems in the management of pressure networks. This complex phenomenon is influenced by several factors. One of them is the impact of the pressure wave downstream of the valve that enforces the phenomenon. This involves concurrent excitation of pressure waves with opposite signs in the upstream and downstream conduit. The research confirmed significant influence of the downstream pipeline length on the hydraulic impact parameters. In particular, this applies to the primary phase of the phenomenon. The effect of the pipeline length on delay of the phenomenon was demonstrated, namely on the time between the start of the valve closure and start of the pressure increase. Furthermore, the pipe length influenced the time of pressure increase and the value of pressure increments. For equal lengths of the pipe upstream and downstream of the valve, delays in the pressure increase were the smallest. The biggest delay time values were observed in the extreme valve positions – near the beginning or the end of the pipe. For a simple water hammer, the measured values differed slightly from those calculated from the Joukovsky formula. For non-simple (complex) impacts, the variations between the pressure increases obtained for different pipe lengths reached up to 40%.

3

Pressure drop related to flow maldistribution in a model minichannel plate heat exchanger

Klugmann M., Dąbrowski P., Mikielewicz D.

Archives of Thermodynamics

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2018

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Vol. 39, no 2

123--146

EN

The paper describes issues related to pressure drop that accompanies the phenomenon of maldistribution of working fluid between the channels of a model minichannel plate heat exchanger. The research concerns a single exchanger’s plate containing 51 (in every geometry) parallel rectangular minichannels of hydraulic diameters 461 µm, 571 µm, 750 µm, and 823 µm. In addition, more complex geometry has been investigated, equipped with additional diagonal channels (so called extended geometry). The moment of the liquid phase transition through the heat exchanger was recorded at the flow rates ranging from 0.83 g/s to 13.33 g/s in the inlet manifold. The paper discusses the total pressure drop as a function of the flow rate and the characteristic dimension of minichannels, as well as the pressure drop as a function of the time of the fluid passage through the main part of the measuring section in which measurements were done. The resulting profiles correlate with the images of the flow distribution between channels recorded using the fast shutter speed camera, that allows to draw a further conclusions about the specifics of the maldistribution process. The impact of the total pressure drop on the actual range of optimum operating conditions of the heat exchanger was analyzed.

4

Numerical modeling of transient flows in load pipes with complex geometry

Ramoul S., Fourar A., Massouh F.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 4

67--78

EN

Changes in the system flow of a fluid in a pipe often cause sudden pressure changes and give rise to so-called transient load flows. So, the study of the phenomenon of transient load flows aims to determine whether the pressure in the whole of a system is within the prescribed limits, following a perturbation of the flow. By defining the scope of a water hammer study, an examination is made of variations in velocity or flow and pressure resulting from poor operation of the hydraulic system, its normal operation and emergency operations. This paper introduces a numerical modeling of the phenomenon of transient flows in load pipes with variable geometries which presents a study of the average pressure and the average velocity of the transient flow in the pipe with quasi-steady term friction. The characteristic method is used to solve the governing equations of “Saint-Venant”. Thanks to the AFT Impulse industrial program, we have obtained very interesting and very practical numerical results to describe the phenomenon of transient flows in variable load pipes.

5

Symulacje numeryczne w analizie wałów przeciwpowodziowych

Urbański A., Grodecki M., Kot A.

Czasopismo Inżynierii Lądowej, Środowiska i Architektury / Journal of Civil Engineering, Environment and Architecture

|

2016

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z. 63, nr 1/II

41--48

PL

Artykuł przedstawia metodologię numerycznego modelowania najważniejszych dla praktyki inżynierskiej zagadnień oceny bezpieczeństwa wałów przeciwpowodziowych. Z uwagi na zakres czasowy zachodzących zjawisk filtracyjnych problem filtracji musi być rozważany jako nieustalony oraz połączony z analizą mechaniczną (deformacji i stateczności). Opisane podejście zostało zilustrowane przykładami wyników analiz rzeczywistych wałów. Obliczenia wykonano systemem Metody Elementów Skończonych (MES) ZSoil.PC.

