numerical simulation of unsteady flow in resonance tube
AIAA 2002-1118
NUMERICAL SIMULATION OF UNSTEADY FLOW IN RESONANCE TUBE A. Hamed*, K. Das**, and D. Basu** Department of Aerospace Engineering and Engineering Mechanics University of Cincinnati Cincinnati, OH 45221-0070
Abstract Computational results are presented for the unsteady flow in a powered Hartmann resonance tube. The results were obtained using the WIND code for the two dimensional viscous flow in a five-block solution domain that includes the CD nozzle supplying the jet to the closed end tube within the ambient surrounding. The computed results are presented for the unsteady flow mechanisms involved in the resonance tube operation. These include the filling and emptying of the tube, which causes the lateral deflection of the jet and the changing shock structure in the under-expanded jet. The time history and Fourier transform of the flow pressure oscillations are also presented. The fundamental resonance frequency is compared to that computed from an approximate analytical formula. Nomenclature CD c D f L Mj SPL
Convergent divergent Speed of sound Tube width Frequency Tube length Jet Mach number Sound Pressure level
technique with a very short exposure time to obtain instantaneous and phase-averaged flow images for the exiting jet from a Hartmann tube. They studied the effect of tube depth, jet Mach number and separation distance on the power spectra of both far acoustic field and near pressure field. Brocher et al.6 explained the oscillation mechanism in the resonance tubes in terms of the compression and evacuation phases. In the compression phase, the jet penetrates into the tube and compresses the air. In the evacuation phase, the compressed gas in the tube expands to the atmosphere and the jet is deviated laterally. These phases are repeated in a high frequency oscillating hydrodynamic perturbation. They carried out experiments for jet Mach numbers ranging between 0.1 and 2.0, using a thin cylindrical body along the axis of the jet to produce a wake, and measured the pressure fluctuations at various locations in the resonance tube. The measured frequencies agreed with those computed from a formula based on linear acoustic theory for Mj1. Experimental results 6 suggest that the highest amplitude oscillations are obtained when the gap between the supply jet and the resonance tube was equal to the diameter of the resonance tube for subsonic flows and twice the diameter for supersonic flows.
Introduction Recent interest in characterizing the performance of Hartmann resonance tubes1 is motivated by their potential application as high frequency actuators2,3,4,5 for suppressing high speed jet noise and supersonic flow oscillations in open cavities. Experimental results in constant area and tapered tubes indicate that the pressure oscillation modes are dependant on the stagnation conditions, speed of sound, and tube length6. Kastner and Samimy7 recently used a planar flow visualization __________ *Professor, AIAA Fellow **Graduate Student, Student Member AIAA
In this work numerical solutions were obtained for the unsteady flow field in a forced Hartmann tube using the WIND code8. To the best of our knowledge this is one of the first reported unsteady flow simulations using WIND. Only an inviscid one-dimensional flow example9 in a shock tube was used to evaluate the overshoot in the computed final pressure distribution, and assess its agreement with theoretical results. In the investigated resonance tube configuration, the jet from a CD nozzle is directed towards the mouth of a constant area duct, which is closed at the opposite end. A wake producing needle at the nozzle center enables the supersonic flow penetration of the duct during the compression phase. Results are presented for the
Copyright ©2002 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
AIAA 2002-1118
which are superimposed on the Mach carpet plots, demonstrate the changing lateral jet deflection in the gap between the nozzle exit and tube entrance and the associated pulsing mass injection. The peak jet lateral deflection coincides with the flow purging from the resonance tube near the center. A shock train is formed in the jet that is more obvious at the higher jet deflections. The vorticity contours and flow velocity vectors inside the tube are shown in Figure 4 at the corresponding times. The vectors show the jet penetration into the tube during the compression phase and the tube evacuation during the expansion phase, which causes the jet deflection. The compression and evacuation phases repeated themselves over the course of the unsteady flow simulations.
unsteady flow inside the tube and across the gap between the nozzle exit and the tube mouth. The pressure fluctuations frequency is compared with linear acoustic theory predictions. Methodology The solution domain shown in Figure 1 extends from the CD nozzle throat through that is 0.82D times the exit width D, across the 1.55 D gap to the closed end of the L/D=1.8 tube in the axial direction, and 5.5 D outside the tube in the lateral direction. Baldwin-Lomax turbulence model was used in all five blocks of the two-dimensional solution domain. Approximately 30,000 points were used in the lower half of the symmetric domain as shown in Figure 2. The grid points were clustered near the walls and in the wake of the 0.0447D central needle. The value of y+ for the first grid point next to the wall was 9.36 inside the nozzle near the exit.
