5.1.1. Linear perturbation

Stationary linear and adjoint perturbation fields were excited by a Gaussian shaped surface roughness

(5.1)\begin{equation} h(r)=\dfrac{1}{\sqrt{2{\rm \pi} c^2}}\exp\left(\dfrac{-(r-r_f)^2}{2c^2}\right),\end{equation}

where $r_f$ denotes the radial location of the roughness and $c^2$ the variance. (In the subsequent discussion $c=2$.) The linear disturbance was established by fixing the roughness about $r_f=200$. This particular radial location was chosen as it is sufficiently far upstream of the onset of the stationary cross-flow instability, which first becomes linearly unstable about $r_c\approx 323$ for $n=32$. The roughness excites a stationary cross-flow vortex that develops radially outwards. After sufficient time has passed, the disturbance has convected across the entire radial domain and is unchanged by further increments in time. (Disturbance development was simulated for both shorter and longer radial domains and the results were unchanged.)

Figure 2 displays several characteristics associated with the linear disturbance established above. The maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|$, is plotted in figure 2(a) as a function of $r$, while figure 2(b) depicts the corresponding contours of $u_{\theta }$ scaled on $\max _z|u_{\theta }|$ in the $(r,z)$-plane. The scaling was implemented here to assist visualisation of the disturbance development, which would otherwise be impossible due to the rapid spatial growth associated with this form of instability. Red and blue colours in figure 2(b) depict positive and negative valued contour levels, respectively. The radial location of the roughness in figures 2(a) and 2(b) is made evident by the respective small peak and red contours located about the radius $r=200$. Immediately downstream of the roughness the amplitude of the disturbance decreases. This behaviour is to be expected, as this particular cross-flow instability does not become linearly unstable until the critical radius $r_c\approx 323$. However, radially downstream of $r_c\approx 323$ the disturbance grows rapidly, and the cross-flow instability assumes the typical structure and develops near the disc surface.

Figure 2.

Characteristics of the linear azimuthal velocity perturbation field $u_{\theta }$ excited about $r_f=200$, for an azimuthal mode number $n=32$. (a) Plot of $\max _z|u_{\theta }|$ as a function of $r$. (b) Contours of $u_{\theta }/\max _z|u_{\theta }|$ in the $(r,z)$-plane. (c) Radial wavenumber $\alpha _r$. (d) Radial growth rate $\alpha _i$. The dot markers in (c,d) indicate those results obtained for fixed Reynolds numbers ${\textit {Re}}$ derived by applying the homogeneous flow approximation.

The spatial development of the linear perturbation $\hat {\boldsymbol {q}}$ (recall (3.9)) may be decomposed as

(5.2a)\begin{equation} \hat{\boldsymbol{q}}(r,z)= \check{\boldsymbol{q}}(r,z)\exp\left(\textrm{i} \int_{r_c}^{r}\alpha (r') \,\mathrm{d} r'\right), \end{equation}

where $\check {\boldsymbol {q}}$ is a slowly varying shape function and $\alpha =\alpha _r+\textrm {i}\alpha _i$ represents the complex-valued radial wavenumber. The real and imaginary parts of $\alpha$ may be interpreted as the respective radial wavenumber and growth rate of the linear disturbance, and may be computed by considering the formula

(5.2b)\begin{equation} \alpha ={-}\dfrac{\textrm{i}}{f}\dfrac{\partial f}{\partial r}, \end{equation}

where $f$ is taken to be a measure of the disturbance amplitude at any given radial and wall-normal location. As before, the azimuthal velocity perturbation field $u_{\theta }$ was selected, with computations based on measurements about the wall-normal height $z=1.5$. (Note that the calculations for $\alpha$ were not noticeably affected by shifting the wall-normal $z$-location.)

The radial wavenumber $\alpha _r$ and growth rate $\alpha _i$, associated with the above disturbance, are plotted in figures 2(c) and 2(d) and indicate that there is an initial transient phase before linearly unstable behaviour emerges about the critical radius $r_c\approx 323$. (Note that a negative valued $\alpha _i$ corresponds to radial growth and unstable behaviour.) The black dot markers in figures 2(c) and 2(d) depict the equivalent results obtained by solving the linear system of governing equations (4.2) with the homogeneous flow approximation, where the radial dependence of the undisturbed flow (2.6) is removed by fixing $r={\textit {Re}}$. To the accuracy of the illustration, the homogeneous flow computations about fixed radii are identical to the inhomogeneous flow solution, with results tabulated in table 1.

