Journal of Sound and Vibration

Localization of aeroacoustic sound sources in open jet wind tunnel experiments requires an accurate prediction of the acoustic propagation time. Most conventional predictions use either a ray-tracer, coupled with a modelled continuous velocity field, or use ray diffraction and a discretization of the velocity field by means of vortex sheets. In this work a novel method is proposed in which the continuous velocity field is discretized into blocks of constant velocity separated by velocity discontinuities, thus removing the requirement for the velocity to be parallel to the surface that separates the blocks. The acoustic ray is solved by minimization of the acoustic propagation time. The computational effort is low compared to ray-tracing methods while maintaining an improvement in accuracy compared to methodologies using vortex sheets. A specific continuous velocity field is derived that models a self-similar shear layer expanding asymmetrically from a rectangular nozzle. Subsequently, this velocity field is discretized to compute the acoustic rays. Experimental results with a loudspeaker source placed in the open jet of a large industrial wind tunnel showed a decrease in source localization uncertainty compared to techniques based on vortex sheets. This is attributed to the inclusion of the shear layer slanting.


Introduction
Society requires more renewable energy production from wind turbines.In addition there is increased demand for transportation of goods and people by aircraft.Both wind turbines and aircraft should be as quiet as possible.To achieve this the design of wind turbines and aircraft is aided by experiments performed in aeroacoustic wind tunnels.The wind tunnel experiments entail the recording of the sound by arrays of microphones which allows to predict and study the noise emitted from the final product.Microphone signals may be processed further in order to localize the sound sources.The localization algorithms rely on accurate predictions of the acoustic propagation time from the sound source to the microphones.Computation of an accurate propagation time with minimal computational effort is therefore an important step towards reduced noise pollution.
Performing the experiment in aeroacoustic wind tunnels is important because these wind tunnels combine aerodynamic and acoustic characteristics essential to the development of silent technologies [1,2] and research concerning aerodynamic sound production phenomena and noise abatement techniques [3].Aeroacoustic facilities support the development of aviation [4][5][6], wind energy [7][8][9] and the emerging industry of urban aerial mobility [10,11].Aeroacoustic testing facilities feature different types of test sections, namely, open, closed and hybrid [12][13][14].Each test section type has specific advantages and limitations [15,16], yielding different results [17,18].In particular, aeroacoustic wind tunnels with an open test section are used to measure the far-field radiation of a sound source under free-field conditions.In the far-field the distance  to a compact source is much larger than the wavelength