EN

The paper presents a FE - based methodology of numerical modeling of the most important aspects of assessing safety and serviceability of anti-flood dikes. In particular, due to the time scale of undergoing processes, pore fluid flow must be formulated as a non-steadystate problem, and coupled with mechanical deformation and stability analysis. Examples, illustrating the effectiveness of the proposed approach on practical cases, performed with the software system ZSoil.PC, are shown.

6

An analysis of the impact of valve closure time on the course of water hammer

Kodura A.

Archives of Hydro-Engineering and Environmental Mechanics

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2016

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Vol. 63, nr 1

35--45

EN

The knowledge of transient flow in pressure pipelines is very important for the designing and describing of pressure networks. The water hammer is the most common example of transient flow in pressure pipelines. During this phenomenon, the transformation of kinetic energy into pressure energy causes significant changes in pressure, which can lead to serious problems in the management of pressure networks. The phenomenon is very complex, and a large number of different factors influence its course. In the case of a water hammer caused by valve closing, the characteristic of gate closure is one of the most important factors. However, this factor is rarely investigated. In this paper, the results of physical experiments with water hammer in steel and PE pipelines are described and analyzed. For each water hammer, characteristics of pressure change and valve closing were recorded. The measurements were compared with the results of calculations perfomed by common methods used by engineers – Michaud’s equation and Wood and Jones’s method. The comparison revealed very significant differences between the results of calculations and the results of experiments. In addition, it was shown that, the characteristic of butterfly valve closure has a significant influence on water hammer, which should be taken into account in analyzing this phenomenon. Comparison of the results of experiments with the results of calculations? may lead to new, improved calculation methods and to new methods to describe transient flow.

7

Designing hydraulic air chamber in water transmission systems using genetic algorithm

Jamal A., Najarchi M., Asadiany Yekta A. H.

Advances in Science and Technology. Research Journal

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2016

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Vol. 10, nr 31

1--7

EN

Transient flow control in Water Transmission Systems (WTS) is one of the requirements of designing these systems. Hence, among control equipment, air chambers offer the best solution to control transient flow effects, i.e. both prevents water column separation and absorbs pressure increase. It is essential to carry out an accurate and optimized design of air chambers, not only due to high costs of their manufacturing but also their important protective role. Accordingly, hydraulic design parameters comprise tank volume, diameter of nozzle and coefficients of inflow and outflow of nozzle. In this paper, it is intended to optimize these parameters in order to minimize manufacturing costs. On the other hand, maximum and minimum pressures in main pipeline are considered as constraints which shall fall in allowed range. Therefore, a model has been developed which is a combination of a hydraulic simulation model of WTS and an optimization model based on genetic algorithm. This model is first applied to WTS of Dehgolan-Ghorveh plain as a case study. Results of this research demonstrate that based on suggested model, negative wave creation and pressure increase in pipeline is prevented as well as decrease in manufacturing costs of air chamber.

8

Hydraulic analysis of leakage from submarine pipe network

Li J. H., Fan Y. T.

Polish Maritime Research

|

2015

|

S 1

48--55

EN

The leakage from submarine pipe network is investigated by a numerical model in this paper. A whole event including of the steady flow before the leakage, the transient flow during the leakage and the steady flow after the leakage is considered in the model. The numerical results of the discharge and the head compare with the theoretical solution under the condition of steady flow. Good agreement is obtained. The numerical results of the discharge and the head show that the discharge and the head in each pipe changes suddenly after the leakage occurring. The abrupt trend of the discharge in each pipe is not same completely. Some change upwards, while others downwards. However, the abrupt trend of the head is always downward. The effects of leakage discharge on the transient flow are also investigated. It is founded that the larger the leakage discharge, the more obvious the abrupt trend of the discharge and the head before or after the leakage is.

