3.1 First-order (linear) effects

The governing dynamics for the first-order flame response was presented earlier in (2.9). The connection between modes in the source disturbances and modes in the first-order flame response are depicted in the left diagram of figure 4. The first-order area response is given by (Acharya et al. Reference Acharya, Shin and Lieuwen2013)

(3.1)$$\begin{eqnarray}\displaystyle \hat{A}_{1}(\unicode[STIX]{x1D714}_{n})=\mathop{\sum }_{m_{\unicode[STIX]{x1D709}1}=-\infty }^{+\infty }\int _{r}r\,\text{d}r\cos \unicode[STIX]{x1D713}(r)\frac{\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1}}(r,\unicode[STIX]{x1D714}_{n})}{\unicode[STIX]{x2202}r}\int _{\unicode[STIX]{x1D703}=0}^{2\unicode[STIX]{x03C0}}\text{e}^{\text{i}m_{uj}\unicode[STIX]{x1D703}}\,\text{d}\unicode[STIX]{x1D703}. & & \displaystyle\end{eqnarray}$$

Here, $\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1}}$ was defined earlier in (1.11). This equation clearly shows that the azimuthal mode number in the area element is given by $m_{uj}$, i.e. $m_{A1}=m_{uj}$. This interaction is shown in the right diagram of figure 4. Here, the source helical mode is the local flame response mode denoted in the red box, which then causes (green arrow) the corresponding local flame area mode denoted in the blue box, through the term indicated in the dashed black box. From the interactions shown in figure 4, we can see that only the azimuthal mode coefficient, $\hat{\unicode[STIX]{x1D701}}_{1,m_{uj}=0}(r,\unicode[STIX]{x1D714}_{n})$, and thus only the axisymmetric ($m_{uj}=0$) flow disturbance modes, contributes to the global flame area at leading order. This was alluded to earlier in § 1, a result that has experimental support (Moeck et al. Reference Moeck, Bourgouin, Durox, Schuller and Candel2012).

Figure 5.

Modal interactions between source terms for the second-order contributions to local flame dynamics (a) and global flame dynamics (b). Here, the first-order flame response modes (red box) and velocity modes (black box) are the source helical modes. The interactions between the different terms are indicated by black arrows and the direction to the resultant mode is indicated by the green arrow. The second-order flame response mode, which is the focus of this figure is indicated by the orange box. The dashed black boxes indicate the interacting terms resulting in the second-order mode and have been described in the text.

3.2 Second-order effects

The second-order local flame dynamics is governed by (2.10) and is indicated by the modal interactions in figure 5(a). The second-order local flame response mode (orange box) originates through:

1. (i) Interactions between velocity modes (black boxes) with first-order flame response (red boxes):

   1. (a) convective nonlinearities (terms of the form $\boldsymbol{u}_{\mathbf{1}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}$), and

   2. (b) flame-motion nonlinearities (terms of the form $(\unicode[STIX]{x2202}\boldsymbol{u}_{\mathbf{1}}/\unicode[STIX]{x2202}z)\unicode[STIX]{x1D709}_{1}$).

2. (ii) Self-interactions within the first-order flame response:

   1. (a) kinematic nonlinearities (terms of the form $\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}$), and

   2. (b) flame-motion nonlinearities (terms of the form $\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}$).

Note that the sources of azimuthal modes in the first-order flame response (red boxes) are determined from the left diagram in figure 4. Using (2.7) and (2.16), we have

(3.2)$$\begin{eqnarray}A_{2}(t)=\int _{r}r\,\text{d}r\sin \unicode[STIX]{x1D713}(r)\int _{\unicode[STIX]{x1D703}=0}^{2\unicode[STIX]{x03C0}}\left(\frac{1}{2}\left(\left(\frac{1}{r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)^{2}+\sin ^{2}\unicode[STIX]{x1D713}(r)\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{1}}{\unicode[STIX]{x2202}r}\right)^{2}\right)+\cos \unicode[STIX]{x1D713}(r)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{2}}{\unicode[STIX]{x2202}r}\right)\text{d}\unicode[STIX]{x1D703}.\end{eqnarray}$$

For discussions on nonlinear effects, the frequency contents in the flow perturbations are assumed to be integer harmonics of a fundamental frequency, $\unicode[STIX]{x1D714}_{0}$; i.e. $\unicode[STIX]{x1D714}_{n}=n\unicode[STIX]{x1D714}_{0}$ where $n$ is a positive integer. this is done for simplicity in expressions. The qualitative arguments for all three integrals are similar, so the ensuing discussion focuses on the first integral to illustrate how these terms contribute to the second-order area fluctuations. From (1.10), we have

(3.3)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}_{1,r}^{2}(r,\unicode[STIX]{x1D703},t) & = & \displaystyle \mathop{\sum }_{n_{1}=1}^{\infty }\mathop{\sum }_{n_{2}=1}^{\infty }[\!\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(n_{2}\unicode[STIX]{x1D714}_{0})\cos (n_{1}\unicode[STIX]{x1D714}_{0}t)\cos (n_{2}\unicode[STIX]{x1D714}_{0}t)\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(n_{2}\unicode[STIX]{x1D714}_{0})\cos (n_{1}\unicode[STIX]{x1D714}_{0}t)\sin (n_{2}\unicode[STIX]{x1D714}_{0}t)\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(n_{2}\unicode[STIX]{x1D714}_{0})\sin (n_{1}\unicode[STIX]{x1D714}_{0}t)\cos (n_{2}\unicode[STIX]{x1D714}_{0}t)\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(n_{2}\unicode[STIX]{x1D714}_{0})\sin (n_{1}\unicode[STIX]{x1D714}_{0}t)\sin (n_{2}\unicode[STIX]{x1D714}_{0}t)\!].\end{eqnarray}$$

Here, the expressions are in the time domain and the superscripts $R$ and $I$ correspond to the real and imaginary parts of the complex quantity. Although not indicated, each of $\hat{\unicode[STIX]{x1D709}}_{1,r}^{R,I}$ are functions of $(r,\unicode[STIX]{x1D703})$ also. Each of the terms can be split into two terms with frequencies: $|n_{1}-n_{2}|\unicode[STIX]{x1D714}_{0}$ and $(n_{1}+n_{2})\unicode[STIX]{x1D714}_{0}$. There are two important observations that can be made.

