7.1 Theory of vortex sound

The theory of vortex sound was first discussed by Powell [1] who manipulated the source term in Lighthill's wave equation to specifically include the vorticity. In this section we extend this concept and show how the linearized Euler equations developed in the previous chapter can be used to identify acoustic source terms that depend on vorticity [2]. In Section 6.1 we derived Goldstein's wave equation, which was given by Eq. (6.1.9) as

$\frac{D_o}{Dt}\left(\frac{1}{c_o^2}\frac{D_o\stackrel{.}{\varphi }}{Dt}\right)-\frac{1}{{\rho }_o}\nabla \left({\rho }_o\nabla \stackrel{.}{\varphi }\right)=\frac{1}{{\rho }_o}\nabla \left({\rho }_o{\stackrel{.}{\mathbf{u}}}^{\left(g\right)}\right)$

where the dot represents a time derivative and the velocity and pressure perturbations were defined such that

$\mathbf{v}=\mathbf{U}+\nabla \varphi +{\mathbf{u}}^{\left(g\right)}\mathbf{,}{\rho }_o\frac{D_o}{Dt}\left(\frac{\partial \varphi }{\partial t}\right)=-\frac{\partial p}{\partial t}$

The velocity perturbation can be specified using Crocco's equation (Eq. 2.5.4) for an inviscid flow

$\frac{\partial \mathbf{v}}{\partial t}=-\nabla H+T_e\nabla s-\mathbf{\omega }\times \mathbf{v}$

and so Eq. (6.1.9) can be written as

$\frac{D_o}{Dt}\left(\frac{1}{c_o^2}\frac{D_o\stackrel{.}{\varphi }}{Dt}\right)+\frac{1}{{\rho }_o}\nabla \left({\rho }_o\nabla H\right)=\frac{1}{{\rho }_o}\nabla \left({\rho }_o{T}_e\nabla s-{\rho }_o\mathbf{\omega }\times \mathbf{v}\right)$

We can then modify this equation to make H the dependent variable of the wave operator on the left side. From the definition of the stagnation enthalpy H we find that

$\frac{\partial H}{\partial t}=T_e\frac{\partial s}{\partial t}+\frac{1}{\rho }\frac{\partial p}{\partial t}+\frac{1}{2}\frac{\partial v_i^2}{\partial t}$

and taking the dot product of Crocco's equation with the velocity v, and ignoring viscous terms, gives

$\mathbf{v}\cdot \frac{\partial \mathbf{v}}{\partial t}=\frac{1}{2}\frac{\partial v_i^2}{\partial t}=-\mathbf{v}\cdot \nabla H+\mathbf{v}\cdot \left(T_e\nabla s\right)$

Rearranging these equations gives

$\frac{DH}{Dt}=\frac{1}{\rho }\frac{\partial p}{\partial t}$

for an isentropic flow where Ds/Dt=0. If this result is linearized about the mean flow then $D_{o}H/Dt=-D_o\stackrel{.}{\varphi }/Dt$, which can be used in Eq. (7.1.1) to obtain Howe's wave equation [2]

$\frac{D_o}{Dt}\left(\frac{1}{c_o^2}\frac{D_{o}H}{Dt}\right)-\frac{1}{{\rho }_o}\nabla \left({\rho }_o\nabla H\right)=\frac{1}{{\rho }_o}\nabla \left({\rho }_o\mathbf{\omega }\times \mathbf{v}-{\rho }_o{T}_e\nabla s\right)$

The significant point here is that the acoustic variable is now defined by the stagnation enthalpy and the source terms are specified in terms of the Lamb vector ω×v, and the gradient of the entropy. The form of the equation is identical to Goldstein's equation but the source terms are specified in a more convenient form, if the vorticity is a well-defined quantity.

For the special case when the flow Mach number is very small, the mean density is constant, there are no entropy fluctuations, and the fluid is at rest outside a bounded region, we can simplify Howe's wave equation to

$\frac{1}{c_{\infty }^2}\frac{{\partial }^{2}H}{\partial t^2}-\frac{{\partial }^{2}H}{\partial x_i^2}=\nabla \cdot \left(\mathbf{\omega }\times \mathbf{v}\right)$

The solution to Eq. (7.1.3) can be obtained using the approaches described in Chapters 4 and 5. In the absence of scattering surfaces we can use the method of Green's functions to obtain

$H\left(\mathbf{x},t\right)=-{\displaystyle {\int }_{-T}^T{\displaystyle {\int }_V{\left(\mathbf{\omega }\times \mathbf{v}\right)}_i\frac{\partial G_o}{\partial y_i}\mathit{dVd\tau }}}$

The source term on the right of Eq. (7.1.4) reveals the role of vorticity as a source of sound. It shows that there is no sound caused by vorticity that is aligned with the local flow. Furthermore, if the source term is linearized about an irrotational mean flow, we can use Eq. (6.3.11) to relate the local vorticity to the vorticity of a harmonic gust on some upstream inflow boundary, defined by $\mathbf{\omega }=\mathrm{R}\mathrm{e}\left({\hat{\mathbf{\omega }}}^{\left(\infty \right)}exp\left(i\mathbf{k}\cdot \mathbf{x}-ik{U}_{\infty }t\right)\right)$, so that