9

Modelowanie stanów nieustalonych w przewodach ciśnieniowych z uwzględnieniem kawitacji oraz zmiennych oporów hydraulicznych. Cz. 1, Modele kawitacji

Urbanowicz K.

Biuletyn Wojskowej Akademii Technicznej

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2015

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Vol. 64, nr 3

49-73

PL

W pracy omówione zostały cztery kluczowe modele matematyczne opisujące przepływ nieustalony z kawitacją w przewodach ciśnieniowych: model rozerwania strumienia cieczy (Column Separation Model - CSM), model kawitacji parowo-gazowej (Gas Column Separation Model - CSMG), model Adamkowskiego (Adamkowski’s Column Separation Model - CSMA) i model kawitacji pęcherzykowej (Bubble Cavitation Model - BCM). We wszystkich tych modelach uwzględnione zostały zmienne opory hydrauliczne. Równania opisujące te modele rozwiązano z wykorzystaniem metody charakterystyk. Naprężenie styczne na ściance przewodu obliczane było z sumy dwóch wyrażeń: quasi-ustalonego i nieustalonego. Wyrażenie nieustalone jest modelowane w postaci całki splotowej z lokalnego przyspieszenia cieczy i funkcji wagi w(t).

EN

The paper presents four key mathematical models of transient cavitating pipe flow, i.e. column separation model (Column Separation Model - CSM), gas cavitation model (Gas Column Separation Model - CSMG), Adamkowski model (Adamkowski’s Column Separation Model - CSMA) and bubbly cavitation model (Bubble Cavitation Model - BCM). In the all investigated models, frequency dependent frictional losses were taken into account. The equations describing all models have been solved using the method of characteristics. In this work, the wall shear stress (defined as an effect of unsteady fluid friction) is presented as a sum of quasi-steady and unsteady components. The unsteady component of the wall shear stress is modelled as an convolution of local fluid acceleration and a weighting function w(t).

10

Modelowanie stanów nieustalonych w przewodach ciśnieniowych z uwzględnieniem kawitacji oraz zmiennych oporów hydraulicznych. Cz. 2, Opór oraz walidacja eksperymentalna

Urbanowicz K.

Biuletyn Wojskowej Akademii Technicznej

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2015

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Vol. 64, nr 3

75-101

PL

Modelowanie zmiennych w czasie oporów hydraulicznych nie jest zagadnieniem łatwym. Jak wykazały liczne badania, naprężenie styczne na ściance przewodu może być wyznaczane jako suma quasi-ustalonego i zmiennego w czasie wyrażenia. Zmienne w czasie wyrażenie jest tzw. całką splotową z lokalnego przyspieszenia cieczy i funkcji wagi. Oryginalna postać funkcji wagi ma przeważnie skomplikowaną strukturę, przez co nie nadaje się do efektywnego symulowania przebiegów dynamicznych. Dlatego też w celu umożliwienia efektywnego wyznaczania niestacjonarnego naprężenia stycznego zaprezentowano nową postać funkcji wagi (będącą skończoną sumą wyrażeń eksponentalnych). W przypadku przepływu turbulentnego wykorzystano procedurę skalowania współczynników efektywnej funkcji wagi zaprezentowaną przez Vitkovskiego i in. Nowe postacie funkcji wagowych charakteryzuje duża zbieżność z funkcjami klasycznymi (nieefektywnymi). Z wykorzystaniem uprzednio omówionych modeli przepływu kawitacyjnego CSM, CSMG, CSMA oraz BCM i powyższych efektywnych funkcji wagi dokonano szeregu badań symulacyjnych, które wykazały, że wprowadzone w modelach przepływu niestacjonarnego z kawitacją zmiany poprawiają stopień zgodności symulacji z wynikami eksperymentalnymi.

