Direct numerical simulation of turbulent pipe flow using ...research.me.udel.edu/~lwang/reprints/Peng_etal_JCP... · Under consideration for publication in J. Fluid Mech. 1 Direct

Direct numerical simulation of turbulent pipe flow using the lattice Boltzmann method

Journal: Journal of Fluid Mechanics

Manuscript ID JFM-16-S-1109

mss type: Standard

Date Submitted by the Author: 07-Sep-2016

Complete List of Authors: Peng, Cheng; University of Delaware, Department of Mechanical Engineering Geneva, Nicholas; University of Delaware, Department of Mechanical Engineering Guo, Zhaoli; Huazhong University of Science and Technology, State Key Laboratory of Coal Combustion Wang, L-P; University of Delaware, Mechanical Engineering Dept.

Keyword: Turbulence simulation < Turbulent Flows, Pipe flow boundary layer <

Boundary Layers, Computational methods < Mathematical Foundations

Under consideration for publication in J. Fluid Mech. 1

Direct numerical simulation of turbulentpipe flow using the lattice Boltzmann

method

Cheng Peng1, Nicholas Geneva1, Zhaoli Guo2 and Lian-Ping Wang1,2†1Department of Mechanical Engineering, 126 Spencer Laboratory, University of Delaware,

Newark, Delaware 19716-3140, USA2State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology,

Wuhan, P.R. China

(Received xx; revised xx; accepted xx)

In this paper, we present a first direct numerical simulation (DNS) of turbulent pipeflow using the mesoscopic lattice Boltzmann method (LBM) on a D3Q19 lattice grid.DNS of turbulent pipe flow using LBM has never been reported previously, perhaps dueto inaccuracy and numerical stability associated with LBM in the presence of a curved

solid surface. In fact, it was even speculated that the popular D3Q19 lattice might beinappropriate as a DNS tool for turbulent pipe flow. In this paper, we show, througha novel implementation, that accurate DNS simulation of turbulent pipe flow using theD3Q19 lattice is achievable. Our implementation makes use of an extended MRT modeland a moving reference frame to enhance numerical stability. An issue associated withGalilean invariance in the moving frame is identified and resolved through coordinatetransformation. The resulting turbulent flow statistics at a friction Reynolds number ofReτ = 180 are compared systematically with both published experimental and otherDNS results based on solving the Navier-Stokes equations. The comparisons cover themean-flow profile, the r.m.s. velocity and vorticity profiles, the mean and r.m.s. pressureprofiles, the velocity skewness and flatness, and spatial correlations and energy spectra ofvelocity and vorticity. Together, we are able to identify various levels of inaccuracy in allprevious results and their likely sources, and clarify which data can be used as benchmarkfor turbulent pipe flow in the future. Furthermore, statistics and flow visualizationsfor three distinct regions (the near-wall region, buffer region, and inertial sublayer)are compared and contrasted, to demonstrate that the velocity field remains stronglyanisotropic throughout the pipe, while the vorticity field becomes nearly isotropic in thecenter region of the pipe.

Key words:

1. Introduction

Over the last two decades, the lattice Boltzmann method (LBM) has been rapidlydeveloped into an alternative and viable computational fluid dynamics (CFD) methodfor simulating viscous fluid flows involving complex boundary geometries or moving fluid-solid / fluid-fluid interfaces (Aidun & Clausen 2010). Unlike conventional (macroscopic)

† Email address for correspondence: [email protected]

Page 1 of 44

2 C. Peng, N. Geneva, Z.L. Guo and L.-P. Wang

CFD methods that are based on directly solving the Navier-Stokes (N-S) equations, LBMis a mesoscopic method that is governed by a discretized version of gas kinetic equation inwhich molecular distribution functions are relaxed locally and then propagated to theirneighboring locations. Since only local data communication is needed in each time step,LBM is extremely suitable for large-scale simulations that require parallel computationusing a large number of processors, e.g., direct numerical simulations (DNS) of turbulentflows. Although still confined to relative low to moderate flow Reynolds (Re) numbers,DNS has been established as an independent research tool to study the physics ofturbulent flows. The data generated from DNS not only agrees well with experimentalresults, but often provides greater details and insights of the flow field to a degree thatmay be very difficult or impossible to achieve experimentally.In recent years, the capabilities of LBM as a DNS tool for turbulent flows have been

explored by a series of studies in homogeneous isotropic turbulence and turbulent channelflows (see ten Cate et al. 2004; Lammers et al. 2006; Peng et al. 2010; Gao et al. 2013;Kang & Hassan 2013; Wang et al. 2014, 2016b). However, DNS of turbulent flow in acircular pipe using LBM has not yet been reported, to our best knowledge. So far allsuccessful DNS studies of turbulent pipe flow have been performed using conventionalnumerical methods based on directly solving the N-S equations. Eggels et al. (1994)presented the first DNS of a fully developed turbulent pipe flow using a second-orderfinite-volume (FV) method based on a uniform grid in a cylindrical coordinates. Thebulk Reynolds number based on the domain-averaged mean flow speed Ub and the pipediameter D of their DNS is Reb = UbD/ν = 5300, which corresponds to a Reynoldsnumber based on the friction velocity u∗ and pipe radius R of Reτ = u∗R/ν = 180. Hereν is the fluid kinematic viscosity. The turbulent statistics from their simulation werereported to be in reasonable agreements with those from Laser Doppler Anemometry(LDA) and Particle Image Velocimetry (PIV) measurements at Reb = 5450. Theyconfirmed that the logarithmic layer deviates from the classical log-law at this low flowReynolds number due to the pipe curvature effect. Later, Orlandi & Fatica (1997)conducted DNS of a fully developed turbulent flow in a pipe rotating about its axis usingthe same second-order FV method but on a non-uniform grid in the radial direction. Theircomparison of the vorticity field in the non-rotating and rotating pipe flows indicatedthat the rotation motion of the pipe wall enhances the transport of streamwise vorticitytowards the center of the pipe. Loulou et al. (1997) performed DNS of a fully developedturbulent pipe flow at Reb = 5600 using a B-spline pseudo-spectral method. Comparedwith the previous FV simulations, Loulou et al.’s pseudo-spectral method is much moreaccurate providing likely a true benchmark for turbulent statistics, although later in thispaper we will find that even the spectral results are not always accurate due to inadequateresolution and limited time duration of the simulation.

