2 Brief review of earlier work

3 Basic equations for the two-dimensional turbulent jet

For boundary layers, jets and wakes the thin shear-layer approximation, which assumes that $x$-derivatives of mean quantities are much smaller than $y$-derivatives and that $V\ll U$, is commonly used. This makes it possible to neglect some terms in the Reynolds averaged Navier–Stokes equations. However, the assumption works better for turbulent boundary layers than for wakes and jets since the spread in the lateral direction is an order of magnitude larger for the latter. However, for jets and wakes a further assumption can be made, namely that for high enough Reynolds numbers, the viscous term can be neglected; see, for instance, Townsend (Reference Townsend1956) and Pope (Reference Pope2000). In order to verify that the thin shear-layer approximation as well as neglecting viscous terms are both valid, these terms can be evaluated a posteriori when the approximate solution has been found. Under the above assumptions the Reynolds-averaged continuity, streamwise ($x$) and transversal ($y$) momentum equations reduce to

(3.1)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}x}+\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}y}=0, & \displaystyle\end{eqnarray}$$

(3.2)$$\begin{eqnarray}\displaystyle & \displaystyle U\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}x}+V\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}y}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}P}{\unicode[STIX]{x2202}x}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\overline{uv}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\overline{uu}, & \displaystyle\end{eqnarray}$$

(3.3)$$\begin{eqnarray}\displaystyle & \displaystyle U\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}x}+V\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}y}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}P}{\unicode[STIX]{x2202}y}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\overline{vv}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\overline{uv}, & \displaystyle\end{eqnarray}$$

where an overbar denotes the time-average operator.

By assuming that both the left-hand side (LHS) and the $x$-derivative of the Reynolds shear stress in (3.3) are negligible, it can be integrated to become

(3.4)$$\begin{eqnarray}P+\unicode[STIX]{x1D70C}\overline{vv}=P_{\infty },\end{eqnarray}$$

simplifying (3.3) and leading to

(3.5)$$\begin{eqnarray}U\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}x}+V\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}y}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\text{d}P_{\infty }}{\text{d}x}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\overline{uv}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}(\overline{uu}-\overline{vv}).\end{eqnarray}$$

Although we can expect the last term involving the $x$-derivative of the normal Reynolds stresses to be much smaller than the $y$-derivative of the Reynolds shear stress we keep it here since, as we will show later, it gives a non-negligible contribution to the momentum flux of the jet. Equations (3.5) and (3.1) are then complemented by the boundary conditions on the jet axis

(3.6a,b)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}y}(x,0)=0\quad \text{and}\quad V(x,0)=0,\end{eqnarray}$$

together with $U(x,y)\rightarrow 0$ as $y\rightarrow \pm \infty$.

Partly following Pope (Reference Pope2000) in his analysis of the planar jet, we now assume a similarity solution of the form

(3.7)$$\begin{eqnarray}\displaystyle & \displaystyle U(x,y)=U_{c}(x)F^{\prime }(\unicode[STIX]{x1D702}), & \displaystyle\end{eqnarray}$$

(3.8)$$\begin{eqnarray}\displaystyle & \displaystyle \overline{uv}=U_{c}^{2}(x)g(\unicode[STIX]{x1D702}), & \displaystyle\end{eqnarray}$$

(3.9)$$\begin{eqnarray}\displaystyle & \displaystyle \overline{uu}=U_{c}^{2}(x)f(\unicode[STIX]{x1D702}), & \displaystyle\end{eqnarray}$$

(3.10)$$\begin{eqnarray}\displaystyle & \displaystyle \overline{vv}=U_{c}^{2}(x)h(\unicode[STIX]{x1D702}), & \displaystyle\end{eqnarray}$$

where a prime denotes differentiation with respect to the similarity coordinate given by $\unicode[STIX]{x1D702}=y/\unicode[STIX]{x1D6FF}(x)$. The centreline velocity of the jet is $U_{c}(x)=U(x,0)$ so that $F^{\prime }(0)=1$ and $F^{\prime \prime }(0)=0$, while away from the jet the condition $F^{\prime }(\pm \infty )=0$ holds. The volume flux per unit width of the jet is denoted by $Q=\int _{-\infty }^{\infty }U\,\text{d}y$ and is expected to increase with $x$ as more fluid from the surroundings is entrained into the jet. We also define the transverse length scale $\unicode[STIX]{x1D6FF}(x)$ from the volume flux such that

(3.11)$$\begin{eqnarray}\unicode[STIX]{x1D6FF}=\frac{1}{2U_{c}}\int _{-\infty }^{\infty }U\,\text{d}y=\frac{Q}{2U_{c}}.\end{eqnarray}$$

Usually a transverse length scale, $\unicode[STIX]{x1D6FF}_{1/2}$, is adopted, defined as the position where $U(x,\unicode[STIX]{x1D6FF}_{1/2})=U_{c}/2$. However, we find the definition in (3.11) a better choice from a physical point of view since it is directly coupled to the entrainment of the fluid outside the jet. These two length scales are related in the self-similar region as $\unicode[STIX]{x1D6FF}_{1/2}=\ln (1+\sqrt{2})\unicode[STIX]{x1D6FF}\approx 0.88\unicode[STIX]{x1D6FF}$.

