RIAM研究集会「壁乱流における大規模構造の統計法則と動力学に果たす役割」

摩擦抵抗低減効果を持つ
機能性船底塗料の数値解析
Numerical simulation for functional
painting of ship hull with friction dragreduction effect
応用流体力学グループ
高木 洋平
Research group of applied hydrodynamics
Y. Takagi

Turbulent drag reduction• Applications in wall turbulence
RibletStenzel et al., Prog. Org. Coat. (2011)

Micro bubble

Turbulent drag reduction
Compliant surface

Requirement for ship hull:• Easy-to-use• Durability• Low cost

Painting is usually used for antifouling.

Functional painting with dragreduction effect
M. O. Kramer, J. American Soc.
Naval Eng. (1960)

Commercial painting• LF-Sea (Nippon Paint)– Hydrogel formation– DR ~ 10%– Biomimetic– Water trapping effect

Water trapped layer

Drag reduction effect with hydrogel
Result of towing tank test

Proposed drag reduction mechanism

Water trapped layer
山盛, TECHNO-COSMOS, 2005

山盛&島田, TECHNO-COSMOS, 2009

Painting condition in sea water
Self-polishing typeHydration typeafter shipping• Green• Stress relaxation

Structures around hydrogel painting

Proposed mechanism for drag reduction
1. Slip velocity due to water trapping
2. Stress relaxation due to compliant gel polymer
3. Surface deformation and roughness
4. Water penetration into hydrated and gel layers(porous-like structure)1. Slip velocity due to water trapping:numerical modeling in plane channel flow

Direct Numerical Simulation(DNS) of turbulent channel flow

Governing equations¶ui ¶(ui u j )1 ¶p 1 ¶ ui+=+¶t¶x jr ¶xi Re ¶x 2j2

¶ui=0
¶xi

SymbolValue

Grid number
Re numberNx×Ny×Nz
ReC (Ret)

128×151×128
4200 (180)

Numerical scheme

Navier-Stokes equation:

Continuity equation:

Property• 2nd-order Finite Difference Method• Crank Nicholson for viscous term
• 3rd-order Runge-Kuttta• Poisson eq. in Fourier space wassolved by TDMA.

Slip velocity condition
Local slip velocity¶u¶w
us = lx , ws = l z¶y¶ywherelx ,l z = f x f zl0

Property

Value

Reference slip length (l0+)
Slip wave length:
x-direction (lx+)
z-direction (lz+)

0.36275, 368, 550, 735, 1100
50, 100, 183, 367

Validation of simulation code
Drag reduction ratio (DR) under uniform slip velocity condition
Case

Slip lengthDR(Min & Kim[1])DR(Present)

Streamwise slip

lx+ = 0.36, lz+ = 05%

5.5%

Spanwise slip

lx+ = 0, lz+ = 0.36-3%-3.6%

Isotropic slip

lx+ = lz+ = 0.36-1%-0.6%[1] T. Min and J. Kim, Phys. Fluids, 16(7), L55 (2004).
Drag reduction ratio

Validation of simulation code
Mean velocity profiles under uniform slip velocity condition

Drag decrease/increasemechanism (Min&Kim, 2004)

Streak structure
Streamwise velocity fluctuation at y+ ~ 5.8

u’ > 0: high speed streaku’ < 0: low speed streak

Scale of streak structures
Two-point correlations at y+ ~ 5.8

In the streamwise direction

In the spanwise direction

Negative correlation is related with the scale of high/low streaks.

Scale of streak structures
xS+ ~ 500

Dependency of local slip velocity

Optimum local slip condition was not specified uniquely.

Suppression of energy dissipation
One-dimensional energy spectra at y+ ~ 5.8

The spanwise intensity of streak structure decreased by local slip.

Vortex interaction
Two-point correlation of vorticity

Velocity fluctuation (streak structure)

Peaks by local slip
No significant change
Vortex interaction was seen in thehorizontal plane.

Summary of topic 1• A turbulent channel flow simulation with local
slip velocity was carried out.
• The slip wavelength compared with the size of
streak structure was effective for drag reduction.
• The spanwise energy dissipation related withthe streak scale was selectively suppressed.
• The vortex interaction was seen in thestreamwise and spanwise directions. However,it did not lead to drag increase.