𝑐 𝑒
Effective speed of sound [-]   Thermodynamic speed of sound [ms − Dimensionless time [-] and the pressure decays as 1∕.In general, aeroacoustic noise sources do not radiate sound equally in every direction.Therefore, an open test section is ideal for measuring the far-field radiation because the wall can be acoustically treated to avoid reflections.This simulates a free-field environment whereby only the noise characteristics of the subject are measured.Furthermore, the microphones can be located sufficiently far from the noise source at different observer angles, making the open jet flow configuration the most adequate for the quantification of the far-field radiation of the aeroacoustic sources.Large-scale industrial wind tunnels are necessary for testing new products at a high Reynolds number, e.g., aeroacoustic tests on aircraft models [19].The dependence of the acoustics on the Reynolds number is important both for Aeolian tones and broadband noise [20].For a simple Helmholtz scaling the frequencies of interest can become so high that it becomes a challenge to accurately measure the acoustics of the source due to the presence of background noise and wave-turbulence interactions.Even for a moderate scaling of 1:10 the frequencies can already span a range up to 50 kHz while the model span is still in the order of meters.Large wind tunnels also allow for full scale tests on many transportation vehicles, e.g., trucks and trains [21], or wing sections from a wind turbine [22].This eliminates any scaling uncertainties or geometry related fidelity uncertainties, e.g., missing bolts or slots.
Correcting the acoustic measurement for the presence of the wind tunnel shear layer is essential for an accurate prediction of the free-field radiation of the full-scale product.Fig. 1(a) shows a schematic of a typical test with an aircraft model placed in the wind tunnel jet and the microphone array standing in the quiescent air.The jet and quiescent air are separated by the shear layer.For large industrial wind tunnels the shear layer thickness can reach up to 1 m.To compare acoustic wind tunnel measurements to free-field radiation emission-angle corrections are applied.Furthermore, phased array techniques, in particular the beamforming method, use predictions of the acoustic propagation path to localize the sound sources and quantify the source power.Generally, more accurate predictions of the acoustic propagation time decrease the uncertainty of the sound source location.
The current state-of-the-art correction methodologies for acoustic propagation can be classified as purely analytical methods, computational aeroacoustics (CAA), and hybrid methods, sometimes coupled with computational fluid dynamics (CFD).Conventional methodologies assume the acoustic pressure to be small and therefore use the theory of linear acoustics.Additional simplifications can be made by assuming a high frequency sound source.Within this high frequency assumption the Eikonal equation [23] describes the propagation of the acoustic wavefront.The characteristics of the Eikonal equation equal the paths of the acoustic rays, hence the approach is also known as geometrical acoustics.The Eikonal equation's characteristics can be solved as an initial value problem in which a numerical solver iteratively searches for an initial wavefront direction at the source location for which the ray passes through the observer location [24,25].This leads to similar methodologies as found in optics relating to Snell's law or Huygens principle, where either the ray or wavefront is traced through the flow.The other possibility is to solve a boundary value problem and solve the ray as the path of stationary action (or minimal acoustic delay time) [26,27], known in optics as Fermat's principle.Depending on the modelling of flow, e.g., continuous or discrete, and the exact formulation both methods may offer benefits over the other in terms of simpleness, computational effort or accuracy.Gottlieb [28] derived analytical solutions to compute the phase of the acoustic pressure generated by a sound source near a velocity discontinuity, specifically a vortex sheet.When the shear layer is thin and approximately planar the flow can indeed be modelled as a vortex sheet, e.g., in the correction methodology of Amiet [29,30].The results of Amiet were confirmed by experimental work performed by Plumbee [31], Ahuja [32], and Bahr [33].In addition to experimental work in a 0.76 m × 1.07 m wind tunnel Plumbee also performed a numerical study to assert that the influence of the shear layer thickness on the refraction angle was within 1 • , and thus neglectable in the academically scaled wind tunnel.For radially symmetric shear flows, with a thin shear layer, Morfey [34] derived an analytical expression for the propagation time based on the vortex sheet assumption.Porteous [35] generalized the vortex sheet approach and derived an analytical solution for any convex vortex sheet.Porteous computed the ray paths using the least acoustic time formulation, which resulted in a computationally inexpensive methodology.The parameterization of the convex vortex sheet was independent from the wind tunnel -coordinate.This limits the extend of the method, e.g., flow models for an expanding shear layer or displaced potential core are inherently dependent on the wind tunnel -coordinate.
A semi-analytical solution for continuous flow with radial symmetry has been derived by Candel [36] and Tam [37].Candel [38] also derived a similar solution for a shear layer with finite (constant) thickness.In order to reduce the computational effort of raytracers Sarradj [24] used a ray-casting method to compute the delay-time at a few points and interpolated the results between these points.Padois [39] compared the Amiet methodology with results obtained from solving the linearized Euler equations.The numerical predictions compared favourably with an experiment in which a loudspeaker was mounted in the wind tunnel wall.A CAA solution was presented by Redonnet [40,41] to compute the refraction effects for a planar shear layer and a radial symmetric shear layer.Casalino [42] performed a finite element method computation, presenting a discretization scheme for the Lilley-Goldstein acoustic analogy for sound propagation through a non-isothermal axial symmetric jet.Jiao [43,44] carried out combined CFD and CAA computations to study multiple propagation phenomena including: the refraction of sound by a finite shear layer, a displaced shear layer, and wavefront distortion due to the shear layer turbulence, using a synthetic turbulence method.
Observing the current state-of-the-art there is a clear spectrum of computational schemes available.One side of the spectrum is focused on detailed acoustic propagation descriptions often aided by CFD and CAA, whereas the other side aims to deliver simple and computationally inexpensive solutions at the cost of lower accuracy.The literature is consistent in the claim that the inclusion of the shear layer thickness results in small angle of arrival differences.However, inherent to the large scale of industrial wind tunnels is the presence of a thick shear layer which is combined with high accuracy standards.Combined, this means that a small error may still be relevant simply due to the large wind tunnel scale.Furthermore, realistic shear layers expand asymmetrically around the lip-line and change shape as function of the wind tunnel -coordinate, which should be included in the flow description.Lastly, a simple mathematical description to model the merging of finite thickness shear layers has not been found in the literature.
In this paper a correction methodology for arbitrary jet flows is developed.With the methodology acoustic delay times are efficiently computed.This is mainly beneficial for beamforming codes.The methodology alleviates the need for vortex sheets and instead subdivides the velocity field into regions with an uniform velocity using velocity discontinuities.The methodology is the used to assess the effect on the acoustic propagation time for experiments in large industrial wind tunnels, with thick merging shear layers, as shown in Fig. 1.A flow model is developed that includes the shear layer thickness, the asymmetric expansion and merging of shear layers.

