Large-eddy simulation of turbulent flow over spanwise-offset barchan dunes: Interdune vortex stretching drives asymmetric erosion

The coupling between turbulent flow physics and barchan dune geometry is important to dune migration, morphology of individual dunes, and the morphodynamics of merging and separating proximal dunes. Large-eddy simulation was used to model turbulent, inertial-dominated flow over a series of static barchan dune configurations. The dune configurations were carefully designed to capture realistic stages of a so-called “offset interaction,” wherein a small dune is placed upflow of a relatively larger dune, thereby guaranteeing interaction since the former migrates faster than the latter. Moreover, as interaction proceeds, the morphology of the small dune is mostly preserved, while the large dune undergoes dramatic transformation with greater erosion downflow of the interdune space. Simulations reveal that the wake centerline—determined here as the spanwise location at which the momentum deficit associated with each dune exhibits a minimum—veers due to dune geometry. Visualization of vortex identifiers reveals that hairpin vortices are produced via separation across the crestline of dunes, and these hairpins are advected downflow by the prevailing, background flow. The legs of hairpins emanating from the upflow dune contain streamwise vorticity of opposite sign, wherein the hairpin leg within the interdune space exhibits positive streamwise vorticity. This positive streamwise vorticity is supplied to the interdune space, where flow channeling induces acceleration of streamwise velocity. An assessment of right-hand side terms of the Reynolds-averaged streamwise vorticity transport equation confirm, indeed, that vortex stretching is the dominant contributor to sustenance of streamwise vorticity. With this, we can conclude that asymmetry of the large, downflow dune is a consequence of scour due to the interdune roller, and scouring intensifies as the spacing between dunes decreases. A structural model outlining this process is presented.

Figure 1

Photographs of offset merger interaction stages [26], observed in mobile-bed flume experiments (images retrieved from Hersen and Douady [31]). Annotations of prevailing mean flow direction and representative times added for illustration; red arrows illustrate downflow trajectory of ejected dune, or the “ejecta” [25].

Figure 2

Visualization of dune configurations considered for this study. Panel (a) shows Cases S1 to S4, Panel (b) shows Case $\mathrm{S}{3}^{\prime }$, and Panel (c) shows Case $\mathrm{S}{4}^{\prime }$ (see Table 1 for summary of simulation and geometric attributes). For visualization purposes, streamwise and spanwise position have been normalized by large dune crest height, $h$. Panel (a) includes annotation of streamwise offset, $s_x/h$, spanwise offset, $s_y/h$, and streamwise asymmetry, $\mathrm{\Delta }x/h$. Cases $\mathrm{S}{3}^{\prime }$ and $\mathrm{S}{4}^{\prime }$ are equivalent to S3 and S4, respectively, with the exception of $\mathrm{\Delta }x/h$ streamwise asymmetry. Recall Fig. 1, which shows how the large downflow dune horn exhibits asymmetric elongation as the upflow dune approaches [38], which is why we have considered $\mathrm{S}{3}^{\prime }$ and $\mathrm{S}{4}^{\prime }$. In all cases (except S1), the upflow dune is equivalent, and a ratio of the small dune volume is computed and provided in Table 1, where $\chi ={\mathcal{V}}_s/{\mathcal{V}}_L$, and where ${\mathcal{V}}_s$ and ${\mathcal{V}}_L$ is the volume associated with the small and large dune, respectively. Panel (b) annotations show local coordinates, $x_s$ and $x_l$, used to quantify dune wakes, ${\delta }_s(x_s;z)$ and ${\delta }_l(x_l;z)$ (red lines). Solid black arrow denotes flow channeling. Panel (c) shows discrete locations (solid circles) used for time-series sampling of flow quantities, where ${\mathit{x}}_L/h=\{1,0.5,0.5\}, {\mathit{x}}_C/h=\{5,0,0.5\}, {\mathit{x}}_E/h=\{2,-2,z/h\}$, and ${\mathit{x}}_F/h=\{2,1.3,z/h\}$. Table 1 gives detailed attributes of each case. Note, finally, that the small dune crest height is denoted by $h_s$. Individual dune digital elevation models provided by K. Christensen, Notre Dame, and used in recent articles [11, 14, 24].

Figure 3

Streamwise-spanwise plane visualization of instantaneous flow over Cases S5 (a) and S6 (b) (see Table 1 for topography details). Visualization shown at wall-normal elevation, $z/h=0.25$. Contour and vectors are swirl strength with sign set by out-of-plane vorticity, ${\lambda }_{c,z}^{*}(x,y,z/h=0.25,t)={\lambda }_c(x,y,z/h=0.25,t){\widehat{i}}_{\omega ,z}(x,y,z/h=0.25,t)$, and instantaneous fluctuating velocity, $\{{\tilde{u}}^{\prime }(x,y,z/h=0.25,t)/u_{*},{\tilde{v}}^{\prime }(x,y,z/h=0.25,t)/u_{*}\}$, respectively.

Figure 4

Color flood contour of Reynolds-averaged vertical vorticity, ${\langle {\tilde{\omega }}_z\rangle }_t(x,y,z/h=0.5)$, at wall-normal elevation, $z/h=0.5$, for Cases S1 (a); S2 (b); S3 (c); S4 (d); $\mathrm{S}{3}^{\prime }$ (e); and $\mathrm{S}{4}^{\prime }$ (f) (see Table 1 for topography details). Included on the color floods are low-pass filtered datapoints for the wake, emanating from the small and large dunes, ${\delta }_s(x_s;z/h=0.5)$ and ${\delta }_l(x_l;z/h=0.5)$, respectively. Low-pass filtered wake profiles emanating from large and small dunes, ${\delta }_l(x_l;z/h=0.5)$ (Panel g) and ${\delta }_s(x_s;z/h=0.5)$ (Panel h), respectively, where local coordinate originates at respective dune crest, where Fig. 2 graphically illustrates the local axes, $x_s$ and $x_l$. Black, gray, and light gray solid lines correspond with Cases S2, S3, and S4, respectively, dashed blue and dotted red lines correspond with $\mathrm{S}{3}^{\prime }$ and $\mathrm{S}{4}^{\prime }$, respectively, while cyan circles and dash-dot magenta line correspond with S5 and S6, respectively.