EN

Modelling of time-depended hydraulic friction is not an easy issue. As numerous studies have shown, wall shear stress in the pipe can be determined as a sum of the quasi-steady and time-dependent expressions. Time-depended expression is an convolution integral of the local acceleration of the liquid and a weighting function. The weighting function, in general, makes allowance for relation of historic velocity changes and unsteady component of wall shear stress. The original weighting function has usually a very complicated structure, and what is more it makes impossible to do an efficient simulation of dynamical runs. In this paper, in order to enable efficient calculation of unsteady component wall shear stress, new weighting functions are presented as a sum of exponential components. To aim this goal in case of turbulent flow, the scaling procedure proposed by Vitkovsky et al. is used. This method makes very easy the estimation of any new turbulent weighting function. Presented approximated weighting functions are compared with the original counterparts, known from literature in case of laminar and turbulent flows. Using the previously discussed models of cavitation flow CSM, CSMG, CSMA, and the BCM with implemented effective weighting function a series of simulation studies has been made, which showed that the introduced changes in models of unsteady flow with cavitation greatly improve the degree of simulation fit in comparison with experimental results.

11

Effect of hydrogen injection into natural gas on the mechanical strength of natural gas pipelines during transportation

Elaoud S., Abdulhay B., Hadj-Taieb E.

Archives of Mechanics

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2014

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Vol. 66, nr 4

269--286

EN

The aim of this paper is to study the effect of hydrogen injection into natural gas transient flows on the mechanical strength of natural gas pipelines. The governing equations of hydrogen-natural gas mixtures are two nonlinear partial differential equations. The fluid pressure and velocity are considered as two principal dependent variables. The fluid is a homogeneous hydrogen-natural gas mixture for which the density is defined by an expression averaging the two gas densities where an adiabatic process is admitted for the two components. The problem has been solved by the nonlinear method of characteristics. By the use of Laplace’s law, the pipe’s circumferential stress has been analyzed for different hydrogen mass fraction in the mixture. It was then compared to the allowable stress of different grade pipeline steels used to transport natural gas. The obtained results have shown that the allowable stress for the natural gas pipelines is exceeded for some fractions of hydrogen in the hydrogen-natural gas mixtures.

12

Experimental Verification of Storm Sewer Transient Flow Simulation

Szydłowski M.

Archives of Hydro-Engineering and Environmental Mechanics

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2014

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Vol. 61, nr 3-4

205–-215

EN

The paper focuses mainly on laboratory investigations of transient and transcritical flow in a single pipe of a sewer system. The aim of this paper is to present a comparison between pressure values calculated by an improved McCormack scheme and those measured at the hydraulic laboratory of the Gdansk University of Technology, which were observed inside a pipe in an experiment for water flow with pressurization. The analysis proves that the numerical approach applied to flow simulation is a sufficiently accurate and reliable technique for predicting the basic parameters of storm sewer flow.

13

Weighting Function Approximation in Transient Pipe Flow

Urbanowicz K.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2012

|

Vol. 16, No 3-4

263--276

EN

A very important problem in the transient liquid pipe flow analysis is accurate and effective modeling of hydraulic resistance. The so called integral convolution of the mean local acceleration of liquid and a weighting function need to be solved in a numerical way in order to simulate unsteady resistance. A necessary condition in effective numerical calculations is that the weighting function needs to be defined as a finite sum of exponential terms. The function keeps a constant shape in a laminar flow, in a turbulent flow, its shape changes and it is dependent on the instantaneous Reynolds number. In this article an easy method is presented to determine a proper weighting function in a very straightforward manner in a quick time. A comparison of the determined functions and function prototypes in a frequency domain will be presented as well.

