3.1 Outline of the asymptotic theory

In this section we seek a time-periodic solution of (2.1) with fixed frequency $\unicode[STIX]{x1D6FA}_{0}$ using a large-Reynolds-number asymptotic approach. The overall asymptotic structure is similar to that found for ASBL by Deguchi & Hall (Reference Deguchi and Hall2014a ); namely we assume that wave-like coherent structures are produced in a ‘production layer’ located a distance $O(\ln R)$ from the centre of the jet. A sketch of the different asymptotic regions is given in figure 1. We denote the basic planar shear flow as $u_{b},v_{b}$ , where the leading-order part is given by (2.2). The key step in finding the location of the production layer is the far-field form of the basic flow:

(3.1a,b ) $$\begin{eqnarray}\displaystyle u_{b}\rightarrow FG\text{e}^{-2\unicode[STIX]{x1D702}},\quad v_{b}\rightarrow -F,\quad \text{as}~\unicode[STIX]{x1D702}\rightarrow \infty , & & \displaystyle\end{eqnarray}$$

where $F(x)=(2/3)x^{-2/3}$ , $G(x)=4x^{1/3}$ . An important point to note here is that the streamwise component approaches a free-stream speed exponentially as is the case for ASBL, and the effective Reynolds number becomes small in the far field. In the production layer, the flow is governed by the Navier–Stokes equations with an effective Reynolds number of unity, and thus wave-like free-stream coherent structures can be generated. The position of the layer can easily be found by a simple order-of-magnitude analysis. We first note that in order to recover the streamwise derivative in the right-hand side of (2.1), we must set $\unicode[STIX]{x2202}_{x}\sim O(R^{1/2})$ . Then for the convective term to balance with the viscous terms in the equations of motion we require $\text{e}^{-2\unicode[STIX]{x1D702}}\unicode[STIX]{x2202}_{x}\sim O(R^{0})$ . Combining these balances, we have

(3.2) $$\begin{eqnarray}\displaystyle \text{e}^{-2\unicode[STIX]{x1D702}}\sim O(R^{-1/2}), & & \displaystyle\end{eqnarray}$$

from which we find that the production layer is located at $\unicode[STIX]{x1D702}\sim O(\ln R)$ .

Of course, since the jet problem is non-parallel, the flow structure beneath the production layer is more complicated than is the case for ASBL. The free-stream coherent structures theory for the spatially growing problem was developed by Deguchi & Hall (Reference Deguchi and Hall2015a ) for a quite general class of two-dimensional boundary layers. However, for the reasons given below a direct application of that theory to the jet problem is not possible.

The first reason is that for the boundary layer flows the decay of the basic flow correction is Gaussian rather than exponential. As found earlier (Deguchi & Hall Reference Deguchi and Hall2014b ) for the Burgers vortex sheet, even for this case we can determine the location of the production layer because locally the behaviour of the basic flow correction is of exponential form in terms of a scaled variable. However, since the basic flow beneath the production layer is not of exponential form, the induced vortex structure there develops in a quite different way than was the case for ASBL. In the adjustment zone between the near-wall boundary layer and the production layer the vertical and spanwise scales are different. Surprisingly, that difference allows us to solve the adjustment problem analytically to find the maximum amplitude of the induced streak there. In contrast, for the jet problem the far-field approximation of the basic flow (3.1) is valid except for the boundary layer near the centre of the jet. This means that there is no need for an adjustment layer, and thus the streak beneath the production layer takes its maximum in the boundary layer as for the ASBL case. The non-parallel effect becomes evident in the boundary layer, because the natural streamwise scale of the flow here becomes $O(1)$ in $x$ . The vertical and spanwise scales are comparable there, and we must therefore use a numerical approach.

The second reason why the jet and growing boundary layer problems differ is the difference between the free-stream speeds in the jet and boundary layer problems. In order to construct a periodic solution in the production layer of non-parallel flows, we describe travelling wave states varying locally using a Wentzel–Kramers–Brillouin (WKB) method. The streamwise wavenumber of the local travelling wave must be chosen appropriately to construct the global solution. In the boundary layer problem, the local wavenumber can be simply fixed by the free-stream speed and the global frequency of the periodic solution. However, since for the jet problem the free-stream speed is zero, clearly the same technique cannot be used. Of course this problem is inherently associated with non-parallel effects. The analysis in the next section shows that the global frequency of the jet problem can be related to the phase speed of the local travelling waves, rather than the wavenumber.

