Hidrológiai Közlöny 1988 (68. évfolyam)

1. szám - Józsa József: Elkeveredési folyamatok részecske szemléletű szimulálása

JÓZSA J.: Elkevéredési folyamatok 23 ut the transport and mixing. In most eases the motion can be separated into two characteristic scales such as convective and diffusive parts. Then the velocity can be considered as the sum of them (Eq. 3), where the large scale gives the convection and the velocity fluctuation is responsible for the diffusion. The first can be'determined by a numerical flow model or by measurements in a number of points, the statistical parameters of the second can be drawn also from measurements. When dealing with the mixing of passive material moving together with the flow, particles are considered independent. For each particle the effective velocity in any time is then calculated by interpolation from the convective velocity field to the particle position, and by generating an additional velocity fluctuation with the given statistical parameters using a suitable Monte-Carlo method. The new particle position can be determined in the simplest way by Eq. 4. At solid boundaries reflection is applied, at outflow boundaries particles leaving the flow domain are simply omitted for further calculation. The more particles are used, the more accurate approximation one obtains. Tn other words, few particles give a qualitative picture though allow longer simulation in relatively short time, but increasing the particle number the results are more and more suitable for drawing quantitative consequences, for ex. concentration distribution. Among the advantages of the method one can mention the absence of numerical diffusion and unrealistic negative concentrations, the satisfaction of the conservation of mass, the need of calculation only at particle positions and the possibility of the direct use of measured turbulence parameters when detailed data are available. The difficulty is that fluctuation generation requires Lagrangian statistics arising measuring technique problems. In most cases one has to convert Eulerian measurements into Lagrangian based on some empirical formulas. The first few examples are to demonstrate the applicability of the quasiparticle method in two dimensions when steep concentration and velocity gradients are to be treated. If we consider the velocity fluctuation as a random Brown-motion, the fluctuating part of the velocity can be generated from a simple tophat distribution, Eq. 5. giving the relation between the band with of the fluctuation, the diffusion coefficient and the time step used in the calculation. Figure 1. shows the 'velocity field around a groin established by a two-dimensional depthintegrated numerical flow model. Different number of particles are put punctually into that rather irregular velocity field and released to investigate their convectiondiffusion. Results obtained in different time levels using 1000 and 10 000 are shown in Figure 2 and 3. It is seen that even few particles can give the tendencies, and increasing the particle number practically one obtains the concentration distribution, linearly proportional to the particle distribution. Releasing 30 000 particles just behind the groin, despite the very strong velocity gradients the results are realistic (Fig. 4.). Application in shallow lake conditions is demonstrated in Fig. 5., where the mixing of as much as 20 particles is simulated released in a nearly steady flow field induced by rather stable wind field. Since in steady flow conditions the path of the cloud statistically gives the plume of a continuous release, displaying all the particle positions Fig. 5. shows that picture. Finally a simple example is given when one has more detailed information about the statistical, and so the eddy structure of the flow. If the standard deviation and the Lagrangian autocorrelation of the velocity is know, fluctuation can be described by a simple Markov-chain (Eq. 6), reflecting the memory of the fluctuation, so the presence of eddies having larger time scale then the applied time step. The purely random component can be generated from a Gaussian distribution with zero mean and standard deviation given by Eq. 7. To obtain Lagrangian parameters for the method is still a problem and has not been solved yet. Figure 6 shows the influence of the velocity autocorrelation on the mixing. Having in mind Taylor (1921) corner-stone theory about diffusion in turbulent flows, one should expect greater diffusion bv increasing the autocorrelation value, so simulating the effect of larger eddies. Taylor's theory is well proved in the figure. For future development one should have advanced measurements to take advantage of the ability of method for finer simulation of mixing phenomena. But the method is also applicable as a numerical tool to check the behaviour of theoretical suppositions. Keywords: quasiparticle simulation, Lagrangian-system, convection-diffusion, turbulence, MonteCarlo method, concentration front [te JÓZSA JÁNOS 1957-ben született Győrben. 1981-ben szerzett építőmérnöki diplomát a Budapesti Műszaki Egyetemen. Végzés után rövid ideig a VITUKI-ban dolgozott, majd 3 évet töltött a VIZITERV-ben, ahol főleg algériai tervezésekben vett részt. 1985-ben tért vissza a VITUKI-ba, az akkor megalakuló Numerikus Hidraulikai Osztályra, ahol jelenleg is dolgozik. Fő tevékenységi területe az áramlási és transzportfolyamatok kétdimenziós numerikus modellezése.