The spatial resolution of velocity and velocity gradient turbulence - PowerPoint PPT Presentation

The spatial resolution of velocity and velocity gradient turbulence statistics measured with multi-sensor hot-wire probes P.V. Vukoslav č evi ć , Univ. of Montenegro N. Beratlis, E. Balaras and J.M. Wallace, Univ. of Maryland

Overview • Background • Operational principles of 12-sensor hot-wire probes • Resolution of a 12-sensor Hot-wire probe • Highly resolved DNS of a Narrow Channel Turbulent Flow at R τ = 200 • Resolution effects on velocity component statistics • Resolution effects on vorticity component statistics • Summary and Conclusions

12-sensor Hot-wire Probe Dimensions in mm 12- sensor probe used to measure velocity and velocity gradient properties of turbulent flows P. Vukoslav č evi ć , J.M. Wallace & J.-L. Balint (1991) J. Fluid Mech. 228 A. Tsinober, E. Kit & T. Dracos (1992) J. Fluid Mech. 242 B. Marasli, P. Nguyen , J.M. Wallace (1993) Exp. Fluids. 15 P. Vukoslav č evi ć & J.M. Wallace (1996) Meas. Sci. Technol. 7 A. Honkan & Y. Andreopoulos (1997) J. Fluid Mech. 350 L. Ong & J.M. Wallace (1998) J. Fluid Mech. 367 R. Loucks (1998) Ph.D. Dissertation, University of Maryland

Operational principles of hot-wire probe The effective cooling velocity is usually defined by Jorgensen’s expression = + + 2 2 2 2 2 2 . U U k U h U e n t b Using this expression, the effective velocities cooling each sensor can be expressed as a function of the three velocity components at the sensor center, = + + + + + 2 2 2 2 . U U a V a W a V U a W U a W V 1 2 3 4 5 eij ij ij ij ij ij ij ij ij ij ij ij ij ij ij The necessary assumption that the velocity variation is linear over probe spacing area leads to a set of 12 equations of the following form, { } = ∂ ∂ ∂ ∂ 2 , , , , ( , , ) / , ( , , ) / , U F a U V W U V W y U V W z 0 0 0 eij ij ijk In terms of the velocity components at the probe centers U 0 , V 0 , W 0 and the six velocity gradients as unknowns. Real probe: calibration proc. Ideal probe: k=0, h=1, α =45 deg. 1.0 2.8 - 1.70 - 0.15 - 0.15 − 1 2 2 0 0 2.8 1.0 0.15 - 1.70 - 0.15 − 2 1 0 2 0 = = a a 1 1 jk 1.0 2.8 1.70 - 0.15 - 0.15 jk 1 2 2 0 0 2.8 1.0 0.15 1.70 - 0.15 2 1 0 2 0

Physical experiment The effective cooling velocity for each sensor, U ij, can be found from King’s Law or from a polynomial fit 5 ∑ − = 1 2 . p b E U = + 2 , n E A BU p e e = 1 p Virtual experiment Virtual probe with S y = 8 ∆ y over the numerical grid where ∆ y is 1 viscous length + = 2 , 4 , 8 , 12 S y

DNS data base ° data from KMM -- present DNS High and low speed streaks at an instant in time, in a plane parallel to the wall at y+=14 Ratio of Kolmogorov to viscous length scale

Velocity Statistics - RMS 3.5 3.5 3 3 2.5 2.5 2 2 u'/u t v'/u t 1.5 1.5 1 1 0.5 0.5 0 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 y + y + 3.5 3 ♦ DNS 2.5 y+ = 15 = 150 2 ■ S+=2 → 1.2 η → 0.6 η w'/u t ▲ S+=4 → 2.4 η → 1.2 η 1.5 x S+=8 → 4.8 η → 2.4 η 1 + S+=12 → 7.2 η → 3.6 η 0.5 0 0 20 40 60 80 100 120 140 160 180 200 y +

Velocity Skewness 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 S(u) S(v) 0 0 0 20 40 60 80 100 -0.2 0 20 40 60 80 100 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 y + y + 1 0.8 ♦ DNS 0.6 y+ = 15 = 150 0.4 ■ S+=2 → 1.2 η → 0.6 η 0.2 ▲ S+=4 → 2.4 η → 1.2 η S(w) 0 0 20 40 60 80 100 x S+=8 → 4.8 η → 2.4 η -0.2 + S+=12 → 7.2 η → 3.6 η -0.4 -0.6 -0.8 -1 y +

Velocity Flatness 8 8 7 7 6 6 F(v) F(u) 5 5 4 4 3 3 2 2 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 y + y + 8 7 ♦ DNS 6 y+ = 15 = 150 F(w) ■ S+=2 → 1.2 η → 0.6 η 5 ▲ S+=4 → 2.4 η → 1.2 η 4 x S+=8 → 4.8 η → 2.4 η + S+=12 → 7.2 η → 3.6 η 3 2 0 20 40 60 80 100 y +

Comparison of ideal and real probe response 3.5 3.5 3 3 2.5 2.5 2 2 v'/u t u'/u t 1.5 1.5 1 1 0.5 0.5 0 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 y + y + 3.5 3 2.5 2 w'/u t S+=8 1.5 ♦ , DNS 1 x, ideal probe 0.5 -, real probe 0 0 20 40 60 80 100 120 140 160 180 200 y +

