The spatial resolution of velocity and velocity gradient turbulence

The spatial resolution of velocity and velocity gradient turbulence statistics measured with multi-sensor hot-wire probes P.V. Vukoslavčević, 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 Rτ = 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. Vukoslavčević, 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. Vukoslavčević & 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

U e2 = U n2 + k 2U t2 + h 2U b2 . Using this expression, the effective velocities cooling each sensor can be expressed as a function of the three velocity components at the sensor center,

U eij2 = U ij2 + aij1Vij2 + aij 2Wij2 + aij 3VijU ij + aij 4WijU ij + aij 5WijVij . The necessary assumption that the velocity variation is linear over probe spacing area leads to a set of 12 equations of the following form,

U eij2 = Fij {aijk , U 0 , V0 , W0 , ∂(U , V , W ) / ∂y, ∂(U ,V ,W ) / ∂z},

In terms of the velocity components at the probe centers U0, V0, W0 and the six velocity gradients as unknowns.

Real probe: calibration proc.

Ideal probe: k=0, h=1, α=45 deg. a1 jk

1 2 = 1 2

2 −2 0 1 0 −2 2 2 0 1 0 2

0 0 0 0

a1 jk

1.0 2.8 = 1.0 2.8

2.8 1.0 2.8 1.0

- 1.70 - 0.15 - 0.15 0.15 - 1.70 - 0.15 1.70 - 0.15 - 0.15 0.15 1.70 - 0.15

Physical experiment The effective cooling velocity for each sensor, Uij, can be found from King’s Law or from a polynomial fit 5

E = A + BU , 2

n e

p −1 2 b E = U ∑ p e. p =1

Virtual experiment

Virtual probe with Sy = 8 ∆y over the numerical grid where ∆y is 1 viscous length

S y+ = 2, 4, 8, 12

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

3.5

3.5

3

3

2.5

2.5

2

2

v'/ut

u'/ut

Velocity Statistics - RMS

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

+

y

y

3.5

3

♦ DNS

2.5

y+ = 15 ■ S+=2 → 1.2 η ▲ S+=4 → 2.4 η x S+=8 → 4.8 η + S+=12 → 7.2 η

w'/u t

2

1.5

1

0.5

0 0

20

40

60

80

100 +

y

120

140

160

180

200

= 150 → 0.6 η → 1.2 η → 2.4 η → 3.6 η

200

Velocity Skewness 1

1

0.8

0.8 0.6

0.6

0.4

0.4

S(v)

S(u)

0.2 0.2 0 0

20

40

60

80

0 0

20

40

60

80

-0.2

100

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1 +

+

y

y

1 0.8

♦ DNS

0.6 0.4

S(w)

0.2 0 0

20

40

60

-0.2 -0.4 -0.6 -0.8 -1 +

y

80

100

y+ = 15 ■ S+=2 → 1.2 η ▲ S+=4 → 2.4 η x S+=8 → 4.8 η + S+=12 → 7.2 η

→ → → →

= 150 0.6 η 1.2 η 2.4 η 3.6 η

100

8

8

7

7

6

6

F(v)

F(u)

Velocity Flatness

5

5

4

4

3

3

2

2 0

20

40

60

80

100

0

10

30

40

50

60

70

80

y

y

8

20

+

+

7

♦ DNS

F(w)

6

y+ = 15 ■ S+=2 → 1.2 η ▲ S+=4 → 2.4 η x S+=8 → 4.8 η + S+=12 → 7.2 η

5

4

3

2 0

20

40

60 +

y

80

100

→ → → →

= 150 0.6 η 1.2 η 2.4 η 3.6 η

90

100

Comparison of ideal and real probe response 3.5

3.5

3

3

2.5

2.5

2

v'/ut

u'/ut

2

1.5

1.5

1

1

0.5

0.5

0

0 0

20

40

60

80

100

120

140

160

180

200

20

40

60

80

100 +

y

y+

3.5

0

3

2.5

w'/ut

2

S+=8 ♦, DNS x, ideal probe -, real probe

1.5

1

0.5

0 0

20

40

60

80

100 +

y

120

140

160

180

200

120

140

160

180

200

0.5

0.5

0.4

0.4

0.3

0.3

w y'n /ut 2

w x'n /ut 2

Vorticity Statistics - RMS

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

+

y

y+ 0.5

♦ DNS

w z'n /ut 2

0.4

y+ = 15 ■ S+=2 → 1.2 η ▲ S+=4 → 2.4 η x S+=8 → 4.8 η + S+=12 → 7.2 η

0.3

0.2

0.1

0 0

20

40

60

80

100 +

y

120

140

160

180

200

→ → → →

= 150 0.6 η 1.2 η 2.4 η 3.6 η

180

200

Vorticity Skewness

1

1

0.75

0.75 0.5

0.25

0.25

0

0

S(w y)

