Reynolds number eあect on compressible cylindrical cavity flow

Kung-Ming CHUNG, Yi-Xun HUANG, Kun-Hung LEE,Keh-Chin CHANG

a Aerospace Science and Technology Research Center, National Cheng Kung University, Tainan 711, China

b Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 701, China

KEYWORDS Cavity flow;Compressible flow;Convection velocity;Open cavity;Oscillation

Abstract Presence of a cavity changes the mean and fluctuating pressure distributions inside and near the cavity. For cylindrical cavity flow, the diameter-to-depth ratio is the dominant factor. In this study,flow is naturally developed along a flat plate with two different lengths,resulting in different incoming boundary layer thicknesses ahead of the cavity. The effect of Reynolds number based on incoming boundary layer thickness on characteristics of mean and fluctuating pressure distributions is addressed.Pressure sensitive paint was also used to visualize the mean surface pressure patterns.The effect of Reynolds number on the classification of compressible cylindrical cavity flow and self-sustained oscillating frequency is not significant.An increase in Reynolds number results in a reduction in the value of differential pressure or momentum flux near the rear edge.

1. Introduction

Cavities occur in many engineering applications and flight vehicles.Flow over such cavities may result in structural loading problems and produce intense tonal pressure fluctuations.1-3For compressible rectangular cavity flow, the static surface pressure distributions mainly depend on the lengthto-depth ratio.4-6A shear layer is formed over an open-type cavity (length-to-depth ratio <6-8). Uniform static pressure distribution occurs along the cavity floor and discrete acoustic tones are generated, in association with the feedback loop between vortex shedding and acoustic disturbance (Rossiter’s empirical formula).7Closed-type cavity flow occurs when the length-to-depth ratio is greater than 9-15.Two distinct separation regions form downstream from the front face and upstream from the rear face. The longitudinal static surface pressure distribution shows an inflection point near the center of the cavity floor, followed by a plateau region. For a transitional-type cavity, the amplitude of the static surface pressure coefficient varies from negative values downstream from the front face to positive values ahead of the rear face.8

For incompressible cylindrical cavity flows, the diameterto-depth ratio, D/H, appears to be the dominant factor. The static surface pressure in the spanwise direction is symmetrical for D/H=5 and there is an asymmetrical flow pattern for D/H=2.0.9-11The frequencies of discrete tones can be accurately predicted using Rossiter’s empirical formula.12Chung et al.13-15examined the effect of compressibility on cylindrical cavity flow. The boundaries for open- and transitional-type cavities are roughly the same as that of compressible rectangular cavity flow.The effect of the freestream Mach number,Ma,is evident only for the static surface pressure upstream from a cavity and the amplitude of the peak pressure fluctuations.The trailing-edge expansion and amplitude of the peak pressure fluctuations decrease as the boundary layer thickness-todepth ratio, δ/H, increases (fixed boundary layer thickness).It has been noted that an increase in δ/H (≥1.0) results in the smoothing of all pressure gradients in cavity flow.16

This study aims to determine the effect of the Reynolds number according to δ,Reδ,on distributions of mean and fluctuating pressures over a range of Ma,D/H and δ/H.Turbulent boundary layer flow was naturally developed along a flat plate with two different lengths ahead of the cavity. Pressure Sensitive Paint (PSP) was also used to visualize the mean surface pressure pattern. The data are used to characterize compressible cylindrical cavity flow.

2. Experimental technique

2.1. Transonic wind tunnel and instrumentation

The transonic wind tunnel at the Aerospace Science and Technology Research Center in National Cheng Kung University is a blowdown type tunnel.It consists of compressors,air dryers,storage tanks, a hydraulic system and tunnel. A rotary perforated sleeve valve controls the stagnation pressure, po. Inside the stilling chamber, there are acoustic baffles, screens and a honeycomb to reduce the noise and turbulence intensity of the flow. The constant-area test section is 600 mm square and 1500 mm long. The present study uses solid sidewalls and perforated top/bottom walls. Downstream from the test section, two choke flaps are employed to monitor Ma under subsonic conditions.

