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Numerical Simulation of Sidewall Effects on the Acoustic Field in Transonic Cavity

Zhisong Li* and Awatef Hamed†

University of Cincinnati, Cincinnati, Ohio, 45220

Debashis Basu ‡

Southwest Research Institute, San Antonio, Texas, 78238

[Abstract] High Reynolds number detached eddy simulations (DES) were performed for the transonic flow over open cavity to study the effect of sidewall boundary conditions on the unsteady flow and acoustic fields. Two kinds of sidewall boundaries: Slip and periodic conditions were examined. The use of wall function on non-slip surface was tested. Computed sound pressure level (SPL) spectra were compared with experiment data, LES and DES without wall function from reduced Reynolds number. Spectra from sidewall points and turbulent kinetic energy in shear layer were also investigated.

NomenclatureCf = friction coefficientD = cavity depth f = frequencyk = turbulent kinetic energy per unit massL = cavity lengthl = turbulent length scaleM = inflow Mach numberRe = Reynolds numberT = static temperatureU = flow velocityui = velocity componentω = dissipation rate per unit turbulent kinetic energyε = turbulent dissipation per unit massγ = gas constantρ = gas densityμ = laminar viscosityΔ = grid spacingΔt = time step length

I. Introductionree stream over open cavity is often encountered in industrial applications. Many of them in aerospace area gave rise to the major research into the cavity flow unsteadiness and associated aero-acoustics. Open under-

carriage wheel wells are the primary airframe noise source during aircraft take-off and landing. In-flight refueling ports on military aircrafts, and pressure vents in the space shuttle’s cargo bay are similar cases. The application of internal weapon storage on the new generation fighter led to the exigency for high-speed cavity flows research. As weapon bay opens, highly unsteady flow develops inside the bay cavity and generates severe vortical oscillations. Such flow fluctuation has adverse effects on the sensitive avionics on weapons; and it can cause premature material fatigue or even airframe structure failure.

F

* Graduate student.† Bradley Jones Professor, AIAA fellow.‡ Research engineer.

American Institute of Aeronautics and Astronautics1

The self-sustained pressure oscillation within the flow passing a cavity is featured by a spectrum of a single or multiple distinct tones (up to 170dB for transonic flows). The explanation is generally contributed to a complex feedback mechanism induced by the amplification and convection of small instability from the shear layer, which impinges the cavity rear wall and generates acoustic waves. The acoustic waves traveling upstream excites further disturbance to the shear layer and finish the self-sustained oscillation process.1

Numerous analytical studies, experiments and computer simulations have been performed over the years to investigate cavity flow inner physics. Vast experimental results were databased to resolve open cavity resonance phenomenon and to validate numerical simulations2. Numerical simulation using RANS and LES also achieved different success for cavity flow mechanism and acoustics studies. To overcome the drawback of RANS for unsteady flow fidelity and limitation of LES on computer resources requirement, and to attempt 3D simulation of higher or even realistic Reynolds number, an effective tool of detached eddy simulation (DES) was chosen by many researchers for cavity flow study. Recently, Hamed3 (Spalart-Allmaras model-LES), Nayyar4 (Menter’s baseline

model-LES), Sinha et al 5( model-LES) and many others performed 3D DES on subsonic/supersonic cavity flow and exhibited reasonable consistency with experiment results.

The aforementioned 3D simulations either implemented slip sidewall boundary or periodic sidewall boundary. Both boundaries do not serve as the non-slip sidewalls in actual experiment but provide a numerical simplicity for computation. The objective of this paper is to examine the difference on acoustics between these two boundary conditions and to judge that which one is better suited to compare with experiment. New attempts also include transonic simulation of real Reynolds number (12106/m) and application of wall function as a simplified method to relieve computational burden for near-wall region.

II. Menter-SST based Detached Eddy SimulationII.1. Menter-SST ModelShear-stress transport (SST) model is a non-algebraic two-equation turbulence model suggested by Menter6 as a

combination of model and model. The standard model excels in modeling flow with strong shear stress, but it over-predicts shear stress under reverse pressure gradients, and brings in numerical stiffness of in resolving viscous sub-layer. Wilcox model has superior performance to model in adverse pressure gradient. It is more robust without any damping function in viscous sub-layer. But model also has weakness as strong dependency on free stream values. To take advantages of both, Menter6 applied Wilcox model in the near wall region and gradually changed it into standard model in the outer wake region. This model is considered to be one of the best two-equation RANS models, particularly suitable for the scenario of flow separation such as cavity flow.