(a) Non-zero time-average contribution (Mechanism TA):

The terms where the frequency $|n_{1}-n_{2}|\unicode[STIX]{x1D714}_{0}=0$ result in time-independent terms which occurs when: $n_{1}=n_{2}=n$. These introduce a change in the time-average flame position, corresponding to ‘Mechanism TA’. This change is expressed as

(3.4)$$\begin{eqnarray}\overline{\unicode[STIX]{x1D709}_{1,r}^{2}}=\frac{1}{2}\mathop{\sum }_{n=1}^{\infty }(\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(n\unicode[STIX]{x1D714}_{0})+\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(n\unicode[STIX]{x1D714}_{0})).\end{eqnarray}$$

Note that not all terms in the above summation contribute to the time average. For example, the first term can be expanded using (1.11) as

(3.5)$$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D709}}_{1,r}^{R}\hat{\unicode[STIX]{x1D709}}_{1,r}^{R} & = & \displaystyle \mathop{\sum }_{m_{\unicode[STIX]{x1D709}1,1}=-\infty }^{+\infty }\mathop{\sum }_{m_{\unicode[STIX]{x1D709}1,2}=-\infty }^{+\infty }\nonumber\\ \displaystyle & & \displaystyle \times \,[\!\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{R}\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{R}\cos (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\cos (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{R}\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{I}\cos (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\sin (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{I}\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{R}\sin (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\cos (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{I}\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{I}\sin (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\sin (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\!].\end{eqnarray}$$

Each term can be split into two terms with mode numbers ($\pm m_{\unicode[STIX]{x1D709}1,1}\pm m_{\unicode[STIX]{x1D709}1,2}$). Thus, the azimuthal integral of these terms is non-zero only if

(3.6)$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}m_{\unicode[STIX]{x1D709}1,1}+m_{\unicode[STIX]{x1D709}1,2}=0,\\ m_{\unicode[STIX]{x1D709}1,1}-m_{\unicode[STIX]{x1D709}1,2}=0.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Thus, the contributing terms from (3.5) can be expressed as

(3.7)$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D703}=0}^{2\unicode[STIX]{x03C0}}\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0})\,\text{d}\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x03C0}\displaystyle \mathop{\sum }_{m_{\unicode[STIX]{x1D709}1}=-\infty }^{+\infty }[\!\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})+\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,-m_{\unicode[STIX]{x1D709}1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})-\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,-m_{\unicode[STIX]{x1D709}1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\!].\end{eqnarray}$$

From the above expression, the following conclusions can be drawn:

1. (i) The first row of terms on the right-hand side involve the interaction of the first-order flame response at frequency $n\unicode[STIX]{x1D714}_{0}$ and helical mode $m_{\unicode[STIX]{x1D709}1}$ with itself. Thus, there will always be a contribution to the time average, $\overline{A_{2}}$, from the second-order dynamics.

2. (ii) The second row of terms on the right-hand side involve the interaction of the first-order flame response at frequency $n\unicode[STIX]{x1D714}_{0}$ and helical mode $m_{\unicode[STIX]{x1D709}1}$ with the first-order flame response at the same frequency $n\unicode[STIX]{x1D714}_{0}$ and helical mode $-m_{\unicode[STIX]{x1D709}1}$. Thus, the flame responds to helical modes whose mode numbers sum to zero, and at the same frequency also contribute to the second-order correction of the time average. Note that this contribution to the time average comes from all excitation frequencies in the disturbance, regardless of the frequency of focus for $\hat{F}$ in (1.8).

In addition to the above second-order effects on the global area, the second-order correction affects the time-average flame shape as well. From (2.10), the time-averaged correction follows from the solution of

(3.8)$$\begin{eqnarray}\displaystyle & & \displaystyle \left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{\mathbf{0}}}{\unicode[STIX]{x2202}z}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{0}+(\boldsymbol{u}_{\boldsymbol{T},\mathbf{0}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\right)\overline{\unicode[STIX]{x1D709}_{2}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\overline{(\boldsymbol{u}_{\mathbf{1}}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\unicode[STIX]{x1D709}_{1}}-\overline{\left(\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{\mathbf{0}}}{\unicode[STIX]{x2202}z}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}+\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{\mathbf{1}}}{\unicode[STIX]{x2202}z}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{0}\right)\unicode[STIX]{x1D709}_{1}}-\left(\frac{\unicode[STIX]{x2202}^{2}\boldsymbol{u}_{\boldsymbol{ 0}}}{\unicode[STIX]{x2202}z^{2}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{0}\right)\overline{\frac{\unicode[STIX]{x1D709}_{1}^{2}}{2}}+s_{L}\overline{\unicode[STIX]{x1D6E5}_{2s}}.\qquad\end{eqnarray}$$

Here, $\unicode[STIX]{x1D6E5}_{2s}$ is as in (2.7). The qualitative arguments for each of the terms are similar to the arguments used for (3.4)–(3.5) (which also happens to be one of the terms on the right-hand side of (3.8)).

Thus, the physics contributing to distortions of the time-average flame position can be summarized as:

1. (i) Convective nonlinearities: interaction of the first-order flame response at frequency $n\unicode[STIX]{x1D714}_{0}$ and helical mode $m_{\unicode[STIX]{x1D709}1}$ with flow disturbances at the same frequency $n\unicode[STIX]{x1D714}_{0}$ and helical mode $m_{uj}$.

2. (ii) Kinematic nonlinearities: interaction of the first-order flame response at frequency $n\unicode[STIX]{x1D714}_{0}$ and helical mode $m_{\unicode[STIX]{x1D709}1,1}$ with itself at the same frequency $n\unicode[STIX]{x1D714}_{0}$ and helical mode $m_{\unicode[STIX]{x1D709}1,2}$.

3. (iii) Flame-motion nonlinearities: flame motion causes the instantaneous imposed velocity disturbance value to differ from its value at the nominal flame position (second and third terms on the right-hand side of (3.8)).

The frequency and modal contributions can be summarized as:

1. (i) Frequency contributions: the contribution to the time average stems from all excitation frequencies (regardless of the frequency of focus) as they interact with themselves.

2. (ii) Modal contributions: in general, a non-axisymmetric correction to the time-average flame shape occurs for

   (3.9)$$\begin{eqnarray}m_{\unicode[STIX]{x1D709}1,1}\neq m_{\unicode[STIX]{x1D709}1,2}.\end{eqnarray}$$
   However, when $m_{\unicode[STIX]{x1D709}1,1}=\pm m_{\unicode[STIX]{x1D709}1,2}=m$, the nonlinear terms are of the form $\cos ^{2}(m\unicode[STIX]{x1D703})$, $\sin ^{2}(m\unicode[STIX]{x1D703})$, $\sin (2m\unicode[STIX]{x1D703})$. In this case, the first two terms can be split into the form ($1+\cos (2m\unicode[STIX]{x1D703})$), which results in an axisymmetric and non-axisymmetric part. Thus, the nonlinear correction to the time average has a purely axisymmetric contribution whenever the axisymmetric mode interacts with itself, or from nonlinear interactions between helical modes of the same or opposite sign.

