2 Review of basic formalism and comparison with the Orr result

As in G78, G79 and GAL the flow is assumed to be inviscid and non-heat-conducting and the fluid is assumed to be an ideal gas so that the entropy is proportional to $\ln (p/\unicode[STIX]{x1D70C}^{\unicode[STIX]{x1D6FE}})$ and the squared sound speed is equal to $\unicode[STIX]{x1D6FE}p/\unicode[STIX]{x1D70C}$ , where $p$ denotes the pressure, $\unicode[STIX]{x1D70C}$ the density and $\unicode[STIX]{x1D6FE}$ the specific heat ratio. Then the pressure $p^{\prime }=p-p_{0}$ and mass flux

perturbations (where $v_{i}^{\prime }$ denotes the velocity perturbation) on a transversely sheared mean flow with pressure $p_{0}=\text{const.}$ , velocity $U(\,\boldsymbol{y}_{T})$ and mean sound speed squared $c^{2}(\,\boldsymbol{y}_{T})$ , are governed by the linearized Euler equations

and

where $\boldsymbol{y}_{T}=\{y_{2},y_{3}\}$ , $\boldsymbol{y}=\{y_{1},y_{2},y_{3}\}=\{y_{1},\boldsymbol{y}_{T}\}$ and $\text{D}_{0}/\text{D}\unicode[STIX]{x1D70F}\equiv \unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}+U\unicode[STIX]{x2202}/\unicode[STIX]{x2202}y_{1}$ denotes the convective derivative.

G79 shows that the solution to these equations can be expressed in terms of the two arbitrary convected functions $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ and $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-(y_{1}/U),\boldsymbol{y}_{T})$ and a potential function $\unicode[STIX]{x1D719}$ that satisfies

where

and the physical variables $p^{\prime }$ and $u_{i}$ are determined by

and

with $\unicode[STIX]{x1D6FF}_{ij}$ denoting the Kronecker delta, $\unicode[STIX]{x1D700}_{ijk}$ the alternating tensor and

denoting a kind of generalized particle displacement.

It is well known that the mass flux perturbation, $u_{i}$ , can be eliminated between (2.2) and (2.3) to show that the pressure fluctuation $p^{\prime }$ satisfies Rayleigh’s equation

where

denotes the usual Rayleigh operator, which is easily shown to be adjoint to the operator  $L_{a}$ .

For reasons given in the introduction, our focus here is on the steady state (i.e. time stationary) solutions (which are assumed to exist) and we suppose that $\unicode[STIX]{x1D719}$ is a stationary random function (Wiener Reference Wiener1938) and therefore that initial conditions imposed in the distant past have all decayed out at the finite time $t$ . A formal steady state solution to (2.4) can then be written as

where $g(\,\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ denotes the Rayleigh operator Green’s function which exhibits incoming wave behaviour as $|\,\boldsymbol{y}|\rightarrow \infty$ and satisfies

the first two arguments of $g(\,\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ denote the dependent variables and the second two denote the source variables, $T$ denotes a very large but finite time interval, $V$ is a region of space bounded by cylindrical (i.e. parallel to the mean flow) surface(s) $S$ that can be finite, semi-infinite or infinite in the streamwise direction and $\hat{\boldsymbol{n}}=\{\hat{n}_{i}\}$ is the unit outward-drawn normal to $S$ . The upper limit $+T$ of the $\unicode[STIX]{x1D70F}$ -integration can be replaced by $t$ since $g(\,\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)=0$ for $\unicode[STIX]{x1D70F}>t$ . The lower limit $-T$ reflects the fact that the initial conditions must be imposed in the distant past in order to ensure that they do not contribute to the steady state solution.

Equation (2.11) expresses the solution to (2.4) in terms of the volume source distribution $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U(\,\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ and the values of the potential $\unicode[STIX]{x1D719}$ on some arbitrary cylindrical surfaces $S$ (some or all of which may be at infinity). The analysis is somewhat unconventional in that the direct Green’s function $g$ now plays the role of an adjoint Green’s function for the solution  $\unicode[STIX]{x1D719}$ .

The surface integrals in (2.11) drop out when any of the surfaces $S$ are at infinity (i.e. when $V$ represents all of space) and they can be eliminated when they are not by requiring that the Green’s function $g$ satisfies certain boundary conditions on these bounding surfaces (since $g$ is not uniquely determined by (2.12)). Equation (2.11) then becomes

Equations (2.6) and (2.13) show that the pressure perturbation $p^{\prime }$ is then given by

while (2.7) and (2.8) show that the corresponding transverse velocity perturbation,

is given by

where

Inserting equation (B.12) of Goldstein, Afsar & Leib (Reference Goldstein and Afsar2013b ) into this result, noting that the integral over the second term vanishes and that the relevant Poisson’s equation Green’s function is self-adjoint (i.e. $g_{0}(\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)=g_{0}(\boldsymbol{x},t\,|\,\boldsymbol{y},\unicode[STIX]{x1D70F})$ ), shows that it reduces to (1.4) for two-dimensional incompressible flows with constant mean shear when the arbitrary convected quantity $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U(\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ is replaced by the renormalized convected quantity

which has dimensions of vorticity (based on the rescaled velocity $u_{i}$ ). Equation (2.16) which, like (2.14), does not depend on the second arbitrary convected quantity $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ is, therefore, a generalization of the Orr result (1.4). The most significant difference is that the convected quantity $\unicode[STIX]{x1D714}_{c}$ is no longer equal to the spanwise vorticity.

GAL show that (2.14) will apply even when solid surfaces and accompanying downstream wakes are present in the flow if $g(\,\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ and $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ are required to satisfy appropriate boundary conditions on these surfaces and $g(\,\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ is required to satisfy appropriate jump conditions across the downstream wakes. The formulae (2.14) and (2.16) for the physical variables $p^{\prime }$ and $u_{\bot }$ can then be viewed as formal solutions to the complete non-homogeneous RDT problem (in the usual case where the solid surfaces are aligned with the constant velocity surfaces). They effectively reduce the RDT problem to the problem of finding the Rayleigh’s equation Green’s function that satisfies the appropriate boundary conditions on the bounding surfaces $S$ . The solution $p^{\prime }(\boldsymbol{x},t)$ will then be independent of the second convected quantity $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ and the acoustic field will only depend on the single convected quantity $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U(\,\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ .

In the absence of scattering surfaces and other external sources, the unsteady flow (2.14)–(2.17) consists entirely of subsonically propagating disturbances when the mean flow is purely subsonic and, therefore, cannot radiate to the far field (Goldstein Reference Goldstein2005, Reference Goldstein2009). This can easily be verified in any particular case by working out the relevant far-field expansion. It is therefore appropriate to identify it with the hydrodynamic component of the motion.

3 Conservation laws for $\tilde{\unicode[STIX]{x1D714}}_{c}$ , $\unicode[STIX]{x1D717}$ , transverse particle displacement and physical variables

This section summarizes the conservation laws derived in Goldstein et al. (Reference Goldstein and Afsar2013b ) and G79 that relate the arbitrary convected quantities $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ and $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ and a quantity, which we refer to as the transverse particle displacement, to the physical variables. The next section shows that this transverse particle displacement vanishes when $y_{1}\rightarrow -\infty$ and § 5 shows how these results can be used to obtain upstream boundary conditions that relate $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ and $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ to the physical (hopefully measurable) flow variables.

The conservation laws, which are given by (3.1) and (3.2) of Goldstein et al. (Reference Goldstein and Afsar2013b ), can be written as

where $\tilde{\unicode[STIX]{x1D714}}_{c}$ is related to the rescaled vortical-like quantity $\unicode[STIX]{x1D714}_{c}$ by (2.18),

and

are source functions and we have used (2.3) to obtain the last member of (3.4).

is the velocity component generated by the second convected quantity $\unicode[STIX]{x1D717}$ , and

is the transverse particle displacement.

Equations (2.7) and (2.15) show that $\unicode[STIX]{x1D702}_{\bot }$ is related to $u_{\bot }$ by

which justifies referring to it as the transverse particle displacement.

Equations (3.1) and (3.4)–(3.6) relate the arbitrary convected quantities $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U(\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ and $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U(\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ to the pressure $p^{\prime }$ , density-weighted velocity $\boldsymbol{u}$ and the transverse particle displacement $\unicode[STIX]{x1D702}_{\bot }$ , while (3.2) and (3.4)–(3.6) relate the arbitrary convected quantity $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U(\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ to the pressure $p^{\prime }$ , density-weighted velocity $\boldsymbol{u}$ and the transverse particle displacement $\unicode[STIX]{x1D702}_{\bot }$ .

The tensor $(\unicode[STIX]{x2202}N_{k}/\unicode[STIX]{x2202}y_{i}-\unicode[STIX]{x2202}N_{i}/\unicode[STIX]{x2202}y_{k})$ is equal to zero and $u_{k}^{(c)}$ drops out of the first term on the right-hand side of (3.4) for planar base flows, where $c^{2}$ and $U$ depend on a single Cartesian coordinate (say $y_{2}$ ) and (3.1) then becomes

which is independent of $u_{i}^{(c)}$ and, therefore, of the second convected quantity $\unicode[STIX]{x1D717}$ . But the divergence $\unicode[STIX]{x2202}N_{i}/\unicode[STIX]{x2202}y_{i}$ is equal to zero for the constant shear-constant $c^{2}$ parallel mean flow (1.1), since $N_{i}$ is a constant in that case, and it follows from (2.18) that (3.9) then reduces to Möhring’s (Reference Möhring1976) result

and to Orr’s equation (1.3) when the flow is incompressible and two-dimensional.

The particle displacement $\unicode[STIX]{x1D702}_{\bot }$ which appears in equations (3.1) and (3.2) is not actually a physical variable in the usual sense, and requires further clarification, which is provided in the next section. However, our interest here is in obtaining a set of upstream boundary conditions that relate $\tilde{\unicode[STIX]{x1D714}}_{c}$ and $\unicode[STIX]{x1D717}$ to the physically measurable variables at upstream infinity, which can be obtained by taking the limit as $y_{1}\rightarrow -\infty$ of these equations. This greatly simplifies the formulae and, as will be shown below, even allows us to obtain an explicit formula for $\tilde{\unicode[STIX]{x1D714}}_{c}$ .

4 Particle displacement and causality

As indicated in the paragraphs above and below equations (2.11) and (2.12), our interest is in time-stationary solutions which are assumed to exist for the physical variables $p^{\prime }$ and $u_{\bot }$ . It is therefore appropriate to work with the temporal Fourier transforms

where the integrals are to be interpreted in a stochastic sense and the limits are to be taken after the statistical quantities are calculated (Wiener Reference Wiener1938). (Laplace transforms would not be appropriate here.) However, the formula (2.13) for the potential $\unicode[STIX]{x1D719}$ is still only formal in that the integrand on the right-hand side has a non-integrable singularity at $\boldsymbol{y}=\boldsymbol{x}$ . But the corresponding integrands in (2.14) and (2.16) for the physical variables $p^{\prime }$ and $u_{\bot }$ remain finite, and these quantities are therefore (stochastically) well defined. In fact, GAL, G78 and G79 show that they are given by

and

respectively, where $\boldsymbol{y}_{T}$ is defined below (2.3), $A_{T}$ denotes the cross-sectional area such that $\int _{A_{T}}\int _{-\infty }^{\infty }\cdot \,\text{d}\boldsymbol{y}_{T}\,\text{d}y_{1}=\int _{V}\cdot \,\text{d}\boldsymbol{y}$ , $\bar{\unicode[STIX]{x1D6FA}}_{c}(\boldsymbol{x}:\unicode[STIX]{x1D714})$ is defined as the limit $T\rightarrow \infty$ of

