2.1. Flow decomposition and governing equations

The instantaneous field $(\boldsymbol{u},p)$ is composed of a mean flow $(\bar{\boldsymbol{U}},\bar{P})$ , quasi-periodic coherent motion $(\tilde{\boldsymbol{u}},\tilde{p})$ representing CS and small-scale turbulence $(\boldsymbol{u}^{\prime },p^{\prime })$ , and thus has the triple decomposition (Reynolds & Hussain Reference Reynolds and Hussain1972)

where a quality with an overbar denotes the time average. CS can be extracted by the phase average, which may, for an instantaneous flow quantity $f(\boldsymbol{x},t)$ , be defined as

where $t_{i}$ denotes the instants at which the signal, or the structure, is deemed to have the same phase with respect to a reference oscillation (Hussain Reference Hussain1983). The second relation holds only when structures appear at regular interval $T$ (Hussain & Reynolds Reference Hussain and Reynolds1970), which is the case where CS are introduced by controlled periodic excitation. The signature of CS, $\tilde{f}$ , is then obtained as $\tilde{f}\equiv \langle f\rangle -\bar{f}$ .

The total time-averaged mean flow is driven by the Reynolds stresses contributed by both CS and turbulence. Unlike the conventional treatment, here we take $(\bar{\boldsymbol{U}},\bar{P})$ to be the ‘partial mean flow’ driven only by the Reynolds stresses of the latter, that is, they satisfy the Reynolds-averaged equations

This implies that the mean-flow distortion caused by nonlinear interactions of CS is to appear as part of CS. As we will explain later, this treatment is necessary.

Substituting (2.2) into the Navier–Stokes equations and performing phase averaging, we obtain, after use is made of (2.4), the equations governing CS:

Here

are the phase-averaged Reynolds stresses of fine-scale turbulence; the latter affects CS through $\tilde{{\it\tau}}_{ij}$ . The system is to be closed by introducing suitable models for $\tilde{{\it\tau}}_{ij}$ and $\bar{{\it\tau}}_{ij}\equiv -\overline{u_{i}^{\prime }u_{j}^{\prime }}$ .

The above formulation presents no difficulty if a reliable model for $\overline{u_{i}^{\prime }u_{j}^{\prime }}$ allows the ‘partial mean flow’ to be computed numerically. A problem does, however, arise if a mean field needs to be specified beforehand. The measured velocity profile may not be appropriate because it is the total mean flow. A simple resolution is to resort to experiments where no artificial excitation is introduced and CS are relatively weak so that the corresponding mean flow may be regarded as approximating the partial mean flow. The measured mean flow may be used in the above formulation to investigate the evolution of CS artificially excited subsequently. In this case, the viewpoint of ‘instability of a turbulent flow’ may be taken, leading to an alternative formulation as follows. Suppose that an unforced turbulent flow is $(\bar{\boldsymbol{U}},\bar{P})+(\boldsymbol{u}^{\prime },p^{\prime })$ . To this ‘turbulent base flow’, coherent perturbations are then introduced in a controlled manner. The resulting perturbed turbulent flow has a triple decomposition

Note that introduction of CS affects also small-scale turbulence, which is now denoted by $(\boldsymbol{u}^{\prime \prime },p^{\prime \prime })$ . Following the same steps, we can derive the equation

where

When CS are weak or have moderate amplitude, introduction of them may not significantly influence small-scale fluctuations, that is, $\boldsymbol{u}^{\prime \prime }$ remains close to $\boldsymbol{u}^{\prime }$ (whose magnitude is of $O(1)$ ), or in a statistical sense $\overline{u_{i}^{\prime \prime }u_{j}^{\prime \prime }}\approx \overline{u_{i}^{\prime }u_{j}^{\prime }}$ , and the two viewpoints, embodied in (2.6) and (2.9), are then equivalent.

Note that, in the formulation of Reynolds & Hussain (Reference Reynolds and Hussain1972), $(\bar{\boldsymbol{U}},\bar{P})$ represents the total mean flow. As a result, $\overline{\tilde{u} _{i}\tilde{u} _{j}}$ is added to $\overline{u_{i}^{\prime }u_{j}^{\prime }}$ in (2.4), and correspondingly $\tilde{u} _{i}\tilde{u} _{j}$ in (2.6) and (2.9) is replaced by $(\tilde{u} _{i}\tilde{u} _{j}-\overline{\tilde{u} _{i}\tilde{u} _{j}})$ , namely,

The three formulations, (2.6), (2.9) and (2.11), are equally valid if suitable models can be provided for both the time- and phase-averaged Reynolds stresses, $-\overline{u_{i}^{\prime }u_{j}^{\prime }}$ and $\tilde{{\it\tau}}_{ij}$ (or $\tilde{{\it\tau}}_{ij}^{\dagger }$ ) of fine-grained turbulence. From the modelling perspective, the first and third seem advantageous since $\tilde{{\it\tau}}_{ij}$ should be less problematic to model than $\tilde{{\it\tau}}_{ij}^{\dagger }$ , which involves small-scale turbulence of both perturbed and unperturbed states. In practice, further progress could be made by using an assumed mean flow, and it is better to predict (1) the mean-flow correction by coherent fluctuations as part of CS, and (2) its impact on the evolution of CS rather than presuming that (1) and (2) are taken into account implicitly by the time-averaged mean flow. We shall therefore proceed with the first viewpoint, i.e. with the system (2.6) and (2.7).

2.2. Closure models for Reynolds stresses of small-scale turbulence

Small-scale fluctuations contribute to time- and phase-averaged Reynolds stresses. No reliable closure model is available for either of them. Given the local nature of fine-grained turbulence, a gradient type of model will be employed. For the time-averaged Reynolds stresses, the model is, in non-dimensional form, written as

where $k=(\overline{\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{u}^{\prime }}/2)$ is the kinetic energy, with ${\it\delta}_{ij}$ being the Kronecker delta, ${\it\nu}_{t}$ is the mean eddy viscosity normalized by a reference value ${\it\nu}_{t}^{\ast }$ , and the mean turbulent Reynolds number $R_{T}$ is defined as

The phase-averaged Reynolds stresses are time-dependent, and they are related to the time-dependent strain rate of CS by a gradient type of model that includes the effect of time relaxation, namely,

where $\tilde{{\it\nu}}_{t}$ is an eddy viscosity accounting for the impact of small-scale turbulence on CS; it is normalized by a reference value $\tilde{{\it\nu}}_{t}^{\ast }$ and the associated Reynolds number

The key difference of (2.14) from a usual eddy-viscosity model (cf. Reynolds & Hussain Reference Reynolds and Hussain1972) is that it allows for a possible time delay $\hat{{\it\tau}}$ between the phase-averaged Reynolds stresses and the strain rate. Let $\widehat{{\it\tau}}_{ij}$ and $\widehat{u}$ denote the Fourier transforms of $\tilde{{\it\tau}}_{ij}$ and $\tilde{\boldsymbol{u}}$ with respect to time $t$ , respectively. Then the model (2.14) in spectral space can be expressed as

which consists of a complex eddy viscosity $\tilde{{\it\nu}}_{t}\text{e}^{-\text{i}{\it\omega}\hat{{\it\tau}}}$ . This generalized eddy viscosity was first introduced by Wu & Zhou (Reference Wu and Zhou1989) in their study of CS in turbulent boundary layers. In general, both ${\it\nu}_{t}$ and $\tilde{{\it\nu}}_{t}$ may depend on space variables and/or the mean flow (Reynolds & Hussain Reference Reynolds and Hussain1972; Kitsios et al. Reference Kitsios, Cordier, Bonnet, Ooi and Soria2010). In what follows, for simplicity they are assumed to be constants, in which case we may set ${\it\nu}_{t}=\tilde{{\it\nu}}_{t}=1$ .

Use of (2.14) in (2.4) and (2.6) leads to

The above closure model accounts for the influence of small-scale fluctuations on CS (and the mean flow), but the fine-grained turbulence itself remains dynamically passive. As a result, the evolution of CS becomes, in a nutshell, analogous to that of waves propagating through a random medium.

