Influence of freestream turbulence on the flow over a wall roughness

The effect of freestream turbulence on the dynamics of an incompressible flow past a cylindrical roughness element in subcritical conditions (i.e., for Reynolds numbers below the onset of linear instability) has been investigated by the joint application of direct numerical simulations, linear modal and nonmodal stability analyses, and dynamic mode decomposition. At first, the influence of the Reynolds number and the ratio of the boundary layer's thickness to roughness height on the three-dimensional spatiotemporal (global) stability of the flow has been investigated. Depending on the operating conditions, the leading instability can either be varicose (symmetric) or sinuous (antisymmetric). In both cases, when the flow is excited by broadband frequency forcing, dynamic mode decomposition extracts only varicose coherent structures even though optimal response analysis predicts a strong amplification of sinuous disturbances having frequency close to that of the marginally stable sinuous eigenmode. This apparent discrepancy is attributed to the fact that the sinuous instability is sensitive to a very limited range of frequencies barely excited by freestream turbulence while varicose disturbances are associated with high amplification in a much wider frequency range. Hence, in this case the flow behaves as an amplifier of varicose perturbations rather than a resonator. Consequences on the subsequent transition to turbulence have been studied, highlighting that varicose perturbations extract energy from the near-wake region, get continuously amplified due to the excitation provided by freestream turbulence, and eventually give rise to a shedding of hairpin vortices.

Figure 1

Reproduction of the von Doenhoff–Braslow diagram, representing the transition region (gray region between the solid lines) and the steady (red zone below the bottom line) and the turbulent regions (blue zone above the top line), depending on the aspect ratio of the roughness element and the roughness Reynolds number ${\mathrm{Re}}_{hh}$. The different symbols report the critical Reynolds numbers computed using three-dimensional global stability analysis: $\blacksquare$ for the cylinder [9], $•$ for the hemisphere [10], $⧫$ for the bump [43]. Red symbols indicate sinuous unstable global modes; blues ones denote varicose instability.

Figure 2

Sketch of the flow case studied.

Figure 3

Eigenspectra for (top) $\mathrm{Re}=700$ and (from left to right) $h/{\delta }^{*}=2.56,2.04,1.70$; (bottom) $h/{\delta }^{*}=2.04$ and (from left to right) $\mathrm{Re}=700,900,1200$. Red crosses indicate sinuous modes, purple stars varicose isolated ones; blue circles represent nonisolated varicose modes.

Figure 4

Stability diagrams: $\mathrm{Re}$ (left) and ${\mathrm{Re}}_{hh}$ (right) versus shear ratio. The sketched black regions indicate the flow conditions in which the flow is prone to a varicose unstable mode, the gray one refers to sinuous instability, whereas in the white striped zone the flow is subcritical. The triangles and the circles indicate the cases studied in Refs. [47] and [9], respectively. In all cases the aspect ratio of the roughness elements has been set at $d/h=1$.

Figure 5

Streamwise evolution of the skin friction coefficient for different values of Tu in the varicose case ($\mathrm{Re}=700$, $h/{\delta }^{*}=2.04$).

Figure 6

Snapshot showing the streamwise evolution of the unsteady perturbation in the varicose case with $\mathrm{Tu}=0.18\%$: isolevels of ${\lambda }_2=-0.02$ colored by values of the streamwise velocity.

Figure 7

Left: Time Fourier transform at several streamwise locations for the varicose case. Right: Associated shape function for the dominant frequencies of the Fourier spectrum for (top) $\omega \approx 1$ and (bottom) $\omega \approx 0.11$. The Fourier transform has been performed on velocity signals extracted every $\mathrm{\Delta }t=0.018$ on several cube of probes of size $dx,dy,dz=0.2$ placed at the streamwise positions reported in the figure. The signals recorded in these probes are averaged in space over $dx,dy,dz$ and rendered periodic before applying the Fourier transform.