When carrying out the similarity analysis, one usually neglects the term $\overline{uu}-\overline{vv}$ in the momentum conservation (3.5) which is assumed to be negligible. However, as will be shown later, the contribution of $m_{x}=\int _{-\infty }^{\infty }(\overline{uu}-\overline{vv})\,\text{d}y$ is of the order of 5 % of the mean momentum flux per unit mass, $M_{x}=\int _{-\infty }^{\infty }U^{2}\,\text{d}y$, and thereby gives a non-negligible contribution to it. The general balance obtained by integrating the streamwise momentum equation gives

(3.12)$$\begin{eqnarray}\int _{-\infty }^{\infty }(\unicode[STIX]{x1D70C}U^{2}+\unicode[STIX]{x1D70C}\overline{uu}+P-P_{\infty })\,\text{d}y\approx \unicode[STIX]{x1D70C}\int _{-\infty }^{\infty }(U^{2}+\overline{uu}-\overline{vv})\,\text{d}y=\text{const}.\end{eqnarray}$$

The assumption that the velocity distributions of $U$, $\overline{uu}$ and $\overline{vv}$ are self-similar gives

(3.13)$$\begin{eqnarray}M_{x}+m_{x}=U_{c}(x)^{2}\unicode[STIX]{x1D6FF}(x)\int _{-\infty }^{\infty }(F^{\prime 2}+f-h)\,\text{d}\unicode[STIX]{x1D702}=\text{const.},\end{eqnarray}$$

showing that the product $U_{c}^{2}\unicode[STIX]{x1D6FF}$ is independent of $x$.

If a similarity solution exists, it is possible to show (see, for example, Pope (Reference Pope2000)) that $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FD}(x-x_{0})$ where $\unicode[STIX]{x1D6FD}=\text{d}\unicode[STIX]{x1D6FF}/\text{d}x$ is the jet spreading rate and $x_{0}$ is the virtual origin of the jet. Equation (3.13) immediately gives that $U_{c}\propto (x-x_{0})^{-1/2}$. By modelling the Reynolds shear stress as $\overline{uv}=-\unicode[STIX]{x1D708}_{T}(\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}y)$, where $\unicode[STIX]{x1D708}_{T}$ indicates the eddy viscosity, and assuming that $\unicode[STIX]{x1D708}_{T}$ is independent of $\unicode[STIX]{x1D702}$, it is possible to obtain

(3.14)$$\begin{eqnarray}\overline{uv}=U_{c}^{2}g=-\frac{\unicode[STIX]{x1D708}_{T}U_{c}}{\unicode[STIX]{x1D6FF}}F^{\prime \prime }\Rightarrow g=-\frac{\unicode[STIX]{x1D708}_{T}}{U_{c}\unicode[STIX]{x1D6FF}}F^{\prime \prime }=-\frac{F^{\prime \prime }}{Re_{T}}.\end{eqnarray}$$

Since both $g$ and $F$ are independent of $x$, this means that the turbulent Reynolds number, $Re_{T}$, is independent of $x$ and consequently that $\unicode[STIX]{x1D708}_{T}\propto (x-x_{0})^{1/2}$. The usual approach to obtain the mean velocity and shear stress profiles (see Pope (Reference Pope2000)) is now to make use of (3.5), assume a zero pressure gradient and neglect the $x$-derivative of the normal stress terms. One then obtains the following equation for $F$:

(3.15)$$\begin{eqnarray}F^{\prime \prime \prime }+2\unicode[STIX]{x1D6FC}^{2}(F^{\prime 2}+FF^{\prime \prime })=0\quad \text{where }\unicode[STIX]{x1D6FC}={\textstyle \frac{1}{2}}(\unicode[STIX]{x1D6FD}Re_{T})^{1/2},\end{eqnarray}$$

and can thereby obtain the streamwise mean velocity distribution. The solution to (3.15) is found to be $F^{\prime }(\unicode[STIX]{x1D702})=\text{sech}^{2}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702})$. By integrating this distribution across the jet, we find (with our choice of length scale) that $\unicode[STIX]{x1D6FC}=1$ and that $Re_{T}=4/\unicode[STIX]{x1D6FD}$, leading to the theoretical velocity profile

(3.16)$$\begin{eqnarray}U_{th}=U_{c}\,\text{sech}^{2}\unicode[STIX]{x1D702}.\end{eqnarray}$$

It is interesting to extend the present theory to non-uniform viscosity in the transversal direction. Indeed, one can assume $\unicode[STIX]{x1D708}_{T}=\unicode[STIX]{x1D708}_{T0}(x)w(\unicode[STIX]{x1D702})$ with $w(0)=1$, so that (3.15) becomes

(3.17)$$\begin{eqnarray}wF^{\prime \prime \prime }+w^{\prime }F^{\prime \prime }+2\unicode[STIX]{x1D6FC}^{2}(F^{\prime 2}+FF^{\prime \prime })=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}=(\unicode[STIX]{x1D6FD}Re_{T0})^{1/2}/2$ and $Re_{T0}=U_{c}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D708}_{T0}$. The integration of (3.17) in the transversal direction leads to

(3.18)$$\begin{eqnarray}wF^{\prime \prime }+2\unicode[STIX]{x1D6FC}^{2}FF^{\prime }=0,\end{eqnarray}$$

where the boundary conditions have been already enforced. Equation (3.18) can, however, not be solved analytically as in the constant eddy-viscosity case.

The transversal-velocity distribution can be calculated directly from the continuity equation (3.1) in the constant eddy-viscosity case as

(3.19)$$\begin{eqnarray}\frac{V_{th}}{U_{c}}=\unicode[STIX]{x1D6FD}\left(\unicode[STIX]{x1D702}F^{\prime }-\frac{F}{2}\right)=\unicode[STIX]{x1D6FD}\left(\unicode[STIX]{x1D702}\,\text{sech}^{2}\unicode[STIX]{x1D702}-\frac{1}{2}\tanh \unicode[STIX]{x1D702}\right),\end{eqnarray}$$

while the expression for $\overline{uv}$ is determined as

(3.20)$$\begin{eqnarray}\frac{\overline{uv}_{th}}{U_{c}^{2}}=-\frac{\unicode[STIX]{x1D708}_{T}}{U_{c}^{2}}\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}y}=-\frac{1}{Re_{T}}F^{\prime \prime }=\frac{\unicode[STIX]{x1D6FD}}{2}\,\text{sech}^{2}\unicode[STIX]{x1D702}\tanh \unicode[STIX]{x1D702},\end{eqnarray}$$

where $F^{\prime \prime }$ has a maximum at $\unicode[STIX]{x1D702}=0.66$ with a value of $F_{max}^{\prime \prime }=0.77$. This gives the maximum $|\overline{uv}|_{max}/U_{c}^{2}\approx 0.19\unicode[STIX]{x1D6FD}$ so that the only empirical input to this expression is the jet spreading rate. From the theoretical solution it is possible to get an analytical expression for the mean momentum flux; i.e. $M_{x}=4U_{c}^{2}\unicode[STIX]{x1D6FF}/3$.