The NEFF 620 system was used to record the test conditions and monitor the experiments through a high-speed interface. Flush-mounted dynamic pressure transducers (Kulite XCS-093-25A, B screen with a pressure-sensitive element that is 0.97 mm in diameter)were used for the mean and fluctuating surface pressure measurements.The sensors were powered by a DC power supply (GW Instek PSS-3203) of 10.0 V. Ectron amplifiers (753 A) at a gain of 20 (roll-off frequency ≈140 kHz) were used to improve the signal-to-noise ratio. A National Instruments (NI-SCXI) system was used to record the output of the sensors. The sampling rate was 5 μs and each sample record had 131,072 data points. The experimental uncertainty is 2.4%for the static surface pressure coefficient, Cp=(pw-p∞)/q∞, and 0.4% for the fluctuating pressure coefficient, Cσp=(σp-σp,∞)/q∞.

2.2. Test models and conditions

Fig. 1 Test configuration.

The test model, as shown in Fig. 1, was supported by a single sting and mounted on the bottom wall of the test section. It consisted of a flat plate and an instrumentation plate with a cylindrical cavity. The flow was naturally developed along a flat plate with two different lengths (225 mm for Case A and 450 mm for Case B). The boundary layer at the measurement locations was fully turbulent.17Nineteen instrumentation plates (150 mm2)were fabricated and the pressure transducers were flush-mounted along the centerline of each cavity in the longitudinal (y/D=0) and spanwise (x/D=0.5) directions.The cavity front face was located at 275 mm or 500 mm from the leading edge of the flat plate. Ma was 0.64, 0.70 and 0.83±0.01. p0was 172±0.5 kPa (25 psia) when the stagnation temperature was ambient temperature with a 3°C drop during a 10 s run. The incoming boundary layer thickness, δ, at 25 mm ahead of the cavity front edge was estimated to be approximately 4 mm and 7 mm for Case A and Case B,respectively.18The geometry of the cavities is summarized in Table 1.For Case A (Reδ=8.04×104-9.32×104), the D/H and δ/H ratios were 3.23-43.00 and 0.42-4.0, respectively. For Case B(Reδ=1.41×105-1.63×105),13,15the D/H ratio ranged from 4.48 to 43.00 and the value of δ/H ratio was 0.73-7.0. Note that D=31, 43 and 59 mm when H=1.0-9.6 mm.

2.3. Pressure sensitive paint

Fluorescent paints are sensitive to pressure or temperature,which is related to the oxygen quenching of a luminescent molecule. By applying PSP, full-field data, rather than point measurements, can be obtained.19-22This study used apolymer-ceramic PSP with a porous material as the supporting matrix to visualize the mean surface-pressure pattern. The response time is on the order of ten to a few hundred microseconds.23A silica gel (SiO2, a mean of particle size of 70-90 A?)and an RTV-118 were chosen as the porous particle and polymer, respectively. To mix these components (3 g:7 g), toluene was used as a solvent. The compound was sprayed onto the model surface with a spray gun.As a luminophore,the absorption and emission spectra of Ru(dpp) (Tris(4,7-diphenyl-1,10-Phenanthroline)ruthenium(II)dichloride) (purchased from ALFA Co.) were 441-467 nm and 597 nm, respectively. Ru(dpp) was dissolved in isopropyl alcohol (IPA) (1 mg:1 mL)and paint was applied on the model surface for mean pressure measurement. Note that fast-responding PSP can be used for full-field unsteady measurements in future study.24-26

Table 1 Geometry of cylindrical cavities.

Fig. 2 Calibration curve for Ru(dpp).

