Scaling Relations and Self-Similarity of 3-Dimensional Reynolds-Averaged Navier-Stokes Equations

Ercan, Ali
Kavvas, M. Levent
Scaling conditions to achieve self-similar solutions of 3-Dimensional (3D) Reynolds-Averaged Navier-Stokes Equations, as an initial and boundary value problem, are obtained by utilizing Lie Group of Point Scaling Transformations. By means of an open-source Navier-Stokes solver and the derived self-similarity conditions, we demonstrated self-similarity within the time variation of flow dynamics for a rigid-lid cavity problem under both up-scaled and down-scaled domains. The strength of the proposed approach lies in its ability to consider the underlying flow dynamics through not only from the governing equations under consideration but also from the initial and boundary conditions, hence allowing to obtain perfect self-similarity in different time and space scales. The proposed methodology can be a valuable tool in obtaining self-similar flow dynamics under preferred level of detail, which can be represented by initial and boundary value problems under specific assumptions.


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The conditions under which depth-averaged two-dimensional (2D) hydrodynamic equations system as an initial-boundary value problem (IBVP) becomes self-similar are investigated by utilizing one-parameter Lie group of point scaling transformations. Self-similarity conditions due to the 2D k-epsilon turbulence model are also investigated. The self-similarity conditions for the depth-averaged 2D hydrodynamics are found for the flow variables including the time, the longitudinal length, the transverse length, the...
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Exact bound state solutions and corresponding normalized eigenfunctions of the radial Schrodinger equation are studied for the pseudoharmonic and Mie-type potentials by using the Laplace transform approach. The analytical results are obtained and seen that they are the same with the ones obtained before. The energy eigenvalues of the inverse square plus square potential and three-dimensional harmonic oscillator are given as special cases. It is shown the variation of the first six normalized wave-functions ...
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Citation Formats
A. Ercan and M. L. Kavvas, “Scaling Relations and Self-Similarity of 3-Dimensional Reynolds-Averaged Navier-Stokes Equations,” SCIENTIFIC REPORTS, vol. 7, pp. 0–0, 2017, Accessed: 00, 2022. [Online]. Available: