Computational Modeling of Jets with Coflow

Computational Modeling of Jets with Coflow Paul Tucker*, Simon Eastwood†, Yan Liu‡, and Richard Jefferson-Loveday§ University of Wales, Swansea, SA2 8PP

An integrated solution approach using Hamilton-Jacobi and Eikonal equations for providing turbulence length scales, sponge, numerical order and also blending control is shown to be promising. For all these aspects essentially distance functions from surfaces can be needed. Although, use of these integrated features focuses on jet noise problems and structured grids perhaps the greater potential is for more complex geometry problems with dense unstructured solution adaptive grids suited for say aircraft landing gear noise modeling. Studies are made exploring optimal scheme options in the high order Large Eddy Simulation (LES) code. Since, for jets, being able to model the initial shear layer instability is important, the propagation of a Tollmien-Schlichting (TS) wave is considered. Comparison with an exact analytical solution of the Orr-Sommerfeld equation for the TS wave is made. On relatively coarse grids (intended to most strongly test the discretization) the code gave excellent agreement with the analytical TS data. The code is also found to propagate well the TS wave through distorted embedded, overset grid sections. A new hybrid Implicit LES (ILES) – Reynolds Averaged Navier Stokes (RANS) method making use of the HamiltonJacobi equation is discussed and applied to jets with co-flow. Using the hybrid ILES-RANS method, the influence of swirl, synthetic chevrons, jet eccentricity, jet external surface taper and width of the co-flow region is explored. Jet centre line velocity decay predictions show encouraging agreement with established data. Results show the co-flow region thickness has a more startling influence on potential core region length than jet eccentricity. As expected, predictions show that with co-flow the acoustic waves are directed more in the downstream direction. Predictions suggest a thick co-flow region has a strong acoustic influence and for a thin co-flow jet eccentricity has a diminished effect on the acoustic field. Mild co-flow angle ( 0. The Laplacian enables a smooth transition between the modeled RANS length scale (that needs an ~ accurate d) and the ILES zone (needing d = 0). The function f( d ) forces the Laplacian to tend to zero near walls. This ensures near wall distances are accurate. For hybrid ILES/RANS the function g(d) controls the RANS length ~ scale in the vicinity of the ILES zone. Figure 5a (to be discussed more later) gives typical Eq. (12) d distributions for various ε0 and ε1 combinations. When Eq. (12) is used for sponges g(d) ensures the influence of the target field smoothly diminishes away from the flow boundaries. Despite the simple appearance of the hyperbolic natured Eikonal and Eq. (12), they are not trivial to solve (see Tucker19). For example, transformed equation metric terms need to be discretized using offset metrics. The offset is governed by the front propagation implied in the Eikonal equation.

D. Synthetic chevron implementation For noise reduction, the use of wires (see Narayanan et al.21) and also chevrons have in the past been explored. Here we consider chevrons and to account for the effect of these we use the emulation procedure of Shur et al. 14. The near chevron tip region could be viewed as a sharp convex feature. As shown in Tucker 19 such features can have a strong anomalous RANS model influence. For example for flow over a small wire the k-l model of Wolfshtein can predict a dramatic over 100% turbulence viscosity increase. The Spalart-Allmaras model (along with many other popular turbulence models) strongly exhibits the opposite trait strongly damping modeled turbulence. In a DES/hybrid RANS-LES context these anomalous influences are likely to be propagated downstream. Hence more 6 American Institute of Aeronautics and Astronautics

research is required before such approaches can be reliably used. Therefore, an emulation procedure, which allows precise vortex strength control seems sensible. Figure 2 gives a schematic showing the source, sink, chevron locations and streamlines arising from use of the emulation procedure of Shur et al. 14

Figure 2: Source and sink locations with corresponding streamlines. Again, following Shur et al.8, to make the chevron modeling more realistic the pipe like geometry that the jet issues from is made non-circular. This is done in such a fashion that at angles where there are sources the radius is decreased and where there are sinks it is increased. In addition the grid has been modified at the nozzle exit plane. Grid lines are radially perturbed. The chevron parameters that have been used are based on preliminary RANS simulations, (see Shur et al. 8).