2. Stress relaxation due to compliant gel
polymer: Nonlinear response ofhydrogel to shear stressResponse to high wall shear stress• Gel: nonlinear stress response
No-slip

¶ui
¶y0

Slip

Simplified nonlinear-responseThreshold of slip condition
Calculated range of velocity gradienton wall (no-slip case)

Nonlinear stress response

LabelType

Case 1

only x-slip

Case 2

only z-slip

Case 3

isotropic (xz-slip)

Threshold0 ~ 20¶u~ 8,¶y¶w~1¶yDependency of drag reduction

Total slip effect was given by the summation of each contribution.

Summary of topic 2• A three-dimensional simulation considering shear stressresponse in hydrogel was carried out.
• Although the stress response was isotropic, significant
drag reduction effect was achieved.

• Isotropic hydrogel painting might be useful for wall
turbulence with the change of streamwise direction.
In oblique flow

Super hydrophobic flow

Ou&Rothstein,2005

3. Surface deformation and roughness:wavy channel flow simulation

Numerical condition
Streamwise wavy channel

Spanwise wavy channel

Case 1

Wave shape of the wall :Case1
Wave type
Wave length(λ) [m]Case2

streamwise

Case3
spanwise2.0

1.33

0.1

148,95,128

128,95,148

129,97,241

Amplitude(a) [m]Grid Number(x, y, z)Code validation
DNS of a turbulent channel flow by using OpenFOAM®
No turbulence modelLabel
WallCyclic2um δRe  5600
ν

um: mean velocityδ : channel-half widthν : kinematic viscosity

Grid number

MethodKim et al. 192×129×1(1987)[1] 60

Fourierspectral

Coarse64×64×64

Middle96×96×96

Fine

128×128×128

FVM with OF[1] J. Kim et al., J. Fluid Mech., 177, pp. 133-166 (1987).

Code validation
Mean velocity profiles

Averaged y+

CoarseMiddle

Fine

0.60

0.51

0.49

Middle and Fine resolution
showed good agreements.

OpenFOAM is available for
DNS on a turbulent channel flow.

Drag reduction ratio (DR)DR dP  dP
 dP : mean pressure gradient in a flat channel 
 dx flat   dx wavy  dx flat dP : mean pressure gradient in a wavy channel
 dP



 dx flat dx wavyCase 1Case 2

Drag increase

DR = -20.4 %DR = -9.4 %Case 3

Drag reductionDR = +9.3 %Discussion ― Case1 : streamwise wavy channel

The peak of RMS value profile

Most random flow regionyδ

z direction

aTwo peaks above and below wave tipwave tip

Case2 and Case 3 : spanwise wavy channel
Case ３

Case 2

Some vortices in one wave

collision

Fluctuationis conserved
Unit vortex
in onewave
in the wall wave

z (spanwise direction )Vortex ?
Advance the dissipation

Drag reductionz (spanwise direction )Median diameter of eddies
Case1
Median diameter [m]0.184

Spanwise wave spacing[m]DR [%]-20.4

Case 2

Case 3

Flat channel

0.272

0.09

0.240

1.333

0.1

―-9.4

+9.30Summary of topic 3• Drag reduction effect was shown when collision of vortices wassuppressed by wall shape.
• Dissipation was advanced by collision of several vortices in one
wave .
Case1
Wave type

Wave length(λ)
Median diameter

DR [%]

Case2

Streamwise

2.0
0.184
-20.4

Case 3
Spanwise

1.33
0.272
-9.4

0.1
0.09
9.3

Soft matter, Hydrogel, has advantage for drag reduction
on ships bottom surface

4. Water penetration into hydrated andgel layers (porous-like structure):channel flow with porous wall

Approaches for fluid flow through porousstructure• Experimental research• Observation ofsweep/ejection nearwall1)• Boundary
Numericalcondition
research approach2)?
Increase of turbulent drag1) K. Suga et al., Int. J. Heat Fluid Flow, 32, 586-595 (2011)2) S. Hahn et al., J. Fluids Mech. vol. 450, pp. 259- 285(2002)
3) W. P. Breugem et al., J. Fluids Mech. vol. 562, pp. 35-72 (2006)

Continuum approach3)