Wagner et al. (2001) investigated the effect of Reynolds number on the turbulentstatistics of fully developed turbulent pipe flow by considering a range of flow Reynoldsnumber from Reb = 5, 300 to 10,300, using a second-order FV method based on anon-uniform grid in the radial direction. The comparisons of turbulent statistics underdifferent Reb suggest the strong influence of flow Reynolds number on both mean andfluctuation flow properties. Their results also confirmed that, under different Reb, thenear-wall velocity fluctuations scaled with the Kolmogorov length ηk = (ν3/ǫw)

1/4 andKolmogorov velocity vk = (νǫw)

1/4 collapse better than those scaled by the wall scaling,as first pointed out in Antonia & Kim (1994), where ǫw is the dissipation rate of turbulentkinetic energy at the wall. More recently, Wu & Moin (2008) conducted a DNS study ofturbulent pipe flows with Reb up to 44,000 with a second-order FV method. Although theclassical log-law was still not followed due to the lack of distinct separation between the

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DNS of turbulent pipe flow using LBM 3

inner and outer scales, under this wide range of Re, the radial gradient of mean velocityprofile was reported to be qualitatively similar with a narrow convex plateau near thewall, followed by an extended concave region at intermediate radial positions and a finalconvex segment in the center region of the pipe.The above studies establish that the boundary layer structures and dynamics in a

circular pipe share common features with a turbulent flow in a rectangular channel.However, due to its curved geometry, distinct differences exist between the two. Forexample, while the mean velocity profile conforms to the log-law with a bulk Reb (definedwith the channel width) as small as 3,000 in a channel flow, the mean velocity profile in apipe still deviates from the classical log-law with Reb as high as 230,000 (see Huffman &Bradshaw 1972; denToonder & Nieuwstadt 1997; Wagner et al. 2001; Wu & Moin 2008).Since turbulent pipe flow represents a more realistic flow often encountered in en-

gineering applications, it is highly desirable to perform DNS of this flow with LBMin order to establish LBM as a viable alternative for turbulent flow simulation. ForLBM simulations, the presence of a curved pipe wall adds a great deal of fundamentaldifficulties. The accurate representation of the curved pipe surface is essential since themajority of turbulent kinetic energy production and viscous dissipation is confined tothe near-wall region. Unlike conventional CFD methods, such as the finite volume (FV)scheme, that can be applied to a computational mesh in cylindrical coordinates whichallows accurate representation of the pipe wall (see Eggels et al. 1994; Loulou et al.

1997; Wagner et al. 2001; Wu & Moin 2008), standard LBM is restricted to a cubicgrid in Cartesian coordinates that does not conform with the cylindrical surface. Thusto resolve the pipe boundary in LBM, local interpolation is used which could introduceboth inaccuracy and numerical instability. Although the boundary treatments on a curvedboundary have received a great deal of attention in LBM (see Bouzidi et al. 2001; Meiet al. 2000; Yu et al. 2003), their impacts on the accuracy and numerical instability in aturbulent pipe flow simulation has not been systematically studied.

Previously, only large-eddy simulations (LES) of turbulent pipe flow have been at-tempted using LBM (see Kang & Hassan 2013; Suga et al. 2015), which confirmedthe existence of so-called lattice effects in the simulation of turbulent flow in circularpipe due to the lack of full isotropy of a D3Q19 lattice (see White & Chong 2011).Kang & Hassan (2013) conducted LES of turbulent pipe flow on both D3Q19 andD3Q27 lattice grids. Their results indicate that with the same resolution, the meanand fluctuation velocities with D3Q27 are in reasonable agreement with the previousDNS benchmark (Eggels et al. 1994), while those with D3Q19 developed an unphysicalsecondary flow pattern and exhibited large deviations (up to 28% relative error) fromthe respective benchmark (Kang & Hassan 2013). Suga et al. (2015) later presented asimilar LES of turbulent pipe flow based on their proposed multi-relaxation time (MRT)LBM model with D3Q27 lattice grid. Both their mean flow velocity and root-mean-squared (r.m.s.) velocity profiles are in reasonable agreement with the correspondingDNS benchmark (Eggels et al. 1994). As a result of the issues associated the D3Q19lattice, these studies have argued that D3Q27 has better isotropy properties (Kang &Hassan 2013; White & Chong 2011; Suga et al. 2015) and thus should be adopted insteadof D3Q19 in DNS of a turbulent pipe flow. However, careful benchmarking of the LESsimulation results is still necessary in order to fairly assess the capabilities of differentlattice models in LBM. As will become clear later, the capabilities and accuracy of alattice model depend largely on the implementation details used to meet the no-slipboundary condition at the fluid-solid interface. In this paper, we will introduce a novelimplementation of LBM, to demonstrate that accurate DNS of a turbulent pipe flow isactually achievable even on a D3Q19 lattice.

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We should note that there are some limited efforts to introduce cylindrical coordinatesinto LBM. One typical approach is the development of LBM models for axisymmetricflows (see Guo et al. 2009; Li et al. 2010; Wang et al. 2011). The idea is to add asource term in the lattice Boltzmann equation (LBE) to reproduce the N-S equationsin the cylindrical coordinates, when the flow has no azimuthal dependence. Althoughsuch methods allows one to simulate three dimensional axisymmetric flow on a two-dimensional lattice grid, which greatly enhances the computational efficiency, it cannotbe used for three-dimensional turbulent pipe flow simulation. Other approaches that couldbe relevant to DNS of turbulent pipe flow is the development of finite-difference (FD) (seeGuo & Zhao 2003; Hejranfar & Ezzatneshan 2014), FV (see Xi et al. 1999; Dubois &Lallemand 2008; Guo et al. 2013, 2015) or spectral implementations (see Min & Lee 2011)of the discrete-velocity Boltzmann equation, by uncoupling the discretization of physicalspace and lattice time. Those alternative Boltzmann-equation based approaches do allowthe use of non-uniform grid in curvilinear coordinates to fit, for example, the circularpipe wall. However, compared with the standard LBM that uses direct streaming, thesemethods introduce additional diffusion and dissipation in the discretization of space andtime, which may affect the accuracy of the simulated turbulent statistics. Moreover, theyare less efficient in terms of parallel scalability when compared to the standard LBM.The present work has three major objectives. First, we will develop a novel LBM

implementation on a D3Q19 lattice and use it as a DNS tool for turbulent pipe flow. Inorder to achieve this goal, we have introduced improved methods both within the LBMscheme itself and at the fluid-solid interface to address issues related to accuracy andnumerical instability in the presence of a curved wall. While the study here is limitedto turbulent pipe flow, the methods we will incorporate have broader implications forsimulating turbulent flows in complex geometries.

Second, we will conduct a careful comparison of the resulting turbulent statisticsbetween our LBM and other conventional methods (Eggels et al. 1994; Wagner et al.

2001; Loulou et al. 1997) as well as those from experimental measurements. Experimentaldata of turbulent pipe flow at moderate flow Re numbers include measurements usingLDA (Eggels et al. 1994; Durst et al. 1995) and PIV (Westerweel et al. 1996). Suchsystematic comparisons will not only provide a fair assessment on the performance of ourLBM, especially its boundary treatment schemes on the simulation of flow in a cylindricalgeometry; but will also reveal inaccuracies in some previous DNS and experimental datawhich should benefit the community in the future.