3. Results and discussion

3.1. Surface pressure distributions

The characteristics of compressible cylindrical cavity flow(Case B) were examined by Chung et al.15Open-type cavity flow appeared when D/H ≤6.14 and a transitional-type cavity was observed when the value of D/H equals 8.60-21.00. The effect of Ma is more pronounced on the amplitude of peak pressure fluctuations near the rear wall. For Case A and D=43 mm, the static surface pressure distributions for Ma=0.83 are shown in Fig. 3. The value of Cpfor D/H=4.48-7.17 (open-type cavities) decreases slightly inside the cavity up to a value of x/D ≈0.5, followed by an adverse pressure gradient in the second half of the cavity. The peak pressure is observed ahead of the rear face (x/D=0.919)and its value (Cp,max=0.031-0.117) increases with a greater value of D/H. Downstream from the rear corner, a drop in the value of Cpand a recovery process were observed. For D/H=8.60-21.50, the flow accelerates slightly over the front face and the amplitude of Cpincreases toward the rear face.This corresponds to transitional-type cavities. The location of the minimum Cpmoves upstream with an increase in the value of D/H.Note that the peak Cpat x/D=0.919 increases along with an increase in the value of D/H.For D/H=43.00,a plateau region in the Cpdistribution is observed,corresponding to a closed-type cavity.For Ma=0.64 and 0.70,the characteristics of Cpdistribution show similar features as those for Ma=0.83.The results show that the effect of Reδon the classification of compressible cylindrical cavity flow is not significant.

Variations of Cpnear the rear face of the cavity with D/H are shown in Fig. 4(a). At x/D=0.919, the value of Cpdecreases initially for open-type cavities (D/H=4.48-6.14),followed by an increase for transitional cavities with greater D/H. The peak Cpis observed at D/H=21.50. Notably, the greatest Cpat x/D=0.919 for Case B(greater Reδ)is observed at D/H=14.33.15The amplitude of Cpat x/D=1.058 decreases with increasing D/H up to a value of 21.50.This indicates a larger favorable pressure gradient near the rear face and a longer downstream influence for transitional-type cavities. The differential pressure near the rear face (ΔCp=Cp,x/D=0.919-Cp,x/D=1.058) corresponds to momentum flux. As shown in Fig. 4(b), the amplitude of ΔCpincreases from an open- to a transitional-type cavity, followed by a decrease for a closed-type cavity. The peak ΔCpis observed at D/H=14.33. The effect of Ma is also evident. At a given D/H, there is a decrease in the value of ΔCpwith a lower Ma.For Case B at Ma=0.83,the data are also shown for comparison. Variation of ΔCpwith D/H shows a feature similar to that seen in Case A. However, an increase in Reδresults in a reduction in the value of ΔCp, implying less momentum flux near the rear face.

Fig. 3 Static surface pressure distributions at Ma=0.83 (Case A).

Charwat et al.16showed that there are smoothing pressure gradients in cavity flow for δ/H ≥1.0. Near the rear face, this corresponds to shear layer deflection and mass removal/addition process. For a given H=9.6 mm (δ/H=0.42 for Case A), the Cpdistributions for Ma=0.83 and D/H=3.23-6.15(open-type cavities) are shown in Fig. 5. The effect of D/H on the variation of Cpin the first half of the cavity is less significant. Near the rear face, the highest Cp(x/D=0.919) is observed for D/H=6.15 and the value of Cpat x/D=1.058 decreases significantly with an increase in D/H.Variation of ΔCpwith D/H is shown in Fig. 6 (Case A). The value of ΔCpwith a fixed H increases linearly with D/H for Ma=0.64-0.83. The effect of Ma is more pronounced for D/H=6.15,i.e.greater ΔCpcan be observed with an increase in Ma. The data for a fixed D=43 mm; δ/H=0.42-0.67 are also shown for comparison. There is a slight increase in ΔCpwith a greater Ma. For a given Ma, variation of ΔCpwith D/H=4.48-6.15 is minimal, following an increase for D/H=7.17. It is also noted that the value of ΔCpis lower than that for a fixed H. Taking H into account, the favorable pressure gradient on the rear face for a given Ma is approximately the same,i.e.ΔCp/H ≈0.05 for D/H=6.15.It implies that mass removal/addition process near the rear face is approximately the same for δ/H ≤0.67.

Fig. 4 Effect of D/H on static surface pressure near rear face.