where

satisfies the reduced inhomogeneous Rayleigh’s equation

with $L_{R}$ being the reduced Rayleigh operator

written in terms of the Laplacian $\unicode[STIX]{x1D735}_{T}$ with respect to the transverse coordinate $\boldsymbol{y}_{T}$ . Appendix A shows that $\bar{G}_{0}(\,\boldsymbol{y}_{T}\,|\,\boldsymbol{x}_{T}:\unicode[STIX]{x1D714},\unicode[STIX]{x1D714}/U(\,\boldsymbol{y}_{T}))$ remains finite and is continuous at $\boldsymbol{y}_{T}=\boldsymbol{x}_{T}$ for two-dimensional mean flows, and a similar analysis would show that this is true in general, but the notation becomes very tedious in that case. Appendix A also shows that

remains finite and continuous at $\boldsymbol{y}_{T}=\boldsymbol{x}_{T}$ for two-dimensional mean flows. It therefore follows from (4.3), the first line of (B 4), (B 6) and inversion of the Fourier transform (4.1) that

and

where the purely convected quantity ${\mathcal{U}}_{\bot }(t-x_{1}/U(x_{2}),\boldsymbol{x}_{T})$ is a function of the indicated arguments and $\bar{{\mathcal{U}}}_{\bot }(\boldsymbol{x}_{T},\unicode[STIX]{x1D714})$ is the Fourier transform of that quantity. The comment below (4.8) suggests that these results, which generalize the behaviour discussed in the introduction, are expected to apply to much more general transversely sheared mean flows (such as those described below) even though they were derived for two-dimensional base flows.

The Fourier transform

of the transverse particle displacement (3.7), which formally satisfies

will become unbounded at $\boldsymbol{y}=\boldsymbol{x}$ since, as shown in appendix A for the two-dimensional case, $\bar{G}_{i}(\,\boldsymbol{y}_{T}\,|\,\boldsymbol{x}:\unicode[STIX]{x1D714},\unicode[STIX]{x1D714}/U(\,\boldsymbol{y}_{T}))$ will usually not vanish when $\boldsymbol{y}_{T}=\boldsymbol{x}_{T}$ . It can be made finite in a number of ways. But there is only one possibility if causality is also imposed. This amounts to assuming that the time-stationary solutions will exist even when $\unicode[STIX]{x1D702}_{\bot }(\boldsymbol{x},t)$ is assumed to be identically zero in the distant past. This can be implemented by using the Briggs (Reference Briggs1964)–Bers (Reference Bers, Dewitt and Perraud1975) procedure, which amounts to letting $\unicode[STIX]{x1D714}$ have a small positive imaginary part, say $\unicode[STIX]{x1D700}$ , and taking the limit as $\unicode[STIX]{x1D700}\rightarrow 0$ of the resulting formula. It is not possible to do this directly in the present case, but (4.13) can represented as the limit of a sequence and this procedure can be used to impose causality on each term of that sequence. (Details are given in appendix C.) It could, however, be argued that $\unicode[STIX]{x1D702}_{\bot }$ need not be causal because it is not actually a physical variable, but the conservation laws (3.1)–(3.2) and, more importantly, the upstream boundary conditions would then also be non-causal. Our primary interest is in the upstream behaviour of $\unicode[STIX]{x1D702}_{\bot }$ , which will be used to derive the upstream boundary conditions referred to in the introduction. The analysis in appendix C shows that

when causality is imposed, which implies that

in this case. Different results would be possible if causality were not imposed.

5 Upstream boundary conditions and relation of $\tilde{\unicode[STIX]{x1D714}}_{c}$ , $\unicode[STIX]{x1D717}$ to the physical variables

It is useful, although not essential, to first split the dependent variables into a hydrodynamic component, which does not directly produce any sound at subsonic Mach numbers, and a non-hydrodynamic component, which accounts for the remaining – including the acoustic – components of the motion, before attempting to derive the relevant boundary conditions. We can then think of the former component as being an upstream ‘input’ that generates a downstream ‘response’ when it interacts with streamwise changes in the boundary conditions.

As is well known, it is impossible to unambiguously decompose the unsteady motion on a transversely sheared mean flow into acoustic and hydrodynamic components. We can, however, require that the hydrodynamic component not radiate any sound at subsonic Mach numbers, with all the acoustic radiation being accounted for by the remaining non-hydrodynamic component. Then, in order to identify the input disturbance with the hydrodynamic component of the motion, we divide the Rayleigh equation Green’s function $g(\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ that appears in the time-dependent solution (2.13)–(2.16) into two components, say

where $g^{(H)}(\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ denotes a particular solution of (2.12) which is defined on all space when the bounding surfaces $S$ are all at infinity or, more generally, satisfies appropriate boundary conditions (given in Goldstein et al. Reference Goldstein and Afsar2013a ,Reference Goldstein and Afsar b ) on a constant mean velocity surface that extends from minus to plus infinity in the streamwise direction. The corresponding solution, which is given by (2.14) and (2.16) with $g(\,\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ replaced by $g^{(H)}(\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ , does not produce any acoustic radiation and can, therefore, be identified with the hydrodynamic component of the unsteady motion. The corresponding ‘scattered solution’, $g^{(s)}(\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ , satisfies the homogeneous Rayleigh’s equation along with appropriate inhomogeneous boundary and jump conditions on the streamwise discontinuous surfaces $S$ and, therefore, accounts for all of the acoustic components of the motion.