3.1. Asymptotic scalings

Prompted by the overwhelming experimental evidence, we subscribe to the view that CS in free shear layers behave as instability modes that develop from some small-amplitude disturbances upstream. We consider a two-dimensional wavetrain whose carrier wave has a frequency ${\it\omega}_{n}$ , say. The carrier wave is locally neutral at a streamwise location $\tilde{x}=x_{n}$ . The reference length ${\it\delta}_{0}^{\ast }$ is taken to be the shear-layer thickness at this location, which is chosen to be the origin of the coordinates for convenience. The precise meaning of ${\it\delta}_{0}^{\ast }$ will transpire in § 5.2, where ${\it\delta}_{0}^{\ast }$ is related to the well-defined and experimentally measurable momentum thickness. The wavetrain becomes, following the initial exponential growth, neutrally stable in the vicinity of $x_{n}$ and enters a nonlinear stage due to enhanced nonlinearity associated with the emergence of a critical layer as in laminar flows (Cowley & Wu Reference Cowley and Wu1994; Goldstein Reference Goldstein1995). Since the critical level is at the inflection point and the perturbation is planar, the critical layer is strongly nonlinear and non-equilibrium, and has a thickness of $O({\it\epsilon}^{1/2})$ , where ${\it\epsilon}\ll 1$ represents the amplitude.

The formulation will be presented for a plane mixing layer. However, the nonlinear critical-layer dynamics to be described is generic, and independent of the detailed profile because the critical layer is thin. The analysis and resulting theory can easily be adjusted to other free shear layers such as wakes and (planar and circular) jets. Indeed, the fact that the same nonlinear critical-layer dynamics operates explains why CS in these flows exhibit similar behaviours.

In the main part of the shear layer, the disturbance can, to leading order, be written as

where ${\it\alpha}$ , $c$ and ${\it\phi}_{1}$ are the wavenumber, phase speed and eigenfunction of the locally neutral mode, respectively, and $A^{\dagger }$ is the amplitude function of the slow spatial and time variables, $\bar{x}$ and ${\it\tau}$ , defined respectively as

to describe the spatial and temporal modulation of the disturbance (Wu & Tian Reference Wu and Tian2012). For $\bar{x}=O(1)$ , the CS corresponds to a nearly neutral instability mode (or more generally a wavetrain) residing on the turbulent mean flow. The effects of (molecular and eddy) viscosities and nonlinearity are negligible in the main part of the shear layer, but they all come into play in the critical layer if we choose the distinguished scaling, $R=O(\tilde{R}_{T})=O({\it\epsilon}^{-3/2})$ . These may be expressed more precisely by writing

where $\bar{{\it\lambda}}=O(1)$ and $\tilde{{\it\lambda}}=O(1)$ are parameters measuring the importance of molecular and eddy viscosities relative to nonlinearity, respectively (cf. Benney & Bergeron Reference Benney and Bergeron1969; Haberman Reference Haberman1972). Crucially, we take $R_{T}=O(R^{2/3})$ , that is,

with ${\it\sigma}$ being $O(1)$ , so that the non-parallel-flow effect appears as well at leading order in the critical layer.

It follows from (3.2), (3.6) and (3.8) that the variables $\bar{x}$ and $\tilde{x}$ are related to each other via

In the vicinity of $x_{n}=0$ , the mean-flow profile, $\bar{U}_{B}$ and $\bar{V}_{B}$ in (3.1), is expanded as

where $(\bar{U}_{1},\bar{V}_{1})\equiv \partial (\bar{U}_{B},\bar{V}_{B})/\partial \tilde{x}$ evaluated at $\tilde{x}=x_{n}=0$ .

The derivation of the evolution system involves using simultaneously the multi-scale method and matched-asymptotic-expansion technique to analyse the disturbance outside and within the critical layer. The procedure is similar to those in Goldstein & Hultgren (Reference Goldstein and Hultgren1988) and Wu & Tian (Reference Wu and Tian2012), but the main difference is that non-parallelism now appears at leading order in the critical layer. The reader who is interested primarily in physical aspects of the problem may skip the detail, and go directly to the final outcome of the analysis, which is the nonlinear evolution system consisting of the amplitude equation (3.30) for $A^{\dagger }$ coupled with a nonlinear equation (3.28) for the critical-layer vorticity. The reader is also reminded that the scalings and critical-layer dynamics are different for three-dimensional disturbances such as a single (Churilov & Shukhman Reference Churilov and Shukhman1994) or a pair of oblique modes (see e.g. Goldstein & Choi Reference Goldstein and Choi1989; Wu, Lee & Cowley Reference Wu, Lee and Cowley1993).

3.2. Outer expansion

In the main part of the shear layer, the streamfunction $\tilde{{\it\psi}}$ expands as

The eigenfunction ${\it\phi}_{1}$ is determined by the eigenvalue problem,

where the Rayleigh operator

Let $y_{c}$ denote the critical level, i.e.  $\bar{U}(y_{c})=c$ . As $\hat{{\it\eta}}\equiv y-y_{c}\rightarrow 0$ , ${\it\phi}_{1}$ has the asymptote

where $a_{1}$ is a constant. The function ${\it\phi}_{2}^{(1)}$ is governed by the system consisting of the inhomogeneous Rayleigh equation and the boundary condition:

where we have introduced the differential operator

As $\hat{{\it\eta}}\rightarrow 0$ , the solution is, by examining the asymptotic properties of both sides of (3.15), found to behave as

where $b_{2}^{(1)}$ and $a_{2}^{(1)\pm }$ are as yet unknown functions of $\bar{x}$ , and

The jump $(a_{2}^{(1)+}-a_{2}^{(1)-})$ is to be determined by analysing the critical-layer dynamics. For the boundary-value problem (3.15) to have a solution, a (modified) solvability condition

must be satisfied, which is derived via the following routine procedure. Multiply both sides of the inhomogeneous Rayleigh equation in (3.15) by ${\it\phi}_{1}$ , and perform integration by parts on the left-hand side but take into account the jump across the critical layer by using (3.17) (cf. Redekopp Reference Redekopp1977). In (3.19),

with $J_{2}$ and $I_{1}$ both being Cauchy principal values.

The solution for ${\it\phi}_{2}^{(1)}$ could, of course, only be determined up to an arbitrary multiple of the eigenfunction ${\it\phi}_{1}$ . The multiplier of ${\it\phi}_{1}$ is a function of $\bar{x}$ , and can be fixed by imposing the condition that ${\it\phi}_{2}^{(1)}$ be orthogonal to ${\it\phi}_{1}$ . This is, however, unnecessary in the present study, as the indeterminacy does not affect the dynamics of the CS, or the Reynolds stresses and the mean-flow distortion to be considered in § 4.

The harmonic components ${\it\phi}_{2}^{(m)}$ satisfy homogeneous Rayleigh equations

and, as $\hat{{\it\eta}}\rightarrow 0$ ,

It should be remembered that ${\it\phi}_{2}^{(m)}~(m\neq 1)$ are not eigenfunctions because there is a forcing from the interaction within the critical layer through the jumps $(a_{2}^{(m)+}-a_{2}^{(m)-})$ , which renders the boundary-value problem inhomogeneous. An important fact of a strong nonlinear critical layer is that all harmonics are generated at the same order, which causes the vortex to roll up.

3.3. Inner expansion: critical layer analysis

The dynamics of CS within the critical layer can be analysed by introducing the local transverse variable

The mean-flow quantities are expanded as Taylor series in $Y$ , namely,

The streamfunction of the perturbation, $\tilde{{\it\psi}}$ , is expanded as

The first three terms are just the straightforward continuation of the outer solution, namely,

and it can easily be verified that they all satisfy the required equations.