Ray-tracing with time minimization
The acoustic pressure waves are assumed to have a low amplitude [45] and a high frequency.With these assumptions the sound propagation can be modelled with the Eikonal equation [23], also known as geometrical acoustics.Within the framework of geometrical acoustics the solution is the acoustic ray travelling from source to observer.The ray can be solved with multiple methods.In this work solutions are based upon the minimization of the acoustic propagation time, also know as Fermat's principle.Fermat's principle defines the acoustic ray trajectory as the path of least travel time between two locations.Fig. 2 shows a sound ray connecting points  and .There are an infinite amount of rays that connect these points.However, the principle of Fermat states that the physical path is the path of least propagation time, i.e., the fastest path between  and .The time minimization scheme offers a simple method to solve for the acoustic rays and the acoustic propagation time in the rest of the paper.This section summarizes how Fermat's principle yields the same solution as obtained from a conventional ray-tracer for continuous velocity fields.The explanation is based on the work of Uginčius [26].

Sound propagation & sound rays
Suppose that a ray r tracks a surface of equal phase Φ as function of time  while interacting with the averaged velocity field  .A reference length  and the thermodynamic speed of sound   , assumed constant, are used to define the following non-dimensional quantities: Fig. 3 shows a surface of constant non-dimensional phase  (the wavefront) with normal vector .The Eikonal equation approximates the wavefront phase function in a fluid and in non-dimensional form is given as: ( Eq. ( 2) is a first order non-linear partial differential equation valid for acoustic waves of small amplitude.The local flow velocity is denoted by the Mach vector , the ray path () depends on  the equivalent time parameter.The Cauchy method of characteristics, see Evans [46](Ch 3, pg.91), describes the transformation of a first order partial differential equation into a set of ordinary differential equations.The Eikonal equation is first written in the standard form as: where the normal vector  = ∇ = ∕.The standard form defines a functional  as function of the location (), the local phase  and the gradient of the phase denoted by , and  must equal zero.For the fully non-linear case the characteristics are computed according to the expressions: Substituting  from Eq. (3) in Eq. ( 4) yields the characteristic curves parameterized by : The modulus of  follows from Eq. ( 2) as: Eq. ( 7) allows to simplify Eqs. ( 5)-( 6) into: which are 6 ordinary differential equations describing the change in ray position and the change in wavefront normal.With a given Mach vector field  the ray path is found by integrating Eqs. ( 8)-( 9) simultaneously.The integration is often performed numerically.