In SST model, the turbulence transport equations are as follows: For turbulence kinetic energy k:

(1)

For specific turbulent dissipation rate:

(2)

Cross-diffusion term is added in the equation from original model.The standard turbulence transport equations are transformed into a formulation, respectively:

(3)

(4)

The turbulent stress tensor is . Equations (1), (2) are then

multiplied by a blending function F1, and equations (3), (4) by (1-F1). Equation (1) and (3), (2) and (4) are each added together to give the complete SST model:


(5)

(6)The turbulent shear-stress is defined based on Bradshaw’s assumption that shear stress is proportional to the turbulent kinetic energy k in boundary layer: , where a1 is a constant. And the eddy-viscosity is given by

, in which is the absolute value of the vorticity, and F1 is given by

. The constants in original Wilcox equations (1), (2) are given as:

, , , , , and . And

the constants in original standard equations (3), (4) are given as: , , ,

, , . If represents any constant in Wilcox equations (1),

(2); and for standard equations (3), (4), then for the constants in SST model equations (5), (6), . For non-slip wall boundary condition, Mentor5 recommended the following:

at y = 0, where is the distance to the next point away from the wall. The flow in this

formulation is compressible, but Menter did not include compressibility corrections in his model. Suzen and Hoffmann7

added compressible dissipation and pressure dilatation terms to the portion of Menter's model. Applying Menter's blending process, the equations are

(7)

(8)where the pressure dilatation term is

(9)

II.2 SST-based DES FormulationThe Menter’s SST based DES model was first introduced by Spalart7 by modifying the length scale in the model. The

length scale in the model is (10)

And the DES length scale replaces this length scale as:

(11)

Is the largest grid spacing in any direction: .Assuming that the eddy viscosity is proportional to the magnitude of the strain tensor, and to the square of the grid spacing. The length scale increases as one moves away from the non-slip surface. When near the surface,

, the model acts as a RANS model. Away from surface, , the RANS formulation acts as a Smagorinski sub-grid-scale LES-like model. Then the dissipation term of k-transport equation (5):

(12) Is modified into the DES mode:


(13)

Parameter CDES is obtained from (14)

Where F1 is the blending function. Strelets selected the values of and as 0.61 and 0.78 respectively. The Menter-SST based DES model implementation was proposed by Bush8 with similar approach. The turbulent

length in RANS is defined as:

(15)

Bush reformulated the LES turbulent viscosity based on the kinetic energy of the unsolved scale (k):

(16)

In the proposed model, the new DES length scale is:

(17)

Therefore the turbulent viscosity of the new model is:

(18)

The coefficient CB and grid size determine the extent of the LES and RANS. Increasing CB enlarges the region where the combined model reduces to standard Menter-SST model. Bush9 and Mani et al10 tested different CB values for several separated flow cases and recommended the CB value range from 0.1~0.5. Hamed et al11 examined the transonic cavity flow cases with CB as 0.1, 0.5 and 1.0, and found that 0.1 is the appropriate value for such simulations to resolve proper scales in SPL spectra.

III. Computational DetailIII.1 Computational SetupThe present open cavity DES simulations were carried out in parallel processing with WIND code developed by

NPARC Alliance. As displayed in Figure 1, the domain has a size of 5D D D (length width depth, D = 0.33ft). The inflow sections extend 5D upstream to be in consistent with the leading edge setting in experiment. And the outflow boundary is 6D behind the cavity to reduce disturbance from the end boundary. A symmetry plane is set 10D above the cavity mouth to avoid any acoustic wave reflection back into the cavity flow field. The inflow/outflow plates and the cavity floor are set as viscous adiabatic walls. Two different sidewall boundary conditions settings are to investigate, namely slip wall and periodic boundaries, corresponding to two simulation cases. The slip boundary is identical to symmetry boundary. While the periodic boundary is treated as normal coupled boundaries, with left sidewall virtually point-to-point connected to the right one in translation. For inflow conditions, Mach number = 1.19, static pressure = 14.7 psi, static temperature T=568 ºR. The flow Reynolds number based on cavity length is 12.3 106/m, as in the experiment by Stanek et al12.

A structure grid containing totally 3.5 106 grid points was applied for the simulations as shown in Figure 2. More than 90% grid points are in within the cavity or concentrate near the inflow/outflow plates. From some rule of thumb in aero-acoustics for the accuracy consideration, cell size in region of interest should range from to , where is the sound wave length to capture. Therefore, a largest grid spacing of =1.78 10-2D will secure a cutoff frequency up to 20000Hz. Concerning the stability of computation, a time step is chosen as 8 10-7s. The domain was split into 56 sub-domains for parallel computing allocation. Most sub-domains consist of approximately equal grid points to obtain better-balanced load on each processor.