Figure 8

Streamwise variation of the wall-normal position of the maximum amplitude mode of the spanwise Fourier transform of the root-mean square of the velocity perturbation, $u_{\text{rms}}$. The color bar indicates the values of the spanwise wave number of the dominant Fourier mode. The red dashed line provides the analytical displacement thickness of the Blasius solution whereas the blue dash-dotted line indicates the height of the cylindrical roughness element. Varicose case with $\mathrm{Re}=700$, $h/{\delta }^{*}=2.04$, $\mathrm{Tu}=0.18\%$.

Figure 9

Top: DMD spectrum providing the dynamic modes with their associated weights. Spatial structure of the modes with (middle row) $\omega \approx 0.11$ and (bottom row) $\omega \approx 1$. On the right column are provided the isolevels of the streamwise velocity perturbation of the DMD modes in the crossflow plane with $x=10$, whereas the left column shows isosurfaces of the streamwise velocity perturbation ($\pm 5\%$ of the maximum velocity perturbation amplitude).

Figure 10

Streamwise evolution of the skin friction coefficient for different intensities of the freestream turbulence for the sinuous case ($\mathrm{Re}=1000$, $h/{\delta }^{*}=1.47$). A negligible sensitivity has been found with respect to the integral length scale (not shown).

Figure 11

Snapshot showing the streamwise evolution of the unsteady perturbation in the sinuous case with $\mathrm{Tu}=0.26\%$: isolevels of ${\lambda }_2=-0.005$ colored by values of the streamwise velocity.

Figure 12

Left: Time Fourier transform at several streamwise locations for the sinuous case. Right: Associated shape function for (top) $\omega \approx 1.2$, (middle) $\omega \approx 0.4$, and (bottom) $\omega \approx 0.15$. The Fourier transform has been performed on velocity signals extracted every $\mathrm{\Delta }t=0.014$ on several cubes of probes of size $dx,dy,dz=0.2$ placed at the streamwise positions reported in the figure. The signals recorded in these probes are averaged in space over $dx,dy,dz$ and rendered periodic before applying the Fourier transform.

Figure 13

Streamwise variation of the wall-normal position of the maximum amplitude mode of the spanwise Fourier transform of the root-mean square of the velocity perturbation, $u_{\text{rms}}$. The color bar indicates the values of the spanwise wave number of the dominant Fourier mode. The red dashed line provides the analytical displacement thickness of the Blasius solution whereas the blue dash-dotted line indicates the height of the cylinder. The case considered is the sinuous case with $\mathrm{Tu}=0.26\%$.

Figure 14

Top panel: Amplitude of the DMD modes as a function of their frequency. The other panels depict the spatial structure of the modes associated with (second row) $\omega \approx 0.15$, (third row) $\omega \approx 0.4$, and (fourth row) $\omega \approx 1.2$. On the right column are provided the isolevels of the streamwise velocity perturbation of the DMD modes in the crossflow plane with $x=10$, while the left column shows isosurfaces of the streamwise velocity perturbation ($\pm 5\%$ of the maximum velocity of the perturbation).

Figure 15

Top: Resolvent norm versus forcing frequency. Bottom: Eigenspectrum for the sinuous case. ($+$) sinuous mode associated with $\omega =0.68$, ($\circ$) varicose modes.

Figure 16

Top: Sinuous mode at $\omega =0.68$ (right) and its adjoint counterpart (left). Bottom: Optimal forcing (left) and response (right) for the same value of $\omega$ in the sinuous case. The isosurfaces show the $\pm 5\%$ of the maximum amplitude of the streamwise velocity perturbation.

Figure 17

Top: Varicose mode for $\omega =1.25$ (right) and its adjoint counterpart (left). Bottom: Optimal forcing (left) and response (right) for $\omega =1.2$ for the sinuous case. The isosurfaces show the $\pm 5\%$ of the maximum amplitude of the streamwise velocity perturbation.