3.2 Production layer analysis

Now let us derive the canonical problem that governs nonlinear free-stream coherent structures in the production layer. Following the discussion in the previous section, we define the scaled vertical coordinate $Y\equiv Fy-\ln (R^{1/2}G)$ . Since $2\unicode[STIX]{x1D702}=Fy$ , the large- $\unicode[STIX]{x1D702}$ form of the basic flow (3.1) then becomes

(3.3a,b ) $$\begin{eqnarray}\displaystyle u_{b}=R^{-1/2}F\text{e}^{-Y}+\cdots \,,\quad v_{b}=-F+\cdots \,. & & \displaystyle\end{eqnarray}$$

Beneath the production layer, we assume that the flow is simply the unperturbed jet flow to leading order with a small correction induced by the flow in the production layer.

In order to have a unit-Reynolds-number Navier–Stokes problem in the production layer, the scaled spanwise variable $Z=Fz$ is now introduced. In order to allow for a wave-like dependence of the flow in the streamwise direction, we introduce a WKB phase function $\unicode[STIX]{x1D6F7}\in [0,2\unicode[STIX]{x03C0}]$ defined by

(3.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}=-R^{1/2}\left(\int ^{x}[F\unicode[STIX]{x1D6FC}_{0}(x)+\cdots \,]\,\text{d}x-R^{-1/2}\unicode[STIX]{x1D6FA}_{0}t\right), & & \displaystyle\end{eqnarray}$$

so that $\unicode[STIX]{x2202}_{x}=-R^{1/2}F\unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6F7}}+\cdots \,$ . Here $\unicode[STIX]{x1D6FC}_{0}$ is the local streamwise wavenumber of the travelling wave, and the minus sign has been introduced to aid comparison with Deguchi & Hall (Reference Deguchi and Hall2014a ). Note that $\unicode[STIX]{x1D6FC}_{0}$ is real, and varies in the streamwise direction, whereas the frequency $\unicode[STIX]{x1D6FA}_{0}$ is constant. We now substitute the expansions

(3.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle u=-R^{-1/2}FU(\unicode[STIX]{x1D6F7},Y,Z)+\cdots \,,\quad v=FV(\unicode[STIX]{x1D6F7},Y,Z)+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle w=FW(\unicode[STIX]{x1D6F7},Y,Z)+\cdots \,,\quad p=F^{2}P(\unicode[STIX]{x1D6F7},Y,Z)+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.6a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{0},\quad \unicode[STIX]{x1D6FD}=\frac{\unicode[STIX]{x1D6FD}_{0}}{F},\quad c_{1}=\frac{\unicode[STIX]{x1D6FA}_{0}}{\unicode[STIX]{x1D6FC}_{0}F^{2}}. & & \displaystyle\end{eqnarray}$$

The leading-order problem then becomes

(3.7a ) $$\begin{eqnarray}\displaystyle ([\boldsymbol{U}+c_{1}\boldsymbol{i}]\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{U}=-\unicode[STIX]{x1D735}P+\unicode[STIX]{x1D6FB}^{2}\boldsymbol{U},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}=0, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D735}=(\unicode[STIX]{x1D6FC}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6F7}},\unicode[STIX]{x2202}_{Y},\unicode[STIX]{x2202}_{Z})$ . Note here that $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},c_{1}$ defined above are functions of $x$ , but since the wave operates on a shorter streamwise length scale than does the basic flow, the variables $U,V,W,P$ depend only parametrically on $x$ . Therefore as the wave system moves downstream at each $x$ we must determine the solution of (3.7a ) in the periodic domain $\unicode[STIX]{x1D6F7}\in [0,2\unicode[STIX]{x03C0}],Z\in [0,2\unicode[STIX]{x03C0}F/\unicode[STIX]{x1D6FD}]$ subject to the far-field conditions