Since acoustic spectra prediction is very sensitive to numerical diffusion, high-order spatial discretization scheme is necessary to track the sound waves. Roe’s 3rd order upwind scheme is applied. Numerical smoothing is also chosen to apply dissipation in the explicit operator and dampen instability. Total-Variation-Diminishing (TVD) operator is not used because it may produce undesirable damping or smearing in the flow field.


III.2 Wall Function ApplicationThe applying of wall function significantly reduces the grid fineness in near-wall regions14. The initial grid point off

wall is selected as y+ = 69 rather than < 10 for common RANS applications. The White-Christoph law of wall13 serves as the near-wall turbulent flow presumption. It reads

(19)

for wall-parallel mean velocity prediction. DalBello et al14 and Dippold15 have taken systematic investigations of this wall function in Wind code. They arrived

on a conclusion that wall function works well for attached flow problems, but not with large amount of separation. And wall function cannot predict the flow properties very near the wall as well as full wall integration. For the cavity flow, no major flow separation will appear on leading and rear floors, but this will probably occur on cavity floor. How will it affect the overall flow field and acoustic performance is still not known and has not been tested before. It would be a challenging and exploratory study to use wall function in this simulation.

III.3 Cases for ComparisonA series of cases of identical cavity configuration are to compare with their critical parameters listed in Table 1.

Experiment data are used to validate the simulation results, and numerical simulation cases will be examined for their prediction accuracy, resources consumption and the effectiveness of utilizing wall function in present cases.

Case ReL (/m) Mach number Grid Number Δt (s) Min Δy+

Experiment (Stanek et al12) 12.3M 1.19 ~ ~ ~

DES slip sidewall 12.3M 1.19 3.5M 8×10-7 68.6

DES periodic sidewall 12.3M 1.19 3.5M 8×10-7 68.6

LES non-slip sidewall (Rizzetta et al16) 0.82M 1.19 20.5M 4.87×10-4 1.6

DES slip sidewall without wall function (Basu et al17) 1.97M 1.19 9.8M 2.5×10-7 1.0Table 1 Comparing cases

IV. Result and DiscussionThe computations were performed on a Unix cluster of 50 AMD Athlon processors. The simulations took around

35000 time steps to purge out the initial flow field and establish steady oscillation. After this, 65000 time steps were experienced for signal processing. Therefore a frequency resolution is about 19Hz for the sampling frequency of 1.25MHz.

IV.1 Flow featuresFigure 3, 4, 5 and 6 show the cavity flow instantaneous features. Both slip sidewall case and periodic sidewall case exhibit analogue characteristics. Mid-plane Mach number contour of figure 3 indicates a clear shear layer and an oblique shock before the cavity leading edge. The shear layer flow directly impinges onto the cavity rear wall, leaving a big low speed stagnant flow in forward cavity. Figure 4 presents a non-dimensional vorticity contour on the mid-span plane. Vortexes originated from shear layer appear very small under high Reynolds number, compared with cavity dimension. From Figure 4 and 5, these initially uniform vortexes quickly break down in all directions and grow larger in size as they travel downstream. Strong and large vortexes tend to stay at the cavity mouth and small and weak vortexes fill up the rear cavity room. With violent and erratic movement of vortexes

, the present cavity flow is undoubtedly of a wake mode. As of pressure field, an instant mid-plane pressure contour of Figure 6 indicates some small low-pressure regions within the high-pressure field behind the shock. As the main high-pressure field is relatively steady and predominant from supersonic aerodynamics, these low-pressure regions are supposed to move back and forth in the cavity to generate pressure signals of featured spectra.

IV.2 Pressure Signal and Spectra AnalysisThe current cavity flow of wake mode is believed to have a broadband and quasi-harmonic spectrum, involving a

more complicated mechanism than shear layer mode. To track the pressure signals, three sensors are set on the cavity floor and rear wall on mid-span plane in experiment and simulation as shown in Figure 7. Simulation data show that pressure variations are highly disordered even after purging out (Figure 8). As the pressure tracking


duration is very short in computation (< 0.1s), the method of maximum entropy was adopted to extract the frequency spectra information.