E. Solution of flow governing equations The flow governing equations are solved using a modified version of the NTS code of Shur et al.8. The problem set-up is virtually identical to that in the paper of Shur et al. Therefore, only brief details are given here. An x-y plane view of the solution grid is shown in Fig. 3a. The grid comprises two blocks. The much smaller inner ‘axis’ block has a radial extent of approximately 0.2. It is intended to avoid an axis singularity. Since, none of this block’s faces are wall adjacent, when solving Eq. (12) for turbulence length/distance scales, it is ignored. The total number of grid points is 500,000. At the Fig. 3a boundaries, labelled (III), sponges are applied in the manner of Ashcroft and Zhang 22. This method is discussed further later. Flow boundaries (I) and (II) use Dirichlet conditions. Figure 3b gives a y-z view of the perturbed solution grid when the synthetic chevron modeling is used. The schematic above the grid indicates the modified nozzle outlet geometry shape. It also gives the location of the counter-rotating vortex pairs defined by the sources that represent the effects of the chevron tips (which point into the jet flow).

Figure 3: Solution grid: (a) x-y plane and (b) perturbed y-z grid for synthetic chevrons. . 7 American Institute of Aeronautics and Astronautics

Simulated resolved flow inlet turbulence levels are zero. When there is co-flow, the RANS modeled k is set to match the measured value. There is no real resolved turbulence in the RANS zone. Hence, saving computational effort, once converged the k field is frozen. This has similarities to the approach used in Tucker23. A hybrid convective flux differencing discretization is used. Following Strelets et al., for jet simulations, fourth order centred differencing (NCD = 4) is primarily used in a blended form with 5th order upwinding (NUP = 5). The two are blended using an exact mathematical equivalent to the following equation J conv = αJ ctr + (1 − α ) J upw

(16)

where Jctr and Jupw are the centred and upwind flux components and α is the blending function. Figure 1a plots the α distribution. Note, although the HJ equation could be used (see later) in this plot the α field is geometrically obtained with a Laplacian based smoothing to link the 0 < X ≤ D linearly increasing α distribution with the constant α = 0.75 field. Broadly α increases linearly between X = 0 and X = D with α = 0 at X = 0. The largest X > D usable value of α is 0.75. For α > 0.75 the solution in the turbulence acoustic interface becomes highly non-monotonous and hence unusable. To damp numerical reflections the mathematical equivalent to the approach of Ashcroft and Zhang22 is used where F (l ) = (1 − s ) Ft arg et (l ) + sF (l )

(17)

FTarget being a target field and

s = 1 − ( l / L)

β

.

(18)

In the above L is the width of the sponge layer and 0 < l < L. Also, l = 0 at the domain boundaries and l = L is inside the domain. Figure 1b gives the distribution of s which again can be generated using the HJ equation (see later). To damp numerical reflections the order of the numerical upwind scheme component is also reduced towards ~ flow boundaries (see Fig. 1c). For this an Eikonal or HJ d field propagated some distance L from boundaries can be used. Use of the following Eikonal linked near wall order control expression will be discussed later ~  NUP , MAX d  NUP = max int  ,1 L  

(19)

where NUP,MAX is the maximum order of the upwind solution component (i.e. 5), int represents the mathematical rounding up of the bracketed quantity to the nearest integer and the operator max takes the maximum value. All the above gives an integrated computational approach for the four distance function based solution requirements of the acoustics modeling approach of Shur et al14. As part of code performance studies a range of time schemes in NTS are tried. For example, backwards differentiation Gear schemes with up to 5th order accuracy are tried. For a right hand side R at a time level n+1 we have the following temporal discretization n +1

nφn

Wf  ∂φ   ∂t  =   BD n − (N BD −1) ∆t

∑

= − R n +1

(20)

where the weighting functions Wf for the different order (NBD) backwards difference schemes are given in Table 1.

8 American Institute of Aeronautics and Astronautics

Table 1: Weighting functions in backwards difference schemes. NBD

W fn + 1

W fn

W fn − 1

W fn − 2

W fn − 3

W fn − 4

1 2 3 4 5

1 3/2 11/6 25/12 137/60

-1 -4/2 -18/6 -48/12 -300/60

1/2 9/6 36/12 300/60

-2/6 -16/12 -200/60

3/12 75/60

-12/60

The Crank-Nicolson (CN) scheme is also tried. For this the time derivative is discretized as below. φ  ∂φ   ∂t  =  CN

n +1

−φ ∆t

n

However, for stability reasons the equation of Shur et al. below is used. For NBD = 2 and family of second order schemes.

