Direct numerical simulation of turbulent channel flow up to $Re_τ \approx 5200$ (1410.7809v2)

Abstract: A direct numerical simulation of incompressible channel flow at $Re_\tau$ = 5186 has been performed, and the flow exhibits a number of the characteristics of high Reynolds number wall-bounded turbulent flows. For example, a region where the mean velocity has a logarithmic variation is observed, with von Karman constant $\kappa = 0.384 \pm 0.004$. There is also a logarithmic dependence of the variance of the spanwise velocity component, though not the streamwise component. A distinct separation of scales exists between the large outer-layer structures and small inner-layer structures. At intermediate distances from the wall, the one-dimensional spectrum of the streamwise velocity fluctuation in both the streamwise and spanwise directions exhibits $k^{-1}$ dependence over a short range in $k$. Further, consistent with previous experimental observations, when these spectra are multiplied by $k$ (premultiplied spectra), they have a bi-modal structure with local peaks located at wavenumbers on either side of the $k^{-1}$ range.

• The paper performs DNS of turbulent channel flow at Reτ ≈5200, establishing a distinct logarithmic region in mean velocity with a Von Kármán constant of 0.384.
• The study identifies a weakly logarithmic growth in streamwise velocity variance, quantified by the fit ⟨u'²⟩⁺₍max₎ = 3.66 + 0.642log(Reτ), and observes clear scale separation in spectral analysis.
• The findings have practical implications for CFD, refining turbulence models and guiding future high-Reynolds number simulations in engineering applications.

Direct Numerical Simulation of Turbulent Channel Flow up to $Re_\tau \approx 5200$

This paper by Myoungkyu Lee and Robert D. Moser presents a pivotal investigation through direct numerical simulation (DNS) of incompressible channel flows. Achieving a friction Reynolds number ($Re_\tau$) of 5186, the paper marks a thorough exploration of flow characteristics representative of higher Reynolds numbers pertinent to both industrial applications and fundamental turbulence studies.

Summary of Findings

The DNS conducted exhibits a pronounced logarithmic region in mean velocity, yielding a Von Kármán constant $\kappa = 0.384 \pm 0.004$, an observation consistent with findings by prior researchers like Osterlund and Nagib but slightly differing from other geometries and experimental findings. This paper strengthens the presence of an overlapping layer within which $\langle w^{\prime2} \rangle^+$ follows Townsend's prediction, though not for $\langle u^{\prime2} \rangle^+$, indicating unachieved convergence to logarithmic behavior within practical Reynolds numbers.

Additionally, the peak values of streamwise velocity variance exhibit a weakly logarithmic growth with $Re_\tau$, approximated well by the fit $$\langle u^{\prime2} \rangle^+_{max} = 3.66 + 0.642\log\left(Re_\tau\right)$$, supporting observations by previous researchers such as Lozano-Duran. In contrast, the wall-normal and spanwise variances of velocity fluctuations do not demonstrate similar consistency with either theoretical or previous experimental expectations.

A separation of scales between the near-wall streaky structures and larger-scale motions in the outer region has been faithfully captured in the paper. This is evidenced by the well-formed double peaks in one-dimensional spectra of streamwise velocity, besides the indication of $k^{-1}$ scaling in both the streamwise and spanwise energy spectra, affirming earlier experimental insights.

Implications and Future Directions

This research significantly contributes to the empirical and theoretical understanding of high Reynolds number turbulence, clarifying aspects such as the eventual personality of the log-law region and its variability with channel flows, aligning DNS results more closely with experimental results than previously achieved. Moreover, the clarification of discrepancies across techniques and methods paves avenues for refining numerical approaches in turbulence modeling.

Practically, this DNS at high Reynolds numbers informs models in computational fluid dynamics software often employed in engineering practice. The findings reinforce the validity of utilized turbulence models and encourage the refinement of sub-grid scale models, with direct applications to the aerospace and automotive industries where such flows are prevalent.

Future studies could seek further higher Reynolds number simulations to distill unresolved features like the saturation of $\langle u^{\prime2} \rangle^+$ peaks, enhancing domain sizes, or experimenting with refined boundary conditions. Continued simulations with varying geometric configurations could also provide insights into turbulence universality and differences across flow types. This paper thus sets the stage for more targeted transitional research bridging theoretical predictions with industrial applications, presenting DNS as an indispensable tool in turbulence research.