Figure 9 (a) (b) (c) compare the frequency spectra from point 1, 2 and 3 between experiment, slip sidewall case and periodic sidewall case. For point 1, slip sidewall case has the closest dominant peak sound pressure level (SPL) to experiment with a lower difference of around 0.8dB, in comparison of a 10 dB disparity for periodic case. Both case have best prediction for the prevailing peak for point 3, only 0.1dB and 3dB lower than experiment. Point 2 has the worst performance with up to 14dB lower than experiment for periodic case. This may suggests point 2 (X/L=0.8) as a location of more complicated flow movement and require smaller time steps to resolve. Exact pressure measurement signified a sharp SPL drop in the range from 2000Hz to 3000Hz for all three points. As a major drawback in simulation, this is only fully replicated for point1and partly for point 2. In all significant spectra peaks, slip sidewall case outperforms periodic case in the SPL prediction.

Table 2 lists the modal frequencies for comparison. The classic Rossiter’s equation revised for compressible flow is also computed as reference, which reads

(20)

with constant α taken as 0.25 and κ as 0.57. From the overall quantitative comparison, we can find that slip sidewall offer a more accurate prediction for modal frequency. The slip sidewall case dominant peak (Mode II) is 13.9 Hz from experiment measurement, in contrast with 33.1 Hz for periodic sidewall case. Mode II, III, and IV are all predicted higher than experimental values. And Mode I is also conjectured to be higher than 253Hz in simulation. It’s under-predicted because the low frequency mode requires a lot more data to have credible accuracy. And data processing proved an increasing frequency for Mode I with doubling volume of data, but almost no change for other modes. Figure 10 plots these modes. It shows that both experiment and simulation spectral peaks are not fully linearly distributed as in Rossiter’s formula.

Mode 1 2 3 4Experiment 253 501 790 1000

Periodic Sidewall 209.8 534.1 858.3 1068.1Slip Sidewall 209.8 514.9 839.2 1049

Rossiter's formula 203.1 489.3 775.41 1061.5Table 2 Model frequency comparison

The frequency spectra of points located on the sidewalls are also examined in simulation (Figure 11), using data length equally long as the previous spectra. Points on left and right sidewalls from slip case, and identical two sidewalls from periodic case are plotted. These points are of the same X, Y coordinates with the measured point on mid-span plane in experiment. So experiment spectra are also presented here for comparison. For slip sidewall case, the left and right measured points are generally identical with predominant peak in nicely consistent with experiment result of mid-span points. For slip sidewall case mode I predictions, the SPL magnitude is weaker and weaker relative to the mid-span point result, from point 1 to point 3. This may implicate a more non-uniform in span-wise pressure distribution or stronger effects from the existence of slip sidewalls in downstream cavity. While for periodic case, the time-series performance on the sidewall is very close to that on mid-span plane. Though its SPL predictions still fall behind slip sidewalls, Mode I magnitude is not damping as measurement moves from point 1 to point 3. A previous spectral match of SPL dropping from 2000Hz to 3000Hz at point 1 disappears at the slip sidewalls, but remains at the periodic sidewall.

To further validate the use of wall function as well as DES methodology, slip sidewall simulation results are compared with LES and DES without using wall function (Figure 12) as listed with simulation parameters in Table 1. Though under reduced Reynolds number, LES and DES without wall function consumed 6 times and 3 times the number of grid cells in our case and resolved very fine scale (minimum Δy+ = 1.6 and 1.0 respectively). LES spectra deviate too far from experiment in terms of frequency and SPL magnitude. DES of no wall function insignificantly under-predicts (-3dB) the predominant peak for point1 and over-predicts for point 2 and point 3 (2~3dB). The two DES with/without wall function are equally capable in frequency estimating, though DES of no wall function has a higher resolution and a little outperforms for high-frequency modes (Mode III, IV). The main question of using wall


function is revealed in point 2 spectrum, where our case has a deformed shape for the dominant peak while the pure DES displays a reasonable curve. This is probably because the flow largely separates at point 2 near the downstream corner and wall function disqualifies.

Figure 13 (a)(b) shows the energy spectra of the turbulent flow in the shear layer. Also plotted is the classic Kolmogorov –5/3 slope. The inertial subrange is located around 10,000 Hz. After that, the energy drops fast, proving that almost all energy containing eddies are well captured by the grid.

V. ConclusionDetached eddy simulations (DES) of exact Reynolds number cavity flow for slip and periodic sidewall boundary

conditions were conducted with applying wall function on the non-slip surface. Simulation flow features were examined and acoustic spectra were compared with experiment, Rossiter’s formula, LES and DES without wall function by other researchers. Comparison proved DES as an effective tool for aeroacoustic simulation with moderate expense and high fidelity to experimental flow field and acoustic spectra. Slip plane is inspected as a favorable sidewall condition against the periodic boundary, from frequency estimate to sound pressure level (SPL) prediction. Examination of sidewall located points reveals similar spectra to those on center plane. But periodic sidewalls exhibit a better flow uniformity in span-wise direction, which is close to two-dimensional scenario. The adoption of wall function achieved great saving in computational cost and basic success in acoustic computation. The limitation of wall function under-predicted the SPL magnitude where flow separation is strong near the cavity rear bulkhead. The overall research verified the practicability of DES for highly unsteady flow study of large Reynolds number, and a provident use of wall function for acoustic prediction only at the places where reverse pressure is not prevailing.

