Impact of harmonic inflow variations on the size and dynamics of the separated flow over a bump

The separated flow over a wall-mounted bump geometry under harmonic oscillations of the inflow stream is investigated with direct numerical simulations. The bump geometry gives rise to streamwise pressure gradients similar to those encountered on the suction side of low-pressure turbine (LPT) blades. Under steady inflow conditions, the separated-flow laminar-to-turbulent transition is initiated by self-sustained vortex shedding due to Kelvin-Helmholtz (KH) instability. In LPTs the dynamics are further complicated by the passage of the wakes shed by the previous stage of blades. The wake-passing effect is modeled here by introducing a harmonic variation of the inflow conditions. Three inflow oscillation frequencies and three amplitudes are considered. The frequencies are comparable to the wake-passing frequencies in practical LPTs. The amplitudes range from 1% to 10% of the inflow total pressure. The dynamics of the separated flow are studied by isolating the flow components that are respectively coherent with and uncorrelated to the inflow oscillation. Three scenarios are identified. The first one is analogous to the steady inflow case. In the second one, the KH vortex shedding is replaced during a part of the inflow period by the formation and release of a large vortex cluster. The third scenario consists solely of the periodic formation and release of the vortex cluster; it leads to a consistent reduction of the separated flow length over the entire period compared to the steady inflow case and would be the most desirable flow condition in a practical application.

Figure 1

Spanwise vorticity field around a wall-mounted bump under steady inflow conditions.

Figure 2

Definition of the wall-mounted bump geometry. The symbols show the location of the control points defining the Bezier curves.

Figure 3

Computational domain and representative mesh for spectral element computations.

Figure 4

PSD of spanwise vorticity at selected probe locations for different spatial and temporal discretizations.

Figure 5

Location of the reference point and sampling probes.

Figure 6

Evolution of total pressure and streamwise velocity at the reference point.

Figure 7

$Q(+)$ isosurface colored by streamwise velocity. $A_{\mathrm{in}}=0.01$ and $f_{\mathrm{in}}^{*}=0.5$.

Figure 8

$Q(+)$ isosurface colored by streamwise velocity. $A_{\mathrm{in}}=0.1$ and $f_{\mathrm{in}}^{*}=2$.

Figure 9

Instantaneous spanwise vorticity. $A_{\mathrm{in}}=0.01$ and $f_{\mathrm{in}}^{*}=0.5$.

Figure 10

Instantaneous spanwise vorticity. $A_{\mathrm{in}}=0.1$ and $f_{\mathrm{in}}^{*}=2$.

Figure 11

Time-averaged (left) and root-mean-square (center) and root-mean-square of the incoherent component (right) of streamwise velocity field.

Figure 12

Phase-averaged spanwise vorticity. $A_{\mathrm{in}}=0.01$ and $f_{\mathrm{in}}^{*}=0.5$.

Figure 13

Phase-averaged spanwise vorticity. $A_{\mathrm{in}}=0.05$ and $f_{\mathrm{in}}^{*}=1$.

Figure 14

Phase-averaged spanwise vorticity. $A_{\mathrm{in}}=0.1$ and $f_{\mathrm{in}}^{*}=2$.

Figure 15

Contours: Phase-averaged streamwise acceleration parameter at $y=0.1$ m. Green lines: angle of the streamline at the separation point $\mathrm{\Gamma }$ corresponding to the phase-averaged component (solid line), the mean component (dashed) and the steady inflow case (dashed-dotted).

Figure 16

Phase-averaged skin friction at the mid-span of the bump surface. Left: $A_{\mathrm{in}}=0.01, f_{\mathrm{in}}^{*}=0.5$. Center: $A_{\mathrm{in}}=0.05, f_{\mathrm{in}}^{*}=1$. Right: $A_{\mathrm{in}}=0.1, f_{\mathrm{in}}^{*}=2$. The location of the phase-averaged separation and reattachment points is shown as a green dot for each discrete phase value.

Figure 17

Probability density function of the phase-averaged recirculation bubble length $\langle L_s\rangle$. (- - -) time-averaged value; (-$·-·$) time-averaged for the steady inflow case, $L_{s,\mathrm{steady}}$.

Figure 18

Triple decomposition of spanwise vorticity. $A_{\mathrm{in}}=0.01$ and $f_{\mathrm{in}}^{*}=0.5$.

Figure 19

Incoherent spanwise vorticity at $\eta =0$ for the steady inflow case and the three representative cases with harmonic inflow.

Figure 20

$Q(+)$ isosurface colored by streamwise velocity. $A_{\mathrm{in}}=0.1$ and $f_{\mathrm{in}}^{*}=1$.

Figure 21

Power spectral density of the streamwise and wall normal velocity at Probe 4, $(x,y)=(0.2,0.05)$.

Figure 22

Power spectral density of the incoherent spanwise vorticity ${\omega }_z^{\prime }$ at the shear layer, $\xi =0.12$.

Figure 23

Descriptive illustrations of the three scenarios. Dashed blue lines: phase-averaged separation streamline; solid blue: KH vortices; orange: small-grained vorticity; red: vortex cluster; green arrows show the instantaneous trajectory of the flow structures.

Figure 24

Scaling of the most amplified frequency along the separated shear layer.

Figure 25

Phase-averaged streamwise acceleration parameter.

Figure 26

Incoherent spanwise vorticity at $\eta =0$ for the all nine cases with harmonic inflow.

Figure 27

Computational domain, representative mesh, and boundary conditions for NASA Hump cases.

Figure 28

Instantaneous spanwise vorticity for the NASA hump. $A_{\mathrm{in}}=0.005$ and $f_{\mathrm{in}}^{*}=0.25$.

Figure 29

Instantaneous spanwise vorticity for the NASA hump. $A_{\mathrm{in}}=0.05$ and $f_{\mathrm{in}}^{*}=0.5$.