Figure 5 presents the history of the pressure oscillations at five points across the gap between the nozzle and tube, shown schematically in Figure 6, and at the closed end of the tube. According to these results the relative duration of the compression and expansion phases within the cycle changes with location. Near the nozzle exit the expansion phase takes much longer than the compression phase. At the closed end wall of the tube, the compression wave lasts slightly longer than the expansion wave. This indicates that the average compression wave speed is lower than the average expansion wave speed inside the tube. On the other hand the compression wave speed is much higher than that of the expansion wave near the nozzle exit.
The unsteady compressible viscous flow solution was obtained for the compressible NavierStokes equations in conservation law form using the WIND based on the methodology highlighted in Table 1. Van-Leer second order scheme and Jacobi implicit operator with 30 sub-iterations in each time step with level of convergence = 0.0001 and fourstage Runge-Kutta time stepping scheme were used. In all five blocks a TVD factor of 1.0 was used to obtain the time-accurate flow solution. The applied boundary conditions in the WIND code are summarized in table 2. They consist of arbitrary inflow boundary conditions at the nozzle inlet with a stagnation temperature of 784.72 oR and stagnation pressure of 50.415 psia, corresponding to a nozzle pressure ratio of 3.37. Viscous no slip boundary conditions were applied at the nozzle, needle and tube surfaces, and symmetric boundary conditions were applied at the plane of symmetry in the tube and gap. At the free-stream boundaries, where the Mach number = 0.03, static pressure = 14.96 psia, and static temperature = 519 oR, extrapolation was applied. The simulations were performed over 400,000 time steps of 4.2345×10-8 seconds. It took 225,000 time steps to purge the transient flow and establish the tube resonance. Five cycles were predicted over the remaining 175,000 time steps.
Figure 7 presents the SPL spectra obtained from the pressure fluctuations at the tube closed-end using 8192 data points. The spectra indicates a first mode frequency of 780 Hz. The following approximate formula for the pressure fluctuations frequency inside the tube is based on linear acoustic theory6: f = c / 4L
(1)
Where f is the frequency of the pressure oscillations in the tube, L is the tube length and c is the speed of sound. The above equation gives a frequency of 950 Hz for our L/D=1.8 tube. Experimental evidence suggests that equation 1 over predicts the frequency for supersonic jets6 and short tubes2. Brocher and Duport10 added a correction ∆L in the equation to account for the end effects.
Results and Discussion A time sequence of the Mach number contours in the gap between the nozzle and tube are presented in Figure 3. The flow velocity vectors,
2
AIAA 2002-1118
References
Summary The computed unsteady flow field in a powered Hartmann resonance tube demonstrated the CD nozzle jet penetration into the closed end tube during the compression phase and its lateral deflection during the tube evacuation phase. The solution also demonstrated the associated lateral pulsation of the jet in the gap between the nozzle exit and tube inlet. The time history of the pressure fluctuations are presented across the gap and at the tube end wall. The dominant frequency determined from the Fourier analysis of the pressure fluctuations inside the tube was compared with an analytical formula based on linear acoustic theory.
1.
Hartmann, J., and Trolle, B., “A new Acoustic Generator,” J. Sci. Instr., Vol. 4, pp. 101-111, 1927. 2. Raman, G., Mills, A., Othman, S., and Kibens, V., “Development of Powered Resonance Tube Actuators for Active Flow Control,” ASME FEDSM 2001-18273, June 2001. 3. Stanek, 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 Paper 2000-1905, June 2000. 4. Stanek, M. J., Raman, G., Kibens, V., Ross, J. A., Odedra, J., and Peto, J. W.,“ Suppression of Cavity Resonance Using High Frequency Forcing – The Characteristic Signature of Effective Devices,” AIAA Paper 2001-2128, June 2000. 5. Raman, G., Kibens, V., Cain, A., and Lepicovsky, J., “Advanced Actuator Concepts for Active Aeroacoustic Control,” AIAA Paper 2000-1930, June 2000. 6. Brocher, E., Maresca, C. and Bornay, M.H., “Fluid Dynamics of the resonance tube,” J. of Fluid mech., Vol. 43, Part 2, pp.369-384, 1970. 7. Kastner, Jeff and Samimy Mo, “ Development and Characterization of Hartmann Tube Base Fluidic Actuators for High Speed Flow Control,” AIAA Paper 2002-0128, January 2001. 8. “WIND: The Prediction Flow Solver of the NPARC Alliance,” AIAA Paper 98-0995, January 1998. 9. http://www.grc.nasa.gov/www/wind/valid/stube/ stube01/stube01.html. 10. Brocher, B. and Duport, E., “Resonance Tubes in a subsonic flowfield,” AIAA J., Vol. 26, No. 3, pp. 548-551, 1988.
