numerical investigation on the influence of anisotropic surface structures on the skin friction

8/19/2019 NUMERICAL INVESTIGATION ON THE INFLUENCE OF ANISOTROPIC SURFACE STRUCTURES ON THE SKIN FRICTION

http://slidepdf.com/reader/full/numerical-investigation-on-the-influence-of-anisotropic-surface-structures 1/12

NUMERICAL INVESTIGATION ON THE INFLUENCE OFANISOTROPIC SURFACE STRUCTURES ON THE SKIN

FRICTION

S. Hohenstein - J. Seume

Institute of Turbomachinery and Fluid DynamicsLeibniz Universit at Hannover, Appelstr. 9, 30165 Hannover, Germany

[email protected]

ABSTRACTThe present study investigates the effect of anistropic surface roughness on ow losses using

Large-Eddy-Simulation (LES) and compares the results to a model based on the equivalentsand grain roughness. For this comparison models of surface roughness are built from mea-surements of real roughness to isolate the effect of anisotropy for which two orthogonal mainow directions with respect to the anisotropy of the surface are investigated. For these two di-rections an equivalent sand grain roughness height is calculated using the roughness elementshape and density parameter. The shift of the velocity distribution is then predicted by usinga roughness function and compared with the results of the LES. Furthermore, the rough andsmooth-wall velocity distributions are compared with each other in velocity defect form to checkthe Reynolds number similarity hypothesis. Both comparisons show a good agreement thus con-rming with LES that the simple model of equivalent sand grain roughness is an appropriatemethod to predict the inuence of anisotropic surface roughness on skin friction.

NOMENCLATURE

As frontal surface area S ij Shear strain tensorAf windward wetted surface ub bulk velocityBF R Nikuradse roughness function u+

e boundary layer edge velocityC + integration constant uτ friction velocityC s Smagorinsky constant u+ dimensionless velocityh local roughness height ∆ u+ roughness functionk roughness height y+ dimensionless wall distanceks equivalent sand grain height Greek symbolsl local roughness length δ boundary layer thicknessN number of nodes δ channel half heightRa arithmetic averaged roughness height ∆ lter widthRe Reynolds number ν kinematic viscosityRe τ friction velocity Reynolds number κ von K arm an constantRz averaged maximum roughness height Λs roughness element shape parameterS reference area ρ densityS f total frontal area µsgs sub-grid-scale viscosity

τ w wall shear stress

INTRODUCTIONThe effect of surface roughness on uid ow has been rst studied by Nikuradse (1933) and

Schlichting (1936) with the result that surface roughness, especially in turbulent boundary layers,

1



Figure 2: Real turbine surface roughness

The investigations carried out so far are of two types. On one hand, experimental investigationswere carried out on real surface roughness. The big disadvantage of this method is that ow close tothe wall and the roughness elements could not be resolved. On the other hand, there are numericalinvestigations using LES and Direct-Numerical-Simulations (DNS) which allow extending the inves-tigation to the whole boundary layer ow down to the wall. These methods however are restrictedto simple generic roughnesses, because spatial discretization is achieved with structured grids. In thepresent study, Large-Eddy-Simulations are carried out on a surface roughness model, which is morerealistic than the roughnesses used in the previous studies of Cui et al. (2003a) and Cui et al. (2003b).

BACKGROUNDThe boundary layer of turbulent ows can be divided into three parts. Near the wall, for y+ ≤ 5,

with y+ = y· u τ

ν , only laminar friction is present. This part is called the viscous sublayer and thevelocity, presented in inner-scaling in Eq. 1, is linearly dependend on the wall distance

u+ = y+ . (1)

In the buffer layer ( 5 < y + ≤ 70) both, laminar and turbulent, stresses exist. Various equations areused to describe the dependency of u+ from y+ which are formulated among others by Schlichtingand Gersten (2005). For values of y+ > 70, only turbulent stresses are present. The course of thevelocity distribution is described by Eq. 2, the ”log law”

u+ = 1κ

· lny · uτ

ν + C + − ∆ u+ (2)

where κ is the von K arm an constant and C + an integration constant. The von K´ arman constant isusually assumed to be κ ≈ 0.41 which is a good assumption for laboratory experiments, but as shownby Frenzen and Vogel (1995) and Cai and Steyn (1996), the von K arm an constant can vary in therange of 0.3 ≤ κ ≤ 0.48. Particularly Frenzen and Vogel (1995) showed that κ is strongly dependendon the wall roughness.

