Session Z22: Turbulence: Wall-Bounded III

Generalised quasilinear approximations of turbulent channel flow: spanwise triadic scale interactions

The generalised quasilinear (GQL) approximation is studied in turbulent channel flow at $Re_\tau \approx 1700$, continuing from the work of {(Hernández \emph{et al.}, \emph{J. Fluid Mech.}, vol. 936, A33, 2022)}. The flow is decomposed into two groups, the former of which contains a set of low-wavenumber spanwise Fourier modes and the latter are composed of the rest high-wavenumber modes. The former group is then solved by considering the full nonlinear equations, while the latter group is obtained from the linearised equations about the former. This is in contradistinction to the flow decomposition employed in our previous work, based on streamwise Fourier modes. This decomposition leads to the nonlinear low-wavenumber group that supports the self-sustaining process within the given integral length scales, whereas the linearised high-wavenumber group is not able to do so, unlike the GQL models in our previous work which place a minimal mathematical description for the self-sustaining process across all integral scales. Finally, a set of numerical experiments suppressing certain triadic nonlinear interactions are carried out with the aim of unveiling key roles of those including energy cascade and inverse energy transfer in the near-wall region.   

A Comparison of Structure and Dynamics Underlying Quasilinear and DNS Turbulence

A POD-based analysis was carried out on numerical data obtained from a DNS of a turbulent Poiseuille flow at Re=1650 and the corresponding quasilinear (QL) simulation. Striking similarity between the velocity fields of the streamwise mean roll-streak structure was seen in the two simulations. Compelling similarity was also seen between the perturbation velocity fields co-located with the streamwise mean roll-streak in the two simulations. This correspondence in the structure of the primary dynamical components underlying turbulence in QL and DNS suggests a similar dynamics is operating to maintain turbulence in these systems. Moreover, it is known that the QL system is a close approximation to a second order statistical state dynamics (SSD) formulation. This consideration invites exploiting the comprehensively characterized dynamics of turbulence in the second order SSD formulation to advance understanding of DNS turbulence. It is concluded that the SSP identified in the second order SSD, which is the same as that operating in QL turbulence, is also operating in DNS turbulence. This fundamental dynamical structure in the streamwise mean flow is identified as the roll-streak while the primary dynamical structure in the perturbation field is found to consist of a large cohort of decaying oblique waves.   

The Josephson-Anderson relation for wall drag in classical turbulent channel flows

Turbulence in pressure-driven channels and pipes is often viewed as a “cascade” of momentum to the wall, but can be viewed also as an “inverse cascade” of vorticity away from the wall. Huggins exploited this idea to derive a “Josephson-Anderson” (JA) relation for incompressible channel flow, recently extended also to flows around solid bodies. These exact relations calculate drag by motion of vorticity relative to the background potential flow originating from the Kelvin minimum energy theorem. Since such Helmholtz-like decompositions into potential and rotational fields are non-unique, a good choice can make computations easier while keeping the two fields orthogonal. We modify Huggins’ JA relation to be more suitable for turbulence. We apply the relation, with spectral accuracy, to periodic channel data obtained from the Johns Hopkins Turbulence Database. Calculations of the spectral contents of vorticity flux reveal that wall-attached eddies are majorly responsible for vorticity flux within the inertial sublayer. Filtering out small scales allows us to identify structures providing more than 50% of the total drag generated in the log layer while occupying only 8% of the volume. 

    

Scaling of near-wall streamwise turbulence intensity: from a viewpoint of attached eddy model

Scaling of near-wall streamwise turbulence intensity has been one of the debating topics at least for the past two decades. There has been emerging evidence that the peak near-wall streamwise turbulence intensity may deviate from the logarithmic scaling with Re_τ predicted by extending the classical attached eddy model to the near-wall region. Recently, a new theory has been proposed by Chen & Sreenivasan (2021, J. Fluid Mech., 908, R3) based on a physical argument on near-wall dissipation deficit in the invisid limit, but two different and mutually independent Navier-Stokes-based models have repeatedly predicted that the near-wall peak streamwise turbulence intensity is proportional to inverse of log Re_τ (Monkewitz & Nagib, J. Fluid Mech., Skouloudis & Hwang, 2021, Phys. Rev. Fluids, 6:034602). In this talk, I will present a new theory obtained by extending the spectrum-based attached eddy model of Perry, Henbest & Chong (1986, J. Fluid Mech. 165:163-199) to the near-wall region with incorporation of viscous wall effect lack in the classcial theory, and show that if the near-wall peak intensity is bounded, it is propoportional to inverse of log Re_τ in the inviscid limit, favouring the predictions by the Navier-Stokes-based models. The relevent evidence from both experimental data and a Navier-Stokes-based model will be presented to support the theory.    

Turbulent Poiseuille flow of two immiscible liquid layers inside a channel

We use pseudo-spectral Direct Numerical Simulation (DNS), coupled with a Phase Field Method (PFM), to investigate the turbulent Poiseuille flow of two immiscible liquid layers inside a channel.  The two liquid layers, which have the same thickness (h₁=h₂=h), are characterized by the same density (ρ₁=ρ₂=ρ) but different viscosities (μ₁ ≠ μ₂). The full problem is described in terms of the following flow parameters: the shear Reynolds number (Re_τ, which quantifies the importance of inertia compared to viscous effects), the Weber number (We, which quantifies surface tension effects compared to inertia) and the viscosity ratio, λ between the two fluids.  In particular, we fix Re_τ=300, We=1, and we consider viscosity ratios in the range 0.1≤ λ=μ₁/μ₂ ≤ 1. We focus on the role of turbulence in initially deforming the interface and on the subsequent growth of capillary waves. Compared to a single phase flow at the same shear Reynolds number (Re_τ=300), in the two-layers case we observe a strong interaction between the turbulent flow and the deformable liquid-liquid interface. A full characterization of the interface deformation, in terms of spatiotemporal specta of wave elevation will be presented and discussed.   

Extracting discrete hierarchies of Townsend's wall-attached eddies

We decompose Townsend's wall-attached eddies from a multi-scale wall-bounded turbulent flow, extract discrete hierarchies of eddies, and study their geometrical characteristics. It is well known that the wall-attached eddies are hierarchical in nature and geometrically self-similar, but very few studies have successfully extracted them from a flow, especially from a decomposed flow. We propose a novel extraction scheme that is based on a spectral linear stochastic estimation methodology and the hierarchical nature of wall-attached eddies. The geometrical characteristics of the intense velocity clusters induced by attached eddies are studied via a clustering method. The extracted clusters are found to be self-similar in geometry that is consistent with Townsend's attached eddy hypothesis. A -2 power law of the population density of attached eddies is reported for the first time from data extraction.