4 Three-component decomposition

The time signal of any velocity component measured in the jet flow can be seen as consisting of the mean velocity, periodic structures related to the jet dynamics and random small-scale turbulence. Through a triple decomposition it is possible to distinguish these different contributions to the signal. We consider a generic physical quantity $q(x,y,z,t)$ in a flow field which has a stationary mean value $\overline{q}$, and furthermore $q$ has both a periodic and a random variation. We can decompose $q$ as (Hussain & Reynolds Reference Hussain and Reynolds1970)

(4.1)$$\begin{eqnarray}q(x,y,z,t)=\overline{q}(x,y,z)+\widetilde{q}(x,y,z,\unicode[STIX]{x1D70F})+\widehat{q}(x,y,z,t),\end{eqnarray}$$

where $\widetilde{q}$ is the periodic component and $\widehat{q}$ the random one. Here, $\widetilde{q}$ is periodic with a periodicity of $T$ which is the duration of one period of the signal and $\unicode[STIX]{x1D70F}=t-nT$ where $n=0,1,2,\ldots ,N-1$ is the period number with respect to a given starting point, and is hence the time measured from a certain phase of the periodic signal. The mean value is obtained as the time-averaged value (over the measurement time $NT$) of the signal as

(4.2)$$\begin{eqnarray}\overline{q}(x,y,z)=\frac{1}{NT}\int _{0}^{NT}q(x,y,z,t)\,\text{d}t,\end{eqnarray}$$

whereas the periodic component is obtained as the ensemble averaged value of $q-\overline{q}$, namely as

(4.3)$$\begin{eqnarray}\langle q\rangle =\widetilde{q}(x,y,z,\unicode[STIX]{x1D70F})=\frac{1}{N}\mathop{\sum }_{n=0}^{N-1}q(x,y,z,\unicode[STIX]{x1D70F}+nT)-\overline{q}(x,y,z).\end{eqnarray}$$

Since $\widetilde{q}$ and $\widehat{q}$ are uncorrelated, the time-averaged values of both need to be equal to zero. In the following we will denote time averages of any quantity $q$ with an overbar $\overline{q}$ and phase averages of $q-\overline{q}$ with angle brackets $\langle q\rangle$ or $\widetilde{q}$: this redundant notation is used to facilitate the writing of the equations. Consequently, the definition (4.3), the phase average of product of two quantities is given by

(4.4)$$\begin{eqnarray}\langle qp\rangle =\overline{q}\langle p\rangle +\langle q\rangle \overline{p}+\langle \widetilde{q}\,\widetilde{p}\rangle +\langle \hat{q}\hat{p}\rangle ,\end{eqnarray}$$

while the time average of the product is given by

(4.5)$$\begin{eqnarray}\overline{qp}=\overline{q}\,\overline{p}+\overline{\langle q\rangle \langle p\rangle }+\overline{\hat{q}\hat{p}},\end{eqnarray}$$

implying that the Reynolds stresses are obtained from the self-interaction of the periodic and random components.

It is possible to obtain an equation for the wave component by substituting the triple decomposition into the incompressible Navier–Stokes equations. By taking the time average and subtracting it from the original, applying the ensemble averaging and then using the definition of a phase average, we obtain an equation for $\widetilde{u}_{i}$ as

(4.6)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{u}_{i}}{\unicode[STIX]{x2202}t}+\overline{u}_{m}\frac{\unicode[STIX]{x2202}\widetilde{u}_{i}}{\unicode[STIX]{x2202}x_{m}}+\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\widetilde{p}}{\unicode[STIX]{x2202}x_{i}}-\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}\widetilde{u}_{i}}{\unicode[STIX]{x2202}x_{m}^{2}}+\widetilde{u}_{m}\frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{m}}=-\frac{\unicode[STIX]{x2202}\langle \widetilde{u}_{i}\widetilde{u}_{m}\rangle }{\unicode[STIX]{x2202}x_{m}}-\frac{\unicode[STIX]{x2202}\langle {\widehat{u}_{i}\widehat{u}}_{m}\rangle }{\unicode[STIX]{x2202}x_{m}}. & & \displaystyle\end{eqnarray}$$

The same operation can be done for the continuity equation resulting in

(4.7a-c)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{i}}=0,\quad \frac{\unicode[STIX]{x2202}\widetilde{u}_{i}}{\unicode[STIX]{x2202}x_{i}}=0,\quad \frac{\unicode[STIX]{x2202}\widehat{u}_{i}}{\unicode[STIX]{x2202}x_{i}}=0. & & \displaystyle\end{eqnarray}$$

All terms on the LHS of (4.6) are first-order for $\widetilde{u}_{i}$ and $\widetilde{p}$. The first term on the right-hand side (RHS) is, however, of second-order for $\widetilde{u}_{i}$, so that it can be ignored when the periodic component is small compared to the mean component. If the second term on the RHS, which is due to the random fluctuations, can also be considered small, equation (4.6) is, together with the continuity equation for $\widetilde{u}_{i}$, a linear system of equations for the periodic components, $\widetilde{u}_{i}$ and $\widetilde{p}$. Another possibility to get linear behaviour is that $\langle {\widehat{u}_{i}\widehat{u}}_{m}\rangle$ scales linearly with the forcing amplitude, a hypothesis that will be verified a posteriori.

