4A.1: Amplitude modulation of streamwise velocityfluctuations in the roughness sublayer: evidence from

large-eddy simulations

William Anderson∗, Ankit Awasthi

Mechanical Engineering Department, The University of Texas at Dallas

American Meteorological Society, Symposium on Boundary Layers and Turbulence

June 20th, 2016

Salt Lake City, Utah

∗wca140030@utdallas.edu

With support from:

• Air Force Office of Scientific Research, Grant # FA9550-14-1-0101PM: Dr. R. Ponnoppan

• The University of Texas at Dallas

*Accepted to Boundary-Layer Meteorology: June 13, 2016

Inner-Outer Interactions: Amplitude Modulation of Roughness Sublayer

x/h

z

yx

(c)

yi

yoU02πδ

2πδ

xc , zc{ }

Virtual hotwire rake

0.52.5

4.56.5

6.5

4.5

2.5

0.5

0

2

4

x/hy/h

z/h

x/h z/h

y/h

(a) (b)

z/h

x/h

y/h

x1 x2 x3

1 2 3 4 5 6 7

1

2

3

4

5

6

7

x4

yiyo

U0

Virtual hotwire rake

Record time-series of velocity across depth of domain (red line)

Time-height contours of u′(xl, y, zl, t) = u(xl, y, zl, t)− 〈u(xl, y, zl, t)〉THomogeneous topography:

tU0/δ

y/δ

400 405 410 415 420 425 4300

0.5

1

−5

0

5

Cubes, position x1:

tU0/δ

y/δ

390 395 400 405 410 415 4200

0.5

1

−4

−2

0

2

4

Cubes, position x4:

tU0/δ

y/δ

390 395 400 405 410 415 4200

0.5

1

−2

0

2

4

Inertial layer momentum excess(deficit) precedes roughness sublayerexcitation(relaxation)

Inner-Outer Interactions: Amplitude Modulation of the RoughnessSublayer

Marusic et al., 2010: Science: up(yi) = u∗(1 + βuL(yO))Amplitude Modulation

+ αuL(yO)Superposition

up(yi): Predicted velocity at inner elevation, yi

u∗: Universal velocity, free of modulation or superposition

β: Amplitude modulation parameter; α: Superposition parameter

αuL(yO): Large-scale (low-pass filtered, uL(yO, t) = Gδ ? u(yO, t)) outer velocity

Marusic et al.: “The simple algebraic form of Eq. 1 [sic] is an ideal basis for anear-wall model for high-Re large-eddy simulations.”

Mathis et al., 2009: J. Fluid Mech.: amplitude modulation decoupling procedure:

1. u′(y, t) = u(y, t)− 〈u(y, t)〉T2. u′L(y, t) = Gδ ? u

′(y, t), where Gδ = Hs(kc − |k|) and kc = 2π/δ (onelarge-eddy turnover)

3. u′S(y, t) = u′(y, t)− u′L(y, t)

4. Hilbert transform complex analytic, Z(t) = u′S(y, t) + iH(t) = A(t)eiφ(t)

5. Low-pass filtered envelope of small scales, EL(u′S(y, t)) = Gδ ? A(t)

Time series: homogeneous roughness

80 100 120 140 160 180 200 220 240 260−4

−2

0

2

4

u′ i,S(y

i,t)

80 100 120 140 160 180 200 220 240 2600

5

E(u

′ i,S(y

i,t))

80 100 120 140 160 180 200 220 240 260

−1

0

1

EL(u

′ i,S(y

i,t))

tU0/δ

Top figure: u′S(yi, t) (black); u′L(yi, t) (orange); yi/δ = 0.004

Middle figure: E(u′S(yi, t)) (black); yi/δ = 0.004

Bottom figure: EL(u′S(yi, t)) (black); u′L(yO, t− δτ(yO; yi)) (orange); yO/δ = 0.1

yO

yi= 25

δτ(y; yRef.) ≡ advective correction [Anderson et al., 2015: J. Turb.]

Correlation: homogeneous roughness

Single-point: R(y; y) =〈u′L(y, t)EL(u′S(y, t))〉T

〈u′2L (y, t)〉1/2T 〈E2L(u′S(y, t))〉1/2T

Large-scale signal and envelope of small scales at same elevation

Two-point: R(y; yRef.) =〈u′L(yRef., t+ δτ(y; yRef.))EL(u′S(y, t))〉T

〈u′2L (yRef., t+ δτ(y; yRef.))〉1/2T 〈E

2L(u′S(y, t))〉1/2T

Large-scale component at fixed outer elevation and envelope of small scales atvarying elevation

10−2

10−1

100

−1

−0.5

0

0.5

1(b)

R

y/δ0 2 4 6 8 10

−1

−0.5

0

0.5

1(c)

y/∆

Thick lines: R(y; yRef.); Thin lines: R(y; y)

Note: R(y; y) > R(y; yRef.)

