2.1. Summation of wakes

The wake of a zither will be considered as the summation of each individual wire’s wake. Unlike Böttcher & Wedemeyer (Reference Böttcher and Wedemeyer1989), each wire will be assumed to have an individual wake. Each wake is given by an unknown function, $W_{n}(z-z_{n})$ , where $z_{n}$ is the spanwise location of the $n$ th wire. Far downstream, the wavelengths are large in comparison to the wire spacing, as observed by Böttcher & Wedemeyer (Reference Böttcher and Wedemeyer1989). They argue that the wake function of a wire can be considered as a point source,

(2.6) $$\begin{eqnarray}\int W_{n}(z-z_{n})=q_{n}{\it\delta}(z-z_{n}),\end{eqnarray}$$

where ${\it\delta}(z-z_{n})$ is the Dirac function and $q_{n}$ is the source strength of the $n$ th wire. Assuming only a drag force acting on the wire (no lateral force component), $q_{n}$ is then related to the drag force ( $D_{n}$ ) by $D_{n}={\it\rho}Uq_{n}$ , where ${\it\rho}$ is density (Batchelor Reference Batchelor2000). Every wire in the zither is perturbed from its perfect position ( $Mn$ ), giving the position and source strength of the $n$ th wire as,

(2.7a,b ) $$\begin{eqnarray}z_{n}=M(n+{\it\delta}_{n}),\quad q_{n}=\overline{q}(1+{\it\epsilon}_{n}),\end{eqnarray}$$

where ${\it\delta}_{n}$ and ${\it\epsilon}_{n}$ are non-dimensional random variables with zero means and ${\it\delta}_{n}\ll M$ , ${\it\epsilon}_{n}\ll 1$ and $\overline{q}$ is the mean source strength. The Fourier coefficients of (2.2) are then,

(2.8) $$\begin{eqnarray}a_{k}=\frac{\overline{q}}{NM}\mathop{\sum }_{n=1}^{N}(2{\rm\pi}\text{i}K{\it\delta}_{n}+{\it\epsilon}_{n}-2{\rm\pi}\text{i}K{\it\epsilon}_{n}{\it\delta}_{n})\exp (-2{\rm\pi}\text{i}Kn),\end{eqnarray}$$

where the first exponential term of (2.2) has been expanded with a power series $(e^{x}\approx 1+x)$ .

2.2. Predicted wake from a zither with imperfect wire spacing and source strength

Assuming all wavelengths are excited relatively equally by the distributions of ${\it\delta}_{n}$ and ${\it\epsilon}_{n}$ gives the wake energy as,

(2.9) $$\begin{eqnarray}\overline{u^{2}}=\frac{\overline{q}^{2}}{N^{2}M^{2}}\mathop{\sum }_{k=-\infty }^{\infty }(4{\rm\pi}^{2}K^{2}N{\it\sigma}_{{\it\delta}}^{2}+N{\it\sigma}_{{\it\epsilon}}^{2})\exp (-8{\rm\pi}^{2}K^{2}\overline{x}),\end{eqnarray}$$

where,

(2.10a,b ) $$\begin{eqnarray}{\it\sigma}_{{\it\delta}}^{2}=\frac{1}{N}\mathop{\sum }_{n=1}^{N}{\it\delta}_{n}^{2},\quad {\it\sigma}_{{\it\epsilon}}^{2}=\frac{1}{N}\mathop{\sum }_{n=1}^{N}{\it\epsilon}_{n}^{2},\end{eqnarray}$$

and terms with higher products of ${\it\delta}_{n}$ and ${\it\epsilon}_{n}$ are assumed negligible. Considering an infinite zither ( $N\rightarrow \infty$ , $k/N\rightarrow K$ ) allows the summation to be replaced with the integral,

(2.11) $$\begin{eqnarray}\overline{u^{2}}=\frac{\overline{q}^{2}}{M^{2}}\int _{-\infty }^{\infty }(4{\rm\pi}^{2}K^{2}{\it\sigma}_{{\it\delta}}^{2}+{\it\sigma}_{{\it\epsilon}})\exp (-8{\rm\pi}^{2}K^{2}\overline{x})\,\text{d}K.\end{eqnarray}$$

The wavenumber with peak energy can be found from the integrand as,

(2.12) $$\begin{eqnarray}K=\frac{1}{2\sqrt{2}{\rm\pi}\overline{x}}\sqrt{\overline{x}-2\frac{{\it\sigma}_{{\it\epsilon}}^{2}}{{\it\sigma}_{{\it\delta}}^{2}}\overline{x}^{2}},\end{eqnarray}$$

giving the dominant wake wavelength as,

(2.13) $$\begin{eqnarray}\frac{{\it\lambda}}{M}=\frac{2\sqrt{2}{\rm\pi}\overline{x}}{\sqrt{\overline{x}-2\displaystyle \frac{{\it\sigma}_{{\it\epsilon}}^{2}}{{\it\sigma}_{{\it\delta}}^{2}}\overline{x}^{2}}}.\end{eqnarray}$$

For small $\overline{x}$ , or when ${\it\sigma}_{{\it\epsilon}}$ is small, the wavelength will be independent of the zither geometry,

(2.14) $$\begin{eqnarray}\frac{{\it\lambda}}{M}=\sqrt{8{\rm\pi}^{2}\overline{x}},\end{eqnarray}$$

as originally derived by Böttcher & Wedemeyer (Reference Böttcher and Wedemeyer1989).

The mean energy of the wake is found as the solution of (2.11). The average wake strength is non-dimensionalised by $U$ , giving the combined wake strength equation as,

(2.15) $$\begin{eqnarray}{\rm\Delta}u=\frac{0.2233\overline{q}{\it\sigma}_{{\it\delta}}}{MU\overline{x}^{0.75}}\sqrt{1+4\frac{{\it\sigma}_{{\it\epsilon}}^{2}}{{\it\sigma}_{{\it\delta}}^{2}}\overline{x}}.\end{eqnarray}$$

where ${\rm\Delta}u$ is defined as ${\rm\Delta}u=\sqrt{\overline{u^{2}}}/U$ .

2.3. Predicted zither wake from only imperfect wire positions

Assuming each wire is identical, and neglecting the change in wire source strength due to an error in position, then the Fourier coefficients of (2.8) are given by,

(2.16) $$\begin{eqnarray}a_{k}=\frac{\overline{q}}{NM}\mathop{\sum }_{n=1}^{N}-2{\rm\pi}\text{i}K{\it\delta}_{n}\exp (-2{\rm\pi}\text{i}Kn).\end{eqnarray}$$

The dominant wavelength of the zither wake is found as (2.14) and the average wake strength of a zither with only imperfect wire position is,

(2.17) $$\begin{eqnarray}{\rm\Delta}u=0.2233\frac{\overline{q}{\it\sigma}_{{\it\delta}}}{MU\overline{x}^{0.75}},\end{eqnarray}$$

as originally derived by Böttcher & Wedemeyer (Reference Böttcher and Wedemeyer1989).