For the purposes of this analysis, those terms that distort the axisymmetric nominal flame to a time-averaged shape with an azimuthal component, $m_{\unicode[STIX]{x1D709}_{0}}$, that matches that of the exciting helical disturbance, $m_{uj}$, are particularly significant, as they can lead to net flame response to helical modes; i.e. the unsteady heat release rate oscillations at third order is analogous to the linear response of a nominally non-axisymmetric flame to helical disturbances, as summarized in (1.13) in § 1.

(b) Contribution to nonlinear dynamics (Mechanism NL):

The second-order fluctuation in flame area, $A_{2}(t)$, has a non-zero frequency component at $\unicode[STIX]{x1D714}_{0}$ when: $n_{1}\pm n_{2}=\pm 1$. The first term in (3.3) when expanded to consider only the response at $\unicode[STIX]{x1D714}_{0}$ is expressed as

(3.10)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}_{1,r}^{2}(r,\unicode[STIX]{x1D703},t) & = & \displaystyle \frac{1}{2}\mathop{\sum }_{n=2}^{\infty }[\!\cos (\unicode[STIX]{x1D714}_{0}t)(\!\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},(n\pm 1)\unicode[STIX]{x1D714}_{0})\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},(n\pm 1)\unicode[STIX]{x1D714}_{0})\!)\nonumber\\ \displaystyle & \pm & \displaystyle \sin (\unicode[STIX]{x1D714}_{0}t)(\!\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},(n\pm 1)\unicode[STIX]{x1D714}_{0})\nonumber\\ \displaystyle & & \displaystyle -\,\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},(n\pm 1)\unicode[STIX]{x1D714}_{0})\!)\!].\end{eqnarray}$$

Note that the interaction of the first-order flame response at the harmonic $n\unicode[STIX]{x1D714}_{0}$ with itself at the next or prior harmonic $(n\pm 1)\unicode[STIX]{x1D714}_{0}$ contributes to $\hat{A}(\unicode[STIX]{x1D714}_{0})$. However, whether this term is non-zero is controlled by the azimuthal integral. Thus, considering the first term in the above equation as an example (and like (3.5)) and expanding

(3.11)$$\begin{eqnarray}\displaystyle & & \displaystyle \hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{m_{\unicode[STIX]{x1D709}1,1}=-\infty }^{+\infty }\mathop{\sum }_{m_{\unicode[STIX]{x1D709}1,2}=-\infty }^{+\infty }[\!\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{R}(r,n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})\cos (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\cos (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{I}(r,n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})\cos (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\sin (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{R}(r,n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})\sin (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\cos (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{I}(r,n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})\sin (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\sin (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\!].\end{eqnarray}$$

Each term can be split into two terms with mode numbers ($m_{\unicode[STIX]{x1D709}1,1}+m_{\unicode[STIX]{x1D709}1,2}$) and ($m_{\unicode[STIX]{x1D709}1,1}-m_{\unicode[STIX]{x1D709}1,2}$). Thus, the azimuthal integral of these terms are non-zero only if

(3.12)$$\begin{eqnarray}m_{\unicode[STIX]{x1D709}1,1}=\pm m_{\unicode[STIX]{x1D709}1,2}.\end{eqnarray}$$

Thus, the contributing terms from (3.11) can be expressed as

(3.13)$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D703}=0}^{2\unicode[STIX]{x03C0}}\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})\,\text{d}\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x03C0}\displaystyle \mathop{\sum }_{m_{\unicode[STIX]{x1D709}1}} [\!\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})+\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,-m_{\unicode[STIX]{x1D709}1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})-\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,-m_{\unicode[STIX]{x1D709}1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0}\pm \unicode[STIX]{x1D714}_{0})\!].\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Thus, the second-order contribution to the global area dynamics at frequency $\unicode[STIX]{x1D714}_{0}$ stems from the nonlinear interaction of the first-order flame response of helical mode $m_{\unicode[STIX]{x1D709}1}=m$ at frequency $n\unicode[STIX]{x1D714}_{0}$ with the first-order flame response of helical mode $m_{\unicode[STIX]{x1D709}1}=\pm m$ at frequency $(n\pm 1)\unicode[STIX]{x1D714}_{0}$.

Finally, consider the last integration term in (3.2). Note that the symmetric mode of $\unicode[STIX]{x1D709}_{2}$ is the only mode that contributes to the second-order flame area. The dynamics of $\unicode[STIX]{x1D709}_{2}$ is governed by (2.10) and, since the left-hand side operator is the same as the first-order dynamics equation, the mathematical properties of the right-hand side in the first-order case apply to the second-order dynamics as well. Thus, only an axisymmetric mode in the right-hand side terms contribute to the axisymmetric mode in the second-order local flame dynamics. Comparing (3.3) and the right-hand side of (2.10), we can see that the terms are similar in nature and thus the same arguments as for (3.5)–(3.13) apply here as well. Note that the right-hand side of (2.10) consists of terms that are due to convective nonlinearities (first term) and kinematic nonlinearities (last term: $\unicode[STIX]{x1D6E5}_{2s}$). Both physics contribute to the mechanism for non-zero area response through similar arguments made for the time-average contribution in the previous sub-section.

Summarizing, a non-zero second-order global flame area response at $\unicode[STIX]{x1D714}_{0}$ occurs if:

1. (i) There is a flow disturbance at either $n\unicode[STIX]{x1D714}_{0}$ and $(n+1)\unicode[STIX]{x1D714}_{0}$ (or) $n\unicode[STIX]{x1D714}_{0}$ and $(n-1)\unicode[STIX]{x1D714}_{0}$ for $n\geqslant 2$.

   and

2. (ii) The flow consists of a symmetric disturbance $(m_{uj}=0)$ (or) at least a symmetric disturbance $(m_{uj}=0)$ and asymmetric disturbance with modes of opposite sign, i.e. both $m_{uj}=+m$ and $m_{uj}=-m$ for $m>0$.

The modal interactions for the global area dynamics are summarized in figure 5 and shall be explicitly illustrated with an example later. We next move to the discussion of the dynamics at third order.

Figure 6.