The results demonstrate that the WIND code can resolve the frequencies and amplitudes involved in the unsteady resonance tube flows. We plan to continue the simulations for Hartmann tubes with axi-symmetric CD as well as convergent nozzles, and for asymmetric configurations with one side shield that direct the pulsating flow to exit from one side for higher pulsating jet velocities. Acknowledgements This research was supported by a DAGSI grant, with AFRL/VA. The authors would like to thank Mr. M. Stanek of AFRL/VA, the project monitor, and Dr. S. Arunajatesan of Combustion Research and Flow Technology, for providing the resonance tube geometry, and Mr. O. Mesalhy for his help with the figures and presentations.
3
AIAA 2002-1118
Table 1. WIND code features (used schemes highlighted) •=
Numerics
Turbulence
Validations
•= •= •=
Finite Difference Discretization of Euler and Reynolds Averaged Navier-Stokes equations. For axisymmetric, two-dimensional and three-dimensional geometric configurations. Provisions for second order Roe/Van Leer mixed upwind and central and Roe upwind modified for stretched grids. Second and fourth order Jameson type smoothing. TVD operator to prevent non-physical instabilities. Runge Kutta time stepping or global Newton iterations for time integration
•= •= •= •=
Baldwin Lomax and Cebeci-Smith algebraic turbulence models. Baldwin-Barth and Spalart-Allmaras one equation models Mentors two-equation SST model and Chien low Reynolds number k-ε model. Spalart Detached Eddy Simulation Model.
•= •= •= •= •= •=
Turbulent Flow over Flat Plate RAE Aerofoil Fraser Duct Axisymmetric jet Back Step Ejector Nozzle
•= •=
Table 2. Boundary Conditions Zone 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5
Boundary I1 IMAX J1 JMAX I1 IMAX J1 JMAX I1 IMAX J1 JMAX I1 IMAX J1 JMAX I1 IMAX J1 JMAX
Condition Free stream Coupled to zone 3, I1 Free stream Viscous Wall Arbitrary Inflow Coupled with Zone 3, I1 Viscous Wall Viscous Wall Viscous wall, coupled to Zone 1, IMAX and Zone 2 IMAX Coupled to Zone 4, I1 and Zone 5 I1 Free stream Reflection Coupled to Zone 3 Free stream Free stream Viscous Wall Coupled to Zone 3 Viscous Wall Viscous Wall Reflection
4
AIAA 2002-1118
Fig. 1. . Schematic diagram of the computational domain
Fig. 2 Computational Grid (shown thinned by a factor of five)
5
AIAA 2002-1118
Fig. 4a. Vorticity and velocity vectors inside tube (T=0.0135 sec)
Fig. 3a. Mach number and velocity vectors between nozzle exit and tube inlet. (T=0.0135 sec)
Fig. 4b. Vorticity and velocity vectors inside tube (T=0.0139 sec)
Fig. 3b. Mach number and velocity vectors between nozzle exit and tube inlet. (T=0.0139 sec)
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AIAA 2002-1118
Fig. 4c. Vorticity and velocity vectors inside tube (T=0.0143 sec)
Fig. 3c. Mach number and velocity vectors between nozzle exit and tube inlet. (T=0.0143 sec)
Fig. 3d. Mach number and velocity vectors between nozzle exit and tube inlet. (T=0.0148 sec)
Fig. 4d. Vorticity and velocity vectors inside tube (T=0.0148 sec)
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AIAA 2002-1118
Fig 5a. Pressure fluctuation history at point 1.
Fig 5b. Pressure fluctuation history at point 2.
Fig 5c. Pressure fluctuation history at point 3.
Fig 5d. Pressure fluctuation history at point 4
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AIAA 2002-1118
Fig 5e. Pressure fluctuation history at point 5.
Fig 5f. Pressure fluctuation history at tube end wall
Fig 7. SPL spectra at the tube end-wall.