We now obtain the relevant upstream boundary conditions for the convected quantities $\tilde{\unicode[STIX]{x1D714}}_{c}$ and $\unicode[STIX]{x1D717}$ by taking the upstream limit of (3.1) and (3.2), but with $g(\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ replaced by $g^{(H)}(\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ . This is most easily done by using the frequency representation discussed in § 3. The reduced Rayleigh equation Green’s function $\bar{G}_{0}(\,\boldsymbol{y}_{T}\,|\,\boldsymbol{x}:\unicode[STIX]{x1D714},k_{1})$ that appears in the frequency domain solutions (4.2), (4.3) and (4.13) then has the decomposition

where $\bar{G}^{(H)}(\,\boldsymbol{y}_{T}\,|\,\boldsymbol{x}:\unicode[STIX]{x1D714},k_{1})$ is either defined on all space when the bounding surfaces $S$ are all at infinity or satisfies

(where $C_{T}$ denotes the bounding curve/curves that generate the doubly infinite surface/surfaces $S$ ) when they are not. The streamwise-homogeneous Green’s functions $g^{(H)}(\boldsymbol{y},\unicode[STIX]{x1D70F}\,|\,\boldsymbol{x},t)$ and $\bar{G}^{(H)}(\,\boldsymbol{y}_{T}\,|\,\boldsymbol{x}:\unicode[STIX]{x1D714},k_{1})$ will then depend on $y_{1}$ and $x_{1}$ only in the combination $x_{1}-y_{1}$ and we, therefore, write

The convected quantity $\tilde{\unicode[STIX]{x1D714}}_{c}$ is determined by (3.1) and (3.2), whose Fourier transforms are given by

and

where $\bar{\unicode[STIX]{x1D702}}_{\bot }^{(H)}(\,\boldsymbol{y}:\unicode[STIX]{x1D714})$ is given by (4.13) with $\bar{G}_{i}(\,\boldsymbol{y}_{T}\,|\,\boldsymbol{x}:\unicode[STIX]{x1D714},k_{1})$ replaced by $\bar{G}_{i}^{(H)}(\,\boldsymbol{y}_{T}\,|\,\boldsymbol{x}:\unicode[STIX]{x1D714},k_{1})$

and $\bar{\unicode[STIX]{x1D6E4}}_{k,i}(\boldsymbol{y}:\unicode[STIX]{x1D714})\equiv \lim _{T\rightarrow \infty }\bar{\unicode[STIX]{x1D6E4}}_{k,i}(\boldsymbol{y}:\unicode[STIX]{x1D714},T)$ for $k=0,1,2,3$ with

where $\unicode[STIX]{x1D6E4}_{k,i}$ , $i=0,1,2,3$ , are defined by (3.4) and (3.5).

Then, since we have shown that the Fourier transform $\bar{\unicode[STIX]{x1D702}}_{\bot }^{(H)}(\,\boldsymbol{y}:\unicode[STIX]{x1D714})$ of the transverse particle displacement $\unicode[STIX]{x1D702}_{\bot }^{(H)}$ vanishes as $y_{1}\rightarrow -\infty$ , and an argument similar to that used to obtain (B 6) shows that $\bar{p^{\prime }}(\boldsymbol{y}:\unicode[STIX]{x1D714})$ should vanish like $y_{1}^{-2}$ as $y_{1}\rightarrow -\infty$ , (5.5) and (5.6) imply that

and

where

These results provide the desired relations between the convected quantities $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U(\,\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ , $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U(\,\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ and the upstream limit $\bar{\unicode[STIX]{x1D6E4}}_{k,i}^{\infty }(\boldsymbol{y}:\unicode[STIX]{x1D714})$ , for $i=0,1,2,3$ of the physically measurable variables that enter through $\unicode[STIX]{x1D6E4}_{k,i}(\boldsymbol{y},\unicode[STIX]{x1D70F})$ in an arbitrary transversely sheared mean flow.

But the focus in the remainder of the paper will be on the two-dimensional mean flows for which $\unicode[STIX]{x2202}N_{k}/\unicode[STIX]{x2202}y_{i}-\unicode[STIX]{x2202}N_{i}/\unicode[STIX]{x2202}y_{k}=0$ and (5.9) becomes

where

is the double Fourier transform of the convected quantity $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D709},\boldsymbol{y}_{T})$ and

is the upstream limit of the double Fourier transform of the physically measurable vorticity derivative $\unicode[STIX]{x1D6E4}_{2,0}$ given by (3.4).

But (3.4) and (3.6) imply that

for two-dimensional mean flows and, therefore that $\unicode[STIX]{x1D6E4}_{2,0}(\boldsymbol{y},\unicode[STIX]{x1D70F})$ and, consequently, $\hat{\unicode[STIX]{x1D6E4}}_{\infty }(y_{2};\unicode[STIX]{x1D714},k_{3},T)$ only depend on the physical variables $u_{2}$ and $p^{\prime }$ for these mean flows. Equation (5.12) therefore provides the desired upstream boundary condition that relates the Fourier transform of the unknown convected quantity $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U(\,\boldsymbol{y}_{T}),\boldsymbol{y}_{T})$ to the physically measurable quantity (5.15) in this case.

But we can go even further than this, since an argument similar to that given at the end of appendix B can be used to show that $p^{\prime }$ should vanish like $1/y_{1}^{2}$ as $y_{1}\rightarrow -\infty$ , and $\lim _{y_{1}\rightarrow \infty }\unicode[STIX]{x1D6E4}_{2,0}$ is, therefore, given by $\unicode[STIX]{x1D735}\boldsymbol{\cdot }[c^{-2}\unicode[STIX]{x1D735}(c^{2}u_{2})]$ . Inserting (4.11) into this result, noting that $u_{\bot }(\boldsymbol{x},t)=u_{2}(\boldsymbol{x},t)$ in this case, shows that

Inserting this into (5.8) and (5.11), and integrating the result by parts shows that

where

6 Relation between the $\tilde{\unicode[STIX]{x1D714}}_{c}$ spectra and measurable turbulence correlations

But only statistical quantities, such as

where $\hat{\unicode[STIX]{x1D6E4}}_{\infty }$ is defined by (5.15) and (5.14), are of interest for the time-stationary turbulent flows that are the main focus of RDT. For simplicity, we only consider mean flows that are uniform in the $y_{3}$ -direction and suppose that the turbulence is statistically homogeneous in the spanwise direction. Then the space–time average

will exist and be independent of $\unicode[STIX]{x1D70F},y_{3}$ and it follows from the convolution theorem that

where $\hat{\unicode[STIX]{x1D6E4}}_{\infty }(y_{2};\unicode[STIX]{x1D714},k_{3},T)$ is given by (5.14) and the asterisk denotes the complex conjugate.