2.4. Predicted zither wake from only imperfect wire source strengths

Assuming only imperfection of wire source strength gives the Fourier coefficients,

(2.18) $$\begin{eqnarray}a_{k}=\frac{\overline{q}}{NM}\mathop{\sum }_{n=1}^{N}{\it\epsilon}_{n}\exp (-2{\rm\pi}\text{i}Kn).\end{eqnarray}$$

The dominant wavenumber is found as $k=0$ , but as the wake has no zero mode, $k=1$ will dominate far downstream. The average wake strength for a zither with imperfect wire source strength is found as,

(2.19) $$\begin{eqnarray}{\rm\Delta}u=0.4466\frac{\overline{q}{\it\sigma}_{{\it\epsilon}}}{MU\overline{x}^{0.25}}.\end{eqnarray}$$

2.5. Differing zither wake decay rates

The wake strength formulas (2.15), (2.17) and (2.19) are compared in figure 2 for zithers with: imperfect wire spacing and source strength ( ${\it\sigma}_{{\it\delta}}={\it\sigma}_{{\it\epsilon}}=0.05$ ); only imperfect wire position ( ${\it\sigma}_{{\it\delta}}=0.05,\,{\it\sigma}_{{\it\epsilon}}=0$ ); and only imperfect source strength ( ${\it\sigma}_{{\it\delta}}=0,\,{\it\sigma}_{{\it\epsilon}}=0.05$ ). The wake decay of the imperfect wire position zither differs significantly to a zither with only imperfect source strength, predominately due to the respective $\overline{x}^{-0.75}$ and $\overline{x}^{-0.25}$ dependencies. For the same standard deviation of imperfection, the zither with only source strength variation will be weaker close to the zither, but stronger far downstream. The wake decay is explained by the Fourier coefficient distributions shown in figure 2. The distribution for a zither with only variation in wire position is approximated by,

(2.20) $$\begin{eqnarray}a_{kP}=\frac{2{\rm\pi}k{\it\sigma}_{{\it\delta}}\overline{q}}{M\sqrt{N}}.\end{eqnarray}$$

The distribution for variation only in source strength is approximated as,

(2.21) $$\begin{eqnarray}a_{kD}=\frac{{\it\sigma}_{{\it\epsilon}}\overline{q}}{M\sqrt{N}}.\end{eqnarray}$$

Variation in wire source strength can create significantly stronger small wavenumbers (large wavelengths), that decay slower and create a stronger far wake. This leads to a hypothesis that significant spanwise variation in the test-section boundary layer is linked to screens that have significant spatial variation of source strength. This spatial variation may be induced by poor quality screens that have larger variations in wire position which, particularly for lower open-area ratios, may lead to nearby wires influencing each other’s source strength (drag) and creating a stronger far wake.

Figure 2. (a) Fourier coefficients at the zither due to: grey solid line, random normal distribution of wire drag; grey dashed, mean distribution of drag, (2.21); black solid, random normal distribution of wire position; black dashed, mean distribution of position, (2.20). (b) Example wake decay: light grey solid, random normal distribution of wire drag calculated with (2.18) and (2.4); light grey dashed, variation of drag, (2.19); black solid, random normal distribution of wire position calculated with (2.16) and (2.4); black dashed, variation of position, (2.17); dark grey solid, random normal distribution of wire drag and position calculated with (2.8) and (2.4); dark grey dashed, variation of drag and position, the combined wake strength equation, (2.15).

2.6. Zither CFD domain and methods

Navier–Stokes simulations (CFD) will be used to study ten zithers with differing combinations of open-area ratio and random wire position error. The steady, pressure-based solver (second-order upwind differencing for pressure, third-order MUSCL differencing for momentum) of the ANSYS Fluent CFD package is used. Solution convergence is judged by the constancy of the far wake with increasing iteration and convergence of global residual monitors. An overview of the zither domain and mesh is shown in figure 3(a). The velocity inlet-boundary-condition is 317 diameters (d) (80.47 mm) upstream of the zither while the outflow boundary condition is located at $2405d$ (610.8 mm). Periodic boundary conditions are used in the spanwise direction.

To reduce the total mesh control volumes (CV), eight regions with varying resolution are used as the wake spanwise scale increases. Each region is composed of structured quadrilateral CV aligned with the uniform flow direction. Hanging nodes are used to halve the number CV in the spanwise direction at each region boundary. Figure 3(b) shows the mesh between wires. There are 216 CV on the surface of a wire. Total mesh size varies between 23.4 and 44.2 million CV, depending on the open-area ratio. Refining the entire ${\it\beta}=60\,\%$ with ${\it\sigma}_{{\it\delta}}=0.05$ mesh by a factor 2 in each direction altered wire forces by less than 0.5 % and has a negligible effect on the far wake (see figure 4 c).

The spanwise domain is the same for all zithers ( $L=243.84$  mm), with up to 576 wires depending on the open-area ratio (see table 1). Numerical tests comparing (2.2) with (2.15) show this is enough wires for the averaging in (2.15) to be representative beyond $\overline{x}=30$ (see Pook Reference Pook2013). The spanwise domain is also more than six times wider than the wavelength for optimal, non-modal streak growth (Andersson et al. Reference Andersson, Berggren and Henningson1999) at the downstream end of the model test section used in § 4.

Figure 3. (a) Overview of the zither mesh domain. Marked regions start at streamwise positions: U2 at $-317d$ , U1 at $-76d$ , Zither at $-7d$ to 7 $d$ , D1 at 7 $d$ , D2 at 67 $d$ , D3 at 205 $d$ , D4 at 405 $d$ , D5 at 805 $d$ . (b) Close-up of the CFD mesh between two zither wires. Every second line shown.

2.7. Zither geometry

Table 1 lists the geometry parameters for all the zithers which are composed of circular wires arranged in a line perpendicular to the free stream. All wires have a diameter of 254  ${\rm\mu}$ m, the same size used by Pook & Watmuff (Reference Pook and Watmuff2014). Four open-area ratios ( ${\it\beta}$ ) are used, ${\it\beta}=40\,\%$ where coalescence of jets is expected, 50 %, 60 % and 66.67 % where coalescence of jets is not expected. The open-area ratio, diameter and the mesh spacing of the perfect zither (uniformly spaced wires) are related by,

(2.22) $$\begin{eqnarray}{\it\beta}=1-\frac{d}{M}.\end{eqnarray}$$

Each wire ( $n$ ) is randomly perturbed from its position in the perfect zither by some distance. A normal distribution is used for the random variation of ${\it\delta}_{n}$ , with (near) zero mean and a given standard deviation ( ${\it\sigma}_{{\it\delta}}$ ) listed in table 1 (see Pook Reference Pook2013 for quantile-quantile plots). The ${\it\beta}=60\,\%$ zithers share the same distribution of ${\it\delta}_{n}$ , i.e. the differing standard deviation is achieved by multiplying by a constant. The same applies for the ${\it\beta}=66.67\,\%$ zithers. The ${\it\beta}=50\,\%$ zithers all have differing distributions. A constant ${\it\sigma}_{{\it\delta}}$ implies a decreasing physical error in wire position with decreasing open-area ratio. The largest standard deviation in physical units for the zithers studied is 57.1  ${\rm\mu}$ m, or 22 % of the wire diameter. The smallest is 10  ${\rm\mu}$ m, or 4 % of the wire diameter. The uniform free stream is $U=1.7$  m s $^{-1}$ giving a Reynolds number based on wire diameter $Re_{d}=29.6$ for all zithers, except the ${\it\beta}=66.67$  % with ${\it\sigma}_{{\it\delta}}=0.05$ zither that uses $U=1.9$  m s $^{-1}$ . The zithers are sub-critical (do not shed eddies).