Third, instantaneous flow visualizations and a variety of flow statistics, both in thephysical space and the spectral space, will be presented to probe the dynamics andrelevant scales of the turbulent pipe flow. The idea is to take advantage of the rathercomplete and multifaceted DNS data to thoroughly explore the physics of this classicalturbulent flow, which may not be feasible in typical experimental measurements.The rest of the paper is arranged as follows. In Sec. 2, we discuss the physical problem

and the numerical method with all novel implementation details. Those details includea MRT model with extended equilibria based on a D3Q19 lattice grid to enhance thenumerical stability, a non-uniform perturbation force field to excite the turbulence and anew bounce back scheme to remove the Galilean-invariance errors when the flow systemis solved in a purposely-designed moving reference frame. In Sec. 3, we first validatethe method against the transient laminar pipe flow and briefly discuss the accuracyof the LBM method. Then, in Sec. 4, the turbulent flow statistics will be discussedsystematically through a careful comparison with existing numerical and experimentalresults. A large number of flow statistics in both physical space and spectral space willbe analyzed. Finally, Sec. 5 contains a summary and the main conclusions of the present

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Page 5 of 44


the present study. Compared with the single-relaxation time (SRT) collision operator, theMRT collision model is preferred due to its better numerical stability, as the relaxationparameters associated with moments irrelevant to the N-S equations can be optimized forthis purpose (see Lallemand & Luo 2000; d’Humieres et al. 2002). The evolution equationof the MRT LBM reads as

f(x+ eiδt, t+ δt)− f(x, t) = −M−1S[

m(x, t)−m(eq)(x, t)]

+M−1Ψ, (2.2)

where f is the distribution function vector, t and x are the time and spatial coordinate,respectively. ei is the lattice particle velocity in the i direction. In D3Q19, nineteenthree-dimensional lattice velocities are used and they are

ei =

(0, 0, 0) c, i = 0,(±1, 0, 0) c, (0,±1, 0) c, (0, 0,±1) c, i = 1, 2, . . . , 6,(±1,±1, 0) c, (±1, 0,±1) c, (0,±1,±1) c, i = 7, 8, . . . , 18.

(2.3)

where c = δx/δt, δx and δt are the grid spacing and time step size, respectively. Nineteenindependent moments are defined by m = Mf , where M is a 19× 19 transform matrix.The diagonal matrix S contains relaxation parameters and is written as

S = diag (0, se, sε, sj , sq, sj , sq, sj , sq, sν , sπ, sν , sπ, sν , sν , sν , sm, sm, sm) . (2.4)

In the standard three-dimensional MRT LBM, the values of the relaxation parametersfor the energy and stress modes, i.e., se and sν , are directly related to the bulk and shearviscosities (νV and ν) of the continuum fluid, as

νV =2

9

(

1

se− 1

2

)

c2δt, ν =1

3

(

1

sν− 1

2

)

c2δt. (2.5)

The values of those relaxation parameters are between 0 and 2 in order to maintainpositive viscosities. Numerical instability may arise when those relaxation parametersapproach the two bounding values.

The vector m(eq) in Eq. (2.2) specifies the equilibrium moments. Normally, in thestandard MRT LBM, m(eq) are functions of only the conserved moments, i.e., densityfluctuation δρ and momentum ρ0u†, as in d’Humieres et al. (2002) and Peng et al.

(2010). However, as indicated by Eq. (2.5), small viscosities in the high Re flows pushthe relaxation parameters se and sν to their upper limit, which sabotages the numericalstability of the simulation. In order to address this problem, stress components areintroduced into m(eq) to modify Eq. (2.5). This idea was firstly proposed by Inamuro(2002) with a SRT collision model. Following the same idea, the equilibrium moments inour extended MRT model is formally written as

m(eq) = m(eq,0) +m(eq,1), (2.6)

† Note that, here we follow the incompressible LBM model proposed by He & Luo (1997),partitioning the density into a local density fluctuation δρ and a constant background densityρ0.

Page 6 of 44


with

m(eq,0)0 = δρ, m

(eq,0)1 = −11δρ+ 19ρ0

(

u2 + v2 + w2)

,

m(eq,0)2 = αδρ+ βρ0

(

u2 + v2 + w2)

, m(eq,0)3 = ρ0u, m

(eq,0)4 = −2ρ0u/3,

m(eq,0)5 = ρ0v, m

(eq,0)6 = −2ρ0v/3, m7 = ρ0w, m8 = −2ρ0w/3,

m(eq,0)9 = ρ0

(

2u2 − v2 − w2)

, m(eq,0)10 = γρ0

(

2u2 − v2 − w2)

,

m(eq,0)11 = ρ0

(

v2 − w2)

, m(eq,0)12 = γρ0

(

2u2 − v2 − w2)

, m(eq,0)13 = ρ0uv,

m(eq,0)14 = ρ0uw, m

(eq,0)15 = ρ0vw, m

(eq,0)16 = m

(eq,0)17 = m

(eq,0)18 = 0.

(2.7)

where α, β and γ are free parameters that are irrelevant to the N-S equations. Thesequence of the moments and their definitions are identical with those in d’Humiereset al. (2002). Furthermore, we choose α = 0, β = −475/63 and γ = 0 as in d’Humiereset al. (2002).

The second term m(eq,1) in Eq. (2.6) contains additional equilibrium moments beingintroduced to modify Eq. (2.5). Only 6 of the 19 elements in m(eq,1) are relevant to thederivation of the N-S equations, and they are

m(eq,1)1 = ρ0ζ (∂xu+ ∂yv + ∂zw) , m

(eq,1)9 = ρ0λ (4∂xu− 2∂yv − 2∂zw) ,

m(eq,1)11 = ρ0λ (2∂yv − 2∂zw) , m

(eq,1)13 = ρ0λ (∂xv + ∂yu) ,

m(eq,1)14 = ρ0λ (∂yw + ∂zv) , m

(eq,1)15 = ρ0λ (∂zu+ ∂xw)

(2.8)

and the others are simply set to zero. Using the Chapman-Enskog expansion, we canshow that this extended model leads to the Navier-Stokes equations with the followingshear and bulk viscosities (see Wang et al. 2016a)

ν =1

3

(

1

sν− 1

2

)

c2δt − λ, (2.9a)

νV =2

9

(

1

se− 1

2

)

c2δt − ζ. (2.9b)

Therefore, the introduction of m(eq,1) into the equilibrium moments modifies the re-lationships between viscosities and corresponding relaxation parameters from Eq. (2.5)to Eq. (2.9). Even for small physical viscosities, there is no need to set the relaxationparameters sν and se too close to 2, due to the presence of non-zero λ and ζ. In the actualimplementation, we find that the best stability can be achieved with 1.8 6 sν 6 1.9, whilethe value of se is found to have no obvious effect on the numerical stability.