It, therefore, follows from (5.12) and (6.3) that

where $\hat{\unicode[STIX]{x1D6FA}}_{c}(y_{2}:\unicode[STIX]{x1D714},k_{3},T)$ is given by (5.13).

7 Application to a large-aspect-ratio rectangular jet

The problem of a two-dimensional jet interacting with the trailing edge of a flat plate is currently of considerable interest because of its relevance to understanding noise production in future aircraft configurations, such as that shown in figure 3, in which the engine exhaust is of a very wide aspect ratio on an almost rectangular jet.

GAL analysed the model problem shown in figure 4 in order to represent the interaction between a jet emanating from a large-aspect-ratio rectangular nozzle with the trailing edge of a flat plate, and compared the results with recent experiments on this configuration that were performed at NASA Glenn Research Center (Zaman et al. Reference Zaman, Brown and Bridges2013; Brown Reference Brown2015).

They considered the general case where the mean flow is non-zero at the surface of the plate, and therefore leaves the trailing edge with different velocities above and below the interface. But, as shown below, the surface velocity is relatively small compared to the maximum velocity in the cases of interest, and will therefore be set to zero in the present computation: in which case their analysis, which minimizes the trailing-edge singularity (i.e. imposes a Kutta condition, Crighton Reference Crighton1985; Ayton, Gill & Peak Reference Ayton, Gill and Peake2016) and uses the Wiener–Hopf method to calculate the Green’s function, shows that the acoustic spectrum

where $\tilde{\boldsymbol{\cdot }}$ denotes the time average, is given by

for $x_{2}\lessgtr 0$ where $S(y_{2},{\tilde{y}}_{2}:k_{3}^{(s)},\unicode[STIX]{x1D714})$ is defined by (6.4),

Figure 3. Proposed aircraft configuration.

Figure 4. Computational model of the jet/surface interaction problem.

$\unicode[STIX]{x1D705}_{\pm }(k_{1},k_{3},\unicode[STIX]{x1D714})$ denote bounded analytic functions in the upper/lower half-planes that satisfy the factorization condition

$P_{\lessgtr }(y_{2}:\unicode[STIX]{x1D714},k_{1},k_{3})$ denote homogeneous solutions to (A 3) that have outgoing wave behaviour as $y_{2}\rightarrow \mp \infty$ , $\unicode[STIX]{x1D703}$ denotes the polar angle measured from the downstream $x_{1}$ -axis, $c_{s}$ denotes the mean speed of sound at the plate surface and $\unicode[STIX]{x1D713}$ denotes the azimuthal angle measured from the plane of the plate.

GAL considered the low-frequency limit $k_{3}=O(k_{\infty }),k_{\infty }\ll 1$ and obtained the result given by equation (6.33) of their paper, which has the advantage of being much more explicit than the exact $O(1)$ result but does not adequately describe the high-frequency sound field produced by the trailing-edge interaction. It does, however, adequately describe the experimentally observed low-frequency spectrum when the negative tail in the transverse velocity correlation is included (Afsar, Leib & Bozak Reference Afsar, Leib and Bozak2017).

The high-frequency spectrum can be described by using the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) method to obtain the high-frequency outgoing wave homogeneous solution

to (A 3) (Goldstein Reference Goldstein1979a ) where

and

and inserting the result into (7.5)–(7.8) to obtain the following

It, therefore, follows that

and

where

and (7.2) then becomes

for $x_{2}<0$ where

$M(y_{2})=U(y_{2})/c_{\infty }$ denotes the local acoustic Mach number at the position $y_{2}$ , $\unicode[STIX]{x1D712}_{0}$ is a positive constant and we have inserted the exponential damping factor $\text{e}^{-\unicode[STIX]{x1D712}_{0}k_{\infty }}$ into (7.19), which leaves the asymptotic expansion unchanged to the order of approximation considered here. In other words, it is asymptotically equivalent to the straightforward result. It reduces to the low-frequency result (6.33) of GAL when $Q(y_{2}\mid \unicode[STIX]{x1D703},\unicode[STIX]{x1D711})=1$ . But $Q(y_{2}\mid \unicode[STIX]{x1D703},\unicode[STIX]{x1D711})$ also $\rightarrow 1$ as $k_{\infty }\rightarrow 0$ and (7.18), therefore, behaves like (but is not identical to) a uniformly valid composite solution that applies at all frequencies.

It is, of course necessary to insert a formula for the source function $S$ into (7.18) before using these results to calculate the acoustic field. GAL used a rather complicated approximate procedure to relate this quantity to an experimentally measurable turbulence correlation. The present analysis allows us to use the much simpler and more general exact relation (6.4), and model the turbulence correlation to obtain the explicit formula (6.13) for  $S$ .

As indicated above, the model problem considered in this section can be used to represent the interaction between a jet emanating from a large-aspect-ratio rectangular nozzle with the trailing edge of a flat plate. The analysis is basically inviscid, but accounts for viscous effects by imposing a Kutta condition at the trailing edge (GAL). Brown & Daniels (Reference Brown and Daniels1975) use high-Reynolds-number asymptotic analysis to show that this condition is consistent with the viscous boundary layer flow at the trailing edge. The importance of imposing a Kutta condition in inviscid analyses involving an edge has been reviewed and discussed by Crighton (Reference Crighton1985) and Ayton et al. (Reference Ayton, Gill and Peake2016).

Recent experiments on this configuration were performed at NASA Glenn Research Center (Zaman et al. Reference Zaman, Brown and Bridges2013; Brown Reference Brown2015). The relevant geometric parameters are shown in figure 5.

Figure 5. Nozzle/plate configuration. Figure courtesy Dr James E. Bridges, NASA Glenn.

We assume that the mean density $\unicode[STIX]{x1D70C}$ is constant and the mean velocity profile $U(y_{2})$ can be represented by the twice differentiable function

with compact support $|y_{2}-y_{d}|\leqslant t_{d}/2$ , where $y_{d}$ is the distance from the plate to the nozzle centreline (see figure 5), $t_{d}$ is the thickness of the jet and $\unicode[STIX]{x1D705}$ controls the profile decay.