Substituting expansions (3.24) and (3.25) with (3.26) into (3.4), we obtain the equation for ${\it\Psi}_{2}$ . That equation can, by introducing

be simplified to

where the nonlinear operator

Matching with the outer solution for the fundamental component determines the jump $(a_{2}^{(1)+}-a_{2}^{(1)-})$ , which is inserted into (3.19) to give the evolution equation for the amplitude $A^{\dagger }$ :

Equations (3.28) and (3.30) form the system that governs the nonlinear evolution of the disturbance.

Matching the harmonic components determines the jump

Once ${\it\Omega}^{\dagger }$ is known, the harmonics in the main part of the shear layer can be obtained by solving (3.21) subject to (3.22) and (3.31).

It is convenient to introduce the normalized dependent and independent variables

as well as the (normalized) local transverse coordinate centred at the moving critical level,

The system (3.28)–(3.30) then becomes

where we have put

As the unknown amplitude function $\bar{A}$ appears in the coefficient of the partial differential equation governing the vorticity, the critical-layer dynamics is ‘strongly nonlinear’.

The parameter ${\it\chi}$ characterizes the effect of a translating critical layer, which was first identified by Cowley (Reference Cowley1985) and Haynes & Cowley (Reference Haynes and Cowley1986) for certain time-dependent flows. For the present problem, this effect arises because non-parallelism causes the critical level to drift with respect to the streamwise variable $\bar{x}$ at the same rate as that at which the amplitude evolves. Non-parallelism is associated with the streamwise variation of the streamwise velocity and with the transverse velocity itself of the mean flow. The former alters directly the growth rate through the terms proportional to $\bar{x}$ in (3.34), while both contribute to the translating critical-layer effect as (3.35) indicates.

3.4. Upstream condition

Let us now consider what condition should be imposed upstream as an ‘initial condition’. In the upstream limit $\bar{x}\rightarrow -\infty$ , the disturbance is small so that the nonlinear term in the first equation of (3.34) can be neglected, and the resulting linear equation admits the solution

where

It follows that

This result is inserted into the second equation of (3.34) to give

where the group velocity

and the coefficient

It can be deduced from (3.41) that

where ${\it\kappa}=\text{i}S_{0}/c_{g}$ . Here $S_{0}$ measures the deviation of the disturbance frequency $({\it\alpha}c+{\it\epsilon}^{1/2}S_{0})$ from that of the neutral mode (and is equivalent to $S_{1}$ in Goldstein & Leib (Reference Goldstein and Leib1988)). Note that because $c_{g}$ is a complex number (which is expected since the mode considered is neutral rather than being locally most unstable), it is impossible to introduce the coordinate moving with the group velocity. As such, the modulation equation is of first order rather than second order as would be the case if the group velocity were real. Note that the linear growth rate is unaffected by the effect of a translating critical layer; the latter remains dormant in the linear regime and only comes into play in the nonlinear stage.

3.5. Fourier decomposition and boundary conditions

The solution for $\bar{A}$ and ${\it\Omega}$ can be written as a Fourier series (cf. Goldstein & Leib Reference Goldstein and Leib1988)

where ${\it\eta}=\bar{{\it\eta}}-S_{0}$ . Then $Q_{n}$ ( $n=0,1,2,\ldots$ ), among which $Q_{0}$ represents the mean-flow distortion generated by the nonlinear interaction of CS, satisfy the system of coupled equations:

where ${\it\delta}_{n1}$ denotes the Kronecker delta, and ${\it\lambda}_{n}$ is a complex quantity,

It follows from (3.46) that, as ${\it\eta}\rightarrow \pm \infty$ ,

where we have put

Owing to the slow decay of $Q_{1}$ , the integral in (3.47) must be interpreted as a Cauchy principal value. The infinite domain in the ${\it\eta}$ direction is truncated to a large but finite interval $-M\leqslant {\it\eta}\leqslant M$ . Then (3.47) can be written as

where we have defined

The second-order derivative with respect to $\bar{x}$ in the above equation is inconvenient for numerical integration. Following a procedure similar to that in Wu & Tian (Reference Wu and Tian2012), we obtain a first-order system with respect to $\bar{x}$ :

The derivation and the expressions for the coefficients are given in appendix A. The equation is subject to the initial condition

4.1. Reynolds stresses of coherent structures

The characteristics of Reynolds stresses are of interest in their own right and are now considered. In order to calculate them, the streamwise and transverse velocities of the CS are obtained, to $O({\it\epsilon}^{3/2})$ accuracy, in terms of the streamfunction:

where only the fundamental component is needed and written out. The general solution for ${\it\phi}_{2}^{(1)}$ , which is governed by (3.15), can be written as (Wu & Tian Reference Wu and Tian2012)

where $B^{\dagger }(\bar{x},{\it\tau})$ is a function of $\bar{x}$ and ${\it\tau}$ , and

In order for ${\it\phi}_{2}^{(1)}$ to satisfy the boundary conditions that ${\it\phi}_{2}^{(1)}\rightarrow 0$ as $y\rightarrow \pm \infty$ , we have to set

Taking the limit $y\rightarrow y_{c}$ , the solution (4.2) assumes the asymptote (3.17) with

where with $\hat{a}\neq y_{c}$ being an arbitrary constant and ${\it\chi}_{1}=(\bar{U}_{c}^{\prime \prime \prime }\bar{U}_{1,c}-\bar{U}_{c}^{\prime }\bar{U}_{1,c}^{\prime \prime })/\bar{U}_{c}^{\prime 2}$ .

Of interest is the shear stress component ${\it\tau}_{12}\equiv -(\tilde{u} \tilde{v}^{\ast }+\text{c.c.})$ . For a nearly neutral mode, the leading-order term vanishes, leaving us with

The first two terms in (4.11) are regular, while the third term, or rather ${\it\phi}_{2,y}^{(1)}$ in it, exhibits a logarithmic singularity at $y_{c}$ . In order to find an expression for the Reynolds stress valid in the entire shear layer, ${\it\phi}_{2,y}^{(1)}$ has to be replaced by a composite solution, $({\it\phi}_{2,y}^{(1)})_{c}$ , which is constructed as follows. First, the inner solution for the $O({\it\epsilon}^{3/2})$ velocity is

By using the additive rule, the composite solution is constructed as

where the last term is the common part of ${\it\phi}_{2,y}^{(1)}$ and $U_{3/2}$ in the limits of $y\rightarrow y_{c}$ and ${\it\eta}\rightarrow \infty$ , respectively. It is important to note that the undetermined complementary solution $B^{\dagger }{\it\phi}_{1}$ in (4.2) does not contribute to the Reynolds stress, and nor does the last term. Taking this into account and inserting (4.2) and (4.13) into (4.11), we can, without the need for the complete solution for ${\it\phi}_{2}^{(1)}$ , determine fully the Reynolds stress,

where we have put ${\hat{C}}^{+}=0$ and ${\hat{C}}^{-}=1$ . Since the last term in (4.14) is confined within the critical layer, in the majority of the shear layer the Reynolds stress ${\it\tau}_{12}$ is proportional to $\partial |A^{\dagger }|^{2}/\partial \bar{x}$ and would therefore change its sign when the CS evolves through its peak amplitude, as was observed in experiments (WW; Marasli et al. Reference Marasli, Champagne and Wygnanski1992).