$\mathbf{\omega }\times \mathbf{U}=\mathrm{R}\mathrm{e}\left({\hat{\omega }}_i^{\left(\infty \right)}\left(\frac{\delta {\mathbf{l}}^{\left(i\right)}}{h_i}\times \mathbf{U}\right)e^{i\mathbf{k}\cdot \mathbf{X}-ik{}_{1}U_{\infty }t}\right)$

where δl⁽ⁱ⁾ are displacement coordinates shown in Fig. 6.3. Since δl⁽¹⁾/h₁=U/U_∞ we can use the relationship given by Eq. (6.2.9) to show that

$\mathbf{\omega }\times \mathbf{U}=\mathrm{R}\mathrm{e}\left(U_{\infty }\left({\hat{\omega }}_3^{\left(\infty \right)}\nabla X_2-{\hat{\omega }}_2^{\left(\infty \right)}\nabla X_3\right)e^{i\mathbf{k}\cdot \mathbf{X}-ik{}_{1}U_{\infty }t}\right)$

where X₂=const and X₃=const are the stream surfaces of the mean flow. It follows that the source term in Eq. (7.1.4) is determined by the two components of the vorticity that are initially normal to the mean flow, and the streamwise component of the vorticity has no impact on the source term, even after distortion by the mean flow. The important consequence of this is that trailing tip vortices or other similar flows where the unsteady vorticity is initially aligned with the mean flow can only cause sound radiation because of nonlinear or self-induced unsteady motion.

7.2 Sound from two line vortices in free space

A simple canonical model of vortex sound [1] is given by two line vortices of equal strength Γ that are separated by the distance 2d in an otherwise stationary fluid, as shown in Fig. 7.1. The vortex pair will spin about the point halfway between them because of the induced flow at each vortex. Because there is no background flow, the motion of each vortex is caused only by its convection by the other vortex in the pair, and so each moves with a speed of U=Γ/4πd.

The location of each vortex is given as y=±y^(v) where y^(v)(τ)=(dcos(Ωτ), dsin(Ωτ), 0), and the angular velocity of the system is Ω=Γ/4πd² and U=Ωd. The vorticity of each vortex is defined as

$\mathbf{\omega }=\mathbf{k}\Gamma \delta \left(y_1\pm y_1^{\left(v\right)}\left(\tau \right)\right)\delta \left(y_2\pm y_2^{\left(v\right)}\left(\tau \right)\right)$

where k is a unit vector pointing out of the page in Fig. 7.1. It follows that the Lamb vector in Eq. (7.1.4) is

$\begin{array}{ll}\mathbf{\omega }\times \mathbf{v}=& -\Omega \Gamma {\mathbf{y}}^{\left(v\right)}\left(\tau \right)\left\{\delta \left(y_1-y_1^{\left(v\right)}\left(\tau \right)\right)\delta \left(y_2-y_2^{\left(v\right)}\left(\tau \right)\right)\right.\hfill \\ & \left.-\delta \left(y_1+y_1^{\left(v\right)}\left(\tau \right)\right)\delta \left(y_2+y_2^{\left(v\right)}\left(\tau \right)\right)\right\}\hfill \end{array}$

Note that, with the vorticity lumped into discrete vortices, the velocity field of the flow v can be replaced with the convection velocity of the vortices. Using Eq. (7.2.2) in Eq. (7.1.4) and integrating over y₁ and y₂ gives the sound field as

$\begin{array}{ll}H\left(\mathbf{x},t\right)=& \Omega \Gamma {\displaystyle {\int }_{-\infty }^{\infty }{\displaystyle {\int }_{-T}^T{\left[y_i^{\left(v\right)}\left(\tau \right)\frac{\partial G_o\left(\mathbf{x},t|\mathbf{y},\tau \right)}{\partial y_i}\right]}_{\mathbf{y}={\mathbf{y}}^{\left(v\right)}\left(\tau \right)+\mathbf{k}{y}_3}}}\hfill \\ \hfill & {\displaystyle {\displaystyle -{\left[y_i^{\left(v\right)}\left(\tau \right)\frac{\partial G_o\left(\mathbf{x},t|\mathbf{y},\tau \right)}{\partial y_i}\right]}_{\mathbf{y}=-{\mathbf{y}}^{\left(v\right)}\left(\tau \right)+\mathbf{k}{y}_3}d{y}_{3}d\tau }}\hfill \end{array}$

The integral is carried out over the infinite length of the two vortices because this is a two-dimensional problem. However, if we consider the sound radiation from a small linear segment of the vortices we can evaluate the far-field sound from that segment. This enables us to use the far-field approximation for the Green's function. The far-field sound from the segment of length b centered on y₃ is then δH, where

$\begin{array}{ll}\delta H\left(\mathbf{x},t\right)=& \Omega \Gamma b{\displaystyle {\int }_{-T}^T{\left[y_i^{\left(v\right)}\left(\tau \right)\frac{\partial G_o\left(\mathbf{x},t|\mathbf{y},\tau \right)}{\partial y_i}\right]}_{\mathbf{y}={\mathbf{y}}^{\left(v\right)}\left(\tau \right)+\mathbf{k}{y}_3}}\hfill \\ & {\displaystyle -{\left[y_i^{\left(v\right)}\left(\tau \right)\frac{\partial G_o\left(\mathbf{x},t|\mathbf{y},\tau \right)}{\partial y_i}\right]}_{\mathbf{y}=-{\mathbf{y}}^{\left(v\right)}\left(\tau \right)+\mathbf{k}{y}_3}d\tau }\hfill \end{array}$