Figure 4. Zither wake strength from CFD simulations compared to prediction: (a) ${\it\beta}=40\,\%$ ; (b) ${\it\beta}=50\,\%$ ; (c) ${\it\beta}=60\,\%$ ; (d) ${\it\beta}=66.67\,\%$ . Lines: - - - - prediction due to only position variation (2.17);  $\cdots \cdots$  prediction of combined wake strength (2.15); 

CFD wake for ${\it\sigma}_{{\it\delta}}=0.025$ ; 

CFD wake for ${\it\sigma}_{{\it\delta}}=0.05$ ; 

CFD wake for ${\it\sigma}_{{\it\delta}}=0.075$ ; 

CFD wake for ${\it\sigma}_{{\it\delta}}=0.1$ . For (b), 

refined mesh for ${\it\sigma}_{{\it\delta}}=0.05$ .

Table 1 shows that the mean wire source strength is (near) invariant to the standard deviation of wire position variation. The wire source strength is obtained from the CFD wire force data and the mean calculated by,

(2.23) $$\begin{eqnarray}\overline{q}=\frac{1}{{\it\rho}UN}\mathop{\sum }_{n=1}^{N}D_{n}.\end{eqnarray}$$

The drag coefficient ( $C_{d}$ ) is defined as $D=0.5{\it\rho}U^{2}dC_{d}$ . The pressure drop coefficient ( $PD={\rm\Delta}p/(0.5{\it\rho}U^{2})$ ) is also listed.

Table 1. Zither geometry, source strength data and wake strength at $\overline{x}=10$ and $x=2300d$ .

2.8. Wake strength

Figure 4 compares the CFD wake strength with the predicted wake strength due only to variation in wire position (2.17), and the combined wake strength equation (2.15). Beyond $\overline{x}\gtrsim 0.1$ , the CFD wakes decay as $\overline{x}^{-0.75}$ . However, at $\overline{x}=1$ , the CFD wake strengths are 20–40 % greater than predicted by (2.17) (see Pook Reference Pook2013), with the error increasing with the standard deviation of wire position variation ( ${\it\sigma}_{{\it\delta}}$ ) and decreasing open-area ratio. By $\overline{x}=10$ , the CFD wake strengths are 20–70 % stronger, indicating a decay-rate slower than $\overline{x}^{-0.75}$ but not nearly as dramatic as predicted by (2.15).

Figure 5. (a) The effect of a downstream initial condition for the ${\it\beta}=40\,\%$ , ${\it\sigma}_{{\it\delta}}=0.025$ zither. —— CFD; - - - - calculated with (2.1) and (2.8); 

calculated with (2.1) from initial condition at $\overline{x}=0.13$ ; 

calculated with (2.1) from initial condition at $\overline{x}=2.5$ . (b) ${\it\beta}=50\,\%$ zithers CFD wake strength compared to calculation with (2.1) and (2.8). Lines and markers as for figure 2.

Figure 6. FFT of streamwise velocity and pressure for the ${\it\beta}=40\,\%$ , ${\it\sigma}_{{\it\delta}}=0.025$ zither: (a) $\overline{x}=0.13$ ; (b) $\overline{x}=2.5$ . Lines: —— streamwise velocity; - - - - pressure.

Table 1 shows ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}$ is a near constant 0.13 for the ${\it\beta}=60\,\%$ and ${\it\beta}=66.67\,\%$ zithers when ${\it\sigma}_{{\it\delta}}\leqslant 0.05$ . The wake strength at a given streamwise distance for the ${\it\beta}=60\,\%$ and 66.67 % zithers scales near linearly with ${\it\sigma}_{{\it\delta}}$ . However, the ${\it\beta}=50\,\%$ zithers show linear scaling only for ${\it\sigma}_{{\it\delta}}=0.025$ and 0.05. The ${\it\sigma}_{{\it\delta}}=0.075$ zither wake strength is significantly greater than a linear scaling, indicating the wake strength due to only variation in wire position (2.17) does not predict the correct wake decay for zithers with large ${\it\sigma}_{{\it\delta}}$ which induces variations of drag across the zither. With decreasing open-area ratio, the ratio ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}$ is seen to increase but only substantially for the ${\it\beta}=40\,\%$ zither. However, its wake decay proportional to $\overline{x}^{-0.75}$ indicates a flawed assumption in the derivation of the combined wake strength equation (2.15).

2.9. Validity of the linear diffusion equation

The ${\it\beta}=40\,\%$ zither CFD wake is sampled and used as a downstream initial condition to (2.1) as validation. Figure 5 shows using an initial condition sampled at $\overline{x}=2.5$ where ${\rm\Delta}u=1\,\%$ , predicts a downstream wake in agreement with CFD. Using a fast Fourier transform (FFT) on the sampled wakes, figure 6, shows an initial condition too close to the zither fails as low-wavenumber variations of pressure have not been recovered into velocity. Fourier coefficients calculated at the zither with both wire drag and drag position are used to predict the wake in figure 5. The predicted wake undershoots the CFD results between $\overline{x}=0.1$ and $\overline{x}=6$ , before showing reasonable agreement further downstream. As wire drag includes the effect of pressure, the linear diffusion equation can be used with known wire drag to predict the far wake.

The relative error between CFD wake strength and (2.1) using known wire drag is less than 10 % for $\overline{x}>1$ , for all zithers. Equation (2.17) under-predicts wake strength primarily because it neglects the variation in wire drag.

Figure 7. Fourier coefficients at the zither due to variation in drag only, calculated with (2.18): (a) ${\it\beta}=40\,\%$ ; (b) ${\it\beta}=50\,\%$ ; (c) ${\it\beta}=60\,\%$ ; (d) ${\it\beta}=66.67\,\%$ .Increasing coefficients is increasing ${\it\sigma}_{{\it\delta}}$ . Lines: black, ${\it\sigma}_{{\it\delta}}=0.025$ ; red, ${\it\sigma}_{{\it\delta}}=0.05$ ; blue, ${\it\sigma}_{{\it\delta}}=0.075$ ; green, ${\it\sigma}_{{\it\delta}}=0.1$ .