At x/D=0.5 and Ma=0.83, the Cpdistributions in the spanwise direction, Cps, are shown in Fig. 7. For a given D/H, there is slight variation in the value of Cps, indicating a symmetrical flow pattern. This agrees with previous studies of incompressible cylindrical cavity flows.9-11For transitionaltype cavities (D/H=8.60-21.50), the Cpslevel increases with a greater D/H. This corresponds to upstream movement for shear layer impingement, as shown in Fig. 3. Decreases in the value of Cpsnear the right and left faces(y/D=±0.5)are also observed.The Cpsdistributions for Ma=0.64 and 0.70 show a similar pattern.The static surface pressure pattern is also visualized using PSP.An example at Ma=0.83 and D/H=14.33(Case A)is shown in Fig.8.A symmetrical flow pattern can be seen. Note that there is an asymmetrical surface pressure pattern for D/H=2.0 in incompressible flow.9-11A horseshoe structure is clearly observed in the first half of the cavity and there is adverse pressure gradient in the second half, followed by favorable pressure gradient downstream from the rear face.The PSP data and the measurements using Kulite sensors are shown in Fig.9.The Cpdistributions show similar characteristics.For other test cases,the PSP data also show similar trend as those using Kulite sensors,in which the deviation is approximately 5%.

Fig.5 Static surface pressure distributions for open-type cavities at Ma=0.83 (Case A).

Fig.6 Differential pressure near rear face for open-type cavities(Case A).

The effect of Reδis of primary interest in this study. The flow was naturally developed along a flat plate with two different lengths (Case A and Case B). The static surface pressure distributions at Ma=0.83 are shown in Fig. 10. For the closed-type cavity of D/H=43.00, the Cpdistributions are identical for δ/H=4.0 and 7.0. For transitional- and opentype cavities,the effect of Reδis evident only at the peak value of Cpand the downstream expansion.A decrease in Reδ(Case A) results in an increase in the peak Cpat x/D=0.919 and a slight reduction at x/D=1.058.Variation of ΔCpwith δ/H for open-type cavities is shown in Fig. 11. The value of ΔCpdecreases when the value of δ/H increases for both cases,indicating less expansion near the rear face. This agrees with the results of Charwat et al.,16in which an increase in δ/H results in the smoothing of all pressure gradients in cavity flow.An increase in Ma results in a slight decrease in the value of ΔCp. The effect of Reδis also evident. For a given δ/H, the value of ΔCpfor Case B is greater than that for Case A(lower Reδ).

Fig. 7 Spanwise static surface pressure distributions at Ma=0.83 (Case A).

3.2. Surface pressure fluctuations

The feedback loop for a cavity flow results in discrete tones.7The distributions of Cσpfor Ma=0.83 (Case A) are shown in Fig. 12. Cσprepresents the relative normalized local surface pressure fluctuations with respect to the undisturbed flow.There is slight variation in the level of Cσpupstream from the cavity for all test cases, indicating minor upstream influence.On the cavity floor,there is a fairly uniform Cσpdistribution for D/H=43.00 (a closed-type cavity). A peak fluctuating pressure, Cσp，peak, is observed near the rear face.For transitional-type cavities (D/H=8.60-21.50), a minor Cσp，peakis observed at x/D=0.36, followed by downstream damping and an increase toward the rear face. A peak value is observed at x/D=1.058. The amplitude of Cσpincreases gradually along the longitudinal direction for open-type cavities (D/H ≤7.17) and the value of Cσp，peakincreases with a decrease in D/H. For a given H=9.6 mm, as shown in Fig.13,the effect in variation of D for open-type cavities is evident.The value of ΔCpfor D/H=3.23 is lower than those for D/H=4.48 and 6.15. Further, variation of Cσp，peakwith D/H is shown in Fig. 14. For a given D=43 mm; D/H=4.48-7.17,open-type cavities, there is slight variation in the value of Cσp，peakwith D/H for Ma=0.64-0.83. The value of Cσp，peakincreases with a greater D/H=3.23-6.15 for a fixed H=9.6 mm. The amplitude of Cσp，peakfor Ma=0.83(D/H=4.48 and 6.15) is lower than that for Ma=0.64 and 0.70. Moreover, variations of Cσp，peakwith D/H appear to be consistent with the trend for ΔCp, as shown in Fig. 6. This implies that momentum flux near the rear face is the dominant factor on the amplitude of Cσp，peakfor compressible cylindrical cavity flow.