Modal interactions between source terms for the third-order contributions to local flame dynamics (a) and global flame dynamics (b). Here, the first-order flame response modes (red box), velocity modes (black box) and second-order flame response modes (orange box) are the source helical modes. The interactions between the different terms are indicated by black arrows and the direction to the resultant mode is indicated by the green arrow. The third-order flame response mode, which is the focus of this figure, is indicated by the purple box. The third-order area response mode is shown by the blue box. The dashed black boxes indicate the interacting terms resulting in the third-order mode and have been described in the text.

3.3 Third-order effects

The third-order local flame dynamics modal interactions are determined from the right-hand side of (2.11) and shown in figure 6(a). As shown, there are several ways in which nonlinear interactions contribute at third order.

1. (i) Interactions between velocity modes (black box) and second-order flame response (orange box):

   1. (a) convective nonlinearities through terms such as $\boldsymbol{u}_{\mathbf{1}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{2}$; and

   2. (b) flame-motion nonlinearities through terms such as $\unicode[STIX]{x1D709}_{2}(\unicode[STIX]{x2202}\boldsymbol{u}_{\mathbf{1}}/\unicode[STIX]{x2202}z)$.

2. (ii) Interactions between velocity modes (black box) and first-order flame response (red boxes)):

   1. (a) flame-motion nonlinearities through terms appearing as $\unicode[STIX]{x1D709}_{1}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}\boldsymbol{\cdot }\unicode[STIX]{x2202}\boldsymbol{u}_{\mathbf{1}}/\unicode[STIX]{x2202}z)$ and $\unicode[STIX]{x1D709}_{1}^{2}(\unicode[STIX]{x2202}^{2}\boldsymbol{u}_{\mathbf{1}}/\unicode[STIX]{x2202}z^{2})$.

3. (iii) Interactions between first-order flame response (red box) and second-order flame response (orange box):

   1. (a) kinematic nonlinearities through terms appearing as $\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{2}$; and

   2. (b) flame-motion nonlinearities through terms appearing as $\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D709}_{2}$, $\unicode[STIX]{x1D709}_{1}\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{2}$ and $\unicode[STIX]{x1D709}_{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}$.

4. (iv) Self-interactions within first-order flame modes (red boxes):

   1. (a) flame-motion nonlinearities through terms appearing as $\unicode[STIX]{x1D709}_{1}^{3}$ and $\unicode[STIX]{x1D709}_{1}^{2}\unicode[STIX]{x1D735}\unicode[STIX]{x1D709}_{1}$.

The modal contributions to the third-order global area dynamics (blue box) are shown in the right in figure 6. Note that the azimuthal mode number $m_{A3}=0$ is the only contributor to the third-order global area response. To understand the modal interactions, consider the expression for the third-order global area response

(3.14)$$\begin{eqnarray}\displaystyle A_{3}(t) & = & \displaystyle \int _{r}r\,\text{d}r\int _{\unicode[STIX]{x1D703}=0}^{2\unicode[STIX]{x03C0}}\left[\frac{\sin \unicode[STIX]{x1D713}(r)}{r^{2}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}-\frac{\cot \unicode[STIX]{x1D713}(r)\sin ^{3}\unicode[STIX]{x1D713}(r)}{2r^{2}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{1}}{\unicode[STIX]{x2202}r}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)^{2}\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{\cot \unicode[STIX]{x1D713}(r)\sin ^{5}\unicode[STIX]{x1D713}(r)}{2}\left(\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{1}}{\unicode[STIX]{x2202}r}\right)^{3}+\cot \unicode[STIX]{x1D713}(r)\sin ^{3}\unicode[STIX]{x1D713}(r)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{1}}{\unicode[STIX]{x2202}r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{2}}{\unicode[STIX]{x2202}r}+\cos \unicode[STIX]{x1D713}(r)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}_{3}}{\unicode[STIX]{x2202}r}\right]\text{d}\unicode[STIX]{x1D703}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

For illustrative purposes, consider the third term in the integrand in the above equation, as similar arguments apply to the other terms. Expanding the integrand in this term and using (3.3)

(3.15)$$\begin{eqnarray}\displaystyle & & \displaystyle \!\!\!\!2\unicode[STIX]{x1D709}_{1,r}^{3}(r,\unicode[STIX]{x1D703},t)\nonumber\\ \displaystyle & & \displaystyle \!\!\quad =\mathop{\sum }_{n_{1}=1}^{\infty }\mathop{\sum }_{n_{2}=1}^{\infty }\mathop{\sum }_{n_{3}=1}^{\infty }[\!\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n_{2}\unicode[STIX]{x1D714}_{0})(\cos ((n_{1}+n_{2})\unicode[STIX]{x1D714}_{0}t)+\cos ((n_{1}-n_{2})\unicode[STIX]{x1D714}_{0}t))\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},n_{2}\unicode[STIX]{x1D714}_{0})(\sin ((n_{1}+n_{2})\unicode[STIX]{x1D714}_{0}t)-\sin ((n_{1}-n_{2})\unicode[STIX]{x1D714}_{0}t))\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n_{2}\unicode[STIX]{x1D714}_{0})(\sin ((n_{1}+n_{2})\unicode[STIX]{x1D714}_{0}t)+\sin ((n_{1}-n_{2})\unicode[STIX]{x1D714}_{0}t))\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},n_{2}\unicode[STIX]{x1D714}_{0})(\cos ((n_{1}+n_{2})\unicode[STIX]{x1D714}_{0}t)-\cos ((n_{1}-n_{2})\unicode[STIX]{x1D714}_{0}t))\!]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,(\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n_{3}\unicode[STIX]{x1D714}_{0})\cos (n_{3}\unicode[STIX]{x1D714}_{0}t)+\hat{\unicode[STIX]{x1D709}}_{1,r}^{I}(r,\unicode[STIX]{x1D703},n_{3}\unicode[STIX]{x1D714}_{0})\sin (n_{3}\unicode[STIX]{x1D714}_{0}t)).\end{eqnarray}$$

From this, we can see that the different frequencies in the time-dependent terms are ($|\pm n_{1}\pm n_{2}\pm n_{3}|\unicode[STIX]{x1D714}_{0}$) corresponding to 4 different possibilities. Thus, any combination that corresponds to:

1. (i) $|\pm n_{1}\pm n_{2}\pm n_{3}|=0$ would result in a contribution to the time average at third order, i.e. Mechanism TA.