4.2. Mean-flow distortion

The mean-flow distortion in the main part of the shear layer has an $O({\it\epsilon}^{2})$ streamwise velocity, whereas that in the critical layer has a larger, $O({\it\epsilon}^{3/2})$ , streamwise velocity due to the larger gradient of the Reynolds stress there. Despite being weaker, the mean-flow distortion in the main part makes the dominant contribution to the change in the shear-layer thickness. The solution takes the form ${\it\epsilon}^{2}(u_{M},{\it\epsilon}^{1/2}v_{M},p_{M})$ with $u_{M}$ and $v_{M}$ being related to ${\it\phi}_{3}^{(0)}$ in (3.11)

The governing equations are found as

From the transverse momentum equation we find that

In terms of the streamfunction ${\it\phi}_{3}^{(0)}$ , the streamwise momentum equation in (4.16) can be written as

where

A numerical approach is needed when the disturbance is a modulated wavepacket. For a simple wave, it turns out that $S$ can be written as

with the shape function

Once again, the expression (4.20) explains the experimental observation that the Reynolds stress (in the main part of the shear flow) switches its sign at the location where the CS reaches its peak amplitude and starts to attenuate. Equation (4.18) reduces to

and the solution is found as

where $a_{3}^{(0)\pm }$ are unknown constants. Let $\bar{U}\rightarrow \bar{U}_{\pm }$ as $y\rightarrow \pm \infty$ . Then

which implies that the mean-flow distortion spreads to the outer inviscid layer, where the streamwise and transverse velocities both become $O({\it\epsilon}^{5/2})$ , and a pressure of the same order of magnitude is induced. On introducing $\bar{y}={\it\epsilon}^{1/2}y/c$ , the perturbation may be written as ${\it\epsilon}^{5/2}(\hat{u} _{M}^{\pm },\hat{v}_{M}^{\pm },\hat{p}_{M}^{\pm })$ , where the scaled velocity $(\hat{u} _{M}^{\pm },\hat{v}_{M}^{\pm })$ and pressure $\hat{p}_{M}^{\pm }$ are functions of $\bar{x}$ and $\bar{y}$ , and satisfy the equations

It follows that

where the boundary condition corresponds to the matching with the transverse velocity $v_{M}=-c^{-1}{\it\phi}_{3,\bar{x}}^{(0)}$ at the upper and lower outer edges of the shear layer. Clearly, in the lower stream, if $\bar{y}$ is changed to $-\bar{y}>0$ , the Laplace equation for $\hat{p}_{M}^{-}$ remains invariant, but the boundary condition changes its sign, that is,

In the main part of the shear layer, there exists an induced mean pressure, ${\it\epsilon}^{5/2}p_{M}^{(2)}$ , with $p_{M}^{(2)}$ satisfying

Integration across the shear layer gives

It follows by matching that $\hat{p}_{M}^{+}=\hat{p}_{M}^{-}$ at $\bar{y}=0$ , and hence

On the other hand, as $y\rightarrow y_{c}^{\pm }$ it follows from (4.23) that

where

with $\hat{a}^{+}>y_{c}$ and $\hat{a}^{-}<y_{c}$ being arbitrary constants.

The jump $(a_{3}^{(0)+}-a_{3}^{(0)-})$ can be found by considering the mean-flow distortion $Q_{0}$ in the critical layer. Let ${\it\Psi}_{2}^{(0)}$ be the streamfunction related to $Q_{0}$ : ${\it\Psi}_{2YY}^{(0)}=\bar{U}_{c}^{\prime \prime \prime }/({\it\alpha}\bar{U}_{c}^{\prime })^{2}Q_{0}$ . Using this in (3.46) for $n=0$ and integrating the resulting equation with respect to ${\it\eta}$ , we obtain

Integrating once more and making use of (3.47), we find that

For a non-modulated disturbance, from the above relation, the jump ${\it\Psi}_{2}^{(0)}(\infty )-{\it\Psi}_{2}^{(0)}(-\infty )$ is obtained, which must, by the matching principle, be equal to ${\it\phi}_{3}^{(0)}(y_{c}^{+})-{\it\phi}_{3}^{(0)}(y_{c}^{-})$ , leading to the relation

It follows from (4.30) and (4.36) that

The mean-flow distortion in the critical layer is given by $({\it\epsilon}^{3/2}U_{M}^{(1)}+{\it\epsilon}^{2}U_{M}^{(2)})$ with

and the $O({\it\epsilon}^{2})$ term $U_{M}^{(2)}$ is contained in ${\it\Psi}_{3}$ in (3.25), which consists, like ${\it\Psi}_{2}$ , of all harmonics. Let the mean-flow component be denoted by ${\it\Psi}_{3}^{(0)}$ . Then $U_{M}^{(2)}={\it\Psi}_{3,Y}^{(0)}$ . As in Wu & Tian (Reference Wu and Tian2012), ${\it\Psi}_{3,YY}^{(0)}\sim O(1/{\it\eta})$ for ${\it\eta}\gg 1$ , and so $U_{M}^{(2)}\rightarrow c_{M}|A|^{2}\ln \!|{\it\eta}|$ with $c_{M}=\bar{U}_{c}^{\prime \prime \prime }/({\it\alpha}^{4}\bar{U}_{c}\bar{U}_{c}^{\prime 3})$ , matching to the corresponding logarithmic term in (4.32). The composite solution for the mean-flow distortion is then given by

The width of the shear layer is usually measured by the momentum thickness, defined as

where $\bar{U}^{\dagger }=\bar{U}+U_{M}$ is the perturbed mean velocity. At leading order, the definition gives the momentum thickness of the ‘partial’ (or unperturbed) mean flow,

The $O({\it\epsilon}^{2})$ change in the momentum thickness is then found as

The contribution of the leading-order inner solution was ignored in Goldstein & Leib (Reference Goldstein and Leib1988) on the ground that $U_{M}^{(1)}$ is trapped in the critical layer of $O({\it\epsilon}^{1/2})$ width, where the integrand, ${\it\epsilon}^{-1/2}(\bar{U}-c)U_{M}^{(1)}$ , is of order one so that the contribution is of $O({\it\epsilon}^{1/2})$ . Here it is retained in the spirit of composition approximation, but the even smaller term $U_{M}^{(2)}$ will be neglected. Using (4.22) and performing integration by parts, we find that

Substitution of (4.36) into the above gives the final expression for ${\rm\Delta}{\it\Theta}$ :

It is interesting that ${\rm\Delta}{\it\Theta}$ can be determined without fixing the individual values of  $a_{3}^{(0)\pm }$ .

Equation (4.39) indicates that, if the full profile is to be obtained accurate up to $O({\it\epsilon}^{2})$ , the critical-layer equation at $O({\it\epsilon}^{5/2})$ has to be solved to calculate ${\it\Psi}_{3}$ and ${\it\Psi}_{3Y}^{(0)}=U_{M}^{(2)}$ . An $O({\it\epsilon}^{2})$ velocity jump across the critical layer is present in the outer solution ${\it\phi}_{3,y}^{(0)}$ , and is smoothed out by the inner solution $U_{M}^{(2)}$ . This can be shown by examining (but without solving) the equation for ${\it\Psi}_{3}$ , which reads

where $\mathscr{G}$ stands for the right-hand side of (2.42) in Wu & Tian (Reference Wu and Tian2012), and $\bar{U}_{2}=\partial ^{2}\bar{U}_{B}/\partial \tilde{x}^{2}$ evaluated at $y_{c}$ and $x_{n}=0$ . Integrating (4.45) with respect to $Y$ for the mean-flow component, and making use of the asymptotic behaviours of ${\it\Psi}_{2,Y}^{(0)}$ and ${\it\Psi}_{3,YY}^{(0)}$ , we obtain

For the non-modulated case, evaluation of the right-hand side by matching with (4.31) shows that

From this relation and (4.32) it follows that

indicating that the jump in the streamwise velocity is smoothed out within the critical layer, as expected.

The expressions for the Reynolds stress (4.14), the mean-flow distortion (4.39) and the change in the momentum thickness (4.44) are valid for a general profile $\bar{U}$ . These quantities are of physical interest as they are relevant for understanding CS and/or their impact on the shear layer, and they will be calculated in § 5 and presented after numerical solutions for the amplitude and vorticity of the CS are obtained. In performing the computation, use is to be made of the relation

5.1. Numerical method

The numerical work consists of solving the coupled evolution system (3.46) and (3.55) subject to the initial condition (3.56) and the boundary conditions (3.49)–(3.51). The system is truncated with $0\leqslant n\leqslant N$ (where $n$ refers to the order of harmonics in (3.45)), and solved in a large but finite domain, $-M\leqslant {\it\eta}\leqslant M$ . A predictor–corrector method is employed to march downstream. The amplitude equation for $A$ is discretized using the fourth-order explicit (Adams–Bashforth) and implicit (Adams–Moulton) schemes to construct the predictor and corrector, respectively. The integrals are evaluated using Simpson’s rule. The vorticity equations for $Q_{n}$ are discretized by the Crank–Nicolson scheme.

