Non-normality and classification of amplification mechanisms in stability and resolvent analysis

Eigenspectra and pseudospectra of the mean-linearized Navier-Stokes operator are used to characterize amplification mechanisms in laminar and turbulent flows in which linear mechanisms are important. Success of mean flow (linear) stability analysis for a particular frequency is shown to depend on whether two scalar measures of non-normality agree: (1) the product between the resolvent norm and the distance from the imaginary axis to the closest eigenvalue and (2) the inverse of the inner product between the most amplified resolvent forcing and response modes. If they agree, the resolvent operator can be rewritten in its dyadic representation to reveal that the adjoint and forward stability modes are proportional to the forcing and response resolvent modes at that frequency. Hence the real parts of the eigenvalues are important since they are responsible for resonant amplification and the resolvent operator is low rank when the eigenvalues are sufficiently separated in the spectrum. If the amplification is pseudoresonant, then resolvent analysis is more suitable to understand the origin of observed flow structures. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The first deals mainly with resonant mechanisms, hence the success of both classical and mean stability analysis with respect to predicting the critical Reynolds number and global frequency of the saturated flow. Both scalar measures of non-normality agree for the base and mean flows, and the region where the forcing and response modes overlap scales with the length of the recirculation bubble. In the case of turbulent channel flow, structures result from both resonant and pseudoresonant mechanisms, suggesting that both are necessary elements to sustain turbulence. Mean shear is exploited most efficiently by stationary disturbances while bounds on the pseudospectra illustrate how pseudoresonance is responsible for the most amplified disturbances at spatial wavenumbers and temporal frequencies corresponding to well-known turbulent structures. Some implications for flow control are discussed.

Figure 1

Cartoon of forcing (left) and response (right) modes corresponding to various amplification mechanisms. (a), (b) Purely normal mechanism where the modes are identical; (c), (d) component-type non-normality due to the lift-up mechanism. (e), (f) A $\pi /2$ phase shift between the modes due to the non-self-adjoint nature of the LNS operator. When coupled with mean shear, this results in the (g), (h) Orr mechanism, where the forcing mode leans upstream against the mean shear and the response mode leans downstream with the mean shear. (i), (j) A convective-type non-normality where mean flow advection results in the forcing being upstream of the response. (k), (l) The vertical dash-dotted lines denote the region where the forcing and response modes overlap in the streamwise direction. The flow is said to be absolutely unstable in this region and convectively unstable elsewhere. Positive (negative) isocontours are denoted by solid (dotted) lines and blue (red) indicates streamwise (transverse) components, in the $x,y$ directions, respectively. Each mode is nonzero in one velocity component only.

Figure 2

Comparison of the pseudospectra and resolvent norm for the operators given in Eqs. (47) and (48), which have the same eigenvalues: (a), (b) normal operator $\mathit{S}$ and (c), (d) non-normal operator $\mathit{P}$. The eigenvalues, i.e., the eigenspectra, are marked by red crosses and color contours outline the bounds of the perturbed spectrum for constant perturbation magnitudes in (a), (c). The dashed contours in (a) reflect that the pseudospectra are circles centered on the eigenvalues, which is not true for the non-normal operator in (c). The resolvent norm in each case (b), (d) reflects the value of these contours along the imaginary axis. Dashed red horizontal lines indicate the resonant frequencies of the operator, i.e., the frequencies corresponding to the eigenvalues, while the solid blue horizontal line represents the most highly amplified frequency in the non-normal case.

Figure 3

Projection of the first resolvent mode ${\widehat{\mathbf{\psi }}}_1$ onto the eigenvectors ${\tilde{\mathit{u}}}_1$ (dash-dotted brown line) and ${\tilde{\mathit{u}}}_2$ (dotted green line) of (a) $\mathit{S}$ and (b) $\mathit{P}$. The product of these projections is the long-dashed black line. Dashed red horizontal lines indicate the resonant frequencies of the operators and the solid blue horizontal line represents the most highly amplified frequency in the non-normal case.

Figure 4

(a) Spectrum (red dots) and pseudospectrum (solid contours, $\epsilon$ increasing as colors change from dark to light) of the LNS operator for the cylinder base flow at $\text{Re}=47$. (b) The resolvent norm, ${\sigma }_1$ (solid line), i.e., the value of ${\epsilon }^{-1}$ along the imaginary axis, second largest singular value ${\sigma }_2$ (dash-dotted line), and inverse distance from the imaginary axis to the nearest eigenvalue (dotted red line).

Figure 5

Comparison of stability modes (left) with resolvent modes (right) at the critical Reynolds number ${\text{Re}}_c=47$ and a temporal frequency of $\omega =0.742$: (a), (b) streamwise component of the forward or response mode, (c), (d) transverse component of the forward or response mode, (e), (f) streamwise component of the adjoint or forcing mode, and (g), (h) transverse component of the adjoint or forcing mode. The eigenmodes and resolvent mode shapes are essentially indistinguishable for this flow.

