Anisotropic Pressure and Acceleration Spectra in Shear Flow

1. Anisotropic Pressure and Acceleration Spectra in Shear Flow Yoshiyuki Tsuji Nagoya University Japan Acknowledgement : Useful discussions and advices were given by Prof. Y. Kaneda

2. Objective T. Ishihara, K.Yoshida, and Y.Kaneda, Anisotropic Velocity Correlation Spectrum at Small Scales in Homogeneous Turbulent Shear Flow, Phys. Rev., Letter, vol.88,154501,(2002) Shear effect on inertial-range velocity statistics are directly investigated . This idea is applied to the pressure field in the uniform shear flow, and the shear effect on pressure and pressure gradient (acceleration) is studied experimentally up to the Reynolds number based on Taylor micro scale is 800.

3. 2. Pressure Measurements

4. Pressure measurement Kolmogorov length scale is for . Φ=0.15mm Φ=0.5mm 2.0 0.4mm d 12mm l δ Φ=0.08mm Φ=0.3mm Microphone Microphone: [Pa] [Hz] [mm] 1/8 inch pressure measurement inside the boundary layer

5. Probability density functions DNS: Kaneda & Ishihara Nearly homogeneous isotropic field.

6. Pressure Spectrum Kolmogorov constant (DNS:Gotoh,2001) Nearly homogeneous isotropic field.

7. Pressure measurement in Boundary layer -7/3 power-law is not observed in the overlap region of smooth-wall boundary layer even if the Reynolds number is very high. Pressure spectrum in the boundary layer

8. 2. Experiment

9. y Experiments: Driving Mixing Layer Mixing layer centerline Transition region d=350mm x/d Potential Core x/d~5 Nozzle exit Mixing layer centerline L=700mm In this region, flow reversals are unlikely and large yaw angles by the flow are infrequent.

10. Driving Mixing Layer x x/d=5 x/d=4 x/d=3 x/d=2 x/d=1 y Nozzle exit Nearly homogeneous shear flow.

11. Reynolds number & Shear parameter Reynolds number Shear parameter Simple uniform shear flow Driving mixing layer is close to the simple uniform shear flow.

12. 3. Theoretical formula

13. Modification due to the existence of mean shear. Shear effect on velocity fluctuation According to the formula presented by Ishihara, Yoshida and KanedaPRL(vol.88,154501,2002),velocity spectrum is defined by :independent of wave number :dependent of wave number :characteristic eddy size :characteristic velocity scale Isotropic part (K41) Anisotropic part for large wave numbers :Simple mean shear

14. Shear effect on velocity fluctuation Velocity spectrum is obtained by the summation with respect to over a spherical shell with radius . Isotropic part (K41) In usual experiments, one-dimensional spectrum is obtained. Anisotropic part is proportional to mean shear

15. Isotropic velocity spectrum Isotropic part (K41)

16. Anisotropic velocity spectrum Anisotropic part is proportional to mean shear even if is changed.

17. Shear effect on pressure According to the formula presented by Ishihara, Yoshioda and KanedaPRL(vol.88,154501,2002),pressure spectrum is defined by Isotropic part (K41) Modification due to the existence of mean shear. Anisotropic part :2nd order isotropic tensor :4th order isotropic tensor :Simple mean shear

18. Shear effect on pressure spectrum appears in the second order of Shear effect on pressure spectrum Pressure spectrum is obtained by the summation with respect to over a spherical shell with radius . Isotropic part (K41) Anisotropic part

19. Shear effect on pressure spectrum Pressure spectrum is obtained by the summation with respect to over a spherical shell with radius . Isotropic part (K41) Anisotropic part

20. Shear effect on pressure spectrum IYK formula is well satisfied in this experiment. Isotropic part (K41) Anisotropic part

21. Shear effect on velocity&pressure According to the formula presented by Ishihara, Yoshioda and KanedaPRL(vol.88,154501,2002),velocity&pressure spectrum is defined by Isotropic part (K41) Anisotropic part :5th order isotropic tensor

22. Shear effect on velocity&pressure spectrum Pressure-velocity spectrum is obtained by the summation with respect to over a spherical shell with radius . Anisotropic part

23. Shear effect on velocity&pressure spectrum

24. Isotropic velocity spectrum Isotropic part (K41)

25. Local mean velocity Acceleration In a usual notation, pressure relates to acceleration vector ;

26. As far as looking for the variance of and , there is no significant effect by shear. Shear effect on acceleration Similar discussion is possible in case of acceleration . Isotropic part (K41) Anisotropic part

27. Kolmogorov scaling for acceleration Following the Kolmogorov’s idea, acceleration is scaled by energy dissipation and kinematic viscosity, and the constant becomes universal. :Universal Constant

28. is not constant but increases as Reynolds number increases. There is no significant difference between and Kolmogorov scaling for acceleration :Mixing layer

29. Shear effect on pressure spectrum appears in the second order of Summary : pressure • In a simple shear flow, shear effect doe not appear clearly in a single-point statistics. • Shear effect can be evaluated by two-point statistics. Anisotropic part Anisotropic part

30. Summary: pressure-velocity correlation • In a simple shear flow, shear effect on pressure velocity correlation is evaluated by the relation. Anisotropic part

31. Summary : Acceleration • In a simple shear flow, shear effect appears on the correlation between and . • The constant defined by Kolmogorov scaling of acceleration variance is not affected clearly by shear. Anisotropic part

32. Wall pressure spectrum Frozen flow hypothesis DNS result by H. Abe for Channel Flow Frozen Flow Hypothesis for Pressure Wall pressure spectra

33. Probability density function of acceleration

34. Pressure measurement in cylinder wake Second invariance of velocity gradient tensor vorticity pressure

35. Spectra of pressure and acceleration Inertial range Inertial range