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Unstructured grid solution of the eikonal equation for acoustics by Paul G. Tucker and Sergey A. Karabasov

reprinted from

aeroacoustics volume 8 · number 6 · 2009

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aeroacoustics volume 8 · number 6 · 2009 – pages 535–554

535

Unstructured grid solution of the eikonal equation for acoustics Paul G. Tucker* and Sergey A. Karabasov† University of Cambridge, Whittle Lab, 1 JJ Thompson Ave., Cambridge, CB3 0DY, United Kingdom [email protected] Received October 9, 2008; Revised February 17, 2009; Accepted for publication March 25, 2009

ABSTRACT An acoustic eikonal equation solution procedure, that is easy to implement in unstructured grid Navier-Stokes equation flow solvers, is outlined. The approach is readily parallelizable. The method is tested for the following canonical point source cases: quiescent flow; subsonic uniform flow; supersonic uniform flow and an idealized jet flow. Then, as further validation, sound propagation of a wave front through a viscous vortex is considered. For these cases, encouraging agreement is found with analytic data and a high-fidelity numerical solution of the Euler equations. Finally, as demonstration cases, the use of the approach to study the shielding of noise from a planar jet is considered along with wave propagation in a complex three-dimensional geometry. At lower Mach numbers (< 0.25), for complex geometries, robust multigrid convergence acceleration is found.

I. INTRODUCTION The eikonal equation is the high frequency limit to the full wave equation. Since it is the high frequencies that can be most annoying to the human ear, eikonal equation solutions are potentially useful. The human ear is most sensitive to frequencies in the range of 2000–4000 Hz. Hence, since wavelength, λ = a/f, where f is frequency and a sound speed, this corresponds to λ in the range of 0.08–0.16 m. The eikonal equation is based on the ray-tracing theory which is valid for λ > 1. The latter, for example, makes the eikonal approach suitable for long range propagation such as in the design of noise shielding for aeroengines. The general limitation of the eikonal approach, however, is that it is not esoecially accurate for sound propagation through thin shear layers, where the mean velocity gradients are in the order of high frequency range of audible sound. Below we provide a short overview of the previous use of the eikonal equation. Given the mean flow distributions, e.g. from a preceding CFD (Computational Fluid Dynamics) calculation, the eikonal equation will directly give wave fronts and thus the normal component of acoustic rays. The latter can be generated by ray tracing. Blokhintsev, who studied sound propagation in turbulent atmospheric flow [1], notes that the ray velocity is equal to the sum of the fluid velocity and a directed along the wave normal. This directionality information need, which is a common feature for *Professor of Engineering, Whittle Laboratory, Department of Engineering. †Royal Society University Research Fellow, Whittle Laboratory, Department of Engineering.