Surface roughness leads to a shift of the logarithmic velocity prole and is described by the lastterm on the right hand side of Eq. 2, ∆ u+ , which is called ”roughness function”. Depending on

the height of the roughness elements, surface roughness can be classied as hydraulically smooth fork+s ≤ 5, transitionally rough for 5 < k +

s ≤ 70 and fully rough for k+s > 70 with

k+s =

ks · uτ

ν . (3)

3



To check the Reynolds number similarity hypothesis the ”law of the wall” can also be rewritten inform of the velocity defect as

u+e − u

+

= −1κ ln(η) + B1 −

Πκ ω(η) (4)

with η = y/δ , B 1 being dened as 2Π/κ and the wake function ω (Schultz and Flack (2007)). TheReynolds number similarity hypothesis ”states that the turbulence beyond a few roughness heightsfrom the wall is independent of the surface condition”(Flack et al., 2005, pg.1). This assumes that theheight of the roughness k is very small compared to the boundary layer thickness δ .

For fully rough surfaces, the roughness function is dened by Schultz and Flack (2007) as

∆ u+ = 1κ

ln (k+s ) + C + − BF R . (5)

In Eq. 5 BF R is the Nikuradse roughness function which in the fully rough ow regime equalsBF R = 8.5. The equivalent sand grain height can be calculated with a correlation of Bons (2005),given in Eq. 6.

logks

k= − 0.43log (Λs ) + 0 .82 (6)

and made dimensionless in units of inner scaling with the friction velocity and the kinematic viscosity.The roughness element shape and density parameter Λs , given in Eq. 6, was introduced by Sigal

and Danberg (1990). It takes into account the reference area S without any roughness and the totalfrontal surface area of the roughness elements S f . Furthermore the windward wetted surface area Af

and the frontal surface area As of a roughness element are taken into account.

Λs = S S f

Af

As

− 1 .6

(7)

Originally Λs was dened for two dimensional homogenous surface roughness. Bons (2005) modiedthe denition of the parameter to be applicable to real surface roughness. The denitions of Bons(2005), Eq. 6 and Eq. 8, are also used in the current study to calculate Λs and determine an equivalentsand grain roughness.

Λs = Σdx i

Σ∆ h i

Σ∆ h i

Σli

− 1 .6

(8)

COMPUTATIONAL SETUPNumerical MethodThe numerical investigations in the current work are performed by using the CFD code LESOCC2

(Large-Eddy Simulation On Curvlinear Coordinates) which was developed for the simulation of com-plex turbulent ows. LESOCC2 is based on a cell centered 3D nite-volume method for block-structured grids. A central differences approach of second-order accuracy is applied for the spatialdiscretization of diffusive and convective uxes. For periodic boundaries, a Fourier-Solver has beenimplemented as described by Fr¨ ohlich (2006). Time marching is achieved by applying a multi-stagelow-storage, explicit Runge-Kutta method of second-order accuracy.

LESOCC2 solves the three-dimensional, unsteady, ltered and incompressible Navier-Stokesequations. Similar to statistically averaged equations, a closure problem arises so that different sub-grid-scale models are implemented in LESOCC2 to solve this problem. In the present work, theSmagorinsky model (Smagorinsky, 1963) is applied with Van Driest damping near solid walls. It isan eddy-viscosity model to parameterize the sub-grid-scale stresses

4



τ aij = τ ij − τ kk13

δ ij (9)

where δ ij is the Kronecker delta. The isotropic part of the sub-grid-scale stresses τ kk is not modeledbut added to the static pressure term using a ”pseudo” pressure Π = p + τ kk / 3 (Fr ohlich, 2006). Therest is modeled using a dimensionless empirical parameter.