2. (ii) $|\pm n_{1}\pm n_{2}\pm n_{3}|=1$ would result in a contribution to the third-order dynamics at $\unicode[STIX]{x1D714}_{0}$, i.e. Mechanism NL.

Next, we focus on the helical modes in these terms. Considering the very first term (from the product in the summation) in (3.15), we have

(3.16)$$\begin{eqnarray}\displaystyle & & \displaystyle \hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n_{2}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D709}}_{1,r}^{R}(r,\unicode[STIX]{x1D703},n_{3}\unicode[STIX]{x1D714}_{0})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{m_{\unicode[STIX]{x1D709}1,1}}\mathop{\sum }_{m_{\unicode[STIX]{x1D709}1,2}}\mathop{\sum }_{m_{\unicode[STIX]{x1D709}1,3}} [\!\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{R}(r,n_{1}\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{R}(r,n_{2}\unicode[STIX]{x1D714}_{0})\cos (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\cos (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\cos (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\sin (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{R}(r,n\unicode[STIX]{x1D714}_{0})\sin (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\cos (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,1},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,2},r}^{I}(r,n\unicode[STIX]{x1D714}_{0})\sin (m_{\unicode[STIX]{x1D709}1,1}\unicode[STIX]{x1D703})\sin (m_{\unicode[STIX]{x1D709}1,2}\unicode[STIX]{x1D703})\!]\nonumber\\ \displaystyle & & \displaystyle \qquad \times \, (\!\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,3},r}^{R}(r,n_{3}\unicode[STIX]{x1D714}_{0})\cos (m_{\unicode[STIX]{x1D709}1,3}\unicode[STIX]{x1D703})-\hat{\unicode[STIX]{x1D701}}_{1,m_{\unicode[STIX]{x1D709}1,3},r}^{I}(r,n_{3}\unicode[STIX]{x1D714}_{0})\sin (m_{\unicode[STIX]{x1D709}1,3}\unicode[STIX]{x1D703})\!).\end{eqnarray}$$

Like the arguments for the frequencies, there are eight different helical modes possible, given by: ($\pm m_{\unicode[STIX]{x1D709}1,1}\pm m_{\unicode[STIX]{x1D709}1,2}\pm m_{\unicode[STIX]{x1D709}1,3}$). Thus, for the integral to be non-zero, we must have

(3.17)$$\begin{eqnarray}\pm m_{\unicode[STIX]{x1D709}1,1}\pm m_{\unicode[STIX]{x1D709}1,2}\pm m_{\unicode[STIX]{x1D709}1,3}=0.\end{eqnarray}$$

To summarize:

1. (a) The third-order contribution to the time average is possible only if there are at least two excitation frequencies. This differs from the response at second order, where even a single frequency contributed to the time average. The interaction between the harmonics must be such that $|\pm n_{1}\pm n_{2}\pm n_{3}|=0$. This is denoted by ‘Mechanism TA’ in § 1.

2. (b) The third-order contribution to the flame dynamics at the frequency $\unicode[STIX]{x1D714}_{0}$ is possible if there are three different harmonics such that $|\pm n_{1}\pm n_{2}\pm n_{3}|=1$ and they contain helical modes such that $\pm m_{\unicode[STIX]{x1D709}1,1}\pm m_{\unicode[STIX]{x1D709}1,2}\pm m_{\unicode[STIX]{x1D709}1,3}=0$. For the special case of a single-frequency excitation, $n_{2}=n_{3}=0$, the fundamental frequency in the source disturbances can still cause a third-order flame response when there exist different helical modes such that $\pm m_{\unicode[STIX]{x1D709}1,1}\pm m_{\unicode[STIX]{x1D709}1,2}\pm m_{\unicode[STIX]{x1D709}1,3}=0$. This is denoted by ‘Mechanism NL’ in § 1.

To further summarize, the flame response at the fundamental frequency occurs for the following:

1. (i) For a single-frequency case, the inter-modal interactions at second order lead to a non-axisymmetric time-average correction to the nominal flame shape. Helical modes interacting with a time-average flame shape with similar azimuthal distributions, $m_{\unicode[STIX]{x1D709}_{0}}=\pm m_{uj}$, result in a non-zero global area response at $\unicode[STIX]{x1D714}_{0}$. Thus, the nonlinear global area response for the single-frequency case occurs at third order with the second-order dynamics contributing to changes in the time-average flame shape. Single-frequency helical modes contribute to the global area dynamics first at third order.

2. (ii) For the multi-frequency case, second-order response consists of flame response azimuthal modes at two different frequencies interacting with each other resulting in a response at the sum and difference of the frequencies. This gives the possibility of flame response at the fundamental frequency at second order itself if the sum/difference in frequencies is equal to this fundamental frequency. If the interacting flame response azimuthal modes is of opposite sign, this results in a symmetric flame response at the fundamental frequency at second order itself, thus resulting in a second-order nonlinear global area response. Thus, for multi-frequency excitation, the global area response at the fundamental frequency happens through inter-modal, inter-frequency interactions at second order itself.

4.1 Single-frequency excitation

For the example calculations in this sub-section, the same model flow field and single-frequency excitation are used as in earlier work to facilitate comparisons (Acharya et al. Reference Acharya, Shin and Lieuwen2013)

(4.1)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle u_{z,0}=1\\ \displaystyle u_{r,0}=0\\ \displaystyle u_{\unicode[STIX]{x1D703},0}=\frac{\unicode[STIX]{x1D6FA}L_{f}}{u_{0}}r=St_{\unicode[STIX]{x1D6FA}}r\end{array}\right\}\;\Longrightarrow \;\left\{\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D709}_{0}(r)=(r-R)\cot \unicode[STIX]{x1D713}=\tilde{r}\\ \displaystyle s_{L}=\sin \unicode[STIX]{x1D713}\end{array}\right.\end{eqnarray}$$

with the following velocity fluctuations:

(4.2)$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle u_{r,1}=\text{Re}\left[\mathop{\sum }_{m_{uz}}\text{e}^{(-\text{i}Stt+\text{i}St_{c}z+\text{i}m_{uz}\unicode[STIX]{x1D703})}\right]\\ \displaystyle u_{z,1}=u_{\unicode[STIX]{x1D703},1}=0\end{array}\right\}\;\Longrightarrow \;\left\{\begin{array}{@{}l@{}}\displaystyle \hat{B}_{z,m_{ur}}=\text{e}^{(\text{i}St_{c}(r-R)\cot \unicode[STIX]{x1D713})}\\ \displaystyle \hat{B}_{r,m_{uz}}=\hat{B}_{\unicode[STIX]{x1D703},m_{u\unicode[STIX]{x1D703}}}=0.\end{array}\right.\end{eqnarray}$$

Here, $u_{\unicode[STIX]{x1D703},0}=St_{\unicode[STIX]{x1D6FA}}r$, the mean swirl flow is a rigid-body rotation flow field, $St=\unicode[STIX]{x1D714}_{0}L_{f}/u_{0}$ is the Strouhal number denoting the non-dimensional frequency and $St_{c}=\unicode[STIX]{x1D714}_{0}L_{f}/u_{c}$ is the non-dimensional convective wavenumber of the helical mode.