References1D. Rockwell and E. Naudascher, “Review: self-sustaining oscillations of flow past cavities,” J. Fluids Eng. 100, 1978,

pp.152.2Hensaw, M. J. De. C. “M219 cavity cases. In verification and validation data for computational unsteady aerodynamics”,

Tech. Rep. RTO-TR-26, AC/323(AVT)TP/19, 2000, pp.453-472.3Hamed, A., Basu, D. and Das, K. [2004] “Assessment of hybrid turbulence model for unsteady high speed separated flow

predictions”, AIAA -2004-0684.4Nayyar, P.,Lawrie, D. Barakos, G. N. Badcock, K. J. and Richards, B. E. “Numerical study of transonic cavity flows using

large-eddy and detached-eddy simulation”, RAeS Aerospace aerodynamics research conference, London, UK, 2004.5Sinha, N., Arunajatesan, S., Shipman, J. D. and Seiner, J. M. “High fidelity simulation and measurements of aircraft

weapons bay dynamics”, AIAA -2001-2125.6Menter, F. R. “Zonal two equation turbulence models for aerodynamic flows”, AIAA-93-2906.7Suzen, Y. B., and Hoffmann, K. A. “Investigation of supersonic jet exhaust flow by one- and two-equation turbulence

models,” AIAA -98-0322.8Spalart, P. R. “Young-person’s guide to detached-eddy simulation grids”, NASA/CR-2001-211032.9Bush, R. H. “A two-equation large eddy stress model for high sub-grid shear”, AIAA -2001-2561.10Mani, M. and Paynter, G. C. “Hybrid turbulence models for unsteady simulation of jet flows”, AIAA -2002-2959.11Hamed, A., Basu, D. and Das, K. “Assessment of hybrid turbulence model for unsteady high speed separated flow

predictions”, AIAA -2004-0684.12Stanek M. J., Raman, G., Kibens, V., Ross, J. A., Odedra, J. and Peto, J. W. “Control of cavity resonance through very high

frequency forcing”, AIAA-2000-1905.13White, F.M. and Christoph, G.H. “A simple theory for two-dimensional compressible turbulent boundary layer”, Journal of

Basic Engineering, 94, 1972, pp. 636-642.14DalBello, T., Dippold, V.III and Georgiadis, N.J. “Computational study of separating flow in a planar subsonic diffuser”,

NASA/TM-2005-213894, 2005.15Dippold, V.III “Investigation of wall function and turbulence model performance within the wind code”, AIAA -2005-1002.16Rizzetta, D. P. and Visbal, M. R. “Large-eddy simulation of supersonic cavity flow fields including flow control”, AIAA

Journal, 41, No.8, 2002, pp. 1452-1462.17Basu, D., Hamed, A., Das, K., Liu, Q. and Tomko, K. “Comparative analysis of hybrid turbulence closure models in the

unsteady transonic separated flow simulations”, AIAA-2006-0117.


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Figure 1 Geometry of the computational domain

Figure 2 Structure grid of the domain


Right Sidewall

Left Sidewall

Figure 3 Mid-plane Mach number contour Figure 4 Vorticity contour on mid-plane

Figure 5 Vorticity contours on slices at X/L= 0.01, 0.2, 0.4, 0.6 and 0.8.

Figure 6 Pressure contour on the mid-plane.

Figure 7 Pressure measurement locations


① ② ③

D

D/2

Figure 8 Pressure variation from point 3 in simulation for slip sidewall case.

(a)

(b)

(c)

Figure 9 Pressure variation spectra comparison for point 1 (a), 2 (b) and 3 (c).


Figure 10 Modal frequency comparisons

(a)

(b)

(c)


Figure 11 Pressure variation spectra comparison at the sidewall locations corresponding to point 1 (a), 2 (b) and 3 (c).

(a)

(b)

(c)Figure 12 Pressure variation spectra comparison for point 1 (a), 2 (b) and 3 (c) between slip case, LES and DES.


(a) (b)

Figure 13 Turbulent kinetic energy spectra with –5/3 slope (red) in cavity shear layer at cavity opening (Y/D = 1.0) at (a) X/L=0.2, (b) X/L=0.8.


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