τ ij = − 2ν sgs S ij ; S ij = 12

∂ ui

∂x j+

∂ u j

∂x i. (10)

The Smagorinsky viscosity ν sgs is modeled with the width of the lter ∆ and the shear-stress rate S ,as shown in Eq. 11, with the Smagorinsky constant being set to C s = 0.065 for channel ows.

ν sgs = ( C s ∆) 2

2S ij S ij (11)

For mass conservation, the SIMPLE algorithm with pressure-correction equation is solved usingthe strongly implicit method of Stone (1968). The velocity-pressure coupling is achieved by applyingthe momentum interpolation method of Rhie and Chow (1983). LESOCC2 is highly vectorized witha vectorization ratio > 99 % and additionally parallelized via the domain decomposition techniquewith the use of ghost cells and MPI for data transfer.

Computational domain and meshThe computational domain in the current work, shown in Fig. 3, consists of a channel with the

dimension 8.68 δ x 2δ x 8.68 δ in the x-, y- and z-direction. The main ow is always directed in apositive x-direction. Cyclic boundary conditions are applied to the inlet and outlet as well as to the

domain boundaries in z-direction, so that a channel of innite dimensions in x- and z-direction issimulated. The upper wall of the channel is always kept smooth whereas surface roughness is appliedto the lower wall. A block structured mesh with 64 blocks is used for spatial discretization. Detailsof the computational grid are listed in Tab. 1. The Reynolds number of all simulations is set toRe = 13750, which leads to an Reτ = 395 for a channel ow with smooth walls. In LESOCC2computations are carried out with non-dimensional parameters. The density is set to ρ = 1 and thebulk-velocity to U b = 1 .

Figure 3: Computational domain

The roughness of the bottom wall is derived from surface roughness measurements. The anisotropic

part of the surface roughness is the subject of the current study. To build a realistic model of it, theisotropic and anisotropic parts of measured surfaces have to be separated. On the left side of Fig. 4and Fig. 5 the measured turbine surface roughness are shown. To build a surface model, a Fast-Fourier-Transformation (FFT) is used to isolate the 50 most dominant frequencies, which capture all

5



Roughness Type A BDomain length x=4.3 y=1 z=4.3N x x N y x N z 315x221x192 192x221x315

Total number of cells ≈ 12.7 Mio∆ x+

c ≈ 10 ∆ x+c ≈ 18

Grid width ∆ y+w ≈ 1 ∆ y+

w ≈ 1∆ y+

c ≈ 10 ∆ y+c ≈ 10

∆ z +c ≈ 18 ∆ z +c ≈ 10Growth ratio in y-direction: 3%, in x-,z-direction: uniform

Table 1: Details of computational grid employed

of the relevant anisotropic structures. By setting all other frequencies to zero and doing an inverse

transformation, the roughness models are obtained, as shown on the right side in Fig. 4 and Fig. 5.The topology of the roughness types A and B in Tab. 1 are the same, but with different ow directions.The main ow direction of the roughness A is perpendicular to the grooves whereas the grooves of roughness B are aligned parallel to the main ow direction.

Figure 4: Real and model roughness type A

A scaling of the surface models is necessary due to the non-dimensionality of LESOCC2. Theparameter used for scaling is the dimensionless arithmetic roughness height Ra + , dened in Eq. 12.It is the arithmetic roughness height made dimensionless with the friction velocity uτ of the smoothwall and the kinematic viscosity ν .

Ra + =1l

N

i=1

|h(xi )| · uτ /ν (12)

The measured roughness topology is stored in a matrix. Ra is determined for each row of thematrix and nally averaged over the number of rows. For the roughness of type A and B, the dimen-sionless arithmetic roughness heights are set to Ra + = 5 .76. These values are chosen because theycorrespond to the Ra + of the turbine blade.

