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

Objective 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. 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)

2. Pressure Measurements

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

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

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

Pressure measurement in Boundary layer Pressure spectrum in the 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.

2. Experiment

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

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

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

3. Theoretical formula

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

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

Isotropic velocity spectrum Isotropic part (K41)

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

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

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

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

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

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

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

Shear effect on velocity&pressure spectrum

Isotropic velocity spectrum Isotropic part (K41)

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

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

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

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

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 Shear effect on pressure spectrum appears in the second order of Anisotropic part

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

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

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

Probability density function of acceleration

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

Spectra of pressure and acceleration Inertial range