We note from this result that if we were to ignore the effect of retarded time then the integrand would be zero, giving no far-field sound. However, if we expand the Green's function using a Taylor series expansion about the point ky₃

${\left[\frac{\partial G_o\left(\mathbf{x},t|\mathbf{y},\tau \right)}{\partial y_i}\right]}_{\mathbf{y}=\pm {\mathbf{y}}^{\left(v\right)}\left(\tau \right)+\mathbf{k}{y}_3}={\left[\frac{\partial G_o\left(\mathbf{x},t|\mathbf{y},\tau \right)}{\partial y_i}\right]}_{\mathbf{y}=\mathbf{k}{y}_3}\pm y_j^{\left(v\right)}\left(\tau \right){\left[\frac{{\partial }^2{G}_o\left(\mathbf{x},t|\mathbf{y},\tau \right)}{\partial y_i\partial y_j}\right]}_{\mathbf{y}=\mathbf{k}{y}_3}+\cdots$

and so the integral reduces to

$\delta H\left(\mathbf{x},t\right)=2\Omega \Gamma \mathrm{b}{\displaystyle {\int }_{-T}^T{y}_i^{\left(v\right)}\left(\tau \right)y_j^{\left(v\right)}\left(\tau \right)\frac{{\partial }^2{G}_o\left(\mathbf{x},t|\mathbf{k}{y}_3,\tau \right)}{\partial y_i\partial y_j}d\tau }$

and we obtain an acoustic field that has the characteristics of a quadrupole. As we did in Section 4.4, in the acoustic far field we can shift the derivative of the free-space Green's function so that they are derivatives with respect to time, and then integrate over source time, so we obtain

$\delta H\left(\mathbf{x},t\right)=\frac{2\Omega \Gamma \mathrm{b}}{c_{\infty }^2}\frac{{\partial }^2}{\partial t^2}{\left[\frac{\left(x_j{y}_j^{\left(v\right)}\right)\left(x_i{y}_i^{\left(v\right)}\right)}{4\pi |\mathbf{x}-\mathbf{k}{y}_3|{}^3}\right]}_{\tau =\tau *}$

The analysis is simplified if we choose the observer location to be at x=(R_o,0,0) which gives x_iy_i^(v)=R_odcos(Ωτ). We then obtain

$\delta H\left(\mathbf{x},t\right)=\frac{\Omega \Gamma b{d}^2{R}_o^2}{2\pi c_{\infty }^2}\frac{{\partial }^2}{\partial t^2}{\left[\frac{{\cos }^2\left(\Omega \tau \right)}{{\left(R_o^2+y_3^2\right)}^{3/2}}\right]}_{\tau =\tau *}$

Since cos²(Ωτ)=(1+cos(2Ωτ))/2 we can simplify the integrand so that it only depends on cos(2Ωτ) because the constant term will not contribute to the acoustic field.

Then since the correct retarded time is

$\tau *=t-r/c_{\infty },r={\left(R_o^2+y_3^2\right)}^{1/2}$

we obtain

$\delta H\left(\mathbf{x},t\right)=-\frac{{\Omega }^3\Gamma b{d}^2{R}_o^2}{\pi c_{\infty }^2{r}^3}cos\left(2\Omega \left(t-r/c_{\infty }\right)\right)$

We can then evaluate the far-field pressure because ∂H/∂t is approximately equal to (1/ρ_o)∂p/∂t in the acoustic far field. To obtain a scalable result we can define a Mach number of the vortex motion as M=Ωd/c_∞, and the speed of the vortex as U=Ωd=Γ/4πd so the far-field pressure signal is given by

$\delta p\left(\mathbf{x},t\right)=-\frac{4{\rho }_o{U}^2{M}^2{R}_o^{2}bcos\left(2\Omega \left(t-r/c_{\infty }\right)\right)}{r^3}$

This is a canonical example of two-dimensional flow noise and as such it is valuable to investigate the important features of the result. It shows that the far-field pressure from an elemental part of the vortex scales as U²M² which is the same as the result obtained in Chapter 4 for free turbulence. The acoustic intensity in the far field I=|p|²/(2ρ_oc_∞) will scale as M⁴U⁴ or the eighth power of the velocity at the vortex. Eq. (7.2.6) also shows that the acoustic radiation occurs at a frequency that is twice the angular speed of the vortex, which is to be expected since the flow field replicates itself after half a revolution of the vortex pair.

Some care needs to be taken when evaluating this result for a line vortex whose span exceeds the acoustic wavelength. The result given by Eq. (7.2.6) needs to be integrated along the length of the vortex and the motion is only specified by the model given above if the vortex is of infinite length. The infinite length vortex results in a two-dimensional problem and the far-field approximations used above will no longer be valid, and the result will scale differently, but the problem can be solved exactly [2]. However, if the vortices are of finite length, and their span is acoustically compact, but their motion is well approximated by the self-induced spinning motion used above, then the far-field sound is given by the integral of Eq. (7.2.6) over y₃.

7.3 Surface forces in incompressible flow

In Chapter 4 we showed that, for acoustically compact bodies in low Mach number flows, the flow noise is dominated by dipole sources whose strength is directly proportional to the unsteady loading on the body surface. In this case the acoustic wavelength is much larger than the size of the body, and so the unsteady loading will be dominated by the incompressible part of the flow perturbations. We can therefore separate the problem into two parts, the incompressible flow that causes the unsteady load and the sound radiation from the loading on the surfaces. When this approximation is made, the acoustic far field can be evaluated from Curle's equation (4.4.7) with the source term defined by the unsteady loading caused by incompressible flow perturbations.