5.1.2. Adjoint perturbation

The corresponding adjoint perturbation field was generated by fixing the Gaussian shaped roughness (5.1) about the radius $r_f=650$. Adjoint disturbances evolve in the spatially opposite sense to their linear counterparts and propagate upstream towards the inner radius. The dashed red line in figure 3(a) depicts the maximum absolute value of the adjoint perturbation field, $\max _z|\tilde {u}_{\theta }|$, while figure 3(b) displays the contours of $\tilde {u}_{\theta }$ scaled on $\max _z|\tilde {u}_{\theta }|$. The adjoint disturbance grows radially inwards, attaining a peak amplitude near the radial location that the linear cross-flow instability first becomes unstable. Further upstream, the magnitude of $\max _z|\tilde {u}_{\theta }|$ decreases, which corresponds to a decay in the adjoint perturbation field.

Figure 3.

Characteristics of the adjoint azimuthal velocity perturbation field $\tilde {u}_{\theta }$ excited about $r_f=650$, for an azimuthal mode number $n=32$. (a) Plot of $\max _z|\tilde {u}_{\theta }|$ as a function of $r$. (b) Contours of $\tilde {u}_{\theta }/\max _z|\tilde {u}_{\theta }|$ in the $(r,z)$-plane. (c) Radial wavenumber $\tilde {\alpha }_r$. (d) Radial growth rate $\tilde {\alpha }_i$. The dot markers in (c,d) indicate those results obtained for fixed Reynolds numbers ${\textit {Re}}$ derived by applying the homogeneous flow approximation.

Analogous to the linear perturbation, the spatial development of the adjoint perturbation $\tilde {\hat {\boldsymbol {q}}}$ (recall (3.10)) may be decomposed as

(5.3a)\begin{equation} \tilde{\hat{\boldsymbol{q}}}(r,z)= \tilde{\check{\boldsymbol{q}}}(r,z)\exp\left(-\textrm{i}\int_{\tilde{r}_c}^{r}\tilde{\alpha} (r') \,\mathrm{d} r'\right), \end{equation}

where $\tilde {\check {\boldsymbol {q}}}$ is a slowly varying shape function and $\tilde {\alpha }=\tilde {\alpha }_r+\textrm {i}\tilde {\alpha }_i$ represents the complex-valued adjoint radial wavenumber. (Note a change in sign has been imposed on the adjoint radial wavenumber as the adjoint disturbance grows radially inwards.) The adjoint radial wavenumber $\tilde {\alpha }$ is determined via the formula

(5.3b)\begin{equation} \tilde{\alpha} = \dfrac{\textrm{i}}{f}\dfrac{\partial f}{\partial r}, \end{equation}

with $f$ taken as the adjoint azimuthal velocity perturbation field $\tilde {u}_{\theta }$, measured about the wall-normal height $z=1.5$.

Figures 3(c) and 3(d) depict the real and imaginary parts of $\tilde {\alpha }$, with the adjoint growth rate $\tilde {\alpha }_i=0$ about the radius $\tilde {r}_c\approx 318$. (A negative valued $\tilde {\alpha }_i$ is matched to an adjoint perturbation growing radially inwards, while $\tilde {r}_c \neq r_c$ as a consequence of marginal differences between the linear and adjoint radial growth rates.) Black dot markers correspond to those solutions obtained by solving the equivalent adjoint system of governing equations (4.9) but with the application of the homogeneous flow approximation, i.e. the radius $r$ is held fixed by setting it equal to the Reynolds number ${\textit {Re}}$. Similar to the analysis of the linear perturbation, homogeneous flow calculations about fixed radii (tabulated in table 1) are indistinguishable from the inhomogeneous flow solutions. Moreover, the real parts of the linear and adjoint radial wavenumbers are identical: $\alpha _r=\tilde {\alpha }_r$. On the other hand, there are marginal differences in the corresponding imaginary parts: $\alpha _i\neq \tilde {\alpha }_i$. However, these small differences are counterbalanced by the normalisation (3.13).

5.1.3. Normalisation

Linear and adjoint disturbances were then normalised using (3.13), with the corresponding azimuthal velocity perturbation fields, $\max _z|u_{\theta }|$ and $\max _z|\tilde {u}_{\theta }|$, plotted in figure 4(a). The line types and axes scales are the same as that given in figures 2(a) and 3(a), to help distinguish differences brought about by the normalisation. The linear solution is equal to unity about the radial location $r_c$ for the onset of the stationary cross-flow instability, which is a requirement of the normalisation (3.13). On the other hand, the adjoint solution is scaled downwards and develops at a significantly smaller amplitude than before the normalisation.