Figure 6

Contours of the transverse velocity for the leading adjoint modes ${\tilde{v}}^{†}$ (left) and forward modes $\tilde{v}$ (middle) of the base flow. The Reynolds numbers $(\text{Re}=15$, 25, 35, 45, and 50) increase from top to bottom. The wavemaker $\mathcal{W}$ (right) is computed using the forward and adjoint modes. Contour levels are not identical for the adjoint modes which are normalized based on the forward modes. Note that the streamwise velocity component has not been plotted even though the wavemaker depends on this quantity.

Figure 7

(a) Spectrum (red dots) and pseudospectrum (solid contours, $\epsilon$ increasing as colors change from dark to light) of the LNS operator for the cylinder mean flow at $\text{Re}=100$. (b) The resolvent norm, ${\sigma }_1$ (solid line), i.e., the value of ${\epsilon }^{-1}$ along the imaginary axis, second largest singular value ${\sigma }_2$ (dash-dotted line), and inverse distance from the imaginary axis to the nearest eigenvalue (dotted red line).

Figure 8

(a) The resolvent norm for the critical (solid red line) and subcritical (solid black lines) base flows. (b) The resolvent norm for supercritical mean flows.

Figure 9

Wavemakers for mean flow at $\text{Re}=100$ computed from (a) stability modes and (b) resolvent modes. Wavemakers for the base flow at (c) $\text{Re}=47$ and (d) $\text{Re}=100$. Blue lines superimpose the mean flow streamlines which delineate the mean recirculation bubble.

Figure 10

(a) Turbulent mean velocity profile for channel flow at ${\text{Re}}_{\tau }=2000$ plotted in inner units alongside (b) the mean shear.

Figure 11

Velocity amplitudes for (a) the optimal forcing mode ${\widehat{\mathbf{\varphi }}}_1$ and (b) the optimal response mode ${\widehat{\mathbf{\psi }}}_1$ corresponding to the wavenumber triplet of $(k_x,k_z,c^{+})=(4\pi ,40\pi ,14)$.

Figure 12

The eigenvalues of the operator $\mathit{L}(k_x=4\pi ,k_z=40\pi )$ in red circles overlaid with contours of the pseudospectrum (left). The resolvent norm is plotted in the solid black line along with the inverse distance from the imaginary axis to the nearest eigenvalue, which is the dotted red line (right). The spatial wavenumbers correspond to the near-wall cycle, and the horizontal, dashed blue line represents the $\omega$ with the largest resolvent norm, which corresponds to a phase speed of $c^{+}\approx 14$.

Figure 13

First 20 singular values of the resolvent operator for $(k_x,k_z,c^{+})=(4\pi ,40\pi ,14)$.

Figure 14

Velocity amplitudes for (a) the optimal forcing mode ${\widehat{\mathbf{\varphi }}}_1$ and (b) the optimal response mode ${\widehat{\mathbf{\psi }}}_1$ corresponding to the wavenumber triplet of $(k_x,k_z,c^{+})=(\pi /9,2\pi /3,22)$.

Figure 15

The eigenvalues of the operator $\mathit{L}(k_x=\pi /9,k_z=2\pi /3)$ in red circles overlaid with contours of the pseudospectrum (left). The resolvent norm is plotted in the solid black line along with the inverse distance from the imaginary axis to the nearest eigenvalue, which is the dotted red line (right). The spatial wavenumbers correspond to the VLSM mode, and the horizontal, dashed blue line represents the $\omega$ with the largest resolvent norm, which corresponds to a phase speed of $c^{+}\approx 22$.

Figure 16

First 20 singular values of the resolvent operator for $(k_x,k_z,c^{+})=(\pi /9,2\pi /3,22)$.

Figure 17

Velocity amplitudes for (a) the optimal forcing mode ${\widehat{\mathbf{\varphi }}}_1$ and (b) the optimal response mode ${\widehat{\mathbf{\psi }}}_1$ corresponding to the wavenumber triplet of $(k_x,k_z,\omega )=(0,2\pi /3,0)$.

Figure 18

The eigenvalues of the operator $\mathit{L}(k_x=0,k_z=2\pi /3)$ in red circles overlaid with contours of the pseudospectrum (left). The resolvent norm is plotted in the solid black line along with the inverse distance from the imaginary axis to the nearest eigenvalue, which is the dotted red line (right). The horizontal, dashed blue line represents the $\omega$ corresponding to the phase speed of $c^{+}=0$.

Figure 19

First 20 singular values of the resolvent operator for $(k_x,k_z,\omega )=(0,2\pi /3,0)$.