The mesoscopic external force vector Ψ in Eq. (2.2) can be designed via an inversedesign analysis in the moment space, as done in Min et al. (2016). The relevant elementsof Ψ in terms of the N-S equations are summarized as

Ψ1 = 38 (1− 0.5se) (uFx + vFy + wFz) ,Ψ3 = (1− 0.5sj)Fx, Ψ5 = (1− 0.5sj)Fy, Ψ7 = (1− 0.5sj)Fz,Ψ9 = 2 (1− 0.5sν) (2uFx − vFy − wFz) , Ψ11 = 2 (1− 0.5sν) (vFy − wFz) ,Ψ13 = (1− 0.5sν) (vFx + uFy) , Ψ14 = (1− 0.5sν) (vFz + wFy) ,Ψ15 = (1− 0.5sν) (uFz + wFx) .

(2.10)

where u = (u, v, w) is the local fluid velocity, and F = (Fx, Fy, Fz) is the local macroscopicforce per unit volume. The other mesoscopic terms in Ψ are irrelevant and can be set tozero for simplicity.

It should be noted that all the relevant m(eq,1) components in Eq. (2.8) can be

Page 7 of 44


calculated mesoscopically as

ρ0 (∂xu+ ∂yv + ∂zw) =G1

(

ζ − 38c2δt3se

) (2.11a)

ρ0 (4∂xu− 2∂yv − 2∂zw) =G9

(

λ− c2δt3sν

) (2.11b)

ρ0 (2∂yv − 2∂zw) =G11

(

λ− c2δt3sν

) (2.11c)

ρ0 (∂xv + ∂yu) =G13

(

λ− c2δt3sν

) (2.11d)

ρ0 (∂yw + ∂zv) =G14

(

λ− c2δt3sν

) (2.11e)

ρ0 (∂zu+ ∂xw) =G15

(

λ− c2δt3sν

) (2.11f )

where Gi = Mijfj −m(eq,0)i +Ψi/(2− si). The above relations ensure that all quantities

can be computed mesoscopically to maintain the second-order accuracy of the extendedLBM.

2.2. Parameter set-up

To resolve the smallest scale in the flow, it usually requires that inside the viscoussublayer ((R− r) /y∗ 6 8), there are at least three grid points in the radial direc-tion (Lammers et al. 2006), i.e., δr+ < 2.5 for uniform grid (hereafter we use thesuperscript + to denote quantities normalized by the wall unit y∗ or friction velocityu∗). With this constraint, we choose R = 148.5 in lattice units, which implies thatδr+ = 180/148.5 ≈ 1.212. This grid spacing is sufficiently small to resolve all scaleseven for the near wall region. On the other hand, the pipe length in the streamwisedirection must be long enough to minimize the effect the periodic boundary condition (seeKim et al. 1987; Chin et al. 2010). In the present work, we set L = 1799, which is12.11R. While the physical problem is stated in a cylindrical coordinate, i.e., r, θ,z coordinates, the LBM simulation is set up in the Cartesian coordinates, i.e., x, yand z. Based on the two aforementioned aspects, the grid resolution is chosen to beNx × Ny × Nz = 300 × 300 × 1799. The whole computation domain is decomposed inthe x and z directions as 90× 15 subdomains using 2D domain decomposition as shownin Fig. 1(b), as we did in our previous studies (Wang et al. 2014, 2016b). The datacommunication between neighboring domains is handled with Message Passing Interface(MPI).The pipe center is located in the x−y plane at rc = (150.5, 150.5), which is slightly off

the domain center to suppress the secondary flow patterns observed in White & Chong(2011), Kang & Hassan (2013) and Suga et al. (2015). The shifting of the pipe centerrelative to lattice nodes breaks the symmetry of the boundary link configuration, thusreducing the spurious secondary flow due to the D3Q19 lattice symmetry. Though it hasbeen suggested that the D3Q27 lattice has a better isotropy leading to a much weakerunphysical secondary flow in a circular pipe, our LBM simulation is still based on theD3Q19 lattice because its capability in simulating turbulent flows has been previously

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Table 1: Physical parameters used for the simulation of turbulent pipe flow

Nx ×Ny ×Nz ν νV R u∗ Reτ δx/y∗

300× 300× 1799 0.0032 1.0 148.5 0.00388 180 1.212

Table 2: LBM model parameters used in the simulations

α β γ se sε sj sq sν sπ sm ζ λ

0 -475/63 0 1.0 1.5 1.0 1.8 1.8 1.5 1.5 -0.889 0.0153

confirmed (see ten Cate et al. 2004; Lammers et al. 2006; Gao et al. 2013; Wang et al.

2014, 2016b). One novel aspect of this paper is to precisely demonstrate that the D3Q19lattice is adequate for DNS of turbulent pipe flow. The kinematic viscosity ν is chosento be 0.0032, which is above the limiting value 0.00254 (d’Humieres et al. 2002). Thebulk viscosity νV is set as 1.0 (both viscosities are in lattice unit) in order to dissipatethe acoustic waves in the flow (Luo et al. 2011). The use of a large bulk viscosity maycontaminate the pressure field, which must be corrected, as explained in Section 4.3. Thekey parameters used in the present study are summarized in Table 1 and Table 2.

2.3. Flow initialization and prescribed excitation at the early stage

The flow is initialized with the following mean velocity profile containing the viscoussublayer profile and inertial sublayer (or logarithmic) profile

U+(

δ+)

=

{

δ+, if δ+ 6 10.8,10.4 ln (δ

+) + 5.0, if δ+ > 10.8.(2.12)

where δ = R− r is the distance from the pipe wall. It should be noted that, although ina fully developed turbulent pipe flow, the mean velocity profile does not strictly conformwith Eq. (2.12) in the logarithmic region (Eggels et al. 1994; Durst et al. 1995; Loulouet al. 1997), the developed flow mean velocity profile should not be affected by this initialcondition. While this initial velocity field is not a solution of the N-S equations, it servesthe purpose of speeding up the flow transition to realistic, fully developed turbulent pipeflow when compared to starting the flow from rest. The initial distribution functions aresimply given as the leading-order equilibrium distributions, i.e., f = M−1m(eq,0), basedon the initial velocity field and zero initial pressure.