Since the factor $A(\,y_{2},{\tilde{y}}_{2})$ in (6.8) must vanish at the jet boundaries and is determined by strength of the turbulence at the source location, we expect $A(y_{2},y_{2})$ to be proportional to the turbulence intensity at $y_{2}$ , which is roughly proportional to the mean velocity gradient at that point. We therefore set

where $B_{0}$ and $A_{0}$ are constants and the factor

is inserted to ensure that the turbulence correlation (6.9) vanishes at the jet boundaries.

Measurements of the noise generated by the interaction of rectangular jets in the vicinity of a flat-plate trailing edge have been carried out at NASA Glenn Research Center (Bridges et al. Reference Bridges, Brown and Bozak2014; Brown Reference Brown2015) in a facility validated for jet noise (Bridges & Brown Reference Bridges and Brown2005; Brown & Bridges Reference Brown and Bridges2006). Flow measurements for essentially the same geometries, but at a lower jet exit Mach number ( $M_{a}=0.22$ ), were carried out by Zaman et al. (Reference Zaman, Brown and Bridges2013). We chose the configuration where the plate was located at 1.2 equivalent diameters from the jet centreline and 5.7 equivalent diameters downstream of the exit of an 8 : 1 rectangular nozzle, for jet exit acoustic Mach numbers $M_{a}=0.5,0.7,0.9$ as test cases for the theory. The arbitrary length scale $D$ was taken to be an equivalent nozzle diameter defined by $\unicode[STIX]{x03C0}(D/2)^{2}=$  nozzle width $\times$ nozzle height with nozzle width  $=8\,\times$ nozzle height, and was approximately equal to $2.12$ inches in the experiments. Any ‘scrubbing noise’ that may have resulted from the flow along the plate was deemed to be negligible for this configuration (Khavaran, Bozak & Brown Reference Khavaran, Bozak and Brown2016). Recall that the source location is assumed to be at a large distance from edge and independent of its location on the scale of the interaction, but not on the longer scale over which the turbulence and mean flow evolve. So the mean flow and turbulence properties must be recalibrated when changes in edge location occur on the latter scale.

Figure 6(a) shows a comparison of the normalized (by the jet exit velocity, $U_{J}$ ) mean velocity profile from the model (7.21) with velocity measurements at a very small distance downstream of the plate trailing edge carried out by Zaman et al. (Reference Zaman, Brown and Bridges2013). Reynolds-averaged Navier–Stokes solutions for the test cases considered in this paper (Afsar et al. Reference Afsar, Leib and Bozak2017) show that the normalized mean velocity profiles for $M_{a}=0.5,0.7,0.9$ are similar to each other and to that measured by Zaman et al. (Reference Zaman, Brown and Bridges2013) at $M_{a}=0.22$ . (There is a very slight mismatch in the transverse distance of the plate to the nozzle centreline between the Zaman et al. (Reference Zaman, Brown and Bridges2013) experiment and the one where the acoustic data were taken, which may partially account for the slightly higher velocity at $y_{2}/D_{J}=0$ .) We therefore use the same normalized mean flow model for all jet exit velocities with the mean flow parameters for a best fit to the data, as indicated in the caption of figure 6. The data show that the mean velocity is small but not equal to zero at the interface. This can, in part, be attributed to the turbulent mixing that occurs upstream of the measuring station which, as noted above, was located slightly downstream of the trailing edge. But as pointed out by one of the referees, it could also be due to weakly nonlinear velocity fluctuations, which causes the mean velocity to leave the trailing edge at different speeds above and below the plate. Hunt et al. (Reference Hunt, Ishihara, Szubert, Asproulias, Hoarau and Braza2016) have recently shown that the mean speeds above and below the trailing edge can differ if the plate is at a small angle to the mean flow, and similar effects could occur in the present case where it is nominally aligned with the flow. However, the interface velocity is relatively small and in our cases is deemed to be insignificant relative to other uncertainties in the data comparisons.

Figure 6(b) compares the turbulent kinetic energy measurements from the same experiment to the amplitude $A(y_{2},y_{2})$ defined by (7.22) and (7.23) with the parameters $A_{0}$ and $B_{0}$ set equal to 0.011 and 0.022, respectively. The normalized turbulent kinetic energy profiles are also relatively independent of jet exit velocity (Afsar et al. Reference Afsar, Leib and Bozak2017), and the models appear to be in reasonable agreement with the flow data. They are therefore used in the following noise predictions.

Figure 6. Comparison of (a) mean velocity profile and (b) turbulent kinetic energy calculated from (7.21) and (7.22) against experiments reported in Zaman et al. (Reference Zaman, Brown and Bridges2013).  $(\,y_{d},t_{d})=(0.98,1.85)$ , $\unicode[STIX]{x1D705}=0.2$ and $(A_{0},B_{0})=(0.011,0.022)$ .

Numerical results for the noise generated by jet–edge interaction are obtained by evaluating the formula (7.18) for the acoustic spectrum, with the double integrals being computed using a standard Simpson’s method. The integrand in (7.18) vanishes outside of the support of the mean flow function (7.21), and the range of integration in this equation is therefore limited to the region where $U(y_{2})\neq 0$ .