In an incompressible flow the force applied to the fluid by a stationary surface can be calculated directly from the vorticity in the flow (see Ref. [2] for a more detailed derivation of the results that follow, in which the effect of viscosity is included). To obtain this relationship consider the problem illustrated in Fig. 7.2 where fluid is bounded by a surface S_∞ at a large distance from the body and the mean flow tends to a constant on this surface. Since the fluid is assumed incompressible its density will be constant and the divergence of the velocity v will be zero. The surface of the body is assumed impenetrable and so v·n on S, and we will assume that |v|² tends to $U_{\infty }^2$ on S_∞ with a difference that tends to zero as 1/r^α, where α>2. This implies that there is no net flux of momentum across S_∞ because in the limit that r tends to infinity (ρv·n)v=ρU_∞²n₁ which integrates to zero on S_∞.

The force applied to the fluid by the body F_i can be equated to the fluid momentum and the pressure on S_∞ using Eq. (2.3.3), with the assumption that there is no net flux of momentum across S_∞ giving

$F_i={\rho }_o\frac{d}{dt}{\displaystyle {\int }_V{v}_{i}dV}-{\displaystyle {\int }_{S_{\infty }}p{n}_{i}dS}$

(where in this case the normal is chosen as n=−n^(o) so it points into the enclosed volume). The volume integral can be changed to a surface integral by making use of the identity

$v_i=\mathbf{v}\cdot \nabla \left(x_i\right)=\nabla \cdot \left(\mathbf{v}{x}_i\right)-\left(x_i\nabla \cdot \mathbf{v}\right)$

and noting that the last term is zero in an incompressible fluid. Substituting this into the volume integral in Eq. (7.3.1) and using the divergence theorem results in

${\rho }_o\frac{d}{dt}{\displaystyle {\int }_V{v}_{i}dV}=-{\rho }_o\frac{d}{dt}{\displaystyle {\int }_{S_{\infty }}{x}_i\mathbf{v}\cdot \mathbf{n}dS}-{\rho }_o\frac{d}{dt}{\displaystyle {\int }_S{x}_i\mathbf{v}\cdot \mathbf{n}dS}$

The boundary condition on the surface of the body requires that v·n=0, so the surface integral over S is zero and we can write Eq. (7.3.1) as

$F_i=-{\displaystyle {\int }_{S_{\infty }}\left(p{n}_i+x_i{\rho }_o\frac{\partial \mathbf{v}}{\partial t}\cdot \mathbf{n}\right)dS}$

(where d/dt has been replaced by ∂/∂t inside the integral because the surface is stationary).

The right side of Eq. (7.3.2) can be related to the vorticity in the flow by making use of Kirchhoff coordinates and the form of the momentum equation given by Eq. (2.5.3). The Kirchhoff coordinates Y_i are defined as equal to the potential of the flow around the body that results from an inflow on S_∞ of unit amplitude in the x_i direction. They have the properties that

$\left.\begin{array}{c}{Y}_i=x_i\\ \mathbf{n}\cdot \nabla Y_i=n_i\end{array}\right\}\mathrm{on}{S}_{\infty },{\nabla }^2{Y}_i=0,\mathbf{n}\cdot \nabla Y_i=0\mathrm{on}S$

If we take the dot product of $\nabla Y_i$ with the momentum equation expressed in the form given by Eq. (2.5.3) and integrate over the volume of the fluid then, ignoring viscous terms for a homentropic flow, we obtain

${\int }_V\nabla Y_i\cdot \left({\rho }_o\frac{\partial \mathbf{v}}{\partial t}\right)dV+{\int }_V\nabla Y_i\cdot \nabla \left(p+\frac{1}{2}{\rho }_o{v}_i^2\right)dV=-{\rho }_o{\int }_V\nabla Y_i\cdot \left(\mathbf{\omega }\times \mathbf{v}\right)dV$

The first volume integral on the left of this equation can be turned into a surface integral over S_∞ in exactly the same way as was done for Eq. (7.3.2). Similarly the integrand of the second volume integral can be turned into a divergence because Y_i is a solution to Laplace's equation. Then making use of the boundary conditions on the surface defined in Eq. (7.3.3) we find that

$\begin{array}{ll}{\int }_V\nabla Y_i\cdot \left({\rho }_o\frac{\partial \mathbf{v}}{\partial t}\right)dV+{\int }_V\nabla Y_i\cdot \nabla \left(p+\frac{1}{2}{\rho }_o{v}_i^2\right)dV& ={\int }_V\nabla \left(Y_i{\rho }_o\frac{\partial \mathbf{v}}{\partial t}\right)dV+{\int }_V\nabla \left(\nabla Y_i\left(p+\frac{1}{2}{\rho }_o{v}_i^2\right)\right)dV\\ =& -{\int }_{S_{\infty }+S}{Y}_i{\rho }_o\frac{\partial \mathbf{v}}{\partial t}\cdot \mathbf{n}dS-{\int }_{S_{\infty }+S}\mathbf{n}\cdot \nabla Y_i\left(p+\frac{1}{2}{\rho }_o{v}_i^2\right)dS\\ & =-{\int }_{S_{\infty }}{Y}_i{\rho }_o\frac{\partial \mathbf{v}}{\partial t}\cdot \mathbf{n}dS-{\int }_{S_{\infty }}\mathbf{n}\cdot \nabla Y\left(p+\frac{1}{2}{\rho }_o{v}_i^2\right)dS\end{array}$

