Amano R.S., Sunden B. (Eds.) Computational Fluid Dynamics and Heat Transfer: Emerging Topics

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Sunden CH009.tex 25/8/2010 10: 57 Page 376

376 Computational Fluid Dynamics and Heat Transfer

and

(y) =

⎧

⎪

⎨

⎪

⎩

−

+ 2y

∗

,ifη

− 4η

= 0,



−η

+ 4η

tan

−1

⎛

⎜

⎝

+ 2y

∗



−η

+ 4η

⎞

⎟

⎠

,ifη

− 4η

< 0,



− 4η



+ 2y

∗

−



− 4η

+ 2y

∗



− 4η



,ifη

− 4η

> 0.

(B.10)

9.6 Nomenclature

: integration constants

c: concentration, mean concentration

: speciﬁc heat capacity at constant pressure



: model constants

–C

: model coefﬁcients

: pressure coefﬁcient

: sum of the convection and the diffusion terms of

the concentration equation

: sum of the convection and the diffusion terms of

the energy equation

: sum of the convection and the diffusion terms of

the momentum equation

: mean particle diameter

D pipe diameter

: integration constants

f : friction coefﬁcient of pipe ﬂows

: drag force

F,F

: Forchheimer correction coefﬁcient, Forchheimer

correction tensor

h,h

∗

: roughness height (equivalent sand grain roughness

height), h

√

/ν

: roughness Reynolds number: hU

/ν

H: ramp height or channel height

k,k

: turbulence energy, k at node P

K,K

: permeability, permeability tensor

: turbulent length scale

L: ramp length

Nu: Nusselt number

P: pressure or cell centre of the wall-adjacent cell

: production term of k equation

Pr, Pr

,Pr

∞

: Prandtl number, turbulent Prandtl numbers

Sunden CH009.tex 25/8/2010 10: 57 Page 377

AWFs of turbulence for complex surface ﬂow phenomena 377

: wall heat ﬂux

: surface concentration ﬂux

Re, Re

,Re

: Reynolds numbers: U

D/ν, U

x/ν, U

θ/ν; θ:

momentum thickness

: permeability Reynolds number: U

1/2

/ν

Sc, Sc

,Sc

∞

: Schmidt number, turbulent Schmidt numbers

St: Stanton number: Nu/(RePr)

: source term of the energy equation

U,U

: mean velocity, U/U

: bulk velocity, friction velocity:

√

/ρ

: Darcy velocity

: free stream velocities

: velocity at the point n

: slip velocity

x: wall-parallel coordinate



: x/L in the ramp ﬂow

y: wall normal coordinate or wall normal distance

: cell height, viscous sublayer thickness, characteristic

dissipation length

∗

: normalised distances: yU

/ν, y

√

/ν

∗

: connection points in two segment diffusivity models

∗

:1+α(y

∗

−y

∗

), 1+α(y

∗

−y

∗

); i =w,n,h; y

=0; y

αT

:1+α

∗

−y

∗

), 1+α

∗

−y

∗

); i =w,n,h; y

=0; y

βT

:1+β

∗

−y

∗

), 1+β

∗

−y

∗

); i =w,n,h; y

=0; y

α: c



or heat transfer coefﬁcient: q

−

inlet

)



,α

: α(y

∗

−y

∗

)/y

∗3

, αβSc/Sc

, α

∗

,α

: α Pr/ Pr

∞

, α Pr/ Pr

β,β

: model coefﬁcients

: C

/(h

∗

∞

)



,

: normalised total viscosity, thermal diffusivity and turbulent

diffusivity

: origin shift

x,y: cell widths in the x,y direction

ϕ: porosity

ε: dissipation rate of k

: mean temperature, | −

|(ρc

)/q

: wall temperature, temperature at the point n

κ,κ

: von Kármán constants

: Y

αT

+β

∗

µ,µ

: viscosity, turbulent viscosity

ν,ν

: kinematic viscosity, kinematic turbulent viscosity

ρ: ﬂuid density

: wall shear stress

φ,φ

: superﬁcial and intrinsic averaged values of φ

φ: Reynolds averaged value of φ

Sunden CH009.tex 25/8/2010 10: 57 Page 378

378 Computational Fluid Dynamics and Heat Transfer

References

[1] Launder, B. E. On the computation of convective heat transfer in complex turbulent

ﬂows,ASME J. Fluid Eng., 110, pp. 1112–1128, 1988.

