Genuine compressibility effects in wall-bounded turbulence

Compressible wall-bounded turbulence is generally assumed to be devoid of genuine compressibility effects, meaning that the effect of finite fluid dilatation is regarded as “small,” at least in the absence of disturbing pressure gradients. In the present paper we attempt to answer the basic question of how small these effects are, by interrogating a DNS database of compressible channel flow and by using Helmholtz decomposition to infer the relative magnitude and correlations between the solenoidal and the dilatational parts of turbulence velocity fields. Not surprisingly, we find dilatational velocity fluctuations to be much smaller than solenoidal ones, but perhaps unexpectedly, we find that finite correlation between the two components accounts for a nonnegligible fraction (about 10%) of the turbulent shear stress near walls, and for up to 4% of the wall skin friction. Quadrant analysis of the dilatational velocity fluctuations shows that the largest contribution to the turbulent shear stress results from significant correlation between positive streamwise solenoidal velocity fluctuations (i.e., high-speed streaks), and positive vertical dilatational velocity fluctuations, which tend to mitigate the intensity of wall-ward sweep events.

Figure 1

Sketch of the flow configuration under scrutiny.

Figure 2

Spanwise spectral densities of streamwise velocity fluctuations at $y/h=0.3$, in local Kolmogorov units. Refer to Table 1 for line legend.

Figure 3

Mean velocity (a) and temperature (b) profiles. $T_c$ is the centerline temperature. Symbols in panel (b) indicate temperature profiles obtained from the GRA relationship in Equation (10). Refer to Table 1 for line legend.

Figure 4

Density-scaled Reynolds stresses distributions, as defined in Eq. (11). The gray line in (d) refers to the asymptotic shear stress distribution, $1-y/h$. Refer to Table 1 for line legend.

Figure 5

Inner-scaled distributions of vorticity fluctuations (a)–(c), and dilatation fluctuations (d). Refer to Table 1 for line legend.

Figure 6

Correlation coefficient between fluctuating dilatation and streamwise (a) and wall-normal (b) velocity fluctuations. Refer to Table 1 for line legend.

Figure 7

Distributions of turbulence Mach number (a) and fluctuating Mach number (b). Refer to Table 1 for line legend. Symbols indicate boundary layer DNS data from Duan et al. [4].

Figure 8

Instantaneous fields of $u_s^{″}$ (a), $u_d^{″}$ (b), $v_s^{″}$ (c), $v_d^{″}$ (d), in a wall-parallel plane at $y^{+}=15$, for case C5.

Figure 9

Variance of solenoidal (a), (d), (g) and dilatational (c), (f), (i) velocity fluctuations, and their covariance (b), (e), (h). (a)–(c) $u$ fluctuations, (d)–(f) $v$ fluctuations, (g)–(i) $w$ fluctuations. Refer to Table 1 for line legend.

Figure 10

Contributions to turbulent shear stress from (a) solenoidal velocity fluctuations (SS), (d) dilatational velocity fluctuations (DD), (b), (c) fluctuations covariances (SD and DS). Refer to Table 1 for line legend.

Figure 11

Cross-correlation coefficients of horizontal and vertical decomposed velocity fluctuations. Refer to Table 1 for line legend.

Figure 12

Premultiplied normalized spanwise spectral densities $\left[k_z{\widehat{E}}_{\phi }\left(k_z\right)\right]$ of streamwise (left) and wall-normal (right) velocity fluctuations: $\phi =u_s^{″}$ (a), $\phi =v_s^{″}$ (b), $\phi =u_d^{″}$ (c), (e), (g), $\phi =v_d^{″}$ (d), (f), (h). Data are shown for all flow cases in panels (a), (b), line style as in Table 1. Dilatational spectra are shown for flow cases C1 (c), (d), C4 (e), (f), C7 (g), (h). Contour levels from 0.1 to 0.5, in intervals of 0.1.

Figure 13

Premultiplied integrands of terms contributing to mean wall friction from Eq. (18), normalized by $C_f/2$,: $C_T$ (a), ${C_T}_{\text{SS}}$ (b), ${C_T}_{\text{SD}}$ (c), ${C_T}_{\text{DD}}$ (d), $C_M$ (e), $C_{M1}$ (f). Refer to Table 1 for line legend.

Figure 14

Quadrant analysis of solenoidal and dilatational velocity fluctuations [(a), SS $(u_s,v_s)$; (b), DD $(u_d,v_d)$; (c) SD $(u_s,v_d)$; (d), DS $(u_d,v_s)]$, at $y^{+}=20$ for flow cases C1, C4, and C7 (refer to Table 1 for line legend). The contour levels 0.002, 0.02, 0.1, and 0.2, are shown.

Figure 15

Conditional average eddies based on condition Eq. (19) for flow case C1 (a), C3 (b), C5 (c), and C7 (d). Isosurfaces of ${\lambda }_{\text{ci}}=0.6$ are shown colored in red and blue (corresponding to positive and negative ${\omega }_x$, respectively), overlaid to isosurfaces of dilatation, in green (a, $\theta =0.03$; b, $\theta =0.15$; c, $\theta =0.3$; d, $\theta =0.6$). Streamwise velocity contours at the conditioning wall distance ($y_{\mathrm{ref}}^{+}=20$) are also shown (levels from $-0.19$ to 0.19 in intervals of 0.02: solid, positive; dashed, negative).