Numerical AccuracyThe methods and grids employed in a LES have a rst order impact on the accuracy of the results,because the lter width ∆ , which is used to calculate the sub-grid-scale viscosity ν sgs , is a functionof the mesh distaces ∆ xi . Therefore the sub-grid model is an explicit function of the grid. In order to

6



Figure 5: Real and model roughness type B

100

101

102

103

0

5

10

15

20

25

30

35

y+

u +

case B smooth wall, Reτ=389

DNS of Moser et al., Reτ=392

(a) Baseline velocity distribution in inner scaling

0 0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

y/ δ

s = µ

s g s

/ ( µ

s g s

+ µ

)

case Acase B

(b) Sub-grid activity parameter s distribution

Figure 6: Verication of grid and methods employed

check the accuracy of the simulations carried out, the velocity distribution of the smooth wall of caseB is compared with the results of Moser et al. (1999) which is shown in Fig. 6. It can be seen, that thevelocity is slightly overpredicted for y+ > 20 which is caused by the application of sub-grid stressmodeling and has also been reported by Cui et al. (2003b). The deviation of the velocity distributionis well within the range reported by Cui et al. (2003b).

The sub-grid activity parameter s dened by Durbin and Pettersson-Reif (2011) as the ratio of sub-grid viscosity to total viscosity

s = µsgs

µsgs + µ. (13)

For values of s < 0.5 the accuracy of the LES is high and within the range of a DNS. The distributionsof the average sub-grid activity parameter s for the cases A and B along the channel height is givenin Fig. 6. For both cases the sub-grid activity parameter is well below s = 0.05, which is one tenth

of the requested value for high a accuracy of Large-Eddy-Simulations. Therefore it can be concludedfrom Fig. 6 that grid resolution and quality is high enough for a good accuracy.It is known that the Smagorinsky model has a weak point when applied to wall resolved LES. The

model leads to an error because the Smagorinsky constant remains C s = 0 in near wall regions where

7



100

101

102

103

0

5

10

15

20

25

30

35

y+

u +

roughsmooththeory smooththeory rough

(a) Roughness A, κ = 0 .34 , C + = 4 .4

100

101

102

103

0

5

10

15

20

25

30

35

y+

u +

roughsmooththeory smooththeory rough

(b) Roughness B, κ = 0 .40 , C + = 5 .2

Figure 7: Mean velocity proles in inner scaling

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

y/ δ

u e + -

u +

roughsmooth

(a) Roughness A

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

y/ δ

u e + -

u +

roughsmooth

(b) Roughness B

Figure 8: Mean velocity proles in form of velocity defect

10



REFERENCESAchary, M., Bornstein, J., Escudier, M. P., 1986. Turbulent boundary layers on rough surfaces. Ex-

periments in Fluids (4), 33–47.

Bons, J. P., 2002. St and cf Augmentation for Real Turbine Roughness With Elevated FreestreamTurbulence. Transactions of the ASME (124), 632–644.

Bons, J. P., 2005. A Critical Assessment of Reynolds Analogy for Turbine Flows. ASME Journal of Heat Transfer (127), 472–485.

Bons, J. P., 2010. A Review of Surface Roughness Effects in Gas Turbines. ASME Journal of Turbo-machinery (132).

Bons, J. P., Taylor, R. P., McClain, S. T., Rivir, R. B., 2001. The Many Faces of Turbine SurfaceRoughness. ASME Journal of Turbomachinery (123), 739–748.

Bradshaw, P., 2000. A note on “critical roughness height” and “transitional roughness”. Physics of Fluids 12 (6), 1611.

Cai, X. M., Steyn, D. G., 1996. The von Karman constant determined by large eddy simulation.Boundary-Layer Meteorology 78, 143–164.

Cui, J., Lin, C.-L., Patel, V. C., 2003a. Use of Large-Eddy-Simulation to Characterize RoughnessEffect of Turbulent Flow Over a Wavy Wall. Journal of Fluids Engineering (125), 1075–1077.

Cui, J., Patel, V. C., Lin, C.-L., 2003b. Large-eddy simulation of turbulent ow in a channel with ribroughness. International Journal of Heat and Fluid Flow 24 (3), 372–388.