It is possible to obtain a transport equation for the phase-averaged random component and to derive this equation one first obtains the equation for $\widehat{u}_{i}$, multiplies by $\widehat{u}_{j}$ and vice versa and sums the two equations. Thereafter one subtracts the time averaged from the ensemble averaged equation, using the relation $\langle \langle \widehat{q}_{1}\widehat{q}_{2}\rangle \widetilde{q}_{3}\rangle =\langle \widehat{q}_{1}\widehat{q}_{2}\rangle \widetilde{q}_{3}-\overline{\langle \widehat{q}_{1}\widehat{q}_{2}\rangle \widetilde{q}_{3}}$ and after some algebra one obtains

(4.8)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\langle \widehat{u}_{i}\widehat{u}_{j}\rangle }{\unicode[STIX]{x2202}t}+\overline{u}_{m}\frac{\unicode[STIX]{x2202}\langle \widehat{u}_{i}\widehat{u}_{j}\rangle }{\unicode[STIX]{x2202}x_{m}} & = & \displaystyle -\langle \widehat{u}_{i}\widehat{u}_{m}\rangle \frac{\unicode[STIX]{x2202}\overline{u}_{j}}{\unicode[STIX]{x2202}x_{m}}-\langle \widehat{u}_{j}\widehat{u}_{m}\rangle \frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{m}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{\unicode[STIX]{x1D70C}}\left\langle \widehat{u}_{i}\frac{\unicode[STIX]{x2202}\widehat{p}}{\unicode[STIX]{x2202}x_{j}}+\widehat{u}_{j}\frac{\unicode[STIX]{x2202}\widehat{p}}{\unicode[STIX]{x2202}x_{i}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}\langle \widehat{u}_{i}\widehat{u}_{j}\rangle }{\unicode[STIX]{x2202}x_{m}\unicode[STIX]{x2202}x_{m}}-2\unicode[STIX]{x1D708}\left\langle \frac{\unicode[STIX]{x2202}\widehat{u}_{i}}{\unicode[STIX]{x2202}x_{m}}\frac{\unicode[STIX]{x2202}\widehat{u_{j}}}{\unicode[STIX]{x2202}x_{m}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\overline{\widehat{u}_{i}\widehat{u}}_{m}\frac{\unicode[STIX]{x2202}\widetilde{u}_{j}}{\unicode[STIX]{x2202}x_{m}}-\overline{\widehat{u}_{j}\widehat{u}}_{m}\frac{\unicode[STIX]{x2202}\widetilde{u}_{i}}{\unicode[STIX]{x2202}x_{m}}\nonumber\\ \displaystyle & & \displaystyle -\,\left\langle \langle {\widehat{u}_{i}\widehat{u}}_{m}\rangle \frac{\unicode[STIX]{x2202}\widetilde{u}_{j}}{\unicode[STIX]{x2202}x_{m}}\right\rangle -\left\langle \langle {\widehat{u}_{j}\widehat{u}}_{m}\rangle \frac{\unicode[STIX]{x2202}\widetilde{u}_{i}}{\unicode[STIX]{x2202}x_{m}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\left\langle \frac{\unicode[STIX]{x2202}\widetilde{u}_{m}\overline{\widehat{u}_{i}\widehat{u}_{j}}}{\unicode[STIX]{x2202}x_{m}}\right\rangle -\left\langle \frac{\unicode[STIX]{x2202}\widetilde{u}_{m}\langle \widehat{u}_{i}\widehat{u}_{j}\rangle }{\unicode[STIX]{x2202}x_{m}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x2202}\langle \widehat{u}_{i}\widehat{u}_{j}\widehat{u}_{m}\rangle }{\unicode[STIX]{x2202}x_{m}}.\end{eqnarray}$$

The terms in this equation can be interpreted as follows. First line: LHS, time-dependent and convective change of the phase-averaged random stresses; RHS, production through interaction of the phase-averaged stresses and mean flow field. Second line: pressure redistribution term. Third line: viscous diffusion and viscous dissipation, respectively. Fourth line: production through interaction between the time-averaged random stresses and the periodic field. Fifth line: phased-averaged production through interaction between the phase-averaged random stresses and the periodic field. Sixth line: transport of the random stresses by the periodic field. Seventh line: nonlinear turbulent transport term obtained from the interaction of the random field on itself.

All terms except the term on the seventh line of (4.8) are quadratic in the random fluctuations. In order for this equation to be linear the phased-averaged production terms on the fifth line and the last term on the sixth line need to be negligible and the triple correlation term $\unicode[STIX]{x2202}\langle \widehat{u}_{i}\widehat{u}_{j}\widehat{u}_{m}\rangle /\unicode[STIX]{x2202}x_{m}$ needs to be either proportional to the perturbation amplitude or negligible. As will be shown in § 7.2, the phase average of the product of whatever power of the random velocities (as well as cross products) is linearly proportional to the perturbation amplitude, so that term must be retained. Consequently, an important observation is that the random field is forced linearly by the periodic field. If this is the case, it should be possible to describe the flow development, at least for scales of the order of the periodic motion, with linear modes. On the other hand, the product of $\langle {\widehat{u}_{i}\widehat{u}}_{m}\rangle$ (which is proportional to the perturbation) with $\widetilde{u}_{j}$ is quadratic in the amplitude and therefore is of lower magnitude in (4.8). By neglecting all the higher-order terms as well as the viscous transport, it is possible to get the following balance:

(4.9)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\langle \widehat{u}_{i}\widehat{u}_{j}\rangle }{\unicode[STIX]{x2202}t}+\overline{u}_{m}\frac{\unicode[STIX]{x2202}\langle \widehat{u}_{i}\widehat{u}_{j}\rangle }{\unicode[STIX]{x2202}x_{m}} & = & \displaystyle -\langle \widehat{u}_{i}\widehat{u}_{m}\rangle \frac{\unicode[STIX]{x2202}\overline{u}_{j}}{\unicode[STIX]{x2202}x_{m}}-\langle \widehat{u}_{j}\widehat{u}_{m}\rangle \frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{m}}-\frac{1}{\unicode[STIX]{x1D70C}}\left\langle \widehat{u}_{i}\frac{\unicode[STIX]{x2202}\widehat{p}}{\unicode[STIX]{x2202}x_{j}}+\widehat{u}_{j}\frac{\unicode[STIX]{x2202}\widehat{p}}{\unicode[STIX]{x2202}x_{i}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x1D708}\left\langle \frac{\unicode[STIX]{x2202}\widehat{u}_{i}}{\unicode[STIX]{x2202}x_{m}}\frac{\unicode[STIX]{x2202}\widehat{u_{j}}}{\unicode[STIX]{x2202}x_{m}}\right\rangle -\overline{\widehat{u}_{i}\widehat{u}}_{m}\frac{\unicode[STIX]{x2202}\widetilde{u}_{j}}{\unicode[STIX]{x2202}x_{m}}-\overline{\widehat{u}_{j}\widehat{u}}_{m}\frac{\unicode[STIX]{x2202}\widetilde{u}_{i}}{\unicode[STIX]{x2202}x_{m}}-\left\langle \frac{\unicode[STIX]{x2202}\widetilde{u}_{m}\overline{\widehat{u}_{i}\widehat{u}_{j}}}{\unicode[STIX]{x2202}x_{m}}\right\rangle \nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x2202}\langle \widehat{u}_{i}\widehat{u}_{j}\widehat{u}_{m}\rangle }{\unicode[STIX]{x2202}x_{m}},\end{eqnarray}$$

which provides the leading-order balance of the perturbation equation.

5 Linear stability analysis

Equations (4.6) and (4.7) allow us to estimate the dynamics of the periodic component. The second last term of the RHS of (4.6) is nonlinear in the periodic perturbation, while the last term relates to quantities that are unknown and correlate to the turbulence closure problem. In order to assess the dynamics of the periodic fluctuation, a linearised analysis can be performed, where it can be assumed that the perturbation component is small compared to the base flow. The nonlinear term in the periodic–periodic interaction can therefore be neglected, while the random–random interaction needs to be modelled. Barkley (Reference Barkley2006) pioneered this problem by performing a linear stability analysis of the mean flow behind a cylinder above the critical Reynolds number, rather than by using the unstable base flow determined from the steady solution of the Navier–Stokes equation, obtaining good agreement between the global mode frequency and the experimental observation. It is noteworthy that Barkley did not account for any Reynolds stress modification for the perturbed flow. Sipp & Lebedev (Reference Sipp and Lebedev2007) provided a theoretical justification by means of a weakly nonlinear global stability analysis, although some counterexamples about the superiority of the mean flow against the base flow were also provided. In later works (Viola et al. Reference Viola, Iungo, Camarri, Porté-Agel and Gallaire2014; Camarri, Trip & Fransson Reference Camarri, Trip and Fransson2017) an eddy-viscosity model has been included to account for the energy transfer from the random part. Here it is assumed that the interaction between incoherent and coherent motion is present and given by the eddy-viscosity model

(5.1)$$\begin{eqnarray}\langle \widehat{u}_{i}\widehat{u}_{j}\rangle =-\unicode[STIX]{x1D708}_{T}\left(\frac{\unicode[STIX]{x2202}\widetilde{u_{i}}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}\widetilde{u_{j}}}{\unicode[STIX]{x2202}x_{i}}\right)\end{eqnarray}$$

as an extension of the eddy-viscosity model (usually developed for the mean flow) to the periodic component. This is not expected to be unreasonable since the periodic component is characterised by a time scale larger than the random fluctuations, thereby following a law similar to the mean component. For the sake of simplicity, a constant eddy-viscosity distribution will be assumed in the transversal direction. In our stability analysis a comparison with zero eddy-viscosity was also made and will be discussed in § 7.3.

Let us consider a two-dimensional base flow $\boldsymbol{U}=(U,\unicode[STIX]{x1D6FD}V)^{\text{T}}$ with a superposed two-dimensional perturbation $\boldsymbol{u}=(u,v)^{\text{T}}$ (the phase-average nomenclature is omitted in this section for the sake of clarity). The base flow is related to the time-averaged velocity field by means of the formulas $U=U_{th}$ and $V=V_{th}/\unicode[STIX]{x1D6FD}$. Since the flow is non-parallel, simple parallel-flow based techniques are subject to errors. However, this problem can be circumvented by assuming that the base-flow development in the streamwise direction is slow compared to the perturbation one. The spreading rate, $\unicode[STIX]{x1D6FD}$, can at this point be used to scale the streamwise coordinate, $x$, such that $X=\unicode[STIX]{x1D6FD}x$ so that $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}X=O(1)$. In scaled coordinates, equations (4.6) and (4.7), together with the ansatz (5.1), become

