Search Results (4)

This paper develops an analytical turbulence model for open-ended squeeze film damper (SFD) application. Prandtl’s mixing length theory is adopted to describe the momentum transfer within the damper for its thin-film turbulent flow. A novel turbulence coefficient function is developed to describe the effective fluid viscosity such that the classical Reynolds equation remains applicable. Model validation is presented by (i) comparing the damping coefficient obtained by several existing empirical formulas and (ii) correlating the rotor dynamic prediction with the experimental measurement of an integrated rotor-SFD test rig. This work provides a reduced form of turbulence coefficient for certain SFD implementations. It quantifies the turbulence effect under different operating conditions, which is valued as a practical tool to assess the significance of turbulence consequences in rotor dynamic applications. Full article

The two-dimensional turbulent thermal free jet is formulated in the boundary layer approximation using the Reynolds averaged momentum balance equation and the averaged energy balance equation. The turbulence is described by Prandtl’s mixing length model for the eddy viscosity ${\nu }_T$ with mixing length l and eddy thermal conductivity ${\kappa }_T$ with mixing length $l_{\theta }$. Since ${\nu }_T$ and ${\kappa }_T$ are proportional to the mean velocity gradient the momentum and thermal boundaries of the flow coincide. The conservation laws for the system of two partial differential equations for the stream function of the mean flow and the mean temperature difference are derived using the multiplier method. Two conserved vectors are obtained. The conserved quantities for the mean momentum and mean heat fluxes are derived. The Lie point symmetry associated with the two conserved vectors is derived and used to perform the reduction of the partial differential equations to a system of ordinary differential equations. It is found that the mixing lengths l and $l_{\theta }$ are proportional. A turbulent thermal jet with ${\nu }_T\ne 0$ and ${\kappa }_T\ne 0$ but vanishing kinematic viscosity $\nu$ and thermal conductivity $\kappa$ is studied. Prandtl’s hypothesis that the mixing length is proportional to the width of the jet is made to complete the system of equations. An analytical solution is derived. The boundary of the jet is determined with the aid of a conserved quantity and found to be finite. Analytical solutions are derived and plotted for the streamlines of the mean flow and the lines of constant mean thermal difference. The solution differs from the analytical solution obtained in the limit $\nu \to 0$ and $\kappa \to 0$ without making the Prandtl’s hypothesis. For $\nu \ne 0$ and $\kappa \ne 0$ a numerical solution is derived using a shooting method with the two conserved quantities as targets instead of boundary conditions. The numerical solution is verified by comparing it to the analytical solution when $\nu \to 0$ and $\kappa \to 0$. Because of the limitations imposed by the accuracy of any numerical method the numerical solution could not reliably determine if the jet is unbounded when $\nu \ne 0$ and $\kappa \ne 0$ but for large distance from the centre line, $\nu >{\nu }_T$ and $\kappa >{\kappa }_T$ and the jet behaves increasingly like a laminar jet which is unbounded. The streamlines of the mean flow and the lines of constant mean temperature difference are plotted for $\nu =0$ and $\kappa =0$. Full article

Even though applications of direct numerical simulations are on the rise, today the most usual method to solve turbulence problems is still to apply a closure scheme of a defined order. It is not the case that a rising order of a turbulence model is always related to a quality improvement. Even more, a conceptual advantage of applying a lowest order turbulence model is that it represents the analogous method to the procedure of introducing a constitutive equation which has brought success to many other areas of physics. First order turbulence models were developed in the 1920s and today seem to be outdated by newer and more sophisticated mathematical-physical closure schemes. However, with the new knowledge of fractal geometry and fractional dynamics, it is worthwhile to step back and reinvestigate these lowest order models. As a result of this and simultaneously introducing generalizations by multiscale analysis, the first order, nonlinear, nonlocal, and fractional Difference-Quotient Turbulence Model (DQTM) was developed. In this partial review article of work performed by the authors, by theoretical considerations and its applications to turbulent flow problems, evidence is given that the DQTM is the missing (apparent) constitutive equation of turbulent shear flows. Full article

River resistance characteristics, which can be reflected by the resistance factor, have an impact on flow and sediment transport. In the classical theory, Prandtl proposed the mixing length model for the simulation of the turbulence, and von Kármán established the logarithmic formula of the flow velocity distribution. Based on that, the expression of the resistance factor can be derived. With the development of the numerical technology, the $k-\epsilon$ model has been widely applied in the channels computation. However, for the different closure ways of the Reynolds stress in turbulence equations, the outcomes of the $k-\epsilon$ model and the Prandtl mixing length model are not exactly identical. In this paper, both qualitative and quantitative studies are carried out on the difference between these two models, with respect to the resistance factor. This difference is evaluated by the ratio of the resistance factor computed with the two models. The result shows that with the increment of the relative flow depth, the ratio first increases and then decreases. Moreover, it is also affected by the bed slope. Therefore, the difference should be taken into account when a comparison is made between the simulation results of the $k-\epsilon$ model and the classical theory of river mechanics. Full article