536

Unstructured grid solution of the eikonal equation for acoustics

hyperbolic problems, makes basic ray tracing rather challenging in a CFD context. Avila and Keller [2] explored rays from a point source when a is constant. Asymptotic expansions were used, the leading term being consistent with the eikonal solution. With aircraft wake and atmospheric vortices in mind, Georges [3] numerically integrated the Haselgrove equation, exploring the propagation of an acoustic wave front through a steady vortex. Colonius, Lele and Moin [4] also consider this case but solving the Navier-Stokes equations and comparing the results with the ray-theory solution constructed by the method of characteristics. Freund [5] solved the eikonal equation in an unsteady form for turbulent jets with the meanflow coefficients based on the results of Direct Numerical Simulation. Khritov et al. [6] and Secundov et al. [7] directly solved the eikonal equation using the method of Landau and Lifshitz [8] for unsteady jet flows representative of those found in aeroengines. Apart from the theoretical studies, the eikonal equation can be a valuable engineering tool in the context of studying acoustic shielding designs. For example, with aircraft, the engines can be located so that the airframe shields observer noise, thus reducing environmental impact (see Agarwal [9], Moore and Mead [10] and Moore [11]). In most of the studies in the literature, either too simplistic geometries are considered or the meanflow effect on sound refraction is completely ignored. In this paper, an easy to implement eikonal solution approach for fully three-dimensional (3D) unstructured grid solvers is outlined that readily admits non-uniform meanflow properties. The main limitation of the method at present is that it neglects the acoustic wave scattering by solid boundaries, which would be important to include, for example, in the problem of sound scattering by a sharp trailing edge. However, despite this drawback, the present method is readily suitable for a wide class of problems of long-range propagation where the effect of sound refraction by non-uniform meanflow dominates over all other effects. Broadly the solution approach borrows ideas from Tucker et al. [12] for the basic eikonal equation and structured grids, which was done in a turbulence modelling context. The current approach allows use of existing, efficiently parallelizable code elements readily available in most unstructured CFD solvers. The same CFD solver can be used, first, to compute the base flow fields and, then, to calculate the acoustic eikonal solution. The novel approach is tested for the following canonical point source cases: quiescent flow; subsonic uniform flow; supersonic uniform flow and an idealized jet. Then the propagation of a plane wave through a viscous compressible vortex and also a typical jet flow are considered, the latter, in a noise shielding context. The base CFD program used is HYDRA (see Lapworth [13]), which is readily suitable for engineering design optimisation studies. The numerical eikonal solution procedure, as discussed in the paper, feeds naturally in the design optimization loop and, therefore, is very attractive for design. II. THE GOVERNING EQUATION AND NUMERICAL SOLUTION A. Governing equation The eikonal equation can be derived in a variety of ways. For example, by performing Fourier transform in space and time of the linearised Euler equations and keeping just the highest order terms in frequency of the acoustic part of the equations (e. g., Colonius at al.) leads to


1 − un L

∂φ% ∂x n

537

2

− L2 a 2

∂φ% ∂x n

2

=0

(1)

~ ~ In the above φ (x) is the dimensionless eikonal where φ = φ L, which is proportional to the propagation time of the sound ray. B. Numerical solution method The eikonal equation solution approach outlined in Tucker et al. [12] is followed. Hence, the eikonal equation is re-written as ∂φ% ∂x n

2

= Sφ%

(2)

where Sφ% =

1 ∂φ% 1 − un L 2 (aL ) ∂x n

2

(3)

Solving the governing Equation (2), which is a non-linear partial-differential equation (PDE), is numerically challenging. Hence, the non-linear PDE is cast into a quasi-linear form, which is more amenable to numerical solution by an iterative time-like method. To do this the auxiliary velocity variable is introduced ∂φ% . ∂x n

(4)

∂φ% = Sφ% ∂x n

(5)

uˆn = Hence (2) becomes uˆn

where the left hand side has a non-conservative advection term. Equation (5) is solved using an iterative time-like scheme, e.g.: ∂φ% ∂φ% ∂φ% + uˆn = Sφ% , → 0. ∂τ ∂x n ∂τ

(6)

For the numerical convergence it is advantageous to move the non-linear term 2  ∂φ%  un M  ∂x  , M = a , n

(7)

which can be stiff for large mean flow velocities and variations of the ray phase, from the right-hand-side source term (5) to the left-hand-side advection operator. Hence (5) becomes

538


1 − M 2 uˆn

∂ % % φ = Sφ% , ∂x n

(8)

where S%φ% =

1 ∂φ% . 1 − 2un L 2 (aL ) ∂x n

(9)

For accelerating numerical convergence when computing flows with high gradients, and to be consistent with most CFD solvers, (8) is cast in the conservative form: ∂ ∂  ∂ % u%nφ% = S%φ% + φ% D φ  ∂x n ∂x n  ∂x n 

( )