[2] Kim, J., Moin, P., and Moser, R. Turbulence statistics in fully developed channel ﬂow

at low Reynolds number, J. Fluid Mech., 177, pp. 133–166, 1987.

[3] Ahmed, A., and Demoulin, M. Turbulence modelling in the automotive industry, in:

EngineeringTurbulenceModellingandMeasurements–5,W.RodiandN.Fueyo(Eds.),

Elsevier Science Ltd.,Amsterdam, pp. 29–42, 2002.

[4] Launder, B. E., and Spalding, D. B. The numerical computation of turbulent ﬂows,

Comp. Meth. Appl. Mech. Eng., 3, pp. 269–289, 1974.

[5] Chieng, C. C., and Launder, B. E. On the calculation of turbulent heat transport down-

stream from an abrupt pipe expansion, Numerical Heat Transfer, 3, pp. 189–207,

1980.

[6] Amano, R. S. Development of turbulence near wall model and its application to

separated and reattached ﬂows, Numerical HeatTransfer, 7, pp. 59–76, 1984.

[7] Launder,B. E. Numerical computation ofconvective heat transferincomplexturbulent

ﬂows: time to abandon wall functions?, Int. J. Heat Mass Transfer, 27, pp. 1485–1490,

1984.

[8] Craft, T. J., Gerasimov, A. V., Iacovides, H., and Launder, B. E. Progress in the gen-

eralization of wall-function treatments, Int. J. Heat Fluid Flow, 23, pp. 148–160,

2002.

[9] Cabot, W., and Moin, P. Approximate wall boundary conditions in the large-eddy

simulation of high Reynolds number ﬂow, Flow Turb. Combust., 63, pp. 269–291,

1999.

[10] Suga, K., Craft,T. J., and Iacovides, H.An analytical wall-function for turbulent ﬂows

and heat transfer over rough walls, Int. J. Heat Fluid Flow, 27, pp. 852–866, 2006.

[11] Suga, K., and Nishiguchi, S. Computation of turbulent ﬂows over porous/ﬂuid

interfaces, Fluid Dyn. Res., 41, pp. 012401:1–15, 2009.

[12] Suga, K. Computation ofhighPrandtlnumberturbulentthermal ﬁelds bytheanalytical

wall-function, Int. J. Heat Mass Transfer, 50, pp. 4967–4974, 2007.

[13] Suga,K., andKubo,M.Modelling turbulenthighSchmidtnumber masstransferacross

underformable gas-liquid interfaces, Int. J. Heat Mass Transfer, 53, pp. 2989–2995,

2010.

[14] Patankar, S. V. Numerical Heat Transfer and Fluid Flow. Hemisphere/McGraw Hill,

Washington, 1980.

[15] Rhie, C. M., andChow, W. L.Numerical study of theturbulent ﬂow pastanairfoil with

trailing edge separation,AIAA J., 21, pp. 1525–1532, 1983.

[16] Lien,F.S.,andLeschziner,M.A.Upstreammonotonicinterpolationforscalartransport

withapplicationincomplexturbulentﬂows, Int. J.Num. Meth. Fluids,19,pp.527–548,

1994.

[17] Gerasimov, A. V. Development and validation of an analytical wall-function strategy

for modelling forced, mixed and naturalconvection phenomena, Ph.D.Thesis, UMIST,

Manchester, UK, 2003.

[18] Suga, K., andBasara, B.Application oftheanalytical wall-function of turbulence toIC

engine ﬂows, Proc. 43rd Nat. Heat Transfer Symp., Japan, Nagoya, CD ROM, 2006.

[19] Suga, K. A study toward improving accuracy of large scale industrial turbulent ﬂow

computations, in: Frontiers of Computational Science, Y. Kaneda, H. Kawamura, M.

Sasai (Eds.), Springer, pp. 99–105, 2007.

Sunden CH009.tex 25/8/2010 10: 57 Page 379

AWFs of turbulence for complex surface ﬂow phenomena 379

[20] Nikuradse, J. Strömungsgesetze in rauhen rohren.VDI-Forschungsheft, 361, 1933.