Since the surface lies at infinity we can use the limiting values given by Eq. (7.3.3) to obtain

$-{\int }_{S_{\infty }}{x}_i\left({\rho }_o\frac{\partial \mathbf{v}}{\partial t}\cdot \mathbf{n}\right)+n_i\left(p+\frac{1}{2}{\rho }_o{v}_i^2\right)dS=-{\rho }_o{\int }_V\nabla Y_i\cdot \left(\mathbf{\omega }\times \mathbf{v}\right)dV$

Then since the velocity tends to its value in the uniform flow with an error proportional to 1/r^α where α>2 and S_∞ increases as r² the integral of the squared velocity term becomes

${\int }_{S_{\infty }}{n}_i\left(\frac{1}{2}{\rho }_o{v}_i^2\right)dS\to {\int }_{S_{\infty }}{n}_i\left(\frac{1}{2}{\rho }_o{U}_{\infty }^2\right)dS=0$

By matching terms in Eq. (7.3.2) we obtain the force applied to the fluid by the body in terms of the volume distribution of the Lamb vector ω×v as

$F_i=-{\rho }_o{\displaystyle {\int }_V\nabla Y_i\cdot \left(\mathbf{\omega }\times \mathbf{v}\right)dV}$

To evaluate this integral, we need to include the vorticity throughout the fluid, including any bound vorticity such as the vorticity in the surface boundary layer and the wake behind the body. However, since the mean vorticity and mean velocity in the boundary layer next to the surface of the body will be parallel to the surface, the Lamb vector near the surface will point in the direction of the surface normal which is orthogonal to $\nabla Y_i$. This greatly reduces the contribution from the blade boundary layer velocity fluctuations in this calculation. However, the vorticity in the wake behind the body, that is an extension of the surface boundary layers, cannot be ignored and may play a dominant role. The other important conclusion from this result is that stationary vorticity does not contribute to the unsteady loading, so we need to only consider vorticity that is convected when evaluating Eq. (7.3.4). Also we note that the streamwise force is given by the F₁ component $\nabla Y_1\cdot \left(\mathbf{\omega }\times \mathbf{v}\right)$. If the vorticity is convected by the potential mean flow around the body then $\mathbf{v}=U_{\infty }\nabla Y_1$, and there will be no streamwise force. The streamwise force is therefore determined by the deviation of the vortex path from the potential flow streamlines, which results from the influence of other regions of vorticity or the image vorticity inside the body that impacts the vortex trajectory in the flow.

When vorticity is close to a rigid surface it will be convected by both the local mean flow (which would still exist if the vorticity was zero) and a self-induced flow that is required to meet the non-penetration boundary condition on the surface. The self-induced flow is often characterized by equivalent image vorticity that is placed inside the surface but is a potential flow outside the surface. The image vorticity can have a big impact if the path of the vortex takes it close to the stagnation point, but little impact if the vortex is relatively weak and some distance from the surface. To assess the importance of this we will consider a line vortex near a surface and estimate the velocity induced by the image vortex to be Γ/4πδ, where δ is the distance of the line vortex from the surface and Γ is its circulation. Then if Γ/4πUδ≪1 (where U is the local mean flow speed) the motion induced by the image vortex will not be important. However, this criterion cannot be met close to the stagnation point at the front of the body where the mean flow speed tends to zero. In this case the motion of the vortex will be controlled by the induced flow and the amplitude of the unsteady loading will be of order ρ_oΓ²/4πδ. However this will only have a small impact on the unsteady loading pulse which will have a peak amplitude of order ρ_oΓU_∞. This leads to a criterion that requires Γ/4πU_∞δ≪1, which is less restrictive than the criterion based on the local flow speed. In real flows one might expect Γ/2π~u_oL, where u_o is a typical gust velocity and L is the lengthscale of the turbulence. Typically, L~δ, so the importance of the self-induced motion of the vortex will depend on u_o/4πU_∞ which is very small in most flows of interest for which u_o≪U_∞. Therefore, we will not consider the impact of the image vortex on the convection speed any further, and we will assume the vortex is simply convected by the mean flow.

7.4 Aeolian tones

A well-known phenomenon of sound caused by flow over solid bodies is the Aeolian tone. This describes the mechanism of sound generation from the wind blowing through boat rigging and over telephone wires, or any other cylindrical body. The mechanism responsible for Aeolian tones has been studied extensively and is a classic example of vortex sound. The flow is illustrated in Fig. 7.3, which shows a stationary cylinder in a uniform flow, which sheds well-defined vortical structures into its wake. The vortical structures form a von Kármán vortex street that consists of equally spaced vortices that are convected downstream at a constant speed. At Reynolds numbers Re_d=U_∞d/v (based on the flow speed U_∞ and the cylinder diameter d) that lie in the range 40<Re_d<5×10⁴ it is found that well-defined vortical structures are shed periodically, and the frequency of vortex shedding f is given by a Strouhal number St=fd/U_∞ of approximately 0.2. This is a weak function of Reynolds number and given by Goldstein [3] as

$\mathit{St}=0.198\left(1-19.7/{Re}_d\right)$

Flow visualization of the wakes behind cylinders have shown that the formation of the wake vortices follows a consistent process. As shown in Fig. 7.3 the vortices are initiated just behind the cylinder on either the upper or lower side of the wake.

