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Eine direkte Lösung der Erhaltungsgleichungen für Impuls, Energie, Masse und Teilchenmassen ist prinzipiell möglich, jedoch, wie oben dargelegt, vollkommen unpraktikabel. Zusätzlich bewirkt die große parametrische Sensitivität turbulenter Prozesse, dass für praktisch relevante Größen, wie z.B. die mittlere Temperatur und mittlere Schadstoffkonzentrationen, eine Mittelung über viele solcher direkten numerischen Simulationen nötig wäre.
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A direct solution of the conservation equations for momentum, energy, mass and particle masses is possible in principle, but as demonstrated above quite impracticable. Moreover, the high parametric sensitivity of turbulent processes means that for practical relevant quantities, such as e.g. the mean temperature and mean pollutant concentrations, averaging over many such direct numerical simulations would be necessary. However, due to the high non-linearity of the chemical kinetics, a ‘naïve’ calculation of the mean chemical source terms from the means of the temperature and of species concentrations leads to totally incorrect results. For this reason, simulations of technical combustion systems are performed using statistical models that rely on a consideration of the joint probability density function (PDF) for velocities and scalars. It turns out, however, that it is very difficult to determine the (space- and time-dependent) PDF. The Institute of Technical Thermodynamics uses a very accurate but also very demanding method for determining the PDF, based on the solution of a transport equation. In this method, the chemical kinetics are treated exactly. The turbulent mixing processes, however, must be modelled. In addition to the problem of modelling the molecular transport processes, the high dimensionality of the transport equation hampers its solution even further, since on top of the velocity components each chemical species goes into the transport equation as an independent variable. Therefore, Monte Carlo solution methods are the usual technique being relied upon.
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