Vorticity Statistics - RMS 0.5 0.5 0.4 0.4 0.3 0.3 2 2 w y 'n/u t w x 'n/u t 0.2 0.2 0.1 0.1 0 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 y + y + 0.5 ♦ DNS 0.4 y+ = 15 = 150 ■ S+=2 → 1.2 η → 0.6 η 0.3 2 w z 'n/u t ▲ S+=4 → 2.4 η → 1.2 η x S+=8 → 4.8 η → 2.4 η 0.2 + S+=12 → 7.2 η → 3.6 η 0.1 0 0 20 40 60 80 100 120 140 160 180 200 y +

Vorticity Skewness 1 1 0.75 0.75 0.5 0.5 0.25 0.25 0 0 S(w x ) S(w y ) -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 -1 -1.25 -1.25 -1.5 -1.5 0 10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100 y + y + 1 0.75 ♦ DNS 0.5 y+ = 15 = 150 0.25 0 ■ S+=2 → 1.2 η → 0.6 η S(w z ) -0.25 ▲ S+=4 → 2.4 η → 1.2 η -0.5 x S+=8 → 4.8 η → 2.4 η -0.75 + S+=12 → 7.2 η → 3.6 η -1 -1.25 -1.5 0 10 20 30 40 50 60 70 80 90 100 y +

Vorticity Flatness 10 10 9 8 8 7 F(w x ) F(w y ) 6 6 5 4 4 3 2 2 0 20 40 60 80 100 0 20 40 60 80 100 y + y + 10 9 ♦ DNS 8 y+ = 15 = 150 7 ■ S+=2 → 1.2 η → 0.6 η F(w z ) 6 ▲ S+=4 → 2.4 η → 1.2 η 5 x S+=8 → 4.8 η → 2.4 η 4 + S+=12 → 7.2 η → 3.6 η 3 2 0 20 40 60 80 100 y +

Comparison of ideal and real probe response 0.5 0.5 0.4 0.4 0.3 0.3 2 w y 'n/ut 2 w x 'n/u t 0.2 0.2 0.1 0.1 0 0 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 y + y + 0.5 0.4 0.3 w z 'n/ut 2 S+=8 0.2 ♦ , DNS x, ideal probe 0.1 -, real probe 0 0 20 40 60 80 100 120 140 160 180 200 y +

Velocity PDFs of real and ideal probe response at y+=12.5 1.6 1.6 1.4 1.4 1.2 1.2 1 1 PDF(u + ) PDF(v + ) 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 -4 -2 0 2 4 u + v + 1.6 1.4 ♦ , DNS 1.2 ■ , s+=4, ideal probe response 1 ▲ , s+=4, real probe response PDF(w + ) 0.8 x, s+=8, ideal probe response 0.6 +, s+=8, real probe response 0.4 0.2 0 -6 -4 -2 0 2 4 6 w +

Vorticity PDFs of real and ideal probe response at y+=12.5 5 5 4.5 4.5 4 4 3.5 3.5 3 3 + ) + ) PDF(w x PDF(w y 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 + w x + w y 5 4.5 4 3.5 ♦ , DNS 3 + ) PDF(w z 2.5 ■ , s+=4, ideal probe response 2 ▲ , s+=4, real probe response 1.5 x, s+=8, ideal probe response 1 +, s+=8, real probe response 0.5 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 + w z

Velocity Spectra @ y+ = 20 Probe scale S+ = 8 __ DNS y+ = 15 = 150 __ S + = 2 → 1.2 η → 0.6 η __ S + = 4 → 2.4 η → 1.2 η __ S + = 8 → 4.8 η → 2.4 η __ S + = 12 → 7.2 η → 3.6 η

Vorticity Spectra @ y + = 20 Probe scale S + = 8 __ DNS y+ = 15 = 150 __ S+ = 2 → 1.2 η → 0.6 η __ S+ = 4 → 2.4 η → 1.2 η __ S+ = 8 → 4.8 η → 2.4 η __ S+ = 12 → 7.2 η → 3.6 η

Local Isotropy of the Vorticity Field in a High Reynolds Number Turbulent Boundary Layer E ω x υ 1/4 / ε 3/4 12- sensor probe scale dissipation range E ω y υ 1/4 / ε 3/4 inertial subrange NASA Ames 80´ x 120´ Wind Tunnel Kim and Antonia (1993) , channel flow DNS, JFM 251 E ω z υ 1/4 / ε 3/4 Kim and Antonia probe isotropic Ong & Wallace (1994), location experiment , Proc. ETC V k 1 η

Summary & Conclusions Spatial resolution of 12-sensor hot-wire probe investigated using highly resolved minimal channel flow DNS. Virtual probe with 12 point sensors varied so that spacing between arrays is 2, 4, 8 and 12 viscous lengths. The velocity component rms values are attenuated les then 10% everywhere in the flow for s + <8. The skewness factor of the wall normal velocity fluctuations, S(v), display stronger dependence on spatial resolution. In the wall layer all the vorticity component rms values are strongly influenced by spatial resolution for S + = 8 and 12. The statistics from the ideal and real probe responses are nearly identical. The shapes of the velocity and vorticity pdfs reflect the resolution effects. Spectra demonstrate the attenuation due to spatial resolution

Comparison of ideal and real probe response 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 S(u) S(v) 0 0 0 20 40 60 80 100 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 y + 0 20 40 60 80 100 y + 1 0.8 0.6 S+=8 0.4 ♦ , DNS 0.2 S(w) x, ideal probe 0 -, real probe -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 100 y +

Comparison of ideal and real probe response 8 8 7 6 6 F(u) F(v) 5 4 4 3 2 2 0 20 40 60 80 100 0 20 40 60 80 100 y + y + 8 7 S+=8 6 ♦ , DNS F(w) 5 x, ideal probe -, real probe 4 3 2 0 20 40 60 80 100 y +