(21)

ϑ

= 0.5 this gives a

 ∂φ   ∂φ  n + 1 + (1 − β )(1 − ϑ )R n + ϑR n + 1  + (1 − β ) ∂t  = βR   ∂ t   BD   CN

β

(22)

With ϑ = 0.5 and β = 0 Eq. (22) gives the pure Crank-Nicolson scheme. Here we use ϑ = 0.5 with β = 0.1. Hence we generally have CN blended with 10% of the NBD = 2 scheme. For jet simulations the time step is ∆t=0.04(d0/U0). To become statistically steady, simulations need t ≅ 2000d0/Uo.

III.

Discussion of Results

~ A. Eikonal/HJ d Results Figure 4a shows the general mode of operation of the Eikonal d equation when fronts are propagated from all boundary surfaces. The vertical axis corresponds to d. Fig. 4b shows how the Eikonal with Equation (19) can be used to reduce the numerical discretization order at boundaries.

(a)

(b)

Figure 4: Examples of uses of Eikonal equation: (a) Eikonal d field and (b) use of Eikonal equation to control NUP at boundaries.

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~

Figure 5a gives the HJ based hybrid ILES-RANS d distributions. The pure ILES zone is set at y+ ≈ 45. The short ~ ~ dashed line gives the Eikonal ε1 = ε0 = 0 d solution. As can be seen, d changes abruptly around y+ ≈ 45. This is undesirable, potentially introducing spurious acoustic waves. The long dashed line is for ε0=0.2 and ε1=0. The maximum-modeled length scale is now centred between the wall and ILES zone at y+ ≈ 22.5. Fig. 5b gives a three~ dimensional view of the hybrid ILES-RANS d field predicted from solution of HJ equation with ε0=0.2 and ε1=1.5. ~ As shown by the full Fig. 5a line, increasing ε1 to 1.5 biases the maximum modeled d towards the ILES zone. Hence the RANS influence in the mixed model region is increased. The mixed model region is reminiscent of the ‘DES buffer layer’ described by Piomelli et al.24. It might seem most sensible to have the RANS zone extend, without any diminishment of the modeled turbulence, beyond the actual buffer layer and well into the inner logarithmic region. However, experience suggests (Temmerman et al., 25) that more realistic turbulence statistic results are gained if the RANS region influence is diminished. This is perhaps because ILES zone activity will force resolved scales in the RANS regions. The sum of these resolved and modeled scales will yield excessive turbulence energy. Reτ = 1050 ‘law of the walls’ and turbulence statistics for the current hybrid ILES-RANS approach show encouraging agreement with benchmark LES data of Piomelli 26 and also measurements (see Tucker 11). Predictions have also been made (not shown here) for all the Fig. 5a ε values when the RANS model and interface selection procedure are as prescribed for standard DES. The interface selection procedure results in a greater near wall modeled component extent. However, mean velocity and turbulence statistics again show encouraging agreement with benchmark data.

(a)

(b)

~ Figure 5: Hybrid ILES-RANS d field predicted from solution of HJ equation : (a) graphs for different ε0 and ε1 combinations and (b) 3D surface plot with ε0 =0.2 and ε1 = 1.5.

Figure 6: Use of HJ equation to predict sponge weighting function.

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~ Figure 6 shows the HJ predicted sponge weighting function/ d fields when ε1 = 10, m = 2, 3, 3.5 and 4 and n = 2 in g(d) and ε0 = 0 (see equations (15)). The full line is for an Eikonal equation solution. The fields are normalized to unity. The polynomial function form used by Shur et al.8 (but with a power of 4) is also included. As can be seen the HJ equation can generate suitable weighting functions especially perhaps for m = 4. However, although an interesting range of functions can be generated with the HJ approach it would be computationally cheaper to apply a polynomial equation similar to that considered by Shur et al114. directly to an Eikonal field.