During the period of growth, the vortex is almost static as it spins up, extracting circulation from the boundary layer on the surface of the cylinder. Once it has reached a critical circulation, it detaches from its point of initiation and is swept downstream by the flow. The initial period of acceleration is quite fast, and once it reaches the wake region it is convected with constant speed U_c. The process is then repeated for a vortex in the lower part of the wake, and it grows and separates in exactly the same way as the vortex in the upper part of the wake. The process is then repeated resulting in a time varying, periodic flow.

The periodic vortex shedding causes an unsteady lift fluctuation on the cylinder that is readily understood by using the results given in Section 7.3. In terms of Eq. (7.3.4) the unsteady lift on the cylinder will be given by

$F_2\left(t\right)=-{\rho }_o{\displaystyle {\int }_V\nabla Y_2\cdot \left(\mathbf{\omega }\times \mathbf{v}\right)dV}$

where the function Y₂ is the velocity potential of a flow of unit free-stream velocity in the i=2 direction around the cylinder and is given by Eq. (2.7.27) as

$Y_2=\mathrm{R}\mathrm{e}\left(z{e}^{-i\pi /2}+\frac{d^2{e}^{i\pi /2}}{4z}\right)=y_2\left(1+\frac{d^2}{4\sqrt{y_1^2+y_2^2}}\right)\mathrm{where}z=y_1+i{y}_2$

The first thing we note from Eq. (7.4.1) is that, since we are assuming that the vorticity is concentrated at a point in the flow, there will be no unsteady load while the vortex is stationary during its period of growth because v=0. However, once it reaches its critical strength and starts to move then it will contribute to the unsteady load. Vortex formation occurs at a point l_o>d downstream from the center of the cylinder, and so when the vortex is in motion we can approximate Y₂~y₂ and the trajectory is almost parallel to the y₁ axis, so we can write

$\nabla Y_2\cdot \left(\mathbf{\omega }\times \mathbf{v}\right)\approx {\Gamma }_{n}u\left(t-t_n\right)\delta \left(y_1-l_o-s\left(t-t_n\right)\right)\delta \left(y_2\pm h/2\right)H\left(\left|y_3\right|-b/2\right)$

where u(t−t_n) is the speed of the vortex after it detaches at time t=t_n and is zero for t<t_n. The location of the vortex in the downstream direction is given by l_o+s(τ), where τ is the time since the vortex was shed and s(τ) is the distance traveled. The vertical location of the vortex is given by y₂=±h/2 depending on whether it is in the upper or lower row, and the spanwise extent of the vortex is given by b. Furthermore the vortices shed in the upper row will have the opposite direction of rotation to the vortices in the lower row, so Γ_n=(−1)ⁿΓ. Using this model in Eq. (7.4.1) for multiple vortices then gives

$F_2\left(t\right)=-{\rho }_o\Gamma b{\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^{n}u\left(t-t_n\right)}$

If the vortex convection speed increases linearly with time until it reaches a steady convection speed U_c and the acceleration time is half a period, then we can model u(τ) such that

$u\left(\tau \right)=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill \tau <0\hfill \\ \hfill 2U_c\tau /T\hfill & \hfill 0<\tau <T/2\hfill \\ \hfill U_c\hfill & \hfill \tau >T/2\hfill \end{array}\right.$

The series in Eq. (7.4.3) then represents a triangular wave with amplitude U_c, and it can be expanded into a Fourier series so that

$F_2\left(t\right)=-{\rho }_o\Gamma b{U}_c\left(\frac{1}{2}+\frac{4}{{\pi }^2}cos\left(\Omega t\right)+\frac{4}{9{\pi }^2}cos\left(3\Omega t\right)+\cdots \right)$

where Ω=2π/T is the fundamental frequency of the fluctuations. We can then define a lift coefficient for the first harmonic based on the frontal area of the cylinder as

$C_L=\frac{\left|F_2\right|}{\left(1/2\right){\rho }_o{U}_{\infty }^{2}bd}=\frac{8}{{\pi }^2}\left(\frac{\Gamma }{U_{\infty }d}\right)\left(\frac{U_c}{U_{\infty }}\right)$

Analysis of a von Kármán vortex street [4] shows that the convection speed of the vortices in the sheet relative to a fixed body in a uniform flow is given by

$U_c=U_{\infty }-\frac{\Gamma }{2a}tanh\left(\pi h/a\right)$

where a is the horizontal distance between vortices and is approximately a=4d for Reynolds numbers above 1000. The stability of the vortex street requires that h/a=0.281. To obtain Γ we note that the Strouhal number is directly related to the convection velocity, so St=U_cd/U_∞a, and since St=0.2 the convection velocity is approximately U_c=0.8U_∞. However, there is also experimental evidence that the convection velocity is higher than this and has a value of U_c=0.9U_∞ [5]. Solving Eq. (7.4.6), with U_c=0.9U_∞, then gives

$\frac{\Gamma }{U_{\infty }d}=\frac{2a}{dtanh\left(\pi h/a\right)}\left(1-\frac{U_c}{U_{\infty }}\right)\approx 1.13$

and a lift coefficient of approximately 0.81, which is in good agreement with the measured values given by Blake [5].