The flow is forced by the body force (Fr, Fθ, Fz) = (0, 0, ρ0g). To accelerate thetransition from the initial laminar flow to turbulence, we add a non-uniform, divergence-free force field to the flow only during the first three large-eddy turnover times of thesimulation, namely,

F ′

r = −gκB0R

r

kzl

Lsin

(

2πt

T

){

1− cos

[

2π (R− r − l0)

l

]}

cos

(

kz2πz

L

)

cos (kθθ)

(2.13a)

F ′

θ = g (1− κ)B0kzkθ

2πR

Lsin

(

2πt

T

)

sin

[

2π (R− r − l0)

l

]

cos

(

kz2πz

L

)

sin (kθθ)

(2.13b)

F ′

z = −gB0R

rsin

(

2πt

T

)

sin

[

2π (R− r − l0)

l

]

sin

(

kz2πz

L

)

cos (kθθ)

(2.13c)

Page 9 of 44


where kz and kθ are the wavenumbers of the perturbation force in streamwise andazimuthal directions, respectively. T is the forcing period, B0 is the forcing magnitude,κ is the weighting parameter that distributes the perturbation in radial and azimuthaldirections. The above forcing is added only to the region l0 6 R − r 6 l0 + l in radialdirection. After the initial period of three eddy turnover times, the perturbation forcingis no longer applied, namely, (F ′

r, F′

θ, F′

z) = (0, 0, 0) for t > 3R/u∗.In this paper, since we focus only on the fully developed stage of the turbulent pipe flow,

how to drive the flow to its stationary turbulence can be flexible. To save computationalresources, the simulation was started in a smaller domain with N0x × N0y × N0z =300× 300× 599, for about 30 eddy turnover times, to generate a preliminary turbulentflow field. For the the starting smaller domain, the parameters in Eq. (2.13) are chosenas: kz = 3, kθ = 2, T = 2000, l0 = 0.2R, l = 0.4R, B0 = 50.0, κ = 0.5. At this stage, theflow is already fully developed, but for a 1/3 pipe length. We then copied this flow twiceto form a new starting flow field. Since the streamwise domain size 1799 is not an exactmultiple of 599, the missing information for two gap layers (with grid indices nz = 600and nz = 1200) between copies is filled by a simple linear extrapolation scheme, as

fi (nx, ny, 600) = 2fi (nx, ny, 599)− fi (nx, ny, 598) , (2.14a)

fi (nx, ny, 1200) = 2fi (nx, ny, 1199)− fi (nx, ny, 1198) . (2.14b)

Once the copy and extension are done, the LBM simulation was run at least another 10eddy turnover times for the flow to reach the stationary stage in the full domain.

2.4. The use of a moving frame and improved implementation at the pipe wall

The interpolated bounce back rule is applied on the pipe wall to maintain the overallsecond-order accuracy in the simulation. While both linear and quadratic interpolatedschemes possess at least the second-order spatial accuracy, they have significant differ-ence in terms of numerical stability of the present simulation. Although the quadraticinterpolation schemes have generally better accuracy than the corresponding linearinterpolation schemes (see Peng et al. 2016c), they are not numerically stable in thepresent pipe-flow simulation. The reason could be related to the appearance of negativecoefficient in front of the distribution functions on the furthest node point in the quadraticinterpolation (Bouzidi et al. 2001; Yu et al. 2003). Furthermore, consider the very thinboundary layer and the large velocity gradient in the near-wall region, linear interpolationschemes are more preferable than the quadratic schemes due to their better localization.For these reasons, in the present simulation, the linear interpolated bounce-back schemesby Bouzidi et al. (2001) and Yu et al. (2003) are applied, and both of them are found tobe numerically stable.

The checkerboard instability was another major problem encountered in the presentsimulation. Unlike the instability results from the quadratic interpolation that originatesfrom the pipe wall, the checkerboard instability appears first in the pipe center regionand quickly propagates to the whole flow field, as shown in Fig. 2. The checkerboardinstability is mainly due to the insufficient discretization of the mesoscopic velocity fieldthat causing the distribution functions at one node point to isolate from its surroundingneighbors (Kandhai et al. 1999). Normally, such checkerboard instability can be elimi-nated by applying odd numbers of grid points in all location, as indicated in Kandhaiet al. (1999) and in Wang et al. (2016b). However, this simple solution does not applyto the current pipe flow simulation. We tested both 599 and 600 lattice points in thestreamwise direction, but the same checkerboard pattern was observed. In our simulation,the checkerboard instability is found to be strongly related to the Mach number (Ma)

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scheme using the idea of coordinate transformation. The essential idea is to alwaysperform the bounce back operation in the frame moving with the wall. The procedurefor this new bounce-back scheme with the moving pipe wall is as follows:

(i) Begin with the distribution functions in the fixed coordinate system (i.e., thecoordinate system attached to the lattice grid) and construct all distribution functionsnecessary to calculate the bounce-back populations at the boundary fluid node xb, in thecoordinate system moving with the wall (the moving coordinate system), as

f ′

i = f(eq)i (u− uw, δρ) + fi − f

(eq)i (u, δρ) (2.18)

where f with the superscript ′ indicates the distribution functions in the moving coordi-nate system, and those without ′ indicate the distribution functions in the fixed coordinatesystem.(ii) Perform the bounce back in the moving coordinate system for all velocity links

crossing the solid boundary. For interpolated bounce back, the distributions necessaryfor the interpolation must also be transformed into the moving frame as in Step 1. Forexample, applying the double linear interpolation scheme of Yu et al. (2003) at t+ δt, wehave

f ′

i (xb) =q

1 + qf ′

i (xw) +1− q

1 + qf ′

i (xb) +q

1 + qf ′

i (xfi) (2.19)

where q is the relative distance from xb to the pipe wall on a boundary link. f ′

i (xb) andf ′

i(xf ) should also be constructed in Step 1.(iii) Next, update the density and velocity at the boundary node xb in the moving

coordinate system

δρ′ (xb) =∑

i

f ′

i (xb) , ρ0u′ (xb) =

∑

i

f ′

i (xb) ei. (2.20)

Before this step, all distributions at xb must have been converted into the moving

coordinate system following Step 1 and have been properly bounced back if necessary.(iv) Finally, transform all distribution functions at xb back to the fixed coordinate

system, as

fi (xb) = f(eq)i (u′ + uw, δρ

′) + f ′

i (xb)− f(eq)i (u′, δρ′) . (2.21)

Here the notation fi denotes the population after streaming but before the bounce-back,while fi denotes the population after the bounce-back and the transformation back tothe fixed frame.The force evaluation on the pipe wall in the present scheme becomes

F (xb, t) δt =∑

B links

[

fi (t+ δt,xw) + fi (t+ δt,xb)]

ei

+∑

others

[

fi (t+ δt,xb)− fi (t+ δt,xb)]

ei

(2.22)

where the first summation is over the boundary links while the second summation isover the other directions at the boundary nodes. The second part is necessary since allthe distribution functions at xb are updated in Step 4. This part of momentum changeis considered in order to obey the Newton’s Third Law: namely, the forcing acting onthe pipe wall is balanced with the momentum change of the fluid due to fluid-solidinteractions.For further details, the readers are referred to Peng et al. (2016a) where several

validation cases are presented for this new implementation based on the coordinate

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u−uw

uc

r/R

u−uw

uc

r/R

(a) (b)

Figure 4: Velocity profiles at six different times in a transient laminar pipe flow simulation:(a) uw = 0, (b) uw = −uc.

transformation. The above transformation method can be incorporated with any in-terpolated bounce-back scheme. In the present simulation, this new bounce back schemeis implemented with the linear interpolation scheme of Yu et al. (2003).