B. Numerical Scheme Performance Studies To explore numerical scheme and sponge performances, the spatial development of a Tollmien-Schlichting (T-S) instability wave in a plane channel is considered. The plane channel flow is linearly stable for Rec< 5772 Orszag27. The Reynolds number is based on the half channel height and the centre-line velocity. For Re < Rec, instabilities show a downstream decay. Through considering Poiseuille flow a T-S wave can be obtained from eigensolution of the Orr-Sommerfeld equation. After Navier-Stokes equation linearisation and assuming a wave-like solution, the resulting Orr-Sommerfeld equation can be written as: 2  2 2 2 ω U − R  d − αˆ 2  − d U + i  d − αˆ 2   vˆ = 0 0  dy 2 0 αˆ Re  dy 2    αˆ  dy 2       

(23)

Here, U0 is a base flow of the form U0(y) = y(2h - y), where h is the half channel height. Also, αˆ is the complex wave number αˆ = αˆ R + iαˆ I and ωR the real frequency. The variables αˆ R and ωR are related by the expression

ˆ from Eq. (23), the remaining velocity component can be easily calculated via the c = ω R αˆ R . After obtaining v continuity equation. For Re = 5000 a range of spatial differencing schemes were tried. At the channel inlet a two-dimensional T-S wave was superimposed on the laminar parabolic profile. For these incompressible flow studies the dual time-step, artificial compressibility method of Rogers and Kwak28 is used. Simulations generally use a grid with 259 nodes in the streamwise ‘x’ direction and 67 in the cross-stream ‘y’ direction. This modest grid allows differences between the various spatial schemes to be observed. Tests are also performed with distorted grids embedded in this background mesh. Also, a simulation is run using a 515 x 131 grid. Grids are refined at solid boundaries using a hyperbolic tangent function. The numerical domain has a height of (h =) 2 unit in the vertical y direction. Two domain lengths in the x direction are considered. One is uniform extending 51 channel heights (X = 51). The other is uniform for X < 40 and then rapidly geometrically expands in the x direction to a distance of about X = 350. The latter, with its coarse downstream grid avoids the need for explicit nonreflecting boundary conditions. Without some sort of non-reflecting boundary condition treatment solutions diverge. A strong reflected wave that grows without limit is observed. For spatial differencing scheme studies a dimensionless time-step of ∆t = 0.05 is used with a 2nd order backwards difference temporal scheme. The time step is small enough to make the temporal discretization truncation errors negligible. For temporal scheme studies this time step is increased to 0.2 and the NUP = 5 spatial scheme used. For stability, all the spatial central difference results are blended with 25% 5th order upwinding i.e. α = 0.75. Also, when the Crank-Nicolson (CN) scheme is used (reminiscent of the spatial scheme blending) this is 10 % blended with the 2nd order backwards difference (BD) temporal scheme. Figure 7 gives T-S wave disturbance streamfunction contours. These are evaluated using the vertical (v) velocity ˆ component. The analytic disturbance kinetic energy decays as e −α I t with a rate αˆ I =-0.0106 and a wave number αˆ R 27 = 1.155676 (Orszag ). The effects of the discretisation scheme order and nature on the wave number αˆ R and the decay rate αˆ I are explored in tables 2 and 3. The former is for spatial schemes and the latter temporal. Schemes with dispersion error will affect αˆ R . Those with false diffusion affect the decay rate constant αI. To estimate the αˆ R error the following is used Error =

v 7′ − v ′A v ′A

11 American Institute of Aeronautics and Astronautics

(24)