(5.2)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D6FD}U\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}X}+\unicode[STIX]{x1D6FD}V\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}y}+\unicode[STIX]{x1D6FD}u\frac{\unicode[STIX]{x2202}\boldsymbol{U}}{\unicode[STIX]{x2202}X}+v\frac{\unicode[STIX]{x2202}\boldsymbol{U}}{\unicode[STIX]{x2202}y}=-\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6FD}}p+\unicode[STIX]{x1D708}_{T}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D6FD}}^{2}\boldsymbol{u}+\unicode[STIX]{x1D6FD}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D708}_{T}}{\unicode[STIX]{x2202}X}\left(\unicode[STIX]{x1D6FD}\frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}X}+\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6FD}}u\right), & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(5.3)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FD}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}X}+\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}y}=0, & \displaystyle\end{eqnarray}$$

with the operators

(5.4a,b)$$\begin{eqnarray}\unicode[STIX]{x1D735}_{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D719}=\left(\unicode[STIX]{x1D6FD}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}X},\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}y}\right)^{\text{T}},\quad \unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D6FD}}^{2}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6FD}^{2}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}X^{2}}+\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}y^{2}}.\end{eqnarray}$$

It is now possible to introduce the normal mode assumption

(5.5)$$\begin{eqnarray}\boldsymbol{q}=\left(\begin{array}{@{}c@{}}u\\ v\\ p\\ \end{array}\right)=\hat{\boldsymbol{q}}(X,y)\exp \left[\text{i}\left(\frac{1}{\unicode[STIX]{x1D6FD}}\int _{X_{0}}^{X}\unicode[STIX]{x1D6FC}(X^{\prime })\,\text{d}X^{\prime }-\unicode[STIX]{x1D714}t\right)\right],\end{eqnarray}$$

where $\hat{\boldsymbol{q}}$ and $\unicode[STIX]{x1D6FC}$ are the modal shape and streamwise wavenumber (both dependent on the streamwise position as the base flow is changing with the downstream distance) while $\unicode[STIX]{x1D714}$ is the pulsation frequency assumed to be constant and real in the spatial approach. Given the modal assumption (5.5), equations (5.2) and (5.3) become

(5.6)$$\begin{eqnarray}\displaystyle & & \displaystyle \left[-\text{i}\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FC}U+\unicode[STIX]{x1D708}_{T}\left(\unicode[STIX]{x1D6FC}^{2}-\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y^{2}}\right)\right]\hat{\boldsymbol{u}}+\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}y}\hat{v}\unicode[STIX]{x1D6FF}_{j1}+\widehat{\unicode[STIX]{x1D735}}\hat{p}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\unicode[STIX]{x1D6FD}\left\{\left[U\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X}+V\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}-\text{i}\unicode[STIX]{x1D708}_{T}\left(2\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x2202}X}\right)-\text{i}\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D708}_{T}}{\unicode[STIX]{x2202}X}\right]\hat{\boldsymbol{u}}\right.\nonumber\\ \displaystyle & & \displaystyle \quad +\left.\left(\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}X}\hat{u} +\frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}X}\right)\unicode[STIX]{x1D6FF}_{j1}+\frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}y}\hat{v}\unicode[STIX]{x1D6FF}_{j2}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D708}_{T}}{\unicode[STIX]{x2202}X}\widehat{\unicode[STIX]{x1D735}}\hat{u} \right\}=0,\end{eqnarray}$$

(5.7)$$\begin{eqnarray}\text{i}\unicode[STIX]{x1D6FC}\hat{u} +\frac{\unicode[STIX]{x2202}\hat{v}}{\unicode[STIX]{x2202}y}+\unicode[STIX]{x1D6FD}\frac{\unicode[STIX]{x2202}\hat{u} }{\unicode[STIX]{x2202}X}=0,\end{eqnarray}$$

where

(5.8)$$\begin{eqnarray}\widehat{\unicode[STIX]{x1D735}}\unicode[STIX]{x1D719}=\left(\text{i}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D719},\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}y}\right).\end{eqnarray}$$

It is now possible to introduce the asymptotic expansion for the modal shape $\hat{\boldsymbol{q}}=\hat{\boldsymbol{q}}_{0}+\unicode[STIX]{x1D6FD}\hat{\boldsymbol{q}}_{1}+O(\unicode[STIX]{x1D6FD}^{2})$. By grouping terms of the same order of magnitude, a set of problems can be obtained as

(5.9)$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}\hat{\boldsymbol{q}}_{0}=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(5.10)$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}\hat{\boldsymbol{q}}_{1}=-{\mathcal{H}}\hat{\boldsymbol{q}}_{0}, & \displaystyle\end{eqnarray}$$

with the linear operators ${\mathcal{L}}$ and ${\mathcal{H}}$ defined as

(5.11)$$\begin{eqnarray}{\mathcal{L}}=\left[-\text{i}\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FC}U+\unicode[STIX]{x1D708}_{T}\left(\unicode[STIX]{x1D6FC}^{2}-\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y^{2}}\right)\right]\unicode[STIX]{x1D64C}+\left(\begin{array}{@{}ccc@{}}0 & \displaystyle \frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}y} & \text{i}\unicode[STIX]{x1D6FC}\\ 0 & 0 & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\\ \text{i}\unicode[STIX]{x1D6FC} & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y} & 0\end{array}\right),\end{eqnarray}$$

(5.12)$$\begin{eqnarray}\displaystyle {\mathcal{H}} & = & \displaystyle \left[U\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X}+V\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}-\text{i}\unicode[STIX]{x1D708}_{T}\left(2\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x2202}X}\right)-\text{i}\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D708}_{T}}{\unicode[STIX]{x2202}X}\right]\unicode[STIX]{x1D64C}\nonumber\\ \displaystyle & & \displaystyle +\,\left(\begin{array}{@{}ccc@{}}\displaystyle \frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}X}-\text{i}\unicode[STIX]{x1D6FC}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D708}_{T}}{\unicode[STIX]{x2202}X} & 0 & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X}\\ \displaystyle -\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D708}_{T}}{\unicode[STIX]{x2202}X}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y} & \displaystyle \frac{\unicode[STIX]{x2202}V}{\unicode[STIX]{x2202}y} & 0\\ \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}X} & 0 & 0\end{array}\right),\end{eqnarray}$$

and the matrix $\unicode[STIX]{x1D64C}$ as

(5.13)$$\begin{eqnarray}\unicode[STIX]{x1D64C}=\left(\begin{array}{@{}ccc@{}}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0\end{array}\right).\end{eqnarray}$$