Figure 4.

(a) Maximum absolute value of the linear and adjoint azimuthal velocity perturbation fields, $\max _z|u_{\theta }|$ and $\max _z|\tilde {u}_{\theta }|$, normalised using (3.13). (b) Corresponding normalised absolute value of the adjoint vorticity perturbation fields, $|\tilde {\omega }_{r,w}|$ and $|\tilde {\omega }_{\theta ,w}|$. Solutions correspond to those parameter settings given in figures 2 and 3.

Figure 4(b) displays the normalised absolute value of the adjoint vorticity perturbation fields $|\tilde {\omega }_{r,w}|$ (solid blue line) and $|\tilde {\omega }_{\theta ,w}|$ (dashed red line) that are required to compute the receptivity amplitude $A$ of the stationary cross-flow instability (recall (4.13)). The adjoint azimuthal vorticity perturbation field is almost an order of magnitude greater than the equivalent radial component. This would suggest that the azimuthal component of vorticity will have a greater influence on the receptivity of the rotating disc boundary layer to surface roughness. Furthermore, both adjoint perturbation fields achieve larger amplitudes about the radial location $r_c$ associated with the onset of the linear disturbance. Thus, we might expect receptivity to be enhanced by roughness elements located within this particular radial region.

The above process was repeated for azimuthal mode numbers $n\in [25,40]$ at unit step intervals. The decision to focus on this particular range of modes was made as a consequence of experimental investigations that typically observe cross-flow disturbances within this parameter range. Figure 5 depicts contours of $|\tilde {\omega }_{r,w}|$ and $|\tilde {\omega }_{\theta ,w}|$ in the $(r,n)$-plane. For azimuthal mode numbers $n\leqslant 32$, the two adjoint vorticity components attain a peak over the radial range $250\leqslant r\leqslant 350$. Beyond this annular region, both perturbation fields are significantly reduced. As $n$ increases, the radial location that the two adjoint vorticity terms achieve a maxima is shifted radially downstream. This particular observation coincides with a corresponding radial shift downstream in the onset of the stationary cross-flow instability that occurs as $n$ increases to larger values. Hence, for each azimuthal mode number $n$, receptivity will be most affected by surface roughness in the region that the stationary cross-flow instability first appears. This behaviour is consistent with that found in the flow over a swept wing. Radeztsky et al. (Reference Radeztsky, Reibert and Saric1999) undertook an experimental study on the flow over a swept wing with isolated roughness elements. Roughness centred about the location for neutral instability established the strongest receptivity, while roughness elements located further downstream had a negligible effect.

Figure 5.

Contours in the $(r,n)$-plane of (a) $|\tilde {\omega }_{r,w}|$; (b) $|\tilde {\omega }_{\theta ,w}|$.

5.3.1. Model

Natural surface roughness is random and will be composed of several peaks and troughs of variable height, depth and wavelength. Mughal & Ashworth (Reference Mughal and Ashworth2013) undertook a receptivity investigation of stationary cross-flow vortices on a swept wing, whereby three-dimensional roughness distributions were modelled using a Fourier cosine series. A simplification was made in the subsequent adjoint based study by Thomas et al. (Reference Thomas, Mughal and Ashworth2017), who assumed the roughness distribution was periodic along the spanwise direction. A similar simplification was implemented here, where due to the natural periodicity of the disc, surface roughness was assumed to be periodic in the azimuthal direction. Two-dimensional roughness distributions were modelled as

(5.4)\begin{equation} h(r)=\sum_{j=N_{min}}^{N_{max}}a_j\cos\left(2{\rm \pi} \left(\frac{jr}{r_{\lambda}}+\phi_j\right)\right),\end{equation}

where $N_{min}$ and $N_{max}$ specify long and short scale wavelength filters, and $r_{\lambda }$ denotes a radial wavelength. Parameters $a_j$ and $\phi _j$ represent randomly generated roughness amplitudes and phase shifts, respectively. Roughness distributions were then implemented on the radial interval $r_a \leqslant r \leqslant r_b$, where $r_a$ and $r_b$ specify the upstream and downstream boundaries of the surface roughness, respectively. Figure 7 displays a diagram of the roughness arrangement, including labels for the radial positions $r_o$, $r_a$ and $r_b$. (Note that in the subsequent analysis, the downstream boundary of the roughness distribution $r_b\leqslant r_{\lambda }+r_a$.)