14

Numerical Simulation of Transient Flow in Storm Sewers Using Standard and Improved McCormack Scheme

Szydłowski M., Machalińska-Murawska J.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2012

|

Vol. 16, No 1-2

53-74

EN

This paper describes the results of the first part of the research project which aims at developing a hydraulic model for simulation of unsteady flows in storm sewers ranging from gravity flows to surcharged flows resulting with water outflow on the ground surface and propagation of inundation in the flooded area. The paper focuses on the development and assessment of a second-order explicit numerical scheme for unsteady flows in sewers, but only in a single pipe at this moment, without any special elements such as manholes or drop shafts and with no water overflowing problem. The problem of water flow in sewer system pipes is associated with some specific phenomena occurring in conduits during storm events. If the pipes start to be fully filled with water, there is a transition from free surface to pressurized flow. Then, the vice versa effect can be observed. Such transitions are also possible in sewers when the discharge is controlled by control devices, such as gates for example. Moreover, the rapidly varied flow with some hydraulic local effects such as hydraulic jumps or bores can appear during extreme rain episodes. The appropriate modeling techniques have to be applied to solve these problems. The ‘Preissmann slot’ concept is implemented to simulate the pressurized flow. The original and improved McCormack scheme is used for transcritical flow simulation. The calculated results obtained for some benchmark tests are compared with numerical solutions and laboratory measurements published in the technical literature.

15

Convolution integral in transient pipe flow

Urbanowicz K., Zarzycki Z.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

|

2012

|

Vol. 16, No 3-4

277--291

EN

This paper is devoted to the modeling of hydraulic losses during transient flow of liquids in pressure lines. Unsteady pipe wall shear stress is presented in the form of a convolution integral of liquid acceleration and a weighting function. The weighting function depends on the dimensionless time and the Reynolds number. In its first revision (Zielke W 1968 J. ASME 90 109) it had a complex and inefficient mathematical structure (featured power growth of computational time). Therefore, further work aimed at developing the so-called efficient models for correct estimation of hydraulic resistance with simultaneous linear loading of the computer's operating memory was needed. The work compared the methods of numerical solving of the convolution integral known from the literature (classic by Zielke W 1968 J. ASME 90 109 and Yardy A E and Brown J M B 2010 J. Hydratd. Eng. 136 (7) 453 and efficient by Trikha A K 1975 J. Fluids Eng. p. 97. Kagawa T et at. 1983 Trans. Jpn. Soc. Mech. Eng. 49 (447) 2638 and Schohl G A 1993 J. Fluids Eng. 115 420). The comparison highlighted the level of usefulness of the analyzed models in simulating the water hammer and revealed the demand for further research for the improvement of efficiency of the solutions.

16

New efficient approximation of weighting functions for simulations of unsteady friction losses in liquid pipe flow

Urbanowicz K., Zarzycki Z.

Journal of Theoretical and Applied Mechanics

|

2012

|

Vol. 50 nr 2

487-508

EN

Most papers dealing with calculations and simulations of the unsteady liquid pipe flow are based on the assumption that the formula for quasi-steady friction (Darcy-Weisbach formula) can be applied. In the case of fast changes, like fast transients e.g. water hammer, it fails. In this work, the wall shear stress is presented as a sum of quasi-steady and unsteady component. The unsteady component of the wall shear stress is modeled as a convolution of local fluid acceleration and a weighting function. The original weighting function has usually a very complicated structure, and what is more, makes impossible to carry out an efficient simulation of dynamical runs. In this paper, in order to enable efficient calculation of the unsteady component of the wall shear stress, new weighting functions are presented as sums of exponential components.

PL

W dużej części prac, dotyczących obliczania oraz symulacji niestacjonarnego przepływu cieczy w przewodach ciśnieniowych, opory wyznacza się jedynie z dobrze znanej formuły Darcy-Weisbacha. Niestety w szybkozmiennych przepływach, takich jak np. uderzenie hydrauliczne, założenie quasi-stacjonarności oporów jest poważnym błędem. Wponiższej pracy naprężenie styczne na ściance przewodu wyznacza się z sumy quasi-stacjonarnego składnika oraz niestacjonarnego. Niestacjonarny składnik naprężenia stycznego modelowany jest jako splot lokalnego przyśpieszenia cieczy oraz funkcji wagi. Klasyczne funkcje wagi mają skomplikowaną postać, co uniemożliwia efektywne wyznaczanie oporów. W tej pracy przedstawione zostały nowe efektywne postacie funkcji wagowych o rozszerzonych zakresach stosowalności.