The sound radiation from the unsteady loading at the fundamental frequency is readily calculated using Eqs. (7.4.4), (4.4.7), and we obtain the far-field sound as

$p\approx \left(\frac{{\rho }_o{U}_{\infty }^{3}b{C}_L\mathit{St}}{4c_{\infty }}\right)\frac{x_{2}sin\left(\Omega \left(t-r/c_{\infty }\right)\right)}{\left|\mathbf{x}\right|{}^2}$

This shows the classic dipole scaling of the far-field sound in which the acoustic intensity I=(p_rms)²/ρ_oc_∞ scales as M³U_∞³ and depends on the sixth power of the flow speed. Also note the directionality that has cosine dependence with a maximum in the direction of the unsteady lift.

7.5 Blade vortex interactions in incompressible flow

An important source of noise in aeroacoustics occurs when a blade moves past a line vortex. This is particularly relevant to helicopter noise when blade vortex interactions occur during maneuvers and cause a loud thumping sound. The source of the vortices in this case is the tips of the blades themselves. Using the formulas developed in Section 7.3 we can derive a simple incompressible flow model for the unsteady load caused by a blade vortex interaction from which we can calculate the acoustic field. More detailed analysis of this problem, which includes the effect of compressibility, will be given in Chapter 14, and here we will focus on the incompressible flow solution, using the approach given by Howe [2].

If we make the assumptions of thin airfoil theory then the blade can be modeled as a flat plate with zero thickness at zero angle of attack to the mean flow, as shown in Fig. 7.4. The vortex is convected past the plate with the mean flow U_∞, and its path takes it a height h above the plate. The coordinate system is chosen to be coincident with the center of the plate and the blade chord is 2a. If we consider a line vortex with circulation Γ then we can calculate the unsteady loading per unit span of the plate on the fluid using Eq. (7.3.4). Since the direction of the vorticity vector is out of the page, and the flow is in the y₁ direction the Lamb vector points in the y₂ direction and the unsteady load per unit span for a vortex located at y₁=U_∞t, y₂=h is

$\frac{F_2}{b}=-{\rho }_o{U}_{\infty }\Gamma {\left[\frac{\partial Y_2}{\partial y_2}\right]}_{y_1=U_{\infty }t,y_2=h}$

where b is the span of the blade. The Kirchhoff coordinate Y₂ for a flat plate can be calculated from potential flow theory and is given by Eq. (2.7.29) with α=π/2 as

$Y_2=\mathrm{R}\mathrm{e}\left(-i\sqrt{z^2-a^2}\right),z=y_1+i{y}_2$

The branch cut of the square root is chosen to lie between z=±a so that there is no flow through the plate. Fig. 7.5 shows the streamlines of the flow defined by $\nabla Y_2$, and it is seen that the function contours around the plate, changing rapidly for locations near the leading and trailing edges. Using this in Eq. (7.5.1) then gives the unsteady loading as a function of time as

$\frac{F_2}{b}=-\frac{1}{2}{\rho }_o{U}_{\infty }\Gamma \mathrm{R}\mathrm{e}\left\{\sqrt{\frac{U_{\infty }t+ih-a}{U_{\infty }t+ih+a}}+\sqrt{\frac{U_{\infty }t+ih+a}{U_{\infty }t+ih-a}}\right\}$

(note: this is the force applied to the fluid that tends to a constant when t=±∞ because the vortex remains in the flow). The time history of the unsteady load therefore has two contributions. The first has a peak when U_∞t=−a, which corresponds to the location of the vortex when it is closest to the leading edge of the blade. The second contribution occurs when U_∞t=a, which corresponds to the location of the vortex when it is closest to the trailing edge. These pulses are shown in Fig. 7.6, and their peak levels are determined by the height of the vortex above the surface, so the results for different values of h/a are shown. The closer the vortex is to the blade the sharper and larger is the unsteady loading pulse, and the leading and trailing edge pulses are of the same magnitude.

However, this model has not included the effect of vorticity in the blade wake, which is required to satisfy the Kutta condition. To fully calculate the impact of the wake vorticity we would need to solve the equations of motion to obtain the shed vorticity as a function of time. However, the impact of the wake vorticity is to eliminate the pressure jump at the trailing edge of the blade, and it was argued by Howe [2] that this reduces the trailing edge pulse. This leads to Howe's approximation which applies to the case when the vortex is close to the blade, so h≪a. It approximates the net unsteady loading by the contribution from the leading edge pulse only, and sets the trailing edge pulse to zero, so the unsteady load is given by the first term in Eq. (7.5.3) in the limit that U_∞t=−a, as

$\frac{F_2}{b}=-\frac{1}{2}{\rho }_o{U}_{\infty }\Gamma \mathrm{R}\mathrm{e}\left\{i\sqrt{\frac{2a}{U_{\infty }t+ih+a}}\right\}$

A comparison between Howe's approximation and the total pulse is shown in Fig. 7.7 for a vortex that passes above the blade at a standoff distance of h/a=0.05.