[21] Cebeci, T., and Bradshaw, P. Momentum Transfer in Boundary Layers, Hemisphere

Pub. Co, Washington, 1977.

[22] Krogstad, P.A.,Antonia, R.A., and Browne, L.W. B. Comparison between rough- and

smooth-wall turbulent boundary layers, J. Fluid Mech., 245, pp. 599–617, 1992.

[23] Tachie, M. F., Bergstrom, D. J., and Balachandar, R. Roughness effects in low-Re

open-channel turbulent boundary layers, Expt. Fluids, 35, pp. 338–346, 2003.

[24] Nagano,Y., Hattori, H., and Houra, T. DNS of velocity and thermal ﬁelds in turbulent

channel ﬂow with transverse-rib roughness, Int. J. Heat Fluid Flow, 25, pp. 393–403,

2004.

[25] Launder, B. E., and Sharma, B. I. Application of the energy-dissipation model of

turbulence to the calculation of ﬂow near a spinning disc, Lett. Heat Mass Transfer,1,

pp. 131–138, 1974.

[26] Craft, T. J., Launder, B. E., and Suga, K. Development and application of a cubic

eddy-viscosity model of turbulence, Int. J. Heat Fluid Flow, 17, pp. 108–115, 1996.

[27] Moody, L. F. Friction factors for pipe ﬂow,Trans.ASME, 66, pp. 671–678, 1944.

[28] Turner, A. B., Hubbe-Walker, S. E., and Bayley, F. J. Fluid ﬂow and heat transfer

over straight and curved rough surfaces, Int. J. Heat Mass Transfer, 43, pp. 251–262,

2000.

[29] van Mierlo, M. C. L. M., and de Ruiter, J. C. C. Turbulence measurements above

artiﬁcial dunes, Delft Hydraulics Lab. Rep., Q789, Delft,The Netherlands, 1988.

[30] Iacovides, H., and Raisee, M. Recent progress in the computation of ﬂow and heat

transfer in internal cooling passages of gas-turbine blades, Int. J. Heat Fluid Flow, 20,

pp. 320–328, 1999.

[31] Song, S., and Eaton, J. K. The effects of wall roughness on the separated ﬂow over a

smoothly contoured ramp, Expt. Fluids, 33, pp. 38–46, 2002.

[32] Zippe, H. J., and Graf, W. H. Turbulent boundary-layer ﬂow over permeable and non-

permeable rough surfaces, J. Hydraul. Res., 21, pp. 51–65, 1983.

[33] Breugem, W. P., Boersma, B. J., and Uittenbogaard, R.E. The inﬂuence of wall

permeability on turbulent channel ﬂow, J. Fluid Mech., 562, pp. 35–72, 2006.

[34] Beavers, G. S., Joseph, D. D. Boundary conditions at a naturally permeable wall, J.

Fluid Mech., 30 pp. 197–207, 1967.

[35] Whitaker, S. The Forchheimer equation: A theoretical development, Transp. Porous

Media, 25, pp. 27–61, 1996.

[36] Ergun,S.Fluidﬂow through packed columns, Chem. Eng. Progr., 48,pp. 88–94, 1952.

[37] Macdonald, I. F., El-Syed, M. S., Mow, K., and Dullien, F. A. Flow through porous

media: The Ergun equation revised, Ind. Eng. Chem. Fundam., 18, pp. 199–208,

1979.

[38] Brinkman, H. C. A calculation of viscous force exerted by a ﬂowing ﬂuid on a dense

swarm of particles,Appl. Sci. Res.A, 1, pp. 27–34, 1948.

[39] Craft, T. J., and Launder, B. E. Principles and performance of TCL-based second-

moment closures Flow, Turb. Combust., 66, pp. 355–372, 2001.

[40] Rogers, M. M., Mansour, N. N., and Reynolds, W. C. An algebraic model for the

turbulent ﬂux of a passive scalar, J. Fluid Mech., 203, pp. 77–101, 1989.