17

Prosta metoda aproksymacji funkcji wagowych umożliwiających modelowanie niestacjonarnych oporów hydraulicznych

Urbanowicz K.

Biuletyn Wojskowej Akademii Technicznej

|

2012

|

Vol. 61, nr 3

233-252

PL

W analizie niestacjonarnego przepływu cieczy zagadnieniem bardzo istotnym jest poprawne i efektywne modelowanie oporów hydraulicznych. W celu modelowania tych oporów rozwiązuje się drogą numeryczną tzw. całkę splotową z chwilowej zmiany prędkości cieczy i funkcji wagi. By obliczenia były efektywne, warunkiem koniecznym jest, by funkcja wagi została zapisana jako skończona suma wyrażeń eksponentalnych. Podczas przepływów laminarnych funkcja wagi zachowuje stały kształt, w turbulentnych natomiast jej kształt jest zmienny i zależny od chwilowej liczby Reynoldsa. W niniejszej pracy przedstawiona została pewna prosta metoda umożliwiająca wyznaczanie poprawnej funkcji wagi w bardzo krótkim czasie.

EN

In transient liquid pipe flow analysis, a very important problem is accurate and effective modelling of hydraulic resistances. In order to simulate unsteady resistances one need to solve, in a numerical way, so-called integral convolution of the mean local acceleration of liquid and a weighting function. In effective numerical calculations, a necessary condition is that the weighting function needs to be defined as a finite sum of exponential terms. In laminar flows, that function keeps a constant shape and in turbulent flows its shape changes in dependence of the instantaneous Reynolds number. In this article, an easy method is presented that enables us to determine proper weighting function very straight forward way in a very short time.

18

Modelowanie stanów nieustalonych podczas uderzenia hydraulicznego z uwzglednieniem kawitacji przejściowej w przewodach ciśnieniowych

Zarzycki Z., Urbanowicz K.

Inżynieria Chemiczna i Procesowa

|

2006

|

T. 27, z. 3/1

915-933

PL

Przedstawiono dwa modele kawitacji parowej: dyskretny model rozerwania słupa cieczy (CSM) i model kawitacji pęcherzykowej BCM. W obu modelach uwzględniono niestacjonarne opory tarcia w postaci całki splotowej z przyśpieszenia i funkcji wagi. Wyniki obliczeń porównano z dostępnymi z literatury wynikami badań eksperymentalnych. Stwierdzono lepszą ich zgodność niż w przypadku modelu quasi-ustalonych strat tarcia.

EN

The two models of vapour cavitation are presented in the paper: the column separation model (CSM) and bubble cavitation model (BCM). Non-stationary friction resistances have been introduced to both models represented by convolution integral with integrand depending on acceleration and certain weighting function. The results of calculation are compared with published experimental results. A better compatibility has been stated compared with the model of quasi-steady friction loses.

19

Transient free convection flow of viscous dissipative fluid past a semi-infinite vertical plate

Soundalgekar V. M., Jaiswal B. S., Uplekar A. G., Takhar H. S.

Applied Mechanics and Engineering

|

1999

|

Vol. 4, no 2

203-218

EN

Transient free convection flow of an incompressible viscous fluid past a semi-infinite vertical plate has been studied by using an implicit finite-difference technique, which is always stable and convergent. The effect of the dissipation parameter ? on time to reach steady-state is studied. It is observed that time required to reach steady-state temperature increases with the increasing Prandtl number of fluid. Also there is a rise in the average skin-friction and a fall in the average Nusselt number due to greater viscous dissipative heat.