Durbin, P. A., Pettersson-Reif, B. A., 2011. Statistical Theory and Modeling for Turbulent Flows;Second Edition, 2nd Edition. Wiley, United Kingdom.

Flack, K. A., Schultz, M. P., 2010. Review of Hydraulic Roughness Scales in the Fully Rough Regime.Journal of Fluids Engineering (132).

Flack, K. A., Schultz, M. P., Shapiro, T. A., 2005. Experimental support for Townsend’s Reynoldsnumber similarity hypothesis on rough walls. Physics of Fluids 17 (3), 035102.

Frenzen, P., Vogel, C. A., 1995. On the magnitude and apparent range of variation of the von Karmanconstant in the atmospheric surface layer. Boundary-Layer Meteorology 72, 371–392.

Frohlich, J., 2006. Large Eddy Simulation turbulenter Str¨ omungen. Teubner, Wiesbaden.

Iacone, G. L., Reynolds, A. M., Tucker, P. G., 2008. Particle Deposition Onto Rough Surfaces. Journalof Fluids Engineering (130).

Ikeda, T., Durbin, P. A., 2007. Direct simulations of a rough-wall channel ow. Journal of FluidMechanics (571), 235–263.

Kunkel, G. J., Allen, J. J., Smits, A. J., 2007. Further support for Townsend’s Reynolds numbersimilarity hypothesis in high Reynolds number rough-wall pipe ow. Physics of Fluids 19, 055109–

1–055109–6.Lee, S.-H., Lee, J. H., Sung, H. J., 2010. Direct Numerical Simulation and PIV Measurement of

Turbulent Boundary Layer over a Rod-Roughened Wall.

11



Moser, R. D., Kim, J., Mansour, N. N., 1999. Direct numerical simulation of turbulent channel owup to Re=590. Journal of Fluids Engineering 11 (4), 943–945.

Nikuradse, J., 1933. Str¨ omungsgesetze in rauhen Rohren. VDI-Forschungsheft 361 (361).

Peet, Y., Sagaut, P., Charron, Y., 2008. Turbulent Drag Reduction using Sinusoidal Riblets withTriangular Cross-Section. Proceedings of the 38th AIAA Fluid Dynamics Conference and Exhibit.

Rhie, C. M., Chow, W. L., 1983. Numerical Study of the Turbulent Flow Past an Airfoil with TrailingEdge Separation. AIAA Journal (Vol. 21, No. 11), 1525–1532.

Schlichting, H., 1936. Experimental Investigation of the Problem of Surface Roughness. Ingenieur-Archiv 1 (3).

Schlichting, H., Gersten, K., 2005. Grenzschichttheorie, 10th Edition. Springer, Berlin-Heidelberg.

Schultz, M. P., Flack, K. A., 2005. Outer layer similarity in fully rough turbulent boundary layers.Experiments in Fluids 38 (3), 328–340.

Schultz, M. P., Flack, K. A., 2007. The rough-wall turbulent boundary layer from the hydraulicallysmooth to the fully rough regime. Journal of Fluid Mechanics 580, 381–405.

Sigal, A., Danberg, J. E., 1990. New Correlation of Roughness Density Effect on the TurbulentBoundary Layer. AIAA Journal (28), 554–556.

Singh, K. M., Sandham, N. D., Williams, J. J. R., 2007. Numerical Simulation of Flow over a RoughBed. Journal of Hydraulic Engineering (133), 386–398.

Smagorinsky, J., 1963. General Circulation Experiments with the Primitive Equations. MonthlyWeather Review 91 (3), 99–164.

Stone, H., 1968. Iterative Solution of Implicit Approximations of Multidimensional Partial Differen-tial Equations. SIAM J. Numer. Anal. (Vol. 5, No. 3).

van Rij, J. A., Belnap, B. J., Ligrani, P. M., 2002. Analysis and Experiments on Three-Dimensional,Irregular Surface Roughness. ASME Journal of Fluids Engineering (124), 671–677.

2

numerical investigation on the influence of anisotropic surface structures on the skin friction

Documents