Figure 7.

Diagram of the roughness model.

Each randomly generated amplitude $a_j$ was then scaled by the factor

(5.5)\begin{equation} \left(\dfrac{r_{\lambda}}{2{\rm \pi} j}\right)^{k} \end{equation}

for a power law coefficient $k$. An identical modelling approach was implemented by Raposo et al. (Reference Raposo, Mughal and Ashworth2018) in their acoustic receptivity study of TS waves in the presence of surface roughness. This particular model for roughness was utilised here as it is well known that the power spectral density (PSD) for natural surface roughness follows a power law of the form (5.5) (Van Deusen Reference Van Deusen1967; Sayles & Thomas Reference Sayles and Thomas1978; Jacobs, Junge & Pastewka Reference Jacobs, Junge and Pastewka2017). For instance, it was shown by Mughal & Ashworth (Reference Mughal and Ashworth2013) that the energy in the PSD of a painted panel or unpainted aluminium plate was primarily found in the longer wavelengths. Furthermore, Van Deusen (Reference Van Deusen1967) noted that the PSD decreased with the square of the angular frequency for many natural and artificial surfaces, while Raposo et al. (Reference Raposo, Mughal and Ashworth2018) set the coefficient $k=1$ in their receptivity investigation to ensure the PSD decayed at the rate found by Van Deusen.

Finally, all roughness distributions were scaled on the r.m.s., which is defined as

(5.6)\begin{equation} h_{{rms}}=\sqrt{\dfrac{1}{r_b-r_a}\int_{r_a}^{r_b}h^2 \,\mathrm{d} r}. \end{equation}

For instance, setting $h_{{rms}}=10^{-3}$ corresponds to a surface roughness with a r.m.s. equal to $0.1\,\%$ of the boundary layer thickness $\delta$.

Figure 8 depicts several randomly generated roughness distributions (solid blue lines), along with their respective gradients $\textrm {d}h/\textrm {d}r$ (dashed red) in the instance the roughness parameter $h_{{rms}}=1$. The long and short scale wavelength filters are set to the respective values $N_{min}=1$ and $N_{max}=100$, while each roughness is fixed about the radial interval $200\leqslant r\leqslant 600$ and $r_{\lambda }=400$. The power law coefficient $k$ increases from $k=0.5$ in figure 8(a) to $k=1.5$ in figure 8(c), at $0.5$ step intervals. The illustration clearly shows that each roughness realisation is distinct and encompasses a variety of wavelengths. Furthermore, the effect of the power law coefficient $k$ on the form of the roughness is demonstrated. For small $k$, shorter wavelengths are significant that establish many peaks and troughs with relatively large gradients. However, as the coefficient $k$ increases, shorter wavelengths have a limited influence on the structure of the roughness, resulting in fewer sharp variations and significantly smaller gradients.

Figure 8.

Examples of surface roughness for $N_{{min}}=1$, $N_{{max}}=100$, $h_{{rms}}=1$ and (a $_{\star }$) $k=0.5$; (b $_{\star }$) $k=1$; (c $_{\star }$) $k=1.5$. The solid blue and dashed red lines depict roughness distributions $h$ and their corresponding gradients $\textrm {d}h/\textrm {d}r$.

Averaging over many thousands of roughness realisations, it was determined that the mean of the absolute value of the surface gradient, $|\textrm {d}h/\textrm {d}r|$, was approximately $0.4$ when $k=0.5$, but reduces as $k$ increases; $|\textrm {d}h/\textrm {d}r|\approx 0.1$ and $0.05$ for $k=1$ and $k=1.5$, respectively. These trends are then scaled linearly with the r.m.s. of the roughness. In the subsequent analysis we assume that any significant spikes or troughs in the surface roughness are not sufficient to bring about boundary layer separation or the nonlinear stages of laminar–turbulent transition.

An additional roughness realisation is plotted in figure 9(a) that is established using those parameter settings given in figure 8(b). The response of the azimuthal velocity perturbation field $u_{\theta }$, to this particular roughness, is then shown in figure 9(b,c). The maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|$, determined via the LNS formulation (4.2) is given by the solid blue line in figure 9(b), while the equivalent adjoint prediction is given by the dashed red line. (Recall that the adjoint formula (4.13) only requires the normalised adjoint vorticity fields $\omega _{r,w}$ and $\omega _{\theta ,w}$, and the roughness shape $h$, as inputs.) The amplitude $A$ of the disturbance established by the given roughness was determined using the adjoint formula (4.13) as $A=0.29$. As with the simpler Gaussian shaped roughness distributions, the LNS solution and adjoint prediction are indistinguishable. Thus, the adjoint approach has again correctly determined the amplitude $A$ of the disturbance.