Figure 5

Streamwise–wall-normal visualization of conditionally-averaged $Q$ criterion for $\widehat{Q}=11$ signed by conditionally-averaged wall-normal rotating direction: Panels (a) and (b) show Cases S1 and S2, respectively. Probability density function (PDF) of normalized streamwise velocity fluctuation at sampling position $\mathit{x}={\mathit{x}}_L$ is showed in panel (a). Black, dark gray, gray, and light gray lines correspond with Cases S1, S2, S3, and S4, respectively, while dashed blue and dotted red datapoints correspond with $\mathrm{S}{3}^{\prime }$ and $\mathrm{S}{4}^{\prime }$, respectively (see Table 1 and Fig. 2 for topography details). Black vertical line notes the conditional sampling threshold used in this work is ${\tilde{u}}^{\prime }({\mathit{x}}_L,t)/u_{*}>2.5$ [57, 58, 59, 60, 61, 62, 63, 64]. Three-dimensional visualization of instantaneous $Q$ criterion for $Q=100$ signed by Reynolds-averaged streamwise velocity: Panels (c) and (d) show Cases S5 and S6, respectively. Note numbered annotation of successive vortex cores emanating from dune brinkline, and vortex core spacing, ${\delta }_s/h$, deduced from high-Reynolds number Strouhal number and advective velocity in vicinity of brinkline.

Figure 6

Global wavelet power spectrum of streamwise velocity fluctuations, for input time series from discrete locations ${\mathit{x}}_L$ (a) and ${\mathit{x}}_C$ (b). Black, dark gray, gray, and light gray lines correspond with Cases S1, S2, S3, and S4, respectively, dashed blue and dotted red lines correspond with $\mathrm{S}{3}^{\prime }$ and $\mathrm{S}{4}^{\prime }$, respectively, while cyan circles and dash-dot magenta line correspond with S5 and S6, respectively. Horizontal orange line denotes $fH{U}_0^{-1}=St=0.25$, the high-Reynolds number asymptote.

Figure 7

Isosurface image of Reynolds-averaged, shear-normalized differential helicity, $h\left(\mathit{x}\right)H{u}_{*}^{-2}=120$ (red) and $h\left(\mathit{x}\right)H{u}_{*}^{-2}=-120$ (blue). Panels (a) to (f) correspond with Cases S1, S2, S3, S4, $\mathrm{S}{3}^{\prime }$, and $\mathrm{S}{4}^{\prime }$, respectively. Panel (g) and (h) are Reynolds-averaged flow over S5 (g) and S6 (h) in spanwise-wall normal plane at $x/h=1.7$, which are showed as black lines in panels (b) and (d), respectively (see Fig. 2 for reference). In panels (g) and (h), contour and vectors are Reynolds-averaged streamwise vorticity, ${\langle {\tilde{\omega }}_x\left(\mathit{x}\right)\rangle }_t$, and components of in-plane velocity, $\{{\langle \tilde{v}(\mathit{x},t)\rangle }_t,{\langle \tilde{w}\left(\mathit{x}\right)\rangle }_t\}$.

Figure 8

Structural model for flow processes associated with dune morphodynamic asymmetry. Panel (a): idealized hairpin vortices shed from dune brinkline; Panel (b): idealized hairpin vortices being simultaneously shed from both dunes, where streamwise vorticity embodied within inner leg of upflow hairpin is stretched by flow channeling (double-headed roller), sustaining the interdune roller and inducing sediment scour on the large dune (green). Red and blue colors denote positive and negative streamwise vorticity directions, respectively [see Figs. 5 and 5 for three-dimensional instantaneous visualization]. On both panels, gray lines denote dune wake centerline (see also Figs. 2 and 4 for details).

Figure 9

Vertical profiles of constituent right-hand side terms from Reynolds-averaged streamwise vorticity transport Equation [Eq. (9)], including vortex stretching ${\langle S_x\rangle }_t$ (a,d), vortex tilting ${\langle T_x\rangle }_t$ (b,e) and turbulent torque ${\langle P_x\rangle }_t$(c,f) at discrete streamwise-spanwise locations collocated with Point ${\mathit{x}}_E$ (a, b, c) and Point ${\mathit{x}}_F$ (d, e, f) (see also Fig. 2). Black, dark gray, gray, and light gray lines correspond with Cases S1, S2, S3, and S4, respectively, while dashed blue and dotted red lines correspond with $\mathrm{S}{3}^{\prime }$ and $\mathrm{S}{4}^{\prime }$, respectively. Horizontal gray line denotes small dune height.

Figure 10

Color flood contour of Reynolds-averaged term responsible for stretching of streamwise vorticity, ${\langle S_x\rangle }_t(x,y,z/h=0.25)$ [see also Eq. (9)]. Panels (a)–(f) correspond with Cases S1, S2, S3, S4, $\mathrm{S}{3}^{\prime }$, and $\mathrm{S}{4}^{\prime }$, respectively. Included on the color floods are low-pass filtered datapoints for the wake, emanating from the small and large dunes, ${\delta }_s(x_s;z/h=0.25)$ and ${\delta }_l(x_l;z/h=0.25)$, respectively.