3. Validation against the transient laminar pipe flow and accuracy

analysis

Before discussing the LBM simulation of the turbulent pipe flow simulation, we firstvalidate our extended MRT LBM and the new bounce back scheme against a transientlaminar pipe flow. In this case, we fix the Reynolds number Re = ucD/ν at 100, whereuc is the flow speed at the centerline when flow reaches the steady state. Three pipediameters in lattice units, D = 45, D = 90 and D = 180 are tested in order to reveal theorder of accuracy of our implementation. For each pipe diameter, we fix the shear andbulk viscosity at 0.025 and 1.0 in lattice units, respectively. The parameters that relatedto the MRT LBM are chosen to be identical to what listed in Table 2 except that λ isadjusted to −0.00648. Two pipe wall velocities uw = 0 and uw = −uc are examined toassess the performance of our implementation with both static and moving boundaries.The flow starts from rest and is driven by a uniform body force given as 16νuc/D

2. WithD = 90, the velocity profiles at different non-dimensional times (normalized by D2/(4ν))for both static and moving wall cases are compared with the theoretical solutions inFig. 4. The LBM results of the flow velocity on the Cartesian nodes are binned with 50equally-spaced bins based on the position relative to the pipe center. As clearly indicatedin Fig. 4, our implementation accurately predicts the flow velocity for both static andmoving wall cases. The L2 norms of the numerical errors for different pipe diametersat several dimensionless times are calculated and presented in Table 3. The L2 norm isdefined as

εL2 =

√∑

x| un (x)− ut (x) |2

√∑

x| uc (x) |2

, (3.1)

where un and ut are the streamwise velocities, respectively, from LBM and the analyticalsolution. The summation in the above equations are over all node points within the piperadius. For both static and moving wall cases, Table 3 shows that the overall accuracy of

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Table 3: The L2 norm and convergence rate of the numerical error of the presentimplementation in a laminar pipe flow simulation (top: uw = 0, bottom: uw = −uc).

D/δx εL2(t∗ = 1/3) order εL2(t

∗ = 2/3) order εL2(t∗ = ∞) order

45 6.270E-4 (-) 7.434E-4 (-) 7.841E-4 (-)90 6.058E-5 3.372 7.780E-5 3.256 8.488E-5 3.207180 2.313E-5 1.389 2.723E-5 1.515 2.865E-5 1.567

overall 2.381 2.386 2.387

D/δx εL2(t∗ = 1/3) order εL2(t

∗ = 2/3) order εL2(t∗ = ∞) order

45 6.227E-4 (-) 7.547E-4 (-) 7.837E-4 (-)90 8.474E-5 2.877 1.103E-4 2.775 1.157E-4 2.760180 1.855E-5 2.192 2.302E-5 2.261 2.405E-5 2.266

overall 2.534 2.518 2.513

the present implementation is of second order. The results based on the L1 norm yieldthe same conclusion.

4. Turbulent pipe flow

In this section, we compare systematically the results from our LBM simulation ofturbulent pipe flow to previous results in the literature. Only the fully-developed stage isconsidered in this paper. The results from the pseudo-spectral method by Loulou et al.

(1997) will be used as the benchmark due to the superior accuracy of the PS method.

4.1. Data post-processing details

All results are presented in cylindrical coordinates. The LBM results on the Cartesiangrid are first transformed into cylindrical coordinate through the following binning andprojection:

(i) Divide the pipe cross section into 100 equal spaced bins according to their radiallocations and number the bins as bin 1, 2, ..., 100 from pipe center to the pipe wall.Since the pipe radius is 148.5, the bin width is 1.485 lattice units. Each lattice node isassociated with a square lattice cell of width equal to one lattice unit and the node atthe center. Clearly, a given lattice cell can at most overlaps partially with two bins sincethe cell diagonal length

√2 = 1.414 is less than 1.485, as illustrated in Fig. 5

(ii) For each square lattice cell (3D cubic lattice cell projected in 2D), identify thecorner points a0 and a1 corresponding to the shortest and longest distances (d0 and d1)from the pipe centerline, respectively.(iii) Find out in which bin/bins the points a0 and a1 are located, then one of the

following cases must apply:(a) Both a0 and a1 are located in the same bin i (Fig. 5, Case 1), then add all thecontribution (i.e., p(i) = 1) from this particular cell into bin i.(b) a0 is located in bin i and a1 is located in bin i + 1 (Fig. 5, Case 2), then thecontribution from this lattice cell is partitioned into Bin i and Bin i+1, accordingto the following relative percentages p(i) and p(i+ 1), respectively,

p(i) =li,i+1 − d0d1 − d0

, p(i+ 1) =d1 − li,i+1

d1 − d0, (4.1)

where li,i+1 is the radial location of the boundary between bin i and bin i+1. Whilethe above is not based on the precise overlap area of the cell and a bin, but test

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Table 4: Physical and simulation parameters in the turbulent pipe flow

Method Reτ Nr ×Nθ ×Nz Lz/D Resolution

Eggels et al. (1994) LDA 185.5 − − −

Durst et al. (1995) LDA 250.0 − − −

Westerweel et al. (1996) PIV 183 − − −

Eggels et al. (1994) 2nd order FV,uniform grid

180 96× 128× 256 5 (∆r+, ∆z+, R∆θ+) ≈(1.88, 7.03, 8.83)

Loulou et al. (1997) b-spline PS 190 72× 160× 192 5 (∆r+, ∆z+, R∆θ+) ≈(0.39− 5.7, 9.9, 7.5)

Wagner et al. (2001) 2nd order FV,non-uniformgrid

180 70× 240× 486 5 (∆r+, ∆z+, R∆θ+) ≈(0.36−4.32, 3.7, 4.7)

Present LBM 180 300×300×1799 6.06∆x+ = ∆y+ =∆z+ = 1.212

Table 5: The domain-averaged flow speed and fitting coefficients of mean velocity profiles

to the logarithmic form u+z = (1/κ) ln δ+ + B0 in the range 30 < δ+ < 150. The values

in the parentheses are the relative errors compared to the corresponding result of the PSsimulation (Loulou et al. 1997).

Data set U+

b κ B0

PS, Loulou et al. (1997) 14.77 0.344 4.477LDA, Eggels et al. (1994) 14.68(0.61%) 0.359(4.23%) 5.027(12.29%)FV, Eggels et al. (1994) 14.73(0.27%) 0.339(1.51%) 4.331(3.26%)FV, Wagner et al. (2001) 14.72(0.34%) 0.340(1.03%) 4.435(0.94%)

LBM, Original 14.12(4.40%) 0.352(2.23%) 4.136(7.62%)LBM, Modified 14.63(0.95%) 0.345(0.41%) 4.473(0.09%)

Log law Kim et al. (1987) 15.63 0.4 5.5

the pipe wall. A well-known fact of the mean velocity profiles at the low Reynolds numbersin a turbulent pipe is the deviation from the classical logarithmic law. Our simulationresults also confirm such phenomenon. The mean bulk flow speed averaged over the wholedomain U+

b for each case is listed in Table 5. It is clear that after addressing the VGIproblem, the relative error, using the PS result of Loulou et al. as the benchmark, reducesfrom 4.4% to less than 1%. Furthermore, in Table 5 we provide the modified logarithmicfit to each data set using the data points in the range 30 < δ+ < 150, along with the well-known logarithmic law in turbulent channel flow. It is obvious that all the simulationsof turbulent pipe flow show deviations from the standard fit for the turbulent channelflow. The two coefficients κ and B0 from our modified LBM are in better agreement withthe PS results, when checked against both FV results. It is noted that a plot digitizerwas used to retrieve data from previous studies, which could contain a minor error. Weassume the data retrieval errors are negligible.