Ray path equation
Eqs. ( 8)-( 9) can also be compacted into a single equation.In this manner the resultant equation can be rewritten into one part that is explicitly changing with time and another part that is depending only on the location.The result, Eq. ( 16), allows to simplify the mathematics necessary to obtain Fermat's principle of acoustic propagation time minimization in Sections 2.3-2.4.Solving  from Eq. ( 8): and then computing the dot product of Eq. ( 10) with  yields: from which the modulus of  is solved explicitly in terms of the ray direction and Mach vector: with the Prandtl-Glauert factor defined as: Substituting Eq. (10) in Eq. ( 9) gives: An equation containing only the ray direction ̇ is obtained by rewriting Eq. ( 14) with the use of Eq. ( 10) and ( 12): The left hand side contains an expression whose value changes with time, whereas the expression on the right hand side is changing with position.

Lagrangian mechanics
In this section the mathematical framework of Lagrangian mechanics is used to derive a minimization problem that yields the acoustic ray path.In Lagrangian mechanics the total action is defined by the functional  as: Fig. 2 shows the ray path  with end points  and .The functional  is the total action along the path.To distinguish the physical path from all possible paths between  and  a further restriction is necessary.For many physical phenomena, including the propagation of sound, the solution to the mathematical models should only depend on the boundary conditions at  and , and not on the path between them.The true path  will then be the path of stationary action, where small changes in the path will cause no change in the total action.This condition is mathematically formulated by equating the functional derivative of the total action from Eq. ( 17) with respect to the change in path equal to zero: The functional derivative is denoted by .Eq. ( 18) constitutes the Euler-Lagrange (EL) equation: The classical example to introduce the EL equation is to derive Newton's equation of motion from the difference between the specific kinetic energy and the specific potential energy for a point mass in a gravity potential: The total specific energy is conserved, hence the difference is zero.Substituting Eq. ( 20) into Eq.( 19) then yields: i.e., the point mass is accelerating towards the centre of gravity.Since the Lagrangian is only dependent on the field variables and time it can also be applied to continuous problems.The EL equation, Eq. ( 19), is similar to Eq. ( 16) and thus the acoustic Lagrangian  is found by equating the left and right hand sides.Starting with the right hand side.
Written in this way the quotient rule for differentiation can be recognized: where By integrating Eq. ( 23) it is found that: Now for the left hand sides the relevant expression becomes: which yields: Upon integration it is found that: Which is indeed equal to the Lagrangian derived by Uginčius.The acoustic energy density is split into the potential energy density and the kinetic energy density [47] (page 4, Eq.(1.6)) [48] (Eqs.( 23)-( 26)).

Fermat's principle
To show that the acoustic rays follow from Fermat's principle the Lagrangian from Eq. ( 29) is substituted into the minimization problem defined by Eq. (18).Secondly, the integration is performed with respect to the arc length  instead of the non-dimensional time parameter .This transform can be found in Appendix A. In the first step the Lagrangian  is written in terms of  such that the ray changes with respect to d, i.e., in terms of: Subsequently, an expression is found that relates d to d.The result is: The integrand on the right hand side of Eq. ( 31) is the path segment d divided by an effective speed of sound equal to: More formally this is can be rewritten with the help of Eq. ( 18) as: where time infinitesimal d is simply the distance infinitesimal d over the effective speed of sound   , and  is the total propagation time.This is Fermat's principle, which states that the total propagation time integrated over the path going from  to  is the minimum propagation time.The rays obtained from a ray-tracer are thus also the rays with minimal propagation time between source and observer.