Figure 9.

(a) Randomly generated roughness distribution $h$ and gradient $\textrm {d}h/\textrm {d}r$. (b) Maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|$. The LNS solution is given by the solid blue line, while the dashed red line depicts the adjoint prediction. (c) Contours of $u_{\theta }$, scaled on $\max _z|u_{\theta }|$, in the $(r,z)$-plane.

The depiction of $\max _z|u_{\theta }|$ in figure 9(b) and the contours of $u_{\theta }$ in figure 9(c) indicate that the disturbance achieves magnitudes greater than the height of the roughness about the radius $r=400$. Beyond this radial location the disturbance grows rapidly, and the response of the linear perturbation is not affected by roughness elements located further outboard. Hence, receptivity is greatest in the radial region that the stationary cross-flow instability first becomes linearly unstable.

5.3.2. Mean receptivity

For the remainder of this investigation, the long wavelength filter $N_{min}=1$ and the wavelength parameter $r_{\lambda }=400$. Additionally, the upstream boundary of the roughness distribution was fixed about $r_a=200$, which is sufficiently far upstream of the radial region that receptivity is expected to be the most significant. The downstream boundary $r_b$ of the roughness distribution and the short wavelength filter $N_{max}$ were then varied to determine the effects, if any, on the receptivity process. Additionally, several values of the power law coefficient $k$ were modelled.

There are two major objectives of the following analysis.

1. (i) Determine those roughness characteristics that are most significant in the receptivity process, and establish the largest disturbance amplitudes $A$.

2. (ii) For a disc with a known roughness r.m.s., compute the mean disturbance amplitudes and determine the radial region that disturbances achieve sufficiently large magnitudes to invalidate the linear stability assumptions. Moreover, these calculations will be used as initial inputs in a future fully nonlinear stability study.

A Monte-Carlo uncertainty quantification analysis was undertaken, whereby several thousand roughness distributions were randomly generated and receptivity amplitudes $A$ computed. Mean amplitudes $\bar {A}$ were then determined via the formula

(5.7)\begin{equation} \bar{A}=\sum_s A/s, \end{equation}

for $s$ roughness realisations. (Calculations were again based on the absolute value of $A$.) Figure 10(a) displays mean amplitudes $\bar {A}$ as a function of the number of roughness realisations $s$, for four downstream boundaries $r_b$. The azimuthal mode number $n=32$ and the low wavelength filter $N_{max}=100$, while the roughness parameter $h_{{rms}}$ and power law coefficient $k$ were both set to unity. A log scaling is used along the horizontal $s$-axis to illustrate the variation in $\bar {A}$ as $s$ increases. In all instances modelled, the size of $\bar {A}$ converges towards a fixed constant after approximately $1000$ roughness iterations. Further iterations were not found to bring about any significant changes in the mean amplitude. Although a different mean amplitude was realised for each $r_b$ considered, results indicate that variations in $\bar {A}$ diminish as $r_b$ increases. Thus, surface roughness located sufficiently far outboard has a negligible effect on the mean amplitude of the stationary cross-flow instability.

Figure 10.

(a) Mean amplitude $\bar {A}$ against the number of roughness realisations $s$. (b) Probability density functions. The azimuthal mode number $n=32$ and roughness parameters $N_{{max}}=100$, $h_{{rms}}=1$, $k=1$ and variable $r_b$.

Probability density functions (p.d.f.s) are plotted in figure 10(b), which were generated using the routines available in MATLAB. Each p.d.f. was constructed using $s=10^4$ roughness realisations and the absolute value of the receptivity amplitude $A$. Solutions indicate that a greater variance in the amplitude $A$ is achieved as the length of the roughness distribution increases. Moreover, the illustration shows that receptivity can vary significantly from one realisation to the next. It is hypothesised that the larger amplitudes are associated with roughness distributions that encompass significant isolated peaks (and troughs) in the radial region that receptivity is most enhanced, i.e. where the cross-flow vortex first becomes linearly unstable.