Figure 18

Top: Optimal energy gain curve, with maximum gain equal to $4.3\times {10}^8$ at $T_{\mathrm{opt}}=50$. Bottom: Optimal perturbation at $t=0$ (left) and $t=T_{\mathrm{opt}}=50$ (right). The isosurfaces represent the $\pm 5\%$ of the maximum amplitude of the streamwise velocity perturbation. Sinuous case with $\mathrm{Re}=1000$, $h/{\delta }^{*}=1.47$.

Figure 19

Fourier transform in time of the wall-normal velocity measured at a probe placed at $(x,y,z)=(10,1,0)$ in a DNS initialized with the optimal perturbation for $T_{\mathrm{opt}}=50$. Sinuous case with $\mathrm{Re}=1000$, $h/{\delta }^{*}=1.47$.

Figure 20

Top left: Kinetic energy budget normalized with respect to the dissipation. Top right: Comparison of the perturbation root-mean-square value $u_{\mathrm{rms}}$ measured experimentally by a probe at $(y,z)=(1,0)$ with the evolution of the crossflow integral of the $I_2$ production term in the streamwise direction. Bottom: Spatial distribution of the lift-up production term. Varicose case $\mathrm{Re}=700$ and $h/{\delta }^{*}=2.04$ forced by freestream turbulence of intensity $\mathrm{Tu}=0.18\%$.

Figure 21

Sketch explaining the physical mechanism for the generation of the rear pair vortices (RP). Points labeled B, C, D represent the successive positions of a particle initially placed in A and HP indicates the horseshoe vortex wrapping the roughness.

Figure 22

Passive particle tracking in the base-flow field: Some of the particles are trapped by the rear pair vortices (RP); the remaining particles are advected by the horseshoe vortices (HP). Varicose case with $\mathrm{Re}=700$ and $h/{\delta }^{*}=2.04$.

Figure 23

Streamwise vorticity ${\mathrm{\Omega }}_x\pm 0.5$ in the plane $y=0.5$ of the (top) base flow and (bottom) mean flow computed from the DNS in the presence of freestream turbulence with $\mathrm{Tu}=0.18\%$. Varicose case at $\mathrm{Re}=700$ and $h/{\delta }^{*}=2.04$.

Figure 24

Streamwise evolution of the streamwise velocity (blue line) and of the module of spanwise vorticity (red line) of the mean flow in the plane $z=0$. The circles indicate the inflectional points according to the Rayleigh Fjørtoft criterion.

Figure 25

Snapshot of transition extracted from the DNS showing the vortical structures (isosurfaces of ${\lambda }_2=-0.02$) and the velocity field (arrows) in the symmetry plane.

Figure 26

Streamwise variation of the (left) amplitude of the streaks and of the (right) area associated with the streaks for the base flow.

Figure 27

Sketch of the optimization loop used for obtaining the optimal forcing/response.

Figure 28

Resolvent norm obtained using the present numerical code (circles) compared to that obtained by a local resolvent analysis for two values of the streamwise wave number ($\alpha =1$, red line, and $\alpha =2$, blue line) for the Poiseuille flow at $\mathrm{Re}=4000$. The resolvent gain obtained with the present analysis is the envelope of the resolvent gain curves obtained with the local theory for different values of the streamwise wave number.

Figure 29

Exponential Tu decay for case 1 in Ref. [2] (left panel) and for the case $(\mathrm{Re},h/{\delta }^{*})=(700,2.04)$ considered in this work. The curve is extracted at $y/{\delta }^{*}=60$.

Figure 30

Skin friction coefficient $C_f$ versus ${\mathrm{Re}}_x$ for case 1 in Ref. [2].

Figure 31

Instantaneous streamwise velocity field at $y/{\delta }^{*}=2$ for case 1 in Ref. [2].

Figure 32

Base flow of the varicose case at $x=10$ used for the bilocal stability analysis (top left) and corresponding varicose (right column) and sinuous (left column) unstable modes.

Figure 33

Base flow of the sinuous case at $x=10$ used for the bilocal stability analysis (left) and corresponding varicose low-frequency unstable mode, $V_1$ (right).