(10)

u%n = 1 − M uˆn , D = 1 − M . 2

2

However, for high-subsonic and supersonic flows the diffusion coefficient, D on the right-hand side becomes small or even non-positive, which slows down the iterative convergence. Then for the actual numerical solution (10) is replaced by (11) % ∂φ% ∂ ∂  ∂ % % ∂  C − D ∂ φ%  , ∂φ → 0, (11) h + + u%nφ% = S%φ% + φ% D φ ε ⋅ ⋅ φ ∂τ ∂x n ∂x n  ∂x n  ∂x n  ∂x n  ∂τ

( )

where C = 0.2, h is cell edge length, and ε is a positive constant O(1) for M > 1 − C . and zero otherwise. Then at convergence the non-homogeneous convection-diffusion equation (11) approximates the governing equation (2) to the same order of accuracy as of the CFD solver (discussed in section C) for M ≤ 1 − C , and to the first order of accuracy for M > 1− C. Since existing optimised program elements are used for solving the convectiondiffusion problem, the solution approach is readily implementable and parallelizable. Equations (11) and (4) need to be solved together in an iterative loop. Hence, an initial ~ guess to φ is needed prior to this loop. This initialization is discussed later. Isosurfaces ~ of the resulting converged φ field gives acoustic wave fronts. The direction cosines of the wave fronts are given by ∂φ% ∂x n

nn =

∑ m

∂φ% ∂x m

2

(12)

The characteristic velocity of ray propagation, Un, is equal to the sum of the fluid velocity and the speed of sound directed along the wave normal (e.g., [1]) as below


539

uˆn = un + a nn

(13)

Hence, ray trajectories can be obtained from dx n = uˆn dt

(14)

A key difficulty with simple direct ray tracing is the evaluation of nn. These issues, in a parallel computing framework are discussed in detail in [14]. Stability measures Since the range of ûn is governed by the characteristic equation uˆn =

1 ∂φ% , = ∂x n L (a ± un )

(15)

for stability in the vicinity of sonic points, the value of ûn is limited in the convection fluxes (11), following the analogy with flux-limiting in shock-capturing schemes (e.g., [15]). Hence, the following condition is used uˆn ≤ k1

(16)

C% aL

(17)

where k1 = ~

Hence, here as a compromise C = 20, which corresponds to clipping activated in (15) for 1 − M < 0.1.

(18)

For more sophisticated clipping procedures, equation (15) could be directly used but tests showed that tight clipping limited strong corrective traits inherent in the iterative solution with negligible stability benefit and slower iterative convergence, and so we haven’t attempted it in this study. It can be noted that close to convergence the limiting (17) doesn’t lead to any convergence stagnation problem. Initialization For initialization, the following is used L φ%init = max aL

(19)

~ ~ where Lmax is the maximum system dimension and φ init the initialized φ distribution. ~ Generally the initialization does not matter and even φ init = 0 yields convergence.

540


C. Numerical discretization Full details of the numerical discretization can be found in [14]. For the convective term a blending between the first-order upwinding and the second-order central differencing is used. Biasing the numerical stencil towards the first-order upwind finite-differences reduces the numerical dispersion error, and the increase of biasing towards the central differences helps in decreasing the dissipation error. The diffusion term is discretised with second-order central differences. D. Convergence acceleration To solve the discretized equations the current eikonal solution makes use of an explicit Runge-Kutta procedure. Due to the CFL (Courant-Friedrichs-Lewy) constraint it is well known that this explicit method results in extremely slow convergence. For a standard Runge-Kutta scheme, a wave front can’t be propagated more than one grid cell at a time for each sweep of all the grid points in the domain. Slow iterative convergence has given rise to specialized techniques for solution of the eikonal equation, e.g. [16]. However, multigrid convergence acceleration is a popular method for dealing with the CFL based limitations arising with explicit flow solvers. Hence, this approach is made use of here. Full details of the multigrid method can be found in Moinier [17]. Also, to improve initialization the solution of equations on a coarse grid, interpolation of the solution to a next finer grid and so on until the finest base solution mesh is reached is used. The solution is then switched to a standard multigrid (V-) cycle. E. Validation setups The following validation cases are considered: I. Point source in quiescent flow (un /a = 0); II. Point source downstream of a line source in subsonic flow (u1 /a = 0.845); III. Point source in supersonic flow (u1 /a = 1.414); IV. Point source in a jet and V. Line source interacting with a vortex. These cases are set around the domain of width W shown in Fig. 1a. The point source, for cases (I–IV), and vortex, for Case (V), are located at the centre of the domain (xn = 0). The case (II) and (V) line source is positioned vertically at Boundary (A) i.e. at x1 = –W/2. For Case (I), the analytical distribution with respect to x1 for x2 = 0 is