Ray refraction in a discretized velocity field
The previous section demonstrated that the acoustic ray follows the path that minimizes the total propagation time.This principle can be used to compute the acoustic path in complex shear flows, e.g., as shown in Fig. 4(a).The acoustic path is used to determine the source location and strength from microphone measurements.The method proposed by Amiet [29] subdivides the flow field using a vortex sheet.This works well for thin shear layers when the flow velocity in -direction is small.When these assumptions are not valid a different approach is proposed.Fig. 4(b) shows a few isocontours for the velocity components corresponding to Fig. 4(a).The flow field is discretized into blocks in which the velocity vector is constant as illustrated in Fig. 5.The geometry of the blocks is based upon the flow topology shown in Fig. 4(b).The discretized flow should approximate the flow in the region where the acoustic rays propagate using a minimum number of blocks.The latter requirement assures a low computational cost.In Section 4.1 a continuous self-similar shear layer flow is discretized into three blocks and the interfaces are determined such that the difference between the discretized flow and the continuous flow model is minimized.The coarse discretization of the flow is used in conjunction with Fermat's principle to obtain discrete approximations of the acoustic ray.The different (non-physical) rays to optimize over are constructed by moving the interface points along the block interface as shown in Fig. 5.The angle of refraction is found implicitly by minimizing the total propagation time.A numerical scheme is constructed as follows: at every interface a new intersection point (  ;   ) is created.The intersection points are then connected to form a ray.The total time  is simply the sum of all the segment lengths divided by the effective speed of sound.
The effective speed of sound, defined in Eq. ( 32), depends on the Mach vector in the block as well as the ray direction.The ray direction equals the ray segment direction.The acoustic ray path is found by changing the ray until  is minimized.For the optimal acoustic path a small change  does not change  (Fermat's principle).A starting point at the source and an end point at the microphone closes the model.A note of caution is that, depending on the velocity field, caustics or reflections may be present.A caustic represents the case where acoustic rays are crossing (the solution becomes multi-valued) and the Eikonal equation is no longer valid at the caustic.Whether caustics are present depends on the specific flow discretization.Using Fermat's principle one effectively solves for the viscosity solution [49,50], i.e., one solves for the minimum acoustic time, which can always be uniquely defined given that the location can be reached by the sound wavefront.

Refraction on a vortex sheet
The theory from the previous section is applied to the simple case where a vortex sheet describes the velocity field.Fig. 6 shows a vortex sheet and ray paths obtained with Fermat's principle, the Amiet theory and Snell's law (in Appendix B).This simple case allows to show that these three solutions are the same mathematical model viewed differently.

Fermat's principle
Starting from Fermat's principle and assuming two media with effective speeds of sound  0 and  1 the total travel time is computed as: with the effective speed of sound defined by Eq. (32).The travel time is minimized by computing the interface locations   and   that give an extremum:

Velocity field discretization
Two particular flow fields are considered to assess the influence of the shear layer thickness, asymmetric expansion and merging shear layer as shown in Figs.1(a)-1(b).The first step is to discretize the simple and well known self-similar shear layer proposed by Görtler [51], which describes a two-dimensional shear layer that grows linear in thickness.Velocity measurements of the shear layer are necessary to measure the two parameters in the flow model.The more complex flow in which four shear layers merge is based on modelling a changing reference contour.The reference contour is rectangular near the nozzle with increasingly rounded corners near the collector.The shortest distance to this reference contour defines the non-dimensional parameter of Görtler's shear layer model.Note that it is not the intent to accurately simulate a shear layer.The main purpose is to provide a simple velocity field description that can be discretized to predict the acoustic propagation time, as an alternative to the planar vortex sheet.Computational effort has been one of the reasons to choose the time minimization scheme, therefore, a table comparing the computational effort of the flow models and solving methods is provided in Appendix C.