5.2. Mean velocity profile

The theory is now applied to a mixing layer that forms between two uniform streams with velocities $U_{1}^{\ast }$ and $U_{2}^{\ast }<U_{1}^{\ast }$ . The reference velocity is $U_{0}^{\ast }=(U_{1}^{\ast }-U_{2}^{\ast })/2$ . As in many previous studies, we will specify a mean-flow profile that is broadly in agreement with experimental data. The latter indicate that the mean-flow profile remains self-similar in the streamwise direction while its local thickness varies, and thus it takes the form

where

the function $F$ characterizes the shape, while $y_{0}$ and ${\it\delta}$ represent the centre and thickness of the shear layer, respectively. The dimensional momentum thickness ${\it\theta}^{\ast }$ , which is often measured in experiments, is related to ${\it\theta}$ (as defined by (4.41)), ${\it\delta}$ and ${\it\delta}^{\ast }$ by

where for the similarity profile (5.1) the constant $C_{1}$ is found as

With a given $F$ , the (rescaled) transverse velocity $V_{B}$ , whose value at the critical level is needed in our theory, can be obtained from the continuity equation, as will be shown below. Before prescribing the profile $F$ , it is informative to discuss useful mean-flow properties that are independent of $F$ . First, the streamwise variations of $y_{0}(\tilde{x})$ and ${\it\delta}(\tilde{x})$ are not independent but satisfy a constraint. We derive it by considering the momentum equation (3.3), which can be rewritten as

where the subtraction of $\bar{U}_{\pm }=\bar{U}_{c}\pm 1$ is for $y>0$ and $y<0$ , respectively. Integration of the above equation with respect to $y$ gives

where $C_{0}$ is a constant. The relation corresponds to conservation of the (renormalized) streamwise momentum flux. Inserting (5.1) into (5.6), we have

On taking a derivative of (5.7) with respect to $\tilde{x}$ , the equation becomes

which relates ${\dot{y}}_{0}$ , the rate of drift of the shear-layer centre, to $\dot{{\it\delta}}$ , the spreading rate of its thickness.

The corresponding transverse velocity $V_{B}$ can be found by introducing a streamfunction ${\it\psi}_{B}$ such that

Integration of the first equation yields

and so

where $d_{0}(\tilde{x})$ is an unknown function of $\tilde{x}$ . In order to determine $d_{0}(\tilde{x})$ , which affects nonlinear dynamics of the CS, we now analyse the impact of the viscous shear layer on the mean flow in the far field. From (5.11), one finds that

where $\bar{V}_{\pm }$ are the so-called ‘transpiration velocities’, i.e. the transverse velocities induced by the viscous motion in the shear layer. Through $V_{\pm }$ , an $O(R_{T}^{-1})$ perturbation $R_{T}^{-1}(\hat{u} ^{\pm },\hat{v}^{\pm },\hat{p}^{\pm })$ is induced in the far field corresponding to ${\tilde{y}}\equiv y/R_{T}=O(1)$ , where the scaled velocity $(\hat{u} ^{\pm },\hat{v}^{\pm })$ and pressure $\hat{p}^{\pm }$ are functions of $\tilde{x}$ and ${\tilde{y}}$ . They satisfy the same equations as (4.25) provided that $\bar{x}$ and $\bar{y}$ are replaced by $\tilde{x}$ and ${\tilde{y}}$ , respectively. It it follows that $\hat{p}^{\pm }$ satisfies the Laplace equation

where the boundary condition corresponds to the matching with the transpiration velocities at the upper and lower outer edges of the shear layer. The solution for $\hat{p}^{\pm }$ may depend on other bounding surfaces that may be present in an experimental set-up. On the assumption that any asymmetry in the upper and lower streams is absent, then $\hat{p}^{\pm }$ are determined by the boundary conditions at ${\tilde{y}}=0$ . A simple order-of-magnitude argument indicates that a mixing layer cannot sustain a pressure difference of $O(R_{T}^{-1})$ , which means that $p^{+}=p^{-}$ on ${\tilde{y}}=0$ . This requires $-\bar{U}_{+}\bar{V}_{+}=\bar{U}_{-}\bar{V}_{-}$ , from which one finds

The relations (5.7), (5.8) and (5.14) hold for any $\tilde{x}$ . Specifically, putting $\tilde{x}=0$ where $y_{0}=0$ and ${\it\delta}=1$ , we have

In our calculations, we choose the profile

which GKW found to fit their low-excitation experimental data for $q_{c}=0.67$ . The special case of $q_{c}=0$ has been the popular choice in theoretical studies, as the neutral mode can be found analytically. Substitution of (5.17) into (5.15) and (5.4) gives the constants

It follows that

For the evaluation of $I_{1}$ , only the odd parts of $\bar{U}_{1}$ and $\bar{U}_{1}^{\prime \prime }$ are required, which are found as

Use of these in (3.20) shows that $I_{1}$ may be written as

The parameters characterizing non-parallelism are found as

where

and

When $q_{c}=0$ , it is found that

but for $q_{c}\neq 0$ , it is necessary to solve the Rayleigh equation numerically to find the neutral mode, and then evaluate $\hat{I} _{1}$ and $J_{1}$ . Here $\hat{I} _{1}$ is a Cauchy principal value but can be computed easily using its definition.

5.3. Numerical results

Flow quantities of interest include the amplitude $A$ of the fundament component and the so-called ‘critical-layer vorticity’ ${\it\Omega}_{c}^{\dagger }$ of CS. The latter is defined as (see Goldstein & Leib Reference Goldstein and Leib1988)

with which $(\bar{U}_{c}^{\prime }+{\it\epsilon}{\it\Omega}_{c}^{\dagger })$ represents the total vorticity within the critical layer. For convenience, the results will be presented by plotting the renormalized vorticity

The choice of the parameters is guided by the experiments of GKW, where $U_{1}^{\ast }=10$  m s $^{-1}$ , $U_{2}=4$  m s $^{-1}$ , $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=1/54$ (see the data set III in table 1 of GKW) and the momentum thickness ${\it\theta}^{\ast }=18.78$  mm, which can be read from figure 1 of GKW. For the purpose of demonstrating qualitative behaviours of the evolution, we take $q_{c}=0$ , for which ${\it\delta}_{0}^{\ast }={\it\theta}^{\ast }/C_{1}$ with $C_{1}=0.5$ . Based on ${\it\delta}_{0}^{\ast }$ and $U_{0}^{\ast }$ , the Reynolds number $R$ turns out to be $7100$ . We choose $S_{0}<0$ such that ${\it\kappa}_{r}=1$ , and set the amplitude ${\it\epsilon}=0.1036$ , for which nonlinear effects are significant, and the parameters characterizing the non-parallel-flow effects, ${\it\chi}$ and ${\it\sigma}_{s}$ , take $O(1)$ values: ${\it\chi}=-0.1180$ and ${\it\sigma}_{s}=-0.4943-0.1349\text{i}$ . The eddy viscosity and phase lag have to be treated as free parameters because no experimental data about them are available. The expression (3.48) implies that, if $\hat{{\it\tau}}\neq 0$ and $\bar{{\it\lambda}}_{2}>\bar{{\it\lambda}}_{1}$ (i.e.  $\tilde{R}_{T}<R$ ), the effective diffusion coefficient $\bar{{\it\lambda}}_{1}+\bar{{\it\lambda}}_{2}\cos (n\hat{{\it\theta}})$  ( $\hat{{\it\theta}}\equiv {\it\alpha}c\hat{{\it\tau}}$ ) will become negative for certain $n$ , rendering the problem ill-posed. This unfavourable feature reflects a shortcoming of the model, which assumes the same relaxation time $\hat{{\it\tau}}$ for the fundamental and all harmonics. Improved models are being pursued, and here the ill-posedness is avoided by restricting $\tilde{R}_{T}\leqslant R$ , i.e. $\bar{{\it\lambda}}_{2}\leqslant \bar{{\it\lambda}}_{1}$ . Specifically, we choose $\tilde{R}_{T}=R$ , for which $\bar{{\it\lambda}}_{2}=\bar{{\it\lambda}}_{1}=0.00422$ , and take phase lag $\hat{{\it\theta}}={\rm\pi}/5$ . These values are rather arbitrary. Fortunately, the evolution of the CS turns out to be rather insensitive to either of $\hat{{\it\theta}}$ and $\bar{{\it\lambda}}_{2}$ , as will be shown later.