φ% =

x1 L (a ± u1 )

(20)

where u1 = 0. For Case (II)

φ% =

(W 2 + x ) n + x 1

1

L (a + u1 )

1

n3

+

x1 n2 L (a − u1 )

(21)

where for x1 > 0, n1 = n3 = 0 and n2 = 1; for x1 < e, n2 = n3 = 0 and n1 = 1 and for e < x1 < 0, n1 = n1 = 0 and n3 = 1


541

(a)

(b) D

C

A X0 = 0

U1

B

W

X2

X1

Figure 1:

Schematics of the domain and grid topology: (a) domain & (b) grid topology.

where e=

W 1 1 − 2(a + u1 ) a + u1 a − u1

(22)

For Case (III), a Mach cone is expected with a cone angle

α = sin −1

a u1

(23)

For Case (IV) the analytical solution of Pierce [18] is used. For this, the invariant with x1, jet velocity is expressed as 2 u 1  u   u1 = a  o −  o + 1 α 2 r 2    a 2 a 

(24)

where uo (=287.45 m/s) is the jet centreline velocity and α is a constant. For this velocity profile the ray path can be approximately expressed (discarding small terms in αr2 and ξ2 - see next equation) as r=

ξ sinh(α x1 ) α

(25)

In the above, ξ is the initial slope of the ray (∂r/∂x1). The modelling embedded in Equation (25) gives a partial explanation of the zone of relative silence found downstream of jets and close to the jet axis.

542


Table 1. Velocity field and boundary conditions.

Case (I) (II) (III) (IV) (V)

un un/a = 0 u1/a = 0.845, u2/a = 0 u1/a = 1.414, u2/a = 0 u1/a (Eqn. (24)) u2/a = 0 See Eqn. (26)

Boundary condition (A) Extrapolation ~ φ =0 ~ φ = 2/aWL Extrapolation ~ φ =0

Boundary conditions (B-D) Extrapolation `` `` `` ``

F. General numerical setup and boundary conditions Figure 1b gives the grid topology for Cases (I–IV). The domain is of side W = 5 m and a = 340.113 m/s. The Frame (b) grid involves around 50000 nodes. For Case (IV) a cylindrical domain might at first suggest itself. However, since the rays have planar behaviours this is not necessary – the result will not change for a cylindrical domain. For Case (V), the grid count is similar to that for Frame (b) but the structure is uniform and Cartesian. Table 1 summarises some velocity field and boundary condition information. Generally, simple linear extrapolation was used at acoustic ray outflow boundaries. For Case (III), at Boundary (A) there is supersonic inflow and the Dirichlet boundary condition ~ φ = 2/aWL is used. This is the exact eikonal equation solution value for un = 0 at x2 = 0. The specification of sound sources is very straightforward. The Dirichlet boundary ~ condition φ = 0 is sufficient. Hence for cases (II) and (V), to express a planar wave front, ~ ~ φ = 0 is used at boundary (A). Similarly, at the point source in cases (I–IV) φ = 0. G. Numerical solution of the Euler equations In order to examine the advantages and limitations of the novel eikonal approach in more detail, the test cases (III) and (V) have been also solved using the full system of Euler equations in two-dimensional Cartesian coordinates. The governing Euler equations are solved with the second-order CABARET method (Karabasov and Goloviznin [15]). The dispersion error of the CABARET scheme is as low, as that of the fourth order optimized – six order central schemes [19], for a wide range of wavenumbers and CFL numbers. III. DISCUSSION OF RESULTS A. Cases (I, II) - point source in quiescent flow and line source in subsonic flow Figure 2, frames (a) and (b), show results for cases (I) and (II), respectively. Recall, Case (II) has two sound sources. These are the point source and an upstream line source. The rays from these sources will collide forming a shock analogous feature. Wave fronts ~ ~ are obtained by directly contouring φ . The differentiation of φ , via Equation (4), and production of stream traces generates the orthogonal ray traces shown in frames (a) and (b).