In order to facilitate the differences in helical mode interactions, the remainder of this section is broken into different sub-sections that shall each focus on different aspects of these interactions.

4.1.1 Interactions between helical modes

Consider the amplitude-dependent response for the flow disturbances containing purely non-axisymmetric helical modes, i.e. $m_{ur}\neq 0$. The flame response function (2.22) can be decomposed into its individual contributions as

(4.3)$$\begin{eqnarray}\hat{F}(\unicode[STIX]{x1D716},St)=\frac{\unicode[STIX]{x1D716}\hat{q}_{1}(St)+\unicode[STIX]{x1D716}^{3}\hat{q}_{3}(St)}{1+\unicode[STIX]{x1D716}^{2}\overline{q}_{2}(St)}.\end{eqnarray}$$

Here, $\hat{q}_{1,3}=\hat{Q}_{1,3}/Q_{0}$ are the normalized flame response at first and third order and $\overline{q}_{2}=\overline{Q}_{2}/Q_{0}$ is the normalized second-order contribution to the time average. Since this example involves purely helical modes, we have $\hat{q}_{1}\equiv 0$ as discussed earlier in § 3.1. We present results here on $|\hat{q}_{3}|$, where the flame angle is set to $\unicode[STIX]{x1D713}=\unicode[STIX]{x03C0}/4$ and the non-dimensional convective wavenumber is set to $St_{c}=0.5$ for the remainder of the example illustrations. As the $O(\unicode[STIX]{x1D716}^{2})$ term in the denominator is small relative to unity, $|\hat{q}_{3}|$ is representative of $|\hat{F}|$, while also allowing us to plot results independent of $\unicode[STIX]{x1D716}$.

Figure 7.

Comparison of flame response amplitude (cubic coefficient $|\hat{q}_{3}|$) variation with swirl ($St_{\unicode[STIX]{x1D6FA}}$) for $|m_{ur}|=1,2,3$ (a) and $|m_{ur}|=1,3,4$ (b) for different signs of the helical modes.

First, consider two example cases with different combinations of helical modes: with $|m_{ur}|=1,2,3$ (a), $|m_{ur}|=1,3,4$ (b) as shown in figure 7. Each of these cases satisfies the ‘Mechanism NL’ condition expressed earlier in (3.17). For this example, the amplitudes of all the modes are assumed to be equal, i.e. $\hat{B}_{r,\pm 1}=\hat{B}_{r,\pm 2}=\hat{B}_{r,\pm 3}$ and $\hat{B}_{r,\pm 2}=\hat{B}_{r,\pm 3}=\hat{B}_{r,\pm 5}$. The purpose of this example is to compare how the changes to the sign of the helical modes affect $|\hat{q}_{3}|$, and how the values of the mode numbers themselves impact the magnitude of $|\hat{q}_{3}|$. The figures show the variation of $|\hat{q}_{3}|$ with swirling strength $(St_{\unicode[STIX]{x1D6FA}})$ for the different helical mode combinations. Similar coloured curves correspond to the same combination of helical mode numbers but of opposite signs. In a nominally axisymmetric flame without swirl (i.e. $St_{\unicode[STIX]{x1D6FA}}=0$), the flame does not differentiate between sign of the velocity disturbance mode, $+m_{u}$ or $-m_{u}$, but only its magnitude, $|m_{u}|$. Thus, these similar coloured curves intersect when $St_{\unicode[STIX]{x1D6FA}}=0$. Since the nominal flame is axisymmetric, $St_{\unicode[STIX]{x1D6FA}}$ is a measure of the degree of symmetry breaking in the flame response. Physically, the swirling strength controls azimuthal transport of disturbances (at all orders) and thus the resulting interference of flame wrinkles. This results in the non-monotonic variation in the response amplitude as shown. Following the above point, the resulting amplitude of the flame response also depends on the sign of the helical modes for non-zero $St_{\unicode[STIX]{x1D6FA}}$. For example, the combination with all modes having positive sign (solid black curves), leads to the highest value of the coefficient $|\hat{q}_{3}|$. In addition, the value of the helical mode also has an effect. For example, the maximum amplitudes attained by the $|m_{ur}|=1,3,4$ cases are higher when compared to the $|m_{ur}|=1,2,3$ cases. This occurs because higher $m_{ur}$ values lead to larger azimuthal derivatives, as can be seen from the nonlinear kinematic restoration terms in (2.8).

In the above example, we used the same amplitude for all helical mode numbers. In real flows, the relative amplitudes of higher helical mode numbers is observed to be significantly lower than the lower mode numbers (see figure 1). This motivates the next example in this sub-section, which is the effect of the relative amplitudes of the different helical modes. In order to illustrate the relative amplitude effect, consider the velocity field with three modes ($m_{1},m_{2},m_{3}$) where mode $m_{3}$ is the mode with a fixed amplitude and either one or both of the other modes have varying amplitudes

(4.4)$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \hat{u} _{r}(r,\unicode[STIX]{x1D703},St)=\frac{\text{e}^{(\text{i}0.5(r-R))}}{(1+2\unicode[STIX]{x1D716}_{hm})}(\unicode[STIX]{x1D716}_{hm}(\text{e}^{\text{i}m_{1}\unicode[STIX]{x1D703}}+\text{e}^{\text{i}m_{2}\unicode[STIX]{x1D703}})+\text{e}^{\text{i}m_{3}\unicode[STIX]{x1D703}}),\\ \displaystyle \hat{u} _{r}(r,\unicode[STIX]{x1D703},St)=\frac{\text{e}^{(\text{i}0.5(r-R))}}{(2+\unicode[STIX]{x1D716}_{hm})}(\unicode[STIX]{x1D716}_{hm}\text{e}^{\text{i}m_{1}\unicode[STIX]{x1D703}}+\text{e}^{\text{i}m_{2}\unicode[STIX]{x1D703}}+\text{e}^{\text{i}m_{3}\unicode[STIX]{x1D703}}),\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Figure 8.