2.10. Distribution of source strength (drag)

Figure 7 shows the Fourier spectra due to only variation in wire drag, calculated with (2.18). The magnitude of the ${\it\beta}=60\,\%$ and 66.67 % zither Fourier coefficients increase from $k=0$ , peak at approximately $k=100$ , before diminishing. Approximating with the mean Fourier coefficient, as $a_{kD}$ does, will significantly over estimate the effect of wire drag variation for large wavelengths. The combined wake equation will over predict the wake strength at large $\overline{x}$ . For ${\it\beta}=50\,\%$ , the Fourier coefficients for small $k$ increase with ${\it\sigma}_{{\it\delta}}$ . Only for the ${\it\sigma}_{{\it\delta}}=0.75$ zither are the Fourier coefficients spread relatively evenly across the spectrum at low wavenumbers, meaning ${\it\sigma}_{{\it\epsilon}}$ should be an accurate representation. The wake decay should reduce from $\overline{x}^{-0.75}$ at large streamwise distances. Indeed, figure 4 does show the wake decay reducing for this zither.

The Fourier spectrum of the ${\it\beta}=40\,\%$ zither is different to the other zithers, with the largest coefficients occurring at wavenumbers $k>200$ , raising ${\it\sigma}_{{\it\epsilon}}$ but not affecting the far wake. A wavenumber of $k=288$ corresponds to a wavelength of $2M$ , i.e. a wavelength that could be expected to be associated with the coalescence of adjacent jets.

The over prediction of wake strength by the combined wake strength equation (2.15) is attributed to the assumed Fourier coefficient distribution due to variation in wire source strength (drag), $a_{kD}$ . Approximating the Fourier coefficients due to wire force variation as constant (independent of $k$ ) is not valid and over predicts the wake strength for large $\overline{x}$ . Equations to predict wake strength more accurately require an understanding of how the drag of a wire varies with an error in its position, particularly at low wavenumbers.

2.11. Wake wavelength

The dominant wavelength at each streamwise position is extracted from the CFD wakes using an FFT of the streamwise velocity. Figure 8 shows the dominant wavelength increases with $\overline{x}^{0.5}$ and is independent of the zither geometry, as predicted by (2.14). Wake profiles can also be calculated from knowledge of the individual wire source strength and position data (see Pook Reference Pook2013).

Figure 8. Wavelength of the peak Fourier mode in the CFD simulation; (a) ${\it\beta}=40\,\%$ ; (b) ${\it\beta}=50\,\%$ ; (c) ${\it\beta}=60\,\%$ ; (d) ${\it\beta}=66.67\,\%$ . Lines: - - - - wavelength predicted by (2.14); marked lines as for figure 4.

2.12. Coalescence of jets

Figures 9 and 10 show contours of streamwise velocity for a limited extent ( ${<}$ 10 %) of the ${\it\beta}=40\,\%$ , 50 % and 60 % zithers. The ${\it\beta}=60\,\%$ zithers share the same wire perturbation pattern. Apart from the longer jets and wakes, the flow pattern does not significantly change over the ${\it\sigma}_{{\it\delta}}$ range investigated. There is minimal deflection of the jets that would be taken to indicate the coalescence of jets in experimental flow visualisation (see Bradshaw Reference Bradshaw1965; Le Gal et al. Reference Le Gal, Peschard, Chauve and Takeda1996), and table 1 shows ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}$ is near constant. For the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither shown in figure 10(c), the jets emanating near the cylinder at $z/d\approx 135$ are visibly deflected away from each other, creating an extended region of low-speed flow. More instances of visible jet deflection can be observed for this zither, and particularly the ${\it\beta}=40\,\%$ zither, but not the other zithers. The ratio ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}$ increases for zithers with instances of jet deflection.

The observed instances of jet deflection in figure 10 are similar to the experimental flow visualisations of Le Gal et al. (Reference Le Gal, Peschard, Chauve and Takeda1996). However, Le Gal et al. (Reference Le Gal, Peschard, Chauve and Takeda1996) observed significant jet deflection for the majority of jets. As previously mentioned, the Fourier coefficients of the ${\it\beta}=40\,\%$ zither wire drag increase rapidly for $k>200$ . This indicates a flow pattern that possibly correlates to the coalescence of jets. However, such small wavelengths have no effect on the far wake strength. For all other zithers, there is no trend towards increased drag variation at wavelengths approaching $2M$ relative to large wavelength, although the Fourier coefficients for the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ are appreciable at short wavelengths. The authors’ opinion is that the coalescence of jet phenomena (large jet deflection visible in contours) appears for a limited number of jets in the ${\it\beta}=40\,\%$ and to a lesser extent the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zithers. This differs from the flow visualisations of Bradshaw (Reference Bradshaw1965) and Le Gal et al. (Reference Le Gal, Peschard, Chauve and Takeda1996) where the majority of jets show large deflection.

Figure 9. Contours of streamwise velocity scaled by free-stream velocity $U$ (see insert for scale) for a limited extent ( ${<}$ 10 %) of the zithers, ${\it\beta}=60\,\%$ with: (a) ${\it\sigma}_{{\it\delta}}=0.025$ , ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}=0.13$ ; (b) ${\it\sigma}_{{\it\delta}}=0.05$ , ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}=0.13$ ; (c) ${\it\sigma}_{{\it\delta}}=0.075$ , ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}=0.13$ ; (d) ${\it\sigma}_{{\it\delta}}=0.1$ , ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}=0.17$ .

Figure 10. Contours of streamwise velocity scaled by free-stream velocity $U$ (see insert for scale) for limited extent ( ${<}$ 10 %) of the zithers: (a) ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.025$ , ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}=0.13$ ; (b) ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.05$ , ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}=0.14$ ; (c) ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ , ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}=0.18$ ; (d) ${\it\beta}=40\,\%$ with ${\it\sigma}_{{\it\delta}}=0.025$ , ${\it\sigma}_{{\it\epsilon}}/{\it\sigma}_{{\it\delta}}=0.29$ . Arrows indicate examples of significant jet deflection.

2.13. The effect of non-uniform inflow

Limited CFD simulations are performed with a random inlet-velocity wake $(U_{r})$ , constructed by the addition of sinusoids, from $k=1$ to 50, with random phase,

(2.24) $$\begin{eqnarray}U_{r}=U+\mathop{\sum }_{k=1}^{50}0.002U\sin \left(\frac{2{\rm\pi}k}{960}\frac{z}{d}+\text{random phase}\right).\end{eqnarray}$$

The non-uniform inflow strength is ${\rm\Delta}u=1\,\%$ and the profile is shown in figure 11(a). The non-uniform inflow is applied to the ${\it\beta}=40\,\%$ , 50 % and 60 % zithers with ${\it\sigma}_{{\it\delta}}=0.025$ . The streamwise domains of the ${\it\beta}=40\,\%$ and 50 % zithers are extended to capture any change of wake decay.