Fig 6. Schematic showing the points of observation
9
NUMERICAL SIMULATION OF UNSTEADY FLOW IN RESONANCE TUBE A. Hamed*, K. Das**, and D. Basu** Department of Aerospace Engineering and Engineering Mechanics University of Cincinnati Cincinnati, OH 45221-0070
Abstract Computational results are presented for the unsteady flow in a powered Hartmann resonance tube. The results were obtained using the WIND code for the two dimensional viscous flow in a five-block solution domain that includes the CD nozzle supplying the jet to the closed end tube within the ambient surrounding. The computed results are presented for the unsteady flow mechanisms involved in the resonance tube operation. These include the filling and emptying of the tube, which causes the lateral deflection of the jet and the changing shock structure in the under-expanded jet. The time history and Fourier transform of the flow pressure oscillations are also presented. The fundamental resonance frequency is compared to that computed from an approximate analytical formula. Nomenclature CD c D f L Mj SPL
Convergent divergent Speed of sound Tube width Frequency Tube length Jet Mach number Sound Pressure level
technique with a very short exposure time to obtain instantaneous and phase-averaged flow images for the exiting jet from a Hartmann tube. They studied the effect of tube depth, jet Mach number and separation distance on the power spectra of both far acoustic field and near pressure field. Brocher et al.6 explained the oscillation mechanism in the resonance tubes in terms of the compression and evacuation phases. In the compression phase, the jet penetrates into the tube and compresses the air. In the evacuation phase, the compressed gas in the tube expands to the atmosphere and the jet is deviated laterally. These phases are repeated in a high frequency oscillating hydrodynamic perturbation. They carried out experiments for jet Mach numbers ranging between 0.1 and 2.0, using a thin cylindrical body along the axis of the jet to produce a wake, and measured the pressure fluctuations at various locations in the resonance tube. The measured frequencies agreed with those computed from a formula based on linear acoustic theory for Mj1. Experimental results 6 suggest that the highest amplitude oscillations are obtained when the gap between the supply jet and the resonance tube was equal to the diameter of the resonance tube for subsonic flows and twice the diameter for supersonic flows.
Introduction Recent interest in characterizing the performance of Hartmann resonance tubes1 is motivated by their potential application as high frequency actuators2,3,4,5 for suppressing high speed jet noise and supersonic flow oscillations in open cavities. Experimental results in constant area and tapered tubes indicate that the pressure oscillation modes are dependant on the stagnation conditions, speed of sound, and tube length6. Kastner and Samimy7 recently used a planar flow visualization __________ *Professor, AIAA Fellow **Graduate Student, Student Member AIAA
In this work numerical solutions were obtained for the unsteady flow field in a forced Hartmann tube using the WIND code8. To the best of our knowledge this is one of the first reported unsteady flow simulations using WIND. Only an inviscid one-dimensional flow example9 in a shock tube was used to evaluate the overshoot in the computed final pressure distribution, and assess its agreement with theoretical results. In the investigated resonance tube configuration, the jet from a CD nozzle is directed towards the mouth of a constant area duct, which is closed at the opposite end. A wake producing needle at the nozzle center enables the supersonic flow penetration of the duct during the compression phase. Results are presented for the
Copyright ©2002 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
AIAA 2002-1118
which are superimposed on the Mach carpet plots, demonstrate the changing lateral jet deflection in the gap between the nozzle exit and tube entrance and the associated pulsing mass injection. The peak jet lateral deflection coincides with the flow purging from the resonance tube near the center. A shock train is formed in the jet that is more obvious at the higher jet deflections. The vorticity contours and flow velocity vectors inside the tube are shown in Figure 4 at the corresponding times. The vectors show the jet penetration into the tube during the compression phase and the tube evacuation during the expansion phase, which causes the jet deflection. The compression and evacuation phases repeated themselves over the course of the unsteady flow simulations.
unsteady flow inside the tube and across the gap between the nozzle exit and the tube mouth. The pressure fluctuations frequency is compared with linear acoustic theory predictions. Methodology The solution domain shown in Figure 1 extends from the CD nozzle throat through that is 0.82D times the exit width D, across the 1.55 D gap to the closed end of the L/D=1.8 tube in the axial direction, and 5.5 D outside the tube in the lateral direction. Baldwin-Lomax turbulence model was used in all five blocks of the two-dimensional solution domain. Approximately 30,000 points were used in the lower half of the symmetric domain as shown in Figure 2. The grid points were clustered near the walls and in the wake of the 0.0447D central needle. The value of y+ for the first grid point next to the wall was 9.36 inside the nozzle near the exit.