Self-similar Görtler solution
The self-similar shear layer flow model as derived by Görtler [51] is equal to: where  and  0 are constants depending on the actual wind tunnel shear layer. describes the linear growth rate, whereas  0 represents the expansion of the potential core.Plumbee [31] also used this expression, however, the variable  0 was set constant equal to  0 = 0.297 and in the numerical part  = 13.5, which is a thin shear layer [52,53].Fig. 7 shows the flow subdivided into three regions.The regions are defined by two straight lines, parameterized by  0 and  1 .For a symmetrically expanding shear layer ( 0 = 0)  0 and  1 are found by solving the following integrals: for η.The integrals equal the difference in velocity between the discretized model and the continuous velocity flow while taking into account the symmetry of the error function.A value of η ≈ 0.6 solves Eq. ( 46), which corresponds to a velocity thickness of 80% with respect to the freestream velocity.For the case  0 ≠ 0 the mean shear layer slope is simply added to  0 and  1 .Modelling the shear layer geometry in this manner differs from, e.g., the approach of Candel [38], who assumed the interfaces parallel to the flow direction.The numerical model is compared to the solution from Amiet who used a vortex sheet to model the velocity field.Furthermore, a ray-tracing code is used to compare the discretized results against the continuous flow model.One set of observers is at ∕ = 0 under the source located at ( = 1;  = 0;  = 0)∕.A second set of observers is located offset from the wind tunnel axis at ∕ = 1.3.With  approaching infinity and  0 equal to 0 the Görtler solution reduces to a vortex sheet.The results presented in Fig. 8 show that for this case the results equal the results computed with the Amiet method (see Section 3.3).The results are made non-dimensional using the propagation time without any flow effects, i.e., the results are divided by ∕.Increasing the complexity by adding a shear layer that grows in thickness the value for  is set to 9. The increase in shear layer thickness causes the acoustic delay time computed with the discretized flow model to differ from the results of Amiet, this relative time difference is shown in Fig. 9.The largest difference with Amiet's method is observed downstream, this is where the shear layer is thickest.The discretized model shows the same trend as the ray-tracer demonstrating an increase in accuracy.Most shear layers tend to expand asymmetrically with respect to the nozzle lip-line.This is accomplished by setting  0 equal to 0.31.Fig. 10 shows the numerical results obtained from these settings.Comparing with Fig. 9 it is noted that the displacement of the shear layer is an order of magnitude more of influence than the shear layer thickness.

Cross-section of the jet potential core
The two-dimensional shear layer assumption is only valid if the acoustic rays go through a two-dimensional shear layer, see e.g., array A in Fig. 1(b).The four shear layers, originating from the nozzle edge, will grow in thickness and start merging.This creates a flow that may no longer be regarded as two-dimensional since the velocity profiles in the corners are expected to blend into each other due to the growing shear layer thickness.This is modelled with an empirical model using a reference contour that changes as function of the -coordinate.The reference contour equals the lip-line at the nozzle and changes shape when moving downstream.The reference contour is modelled by the expression: where  is a parameter that controls the shear layer merging (velocity contour rounding), and  and  are coordinates located in a plane at a specific  location.The constants  and  are scaling parameters dependent on the wind tunnel nozzle dimensions, half the width and height respectively.Fig. 11(a) shows that varying the value of  smoothly as function of the downstream coordinate allows the curves to span a smooth surface.For small values of  the shape is approximately rectangular, e.g., near the nozzle, while   for  = 1 the shape is an ellipse.A distance  is now defined as the shortest distance to the reference contour.Fig. 11(b) shows iso-lines for the distance  with a particular reference contour.
The shortest distance  is then used as the  coordinate in the self-similar solution (compare Eq. ( 45)): The definition of  in Eq. ( 48) is similar to the two-dimensional self-similar flow definition.However, as the shear layer thickness grows it allows for the velocity contours to be curved.The reference contour (where  = 0) can be parameterized explicitly by a non-dimensional distance from the nozzle  with range [0; 1] and the in-plane tangential parameter  with range [0; 2].Using  = sin  and  = cos  the definition of the contour in Eq. ( 47) is rewritten as: where (, ) is the surface and  is the characteristic wind tunnel length.This formulation is more convenient if the flow is to be subdivided into regions of constant velocity.Fig. 12 shows the resulting interfaces after discretizing the velocity field.The expression for  should reflect the actual wind tunnel velocity profiles.One may chose  equal to: which yields a smooth surface (, ).For practical applications the   are fitted to velocity profiles obtained from experiments, e.g., with a Pitot-static tube as performed by Biesheuvel [54].The work also provides a numerically robust manner of describing   and   in Appendices A and B. Fig. 13 shows acoustic delay times obtained by letting  approach infinity and setting  0 equal to 0. Below the wind tunnel axis this setup is effectively equal to Amiet.Since the vortex sheet does not depend on the -coordinate the approach is equal to that of Porteous [35], with the shape of the convex vortex sheet defined by Eq. (49).The largest difference with respect to Amiet is found when the observers are placed offset from the wind tunnel axis and the acoustic rays travel through a curved shear layer.Modelling the linear growth of the shear layer by setting  equal to 9 gives results as shown in Fig. 14.The results at ∕ = 0 equal the two-dimensional shear layer approach.The results at ∕ = 1.3 are now representing the influence of both the finite thickness as well as the merging of the shear layers.Setting  0 = 0.31, as in Fig. 15, the numerical results now include all three effects; finite thickness, asymmetric shear layer expansion and merging.The differences due to the asymmetric expansion and merging are of the same order.The asymmetric expansion increases the acoustic delay, whereas the merging effect decreases the acoustic delay, and partly cancel each other.