From the analysis of (5.9), the solution must be an eigenfunction of the form $\hat{\boldsymbol{q}}_{0}=A(X)\unicode[STIX]{x1D731}(y;X)$. The unknown function $A(X)$ can be found from the solvability condition of the problem (5.10) as (Gaster Reference Gaster1974; Saric & Nayfeh Reference Saric and Nayfeh1975; Segalini & Garrett Reference Segalini and Garrett2017; Segalini & Camarri Reference Segalini and Camarri2019)

(5.14)$$\begin{eqnarray}\int _{\mathbb{R}}\tilde{\unicode[STIX]{x1D731}}^{\ast }{\mathcal{H}}[A(X)\unicode[STIX]{x1D731}]\,\text{d}y=0,\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D731}}^{\ast }$ is the complex conjugate of the associated mode of the adjoint parallel operator

(5.15)$$\begin{eqnarray}{\mathcal{L}}^{+}=\left[\text{i}\unicode[STIX]{x1D714}-\text{i}\unicode[STIX]{x1D6FC}U+\unicode[STIX]{x1D708}_{T}\left(\unicode[STIX]{x1D6FC}^{2}-\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}y^{2}}\right)\right]\unicode[STIX]{x1D64C}-\left(\begin{array}{@{}ccc@{}}0 & 0 & \text{i}\unicode[STIX]{x1D6FC}\\ \displaystyle -\frac{\unicode[STIX]{x2202}U}{\unicode[STIX]{x2202}y} & 0 & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y}\\ \text{i}\unicode[STIX]{x1D6FC} & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}y} & 0\end{array}\right).\end{eqnarray}$$

Equation (5.14) can be written in the form

(5.16)$$\begin{eqnarray}\frac{\text{d}A}{\text{d}X}\int _{\mathbb{R}}\tilde{\unicode[STIX]{x1D731}}^{\ast }{\mathcal{H}}_{1}\unicode[STIX]{x1D731}\,\text{d}y+A\int _{\mathbb{R}}\tilde{\unicode[STIX]{x1D731}}^{\ast }{\mathcal{H}}\unicode[STIX]{x1D731}\,\text{d}y=0,\end{eqnarray}$$

where the operator ${\mathcal{H}}_{1}$ is

(5.17)$$\begin{eqnarray}{\mathcal{H}}_{1}=[U-2\text{i}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D708}_{T}]\unicode[STIX]{x1D64C}+\left(\begin{array}{@{}ccc@{}}0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 & 0\end{array}\right).\end{eqnarray}$$

Equation (5.16) is a linear first-order homogeneous equation, so that the function $A(X)$ is obtained as

(5.18)$$\begin{eqnarray}A(X)=A(X_{0})\exp \left[-\int _{X_{0}}^{X}\frac{G_{2}(X^{\prime })}{G_{1}(X^{\prime })}\,\text{d}X^{\prime }\right],\end{eqnarray}$$

where

(5.19a,b)$$\begin{eqnarray}G_{1}(X)=\int _{\mathbb{R}}\tilde{\unicode[STIX]{x1D731}}^{\ast }{\mathcal{H}}_{1}\unicode[STIX]{x1D731}\,\text{d}y,\quad G_{2}(X)=\int _{\mathbb{R}}\tilde{\unicode[STIX]{x1D731}}^{\ast }{\mathcal{H}}\unicode[STIX]{x1D731}\,\text{d}y.\end{eqnarray}$$

As typical in a weakly divergent approach, an analytical expression for the amplitude modulation of the eigenfunctions (5.18) is obtained, and the evolution of it is obtained from integrals involving the local eigenfunctions obtained from the parallel stability analysis. Therefore, the determination of the correction does not require additional intensive calculations and both analyses will be compared later in the following.

6 Experimental set-up

The experiment was performed in a room (of volume $8.6~\text{m}^{3}$) where walls and ceiling were covered with sound-absorbing fibreglass foam of 40 mm thickness. As shown in figure 1, the air outside of the room is sucked into a settling chamber with honeycomb and wire screens, followed by a rectangular duct. The duct width, $d$, length, $L$, and spanwise width are 10 mm, 1000 mm and 300 mm, respectively. The length-to-width ratio of the channel is hence $L/d=100$ and the Reynolds number is $Re=U_{0}d/\unicode[STIX]{x1D708}=14\,500$, where $U_{0}=22~\text{m}~\text{s}^{-1}$ is the centreline velocity. This ensures that at the end of the duct the flow is a fully developed turbulent channel flow that issues into the room. The coordinates are chosen such that $x$ measures the streamwise distance from the duct exit and $y$ is the transverse distance from the centreplane of the jet.

Figure 1.

Experimental set-up of the two-dimensional turbulent jet.

The excitation can be introduced via two slot devices that are installed close to the exit. Each slot device has a chamber connected to a loudspeaker by plastic tubes of 750 mm length. Both speakers are fed with a sine wave signal of the same frequency and amplitude, but generates out-of-phase forcing. The amplitude, $v_{0}$, and frequency, $f$, are controlled by a computer and the signals are amplified by separate amplifiers before going to the loudspeakers. An accurate condenser microphone (Rion UC-57, NH22A) and a MEMS microphone (TDK T4020) monitor pressure fluctuation in each slot chamber and the root mean square value, $p_{0}$, measured by the condenser microphone is used as reference for the excitation intensity. The forcing amplitude is defined as $v_{0}=p_{0}/(\unicode[STIX]{x1D70C}a_{s}U_{0})$, where $a_{s}$ is the speed of sound at room temperature. The forcing frequency was varied between 50 and 165 Hz, corresponding to Strouhal numbers $St=fd/U_{0}$ between 0.023 and 0.075.