Fig. 8 Visualization for static surface pressure distributions at Ma=0.83 and D/H=14.33 (Case A).

Fig. 9 Cp distributions at Ma=0.83 and D/H=14.33 (Case A).

An example of Cσpdistributions for Ma=0.83 is shown in Fig. 15 to address the effect of Reδ. Variations of Cσpin the longitudinal direction show similar trends for both Case A and Case B. For a given D/H, the value of Cσp，peakat x/D=1.058 increases when there is a reduction in δ/H (or Reδ). Note that the effect of Reδon the distributions of Cσpfor Ma=0.64 and 0.70 are similar to those for Ma=0.83.The effects of Ma and δ/H (or Reδ) on the variation in Cσp，maxare shown in Fig. 16, in which D/H=4.48-8.60. An increase in the value of δ/H for both Case A and Case B results in a decrease in the value of Cσp，max.This results from the lower ΔCpnear the rear face. The effect of Reδappears to be more significant at lower values of δ/H (<1.00).

Fig. 10 Static surface pressure distributions at Ma=0.83 (δ/H effect).

Previous studies12,27,28have demonstrated that high intensity acoustic tones are well predicted by the semi-empirical Rossiter’s formula for open-type cavities.7The steepest descent optimization algorithm is adopted to evaluate the optimal values of the empirical parameters.29,30The effective streamwise lengthis used as the characteristic length.31Note that the uncertainty of the Strouhal number, St, is estimated to be±0.007.Variations of St(the first three modes n)with Ma for open-type cavities are shown in Fig. 17. For a given D=43 mm and D/H=4.48-7.17, there is a slight decrease in the values of St at higher value of Ma for all three modes. For a given H=9.6 mm and D/H=6.14-6.20, a similar trend is observed. The optimized α (phase lag) and kc(convection velocity)are 0.11 and 0.53,respectively.Note that the values are 0.12 and 0.48 for Case B. This implies that the effect of Reδon the self-sustained oscillations of compressible cylindrical cavity flow is minimal.

Fig. 11 Differential pressure near rear face for open-type cavities (δ/H effect).

Fig. 12 Fluctuating pressure distributions for Ma=0.83 (Case A).

Fig.13 Fluctuating pressure distributions for open-type cavities at Ma=0.83 (Case A).

Fig. 14 Peak pressure fluctuations near rear face for open-type cavities (Case A).

4. Conclusions

This study aims to characterize compressible cylindrical cavity flow.The effect of δ and Reδare presented.The results indicate that variation in δ/H affects the amplitude of peak pressure fluctuations. For mean surface pressure distributions, the effect is more significant for an open-type cavity, than for a transitional-type or closed-type cavity. However, variation in favorable pressure gradient, corresponding to mass removal/addition, for open-type cavities is minimized when δ/H ≤0.67. The effect of Reδon the classification of compressible cylindrical cavity flow is not significant and it is evident on peak value of Cpfor transitional- and open-type cavities.There is an increase in peak Cpwhen there is a decrease in Reδ. The visualization by pressure sensitive paint shows that the pressure pattern is symmetrical and curved in the spanwise direction. Variation in Reδat self-sustained oscillating frequency is minimized and there is only a minor effect on empirical constants in the Rossiter equation.

Fig. 15 Fluctuating pressure distributions at Ma=0.83 (δ/H effect).

Fig. 16 Peak pressure fluctuations (δ/H effect).

Fig. 17 Mach number effect on Strouhal number for open-type cavities.

Acknowledgement

The authors acknowledge the financial support from the Ministry of Science and Technology, Taiwan, China (MOST 103-2923-E-006-006-MY3).