Figure 11. (a) Non-uniform velocity inlet profile. (b) Wake strength, ${\it\beta}=40\,\%$ , 50 %, 60 % with ${\it\sigma}_{{\it\delta}}=0.025$ zithers. Dark solid line (stronger far wake) is non-uniform inflow. Light solid line is uniform inflow. Square makers indicated strength downstream calculated with (2.2). Dashed lines are (2.17). (c) Fourier coefficients at $\overline{x}=10$ for ${\it\beta}=40\,\%$ . (d) Fourier coefficients at $\overline{x}=10$ for ${\it\beta}=50\,\%$ . (e) Fourier coefficients at $\overline{x}=10$ for ${\it\beta}=60\,\%$ . Lines: black solid, extracted from CFD with non-uniform inflow; grey dash-dot, extracted from CFD with uniform inflow; black long-dash, calculated with (2.8) at zither and transferred to $\overline{x}=10$ with (2.5); red short-dash, calculated with (2.25) at zither and transferred to $\overline{x}=10$ with (2.5).

Velocity contours near the zither (not shown) are not discernibly different to the uniform inflow simulations. For small $\overline{x}$ , figure 11(b) shows the non-uniform inflow wake decay changes from $\bar{x}^{-0.75}$ towards $\bar{x}^{-0.25}$ . This is readily apparent for the ${\it\beta}=60\,\%$ zither. For $\overline{x}>13$ , the ${\it\beta}=60\,\%$ zither wake is stronger than the ${\it\beta}=50\,\%$ zither.

At $\overline{x}=10$ , the ${\it\beta}=60\,\%$ zither non-uniform inlet wake strength is 50 % greater relative to a uniform inflow. The ${\it\beta}=50\,\%$ zither wake is 16 % greater and the ${\it\beta}=40\,\%$ zither is only 9 % greater. Standard deviations of wire source strength ( ${\it\sigma}_{{\it\epsilon}}$ ) for the ${\it\beta}=40\,\%$ , 50 % and ${\it\beta}=60\,\%$ zithers are 0.0093, 0.0045 and 0.0058, respectively. As ${\it\sigma}_{{\it\delta}}$ is the same, the combined wake-strength equation (2.15) would predict the ${\it\beta}=40\,\%$ zither wake to show a greater deviation from $\overline{x}^{-0.75}$ , opposite to what is observed in figure 11(b).

Wake Fourier coefficients at $\overline{x}=10$ are extracted from the CFD simulations and shown in figure 11(c–e) for the uniform and non-uniform inlet simulations. Also shown are the Fourier coefficients calculated from the individual wire drag and position data, and transferred to $\overline{x}=10$ by (2.5). The extracted Fourier coefficients at low wavenumbers are significantly smaller than calculated with wire position and drag data. This increases the calculated value of ${\it\sigma}_{{\it\epsilon}}$ relative to that observed. Decreasing open-area ratio reduces the difference between uniform and non-uniform inflow Fourier coefficients. The drag variation induced at the zither by the non-uniform inflow, creates an additional wake component tending to cancel the non-uniform inflow. The removal of a non-uniform inflow with increasing screen pressure drops is predicted by the Taylor & Batchelor (Reference Taylor and Batchelor1949) analysis. Hence, the far wake strength of the lower-open area ratio zithers with a non-uniform inflows differs less compared to a uniform inflow.

Prediction of the far wake with known wire position and drag data can be improved by linearly adding the non-uniform inlet velocity. The Fourier coefficients at the zither are then found as,

(2.25) $$\begin{eqnarray}a_{k}=\frac{\overline{q}}{L}\mathop{\sum }_{n=1}^{N}(-2{\rm\pi}\text{i}K{\it\delta}_{n}+{\it\epsilon}_{n}-2{\rm\pi}\text{i}K{\it\epsilon}_{n}{\it\delta}_{n})\exp (-2{\rm\pi}\text{i}Kn)-a_{kw},\end{eqnarray}$$

where $a_{kw}$ is the $k$ th Fourier coefficient of the non-uniform inflow at the zither, assuming no influence of the zither. It is subtracted due to the convention of a wake being positive. Figure 11(c–e) reveals the addition of the non-uniform inflow term produces good agreement between the calculated and CFD Fourier coefficients.

4.1. Measures of streak strength

Various measures of streak strength have been favoured in the literature with no direct relations provided between them. Figure 17(a) documents streak strength ( ${\rm\Delta}{\it\delta}^{\ast }$ ) as measured by the maximum and minimum displacement thickness on a given cross-sectional plane,

(4.1) $$\begin{eqnarray}{\rm\Delta}{\it\delta}^{\ast }=\frac{\displaystyle \max _{y,z}({\it\delta}^{\ast })-\displaystyle \min _{y,z}({\it\delta}^{\ast })}{{\it\delta}_{b}^{\ast }},\end{eqnarray}$$

where the subscript $b$ indicates the two-dimensional flow with no streaks. Figure 17(b) shows the corresponding streak amplitude ( $A$ ) defined by,

(4.2) $$\begin{eqnarray}A=\frac{\displaystyle \max _{y,z}(U-U_{b})-\displaystyle \min _{y,z}(U-U_{b})}{2U}.\end{eqnarray}$$

Both measures are linear with $x^{0.5}$ , downstream of an initial receptivity region, until $R\approx 1000$ .

Figure 17. (a) Streak strength ${\rm\Delta}{\it\delta}^{\ast }$ versus $R$ . (b) Streak amplitude ( $A$ ) versus $R$ . Lines and markers as for figure 15.

The ratio of ${\rm\Delta}{\it\delta}^{\ast }$ to $A$ beyond $R>400$ is approximately 3.6 for all zithers except the ${\it\beta}=40\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ and ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zithers. For these zithers, the ratio varies between 3.5 and 3.8 with increasing $R$ . These zithers have larger amplitude Fourier modes at low wavenumbers. The optimal non-modal growth theory of Andersson et al. (Reference Andersson, Berggren and Henningson1999) and Luchini (Reference Luchini2000) explains that larger spanwise wavelength disturbances will exhibit greater growth and peak further downstream. Indeed, the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither generates the largest streak growth by a substantial margin with a peak beyond the streamwise domain ( $R>1600$ ). The growth of the weaker zither streaks can be seen to be slowing towards the end of the streamwise domain.

The streak growth qualitatively agrees with the compiled Klebanoff streak data of Westin et al. (Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994), which shows linear growth with $x^{0.5}$ . Only three data-sets extend beyond $R=1500$ , with one set showing a peak and then decrease downstream while the other two show a rapid increase which probably indicates transition. Note the experimental measures of Klebanoff streaks are $u_{rms}$ and not directly the streak amplitude measured here.

Figure 18. (a) Standard deviation of streak strength ${\it\sigma}_{{\rm\Delta}{\it\delta}^{\ast }}$ versus $R$ . (b) Standard deviation of streak amplitude ${\it\sigma}_{A}$ versus $R$ . (c) Standard deviation of normalised skin friction ${\it\sigma}_{cf}/\overline{cf}$ versus $R$ . (d) Streak strength scaled to zither wake before contraction ( $As$ ) versus $R$ . Lines and markers as for figure 15.