Figure 1 shows the amplitude evolution (curve a) under the influence of both nonlinearity and non-parallelism. In order to see their roles, the results when either of these is artificially suppressed are also plotted. Without nonlinearity, the development is Gaussian, featuring a broad peak and rapid monotonic decay (curve c). On the other hand, with the nonlinear effect included but non-parallelism neglected (curve b), the amplitude undergoes oscillatory saturation (and actually slight algebraic amplification; see later). The simultaneous effects of non-parallelism and nonlinearity cause the amplitude to decay in an oscillatory manner, which is obviously different from the linear approximation and parallel–nonlinear theory. The neutral position of the mode with frequency ${\it\omega}_{n}={\it\alpha}c$ corresponds to $\bar{x}=0$ , but the disturbance under consideration has the frequency ${\it\alpha}c+{\it\epsilon}^{1/2}S_{0}<{\it\alpha}c\equiv {\it\omega}_{n}$ , and so it becomes linearly neutral and attains its maximum somewhat downstream of $\bar{x}=0$ .

Figure 1. Nonlinear development of the amplitude $A$ : solid line (curve a), non-parallel–nonlinear theory with $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=1/54$ ; dashed line (curve b), parallel–nonlinear theory $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=0$ ; dotted line (curve c), non-parallel–linear theory.

Figure 2. Nonlinear development of the amplitude $A$ for different spread rates of the momentum thickness $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=0$ (curve a), $1/108$ (curve b), $1/54$ (curve c), $1/27$ (curve d) and $1/18$ (curve e).

In the present theory, non-parallelism manifests in two ways: (1) it alters the growth rate of the CS, which is well known, and (2) it induces a translating critical layer, the effect of which becomes active in the nonlinear stage of the evolution. Non-parallelism is controlled by the spreading rate of the shear-layer thickness $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }$ . The amplitude evolution for a range of $0\leqslant \text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }\leqslant 1/18$ is displayed in figure 2. As is illustrated, non-parallelism tends to inhibit CS in the sense that the decay starts earlier and becomes faster, and the peak value decreases as $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }$ increases. However, the overall feature of oscillatory attenuation remains. Such oscillatory attenuations have been observed in a number of experiments (Hussain & Thompson Reference Hussain and Thompson1980; Zaman & Hussain Reference Zaman and Hussain1980; Fiedler & Mensing Reference Fiedler and Mensing1985; WW). It is worth noting that the amplitude ${\it\epsilon}$ influences the evolution through ${\it\chi}$ , ${\it\Lambda}_{0}$ , ${\it\sigma}_{s}$ , $\bar{{\it\lambda}}_{1}$ and $\bar{{\it\lambda}}_{2}$ . Since the first three of these parameters are proportional to $b_{s}=(\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast })/(C_{1}{\it\epsilon})$ (see (5.23)) while $\bar{{\it\lambda}}_{1}$ and $\bar{{\it\lambda}}_{2}$ are practically zero, curves b and d would represent approximately the evolution for ${\it\epsilon}=0.2072$ (doubled) and 0.0518 (halved), respectively, with a fixed $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=1/54$ .

Figure 3. The development of the amplitude $A$ for different viscosity parameters $\bar{{\it\lambda}}_{1}$ and $\bar{{\it\lambda}}_{2}$ . Parallel–nonlinear theory ( $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=0$ ) for $\bar{{\it\lambda}}_{1}=\bar{{\it\lambda}}_{2}=0.5$ (curve a), 0.1 (curve b) and 0 (curve c). Non-parallel–nonlinear theory ( $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=1/54$ ) for $\bar{{\it\lambda}}_{1}=\bar{{\it\lambda}}_{2}=0$ (curve d), 0.0442 (curve e), 0.05 (curve f) and 0.1 (curve g).

We now examine the role of molecular and eddy viscosities, represented by $\bar{{\it\lambda}}_{1}$ and $\bar{{\it\lambda}}_{2}$ , respectively. The amplitude evolution for different values of these parameters is shown in figure 3, where the corresponding result with non-parallelism ignored is also presented. Neither of the viscosities affects the linear stage of the development, and their impact on the nonlinear evolution is opposite depending on whether non-parallelism is included. When non-parallelism is neglected, the amplitude continues to amplify and eventually follows an algebraic growth (Goldstein & Hultgren Reference Goldstein and Hultgren1988). The amplitude is greater for larger values of $\bar{{\it\lambda}}_{1}$ and $\bar{{\it\lambda}}_{2}$ , indicating that viscosities promote the growth of the perturbation, the reason being that nonlinearity plays a stabilizing role and a larger viscosity corresponds to weaker nonlinearity and therefore a less stabilizing effect. On the contrary, when non-parallelism is included, increasing viscosities tend to enhance decay and reduce the oscillation, as is expected intuitively. The quantitative difference indicates the importance of accounting for non-parallelism. In comparison, viscous effects are rather weak, affecting only the later stage of the development, since typical values of $\bar{{\it\lambda}}_{1}$ and $\bar{{\it\lambda}}_{2}$ turn out to be fairly small. This is entirely consistent with the assertion made by Fiedler & Mensing (Reference Fiedler and Mensing1985) based on their experimental observations.

The effect of the phase lag $\hat{{\it\theta}}\equiv {\it\alpha}c\hat{{\it\tau}}$ is also considered. Figure 4 shows the amplitude evolution for different $\hat{{\it\theta}}$ with typical values of $\bar{{\it\lambda}}_{1}=\bar{{\it\lambda}}_{2}=0.004$ as well as with artificially elevated values, $\bar{{\it\lambda}}_{1}=\bar{{\it\lambda}}_{2}=0.05$ . In both cases, increasing $\hat{{\it\theta}}$ only slightly alleviates the decay, which may be attributed to a slightly smaller total viscosity $(\bar{{\it\lambda}}_{1}+\bar{{\it\lambda}}_{2}\cos \hat{{\it\theta}})$ . The results in figures 3 and 4 indicate that the phase lag and the magnitude of the eddy viscosity have a negligible influence on the CS, suggesting that the latter evolves practically in the same manner as an instability wave on a laminar shear layer. Indeed, the dynamics is essentially inviscid since the effect of molecular viscosity is more likely to be even smaller and negligible as well.

Figure 4. The development of the amplitude $A$ for different phase lags $\hat{{\it\theta}}={\it\alpha}c\hat{{\it\tau}}$ : (a) for $\bar{{\it\lambda}}_{1}=\bar{{\it\lambda}}_{2}=0.00422$ and $\hat{{\it\theta}}=0$ (curve a), ${\rm\pi}/5$ (curve b) and ${\rm\pi}/2$ (curve c); (b) an enlarged view of (a) for $4<\bar{x}<8$ ; (c) for $\bar{{\it\lambda}}_{1}=\bar{{\it\lambda}}_{2}=0.05$ and $\hat{{\it\theta}}=0$ (curve a), ${\rm\pi}/5$ (curve b) and ${\rm\pi}/2$ (curve c); (d) an enlarged view of (c) for $2<\bar{x}<8$ .