Effect of relative amplitude $\unicode[STIX]{x1D716}_{hm}$ on the flame response amplitude for different interacting helical modes in a (a) non-swirling and (b) swirling flow. Legend indicates combination of helical modes considered with the mode(s) whose amplitude is varied indicated within the curved braces.

Here, the relative amplitude of mode $m_{1}$ (and/or $m_{2}$) with respect to the other modes is denoted by $\unicode[STIX]{x1D716}_{hm}$ and the scaling factor $(1+2\unicode[STIX]{x1D716}_{hm})$ or $(2+\unicode[STIX]{x1D716}_{hm})$ is to ensure that the maximum amplitude of the expression is 1 and the amplitude of the disturbance is $\unicode[STIX]{x1D716}$ as defined in (1.1). Figure 8 shows the variation of $|\hat{F}|=|\hat{q}_{3}|$ for a non-swirling (a) and a swirling case (b) for different helical mode combinations. For this example, we fix a reference mode $m_{3}=+1$ and consider the amplitude variations for the other modes. First consider the non-swirling case on the left. The dashed curves correspond to amplitude variations for different combinations of $|m_{ur}|=2,3$. For these cases, when $\unicode[STIX]{x1D716}_{hm}=0$, only mode $m_{3}=+1$ exists and thus the curves start from $|\hat{q}_{3}|=0$ since (3.17) is not satisfied. As $\unicode[STIX]{x1D716}_{hm}$ is increased, equation (3.17) is now satisfied and thus $|\hat{q}_{3}|$ increases to non-zero values. In contrast, for the case where the amplitude of mode $m_{ur}=+3$ is varied, the curve starts at a non-zero value since the remainder of the modes $m_{ur}=+1,+2$ satisfy (3.17). Note that, depending on the sign of the interacting modes, the quantitative variations are different indicating that the sign of the helical modes are important for the nonlinear global flame response. These differences stem from the azimuthal derivatives in the nonlinear kinematic restoration terms in (2.8). Note that, qualitatively, all curves increase to a local maximum before decreasing further, which can be attributed to changes in interference of different nonlinear terms as $\unicode[STIX]{x1D716}_{hm}$ is varied. A key takeaway from this example is that while $m_{ur}=+1$ is typically present as a mode in the unsteady flow, the introduction of the requisite higher helical modes, even at much smaller magnitudes, can lead to appreciable heat release rate oscillations through ‘Mechanism NL’. This is evident, for example, from the dashed black curve where even at $\unicode[STIX]{x1D716}_{hm}=0.25$, the flame response amplitude is comparable to that when all modes have equal amplitudes. In addition to the above aspects of the helical modes, swirl plays an important role in the nonlinear interactions through the bias that is introduced between positive and negative modes. Having considered purely helical mode interactions, we next consider the relative roles of the helical modes in the presence of the symmetric mode.

4.1.2 Superposition effects of $m_{ur}=0$ and helical modes $m_{ur}\neq 0$

As noted earlier, the linear response of the flame is controlled by the $m_{ur}=0$ mode, but in many cases, helical modes have larger amplitudes. This section considers the effect of superposition of axisymmetric and non-axisymmetric velocity disturbances on the flame response. We characterize their relative strengths with the coefficient $\unicode[STIX]{x1D716}_{hm}$ as follows:

(4.5)$$\begin{eqnarray}u_{r,1}(r,\unicode[STIX]{x1D703},t)=\text{Re}\left[\frac{\text{e}^{(-\text{i}Stt+\text{i}St_{c}(r-R))}}{1+\unicode[STIX]{x1D716}_{hm}}\left(1+\unicode[STIX]{x1D716}_{hm}\frac{\displaystyle \mathop{\sum }_{m_{ur}\neq 0}\text{e}^{\text{i}m_{ur}\unicode[STIX]{x1D703}+\text{i}\unicode[STIX]{x1D719}_{m_{ur}}}}{\text{max}\left(\displaystyle \mathop{\sum }_{m_{ur}\neq 0}\text{e}^{\text{i}m_{ur}\unicode[STIX]{x1D703}+\text{i}\unicode[STIX]{x1D719}_{m_{ur}}}\right)}\right)\right].\end{eqnarray}$$

Here, $\unicode[STIX]{x1D719}_{m_{ur}}$ is the phasing for helical mode $m_{ur}$ with respect to the axisymmetric mode $m_{ur}=0$. The factor $(1+\unicode[STIX]{x1D716}_{hm})$ is a normalization factor such that the overall maximum amplitude of this expression is 1, with the overall forcing amplitude set to $\unicode[STIX]{x1D716}$, as defined in the perturbation expansion in (1.1). For this example, we do not consider phase effects ($\unicode[STIX]{x1D719}_{m_{ur}}=0$) and we consider the helical modes, $m_{ur}=0,-1,+2$. This choice of the helical mode number magnitudes (i.e. $|m_{ur}|=1,2$) is based upon dominant modes typically observed in high Reynolds number swirl flows (see figure 1). Figure 9 shows the effect of $\unicode[STIX]{x1D716}_{hm}$ on the flame response for different frequencies and swirling strengths. The pure axisymmetric only case, $\unicode[STIX]{x1D716}_{hm}=0$, is used as a reference and is denoted by the black curves. Note the non-zero linear response for all cases. As mentioned earlier, the linear flame response depends only on the $m_{ur}=0$ mode (i.e. the slope from small $\unicode[STIX]{x1D716}$). However, note that the slopes for the different cases (red, blue, black curves) are different because of the amplitude scaling approach, see (4.5). Here, the amplitude of the $m_{ur}=0$ mode is scaled by $(1+\unicode[STIX]{x1D716}_{hm})$ and then for increasing $\unicode[STIX]{x1D716}_{hm}$, the slope decreases as shown. An important qualitative feature seen in the curves is that the flame response initially increases and then reaches a local maximum before decreasing to a local minimum and then continuing to monotonically increase. This non-monotonic variation has been observed in several experiments of both bluff-body and swirling flames excited by high amplitude disturbances (Thumuluru & Lieuwen Reference Thumuluru and Lieuwen2009; Bellows Reference Bellows2006; Balachandran et al. Reference Balachandran, Ayoola, Kaminski, Dowling and Mastorakos2005; Kim & Hochgreb Reference Kim and Hochgreb2011; Schimek et al. Reference Schimek, Ćosić, Moeck, Terhaar and Paschereit2015). For example, this behaviour can be seen in the data reproduced in figure 10. This behaviour occurs at this condition because of the nearly out of phase relationship between the third order, $\hat{q}_{3}$, and first-order flame response, $\hat{q}_{1}$ contributions. The $\hat{q}_{3}$ term is negligible relative to $\hat{q}_{1}$ at low $\unicode[STIX]{x1D716}$, but grows in importance as $\unicode[STIX]{x1D716}$ increases thus resulting in destructive interference effects and decreases in flame response. While the choice of helical mode magnitudes in this example (i.e. $|m_{ur}|=1,2$) is merely for illustration, note that such non-monotonic behaviour can occur with a range of helical mode interactions. An important takeaway from this example is that while the axisymmetric mode controls the leading-order flame response, the inclusion of the non-axisymmetric modes can significantly alter the flame response for higher amplitudes.