(3.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle (U,V,W)\rightarrow (0,-1,0)\quad \text{as}~Y\rightarrow \infty , & \displaystyle\end{eqnarray}$$

(3.7c ) $$\begin{eqnarray}\displaystyle & \displaystyle (U,V,W)\rightarrow (-\text{e}^{-Y},-1,0)\quad \text{as}~Y\rightarrow -\infty . & \displaystyle\end{eqnarray}$$

The nonlinear eigenvalue problem specified by (3.7) is identical to that defined by (4.5)–(4.10) in Deguchi & Hall (Reference Deguchi and Hall2014a ), except for a typo (the minus sign in front of $c_{1}$ ) in that paper. Therefore, the nonlinear eigenvalue $c_{1}$ is to be determined as a function of the wavenumbers $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ solving ASBL. However, the numerical results obtained there cannot be directly used here because, when the boundary layer is growing in $x$ , the wavenumbers and wavespeed satisfy a further constraint, and interestingly it turns out that the constraint for the jet problem is quite distinct from that for the growing boundary layer problem discussed by Deguchi & Hall (Reference Deguchi and Hall2014a , Reference Deguchi and Hall2015a ). In the boundary layer problem the free-stream speed is not zero and so $\unicode[STIX]{x1D6FC}$ is determined locally by the condition that the structure in the production layer moves downstream with the speed of the unperturbed flow in the production layer. The spanwise and streamwise wavenumbers are then related by the fact that the boundary layer grows in the streamwise direction. Thus, in the Blasius case it turns out that moving downstream, $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{-1}$ must be a constant. That constraint enabled the direct use of the results of Deguchi & Hall (Reference Deguchi and Hall2014a ) which were computed with $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{-1}=0.5$ . The corresponding constraint here follows directly from (3.6) by eliminating $x$ to give $c_{1}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{-2}=\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D6FD}_{0}^{-2}$ . Thus as a wave of constant frequency $\unicode[STIX]{x1D6FA}_{0}$ and constant spanwise wavenumber $\unicode[STIX]{x1D6FD}_{0}$ moves downstream we must solve (3.7) subject to $c_{1}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{-2}=\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D6FD}_{0}^{-2}$ at each value of $x$ . Since we know that $\unicode[STIX]{x1D6FD}$ is given as a function of $x$ by (3.6) the extra constraint determines $\unicode[STIX]{x1D6FC}$ as a function of $x$ . The effect of this is to make the solution of (3.7) significantly more difficult than the case discussed by Deguchi & Hall (Reference Deguchi and Hall2014a , Reference Deguchi and Hall2015a ).

Figure 2.

The travelling wave solutions of the ASBL problem. Note that the definition of the Reynolds number and the scale of the coordinates used here are different to those defined for the jet problem. (a) The wavenumbers selected in the computation. The red solid curve corresponds to the jet problem for $c\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{-2}=120$ , whilst the dashed line corresponds to the results computed by Deguchi & Hall (Reference Deguchi and Hall2015a ) for Blasius problem $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FD}=0.5$ . Asymptotic convergence is checked in a range of Reynolds numbers from 100 000 to 200 000. The numbers of Fourier modes used in $x$ and $z$ are 14 and 26, respectively. In the vertical direction 180 Chebyshev modes are used. (b,c) The flow field of the solutions at the corresponding points indicated in (a). The interval $x\in [0,2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FC}]$ , $z\in [0,2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FD}]$ is used. The green surfaces are 50 % maximum absolute value of streamwise vorticity. The red/blue surfaces are 25 % maximum/minimum of streak, namely perturbation of streamwise velocity to the streamwise laminar flow, $1-\text{e}^{-y}$ . Reynolds numbers used are (b) $100\,000$ , (c) $200\,000$ .

Figure 3.

The value of $K$ along the solid red curve in figure 2(a).