Figure 5 presents the history of the pressure oscillations at five points across the gap between the nozzle and tube, shown schematically in Figure 6, and at the closed end of the tube. According to these results the relative duration of the compression and expansion phases within the cycle changes with location. Near the nozzle exit the expansion phase takes much longer than the compression phase. At the closed end wall of the tube, the compression wave lasts slightly longer than the expansion wave. This indicates that the average compression wave speed is lower than the average expansion wave speed inside the tube. On the other hand the compression wave speed is much higher than that of the expansion wave near the nozzle exit.
The unsteady compressible viscous flow solution was obtained for the compressible NavierStokes equations in conservation law form using the WIND based on the methodology highlighted in Table 1. Van-Leer second order scheme and Jacobi implicit operator with 30 sub-iterations in each time step with level of convergence = 0.0001 and fourstage Runge-Kutta time stepping scheme were used. In all five blocks a TVD factor of 1.0 was used to obtain the time-accurate flow solution. The applied boundary conditions in the WIND code are summarized in table 2. They consist of arbitrary inflow boundary conditions at the nozzle inlet with a stagnation temperature of 784.72 oR and stagnation pressure of 50.415 psia, corresponding to a nozzle pressure ratio of 3.37. Viscous no slip boundary conditions were applied at the nozzle, needle and tube surfaces, and symmetric boundary conditions were applied at the plane of symmetry in the tube and gap. At the free-stream boundaries, where the Mach number = 0.03, static pressure = 14.96 psia, and static temperature = 519 oR, extrapolation was applied. The simulations were performed over 400,000 time steps of 4.2345×10-8 seconds. It took 225,000 time steps to purge the transient flow and establish the tube resonance. Five cycles were predicted over the remaining 175,000 time steps.
Figure 7 presents the SPL spectra obtained from the pressure fluctuations at the tube closed-end using 8192 data points. The spectra indicates a first mode frequency of 780 Hz. The following approximate formula for the pressure fluctuations frequency inside the tube is based on linear acoustic theory6: f = c / 4L
(1)
Where f is the frequency of the pressure oscillations in the tube, L is the tube length and c is the speed of sound. The above equation gives a frequency of 950 Hz for our L/D=1.8 tube. Experimental evidence suggests that equation 1 over predicts the frequency for supersonic jets6 and short tubes2. Brocher and Duport10 added a correction ∆L in the equation to account for the end effects.
Results and Discussion A time sequence of the Mach number contours in the gap between the nozzle and tube are presented in Figure 3. The flow velocity vectors,
2
AIAA 2002-1118
References
Summary The computed unsteady flow field in a powered Hartmann resonance tube demonstrated the CD nozzle jet penetration into the closed end tube during the compression phase and its lateral deflection during the tube evacuation phase. The solution also demonstrated the associated lateral pulsation of the jet in the gap between the nozzle exit and tube inlet. The time history of the pressure fluctuations are presented across the gap and at the tube end wall. The dominant frequency determined from the Fourier analysis of the pressure fluctuations inside the tube was compared with an analytical formula based on linear acoustic theory.
1.
Hartmann, J., and Trolle, B., “A new Acoustic Generator,” J. Sci. Instr., Vol. 4, pp. 101-111, 1927. 2. Raman, G., Mills, A., Othman, S., and Kibens, V., “Development of Powered Resonance Tube Actuators for Active Flow Control,” ASME FEDSM 2001-18273, June 2001. 3. Stanek, 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 Paper 2000-1905, June 2000. 4. Stanek, M. J., Raman, G., Kibens, V., Ross, J. A., Odedra, J., and Peto, J. W.,“ Suppression of Cavity Resonance Using High Frequency Forcing – The Characteristic Signature of Effective Devices,” AIAA Paper 2001-2128, June 2000. 5. Raman, G., Kibens, V., Cain, A., and Lepicovsky, J., “Advanced Actuator Concepts for Active Aeroacoustic Control,” AIAA Paper 2000-1930, June 2000. 6. Brocher, E., Maresca, C. and Bornay, M.H., “Fluid Dynamics of the resonance tube,” J. of Fluid mech., Vol. 43, Part 2, pp.369-384, 1970. 7. Kastner, Jeff and Samimy Mo, “ Development and Characterization of Hartmann Tube Base Fluidic Actuators for High Speed Flow Control,” AIAA Paper 2002-0128, January 2001. 8. “WIND: The Prediction Flow Solver of the NPARC Alliance,” AIAA Paper 98-0995, January 1998. 9. http://www.grc.nasa.gov/www/wind/valid/stube/ stube01/stube01.html. 10. Brocher, B. and Duport, E., “Resonance Tubes in a subsonic flowfield,” AIAA J., Vol. 26, No. 3, pp. 548-551, 1988.