Decrease of skin friction
due to slip velocity

Numerical conditions
Case

5%D

10%D

20%D

Flat
―Total channel width
149,(129+16),129

Porousregion149,(129+20),129

149,(129+20),129

129,129,129

4f2(f  p )

 f : Width of fluid region
 p : Porous wall thickness

Fluidregion

Re m 2um ( f   p )Non-uniform grid2f

ymin  0.5

 5600DNS of turbulent channel flow
Governing equations

Continuity equationu  0Momentum equationDu  p ' 2 u  (1   ) SDt

Porous model
Darcy-Forchheimer equationcfS   um  2um um
KK

Numerical scheme
Spatial discretization scheme
Finite Volume Method
Time advancement
Implicit Euler method ( 1 st)
Poisson equation
PISO algorithm

OpenFOAM®
 : Porosity

0.80

K : Permeability

0.02 [mm2]cf : Forcheheimer
coefficient

0.17Drag reduction ratio

FIK identity[1]Viscous term and Turbulent term
DR [%]

Case

5%D

6.55

+9.6

910%D

7.16

+1.3

1820%D

7.56-4.3

37Flat7.25[1 ]Fukagata, K., Iwamoto, K. and Kasagi, N., Phys. Fluids, 14 (2002), pp.73-76

Turbulent statistics
Mean velocity profiles

Turbulence intensity

δpSlip velocity occurred
over porous walls.

Turbulent intensity
became stronger.

Turbulent structure – Streak structure
Flat Smooth Channel

5%D (DR=+9.6%)

10%D (DR=+1.3%)

20%D (DR=-4.3%)

Longer streakystructureEffect of wall thickness
The mechanism of vortex generation
Flat wall

Porous wall

UnstableEffect of wall thickness
The mechanism of vortex generation and decaying
Thick Porous wall (Drag increasing)Thin porous wall (Drag reducing)

UnstableWeak source flow

Joint Probability Density Function (JPDF)
The mechanism of vortex generation and decaying
Thin porous wall 5%D (Drag reducing)Thick Porous wall 20%D (Drag increasing)Summary of topic 4• The direct numerical simulation on a channel with porous walls
which had varying wall thickness were carried out.
• The drag reduction was achieved over the thinnest porous wall,and in that case, turbulent intensity became weaker.
• The turbulent structures suggested that the relationship between
the intensity of sweep (or ejection) and the thickness of porous
wall was an important factor for drag reduction.

5. Experimental evaluation of dragreduction: Taylor-Couette flow

Experimental setup
Rotating cylinder facility
Motor
Torque Meterri  48 mm

Stainlessinner cylinder
Acrylic
outer cylinderro  64 mm
ωh  128 mm

CameraReynolds number: R e ri  d

ri : radius of inner cylinder [m]ro : radius of outer cylinder [m]L: height [m]ω: angular velocity [m]d: gap between the cylinders [m]ν: kinetic viscosity [m2/s]
ρ: density [kg/m3]
T: toque of inner cylinder [Nm]ri = 48 mm
ro = 64 mm
L = 128 mm

T

Nondimensionalized G torque: 2L

Flow pattern
Rotating cylinder facility

Re = 4.2×103

Motor
Torque Meterri  48 mm

Stainlessinner cylinder
Acrylic
outer cylinderro  64 mm
ωh  128 mm

CameraReynolds number: R e 

Steady vortexes

Re = 8.8×103

ri  dWavy vortexes

Re = 4.4×104

Validity of measurement
109

Wendt’s empirical relations

G=108

 3/ 2
1.45
Re1.5 for 4 10 2  Re  10 47/4
(1   ) 3/ 21.7
0.23
Refor
7/4
(1   )10 4  Re  10 5

 : radius ratio ri / roG

F. Wendt, Ingenieur-Archiv. 4, 577 (1933).

107

106 310104

Re

105

Experimental conditions
Test cylinders
Thickness of primer
Total : 220 μm

1st layer : 140 μm
SmoothCured

HydrogelThickness of coating
ThickThinCured

380 μm240 μmHydrogel

380 μm240 μm

2nd layer : 80 μm
Range of Re
4.4×104 < Re < 8.1×104

Drag reduction effectReynolds number109ri  dRe 
Nondimensionalized torqueGTG 2L108104Torque G of hydrogel coating
was smaller than that of Curedcoating.

Re105

Drag reduction effect109

Reynolds numberri  dRe G

Nondimensionalized torque108104TG 2LRe105

Relation with DR and Re
403530Drag Reduction Ratio

ThickDR Thin

GCured  GHydrogel
GCured

100

DR2520DR ~ 7%151050104

Re105

Drag reduction effectof thin hydrogel waslarger than thick one.