Figures 8–10 show quantitative comparisons of measurements of the far-field pressure fluctuations’ power spectral density per unit Strouhal number $=fD/U_{J}$ , in dB scale $PSD=10\log (4\unicode[STIX]{x03C0}I_{\unicode[STIX]{x1D714}}U_{J}/Dp_{ref}^{2})$ (referenced to $p_{ref}=20~\unicode[STIX]{x03BC}\text{Pa}$ ) taken by Brown (Reference Brown2015) with predictions obtained by inserting the spectrum (6.13) with $f=|\unicode[STIX]{x1D702}_{2}/l_{2}|=|(y_{2}-{\tilde{y}}_{2})/l_{2}|$ into the composite RDT solution (7.18). Results are shown at observer locations directly below the plate $(\unicode[STIX]{x1D713}=-90^{\circ })$ and at several observer polar angles, $\unicode[STIX]{x1D703}$ , measured from the downstream jet axis. The experimental trailing-edge noise was educed by subtracting the noise measured in the corresponding free jet (i.e. in the absence of a plate) from the total measured noise. The parameters used in the predictions shown in figures 8–10 are $\unicode[STIX]{x1D70F}_{0}=2.5$ and $(l_{2},l_{3})=(0.67,0.25),\unicode[STIX]{x1D712}_{0}=1$ . Setting the coefficient $a_{1}$ equal to 0.75 in (6.8) produces a normalized turbulence correlation $\langle U_{\bot }(t-y_{1}/U(y_{2}),\boldsymbol{y}_{T})U_{\bot }(t+\tilde{\unicode[STIX]{x1D70F}}-y_{1}/U(y_{2}),{\tilde{y}}_{2},y_{2}+\unicode[STIX]{x1D702})\rangle$ , shown in figure 7, which exhibits the experimentally observed cusp behaviour at zero spatial and temporal separations (figure 7 b) and the small but definite negative region at larger time delays.

Figure 7. Normalized transverse turbulence correlation (6.8) at $\unicode[STIX]{x1D702}_{3}=0$ with parameters $\unicode[STIX]{x1D70F}_{0}=2.5$ , $(l_{1},l_{2})=(0.67,0.25)$ and $a_{1}=0.75$ (a) various fixed $\unicode[STIX]{x1D702}_{2}$ (b)  $\unicode[STIX]{x1D702}_{2}=0$ .

Figure 8. Power spectral density (PSD) of the far-field pressure fluctuations at 100 equivalent diameters from nozzle exit (lossless in dB scale referenced to $20~\unicode[STIX]{x03BC}\text{Pa}$ ) as a function of Strouhal number, for $M_{a}=0.9$ . Predicted (solid line): measured data below the plate at $\unicode[STIX]{x1D713}=-90^{\circ }$ . (Total noise: red; difference between the total noise and noise measured in the free jet: green.) Plate trailing edge at $(x_{d},y_{d})/D=(5.7,0.98)$ . (a) $\unicode[STIX]{x1D703}=90^{\circ }$ , (b) $\unicode[STIX]{x1D703}=75^{\circ }$ , (c) $\unicode[STIX]{x1D703}=60^{\circ }$ , (d) $\unicode[STIX]{x1D703}=45^{\circ }$ .

Figure 9. Power spectral density (PSD) of the far-field pressure fluctuations at 100 equivalent diameters from nozzle exit (lossless in dB scale referenced to $20~\unicode[STIX]{x03BC}\text{Pa}$ ) as a function of Strouhal number, for $M_{a}=0.7$ . Predicted (solid line): measured data below the plate at $\unicode[STIX]{x1D713}=-90^{\circ }$ . (Total noise: red; difference between the total noise and noise measured in the free jet: green.) Plate trailing edge at $(x_{d},y_{d})/D=(5.7,0.98)$ . (a) $\unicode[STIX]{x1D703}=90^{\circ }$ , (b) $\unicode[STIX]{x1D703}=75^{\circ }$ , (c) $\unicode[STIX]{x1D703}=60^{\circ }$ , (d) $\unicode[STIX]{x1D703}=45^{\circ }$ .

Figure 10. Power spectral density (PSD) of the far-field pressure fluctuations at 100 equivalent diameters from nozzle exit (lossless in dB scale referenced to $20~\unicode[STIX]{x03BC}\text{Pa}$ ) as a function of Strouhal number, for $M_{a}=0.5$ . Predicted (solid line): measured data below the plate at $\unicode[STIX]{x1D713}=-90^{\circ }$ . (Total noise: red; difference between the total noise and noise measured in the free jet: green). Plate trailing edge at $(x_{d},y_{d})/D=(5.7,0.98)$ . (a) $\unicode[STIX]{x1D703}=90^{\circ }$ , (b) $\unicode[STIX]{x1D703}=75^{\circ }$ (c) $\unicode[STIX]{x1D703}=60^{\circ }$ (d) $\unicode[STIX]{x1D703}=45^{\circ }$ .

The results for the downstream polar angles (in figures 8–10) show that the present RDT-based edge-noise predictions are in much better agreement with the data than those given in Goldstein et al. (Reference Goldstein and Afsar2013a ) and Afsar et al. (Reference Afsar, Leib and Bozak2017). The agreement is now very good over the entire frequency range where the total measured noise (red symbols) is dominated by that generated by the jet–surface interaction alone (green symbols) for all jet exit Mach numbers at the downstream polar angles shown. The results of GAL were limited to $St<0.4$ and $Ma>=0.7$ . The predictions shown in figure 11 for upstream polar angles are also in very good agreement over the entire Mach number range.

Figure 11. Power spectral density (PSD) of the far-field pressure fluctuations at 100 equivalent diameters from nozzle exit (lossless in dB scale referenced to $20~\unicode[STIX]{x03BC}\text{Pa}$ ) as a function of Strouhal number. Predicted (solid line): measured data below the plate at $\unicode[STIX]{x1D713}=-90^{\circ }$ . (Total noise: red; difference between the total noise and noise measured in the free jet: green.) For plate trailing edge at $(x_{d},y_{d})/D=(5.7,0.98)$ . (a) $M_{a}=0.9,\unicode[STIX]{x1D703}=95^{\circ }$ , (b) $M_{a}=0.9,\unicode[STIX]{x1D703}=105^{\circ }$ , (c) $M_{a}=0.7,\unicode[STIX]{x1D703}=95^{\circ }$ , (d) $M_{a}=0.7,\unicode[STIX]{x1D703}=105^{\circ }$ , (e) $M_{a}=0.5,\unicode[STIX]{x1D703}=95^{\circ }$ , (f) $M_{a}=0.5,\unicode[STIX]{x1D703}=105^{\circ }$ .

