The role of coherent structures in The role of coherent structures in low-Reynolds-number turbulent low-Reynolds-number turbulent

wall flowswall flows

Genta KawaharaGenta KawaharaGraduate School of Engineering ScienceGraduate School of Engineering Science

Osaka UniversityOsaka University

The role of coherent structures in The role of coherent structures in low-Reynolds-number turbulent low-Reynolds-number turbulent

square duct flowsquare duct flow

Genta KawaharaGenta KawaharaGraduate School of Engineering ScienceGraduate School of Engineering Science

Osaka UniversityOsaka University

M. Uhlmann, A. Pinelli, A. SekimotoM. Uhlmann, A. Pinelli, A. Sekimoto

Role of coherent structures in plane channel

Periodic solutions in plane Couette flow (K. & Kida 2001)periodic

gentle periodic

Regeneration cycle (Jiménez & Moin1991; Hamilton, Kim & Waleffe 1995)

No regeneration cycle

contours

surfaces

x

contours

surfaces

u

u

Role of coherent structures in plane channel

Periodic solutions in plane Couette flow (K. & Kida 2001)

turbulent(Moser, Kim & Mansour 1999)

laminar

turbulentperiodic

gentle periodic

Coherent structures Prandtl’s wall law⇒(regeneration cycle) (buffer layer)

Secondary flow of Prandtl's second kind

Generation mechanism

・ Statistical budget　 Kajishima, Miyake, Nishimoto

（ 1991 ）

・ Transient growth　 Biau, Soueid & Bottaro (2008)

Other roles of coherent structures

Coherent structures in square-duct turbulenceUhlmann, Pinelli, Sekimoto & K. (2007, 2008)

1/4 cross-section ofsquare-duct

Reb=1100Re=80

Reb=2200Re=150

Near-wall coherent structures ⇒ Secondary flow

Reb=Ubh/> 1100Ub: bulk mean velocity

Coherent structures in square-duct turbulence

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

h: duct half width

u, v, w xReb=1100

z/h z/h

yh

yh

Velocity and vorticity of mean secondary flow

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)1/4

cross-section (Lx=4h)

u, v, w xReb=1500

z/h z/h

yh

yh

Velocity and vorticity of mean secondary flow

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)1/4

cross-section (Lx=4h)

u, v, w xReb=2200

z/h z/h

yh

yh

Velocity and vorticity of mean secondary flow

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)1/4

cross-section (Lx=4h)

u, v, w xReb=3500

z/h z/h

yh

yh

Velocity and vorticity of mean secondary flow

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)1/4

cross-section (Lx=4h)

Positions of secondary-flow-vortex center

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

maximum pointof x

z/h

y

h

1/4cross-section

Positions of secondary-flow-vortex center

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

innerscaling

maximum pointxof

Reb

z+

y+

1/4cross-section

Local wall shear stress

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008) w/

w

(z+h)+

z+h

localmaximum

localminimum

Positions of local maximum and minimum

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)(z+

h)+

Reb

local maximum

local minimum

inner scaling

Identification of center of streamwise vortices

Local maximum point ofLaplacian of pressure

in cross-streamwise plane

Swirl conditionKida & Miura (1998)

streaks

vortices

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

PDF of position of streamwise-vortex center

Reb=1500

Secondaryflow

yh

yh

z/h

z/h

PDF

x<0

x>0PDF

anti-clockwise

clockwise

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

Reb=2200

Secondaryflow

yh

yh

z/h

z/h

PDF

x<0

x>0PDF

anti-clockwise

clockwise

PDF of position of streamwise-vortex center

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

Identification of position of low-velocity streaks

Reb=2200

y

zx

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

wx / w

wx / w

Position and number of low-velocity streaks

Reb=1500

wall y/h=1

t+

yh

z/h

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

maximum minimumwall shear stress

Position of streaks

PDF of vortices

Position and number of low-velocity streaks

Reb=2200

wall y/h=1

t+

yh

z/h

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

maximum minimumwall shear stress

Position of streaks

PDF of vortices

Reynolds-number dependence of the roles

highReb=2000

Reb

Uhlmann, Pinelli, Sekimoto & K. (2007, 2008)

Concluding remarks

Plane channel　 1. Prandtl’s wall law　　 regeneration cycle (periodic solution)　　⇒ turbulent velocity profileSquare duct at low Reynolds number　 2. Mean secondary flow　　 constrained streamwise vortices　　⇒ mean-secondary-flow vortices　 3. Wall shear stress　　 constrained low-velocity streaks　　⇒ local maximum, minimum of wall shear stress

Roles of near-wall coherent structures

Flow configuration

Dimensionless parameters

Friction velocity

Direct numerical simulation• Time integration

velocity ， pressure 　→　 fractional-step method semi-implicit 3-stage Runge-Kutta method（ Verzicco & Orlandi 1996 ）

• Spatial discretization

pseudo-spectral method

streamwise ( 　　 ) ： Fourier

cross-streamwise ( 　　　　 ) ： Chebyshev

Positions of secondary-flow-vortex center

maximum point（ elliptic-type ）stagnation point of x

z/h z/h

yh

yh

Positions of secondary-flow-vortex center

maximum point（ elliptic-type ）stagnation point of x

outerscaling

innerscaling

Sign selection of coherent vorticestrajectoriesVortex filament

Viscous (Lamb-Oseen) vortexvorticity (red, anti-clockwise; blue, clockwise)