Figure 9.

Relative role of asymmetric helical modes: effect of asymmetric helical mode relative amplitude $\unicode[STIX]{x1D716}_{m}$ on the flame response amplitude.

Figure 10.

Effect of velocity forcing amplitude on the flame response amplitude of a swirling flame. Reproduced from Thumuluru & Lieuwen (Reference Thumuluru and Lieuwen2009).

4.1.3 Effect of relative phasing between the helical modes and the symmetric mode

In the prior examples, the different helical modes were all in phase with each other. The focus of this example is to consider the effect of introducing a phasing between the different modes since this affects the destructive/constructive interference effects in the nonlinear interaction mechanisms. For this example, we use (4.5) with $\unicode[STIX]{x1D716}_{hm}=0.5$, $St_{c}=0.5$ and $\unicode[STIX]{x1D719}_{m_{ur}}\neq 0$ to illustrate phase effects. Specifically, we consider only $m_{ur}=-1,+2$ and assume $\unicode[STIX]{x1D719}_{(m_{ur}=-1)}=0$ to consider the effects of varying $\unicode[STIX]{x1D719}_{(m_{ur}=+2)}=\unicode[STIX]{x1D719}_{hm}$. The variation in flame response is considered for $0\leqslant \unicode[STIX]{x1D719}_{hm}<2\unicode[STIX]{x03C0}$.

Figure 11.

Effect of relative phasing $\unicode[STIX]{x1D719}_{hm}$ between the helical modes on the nonlinear flame response amplitude.

Figure 11 shows the surface plots for the variation of the flame response amplitude with respect to forcing amplitude $\unicode[STIX]{x1D716}$ for different values of $\unicode[STIX]{x1D719}_{hm}$. For the non-swirling cases $(St_{\unicode[STIX]{x1D6FA}}=0)$, phasing exerts minimal influence. However, for increasing swirl, the effect of phasing becomes important, due to the effect of azimuthal interference effects introduced by azimuthal convection of flame wrinkles. Note that the qualitative nature of the curves also changes with swirl and Strouhal number, as can be seen when comparing the cases for $St=0.7,St_{\unicode[STIX]{x1D6FA}}=3$ and $St=0.7,St_{\unicode[STIX]{x1D6FA}}=6$. Also, as the phasing varies, the maxima/minima seen in the curves greatly shift, indicating the overall effect of the constructive/destructive interference of flame response due to the differing phases between the helical modes. In summary, phasing has both a qualitative and quantitative effect on the flame response and is important to consider as swirling flow hydrodynamics would result in different helical modes having differing phases, often with a spatial dependence.

4.2 Multi-frequency excitation

Next, consider multi-frequency content in the disturbances to illustrate the effects of nonlinear inter-frequency interactions. Consider the case where there are two frequencies in the velocity disturbances: $St$ and $2St$. We consider two forms of the velocity disturbances. First, where all modes have both frequencies

(4.6)$$\begin{eqnarray}u_{r,1}=\text{Re}\left[\frac{\text{e}^{-\text{i}Stt}+\text{e}^{-2\text{i}Stt}}{2}\left(1+\frac{\displaystyle \mathop{\sum }_{m_{ur}\neq 0}\text{e}^{\text{i}m_{ur}\unicode[STIX]{x1D703}}}{\displaystyle \text{max}\left(\mathop{\sum }_{m_{ur}\neq 0}\text{e}^{\text{i}m_{ur}\unicode[STIX]{x1D703}}\right)}\right)\right].\end{eqnarray}$$

Second, where only the asymmetric modes ($m_{ur}\neq 0$) have both frequencies

(4.7)$$\begin{eqnarray}u_{r,1}=\text{Re}\left[\frac{\text{e}^{-\text{i}Stt}}{2}+\frac{\text{e}^{-\text{i}Stt}+\text{e}^{-2\text{i}Stt}}{4}\left(\frac{\displaystyle \mathop{\sum }_{m_{ur}\neq 0}\text{e}^{\text{i}m_{ur}\unicode[STIX]{x1D703}}}{\displaystyle \text{max}\left(\mathop{\sum }_{m_{ur}\neq 0}\text{e}^{\text{i}m_{ur}\unicode[STIX]{x1D703}}\right)}\right)\right].\end{eqnarray}$$

In both expressions, the normalization factors are such that the maximum amplitude is 1. The velocity amplitude for forcing is $\unicode[STIX]{x1D716}$.

Figure 12.

Effect of multi-frequency content in velocity disturbances, on the nonlinear flame response for helical mode combination $m_{ur}=0,-1,+2$. Black: single-frequency (baseline), red: multi-frequency for all modes, blue: multi-frequency only for $m_{ur}=-1,+2$.

Figure 12 shows the variation in flame response amplitude ($|\hat{F}|$) with velocity amplitude ($\unicode[STIX]{x1D716}$) for a radial velocity forcing with modal content $m_{ur}=0,-1,+2$. The baseline single-frequency case is denoted by the black curves. The plots show variations for two different frequencies: $St=0.5,0.7$ and three different swirling strengths: $St_{\unicode[STIX]{x1D6FA}}=0,3,6$. The velocity disturbances for both frequencies contain the fundamental and first harmonic, i.e. $St=0.5,1.0$ and $St=0.7,1.4$. Note that the presence of the second frequency alters the flame response noticeably. Specifically, in the case of $St=0.7$, the baseline case shows qualitative changes in the variation of the flame response with forcing amplitude. The local maxima and minima in the single-frequency case are absent when all the modes have multiple frequencies. Similarly, the important effect of non-uniform modal distribution across the two frequencies can be seen when comparing the blue and red curves. This is important since multiple modes and frequencies are present under large amplitude excitation conditions, as harmonics of the fundamental forcing are excited by nonlinear effects.