Mean amplitudes $\bar {A}$ are plotted as a function of the low wavelength filter $N_{max}$ (for $r_b=600$) in figure 11(a) and the downstream boundary $r_b$ of the roughness (for $N_{max}=100$) in figure 11(b). Solutions again correspond to the stationary cross-flow instability established for the azimuthal mode number $n=32$, while roughness parameters $h_{{rms}}=1$ and $k\in [0.5,1.5]$. Moreover, computations are based on $s=10^4$ roughness realisations, which was more than sufficient for $\bar {A}$ to achieve a converged state in all cases modelled. The size of the mean amplitude $\bar {A}$ increases sharply in figure 11(a) when $N_{max}$ exceeds 20. This particular observation coincides with the shortest roughness scale matching the wavelength of the cross-flow instability. The radial wavenumber $\alpha _r$ of the stationary cross-flow instability is within the range $0.25\leqslant \alpha _r\leqslant 0.5$ (recall figure 2c), while the wavenumber associated with the shortest roughness scale in (5.4) is given as $2{\rm \pi} N_{max}/r_{\lambda }\approx 0.31$ for $N_{max}=20$. Similar observations were observed by Thomas et al. (Reference Thomas, Mughal and Ashworth2017) in their receptivity study of cross-flow vortices over a swept wing. With the exception of the case $k=0.5$ (solid blue line), mean amplitudes $\bar {A}$ approach a fixed constant for $N_{max}\approx 40$, which corresponds to a roughness length scale that is approximately half of the wavelength of the cross-flow instability. Hence, shorter roughness scales have a negligible effect on the receptivity process, which is to be expected given the power law scale factor (5.5). In the instance the power law coefficient $k=0.5$, $\bar {A}$ is observed to decrease for $N_{max}>40$. However, if results were extended to larger valued $N_{max}$, than that plotted in figure 11(a), results would eventually converge towards a fixed value as the shorter roughness scales would become insignificant. Finally, as $k$ increases, the size of the mean amplitude $\bar {A}$ decreases. Indeed, there is an order of magnitude difference in $\bar {A}$ between the smallest and largest valued $k$ modelled. The significance of the short roughness scales of commensurate size to the wavelength of the cross-flow instability, diminish for larger valued $k$.

Figure 11.

Mean amplitude $\bar {A}$ as a function of (a) $N_{max}$ for $r_b=600$ and (b) $r_b$ for $N_{max}=100$. Computations are based on $s=10^4$ roughness iterations for $n=32$, and roughness parameters $h_{{rms}}=1$ and variable $k$.

Results plotted in figure 11(b) indicate that the mean amplitude $\bar {A}$ increases rapidly for radii $r_b>250$, and in most instances asymptotes towards a fixed constant for $r_b>400$. (There is a noticeable exception for cases $k>1$, where $\bar {A}$ attains a maximum value near $r_b\approx 330$ and then decreases towards a fixed constant for larger valued $r_b$. This behaviour can again be attributed to the diminished effect of the short roughness scales that occurs for larger valued $k$.) These observations are consistent with our earlier predictions that receptivity is most affected by roughness distributions located about the radial region that the disturbance first becomes linearly unstable and where the adjoint vorticity terms were most significant (recall figures 4(b) and 5).

The above analysis was extended to azimuthal mode numbers $n\in [25,40]$, for roughness parameter settings $N_{max}=100$, $r_b=600$ and $h_{{rms}}=1$. These particular settings were chosen as receptivity calculations were unchanged by further increases in the short wavelength filter $N_{max}$ or the downstream boundary $r_b$ of the roughness. Figure 12 displays the mean amplitude $\bar {A}$ and corresponding standard deviations $\sigma$ as functions of $n$, for five values of the power law coefficient $k$. Calculations are again based on $s=10^4$ roughness realisations. For each $k$ considered, the standard deviation $\sigma$ is approximately half of the corresponding mean amplitude $\bar {A}$. Thus, there is a significant variation in the initial amplitude of the stationary cross-flow instability. Additionally, there is a small increase in $\bar {A}$ as the azimuthal mode number $n$ increases. However, this does not necessarily imply that these larger valued $n$ modes will play a greater role in the laminar–turbulent transition process, as smaller valued $n$ modes become unstable earlier. The cross-flow mode that is ultimately responsible for bringing about the latter stages of convective driven transition is governed by the amplitude $A$, the radial growth rate $\alpha _i$ and the size of the roughness that is characterised in this investigation by the parameter $h_{{rms}}$.

Figure 12.