The velocity components were measured with constant-temperature hot-wire anemometry, using an in-house built X-type probe with $2.5~\unicode[STIX]{x03BC}\text{m}$ diameter platinum wires of 1 mm length, where the intersection angle and spacing of the wires were approximately $90^{\circ }$ and $1~\text{mm}$, respectively. A low-pass filter was used for the anemometer signals with a cutoff frequency of 5 kHz.

The sampling time was 60 s for measurements of $xy$-distributions and 300 s for measurements at $\unicode[STIX]{x1D702}=0$ and $\unicode[STIX]{x1D702}=1$. The sampling frequency was set so that the number of temporal points in one period of the initial forcing was 200. For the unexcited measurements the sampling frequency was 33 kHz.

The probe was positioned using a three-axis traversing system equipped with three stepper motors. In the $x$-direction, measurements for the first three stations close to the nozzle were at 5, 10 and 15 mm from the exit, followed by 30 mm up to 390 mm in 15 mm steps. In the $y$-direction 50 positions were taken at each $x$-station but with a linearly increasing spacing with increasing $x$.

The calibration of the hot-wire probe was performed in a jet formed from a circular nozzle installed next to the planar jet in the sound-absorbing room. During both calibration and measurements the air was sucked from the outer room where the temperature was kept within $\pm 1\,^{\circ }\text{C}$. The calibration grid, that relates the hot-wire voltages to the flow speed and flow angle, was obtained by fitting fifth-order polynomial functions to the calibration data (using the same code as developed by Österlund (Reference Österlund1999)). During the calibration the X-probe response was obtained at 21 different angles with 3.6 degrees step in the range $\pm 36^{\circ }$ and 13 different velocities in the range $0.5{-}25~\text{m}~\text{s}^{-1}$, in all 273 calibration points. The accuracy of the hot-wire measurements can be estimated by how well the calibration points adhere to the calibration grid and the standard deviations of the fitting to the fifth-order polynomial functions were approximately $0.1~\text{m}~\text{s}^{-1}$ for the velocity magnitude and 2 degrees for the angle.

The velocity profile at the exit resembled a channel flow where the shear stress is still linear near the centre of the jet and became rapidly perturbed at the edges, losing the typical structure observed for turbulent channel flows. According to the linear region of the shear stress profile, the friction velocity was determined, corresponding to a maximum velocity ratio of $U_{0}/u_{\unicode[STIX]{x1D70F}}=22.8$ and a friction-based Reynolds number of $Re_{\unicode[STIX]{x1D70F}}=u_{\unicode[STIX]{x1D70F}}d/(2\unicode[STIX]{x1D708})=350$. The value of the streamwise velocity variance (scaled with $u_{\unicode[STIX]{x1D70F}}^{2}$) at the centre was 1.03, in good agreement with the data collection of Alfredsson, Örlü & Segalini (Reference Alfredsson, Örlü and Segalini2012).

8 Discussion and conclusions

In the present work a planar jet exiting from a fully developed turbulent channel flow has been acoustically excited in order to be able to identify coherent structures embedded in uncorrelated random turbulence. This is done through measurements of the streamwise and lateral velocity components which are phase averaged with respect to the excitation. The mean flow was characterised first, and it was in good agreement with the literature in terms of the streamwise velocity profile and spreading rate (although slightly larger). Discrepancies between the similarity solution of the mean transverse velocity and shear stress were observed; however, our results seem to correspond well with the other recent experiments (Gordeyev & Thomas Reference Gordeyev and Thomas2000; Cafiero & Vassilicos Reference Cafiero and Vassilicos2019). In the similarity analysis the assumption of a constant eddy viscosity is needed and used in order to obtain an analytical solution, but the discrepancy between experiments and theory may indicate that this assumption is not fully valid.

Several excitation frequencies and amplitudes were tested and it was possible to extract the periodic mode through phase averaging even for small excitation amplitudes. The analysis of the filtered Navier–Stokes equations provided the governing equation of the perturbation with several linear terms, a quadratic term (related to the nonlinear interaction of the periodic perturbation with itself) and an unknown term (related to the phase average of the product of the incoherent fluctuations). While the nonlinear term is likely to be negligible in the evolution of the perturbation, the last term of (4.8) is instead scaling linearly with the perturbation amplitude, suggesting that the phase average of the incoherent-turbulence moments follows a linear behaviour.

It is interesting to compare the perturbation fields obtained at different frequencies, which show significant differences if physical coordinates are used. Here a coordinate scaling is proposed based on the excitation frequency, that is able to universally scale the perturbation fields. This observation has been further verified by the linear stability analysis, where the same eigenmode was obtained. Some discrepancies between the theoretical predictions and the experimental results were present in the initial stage of the jet development, probably due to the fact that the stability analysis used a velocity profile from the theory rather than from the measurements, but the agreement in the perturbation decay became very good in the later self-similar stage.

When the acoustic forcing is not too strong the periodic components scale linearly with the forcing amplitude. The successful prediction for the structure and growth of the periodic components by linear theory implies that the linear dynamics of modal perturbations are embedded in the turbulent environment. This is in support of the idea that the large-scale perturbations in the jet could be regarded as an incoherent set of linear modes and suggests that a successful strategy for control of turbulent jets should focus on the linear-mode dynamics.

If the linear behaviour of phase averages of velocities and moments of random components can be observed also in wall-bounded flows (boundary layers and channel flows), then it should be possible to analyse coherent structures, such as streaks and wall-attached eddies, based on the phase-average equations such as (4.6).