The results demonstrate that the WIND code can resolve the frequencies and amplitudes involved in the unsteady resonance tube flows. We plan to continue the simulations for Hartmann tubes with axi-symmetric CD as well as convergent nozzles, and for asymmetric configurations with one side shield that direct the pulsating flow to exit from one side for higher pulsating jet velocities. Acknowledgements This research was supported by a DAGSI grant, with AFRL/VA. The authors would like to thank Mr. M. Stanek of AFRL/VA, the project monitor, and Dr. S. Arunajatesan of Combustion Research and Flow Technology, for providing the resonance tube geometry, and Mr. O. Mesalhy for his help with the figures and presentations.
3
AIAA 2002-1118
Table 1. WIND code features (used schemes highlighted) •=
Numerics
Turbulence
Validations
•= •= •=
Finite Difference Discretization of Euler and Reynolds Averaged Navier-Stokes equations. For axisymmetric, two-dimensional and three-dimensional geometric configurations. Provisions for second order Roe/Van Leer mixed upwind and central and Roe upwind modified for stretched grids. Second and fourth order Jameson type smoothing. TVD operator to prevent non-physical instabilities. Runge Kutta time stepping or global Newton iterations for time integration
•= •= •= •=
Baldwin Lomax and Cebeci-Smith algebraic turbulence models. Baldwin-Barth and Spalart-Allmaras one equation models Mentors two-equation SST model and Chien low Reynolds number k-ε model. Spalart Detached Eddy Simulation Model.
•= •= •= •= •= •=
Turbulent Flow over Flat Plate RAE Aerofoil Fraser Duct Axisymmetric jet Back Step Ejector Nozzle
•= •=
Table 2. Boundary Conditions Zone 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5
Boundary I1 IMAX J1 JMAX I1 IMAX J1 JMAX I1 IMAX J1 JMAX I1 IMAX J1 JMAX I1 IMAX J1 JMAX
Condition Free stream Coupled to zone 3, I1 Free stream Viscous Wall Arbitrary Inflow Coupled with Zone 3, I1 Viscous Wall Viscous Wall Viscous wall, coupled to Zone 1, IMAX and Zone 2 IMAX Coupled to Zone 4, I1 and Zone 5 I1 Free stream Reflection Coupled to Zone 3 Free stream Free stream Viscous Wall Coupled to Zone 3 Viscous Wall Viscous Wall Reflection
4
AIAA 2002-1118
Fig. 1. . Schematic diagram of the computational domain
Fig. 2 Computational Grid (shown thinned by a factor of five)
5
AIAA 2002-1118
Fig. 4a. Vorticity and velocity vectors inside tube (T=0.0135 sec)
Fig. 3a. Mach number and velocity vectors between nozzle exit and tube inlet. (T=0.0135 sec)
Fig. 4b. Vorticity and velocity vectors inside tube (T=0.0139 sec)
Fig. 3b. Mach number and velocity vectors between nozzle exit and tube inlet. (T=0.0139 sec)
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AIAA 2002-1118
Fig. 4c. Vorticity and velocity vectors inside tube (T=0.0143 sec)
Fig. 3c. Mach number and velocity vectors between nozzle exit and tube inlet. (T=0.0143 sec)
Fig. 3d. Mach number and velocity vectors between nozzle exit and tube inlet. (T=0.0148 sec)
Fig. 4d. Vorticity and velocity vectors inside tube (T=0.0148 sec)
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AIAA 2002-1118
Fig 5a. Pressure fluctuation history at point 1.
Fig 5b. Pressure fluctuation history at point 2.
Fig 5c. Pressure fluctuation history at point 3.
Fig 5d. Pressure fluctuation history at point 4
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AIAA 2002-1118
Fig 5e. Pressure fluctuation history at point 5.
Fig 5f. Pressure fluctuation history at tube end wall
Fig 7. SPL spectra at the tube end-wall.
Fig 6. Schematic showing the points of observation
9