Streak strength and amplitude as defined are measures of minimum and maximum deviation from the two-dimensional layer. The standard deviation of these quantities for steady streaks is likely to have a closer relation to the experimental measures of unsteady streaks (e.g.  $u_{rms}$ ) and any possible correlation with TS-wave suppression. The standard deviation of streak strength and amplitude are shown in figure 18. The standard deviation of streak strength ( ${\it\sigma}_{{\rm\Delta}{\it\delta}^{\ast }}$ ) at a given streamwise plane is defined by,

(4.3) $$\begin{eqnarray}{\it\sigma}_{{\rm\Delta}{\it\delta}^{\ast }}=\frac{1}{{\it\delta}_{b}^{\ast }}\sqrt{\frac{1}{L}\int _{0}^{L}{\it\delta}^{\ast 2}\,\text{d}z},\end{eqnarray}$$

and the standard deviation of streak amplitude ( ${\it\sigma}_{A}$ ) by,

(4.4) $$\begin{eqnarray}{\it\sigma}_{A}=\sqrt{\frac{1}{U^{2}L}\int _{0}^{L}\max _{y,z}(U-U_{b})^{2}\,\text{d}z}.\end{eqnarray}$$

The standard deviation measures are significantly less than the maximal measures but the same relative trends are observed.

Skin friction also provides a measure of streak growth. Figure 18 shows the variation in skin friction defined by,

(4.5a−c ) $$\begin{eqnarray}{\it\sigma}_{c_{f}}=\frac{2{\it\nu}}{U^{2}}\sqrt{\frac{1}{L}\int _{0}^{L}\left(\frac{\partial U}{\partial y}-\overline{\frac{\partial U}{\partial y}}\right)^{2}\text{d}z},\quad \overline{c_{f}}=\frac{2{\it\nu}}{U^{2}}\overline{\frac{\partial U}{\partial y}},\quad \overline{\frac{\partial U}{\partial y}}=\frac{1}{L}\int _{0}^{L}\frac{\partial U}{\partial y}\,\text{d}z,\end{eqnarray}$$

(4.6) $$\begin{eqnarray}{\rm\Delta}c_{f}=\frac{{\it\sigma}_{c_{f}}}{\overline{c_{f}}},\end{eqnarray}$$

where ${\rm\Delta}c_{f}$ is the ratio of skin-friction standard deviation to the spanwise mean value, and the wall-normal velocity gradients are measured at the wall. The ratio of ${\rm\Delta}c_{f}$ to ${\it\sigma}_{A}$ is not constant, varying between 1.65 and 1.9 for the current zither streaks. Skin friction varies between 1.5 % and 18 % at $R=1550$ , an order of magnitude variation. Bradshaw (Reference Bradshaw1965) found turbulent skin-friction variation exceeding 10 % when screens with ${\it\beta}<57\,\%$ were installed. Only the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither exceeds 10 %.

Based on the magnitude of skin-friction variation for the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither, it seems likely that the cause of the spanwise variation in the results of Bradshaw (Reference Bradshaw1965) is also present in the current simulations. Bradshaw (Reference Bradshaw1965) attributed the variation to a spatial ‘instability’ downstream of the screen, leading to the coalescence of jets. The current simulations suggest the cause can be interpreted as being due to the variation of wire drag and wire position across the zither.

4.2. Streak amplitude related to the zither

Ideally, a simple relation can be found to relate streak amplitude to the zither geometry. Relating streak growth to a single measure of velocity variation in the wake will fail, as streak growth is strongly dependent on spanwise wavelength. The relation must be made in terms of vorticity and consider the spectral distribution. However, the current zither wakes do share a similar spanwise wavelength as the zither position is constant and the zither wake wavelength formula was found to be accurate (see figure 8). The contraction and leading edge geometry are also identical. Thus, the streak amplitude can be scaled by the wake strength entering the contraction. If all zither wakes maintain the same wake decay rate, and the streak growth in the test-section layer is linear with amplitude, then a measure of streak amplitude scaled by a measure of the zither wake entering the contraction should collapse to a single curve. Figure 18(d) shows the standard deviation of streak amplitude ( ${\it\sigma}_{A}$ ) scaled by the standard deviation of the zither wake strength entering the contraction ( ${\it\sigma}_{{\rm\Delta}u/U\,\mathit{in}}$ ),

(4.7) $$\begin{eqnarray}As=\frac{{\it\sigma}_{A}}{{\it\sigma}_{{\rm\Delta}u/U\,\mathit{in}}}.\end{eqnarray}$$

The ${\it\beta}=66.67\,\%$ and ${\it\beta}=60\,\%$ zithers with ${\it\sigma}_{{\it\delta}}=0.025$ and ${\it\sigma}_{{\it\delta}}=0.05$ streak growth collapses with this scaling. However, the ${\it\beta}=60\,\%$ zithers with ${\it\sigma}_{{\it\delta}}=0.075$ and ${\it\sigma}_{{\it\delta}}=0.1$ show an increased scaled streak growth. This indicates a stronger wake, due to a reduced wake decay rate far downstream of the zither. Decreasing ${\it\beta}$ for a given ${\it\sigma}_{{\it\delta}}$ also increases the scaled streak growth. This is highlighted by the ${\it\beta}=40\,\%$ with ${\it\sigma}_{{\it\delta}}=0.025$ and ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zithers that have nearly identical scaled-streak-growth. Their scaled-streak-amplitude is more than double that for the ${\it\beta}=66.67\,\%$ with ${\it\sigma}_{{\it\delta}}=0.05$ zither at $R=1500$ . Although the dominant Fourier mode in the wake is of a similar wavelength, the spectral distribution is not. Figure 14 shows they have larger amplitude Fourier coefficients below $k=10$ , creating larger streaks further downstream. A simple relation between the zither wake strength and the streak amplitude cannot be found.

4.3. Spanwise mean quantities

Figure 19(a) shows the zither streaks have a negligible effect on the spanwise mean displacement thickness, consistent with experimental observations of Klebanoff streaks (Westin et al. Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994). However, figure 19(b) reveals the mean shape factor decreases with streamwise distance relative to the two-dimensional layer. Stronger streaks produce a greater decrease. Westin et al. (Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994) reported a decreasing shape factor when FST induced Klebanoff streaks are present, observing a shape factor of 2.41 at $R=1260$ . Only the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither produces a comparable mean shape factor but it occurs further downstream.

Figure 19. (a) Spanwise mean displacement thickness versus $R$ . (b) Spanwise mean shape factor ( $H$ ) versus $R$ . – ⋅ – ⋅ – Two-dimensional base-flow. Other lines and markers as for figure 15.