For low-Re turbulent pipe flow, due to the insufficient separation of inner and outerscales, the mean velocity result can be normalized by either the inner scale (wall unit, as inFig. 6) or the outer scales. The mean velocity profiles normalized by the bulk velocity (therespective domain-averaged flow speed and pipe radius) are shown in Fig. 7. Interestingly,under this scaling, both LBM results match the other results well, indicating that thetwo LBM profiles have the same shape under the outer scaling. This result confirms

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u+z

δ+

Figure 6: The mean velocity profiles normalized by the wall scales.

uz/Ub

r/R

Figure 7: The mean velocity profiles normalized by the bulk flow velocity.

that the un-modified interpolation method overestimates the friction velocity, which isan outcome of the VGI error contaminating the wall shear stress.

4.3. Basic turbulence statistics

The Reynolds stress profiles are compared in Fig. 8. The straight line indicates thetotal stress −u′

z+u′

r+ + dU+

z /dr+, which can be obtained from the momentum balanceequation in the streamwise direction. As shown in Fig. 8, both LBM results are inexcellent agreement with the other DNS results, implying the VGI error does not havean obvious impact on the turbulent fluctuations. This makes sense because the turbulent

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−u′z+u′

r+

r/R

Figure 8: The Reynolds stress profiles as a function of radial location.

stress measures the level of velocity fluctuation which is independent from the frame ofreference. The two experimental results, however, exhibit significant differences from theDNS counterparts, especially in the near wall region, this could be due to the difficultyof conducting accurate measurements in the near-wall region and other measurementerrors. This is perhaps one obvious example where experimental data should not be usedas the benchmark.Root mean squared (r.m.s.) fluctuation velocity profiles in the streamwise, radial, and

azimuthal directions are shown in Fig. 9, Fig. 10, and Fig. 11, respectively. Again, theVGI error only introduces a minor effect on the r.m.s velocity profiles. Only in the bufferregion where the peaks in these profiles occur, and the pipe center region, small visibledifferences can be observed. The peaks in the streamwise, radial, and azimuthal velocityfluctuations are located at 15.3, 56.7, 37.8, respectively, in LBM, compared to 14.7,57.2, 36.3, respectively, in PS. The modified boundary treatment always leads to slightlysmaller velocity fluctuations compared to the original treatment, which yields a slightlybetter match with the PS results in the buffer region but slightly worse agreement near

the pipe center. At the pipe center, we observe u+z,rms = 0.862, u+

r,rms = 0.664 and

u+θ,rms = 0.664, yielding the relative radios of velocity fluctuations at 1.30 : 1 : 1. This

result indicates that at the pipe center, the turbulence is nearly isotropic, especially inthe cross-sectional planes normal to the pipe axis.It is worth noting that the FV profiles of Eggels et al. (1994) fail to match the other

results, especially for the radial and tangential components, with relative errors in therange of 10% to 15% when compared to the results of Loulou et al. (1997), indicatingclearly that Eggels’ FV data is inaccurate. This could be mainly due to their use ofa uniform grid in the radial direction causing the near wall region to be inadequatelyresolved (see Table 4). The grid resolution in our LBM simulations is δx+ = δy+ = δz+ =1.212, compared to δr+ = 1.88 in Eggels et al. (1994). It has been suggested that theminimum resolution should be δx+ = δy+ = δz+ = 2.25 for DNS of a turbulent channelflow (Lammers et al. 2006; Wang et al. 2016b), the curved boundary in the turbulentpipe could impose a more demanding grid-resolution requirement. Furthermore, since the

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u+z,rms

δ+

Figure 9: Profiles of the r.m.s. velocity fluctuations in the streamwise direction.

advection term in LBM is treated exactly (except at the pipe wall where the interpolatedbounce back is introduced), leading to a much smaller numerical dissipation in LBM whencompared to the second-order FV method used by Eggels et al. (1994).It is interesting to note that while our LBM simulation based on D3Q19 lattice

grid presents highly accurate mean and r.m.s velocity results compared to PS results,significant (up to 30%) deviations were reported in previous LES study (Kang & Hassan2013). Such deviations were attributed to the lack of isotropy of D3Q19 in representingcircular pipe boundary, since the results from the simulation based on D3Q27 latticethat has better isotropy showed good agreement with the FV results by Eggels et al.

(1994). Based on our comparisons above, we conclude that the FV simulation on uniformgrid by Eggels et al. (1994) should not be used as the benchmark. Also, it is more likelythat the large deviations reported in the previous LES study (Kang & Hassan 2013)are partially due to the insufficient resolution (the highest resolution in that study wasδx+ = δy+ = δz+ = 3.6) for the near wall turbulence. Due to the lack of grid resolution,the general good results based on the D3Q27 lattice grid in Kang & Hassan (2013) arealso questionable.According to the radial momentum balance equation, the mean pressure can be related

to the r.m.s. velocities in the radial and azimuthal directions as

1

ρP (r) + u2

r (r) +

∫ R

r

u2θ(r)− u2

r(r)

rdr =

1

ρP (R) = const. (4.3)

This balance is well captured by our LBM results for most of the region as shown inFig. 12, where 〈P 〉, Φ and ζ correspond to the first, second and third terms on the LHS ofEq. (4.3), respectively. It is worth emphasizing that LBM solves the weakly compressibleN-S equations with a divergence term that on the order of O(Ma2). Compared withthe fully incompressible N-S equations, the incompressible-equivalent definition of thepressure from LBM should be

P = P + ρ

[

2

3ν (∇ · ~u)− νV (∇ · ~u)

]

(4.4)

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u+r,rms

δ+

Figure 10: Profiles of the r.m.s. velocity fluctuations in the radial direction.

u+

θ,rms

δ+

Figure 11: Profiles of the r.m.s. velocity fluctuations in the azimuthal direction.

where P = c2sδρ, and P is the corrected pressure that resembles the incompressible flow.In our simulation, due to the use of the large bulk viscosity to enhance numerical stability,the error caused by the third term on RHS of Eq. (4.4) is significant and must be removedusing the above pressure correction in order to satisfy Eq. (4.3). Very close to the pipewall, the LBM pressure profile exhibits small fluctuations and slight deviation of the sumfrom the constant in the region r/R > 0.92. This results from two factors: the pressurenoise associated with the moving boundary in LBM (as demonstrated later), and thenumerical error when calculating ζ. In the post-processing, we computed this integral by