As was mentioned earlier, the most significant feature of the strongly nonlinear critical-layer theory is that all harmonics appear simultaneously at the same order. They are measured by the streamwise velocity jump

Figure 5 shows the development of $H_{n}$ for $n=2,3,4$ . The size of $H_{n}$ decreases with $n$ . In the earlier nonlinear phase, the first harmonic $H_{2}$ is dominant, but all three become more or less comparable in the later stage $2.2<\bar{x}<5$ .

Figure 5. The development of the harmonics as shown by $H_{n}\equiv a_{2}^{(n)+}-a_{2}^{(n)-}$   $(n=2,3,4)$ for $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=1/54$ , $\bar{{\it\lambda}}_{1}=\bar{{\it\lambda}}_{2}=0.0422$ and $\hat{{\it\theta}}={\rm\pi}/5$ .

The simultaneous appearance of harmonics leads to roll-up of vortices. This process is illustrated in figure 6 by plotting contours of the normalized critical-layer vorticity of the CS at representative streamwise locations. At the same time, distributions of the fundamental $Q_{1}$ , the first harmonic $Q_{2}$ and the mean-flow distortion $Q_{0}$ are monitored. In the initial linear stage $\bar{x}=0$ , the fundamental is dominant and the vorticity exhibits a simple pattern of a ‘cat’s eye’ associated with a simple sinusoidal perturbation. By $\bar{x}=1$ , vorticity lines begin to fold and a vortex roller starts forming. The first harmonic acquires an amplitude approximately half of that of the fundamental, whereas the mean-flow distortion becomes comparable with the latter. The harmonic and the mean-flow distortion remain trapped in the critical layer as they decay rapidly towards its outer edges. When $\bar{x}=2$ , a fully developed vortex roller has emerged. The relative magnitudes of the three components remain similar as at $\bar{x}=1$ . However, the profile of the first harmonic exhibits rapid oscillations. A further calculation with the mesh size halved indicates that they are well resolved. At $\bar{x}=4$ , vortices become elongated in the streamwise direction and there is considerable overlapping between two adjacent vortices. This might be related to ‘partial pairing’ observed in experiments (Hussain & Clark Reference Hussain and Clark1981). In this case, vortices appear to merge without involving the subharmonic.

Figure 6. Contours of the critical-layer vorticity ${\it\Omega}_{c}$ (a,c,e,g), and profiles of the fundamental $Q_{1}$ , first harmonic $Q_{2}$ and the mean distortion $Q_{0}$ (b,d,f,h) at $\bar{x}=0$ , $\bar{x}=1$ , $\bar{x}=2$ and $\bar{x}=4$ .

The Reynolds stress and the mean-flow distortion are calculated. For the chosen profile (5.17) with ( $q_{c}=0$ ), it is found that

Substitution of (5.29) into (4.14) gives

where the jump in the outer solution is smoothed out by the contribution of the inner solution $U_{3/2}$ .

Substituting (5.30) and (5.31) into (4.44), we find that

The result reduces to that of Goldstein & Leib (Reference Goldstein and Leib1988) when the contribution from the critical layer is ignored. Finally, the indefinite integral in (4.23) can be evaluated analytically. Inserting the result along with (5.30) into (4.39), we obtain the streamwise velocity of the mean-flow distortion,

As was mentioned earlier, (4.45) coupled with an amplitude equation for $B^{\dagger }$ has to be solved to find $U_{M}^{(2)}$ . As the calculation is highly complex, we shall ignore $(U_{M}^{(2)}+(2/U_{c})|A|^{2}\ln \!|{\it\eta}|)$ , which decays like $1/{\it\eta}$ as ${\it\eta}\rightarrow \infty$ . The resultant $O({\it\epsilon}^{2})$ error causes a small discontinuity at $y=y_{c}$ , but does not change the overall feature of the solution. In calculations, use is made of the relation

Figure 7. Distribution of the CS-induced Reynolds stress ${\it\tau}_{12}$ (dashed lines) and mean-flow distortion $U_{M}$ (solid lines) at $\bar{x}=0$ (a), 1 (b), 2 (c) and 3 (d). A small, $O({\it\epsilon}^{2})$ , discontinuity at $y=y_{c}=0$ arises due to neglecting $(U_{M}^{(2)}-c_{M}|A|^{2}\ln \!|{\it\eta}|)$ . The dashed lines with symbols denote $U_{M}^{(1)}$ , the leading-order inner solution.

Figure 8. The change of the momentum thickness ${\rm\Delta}{\it\Theta}/{\it\epsilon}^{2}$ versus $\bar{x}$ : solid line, full result calculated using (5.33); dashed line, the result with the contribution from the critical layer omitted.

Figure 7 displays the distributions of the Reynolds stress ${\it\tau}_{12}$ and the mean-flow distortion $U_{M}$ at four representative streamwise locations. At $\bar{x}=0$ and 1, the disturbance is amplifying, and the Reynolds stress ${\it\tau}_{12}>0$ almost everywhere. The amplitude of the CS peaks at $\bar{x}\approx 1.3$ , downstream of which (e.g. at $\bar{x}=2$ ) ${\it\tau}_{12}<0$ for all $y$ , giving rise to a negative production. As a result, the energy is transferred from the perturbation to the mean flow in this region. At $\bar{x}=3$ , ${\it\tau}_{12}$ becomes positive again, but its value has almost vanished. The overall features of the mean-flow distortion, which concentrates in the critical layer, are well represented by the leading-order inner solution $U_{M}^{(1)}$ . Interestingly, the mean-flow distortion still persists and retains a significant magnitude farther downstream due to the non-equilibrium effect. That the Reynolds stress changes its sign twice is consistent with the experimental observations shown in figures 8(a) and 15 of WW.

Figure 8 shows the change of the momentum thickness. That ${\rm\Delta}{\it\Theta}>0$ at all locations indicates that the shear layer thickens in the presence of CS, consistent with experimental observations. Before the peak position, the contribution from the critical layer is small but appreciable. Its effect becomes rather significant in the attenuating phase as the contribution from outside of the critical layer vanishes quickly. The slow relaxation of the mean-flow distortion and the shear-layer thickness implies that the imprint of the CS in these two related quantities lasts longer than the CS itself, as has been observed in experiments (e.g. Wygnanski et al. Reference Wygnanski, Champagne and Marasli1986). This long memory must be associated with the non-equilibrium effect in the critical layer, which causes the mean-flow distortion to be history-dependent, whereas the distortion outside is proportional to $|A|^{2}$ (i.e. memoryless) as (4.23) indicates.

5.4. Comparison with experiments

Let the dimensional variables/quantities be indicated by the superscript $\ast$ . In order to facilitate the comparison with experiments, note that there exist the following relationships with the non-dimensional variables/quantities adapted in the present paper:

where $C_{1}$ is a profile-dependent constant defined by (5.4). The dimensional frequency $f^{\ast }$ is related to ${\it\omega}\equiv {\it\alpha}c$ (with $S_{0}=0$ ) by the relation

In the experiment of WW, the measured amplitude of CS, shown in figure 8(b) of their paper, is represented by the integrated intensity

where $U_{c}^{\ast }=(U_{1}^{\ast }+U_{2}^{\ast })/2$ and $\langle \cdot \rangle$ represents the amplitude of the time-dependent signature. Application of the relations in (5.36)–(5.38) yields

In the present work,

It follows from (5.39) and (5.40) that

where

In the linear regime, the amplitude function has the solution

where

The parameter ${\it\epsilon}$ is determined by using (5.41), i.e. by requiring the predicted amplitude to be the same as the measured one, at a suitable location $x^{\ast }$ , e.g. at the peak position or in the linear regime upstream. For the latter case, the expression (5.43) may be used.

For the profile (5.17) with $q_{c}\neq 0$ , the Rayleigh equation is solved numerically to find the neutral mode and to evaluate the constants related to the eigenfunction. The results for several $q_{c}$ are listed and contrasted with those for $q_{c}=0$ in table 1.