4.4. Streak wall-normal velocity profiles

Streaks cause a deviation in the velocity profile relative to the undisturbed two-dimensional base flow. The deviation at a given spanwise position is defined by,

(4.8) $$\begin{eqnarray}\frac{{\rm\Delta}U}{U_{e}}=\frac{U-U_{b}}{U_{e}},\end{eqnarray}$$

where $U_{e}$ is the edge velocity of the layer. Figure 20 shows zither streak profiles at $R=1350$ and a given spanwise position are either all in excess or decrement. The spanwise average of the profiles exhibits an ‘s’-shape, except for the ${\it\beta}=60\,\%$ with ${\it\sigma}_{{\it\delta}}=0.025$ and ${\it\sigma}_{{\it\delta}}=0.05$ zither streaks that have a slight excess across the layer. The ‘s’-shape develops further downstream when the streak amplitude increases. From figure 20, increasing ${\it\sigma}_{{\it\delta}}$ (which increases streak amplitude) increases the magnitude of the ‘s’-shape maximum (lower part of ‘s’) and minimum (upper part of ‘s’), approximately 30-fold for the ${\it\beta}=50\,\%$ streaks. With increasing ${\it\sigma}_{{\it\delta}}$ , the ‘s’-shape maximum moves towards the wall, and the minimum away. This is due to the maximum of local profiles in excess shifting towards the wall, and the minimum of profiles in decrement away. This agrees with the simulations by Jacobs & Durbin (Reference Jacobs and Durbin2001) where backwards jets (unsteady streaks) are lifted to the boundary-layer edge prior to breakdown, and the PIV of Nolan & Walsh (Reference Nolan and Walsh2012) that observed the excess maximum to move towards the wall while the low-speed streak moves away.

The ‘s’-shaped mean profile is similar to the observations of time-averaged, Klebanoff streaks made by Westin et al. (Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994) using a grid in the test section to elevate the FST. At $R=1350$ the maximum of the ‘s’-shape for the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither streaks is approximately $\max ({\rm\Delta}U/U_{e})\approxeq 0.01$ and the mean shape factor is 2.52, similar to Westin et al. (Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994) ( $\max ({\rm\Delta}U/U_{e})\approxeq 0.011$ when $H=2.53$ ). However, Kendall (Reference Kendall1985, Reference Kendall1998) measured time-averaged velocity profiles in the presence of Klebanoff streaks at $R=980$ , with FST elevated by a grid upstream of the contraction, to be in decrement across the entire layer ( $\min ({\rm\Delta}U/U_{e})\approxeq -0.02$ ). The zither streak results combined with Westin et al. (Reference Westin, Boiko, Klingmann, Kozlov and Alfredsson1994) suggest the difference cannot be attributed solely to a grid location upstream of the contraction.

Figure 20. Wall-normal profiles of ${\rm\Delta}U/U_{e}$ , defined by (4.8), at various spanwise positions and $R=1350$ . Lines: ——  ${\rm\Delta}U/U_{e}$ at some spanwise position; - - - - spanwise mean. (a) ${\it\beta}=40\,\%$ , ${\it\sigma}_{{\it\delta}}=0.025$ ; (b) ${\it\beta}=50\,\%$ , ${\it\sigma}_{{\it\delta}}=0.025$ ; (c) ${\it\beta}=50\,\%$ , ${\it\sigma}_{{\it\delta}}=0.050$ ; (d) ${\it\beta}=50\,\%$ , ${\it\sigma}_{{\it\delta}}=0.075$ ; (e) ${\it\beta}=60\,\%$ , ${\it\sigma}_{{\it\delta}}=0.025$ ; (f) ${\it\beta}=60\,\%$ , ${\it\sigma}_{{\it\delta}}=0.050$ ; (g) ${\it\beta}=60\,\%$ , ${\it\sigma}_{{\it\delta}}=0.075$ ; (h) ${\it\beta}=60\,\%$ , ${\it\sigma}_{{\it\delta}}=0.100$ ; (i) ${\it\beta}=66.67\,\%$ , ${\it\sigma}_{{\it\delta}}=0.025$ .

5.1. Two-dimensional base-flow linear stability

The linear stability of the two-dimensional base-flow can be measured by the $N$ -factor, the ratio of disturbance amplitude growth from Branch 1, which is the streamwise position when the disturbance first becomes unstable. The $N$ -factor is defined by,

(5.1) $$\begin{eqnarray}N\text{-}\text{factor}=\int _{Branch\,1}^{x}{\it\alpha}_{i}({\it\zeta})\,\text{d}{\it\zeta}.\end{eqnarray}$$

The CFD base flow is of sufficient accuracy to provide results of at least qualitative accuracy (see Pook Reference Pook2013). A resolution check is performed later.

5.2. BiGlobal resolution of the eigenvalue spectrum

The ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither streak base flow at $R=940$ ( $Re_{{\it\delta}^{\ast }}=1615$ ) is used to examine spanwise resolution requirements. The streak peak amplitude ( $A$ ) is 12.5 % and the standard deviation of amplitude ( ${\it\sigma}_{A}$ ) is 5.4 %. The BiGlobal eigenvalue spectrum is investigated for a disturbance non-dimensional frequency, $F=65$ where,

(5.2) $$\begin{eqnarray}F=\frac{{\it\omega}{\it\nu}}{U^{2}}\times 10^{6}.\end{eqnarray}$$

The dimensional circular frequency is 326.8 rad $^{-1}$ . For this frequency and position, the two-dimensional base flow is unstable (slightly upstream of Branch 2).

The BiGlobal spectrum, calculated using the modes $k=-5$ to 5, is shown in figure 21 ( ${\it\alpha}_{i}<0$ indicates streamwise growth). The streaky base flow is not symmetric, which prevents using the even/odd assumption to reduce the eigenvalue problem size. The QZ eigenvalue solver (Matlab eig function) is used. The spectrum is similar in appearance to that of two-dimensional Blasius flow. The discrete approximation to the continuous modes can be seen to the left of the spectrum. The spread of the continuous modes is due to the wall-normal differencing not capturing the oscillations of the eigenvectors. Four unstable, discrete eigenvalues can be seen near ${\it\alpha}_{r}{\it\delta}_{b}^{\ast }\approxeq 0.3$ .

The eigenvalue spectrum in the region of the unstable eigenvalues is resolved more accurately using an increasing number of spanwise modes and the Arnoldi eigenvalue solver (Matlab eigs function). The increased resolution produces a ‘curved line’ of discrete eigenvalues. Similar to the spectrum for TS modes with varying spanwise wavenumber on a two-dimensional base flow. The most unstable eigenvalue appears by itself. In the limit of two-dimensional flow, without providing direct proof, this mode would correspond to the two-dimensional TS mode. Tracking this mode back towards the leading edge (streak strength decreasing) produces an eigenvector with minimal spanwise variation and an eigenvalue close to that for the two-dimensional layer. It will be called the streaky TS mode following Cossu & Brandt (Reference Cossu and Brandt2004) and labelled A in figure 21. The remaining eigenvalues can be seen to appear in pairs. In the limit of two-dimensional flow they would be the oblique TS modes. They are split because the streaky zither base flow is not spanwise symmetric. They will be called the streaky oblique TS modes and two examples are labelled B and C in figure 21.

Figure 21. The effect of spanwise resolution (Fourier modes $k=-N$ to $N$ ) on the eigenvalues for the ${\it\beta}=50\,\%$ , ${\it\sigma}_{{\it\delta}}=0.075$ zither at $Re_{{\it\delta}^{\ast }}=1615$ ( $R=940$ ). Perturbation frequency is $F=65$ , ${\it\omega}=326.8~\text{rad}^{-1}$ . A is streak TS mode. B and C are first streaky oblique TS modes. Markers: ●  $N=5$ and QZ solver; $+$ $N=10$ and Arnoldi Solver; ◻  $N=15$ and Arnoldi Solver; ○  $N=20$ and Arnoldi Solver; $\times$ $N=25$ and Arnoldi Solver.

