Symmetry based scaling of the multi-point statistics of a turbulent Couette flow extended by wall-transpiration
Final Report Abstract
In this project, two flow configurations were investigated: shear-driven Couette flow with wall transpiration and pressure-driven Poiseuille flow. Direct numerical simulations (DNS) and symmetry-based turbulence theory (SBTT), more precisely scaling laws derived in the framework of SBTT, were applied. The original aims of the project were (i) to calculate DNS of Couette flow with wall transpiration and (ii) to derive scaling laws out of the multi-point momentum equations (MPME) using Lie symmetry methods and subsequently validate these scaling laws using the DNS data obtained in step (i). The first milestone (i) was fully achieved and the results have successfully been published in [7]. To investigate the effects of wall transpiration, we set up a parametric study of Couette flow at various Reynolds numbers Re and different transpiration velocities V0+ . Key findings include the occurrence of an inflection point in the mean streamwise velocity U and locally associated butterfly-shaped spectra due to transpiration. Furthermore, we observed that the large-scale roll-like structures typically present in Couette flow are not destroyed by the cross-flow of transpiration. However, transpiration leads to a general reduction of turbulence in the flow, which can be observed through the reduction of corresponding terms in the turbulent budget equation. To obtain adequate results for the second project goal (ii), the original plan had to be adjusted. Instead of using the DNS data of Couette flow with transpiration obtained in section (i), data from a newly computed large-scale DNS of Poiseuille flow at Reτ = 10000 was used. The decision for this step is justified as follows: The first project phase showed that in order to obtain sufficient convergence in the higher-order moments, a higher amount of flow fields would be needed for the cases of phase (i), in other words, a non-predictable higher amount of computing time to obtain more turnovers of the flow. It should be noted that the insufficient convergence of higher-order moments is due to the large roll-like structures that persist in Couette flow despite transpiration. Eventually, we decided to contribute to the described Poiseuille DNS to obtain reliable data for verifying the scaling laws. The derivation of these scaling laws was carried out by applying Lie symmetry methods to the MPME. This hierarchical system of equations can be derived directly from the Navier-Stokes equations and provides a statistical description based on instantaneous velocities (as opposed to the well known Reynolds-averaged equations, where instantaneous velocities are split into mean and fluctuating components). This reveals two additional symmetries, the so-called statistical symmetries. For the near-wall region, the well-known von Kármán logarithmic law for the first moment U can be derived using Lie symmetry methods and thus is confirmed by first principles. For higher-order moments, a power law applies, with the exponent determined by the second moment. For the channel center region, the deficit law can be generalized to all higher-order moments, with the exponent determined by the first and second-order moments.
Publications
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DNS of a turbulent Couette flow at constant wall transpiration up to. Journal of Fluid Mechanics, 835, 421-443.
Kraheberger, S.; Hoyas, S. & Oberlack, M.
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Letter: The link between the Reynolds shear stress and the large structures of turbulent Couette-Poiseuille flow. Physics of Fluids, 30(4).
Gandía-Barberá, Sergio; Hoyas, Sergio; Oberlack, Martin & Kraheberger, Stefanie
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Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach. Physical Review Letters, 128(2).
Oberlack, Martin; Hoyas, Sergio; Kraheberger, Stefanie V.; Alcántara-Ávila, Francisco & Laux, Jonathan
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Wall turbulence at high friction Reynolds numbers. Physical Review Fluids, 7(1).
Hoyas, Sergio; Oberlack, Martin; Alcántara-Ávila, Francisco; Kraheberger, Stefanie V. & Laux, Jonathan
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Turbulent Couette flow up to. Journal of Fluid Mechanics, 987.
Hoyas, Sergio & Oberlack, Martin