[41] Suga, K., and Abe, K. Nonlinear eddy viscosity modelling for turbulence and heat

transfer near walland shear-free boundaries, Int. J. Heat Fluid Flow, 21(1), pp. 37–48,

2000.

[42] So, R. M. C., and Sommer, T. P. A near-wall eddy conductivity model for ﬂuids with

different Prandtl numbers,ASME J. HeatTransfer, 116, pp. 844–854, 1994.

Sunden CH009.tex 25/8/2010 10: 57 Page 380

380 Computational Fluid Dynamics and Heat Transfer

[43] Kasagi, N., and Shikazono N. Contribution of direct numerical simulation to under-

standing and modelling turbulent transport, Proc. R. Soc. Lond. A, 451, pp. 257–292,

1995.

[44] Kader, B.A.Temperature andconcentration proﬁles infully turbulentboundary layers,

Int. J. Heat and MassTransfer, 24, pp. 1541–1544, 1981.

[45] Kays, W. M., and Crawford, M. E. in: Convective Heat and Mass Transfer, 3rd ed.,

McGraw-Hill, NewYork, pp. 298–301, 1993.

[46] Calmet, I., and Magnaudet, J. High-Schmidt number mass transfer through turbulent

gas-liquid interfacesInt. J. Heat Fluid Flow, 19, pp. 522–532, 1998.

[47] Hasegawa, Y., and Kasagi, N. Effects of interf acial velocity boundary condition on

turbulentmasstransferat highSchmidtnumbers,Int. J.Heat Fluid Flow,28,pp.1192–

1203, 2007.

Sunden CH010.tex 10/9/2010 15: 22 Page 381

IV. Advanced Simulation Modeling

Technologies

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Sunden CH010.tex 10/9/2010 15: 22 Page 383

10 SPH–aversatile multiphysics

modeling tool

Fangming Jiang

andAntonio C.M. Sousa

2,3

The Pennsylvania State University, USA

Universidade deAveiro, Portugal

University of New Brunswick, Canada

Abstract

Inthischapter, thecurrentstate-of-the-artand recentadvancesofanovelnumerical

method – the Smoothed Particle Hydrodynamics (SPHs) will be reviewed through

case studies with particular emphasis to ﬂuid ﬂow and heat transport. To provide

sufﬁcient background and to assess its engineering/scientiﬁc relevance, three par-

ticular case studieswill be used toexemplifymacro- and nanoscale applicationsof

this methodology. The ﬁrst application deals with magnetohydrodynamic (MHD)

turbulence control. Effective control of the transition to turbulence of an electri-

callyconductive ﬂuidﬂowcan beachievedbyapplyinga stationary magneticﬁeld,

which is not simply aligned along the streamwise or transverse ﬂow direction, but

along a direction that forms an angle with the main ﬂuid ﬂow in the range of 0

◦

(streamwise) to 90

◦

(transverse).TheSPH numerical technique is used tointerpret

this concept and to analyze the magnetic conditions. The second application deals

with non-Fourier ballistic-diffusive heat transfer, which plays a crucial role in the

development ofnanotechnology andthe operation ofsubmicron- and nanodevices.

Theballistic-diffusiveequationto heattransportinathinﬁlmissolvednumerically

via the SPH methodology. The third application deals with mesoscopic pore-scale

model for ﬂuid ﬂow in porous media. SPH simulations enable microscopic visu-

alization of ﬂuid ﬂow in porous media as well as the prediction of an important

macroscopic parameter – the permeability.

Keywords: CFD, Numerical methods, Smoothed particle hydrodynamics

10.1 Introduction

A novel numerical method – the smoothed particle hydrodynamic (SPH) offers a

relatively ﬂexible tool for heat and ﬂuid ﬂow computations, as it can cope with a

wide rangeof spacescales andof physical phenomena. SPHis a meshlessparticle-

based Lagrangian ﬂuid dynamics simulation technique, in which the ﬂuid ﬂow is

Sunden CH010.tex 10/9/2010 15: 22 Page 384

384 Computational Fluid Dynamics and Heat Transfer

represented by a collection of discrete elements or pseudo-particles. These par-

ticles are initially distributed with a speciﬁed density distribution and evolve in