To justify this approximation, we need to compare the results using Howe's approximation to the solution that would have been obtained using an exact incompressible flow calculation that includes the Kutta condition. For a harmonic gust this is given by Sears function, which was discussed in Chapter 6. The comparison is most easily carried out by considering a gust that is caused by a harmonic vortex sheet at distance h above the blade. This causes a velocity field given by

$$\begin{gathered} {\hat{v}}_1{e}^{-i\omega t}=U_{\infty }-{\hat{a}}_{1}sgn\left(y_2-h\right)e^{-k_1|y_2-h|+i{k}_1\left(y_1-U_{\infty }t\right)}, \\ {\hat{v}}_2{e}^{-i\omega t}=-i{\hat{a}}_1{e}^{-k_1|y_2-h|+i{k}_1\left(y_1-U_{\infty }t\right)} \end{gathered}$$

It is readily verifiable that this flow is incompressible, and the vorticity is only nonzero on the vortex sheet and has a single component in the i=3 direction given by ${\hat{\omega }}_3{e}^{-i\omega t}=2{\hat{a}}_1\delta \left(y_2-h\right)e^{i{k}_1\left(y_1-U_{\infty }t\right)}$.

Using this result in Eq. (7.3.4), we assume that the amplitude of the velocity fluctuations is very much less than the mean flow speed, giving the unsteady load as

$\frac{{\hat{F}}_2}{b}{e}^{-i\omega t}=-2{\rho }_o{U}_{\infty }{\hat{a}}_1{\int }_{-\infty }^{\infty }{\left[\frac{\partial Y_2}{\partial y_2}\right]}_{y_2=h}{e}^{i{k}_1\left(y_1-U_{\infty }t\right)}d{y}_1$

Using Howe's approximation this becomes

$\frac{{\hat{F}}_2}{b}=-{\rho }_o{U}_{\infty }{\hat{a}}_1{\int }_{-\infty }^{\infty }\mathrm{R}\mathrm{e}\left\{i\sqrt{\frac{2a}{y_1+ih+a}}\right\}{e}^{i{k}_1{y}_1}d{y}_1$

This integral can be evaluated using tables of Fourier transforms, and it is found that

${\hat{F}}_2=-2\pi {\rho }_o{U}_{\infty }{\hat{a}}_1\mathit{ab}\left(\sqrt{\frac{1}{2\pi k_{1}a}}\right)e^{-i{k}_{1}a-k_{1}h-i\pi /4}$

We can compare this with the high-frequency approximation to the unsteady loading calculated using Sears function using Eqs. (6.4.3), in the limit that k₁a=σ≫1. In Eq. (6.4.3) the gust was defined relative to the upwash on the blade, and so the equivalence between the magnitude of the gust response given by Eqs. (6.4.3), (7.5.6) for σ≫1 requires that the upwash gust amplitude is replaced by exp(−k₁h) in Eq. (7.5.6). Comparing Eqs. (6.4.3), (7.5.6) shows that they give the same gust magnitude in the limit that σ≫1. Consequently Howe's approximation correctly accounts for the effect of vortex shedding in the wake and the Kutta condition and leads to the conclusion that the unsteady loading from a blade vortex interaction is dominated by the response of the blade as the vortex passes leading edge of the blade, hence the term Leading Edge Noise which we will discuss in more detail in subsequent chapters.

Another example that is of interest and also verifies Howe's approximation is the response of a flat plate to a step gust. In this case the incident disturbance is specified by the velocity and vorticity distribution where the gust reaches the leading edge at t=0. We assuming w_o≪U_∞, the unsteady loading per unit span is

$v_1=U_{\infty },v_2=w_{o}H\left(U_{\infty }t-y_1-a\right),{\omega }_3=-w_o\delta \left(y_1+a-U_{\infty }t\right)$

$\frac{F_2}{b}={\rho }_o{U}_{\infty }{w}_o{\int }_{-R}^R{\left[\frac{\partial Y_2}{\partial y_2}\right]}_{y_1=U_{\infty }t-a}d{y}_2$

At first sight this integral would appear trivial and gives the load as 2ρ_oU_∞w_oR which does not vary in time. However, there is an additional contribution when the step gust passes the leading edge of the blade because Y₂ has a discontinuity on y₂=0 between y₁=±a. The contribution from the discontinuity is given by the jump in the value of Y₂ across the surface. Applying Howe's approximation to Eq. (7.5.2) gives the jump in Y₂ as

$2\mathrm{R}\mathrm{e}\left(\sqrt{2a}\sqrt{y_1+i\varepsilon +a}\right)H\left(y_1+a\right)$

where ɛ tends to zero. The response to a step gust is then given by

$\frac{F_2}{b}=-2\pi {\rho }_o{U}_{\infty }{w}_{o}a\left(\frac{\sqrt{2U_{\infty }t/a}}{\pi }\right)H\left(U_{\infty }t/a\right)$

This can be compared to the approximate form of Kussner's function [6] for the response of a plate to a step gust, which is given by

$\frac{F_2}{b}=-2\pi {\rho }_o{U}_{\infty }{w}_{o}a\left(\frac{\left({\tilde{\tau }}^2+\tilde{\tau }\right)}{{\tilde{\tau }}^2+2.82\tilde{\tau }+0.8}\right)H\left(U_{\infty }t/a\right),\tilde{\tau }=\frac{U_{\infty }t}{a}$

These results are compared in Fig. 7.8 and there is good agreement when the nondimensional time is less than one.

In conclusion the response of a blade to different types of gusts including a blade vortex interaction, a harmonic upwash gust and a step gust, is well approximated using Eq. (7.3.4) and Howe's approximation, which assumes the unsteady load is defined by the pulse that occurs when the disturbance is close to the leading edge of the blade.