where v7′ is the amplitude of the 7th wave peak along the channel and v ′A the analytical amplitude. Since the error trait is consistent, although due to the crude estimation procedure perhaps not flattering, the tabulated αˆ R values are useful in showing trends. An ultimate aim is to contrast performances of several solvers, including unstructured ones. Hence the current approach is more convenient when dealing with a range of solvers than performing complete integrations through time and across the channel. Clearly αˆ I is more easily accurately evaluated and is taken as an average of the wave lengths along the channel. The wavelengths are calculated using the points where v′ = 0 . Figure 8 plots the variation of v ′ with x along the channel centre line. The NUP = 1, 2, and 5, and NCD = 2 and 4 results are presented. To avoid graph overcrowding the NUP = 3 and 4 results are not included. Like Table 2, Fig. 8 shows that the choice of spatial differencing scheme has a strong solution influence. Clearly, and as would be expected, the first order upwind scheme result is poor. The NUP = 2 result is much better but by X ≈ 45 there is a clear amplitude error. At X = 45 the NCD = 4 has the lowest amplitude error relative to the UP schemes. Away from the channel inlet the NCD = 2 has clear amplitude and phase errors. These traits are reflected in Table 2. The hybrid result entry, in this table, uses the NUP = 5 scheme in the streamwise direction and 4th order CD (blended with 25% 5th order UP) in the cross stream. This scheme (reminiscent of that used by Mahesh et al.29) is intended for when there are shocks- the high order upwind being applied/directed, as much as possible, across the shock. We hope to test this scheme as part of future under-expanded jet work. However, ensuring that the NUP component is normal to the shock direction could need careful grid generation. The final Table 2 entry is for the fine grid. It is important to note that there is no Table 2 entry for αˆ I with NUP = 1. The lack of flow structure for this scheme prevented this being defined. The phase error result for NCD = 6 is worse than those for NCD = 4 and this is attributed to iterative convergence error. The NCD = 6 result proved especially difficult to converge. It is worth noting that the code also performs well when distorted grids are embedded in an overset manner in the channel. These results are not shown here. Table 3 clearly shows the CN scheme has greatest temporal accuracy, even being more accurate than the 5th order BD scheme. However, relative to the 2nd order BD scheme CN proved more unstable. Hence for the jet results CN is not currently used but will be tried in the future.

Figure 7: T-S wave disturbance stream function contours.

Figure 8: Amplitude decay plot (extended x domain).

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Figure 9: Results with sponges: (a) amplitude decay plot; (b) s distribution and (c) NUP distribution. Figure 9 gives data for when the shorter domain is used with sponges. Frame (a) is the amplitude velocity decay for NUP = 5. Frame (b) gives the s distribution (i.e. factor for sponge weighting function) with a HJ, m ≈ 3.25 representation and (c) NUP evaluated using Eq. (19). Since no central differencing is used there is no α contour plot. Essentially, the domain is too short and hence a strong wave arrives at the right hand outlet. However, the sponges deal with this well and the effects of the excessively truncated domain on disturbance amplitude are encouraging. The convective boundary condition of Pauley et al.30 where ∂ui ∂u + Uc i = 0 ∂t ∂x

(25)

is also tested. In this study Uc is taken as the bulk mean velocity. Use of Eq. (25) gave a slightly greater decay than the sponge but this test was carried out in a different solver.

Table 2: Accuracies of spatial differencing schemes. N

Scheme

2 4 6 1 2 3 4 5 Hybrid 5(fine grid)

CD CD CD UP UP UP UP UP CD/UP UP

αˆ R Error (%) 11.9 4.4 4.2 69.4 31.0 25.1 6.4 5.1 5.4 1.39

αˆ I Error (%) 0.890 0.209 0.348 0.578 0.017 0.260 0.244 0.234 0.162

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Table 3: Accuracies of temporal differencing schemes. N

Scheme

1 2 5 2

BD BD BD CN

αˆ R Error (%) 41.0 7.2 3.4 2.5

αˆ I Error (%) 0.0585 0.4607 0.2876 0.2789

The results discussed above show the NCD = 4 scheme blended with 5th order upwinding along with the 2nd order backwards difference scheme is a good choice for the jet simulations.

C. Flow/Acoustic Field Results The Fig. 10 schematic helps define some nomenclature. The outer co-flow diameter is defined as d1 and the inner core flow diameter d0. The eccentricity between the co and core flows is given by a. Where there is swirl, the swirl number is conventionally defined as

S0 =

∫

d0

uwr 2 dr 2 d0 d0 2 u r .dr 2 ∫0 2 0

(26)

In the above w is the swirl velocity, u axial velocity and r radius. For a linear w distribution, where w = Ω r (Ω being the fluid’s angular velocity), Eq. (26) reduces to S0 = ΩD/(4Uo). For all cases considered here S1 = 0. To explore the influence of chevrons/serrations the synthetic source/sink approach of Shur et al. 8 described earlier is used. The source/sink strength is defined as Sc and the number of chevrons Nc. The difference between the core and co flow velocity, ∆U = U0 – U1. Bardbury & Khadem31 studied the influence of internal jet geometry tapering on potential core length for jets with no co-flow. The potential core region length was found to be reduced by about 15% for a 450 taper. Although this deviation is significantly less than that produced by chevron like geometry elements, it seems sensible, for jets with co-flow, to explore the influence of external jet tapering. To explore the influence of geometry on the core/coflow interaction the coflow approach angle, indirectly controlled by θ, is varied between 150 and 450