(a) Mean amplitude $\bar {A}$ and (b) standard deviation $\sigma$ as a function of the azimuthal mode number $n$ and variable $k$. Computations are based on $s=10^4$ roughness iterations, and roughness parameters $h_{{rms}}=1$, $N_{max}=100$ and $r_b=600$. Line types are the same as those given in figure 11.

5.3.3. Predicting the onset of nonlinear behaviour and other processes

Mean amplitudes $\bar {A}$ are coupled to the radial growth of the normalised azimuthal velocity perturbation field $u_{\theta }$ (as plotted in figure 4(a) for $n=32$) to predict the radial location that we might expect the linear stability assumptions to no longer be appropriate, and where nonlinear effects and other transition processes to have emerged. Since the analysis is linear, scaling the surface roughness on the parameter $h_{{rms}}$ equates to an equivalent scaling on the disturbance amplitude. Thus, fixing $h_{{rms}}=10^{-3}$ corresponds to setting the mean amplitude of the disturbance to $10^{-3}\bar {A}$, with $\bar {A}$ as given in figure 12. Furthermore, we assume that the linear stability assumptions are no longer appropriate once $\max _z|u_{\theta }|=10^{-1}$. This particular assumption was based on earlier observations within the literature. For instance, Balachandar, Streett & Malik (Reference Balachandar, Streett and Malik1992) reported that the primary instability has to attain a threshold value of $9\,\%$ of the undisturbed flow (2.6) in order for secondary instabilities to grow. Hence, $r_{u_{\theta }=0.1}$ denotes the radial location used here to distinguish between the linear and nonlinear stages of convectively forced laminar–turbulent transition. (In experiments, the threshold between linear and nonlinear effects may arise at smaller disturbance amplitudes than that implemented here. Moreover, travelling waves, the absolutely unstable mechanism or secondary instabilities may develop, and establish nonlinear behaviour and the onset of laminar–turbulent transition at earlier radii than that presented in the subsequent analysis. Thus, the radial location $r_{u_{\theta }=0.1}$ should only be viewed as an upper bound for when we might expect disturbance development to be no longer governed by linear stability theory.)

Figure 13 displays the maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|$, as a function of the radius $r$ for the azimuthal mode number $n=32$. Solution curves were obtained by scaling the normalised solution depicted in figure 4(a), by the adjoint predictions for the mean amplitude $\bar {A}$ presented in figure 12, with $k$ set to unity. The horizontal dotted line indicates when $\max _z|u_{\theta }|=10^{-1}$. (Computations based on other $k$ values brought about a small change in the radial interval that $\max _z|u_{\theta }|=10^{-1}$ was realised.) The r.m.s. of the roughness decreases by an order of ten monotonically down figure 13, with $h_{{rms}}=10^{-2}$ and $10^{-4}$ in figures 13(a) and 13(c), respectively. Solid blue line-types depict results based on the mean amplitude $\bar {A}$, while the remaining two line-types give an indication of the spread of solutions, with the disturbance amplitude set to $\bar {A}\pm \sigma$. Results indicate that $\max _z|u_{\theta }|=10^{-1}$ is realised about the successive radial intervals $r\in [418,450]$, $[460,485]$ and $[495,520]$, for the three respective $h_{{rms}}$ values modelled. This was extended to $r\in [528,550]$ and $[560,583]$ for $h_{{rms}}=10^{-5}$ and $10^{-6}$, respectively. Hence, for a fixed $h_{{rms}}$, there is a short annular region in which we might expect the stationary cross-flow vortex, established by surface roughness, to achieve amplitudes sufficient to invalidate the linear stability assumptions and bring about nonlinear effects. Furthermore, results clearly demonstrate that a smooth disc will delay the onset of convective induced transition to larger radii, which may allow the absolutely unstable mechanism to develop and play a role in the laminar–turbulent transition process.

Figure 13.

Maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|$, as a function of the radius $r$ for $n=32$ and (a) $h_{{rms}}=10^{-2}$; (b) $h_{{rms}}=10^{-3}$; (c) $h_{{rms}}=10^{-4}$. Calculations are based on those roughness parameters given in figure 12 when $k=1$. The horizontal dotted line corresponds to $\max _z|u_{\theta }|=10^{-1}$.