(a)

2

543

(b) 1

1.5

0.5 Y

Y

1

0

−0.5

0.5

−1 0

0

0.5

1

1.5

−1

2

−0.5

X

Figure 2:

0.5

0

1

1.5

X

Results for cases (I) and (II): (a) Case (I) wave fronts and (b) Case (II) wave fronts.

0.3 Analytical U/a = 0. Analytical U/a = 0.845 Numerical, U/a = 0.845 Numerical, U/a = 0.845

0.25

0.2

0.15

0.1

0.05

0

Figure 3:

−4

−2

0 X

2

4

Plot of φ~ against x1 at x2 = 0 for cases (I) and (II).

In Frame (a) the expected Case (I) circular wave fronts can be observed. In Frame (b), the skewing of these point source fronts by the mean flow is shown. Also, in Frame (b), the zone where the left hand specified plane wave front collides with the point source can ~ be seen. This collision creates a compression shock analogues zone. Figure 3 plots φ ~ against x1 at x2 = 0. Comparison is made with analytical φ values given by equations (20–22). The full symbols and long dashed lines represent Case (I) analytical and numerical results, respectively. The full line and open symbols correspond to Case (II) results. As can be seen, there is encouraging agreement.

544


(a)

3

(b)

1.8 1.6

2

1.4 1.2

0

y

Y

1

1 0.8

−1

0.6 −2

0.4 0.2 −1

Figure 4:

0

1 X

2

3

0.2 0.4 0.6 0.8 x

1

Results for Case (III) – point source in supersonic flow: (a) twodimensional view of wave fronts and ray traces and (b) r.m.s. pressure field of the numerical Euler solution for a point source located at (0.2, 1) and radiating at the reduced frequency lsource fsource / a∞ = 0.08 and 30 linear contour lines are shown.

B. Case (III) point source in supersonic flow Figure 4 shows the Case (III) results. Again Frame (a) gives wave fronts and acoustic rays. Frame (b) shows the r.m.s. pressure field computed from the Euler equations with an acoustically compact source. This has a reduced frequency k = lsource fsource / a∞ = 0.08. It is placed in the supersonic free stream of M∞ = 2, where lsource is source radius, fsource is frequency and a∞ is sound speed at infinity. The grid used for the Euler calculation is 61 × 121 (axial x vertical). For the chosen u1 and a, Equation (23) suggests that a 45° ‘Mach cone’ reminiscent solution is expected. The eikonal results show this to be approximately the case, which is also in a good agreement with the numerical Euler solution. However, the problem is highly challenging. A steep, discontinuous shock front is reproduced in the eikonal solution. Checks have confirmed that the shock front is sharpened with grid refinement. Multigrid convergence acceleration has not been obtained for this shock-dominated case, since the largest residuals are localized within a small vicinity around the shock. C. Case (IV) point source in a jet Figure 5 gives results for Case (IV). Frame (a) gives a two-dimensional view of wave fronts and ray traces. Frame (b) gives a comparison of computed ray trajectories with the analytical solution of Pierce [18]. This solution is given by Equation (25). It needs to be stressed that this solution is approximate. It is only accurate for small α and ξ, the latter being the rays initial angle to the x axis. Unfortunately, making these values small


(a)

(b)

4

0

r

Y

2

−2 −4 0

Figure 5:

2

4

6 X

8

10

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

545

0

2

4

6 X

8

10

12

Results for Case (IV) – point source in a jet: (a) two-dimensional view of wave fronts and ray traces and (b) comparison of computed ray trajectories with analytical solution of Pierce (1994).

results in a modest ray deviation and hence a less apparent zone of silence. For the current simulations α = 0.14 chosen. This gives a jet velocity profile with a reasonable spatial variation and hence zone of silence. The Frame (b) curves are nominally for ξ = 0.0125, 0.025, 0.0375, 0.05, 0.0625 and 0.075. From Frame (a), the jet directivity pattern corresponding to the zone of silence can be seen. For Frame (b), ξ could not be directly set for the eikonal computations. Instead, to assess predictive accuracy, ξ values that matched the r values given by Equation (25) around X ≈ 11 are used. For ξ ≤ 0.05 the computations and analytical solution are in reasonable accord. However, for larger ξ, deviation is apparent. Because r is matched to the analytical at X ≈ 11 this deviation becomes apparent nearer to the acoustic source. D. Case (V) plane wave propagating through a vortex Next acoustic ray deflection for a plane wave passing through a vortex is considered. The vortex is specified using vθ vθ , max

= 1.4

ro −1.26r 2 1 − exp r r02

(26)

given in Georges [3]. In the above vθ and vθ, max are the local and maximum tangential velocity of the fluid and ro is the location where vθ, max occurs. The radial centripetal force induced pressure gradient is given by the equation below ∂p ρ vθ2 = ∂r r

(27)

The isentropic relation relating pressure and density below is used where γ is the ratio of specific heats, taken here to be 1.4 p = constant ργ

(28)

546


180 160 140

Angle

120 100 80 60

Georges Colonius et al. Best fit Current predictions Euler

40 20 0

Figure 6:

0

0.1

0.2

0.3 Ma

0.4

0.5

0.6

Plane wave front propagating though a viscous vortex for Mamax = 0.075, 0.15, 0.3 and 0.55.

The above two equations are together integrated to yield p and ρ and thus a locally within the vortex. A wave front moving from left to right is considered with a vortex moving anticlockwise. Based on vθ, max the four rotational Mach numbers (Mamax) of 0.075, 0.15, 0.33 and 0.55 are considered. Figure 6 plots the maximum ray deflection against Mamax. The open circles represent the predictions of Georges [3]. The triangles give those of Colonius et al. [4]. The line is a best fit to their data. The square symbols are for high resolution Euler simulations to be discussed further later. The closed circles are for the current eikonal solutions. Estimation of the current Euler and eikonal ray deflections can be found in [14]. With regards to the eikonal solution, vortical flows are frequently used to stringently test the performance of convective schemes. As might be expected the current case places strong demands on the advection scheme. Figure 7 gives a three-dimensional plot of u2 (4) for Mamax = 0.55. As can be seen the velocity field is varying very rapidly in the vicinity of ray focusing (caustics). Away from the caustics, the velocity variation is smooth. The problem is next considered further with the Euler method. This has excellent agreement with the reference Navier-Stokes and ray-theory solutions from Colonius et al. [4] for a range of acoustic wave frequencies. This is illustrated in Figure 8. This gives r.m.s. directivity of the scattered pressure field normalised by the amplitude of impinging acoustic wave. Frame (a) gives comparison with the 6-th-order compact


547

200

0 V

X

100

0

−100 −200 200

100

0

−200

−100

Y

Figure 7:

Vertical velocity component (see Equation (5)) governing ray trajectory for Mamax = 0.55.