Experimental results
This section first describes the validation of the simplified flow model that describes the shear layer merging.For practical reasons these measurements are performed in an academically scaled wind tunnel.The effect of the merging shear layers, as presented in Fig. 1, on acoustic images is subsequently assessed for an industrial scale facility.In the latter test a custom speaker source was placed in the wind tunnel.

Anechoic wind tunnel with an open jet test section
Experiments performed in the Anechoic Wind Tunnel (NLR-AWT) of the Royal Netherlands Aerospace Centre support the validation of the flow model presented in this paper.The nozzle dimensions are 0.95 × 0.95 m 2 .The velocity profile in the shear layer flow is mapped by a Pitot-static tube traversing multiple sections of the wind tunnel.Subsequently, the interaction of the four shear layers originating from the nozzle is analysed.Fig. 16(a) shows the measured velocity profile of the shear layer.Results were obtained for Mach numbers  ∞ = 0.12 and  ∞ = 0.17 at 0.5, 1.25, 2.0 and 2.75 m downstream from the nozzle.The self-similar parameters are listed in Table 1 and computed by fitting the velocity field model to the measured shear layer profiles.On a course J. Biesheuvel et al.   grid the flow was scanned with the Pitot-static tube for Mach number  ∞ = 0.12.This measurement is performed at the same 4 downstream locations.Resulting flow contours are shown in Fig. 17.For each section the optimal value of  is found that minimizes the difference between the model velocity and the experimental velocity field.The results showed that the effective width  and height  are not identical to the geometrical width and height of the nozzle.Following, Eq. ( 50) is fitted to the optimal p values shown in Fig. 16(b).Fig. 17 also shows the velocity distribution produced by the fitted model.It is observed that for this wind tunnel the rounding of the velocity contours is mainly due to the spreading of the shear layer and the iso-contours as described in Fig. 11(a) remain mostly square.The resulting coefficients are listed in Table 1.The Pearson's correlation coefficient between the merging shear layer flow model and the experimentally measured velocity distribution was 0.98.