Blue lines: LES with Nx = Ny = Nz = 96; Black lines: Nx = Ny = Nz = 128

Black squares: smooth wall turbulent channel (Mathis et al., 2009: Physics ofFluids)

Anderson, 2016: J. Fluid Mech. 789 567—588

Time series: cubic topography: Position x4

400 420 440 460 480 500 520

−5

0

5

u′ S(y

i,t)

400 420 440 460 480 500 5200

5

E(u

′ S(y

i,t))

400 420 440 460 480 500 520

−2

0

2

EL(u

′ S(y

i,t))

U0t/δ

40 60 80 100 120 140 160 180

−5

0

5

u′ S(y

i,t)

40 60 80 100 120 140 160 1800

5

E(u

′ S(y

i,t))

40 60 80 100 120 140 160 180

−2

0

2

EL(u

′ S(y

i,t))

U0t/δ

Top figure: u′S(yi, t) (black); u′L(yi, t) (orange); yi/δ = 0.254

Middle figure: E(u′S(yi, t)) (black); yi/δ = 0.254

Bottom figure: EL(u′S(yi, t)) (black); u′L(yO, t− δτ(yO; yi)) (orange); yO/δ = 0.5

yO

yi≈ 2

δτ(y; yRef.) = advective correction

Now: correlate envelope of small scales with large-scale signal

Correlation: cubes

R(y; y) =〈u′L(y, t)EL(u′S(y, t))〉T

〈u′2L (y, t)〉1/2T 〈E2L(u′S(y, t))〉1/2T

R(y; yRef.) =〈u′L(yRef., t+ δτ(y; yRef.))EL(u′S(y, t))〉T

〈u′2L (yRef., t+ δτ(y; yRef.))〉1/2T 〈E

2L(u′S(y, t))〉1/2T

10−3

10−2

10−1

100

−1

−0.5

0

0.5

1(a)

R

10−3

10−2

10−1

100

−1

−0.5

0

0.5

1(b)

10−3

10−2

10−1

100

−1

−0.5

0

0.5

1(c)

R

y/δ 10−3

10−2

10−1

100

−1

−0.5

0

0.5

1(d)

R

y/δ

x/h

y/h

x1 x2 x3

1 2 3 4 5 6 7

1

2

3

4

5

6

7

x4

x/hy/h

x1 x2 x3

1 2 3 4 5 6 7

1

2

3

4

5

6

7

x4

x/h

y/h

x1 x2 x3

1 2 3 4 5 6 7

1

2

3

4

5

6

7

x4

x/h

y/h

x1 x2 x3

1 2 3 4 5 6 7

1

2

3

4

5

6

7

x4

Thick lines: R(y; yRef.); Thin lines: R(y; y)

Note: R(y; y) > R(y; yRef.)

Gray lines: LES with N = 64; blue lines: N = 96; black lines: N = 128

Conclusion:

Under inertia-dominated ABL conditions

...Inertial layer amplitude modulates roughness sublayer

θ

u′ < 0

Q4

Mixing-layer-like roughness sublayer: positively correlated to inertial layer dynamics

But, location in canopy (secondary canopy flows) greatly influences correlation

2016 AGU Fall Meeting sessions of interest to AMS BLT community:

Session ID 12480: A014 – Atmospheric Boundary Layer Processes andTurbulence (Atmospheric Sciences)Primary Convener: David Richter; Co-Conveners: Chag Higgins, Will Anderson

Session ID 12334: EP004 – Aeolian Research at the Interface of Biophysical,Sedimentary, and Atmospheric Processes (Earth and Planetary Surface Processes)Primary Convener: Will Anderson; Co-Conveners: Gianluca Blois, Kenzie Day,Jonathan Perkins

Support:

• Air Force Office of Scientific Research, Turbulence and TransitionProgramGrant # FA9550-14-1-0101; PM: Dr. R. Ponnoppan

• Army Research Office, Environmental Sciences DirectorateGrant # W911NF-15-1-0231; PM: Dr. J. Parker

• The University of Texas at Dallas

• Computational Resources: Texas Advanced Computing Center at TheUniversity of Texas at Austin, NSF award OCI-1134872

Acknowledgment:

• Ken Christensen, Gianluca Blois: University of Notre Dame

• Ankit Awasthi: University of Texas at Dallas