The streaky oblique modes require increased resolution to resolve compared to the streaky TS mode. The convergence of the streaky TS-mode eigenvalue is shown in table 3. It does not change significantly beyond $k=20$ . The majority of the streak base-flow energy is in modes $k\leqslant 15$ .

Table 3. Streaky TS-mode convergence with BiGlobal spanwise resolution.

The eigenvectors of labelled eigenvalues A, B and C are shown in figure 22. The $u_{rms}$ maxima of the streaky TS is located in high-speed streak regions, while regions of increased $u_{rms}$ for the streaky oblique TS are offset from each other in the spanwise direction but remain located in high-speed streak regions. The linear summation of the oblique modes is also shown in figure 22. The eigenvectors have been scaled to have the same total energy prior to addition. The majority of elevated $u_{rms}$ for the combined streaky oblique modes is located to the left of $z/d=500$ while the streaky TS mode is most elevated in regions to the right of $z/d=500$ . The maximum spanwise velocity component of the streaky oblique modes, ${\sim}0.6u_{rms}$ , is considerably higher than that for the streaky TS mode, ${\sim}0.4u_{rms}$ (not shown). Increasing streak amplitude by decreasing zither open-area ratio or increasing ${\it\sigma}_{{\it\epsilon}}$ , increases the distortion of the streaky TS relative to two-dimensional TS.

Figure 22. (a) Contours of streamwise velocity for the ${\it\beta}=50\,\%$ , ${\it\sigma}_{{\it\delta}}=0.075$ zither base flow at $Re_{{\it\delta}^{\ast }}=1615$ ( $R=940$ ). (b) Streaky TS mode, labelled A, $u_{rms}$ . (c) Streaky oblique TS mode, labelled B, $u_{rms}$ . (d) Streaky oblique TS mode, labelled C, $u_{rms}$ . (e) Addition of modes B and C, $u_{rms}$ .

Figure 23. $N$ -factor curves for the zither streak base flows. Lines: —— two-dimensional base flow; 

$F=30$ disturbance for ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.025$ zither CFD base flow with increased mesh resolution; Other lines and markers as for figure 15.

5.3. Zither streak $N$ -factor

The effect of the zither wake on streaky TS growth and predicted transition is evaluated with $N$ -factor curves. Disturbance frequencies $F=30$ , 45, 65 are computed and shown in figure 23. The $F=30$ disturbance is also calculated using a base-flow mesh refined in each spatial direction by a factor of 1.5 in the boundary-layer region (total mesh size ${\sim}125$ million CV, ${\sim}60$ CV across the boundary layer). The mesh refinement modifies the calculated $N$ -factor at $R=1550$ by approximately 1 %, an inconsequential error for the current level of analysis. Increasing the resolution of the PSE-3D to 95 collocation points and $k=32$ Fourier modes changes the $N$ -factor by 0.5 %.

The $F=65$ disturbance reaches Branch 2, the streamwise position where the disturbance becomes stable again, just downstream of $R=1000$ . The greatest suppression of the streaky TS is observed for the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither. However, the decrease in the Branch 2 $N$ -factor is only 0.1.

The $F=45$ TS mode reaches Branch 2 just downstream of $R=1300$ on the two-dimensional base flow. At $R=1300$ , the standard deviation of streak amplitude ( ${\it\sigma}_{A}$ ) for the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither has reached 7.8 %. The greater streamwise extent of TS and streak growth leads to greater suppression. For the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither, the reduction in the Branch 2 $N$ -factor is 0.65. If transition were to occur due to this frequency, $N\text{-}\text{factor}=5$ for the two-dimensional base flow, then the predicted position to reach the same mode amplitude ratio would be delayed from $R=1170$ to $R=1250$ . For a wind-tunnel velocity of $8.6$  m s $^{-1}$ and air, this corresponds to a shift from $x=2.33$  m to $x=2.66$  m; a delay of 15 %. For the weaker zither streaks, the delay would be less than 3.5 %.

The $F=30$ disturbance Branch 2 $N$ -factor is not observed in the streamwise domain. However, the suppression of TS growth is considerable. At $R=1550$ , the maximum reduction in $N$ -factor is 1. If transition were to occur due to this frequency for the two-dimensional base flow, $N\text{-}\text{factor}=6.5$ at $R=1470$ , then the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither would reach the same amplitude ratio at $R=1560$ . For the stated tunnel conditions, this corresponds to $x=3.68$  m and $x=4.15$  m. A delay of nearly 500 mm, or 12.6 %. For all other zither wakes, the predicted delay is less than 100 mm, a delay of 2.7 %.

The plausibility of a significant shift in the transition location due to a zither more than 2 m upstream is supported by experimental observations. Watmuff (Reference Watmuff1998) replaced screens in his wind tunnel and observed a 3-fold reduction in $u_{rms}$ in the layer, indicating significantly weaker streaks. For this intermediate configuration (further flow quality improvements were made), turbulent bursting was observed to shift from $R\approx 1500$ upstream to $R\approx 1220$ . For the two-dimensional layer, this corresponds to a reduction in $N$ -factor from $7.3$ ( $F=30$ ) to 5.4 ( $F=45$ ). Assuming transition at $R=1500$ for the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither gives an $N$ -factor of 6. Removing the streaks, an $N$ -factor of 6 would predict transition at approximately $R=1300$ due to $F=42.5$ .

The streak influence on TS-wave secondary instability may be important. Liu, Zaki & Durbin (Reference Liu, Zaki and Durbin2008a ,Reference Liu, Zaki and Durbin b ) found that increasing streak width would promote the growth of the TS-wave secondary instability and transition, while suppressing growth of the TS wave. However, for TS-wave amplitudes of 1 %, streak amplitudes in excess of 13 % were required to destabilise the TS-wave secondary instability. Considering that the zither streaks are weak, and 75–85 % of the pre-transitional region is governed by linear stability theory (Reed, Saric & Arnal Reference Reed, Saric and Arnal1996), i.e. TS amplitudes ${\lesssim}$ 1 %, then the $N$ -factor analysis which describes this region may provide a reasonable transition prediction.

5.4. Streak secondary instability

An extensive search for a streak secondary instability of the ${\it\beta}=50\,\%$ with ${\it\sigma}_{{\it\delta}}=0.075$ zither streak base flow at $R=1550$ was conducted. The QZ method was used with up to 25 Fourier modes, and the Arnoldi method was used with up to 45 Fourier modes, searching the phase velocity region of 0.6–0.85. Streak secondary instability is known to occur at these phase velocities (Andersson et al. Reference Andersson, Brandt, Bottaro and Henningson2001). No secondary instability was detected. The streaky base flow is still quite weak and for steady streaks, only the results of Vaughan & Zaki (Reference Vaughan and Zaki2011) would suggest a streak secondary instability may be present at the current streak amplitudes.