Results of our computations for the case $c_{1}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{-2}=\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D6FD}_{0}^{-2}=120$ are denoted by the red solid curve in figure 2(a). At any point on the curve the local value of $c_{1}$ is found from $c_{1}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{-2}=120$ . This solution curve was computed starting from the results for the Blasius problem given in Deguchi & Hall (Reference Deguchi and Hall2015a ), and shown by the black dashed line in the figure. Using Newton’s method we can continue the solution branch in a certain range of wavenumbers to find the corresponding locus of $c_{1}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}^{-2}$ . The detail of the computational method is given in Deguchi & Hall (Reference Deguchi and Hall2014a ). The points (b) and (c) on the red solid curve are close to the saddle-node points where solutions appear; the flow visualisations at these points are given in figure 2(b,c). Throughout the paper the lower branch states are chosen to visualise the flow. The green surfaces represent the isosurfaces of the streamwise vorticity that shows the presence of free-stream coherent structures in the production layer. We have extracted that structure from the numerical result to construct the production layer solution of the jet problem. The blue and red surfaces show the induced near-wall streak by the free-stream coherent structures; these are the isosurfaces of the streamwise velocity deviation from the basic flow. Although these surfaces show the presence of the streak growth beneath the production layer, it should be noted that the near-wall part of the ASBL solution cannot be a good approximation of the jet problem because it takes no account of non-parallel effects.

3.3 Boundary layer analysis

Deguchi & Hall (Reference Deguchi and Hall2014a ) showed that as $Y\rightarrow -\infty$ the production layer solution behaves like

(3.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle U\rightarrow -\text{e}^{-Y}-K(2\unicode[STIX]{x1D714})^{-1}\text{e}^{(\unicode[STIX]{x1D714}-1)Y}\cos (2\unicode[STIX]{x1D6FD}Z)+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle V\rightarrow -1+K\text{e}^{\unicode[STIX]{x1D714}Y}\cos (2\unicode[STIX]{x1D6FD}Z)+\cdots \,. & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D714}=(\sqrt{1+16\unicode[STIX]{x1D6FD}^{2}}-1)/2>0$

$\unicode[STIX]{x1D6FD}<1/\sqrt{2}$

$K$

(3.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle u\rightarrow FG\text{e}^{-2\unicode[STIX]{x1D702}}+KF(GR^{1/2})^{-\unicode[STIX]{x1D714}}\frac{G}{2\unicode[STIX]{x1D714}}\text{e}^{(\unicode[STIX]{x1D714}-1)2\unicode[STIX]{x1D702}}\cos (2\unicode[STIX]{x1D6FD}_{0}z)+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle v\rightarrow -F+KF(GR^{1/2})^{-\unicode[STIX]{x1D714}}\text{e}^{\unicode[STIX]{x1D714}2\unicode[STIX]{x1D702}}\cos (2\unicode[STIX]{x1D6FD}_{0}z)+\cdots \,. & \displaystyle\end{eqnarray}$$

$x$

$x$

In the boundary layer $u_{b}$ becomes $O(1)$ and thus the $x$ derivative in the convective term is no longer negligible. The leading-order equations are the Görtler vortex equations derived in Hall (Reference Hall1983, Reference Hall1988) with zero Görtler number:

(3.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x2202}_{t}+u\unicode[STIX]{x2202}_{x}+v\unicode[STIX]{x2202}_{y}+w\unicode[STIX]{x2202}_{z})u=u_{yy}+u_{zz}, & \displaystyle\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x2202}_{t}+u\unicode[STIX]{x2202}_{x}+v\unicode[STIX]{x2202}_{y}+w\unicode[STIX]{x2202}_{z})v=-p_{y}+v_{yy}+v_{zz}, & \displaystyle\end{eqnarray}$$

(3.10c ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x2202}_{t}+u\unicode[STIX]{x2202}_{x}+v\unicode[STIX]{x2202}_{y}+w\unicode[STIX]{x2202}_{z})w=-p_{z}+w_{yy}+w_{zz}, & \displaystyle\end{eqnarray}$$

(3.10d ) $$\begin{eqnarray}\displaystyle & \displaystyle u_{x}+v_{y}+w_{z}=0. & \displaystyle\end{eqnarray}$$

$x$

If we expand

(3.11a ) $$\begin{eqnarray}\displaystyle & \displaystyle u=u_{b}+R^{-\unicode[STIX]{x1D714}/2}\widetilde{u}\cos (2\unicode[STIX]{x1D6FD}_{0}z)+\cdots \,, & \displaystyle\end{eqnarray}$$