The radial location $r_{u_{\theta }=0.1}$ is plotted in figure 14 as a function of $h_{{rms}}$. Calculations are again based on the adjoint predictions presented in figure 12 for $n=32$. The first illustration, figure 14(a), gives an indication of the spread of $r_{u_{\theta }=0.1}$ for the case where the power law coefficient $k=1$. The second plot, figure 14(b), depicts the effect of varying the power law coefficient $k$. As expected, results show that the radial location $r_{u_{\theta }=0.1}$ emerges further downstream as $h_{{rms}}$ decreases. Furthermore, for $h_{{rms}}\leqslant 10^{-2}$, the radius $r_{u_{\theta }=0.1}$ is approximated by the formula

(5.8)\begin{equation} r_{u_{\theta}=0.1}=c-d\log_{10}h_{{rms}} \end{equation}

for positive valued constants $c$ and $d$ that are dependent on $k$. (Similar depictions of $r_{u_{\theta }=0.1}$ were realised for other stationary cross-flow disturbances with different azimuthal mode numbers.) Assuming that experimental roughness characteristics are known and that the $n=32$ cross-flow mode is responsible for bringing about convective driven transition, figure 14 and this particular logarithmic relationship may be used to predict the radial range when transition in the rotating disc boundary layer is likely to occur, or at least when nonlinear effects and other processes must be accounted for. Finally, results imply that $r_{u_{\theta }=0.1}$ occurs further downstream for larger valued $k$, which is to be expected given our earlier observations: mean amplitudes $\bar {A}$ decrease as $k$ increases.

Figure 14.

The radius $r$ that the maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|=10^{-1}$, for $n=32$ and those roughness parameters given in figure 12. (a) Calculations are based on the mean amplitude $\bar {A}$ and standard deviation $\sigma$ for $k=1$. (b) Calculations are based on $\bar {A}$ and variable $k$.

The maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|$, is plotted in figure 15 for several azimuthal mode numbers $n$. Computations are again based on the mean calculations presented in figure 12 for $k=1$, while the roughness parameter $h_{{rms}}$ is as indicated in the caption. The horizontal dotted line again specifies when $\max _z|u_{\theta }|=10^{-1}$. Solutions plotted in figure 15(a) indicate that the $n=26$ cross-flow mode (solid blue line) attains this threshold value about $r\approx 400$ when $h_{{rms}}=10^{-2}$, which is radially inboard of the locations that the other modes achieve this particular amplitude. Thus, the $n=26$ mode will be responsible for engineering the early stages of transition in this instance. As $h_{{rms}}$ decreases, the stationary cross-flow mode $n$ that first achieves the amplitude $\max _z|u_{\theta }|=10^{-1}$, increases. For $h_{{rms}}=10^{-3}$ and $10^{-4}$, cross-flow modes $n=28$ (dashed red) and $30$ (chain yellow) are the first to attain the linear-nonlinear threshold amplitude about the respective radial positions $r\approx 450$ and $500$.

Figure 15.

Maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|$, as a function of the radius $r$ for variable azimuthal mode numbers $n$ and (a) $h_{{rms}}=10^{-2}$; (b) $h_{{rms}}=10^{-3}$; (c) $h_{{rms}}=10^{-4}$. Mean amplitudes $\bar {A}$ are based on those roughness parameter settings given in figure 12 when $k=1$. The horizontal dotted line corresponds to $\max _z|u_{\theta }|=10^{-1}$.

Contours of the radius $r_{u_{\theta }=0.1}$ are plotted in figure 16 in the $(n,h_{{rms}})$-plane, for those roughness settings given in figure 12 and $k=1$. Blue contour levels correspond to radii $r\sim 400$, while red contours are matched to $r\sim 600$. The illustration further supports our above comments that azimuthal mode numbers on the interval $25\leqslant n\leqslant 30$ will most likely bring about nonlinear effects when the roughness parameter $h_{{rms}}\geqslant 10^{-4}$. However, for smaller valued $h_{{rms}}$, larger valued azimuthal mode numbers (such as $n=32$) grow at a faster rate and attain amplitudes $\max _z|u_{\theta }|=10^{-1}$ earlier. Hence, these particular stationary cross-flow instabilities will play a greater role in the early stages of laminar–turbulent transition on a clean rotating disc.

Figure 16.

Contours of the radius $r_{u_{\theta }=0.1}$ in the $(n,h_{{rms}})$-plane that the maximum absolute value of the azimuthal velocity perturbation field, $\max _z|u_{\theta }|=10^{-1}$. Mean amplitudes $\bar {A}$ are based on those roughness parameter settings given in figure 12 when $k=1$.