(a)

(b)

M = 0.25

2

1.25

Prms/PI

1 0.75 0.5 0.25

1 0.5

0 −200 −100

Figure 8:

1.5 y/L

Current prediction, 400 × 400 Colonius et al, JFM 1994

0 100 θ (deg)

200

0

0

0.5

1

1.5

2

x/L

Validation of the Euler solution in comparison to Colonius at al [4]; r.m.s. directivity of the scattered pressure field normalised on the amplitude of impinging acoustic wave for Mamax = 0.25: (a) - comparison with the reference 6-th-order compact scheme Navier-Stokes solution for the dimensionless acoustic frequency 2π r0 /λ = 2.5, (b) - comparison with the reference ray-theory characteristic solution (dotted lines shows the caustics locations) for the dimensionless acoustic frequency 2π L/λ = 82.5.

scheme Navier-Stokes solution of Colonius for the dimensionless acoustic frequency 2π r0 /λ = 2.5. Frame (b) compares the current Euler results with the ray-theory characteristic solution (dotted lines shows the caustics locations) of Colonius for the dimensionless acoustic frequency 2π r0 /λ = 82.5. We note that because of the low

548


dispersion and low dissipation properties of the CABARET scheme the computational grids required for these test cases are very modest. For example, 5 points per vortex radius for the medium frequency case, 2π r0 /λ = 2.5 and 7 points per wavelength for the high frequency case 2π r0 /λ = 82.5. Note λ is the wavelength of the incoming acoustic wave. Colonious et al. and Georges use Lagrangian type techniques, which allow the crossing of acoustic rays. Hence the front interactions giving rise to direct shock analogous features are avoided. In the current eikonal approach the consistent inclusion of such interactions, for example, to obtain the scattered sound field, would require an intricate implementation of appropriate jump conditions across the caustics in the numerical solution, as discussed in Colonius at al. Alternatively, solution superposition could be used. We have not attempted to account for these effects in the simple Eulerian eikonal solution method described here. Therefore, some deviation of the caustic locations can be expected. Of course, the full Euler model, for the expense of solving additional and more complex equations, does not require any additional modelling to account for caustics. Consequently, with an appropriate numerical method it will pick up the correct caustics behaviour including the location of the bifurcation point, automatically. Figure 9 superimposes the current eikonal ray traces for Case (V) on the nonlinear Euler solutions (r.m.s. scattered pressure fields). The differences resulting from the ability of the current Eulerian approach to allow ray crossing are evident. The loci of caustics obtained by the eikonal method lie between the two Euler predicted caustic branches. For the eikonal solution, the two caustics become ‘averaged’ into a single caustic, as for other Eulerian approaches, e.g. [9–12]. Away from the immediate vicinity of the caustics the eikonal solution is in a good agreement with the Euler model: the ray trajectories are almost everywhere normal to the scattered sound fronts. E. Basic conceptual design Figure 10 is intended to show the potential use of the eikonal equation for conceptual noise shielding design. Frame (a) gives a two-dimensional idealization of a noise shielding (see Agarwal [9]) concept based, for convenience, on a NACA0012 profile. Since surface reflection is not being considered, at solid boundaries generally linear ~ extrapolation is applied to φ . Frame (b) gives the eikonal solution mesh, wave fronts and rays. A point source has been placed at the lower outer shear layer zone (see Frame (a)). ~ As usual this is achieved through the φ = 0 Dirichlet boundary condition. The Frame (b) ray traces show the zone of silence below the airfoil. This simulation is more proof of concept, a uniform axial velocity corresponding to Ma ≈ 0.7 has been imposed over the whole solution domain. This would approximate a very high by pass ratio engine. There is no reason why an initial full turbulent flow CFD solution for un could not have been used instead of the constant velocity field. Indeed this has been carried out for a full aircraft configuration, but these results are not reported here. Since the present method doesn’t attempt to model the sound scattering from solid surfaces and only models the effect of mean flow on sound refraction, numerical boundary conditions which don’t contaminate the eikonal solution by spurious reflections (as opposed to the physical reflection of acoustic waves), which are generated from the solid boundary of the


549

2 2 1.5

1 y/L

y/L

1 0

0.5 0 −0.5

−1

−1 −1.5

−2 −2

−1

0 x/L

1

−2 −2

2

1

1

0.5

0.5

y/L

y/L

1

2

1.5

1.5

0

0

−0.5

−0.5

−1

−1

−1.5

−1.5 −1

0

1

−2 −2

2

−1

0

1

2

x/L

x/L

Figure 9:

0 x/L

2

2

−2 −2

−1

Plane wave front propagating though a viscous-core vortex computed from non-linear Euler solution (r.m.s. scattered pressure fields) superimposed with the ray trajectories of the eikonal solution for Mamax = 0.075, 0.15, 0.3 and 0.55 (from top left to bottom right).