time according to the constitutive conservation equations (e.g., mass, momentum,

energy). Flow properties are determined by an interpolation or smoothing of the

nearby particle distribution using a special weighting function – the smoothing

kernel. SPH was ﬁrst proposed by Gingold and Monaghan [1] and by Lucy [2]

in the context of astrophysical modeling. The method has been successful in a

broad spectrum of problems, among others, heat conduction [3, 4], forced and

natural convective ﬂow [5, 6], low Reynolds number ﬂow [7, 8], and interfacial

ﬂow [9–11]. Many other applications are given by Monaghan [12] and Randle and

Libersky [13], which offer comprehensive reviews on the historical background

and some advances of SPH. In comparison to the Eulerian-based computational

ﬂuid dynamics (CFD)methods, SPHis advantageous inwhat concerns the follow-

ing aspects: (a) particular suitability to tackle problems dealing with multiphysics;

(b) ease of handling complex free surface and material interface; and (c) relatively

simplecomputercodesandeaseofmachineparallelization.Theseadvantagesmake

it particularly well suited to deal with transient ﬂuid ﬂow and heat transport.

This chapter is organized in six main sections, namely: 1. Introduction; 2. SPH

theory, formulation, and benchmarking; 3. Control of the onset of turbulence in

MHD ﬂuid ﬂow; 4. SPH numerical modeling for ballistic-diffusive heat conduc-

tion; 5. Mesoscopic pore-scale SPH model for ﬂuid ﬂow in porous media, and 6.

Concluding remarks.

10.2 SPHTheory, Formulation, and Benchmarking

SPH, despite its promise, is far from being at its perfected or optimized stage.

ProblemswiththeSPH,suchashandlingofsolidboundaries[14],choiceofsmooth-

ing kernel and speciﬁcation of the smoothing length [15], treatment of heat/mass

conducting ﬂow [5, 6], and modeling of low Reynolds number ﬂow, where the

surface/interface effects act as a dominating role relative to the inertial effect [10],

have prevented its application to some practical problems. This section reports

an effort to enhance the SPH methodology and develop appropriate computer

programmable formulations.

10.2.1 SPH theory and formulation

In the SPH formulation, the continuous ﬂow at time t is represented by a collec-

tion of N particles each located at position r

(t) and moving with velocity v

(t),

i =1,2,......,N. The “smoothed” value of any ﬁeld quantity q(r, v) at a space

point r (bold text designatesvectorortensor) isaweightedsumof allcontributions

from neighboring particles, namely:

q(r,v)=



ρ(r

)

q(r

)w(|r −r

|,h) (1)

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SPH – a versatile multiphysics modeling tool 385

∇q(r,v)=



j=i

ρ(r

)

q(r

)∇w(|r − r

|,h) (2)

where,m

andρ(r

),respectively, denotethemassanddensityofparticlej. w(|r|,h)

is the weight or smoothing function with h being the smoothing length.

Density update

Based on equation (1), the local density at position r

(t) is:



(3)

The summation is over all neighboring particles j including particle i itself. Equa-

tion (1) conserves mass accurately if the total SPH particles are kept constant. In

terms of equation (2), another formulation for density calculation can be derived

from the continuity equation, namely

dρ



j=i

·∇

(4)

The summationis overall neighboringparticles j withexception ofparticle i itself.

Here, v

−v

, r

−r

, and w

=w(|r

−r

|,h); the subscript in the oper-

ator ∇

indicates the gradient derivatives are taken with respect to the coordinates

at particle i

∇



∂w

∂r

(5)

Equation (4) is the Galilean invariant, which is more appropriate for interface

reconstruction in free surface ﬂow simulations. The use of this equation has the

computational advantage over equation (3) of calculating all rates of change in

one pass over the particles; whereas the use of equation (3) requires one pass to

calculate the density, then another one to calculate the rates of change for velocity,

and different scalars, such as temperature.

Equation (4) is employed to perform the density update in the case studies to

be presented.

Incompressibility

A quasi-compressible method is implementedtodetermine the dynamic pressurep

[8], in which an artiﬁcial equation of state is used

p = p





− 1.0



(6)

where, p

is the magnitude of the pressure for material state corresponding to the

reference density ρ

. The ratio of the speciﬁc heats γ is artiﬁcially a large value