Figure 10: Jet geometry. For all cases Re=1x104 (based on U0, the jet outlet velocity and the jet diameter) and the Mach number, Ma = Uo/ao = 0.9 (the ‘o’ subscript on a is used to indicate that this is an ambient condition value). When there is co-flow conditions are such that ∆U/U0 = 0.27. When S0 > 0 w = C0uC1r r n cos(πr )

14 American Institute of Aeronautics and Astronautics

(27)

where Co=54x104, C1=2.7182818. When ∆U > 0, the co-flow velocity and hence turbulence levels are low (0.3%). Therefore, near wall modeling is not strongly tested. However, according to Morse1 the co-flow nature at the jet exit exerts a key influence and so it seems sensible to characterize the co-flow boundary layer. It is also worth noting, beyond near wall turbulence modeling in the hybrid ILES-RANS approach the HJ equation and Eikonal equations are not further used in these results.

(a)

(b)

(c)

Figure 11: Vorticity isosurfaces for ∆U = 0: (a) S0=S1=Sc= 0, (b) S0 > 0, S1=Sc=0 and (c) S0=S1=0, Sc = 0.03 Figure 11 gives vorticity isosurfaces for ∆U=0. Frame (a) is for S0=S1=Sc= 0, (b) for S0 > 0, S1=Sc=0 and (c) for S0=S1=0, Sc=0.03. As can be seen, for swirl ‘fingers’ of relatively high vorticity tend to be convected by the mean flow. With chevrons, streak like structures are formed, with fingers of relatively high vorticity again being convected into the mean flow.

(a)

(b)

(c)

Figure 12: Pressure isosurfaces for ∆U=0: (a) S0=S1=Sc=0, (b) S0>0, S1=Sc=0 and (c) S0=S1= 0, Sc = 0.03 Figure 12 repeats 11 but for pressure rather than vorticity. Frame (b), for S0 > 0 and (c), for Sc=0.03 suggest that swirl and chevrons both break down more coherent torodial vortex structures evident in Frame (a).

Figure 13: y-z plane views of the flow at x/d0 ≈ 0.5: (a) computation for ∆U = 0, S0 ≈ 0.225, S1 = 0; (b) computation for ∆U/U0 = 0.27 (d1/do = 2), Sc > 0, Nc = 8; (c) flow visualisation of Ng (2000) for a lobed nozzle with ∆U/U0 ≈ 0.25, Nc = 8. Figure 13 gives z-y plane flow structure information at x/D ≈ 0.5. Frame (a) gives predicted streak lines and ~ p contours for ∆U = 0, S0 ≈ 0.225 and S1 = 0. Frame (b) repeats (a) for ∆U/U0 = 0.27, d1/do = 2, Sc > 0 and Nc = 8. Frame (c) shows the flow visualisation of Ng32 for a lobed nozzle with ∆U/U0 ≈ 0.25 and Nc = 8. Since the flow 15 American Institute of Aeronautics and Astronautics

visualisation is not especially clear, Ng has also included lines sketching the sense of rotation of the vortices generated by the nozzle lobes. It is clear from frames (a) and (b,c) that both swirl and chevrons produce streamwise vorticity. Comparison of frames (b) and (c) show that the current chevron emulation procedure gives a vortex field that is broadly consistent with the observations of Ng. For chevrons, the vortex generation mechanism is reminiscent of that for delta wings.