S(w x)

0.5

-0.25

-0.25

-0.5

-0.5

-0.75

-0.75

-1

-1

-1.25

-1.25 -1.5

-1.5 0

20

40

60

80

100

0

10

20

30

40

50

60

70

80

90

+

y

+

y 1 0.75

♦ DNS

0.5

y+ = 15 ■ S+=2 → 1.2 η ▲ S+=4 → 2.4 η x S+=8 → 4.8 η + S+=12 → 7.2 η

0.25

S(w z)

0 -0.25 -0.5 -0.75 -1 -1.25 -1.5 0

10

20

30

40

50 +

y

60

70

80

90

100

→ → → →

= 150 0.6 η 1.2 η 2.4 η 3.6 η

100

Vorticity Flatness 10

10 9

8

8

F(w y)

F(w x)

7 6

6 5

4

4 3

2 0

20

40

60

80

100

2 0

y+

10

20

40

60

80

+

y

9

♦ DNS

8

y+ = 15 ■ S+=2 → 1.2 η ▲ S+=4 → 2.4 η x S+=8 → 4.8 η + S+=12 → 7.2 η

F(w z)

7 6 5 4 3 2 0

20

40

60 +

y

80

100

→ → → →

= 150 0.6 η 1.2 η 2.4 η 3.6 η

100

0.5

0.5

0.4

0.4

0.3

2

0.3

0.2

w y'n /ut

w x'n /ut 2

Comparison of ideal and real probe response

0.1

0.2

0.1

0 0

20

40

60

80

100

120

140

160

180

200

+

y

0 0

20

40

60

80

100 +

y

0.5

w z'n /ut

2

0.4

0.3

S+=8 ♦, DNS x, ideal probe

0.2

0.1

-, real probe 0 0

20

40

60

80

100 +

y

120

140

160

180

200

120

140

160

180

200

1.6

1.6

1.4

1.4

1.2

1.2

1

1

PDF(v+)

PDF(u+)

Velocity PDFs of real and ideal probe response at y+=12.5

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2 0

0 -10

-8

-6

-4

-2

0

u

2

4

6

8

10

-4

-2

0

2

+

+

v

1.6 1.4

♦, DNS ■, s+=4, ideal probe response ▲, s+=4, real probe response x, s+=8, ideal probe response

1.2

PDF(w+)

1 0.8 0.6

+, s+=8, real probe response

0.4 0.2 0 -6

-4

-2

0

w

+

2

4

6

4

5

5

4.5

4.5

4

4

3.5

3.5

3

3

+

PDF(w y )

PDF(w x+)

Vorticity PDFs of real and ideal probe response at y+=12.5

2.5 2

2.5 2

1.5

1.5

1

1

0.5

0.5

0

0 -2

-1.5

-1

-0.5

0

wx

5

0.5

1

1.5

2

+

-2

-1.5

-1

-0.5

0

0.5

1

w y+

4.5 4

PDF(w z+)

3.5

♦, DNS ■, s+=4, ideal probe response ▲, s+=4, real probe response x, s+=8, ideal probe response

3 2.5 2 1.5 1

+, s+=8, real probe response

0.5 0 -2

-1.5

-1

-0.5

0

w z+

0.5

1

1.5

2

1.5

2

Velocity Spectra @ y+ = 20

Probe scale S+ = 8

__

DNS

y+ = 15 __ S+ = 2 → 1.2 η __ S+ = 4 → 2.4 η __ S+ = 8 → 4.8 η __ S+ = 12 → 7.2 η

= 150 → 0.6 η → 1.2 η → 2.4 η → 3.6 η

Vorticity Spectra @ y+ = 20

Probe scale S+ = 8

__

DNS

y+ = 15 = 2 → 1.2 η __ S+ = 4 → 2.4 η __ S+ = 8 → 4.8 η __ S+ = 12 → 7.2 η __ S+

= 150 → 0.6 η → 1.2 η → 2.4 η → 3.6 η

Eωx υ1/4/ε3/4

Local Isotropy of the Vorticity Field in a High Reynolds Number Turbulent Boundary Layer

12- sensor probe scale

NASA Ames 80´ x 120´ Wind Tunnel

Eωy υ1/4/ε3/4

dissipation range

inertial subrange

probe location

Eωz υ1/4/ε3/4

Kim and Antonia (1993) , channel flow DNS, JFM 251

Ong & Wallace (1994), experiment, Proc. ETC V

k1η

Kim and Antonia isotropic

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+