(3.11b ) $$\begin{eqnarray}\displaystyle & \displaystyle v=v_{b}+R^{-\unicode[STIX]{x1D714}/2}\widetilde{v}\cos (2\unicode[STIX]{x1D6FD}_{0}z)+\cdots \,, & \displaystyle\end{eqnarray}$$

$(\widetilde{u},\widetilde{v})$

(3.12a ) $$\begin{eqnarray}\displaystyle u_{b}\widetilde{u}_{x}=(\unicode[STIX]{x2202}_{y}^{2}-4\unicode[STIX]{x1D6FD}_{0}^{2}-v_{b}\unicode[STIX]{x2202}_{y}-u_{bx})\widetilde{u}-u_{by}\widetilde{v}, & & \displaystyle\end{eqnarray}$$

(3.12b ) $$\begin{eqnarray}\displaystyle & & \displaystyle (u_{b}(\unicode[STIX]{x2202}_{y}^{2}-4\unicode[STIX]{x1D6FD}_{0}^{2})-u_{byy})\widetilde{v}_{x}-2(u_{bx}\unicode[STIX]{x2202}_{y}+u_{bxy})\widetilde{u}_{x}\nonumber\\ \displaystyle & & \displaystyle \qquad =(\unicode[STIX]{x2202}_{y}^{2}-4\unicode[STIX]{x1D6FD}_{0}^{2}-v_{b}\unicode[STIX]{x2202}_{y}+u_{bx})(\unicode[STIX]{x2202}_{y}^{2}-4\unicode[STIX]{x1D6FD}_{0}^{2})\widetilde{v}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad +\,(u_{bxyy}+u_{bxy}\unicode[STIX]{x2202}_{y})\widetilde{v}+(v_{bx}(\unicode[STIX]{x2202}_{y}^{2}+4\unicode[STIX]{x1D6FD}_{0}^{2})+u_{bxxy})\widetilde{u}.\end{eqnarray}$$

$(x,\unicode[STIX]{x1D702})$

$x$

(3.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}\rightarrow KFG^{1-\unicode[STIX]{x1D714}}(2\unicode[STIX]{x1D714})^{-1}\text{e}^{(\unicode[STIX]{x1D714}-1)2\unicode[STIX]{x1D702}},\quad \widetilde{v}\rightarrow KFG^{-\unicode[STIX]{x1D714}}\text{e}^{\unicode[STIX]{x1D714}2\unicode[STIX]{x1D702}},\quad \text{as}~\unicode[STIX]{x1D702}\rightarrow \infty , & \displaystyle\end{eqnarray}$$

(3.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{y}=\widetilde{v}=0,\quad \text{at}~y=0, & \displaystyle\end{eqnarray}$$

$y$

$K(x)$

$10^{-6}$

Figure 4.

The streak amplitude $\widetilde{u}$ computed by the boundary layer asymptotic problem.

We start the production layer interaction from the point (b) in figure 2(a). Without loss of generality we can choose the starting point at $x=1$ . We assume that before this point the flow is laminar. That choice of the starting point requires us to select the parameters of the global problem as $\unicode[STIX]{x1D6FD}_{0}=0.21\times 2/3=0.14$ ; see (3.6). Also, since we fix $\unicode[STIX]{x1D6FA}_{0}\unicode[STIX]{x1D6FD}_{0}^{-2}=120$ to solve the production layer problem, $\unicode[STIX]{x1D6FA}_{0}=120\times 0.14^{2}=2.352$ . The resultant streak amplitude solution $\widetilde{u}$ is shown in figure 4. Here we used 400 points in $\unicode[STIX]{x1D702}\in [0,8]$ , but the result is insensitive to the number of points and the upper bound of the vertical domain size. Consistent with the theory we see that a large-amplitude streak is generated. The origin of the streak growth is the last term in (3.12a ), where the small vertical perturbation velocity interacts with the basic flow. The vertical perturbation decays exponentially towards the centre of the jet, but since the basic flow grows at faster rate, the forcing term becomes large in the boundary layer. However, the basic flow growth is suppressed when $\unicode[STIX]{x1D702}$ becomes small, and thus the growth of the streak is also suppressed there as well.