Sensitivity gradients of surface geometry modifications based on stability analysis of compressible flows

We derive a discrete framework for the calculation of eigenvalue sensitivity to geometric deformations. We apply the technique to the steady compressible Navier-Stokes and Reynolds-averaged Navier-Stokes (RANS) flows. The analysis enables one to control (reduce or increase) the amplification rate or frequency associated to the least stable global mode, which is identified using stability analysis. A methodology using a discrete framework is proposed, allowing one to recover the gradients in compressible and turbulent flows. The potential of the resulting shape gradients is evaluated on the well-known circular cylinder flow problem and on a RANS turbulent flow scenario to control the buffet onset on a NACA0012 airfoil. The predicted deformations show excellent performance on the stabilization or excitation, and frequency control, of the global modes and for the two cases tested.

Figure 1

Compact second-order stencil used on the calculation of the Jacobian and the residuals. The dark-gray cell indicates the wall cell that is deformed, gray cells correspond to the immediate neighbors, and light-gray cells are the immediate next neighbors.

Figure 2

Mathematical procedure for the calculation of the sensitivity to mesh node displacements.

Figure 3

Comparison between the analytic and numerical sensitivity of the eigenvalue to changes in ${\epsilon }_2$ and the parameter $p$.

Figure 4

(a) Unstructured H-grid used for the simulations, overlapped with vorticity magnitude contours ($\mathrm{Re}=60$). Only half of the mesh nodes are shown. (b) Eigenvalue distribution for the laminar flow over a cylinder, $\mathrm{Re}=60$.

Figure 5

Sensitivity of the eigenvalue to outward normal displacements of the surface nodes, ${\mathbf{\nabla }}_{\mathbf{X}}\sigma$, as a function of the circular cylinder azimuth (left). Node distribution of the sensitivity of the amplification rate (center) and the sensitivity of the pulsation (right) of the eigenvalue.

Figure 6

(a) Comparison of eigenvalue sensitivity for different Reynolds numbers, $M=0.2$. Eigenvalue (b) growth rate and (c) pulsation sensitivity, split into feedback (dashed line) and base flow (continuous line) contributions ($\mathrm{Re}=50$).

Figure 7

Drift of the eigenvalue as the cylinder shape is deformed. Linear estimation and stability analysis results of the deformed body shape are compared: $-\circ$: cross-stream (vertical) deformations; $-▵$: streamwise (horizontal) deformations. The eigenvalue corresponding to the unperturbed body shape is signalized with dotted lines.

Figure 8

Deformed body shapes for (a) more stable configuration, (b) higher associated frequency, and (c) lower associated frequency.

Figure 9

Pressure coefficient distribution over the cylinder surface for the three optimized configurations. (a) Lower eigenvalue amplification rate, (b) higher frequency, and (c) lower frequency.

Figure 10

Illustration of the mesh discretization used on the NACA0012 analysis (left) and Mach number contours of the converged flow solution (right).

Figure 11

Direct (left) and adjoint (right) eigenmodes of the NACA0012 buffet global mode. Streamwise momentum perturbations at $\mathrm{Re}={10}^7$, $M=0.76$, and $\alpha =3.2°$

Figure 12

Contours of the sensitivity of the eigenvalue related to the transonic buffet instability to modifications of the turbulent viscosity, ${\mu }_t$.

Figure 13

Sensitivity of the eigenvalue amplification rate to the application of a steady forcing, ${\mathbf{\nabla }}_f{\sigma }_r$, at $\mathrm{Re}={10}^7$, $M=0.76$, and $\alpha =3.2°$.

Figure 14

Sensitivity of the eigenvalue to outward normal displacements of the surface nodes, ${\mathbf{\nabla }}_{\mathbf{X}}\sigma$. (a) Predicted deformed shape to suppress the instability, with exaggerated height for illustration purposes. Node distribution of the (b) sensitivity of the amplification rate and (c) sensitivity of the pulsation of the eigenvalue. The zoomed area of image (a) is framed by dotted points in (b) and (c).

Figure 15

Comparison of the optimal position of local deformations for different flow conditions, superimposed into local Mach number contours. (a) $M=0.76$, $\alpha =3.2°$; (b) $M=0.75$, $\alpha =3.3°$; (c) $M=0.73$, $\alpha =3.8°$.

Figure 16

(a) Changes in the real and imaginary parts of the eigenvalue with respect to the deformation amplitude. (b) Evolution of the pressure coefficient ($C_p$) near the shock foot location for different surface deformation amplitude.