Figure 14: Contour plots of streamwise velocity in y-z plane for (a) x/d0 ≈ 0, ∆U = 0; (b) x/d0 ≈ 1, ∆U = 0 and (c) x/d0 ≈ 1, ∆U > 0, d1/do = 2. Figure 14 gives contour plots of streamwise velocity in y-z plane. Frame (a) is at x/d0 ≈ 0 (∆U = 0). Frame (b) is at x/d0 ≈ 1 and again ∆U = 0. Frame (c) is at x/d0 ≈ 1 with ∆U > 0 and d1/do = 2. Cleary the co-flow, which reduces the shear layer strength and more strongly convects the imposed vortices, makes the coherence of the imposed vortex structures persist for longer. Figure 15 compares (normalised by Uo) centre line velocity decay predictions for a/d0 = 0 with the measurements of Morse1. Moving left to right, the three comparison sets correspond to ∆U = 0 with S0 ≈ 0.225 (short dashed line and squares), ∆U = S0 = 0 (long dashed line and circles), and ∆U/U0 = 0.27, S0=0 (full line and triangles). For ∆U > 0 to match the experimental set-up of Morse1 d1/d0 = 8. Clearly, S0 > 0 reduces the jet potential core region and coflow extends it. The larger S0 > 0 discrepancy is partly attributed to the fact that the swirl in the experiment would tend to stagnate the jet axis region flow. The predictions, however, use u ≠ f(r). Also, for S0 > 0, there are larger velocity gradients and important small scale streamwise vortices (see Fig.13a). These will place stronger grid demands. Therefore, for this more challenging turbulence physics case a lower level of agreement is perhaps to be expected. However, it is encouraging to note that, at the very least the current numerical approach and grid is capable of capturing centre-line decay velocity trends. Also, as shown by Shur et al.14 and Tucker11 the approach reasonably well captures turbulence statistics and, when used with an acoustic analogy, far field sound data. Hence, the novel hybrid ILES-RANS approach is further used to make co-flow studies with especial focus being paid to the centreline velocity decay and hence potential core region extent. Since the tip of the potential core region is a major noise source focusing on this aspect seems sensible.

Figure 15: Comparison of centre line velocity decay for a/d0=0

Figures 16: Variation of centreline velocity decay with a/do and d1/d0.

16 American Institute of Aeronautics and Astronautics

Figure 16 gives the variation of centreline velocity decay with a/do and d1/d0 for S0 = S1 = θ = 0 and ∆U > 0. The full line repeats the Fig. 15 d1/d0 = 8 line. The chain dashed line is also for d1/d0 = 8 but with eccentric jets. Clearly, as noted by Papamoschou and Debiasi3, eccentricity reduces the potential core region length. However, the long and short dashed lines show that reducing d1/d0 has a more startling influence in reducing the potential core region length. Also, comparison of these dashed lines shows that at low d1/d0 eccentricity has a weaker potential core region influence. Similar analysis suggests for θ less than 300, θ has little influence on potential core region length. Analysis also shows that the addition of chevrons reduces the potential core region length by approximately 10%.

Figure 17: Pressure time derivative contours in the x-y and y-z plane for S0 = S1 = 0. Figure 17 gives near jet pressure time derivative contours in the x-y and y-z plane for S0 = S1 = 0. For the x-y plane, contours extend to x/d0 = 20 and the y-z plane contours are shown at x/d0 = 20. Comparison of the ∆U = 0 frame (a) plot with the others shows the co-flow alters the sound wave directivity. As expected, with co-flow the acoustic waves are directed more in the downstream direction. Comparison of frames (c,e) with (b,d) – especially in the y-z plane – suggests a thick co-flow region has a strong acoustic influence. The contour fields in frames (b) and (d), for a thin coflow, are similar. This suggests that for a thin co-flow jet eccentricity has a diminished effect on the acoustic field.

IV.

Conclusions

The integrated solution approach using the HJ/Eikonal equations for providing turbulence length scales, sponge, numerical order also blending control appears promising. In numerical performance studies the code gave good agreement with the analytical Tollmien-Schlichting data. The 2nd order backwards difference scheme with primarily 4th order central differences was shown to perform well. Jet centre line velocity decay predictions show encouraging agreement with established data. Results show the coflow region thickness has a more startling influence on potential core region length than jet eccentricity. As expected, predictions show that with co-flow the acoustic waves are directed more in the downstream direction. Predictions suggest a thick co-flow region has a strong acoustic influence and for a thin co-flow jet eccentricity has a diminished effect on the acoustic field. Mild co-flow angles (< 300 and produced by nozzle external surface taper) have little effect on centerline velocity decay.

V.

Acknowledgments

We would like to thank M. Strelets, M. Shur and P. Spalart for the use of their excellent code. The funding of the UK Engineering and Physical Sciences Research Council (EPSRC) under grant number GR/T06629/01 and an International Travel Grant from the Royal Academy of Engineering are gratefully acknowledged. 17 American Institute of Aeronautics and Astronautics

VI.

References

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18 American Institute of Aeronautics and Astronautics