Figure 12 is a comparison of the acoustic predictions obtained by inserting the present source function model (6.13) into the low-frequency solution used in GAL and Afsar et al. (Reference Afsar, Leib and Bozak2017) for the parameter values used in figures 8–10. As expected, the present approach converges to the low-frequency result at very low frequencies and, therefore, represents a much more robust mathematical model of trailing-edge noise than either of the two previous studies since (for reasons indicated below (7.20)) it is now applicable to $O(1)$ frequencies. And our numerical tests show that the low-frequency roll-off is now much less sensitive to the magnitude of the negative loop in the correlation function $\langle U_{\bot }(t-y_{1}/U(y_{2}),\boldsymbol{y}_{T})U_{\bot }(t+\tilde{\unicode[STIX]{x1D70F}}-y_{1}/U(\,y_{2}),{\tilde{y}}_{2},y_{2}+\unicode[STIX]{x1D702})\rangle$ than the Afsar et al. (Reference Afsar, Leib and Bozak2017) model – although it is necessary to include this feature in the model in order for the transverse turbulence correlation to be physically realizable. In the present model the negative (anticorrelation) region enables the correct prediction of the absolute level of the very low frequency sound (i.e. for $St<0.1$ ) rather than the roll-off per se.

The improved predictions of the present result (relative to that obtained in GAL) is largely due to the $\exp (\text{i}k_{\infty }\int _{0}^{y_{2}}\sqrt{q(y\mid y_{2})}\,\text{d}y)$ factor in (7.19), which damps out the high frequencies and, therefore, increases the high-frequency roll-off, since the argument of the exponential $\text{i}k_{\infty }\int _{0}^{y_{2}}\sqrt{q(y\mid y_{2})}\,\text{d}y$ is always negative. It accounts for the bending of the sound waves away from the downstream axis and, therefore, represents a kind of ‘zone of silence’.

The present calculations are based on (6.4), which is obtained by using causality to interpret the singular integral (4.13) for the transverse particle displacement $\unicode[STIX]{x1D702}_{\bot }(\boldsymbol{y},\unicode[STIX]{x1D70F})$ . But the causality condition results from an initial condition imposed in the distant past and, as argued in the introduction, the long-time solutions to the initial value problem are not necessarily relevant to the time-stationary turbulent flows being considered here. (Similar arguments can be found in Mani Reference Mani1976; Dowling, Ffowcs Williams & Goldstein Reference Dowling, Ffowcs Williams and Goldstein1978.) However, the singular integral in (4.13) will also be well defined if it is interpreted as a Cauchy principal value. The resulting formulae turn out to be more complicated than the present results, and our computations (not shown here) indicate that the acoustic predictions based on these formulae do not differ significantly from the present results – at least in the low-frequency limit where comparisons were carried out. Data comparisons, such as those given in this section, therefore, cannot be used to distinguish between the two approaches.

Figure 12. Convergence to the GAL solution. Same legend as figures 8–11. (a) $M_{a}=0.9,\unicode[STIX]{x1D703}=90^{\circ }$ , (b) $M_{a}=0.5,\unicode[STIX]{x1D703}=45^{\circ }$ .

Figures 8–10 show that the predictions are better for larger polar angles (near ninety degrees) and higher Mach numbers, as they were in GAL and Afsar et al. (Reference Afsar, Leib and Bozak2017). The former is due to reduction in edge noise relative to the jet noise for shallow polar observation angles and the latter (the deterioration in prediction for $M_{a}=0.5$ near $St\sim 0.1$ ) may be a particular feature of the experimental data (Bridges Reference Bridges2014) or may be associated with a change in the interference between the non-convecting jet noise and edge noise (Afsar et al. Reference Afsar, Leib and Bozak2017, p. 202) at lower Mach numbers. There are four free parameters: $(l_{2},l_{3},\unicode[STIX]{x1D70F}_{0},a_{1})$ that determine the source function $S$ in the present model, with all other parameters determined by matching to the turbulence or mean flow data, which makes the predictions much less empirical than those of GAL and Afsar et al. (Reference Afsar, Leib and Bozak2017). No empirical coefficients would be required if there were an experimental database for the transverse velocity correlation $\langle U_{\bot }(t-y_{1}/U(y_{2}),\boldsymbol{y}_{T})U_{\bot }(t+\tilde{\unicode[STIX]{x1D70F}}-y_{1}/U(y_{2}),{\tilde{y}}_{2},y_{2}+\unicode[STIX]{x1D702})\rangle$ for a rectilinear jet in the vicinity of a plate trailing edge. It could also be obtained computationally using large eddy simulation (LES) which would be much less expensive than a jet-noise experiment. The parameters could, in principle, also be obtained by further optimizing the agreement with the measured spectra.

8 Concluding remarks

This paper is based on the formal solutions (2.14)–(2.16) to the linearized Euler equations for transversely sheared mean flows which, like the Kovasznay results for the unsteady motion on uniform flows, involve two arbitrary convected quantities $\unicode[STIX]{x1D717}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ and $\tilde{\unicode[STIX]{x1D714}}_{c}(\unicode[STIX]{x1D70F}-y_{1}/U,\boldsymbol{y}_{T})$ that can be associated with the hydrodynamic component of the flow and can, therefore, be used to specify upstream boundary (i.e. initial) conditions for RDT problems that involve the interaction of turbulence with solid surfaces. This paper derives a new relation between these quantities and the physically measurable variables that is much simpler and more general than the one given in Goldstein et al. (Reference Goldstein and Afsar2013a ,Reference Goldstein and Afsar b ).

This relation was used to relate the source term $S$ that appears in the formula (7.2) for the noise generated by the interaction of a two-dimensional jet with a semi-infinite flat plate derived in Goldstein et al. (Reference Goldstein and Afsar2013a ) to the physically measurable second-order velocity correlations in the jet. The result was combined with a modified high-frequency solution to obtain a specific formula for the acoustic spectrum that applies over a broad range of frequencies. This result was then compared with experimental measurements carried out at Glenn Research Center, and excellent agreement was obtained. The general results can of course, be applied to many other RDT problems involving the interaction of turbulence with surfaces embedded in transversely sheared base flows or, more generally, in vortical base flows that asymptote to transversely sheared mean flows in the upstream region.