Speaker in a large-low velocity facility
A large scale wind tunnel test was performed in the DNW-LLF wind tunnel with a test section of 8 × 6 m 2 to demonstrate the applicability of the correction methodology and assess the effects of shear layer thickness, asymmetric expansion and shear layer merging.The measurement setup can be seen in Fig. 18.A custom speaker source has been placed near the centre of the test section of the wind tunnel instead of an aircraft model as in Fig. 1.  , 0 • and +30 • direction relative to the negative -axis.For each loudspeaker the input signal was controlled independently.The geometric source location was determined using a theodolite.With the geometric location a beamforming grid was defined.Subsequently, the reference source location was determined using beamforming and a measurement with the wind tunnel turned off.A small difference of 2.5 cm between the geometric location of the source and the beamforming location was found in the -direction.This difference was the identical for all loudspeakers.The speaker source was driven with a broadband white noise signal filtered to the range of 5 kHz to 20 kHz.Acoustics measurements were done with an out-of-flow microphone array.The acoustic array (4 × 4 m 2 ) contained 140 microphones sampled at a rate of 65 kHz (for a schematic representation see Fig. 18(a)).The acquisition time was 45 s per data point.The array was traversable over a distance of −4 m to +4 m.The exact position of the array with respect to the wind tunnel reference is known with a precision of a few centimeters by use of a theodolite.The  parameters ,  0 , and the coefficients in Eq. ( 50) are based on Pitot-static measurements obtained in a 1:10 scale wind tunnel of the DNW-LLF.The numerical values are given in Table 2.The Pearson's correlation coefficient between the merging shear layer flow model and the experimentally measured velocity distribution was 0.99.Fig. 19 summarizes the beamforming results for the experiment.The beamforming process used data blocks of 4096 samples.The 5 kHz band is used for the beamforming process.The array was positioned at three positions: at −4 m, 0 m, and +4 m with respect to the wind tunnel centre.For all three measurements the loudspeaker directed downstream was used.The steering vector used for the beamforming is based on the propagation time correction.This definition ensures that the location of maximum SPL coincides with the source location [55] (formulation I).Since a pressure is not included in the steering vector the relative source power is presented.The acoustic propagation time is computed using Amiet (a) and the method presented, including the effects of expanding merging shear layers.The two-dimensional flow is into 3 regions (b).The three-dimensional (with layer merging) is discretized into 2 regions (c) and 3 regions It is that for the array measurements −4 m and 0 m the methods offer similar results, with the beamformed source power sufficiently close to the reference location.For the array measurement at +4 m the, the processed results with steering vectors based on Amiet's method show a clear difference between the maximum beamformed source power and the reference location.The difference decreases by 3 cm when the shear layer thickness is included.Including the shear layer merging in (c) and (d) shows the maximum beamforming response moving 6 cm closer to the reference location compare to Amiet.When the array is at +4 m the acoustic rays travel through a shear layer that is sufficiently different from the vortex sheet as modelled by Amiet to be noticeable.The difference due to shear layer thickness appears to be neglectable.A small difference can be attributed to the asymmetric expansion and the merging of the shear layers.Furthermore, it is noted that the thick shear layer does not only change the time-invariant wavefront propagation but also the time-dependent wavefront propagation.The interaction of the wavefront with turbulence creates blurred acoustic images due to a loss of microphone signal coherence.This creates further uncertainty on the true source location.

Conclusion
A method based on the discretization of the velocity vector field and acoustic time delay minimization is presented.This method extends the range of experimental configurations for which acoustic time-delays can be computed compared to the current state-ofthe-art.Furthermore, it is able to predict the significance of flow characteristics on the acoustics corrections, e.g., the effects of shear layer thickness and the merging of shear layers.Sound rays are efficiently computed by minimizing the travel time between source and observer.proposed approach is equal Amiet's theory for the baseline of a flow approximated by a flat vortex sheet.However, the novel method allows the acoustic rays to be computed for complex flow fields which cannot be approximated by a vortex sheet.The proposed method is fast when compared to ray-tracers and requires similar computation time as Amiet's approach.
Two flow fields commonly encountered in wind tunnels elaborated.The first represents the finite thickness shear layer and the second a three-dimensional flow that captures the effects of merging shear layers.Pitot-static tube measurements showed that both models adequately represent the flow in an open jet wind tunnel.The solutions produced from Fermat's principle are compared with the numerical ray-traced solutions.The differences in acoustic delay-time compared with Amiet's method are in general small for the wind tunnel parameters adopted in the numerical comparison.The main differences are found when the shear layer location deviates from the flat vortex sheet.Sound source localization inaccuracies arise mainly due to the asymmetric expansion and shear layer merging.An experiment performed in a large industrial wind tunnel, with a nozzle of 8 × 6 m 2 , showed that including the finite thickness and rounding shear layer increases the accuracy.This improved the localization accuracy by 6 cm.

Fig. 5 .
Fig. 5. Flow decomposed in blocks of uniform velocity.The acoustic ray refracts at the interfaces.Of all rays that connect the source and observer (dotted line) the physical ray (solid line) follows from Fermat's principle.

Fig. 18 (
b) shows the speaker source consisting out of an aluminum shell with 3 actual loudspeakers.The loudspeakers face in the −30

Fig. 19 .
Fig. 19.Conventional beamforming with different acoustic delay prediction models for different array locations.Frequency of interest is 5 kHz.

Table 1
Parameters for the NLR AWT.
Fig. 17.Flow velocity in NLR's Anechoic Wind Tunnel, measured with a Pitot-static tube.