Stable channel flow with spanwise heterogeneous surface temperature

Abstract

Recent studies have demonstrated that large secondary motions are excited by surface roughness with dominant spanwise length scales of the order of the flow's outer length scale. Inspired by this, we explore the effect of spanwise heterogeneous surface temperature in weakly to strongly stratified closed channel flow (at $Ri_\tau =120$ , 960; $Re_\tau = 180$ , 550) with direct numerical simulations. The configuration consists of equally sized strips of high and low temperature at the lower and upper boundaries, while an overall stable stratification is induced by imposing an average temperature difference between the top and bottom. We consider the influence of the width of the strips ( ${\rm \pi} /8 \leq \lambda /h \leq 4{\rm \pi} $ ), Reynolds number, stability and upper boundary condition on the mean flow structure, skin friction and heat transfer. Results indicate that secondary flows are excited, with alternating high- and low-momentum pathways and vortices, similar to the patterns induced by spanwise heterogeneous surface roughness. We find that the impact of the surface heterogeneity on the outer layer depends strongly on the spanwise heterogeneity length scale of the surface temperature. Comparison to stable channel flow with uniform temperature reveals that the heterogeneous surface temperature increases the global friction coefficient and reduces the global Nusselt number in most cases. However, for the high-Reynolds cases with $\lambda /h \geq {\rm \pi} /2$ , we find a reduction of the friction coefficient. At stronger stability, the vertical extent of the vortices is reduced and the impact of the heterogeneous temperature on momentum and heat transfer is smaller.

Secondary flows induced by two-dimensional surface temperature heterogeneity in stably stratified channel flow

“…The two sets at Re τ = 180 are distinguished by their spanwise wavelengths λ y /h, namely π/4(≈ 0.79) and π/2(≈ 1.57). In our earlier study, we found that these heterogeneity lengths have the largest global impact on the flow (Bon & Meyers 2022). This is in agreement with a vast number of roughness studies showing that flow response to spanwise heterogeneous roughness is strongest for wavelengths of approximately 1 to 1.5 times the outer scale (e.g.…”

“…To make sure that our simulations with low and higher Reynolds number are comparable and both belong to the weakly stratified turbulent regime, all simulations have Ri τ = 120. This also enables easy comparison to cases from Bon & Meyers (2022) and allows to limit our parameter space.…”

“…The computational domain that was used for each case is indicated in the second column of table 1, with details of the grid resolution and domain size provided in table 2. These specifications match those used by Bon & Meyers (2022). At Re τ = 180, a rather large domain size of 8πh × 4πh (domain A) is required to avoid full laminarization caused by damping of turbulence by the stable stratification (García-Villalba & del Álamo 2011).…”

“…When the same roughness elements were placed in a staggered arrangement by Viggiano et al (2022), the effect on the global velocity defects in the outer layer was only minimal compared with a smooth wall and there were no large-scale secondary flows. We could roughly view the relationship between these two configurations as a 'roughness equivalent' to the link between the present subject of two-dimensional heterogeneous temperature and the study of temperature variations in the spanwise direction by Bon & Meyers (2022) (compare figures 1a and 1b).…”

“…Furthermore, Salesky et al (2022) defined the circulations generated by thermal surface heterogeneity as 'high and low momentum pathways' (HMPs and LMPs). In these thermally induced secondary flows, the upward motions and LMPs are consistently centred above the strips with higher temperature, while downward motions and HMPs appear above strips with lower temperature (Bon & Meyers 2022;Salesky et al 2022). By contrast, the rotational direction of roughness-induced secondary motions depends on how the heterogeneity is modelled, commonly with either 'strip-type' or 'ridge-type' roughness (see e.g.…”