The experiment of WW was performed for $U_{1}^{\ast }=10~\text{m}~\text{s}^{-1}$ , $U_{2}^{\ast }=6~\text{m}~\text{s}^{-1}$ and $f^{\ast }=44.5~\text{Hz}$ . Assuming that the profile is approximated by (5.17), we need to provide three parameters characterizing the unperturbed mean flow: $q_{c}$ , $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }$ and ${\it\theta}^{\ast }$ . Unfortunately, they are not available from the data of WW. Noting that the experiments of WW and GKW were performed in the same apparatus, we decided to estimate these parameters by appealing to the latter. Curiously, for $q_{c}=0.67$ , the frequency of the neutral mode in GKW does not satisfy the constraint (5.37) despite the fact that the profile was in agreement with the measurement. Our calculation shows that, if $q_{c}=0.9$ is taken instead, the constraint is satisfied in all three cases of GKW. For the same experimental set-up, the spreading rate of the unperturbed flow depends primarily on $U_{1}^{\ast }/U_{2}^{\ast }$ (Oster & Wygnanski Reference Oster and Wygnanski1982; Weisbrot et al. Reference Weisbrot, Einav and Wygnanski1982). This ratio in WW is the same as in case I of GKW, where $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=1/85$ was found. For controlled disturbance with $f^{\ast }=20$  Hz and $U_{1}^{\ast }=5$  m s $^{-1}$ and $U_{2}=3$  m s $^{-1}$ , the measured momentum thickness is approximately 11.5 mm, which converts to a corresponding ${\it\theta}^{\ast }\approx 10$  mm in WW by using (5.37), which indicates that the non-dimensional frequency $f^{\ast }{\it\theta}^{\ast }/(U_{1}^{\ast }-U_{2}^{\ast })$ must be equal in the two experiments. Therefore, we take $q_{c}=0.9$ , $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=1/85$ and ${\it\theta}^{\ast }=10~\text{mm}$ in our calculations.

The amplitude development predicted by the theory is shown in figure 9 and compared with the experimental data of WW. The agreement is good in the linear and earlier nonlinear stages up to the peak position. Further downstream, the theory mimics the general trend, e.g. both the theoretical result and the measurement feature a local maximum downstream, but considerable discrepancy emerges: the decay predicted by the theory is much faster and the maximum is more pronounced than those measured. As $V_{B}$ and hence ${\it\chi}$ may be influenced by the detailed arrangements of the experiments, the value of ${\it\chi}$ is doubled, halved (not shown) and set to zero. None of these leads to significant change in the overall features of the solution. However, if the non-parallel-flow effect is suppressed, the result (represented by the dashed line) bears little resemblance to the measurement. Even in the linear stage, reasonable agreement comes only when non-parallelism is accounted for. However, purely linear but non-parallel theory does not capture the characteristics of attenuation.

Calculations were also performed pertaining to the experiment of Fiedler & Mensing (Reference Fiedler and Mensing1985), where the free-stream velocities are $U_{1}^{\ast }=11~\text{m}~\text{s}^{-1}$ and $U_{2}^{\ast }=0$ , and a disturbance with frequency $f^{\ast }=30~\text{Hz}$ was introduced through controlled excitation. Fiedler & Mensing provided the spreading rate and thickness of the mixing layer for what they referred to as ‘neutral flow’, i.e. the state without artificial excitation. The mean flow is thus primarily influenced by small-scale turbulence and its spreading rate, $\text{d}{\it\theta}^{\ast }/\text{d}x^{\ast }=0.0354$ , is used in our calculations. The experimental data for two weakest excitations, shown in figure 13 of their paper, indicate that the amplitude of the fundamental component saturates at $x^{\ast }$ , where $S_{x}=f^{\ast }(x^{\ast }-x_{0}^{\ast })/U_{1}^{\ast }\approx 1.25$ ( $x_{0}^{\ast }\approx -88~\text{mm}$ ), which corresponds to $x^{\ast }\approx 370~\text{mm}$ and ${\it\theta}^{\ast }\approx 16~\text{mm}$ , as can be inferred from figure 28 of Fiedler & Mensing (Reference Fiedler and Mensing1985). This position is taken as an approximation for the neutral location of the imposed disturbance, and hence the value ${\it\theta}^{\ast }=16~\text{mm}$ was selected. The relation (5.37) is then satisfied for the profile (5.17) with $q_{c}\approx 0.1$ . The amplitude ${\it\epsilon}$ was chosen such that the prediction and the measurement are the same at the peak position.

Figure 9. Comparison with the experimental data of WW. Curves a–e represent theoretical results for $q_{c}=0.9$ : (a)  ${\it\chi}=-0.0306$ , the value calculated for the experimental condition; (b) parallel–nonlinear theory; (c) linear solution including non-parallelism; (d)  ${\it\chi}=-0.0612$ , twice the experimental value; and (e)  ${\it\chi}=0$ . Curve f shows experimental data.

Fiedler & Mensing (Reference Fiedler and Mensing1985) measured the root-mean-square velocity of the fundamental component at $y=0$ , normalized by $U_{1}^{\ast }$ . Interestingly, the disturbance amplitude, as displayed in figures 18 and 22 in their paper, experiences more reduction and exhibits more pronounced oscillation than that in the experiment of WW. The measured quantity, denoted here by $A_{exp}$ , is related to $A$ in the present theory by

The comparison is shown in figure 10. There is a fairly good degree of quantitative agreement. With non-parallelism being included, the prediction for the initial growth in the linear regime is quite accurate. The theory not only captures the overall nonlinear development, but also mimics the rather detailed feature of non-monotonic attenuation. The measurement is less oscillatory than the theoretical result, which is also the case for the experiment of WW. The reason for this remains unclear. It might be that the closure model for fine-scale turbulence is not sophisticated enough, or that the disturbances in experiments have narrow spectral bands rather than being monochromatic as assumed in calculations. The possibility of the smoothing effect of phase averaging experimental data cannot be ruled out either.

Figure 10. Comparison with the experimental data of Fiedler & Mensing (Reference Fiedler and Mensing1985). Curves a–e are for $q_{c}=0.1$ : (a)  ${\it\chi}=-0.01729$ , the value calculated for the experimental condition; (b) parallel–nonlinear theory; (c) linear solution including non-parallelism; (d)  ${\it\chi}=-0.03458$ , twice the experimental value; (e)  ${\it\chi}=0$ . Curve f is for $q_{c}=0.3$ . Curve g shows experimental data.

In order to examine the sensitivity of the prediction to uncertainty of the parameters, further calculations were carried out with the value of ${\it\chi}$ being doubled, halved (not shown) and set to zero, and also with $q_{c}=0.3$ . In all these cases, the evolution retains its overall features. While the behaviour of the solution is remarkably robust with respect to ${\it\chi}$ and $q_{c}$ , nonlinearity and non-parallelism are both instrumental. With the former being ignored, the disturbance would feature a too broad peak and would decay monotonically as indicated by the linear solution in the figure. On the other hand, if non-parallelism is neglected, the theory would fail to predict attenuation.

Finally, figure 11 displays contours of the critical-layer vorticity of the CS at four representative streamwise locations. They illustrate the roll-up process leading to the formation of Kelvin–Helmholtz rollers.

Figure 11. Vortex roll-up pertaining to the experimental condition of Fiedler & Mensing (Reference Fiedler and Mensing1985), illustrated by contours of ${\it\Omega}_{c}$ , the normalized critical-layer vorticity of CS, at $x^{\ast }=0.1$ , 0.2, 0.3 and 0.5 m from the leading edge.

The different performance of the theory, i.e. a fairly good quantitative agreement with the experiment of Fiedler & Mensing (Reference Fiedler and Mensing1985) versus a merely qualitative agreement with that of WW, requires some comments. For the latter case, we took the parameter $q_{c}$ in the mean-flow profile (5.17) to be 0.9 rather than 0.67 suggested by experiments because the latter value led to inconsistency in the neutral frequencies. However, it is possible that the assumed profile (5.17) is not entirely appropriate, thereby causing the inaccuracy of the prediction.