Characterisation of rough-wall drag in compressible turbulent boundary layers

Abstract

In compressible turbulent boundary layers (TBLs), roughness drag is typically characterised by first applying a velocity transformation to account for compressibility, after which the momentum deficit $ΔU^+$ (Hama, 1954) and the equivalent sand-grain roughness $k_s$ are inferred. In practice, $k_s$ is often obtained from measurements at a single Mach number $M$ and Reynolds number $Re$, effectively forcing the roughness into the $ΔU^+$--$\log(k_s)$ relation of Nikuradse (1933). This raises a key question: if a rough surface has a known $k_s$ in incompressible flow, under what conditions can this value be used in compressible flows? This question is explored using data obtained through a series of experiments of TBLs on rough walls (P60- and P24-grit sandpapers) over $0.3 \leq M \leq 2.9$ and $7427 \leq Re_τ \leq 30292$, including independent variation of $Re_τ$ at $M=2$. Results show that $ΔU^+$ is largely insensitive to the velocity transformation, but the fully rough regime exhibits a Mach-number-dependent shift in the logarithmic relation. Three empirical scalings are examined: an equivalent incompressible $k_s$, a viscosity-scaled roughness $k_{*} = k/ν_\infty^+$ with $ν_\infty^+ = ν_\infty/ν_w$, and a correction factor $\sqrt{1/F_c}$ where $F_c$ depends on $T_\infty/T_w$. The last provides the most consistent improvement across datasets, although all corrections remain empirical and rely on smooth-wall compressibility transformations. This paves the way for future work to develop custom transformation for a rough-wall TBL that can account for roughness properties and other parameters including wall conditions.

Figures (7)

• Figure 1: Established framework for drag characterisation of rough-wall turbulent boundary layers in both incompressible and compressible flow regimes.
• Figure 2: Coloured contours of various roughness topologies relative to the roughness element height $k$. Types of roughnesses from existing studies listed in table \ref{['tab:ref']}: type 'A' (sandpaper), 'B' (2D traverse bars), 'C' (2D sine waves), 'D' (cubes), 'E' (egg carton), 'F' (expanded mesh), and 'G' (diamonds). Black arrow indicates the direction of the flow.
• Figure 3: (a) Top view ($x$--$z$ plane) and (b) side view ($x$--$y$ plane) of the baseplate with 8 inserts where the rough surfaces are attached. Red circles and solid lines mark 'A' and 'B' measurement stations. Black solid lines mark the 99% boundary-layer thickness growth $\delta(x)$, while $\theta_{LE}$ and $\theta_{TE}$ are the leading and trailing edge ramp angles relative to the $x$-direction. Inset: cross-sectional view of the baseplate with an insert (gray-shaded), showing the insert's thickness lowered by $\Delta h$ to approximately match the height of the baseplate and the rough walls (black-shaded). (c) Coloured contours of the normalised instantaneous streamwise velocity $u/U_{\infty}$ of the reference case (SW at $M = 0.3$). White solid line shows the $\delta(x)$ obtained from PIV measurements. Black solid line marks the power law fit $0.194 x^{0.714}$ (in mm), obtained by fitting $\delta(x)$ of the reference case. White dashed lines mark the FOVs from the two PIV cameras, with 10 mm overlap in $x$-direction. $\delta_0$ is the $\delta$ averaged over $x = 407 \pm 5$ mm (station 'A').
• Figure 4: Coloured contours of the scanned sandpaper surface deviation from its mean height, $h' = h - \overline{h}$, shown for 10 mm $\times$ 10 mm coupons of (a) P60- and (b) P24-grit sandpapers. Probability density function (p.d.f.) of $h'$ of the scanned (c) P60- and (d) P24-grit sandpapers. Solid red line () is the fitted Gaussian distribution of the p.d.f.
• Figure 5: (a) Mean temperature profile $T/T_w$ as function of $y/\delta$ (colours correspond to various test surfaces and $M$ sweep shown in table \ref{['tab:cases']}). Inner-scaled mean streamwise velocity profile for cases (b) SW and P60, (c) SW and P24 stretched by VD transformation (table \ref{['tab:transform']}). : $1/\kappa \log y^+ + B$. (d) Skin friction coefficients $C_f$ vs $Re_x$ estimated from VD-transformed profiles. SW data are multiplied by $F_c$ (equation \ref{['eq:hopkins']}), : $[2\log_{10} Re_x - 0.65]^{-7/3}$schlichting1960. Lines correspond to constant $C_f$, : $M < 1$, : $M = 2$, and : $M = 2.9$. $\Delta U_{VD}^+$ as a function of (e) $k_s^+$ and (f) $k^+$ for all rough-wall cases. Error bars in (d--f) denote the uncertainties estimated in §\ref{['sub:mean']}. Lines correspond to log-relation, : $M < 1$, $1/\kappa \log{k_s^+} + B - B_{FR}$, : $M = 2$, and : $M = 2.9$. Colours corresponds to the test surfaces, $M$ and $Re$ sweeps (table \ref{['tab:cases']}).