(a)

(b)

Noise source

Figure 10:

Use of eikonal equation for conceptual noise shielding design: (a) twodimensional idealization of shielding concept and (b) eikonal solution mesh, ray fronts and rays.

550


(b)

10

5

5

0

0

Z

10

−5 −10

−5

0 Y

Figure 11:

5 10 10

5

−5

0 X

−10 −10

Z

(a)

−5 −10

−5

0 Y

5 10 10

5

−5

0

−10 −10

X

Three-dimensional geometry: (a) mesh and (b) φ~ contours.

computational domain, must be used. The development of robust and accurate numerical non-reflecting boundary conditions on the solid surfaces, which satisfy this condition consistently with the numerical scheme used inside the computational domain, is a stand-alone task. The numerical boundary conditions used in the present paper lead to reasonable results without much contamination of the sound field, but further work in this direction is underway. In future we envisage that the Eulerian eikonal method will be extended to modeling of reflection from the solid boundaries. For the former, the superposition of eikonal ~ solutions would be needed. In this process the surface φ values from the current solution could be stored and used as Dirichlet conditions to model the reflected ray path, for which the numerical framework developed in this paper will be of use. F. Three-dimensional complex geometry As a final case, the approach is tested for a cube containing a cone, hoop, sphere and high and low aspect ratio cylinders at various orientations with M = 0. This time all surfaces are assumed to constitute planar acoustic wave sources. Figure 11 shows the ~ grid (Frame (a)) and isosurfaces of φ (Frame (b)). Figure 12 plots the log of residual against iteration number. A multigrid V-cycle is used. The key thing to note is that, in spite of the multiple shock analogous zones (see Figure 11b) the convergence is robust. The residual has been dropped by three orders of magnitude in about 9 iterations. For M = 0 with Dirichlet boundary conditions the approach has been tested for a range of complex geometries (not shown here) and found to have surprisingly fast and robust multilevel convergence acceleration. Also, it has been tested for a full aircraft configuration for a range of subsonic M and found to give robust convergence. Hence, the approach has promise. However, at high relative M convergence becomes slower.


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1 0.5 0

Log (R)

−0.5 −1 −1.5 −2 −2.5 −3 0

Figure 12:

2

4

6 8 No. of iterations

10

12

Iterative multigrid convergence for three-dimensional geometry.

CONCLUSION An acoustic eikonal equation solution procedure, that is easy to implement in unstructured grid Navier-Stokes equation flow solvers, has been outlined. The approach is readily parallelizable. The method was tested for the following canonical point source cases: quiescent flow; subsonic constant flow; supersonic constant flow and an idealized jet flow field. Then, as further validation, sound propagation through a viscous vortex was considered. For these cases, generally, encouraging agreement was found with analytic data and a full Euler solution. As a demonstration case, a planar jet with noise shielding was considered along with a complex three-dimensional geometry. At lower Mach numbers (< 0.25), and complex geometries, robust multigrid convergence acceleration was found. ACKNOWLEDGMENTS Funding from the Royal Society is greatly appreciated along with the kind support of Rolls-Royce plc. Some of this work also took place as part of EPSRC Grant GR/T06629/01. This funding is gratefully acknowledged. REFERENCES [1] Blokhintsev, D.I. Acoustics of a nonhomogenous moving medium, In Russian (1946), Translated by Natl. Adv. Comm. Aeron. NACA T.M.1399 (1956).

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