Generating wall-bounded turbulent inflows at high Reynolds numbers

TL;DR

The paper tackles the challenge of generating accurate inflow conditions for high-$Re$ wall-bounded turbulence without incurring prohibitive development lengths. It introduces a structure-based inflow method that decomposes cross-stream slices into inner and outer spectral components, upscales the outer content from a base Reynolds number to a higher target using spanwise wavenumber shifting, and reconstructs full velocity fields by combining scaled outer modes with inner information and adjusted timing and energy. Validation shows that inflow slices upscaled from $Re_ heta=2240$ and $4430$ to $Re_ heta=8000$ yield DNS results with skin-friction and shape-factor within $\\pm3.5\%$ and $\\\\\pm0.5\%$, and Reynolds stresses aligning by about $8\, δ_{99,0}$, representing an order-of-magnitude reduction in development length compared to other methods. The approach relies on measurable scaling laws of turbulent boundary layers and demonstrates robust performance in DNS of spatially developing TBLs, with potential applicability to pipes, channels, and atmospheric boundary layers.

Abstract

One of the main challenges in simulating high Reynolds number ($Re$) turbulent boundary layers (TBLs) is the long streamwise distance required for large-scale outer-layer structures to develop, making such simulations prohibitively expensive. We propose an inflow generation method for high $Re$ wall turbulence that leverages the known structure and scaling laws of TBLs, enabling shorter development lengths by providing rich input information. As observed from the inner-scaled pre-multiplied spectra of streamwise velocity, with an increase in $Re$ the outer region grows and occupies more of the spanwise wavenumber space in proportion to the increase in $Re$; while the inner region remains approximately the same. Exploiting this behavior, we generate high-$Re$ inflow conditions for a $\textit{target}$ $Re$ by starting from cross-stream velocity slices at a lower $\textit{base}$ $Re$. In spectral space, we identify the inner and outer region wavenumbers, and shift the outer-region components proportionally to the desired $Re$ increase. We closely examine the capability of this method by scaling a set of velocity slices at $Re_θ=2240$ and $4430$ to $Re_θ=8000$, and using them as inflow conditions for direct numerical simulations (DNS) of spatially developing TBLs growing from $Re_θ=8000-9000$. The skin friction coefficient and shape factor predicted by the new method, regardless of the $\textit{base}$ $Re$ tested, is within $\pm3.5\%$ and $\pm0.5\%$, respectively, of that of a precursor simulation right from the inlet. Reynolds stresses match very well after approximately $8~δ_{99_0}$. This gives an order of magnitude reduction in development length compared to other methods proposed in the literature.

Figures (8)

• Figure 1: Pre-multiplied spectra of streamwise velocity $u$ at increasing $\Rey$ ($\Rey_\theta=2240, 4430, 8000$; contour lines of increasing darkness show increasing $\Rey$), i.e.,$E_{uu}(k_z)\cdot k_z /u_\tau^2$. $^{+}$ indicates normalization using inner units of the respective $\Rey$ and $y$ is the wall-normal coordinate. Notice that the inner region (left of the vertical dashed line) remains the same, whereas the outer region (right of the vertical dashed line) grows with increasing $\Rey$. For each level of darkness, contour lines are plotted for levels $[0.5, 1.5, 2.5, 4.0]$. Data from EitelAmor_2014.
• Figure 2: Schematic of the proposed method highlighting its potential to reduce computational cost by up-scaling low-$\Rey$ precursor data to high $\Rey$. (a) In classic precursor or existing methods, the precursor domain must grow with the target$\Rey$. (b) The proposed scaling method enables a small, fixed-size precursor domain regardless of target$\Rey$. Black dashed boxes show precursor domains; grey dashed boxes show main TBL simulation domains. Single-lined arrows indicate flow direction, double-lined arrows indicate Dirichlet inflow application. The jumping horse symbolizes the proposed scaling-leap bypassing the costly development of large-scale structures.
• Figure 3: Effect of scaling in $k_z$ and $y$ shown using the real part of the 0th POD mode of $k_z=10$ of streamwise component of velocity of re4k_sc, $\emph{i.e.,}~\text{Real}\{\varphi_1^{(k_z=10,n=0)}\}$. In both (a) and (b), the lighter shade of blue shows the state before scaling, and the darker shade shows the state after scaling. For both, contour lines are plotted for levels $[\pm0.0015, \pm0.0030, \pm0.0045]$.
• Figure 4: (a) 1D pre-multiplied spectra of streamwise velocity $u$, i.e.,$E_{uu}(k_z)\cdot k_z /u_\tau^2$, and (b) 2D pre-muliplied PSD of $u$ in terms of $\lambda_z^+$ and $\lambda_t^+$, i.e.,$E_{uu}(k_t k_z)\cdot k_t k_z /u_\tau^2$, at $y\approx 0.2\delta_{99_0}$ (this height is marked using a dashed horizontal line in (a)). In both figures, the reference data re8k is shown in the background as filled contours using shades of gray with levels as shown in the color bar. re2k_sc, re4k_sc and re8k_onlyIO are shown using green, blue, and red contour lines respectively. All three showing the same contour levels as re8k, but without using any shades of the respective color. $^{+}$ indicates normalization using inner units at the target$\Rey$.
• Figure 5: Reynolds stress profiles of the inlet velocity slices. (a) is the raw data whereas, (b) is obtained after passing it through a median filter, without altering the general trend, to smoothen the noisy parts to facilitate easier visual inspection here. The noise results from the coarse $y$-sampling in the precursor data. $^{+}$ indicates normalization using inner units at the inlet.