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P+

r/R

Figure 12: The mean pressure profile and the balance againgst the transverse (radial andazimuthal) rms velocity fluctuations.

trapezoidal rule, which may not be accurate since the r.m.s velocity in both radial andtangential region are changing rapidly near the wall.The r.m.s. pressure profiles are presented in Fig. 13. For the LBM result, the sudden

jump near wall is mainly due to noises associated with the acoustic waves at the movingcurved boundary, as indicated by an instantaneous snapshot of the pressure contour at arandom cross section (here z = (Nz−1)/2) in Fig. 14. Other than that, the r.m.s pressureprofile of LBM is in better agreement with the PS profile than the FV results of Eggelset al. (1994). Our result is also slightly better than that of Wagner et al. (2001) near thepeak of r.m.s. pressure. Specifically, the FV results of Eggels et al. (1994) contain roughly10% relative error when compared with the PS result of Loulou et al. (1997). Althoughit is well known that LBM only has a first-order accuracy for the pressure calculation,the results here show that LBM can provide better results on the simulated velocity andshear stress over the second-order FV method. The latter method solves the pressurePoisson equation to update the pressure field.Finally, the r.m.s. vorticity profiles in the streamwise, radial, and azimuthal directions

are exhibited in Fig. 15. In LBM, the vorticity is calculated in the Cartesian coordinatesbased on a second-order central finite-difference approximation and then projected tothe cylindrical coordinates. For the boundary nodes, special treatment is necessary tomaintain the same accuracy of vorticity calculation in the whole field when straightforward central differencing does not apply (see the Appendix). As indicated in Fig. 15,in general the r.m.s. vorticity profiles from LBM are in better agreements with the PSbenchmark results than their FV counterparts. In terms of the high-order statistics,LBM in the Cartesian coordinates could have better performance than the FV with auniform grid in cylindrical coordinates. The local minimum and maximum locations inthe streamwise r.m.s. vorticity profile are at δ+ ≈ 5 and δ+ ≈ 22, respectively. In theprevious channel flow simulations, similar local minimum and maximum were reportedat about the same locations of y+ = 5 and y+ ≈ 20 (Kim et al. 1987; Moser & Moin1984). In these studies, such local minimum and maximum were related to the streamwise

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ω+z,rms

δ+

ω+r,rms

δ+

(a) (b)

ω+

θ,rms

δ+

(c)

Figure 15: Profiles of the r.m.s. vorticity fluctuations: (a) streamwise vorticity, (b) radialvorticity, (c) tangential vorticity.

vortices in the near wall region. The local maximum occurs around the average locationof centers of near wall streamwise vortices; while the local minimum corresponds to thelocation where the average vorticity is zero, resulting from the mirrored vortex pairs dueto the no-slip condition on the pipe wall (Moser & Moin 1984; Kim et al. 1987; Loulouet al. 1997). The peak location in the radial r.m.s. vorticity profile is at δ+ ≈ 17, whichis very close to the peak location of the streamwise r.m.s. velocity (δ+ ≈ 16, Fig. 9)corresponding to where most intensive ejection and sweeping events take place. Ejectionand sweeping create radial vorticity by tilting the streamwise vortices, which explains whythe location of r.m.s. radial vorticity is aligned with the peak location of the streamwiser.m.s. velocity. In Fig. 15 we also present the r.m.s. vorticity profiles from the classicalPS DNS of turbulent channel flow (Kim et al. 1987). The very similar profiles imply thesimilar vortical structures in the two flows. It is interesting to obverse that the numericalvalues of r.m.s. vorticity in three directions are similar in magnitude in the region ofδ+ > 60. The relative radios at δ+ = 60 for the streamwise, radial and tangential r.m.s.vorticities are 1.016 : 1.012 : 1. At the pipe center, these ratios become 0.863 : 1 : 1.This result implies that the vortex structure is nearly isotropic in the center region ofthe pipe.

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4.4. High-order statistics in physical space

The skewness and flatness profiles of the velocity fluctuation in each direction are alsocalculated. The skewness S (u′

α) and flatness F (u′

α) are defined as

S (u′

α) =u′

α3

u′

α23/2

, F (u′

α) =u′

α4

u′

α22 . (4.5)

As shown in Fig. 16, the LBM skewness profiles match well with the PS and LDA data,especially for the near wall region. It is particularly interesting that in the skewness ofradial velocity, the PS result demonstrates the existence of a negatively skewed thin layervery close to the wall, while the PIV measurement and the FV data of Eggels et al. do notexhibit such behavior. A similar region is also reported in the DNS of turbulent channelflow (Kim et al. 1987). A negative skewness indicates larger negative ur events happenmore frequently than large positive ur events, noting that the radial velocity towards thewall is positive. Therefore, very close to the wall, the ejection events are stronger than thesweeping events, but the sweeping events are more frequent (Kim et al. 1987). The LBMresult shares the similar qualitative trend of reduced skewness as the PS and LDA results,near the wall; but the LBM results have an unphysical jump right at the pipe wall. Suchunphysical jump is likely associated with the acoustic waves generated due the movingpipe wall. Removing this contamination by acoustic waves, the negative skewness regionvery close to the wall could have been captured by our LBM simulation. The reason thatthe FV result of Eggels et al. fails to predict the negative skewness near the wall couldagain be due to the inadequate resolution near the wall. In the azimuthal direction, theskewness should be identically zero since the flow has no preference in the azimuthaldirection. In that sense, our LBM results could be even more trustworthy than the PSresult as the PS skewness in the azimuthal direction oscillates more significantly fromzero. This could be a result of insufficient time duration to average out the deviations.According to Loulou et al. (1997), the PS statistics are averaged over 46 fields at differenttimes spanning 5.82 large eddy turnover times (defined as R/uτ ). In our simulation, amuch longer period (with 2300 snapshots covering 60.1 large eddy turnover times) wasused. Due to the stronger fluctuations of high order statistics, it usually requires a longeraveraging period to reduce the uncertainty. The LBM flatness profiles are also in goodagreements with the PS benchmark data, except for a thin layer very close to the wall.Both the flatness of streamwise and radial velocity in this region exhibit a small jump,again likely a result of contamination by the acoustic waves near the moving pipe wall.

4.5. Instantaneous flow visualization

Next, we investigate the instantaneous flow structures in the turbulent pipe flow. Threerepresentative circular shells in the viscous sublayer (δ+ = 3.5 ± 1), the buffer region(δ+ = 22.0± 1) and the logarithmic region (δ+ = 122.0± 1) are chosen for visualization.The velocity and vorticity at the lattice nodes located inside each shell are projectedinto the cylindrical coordinate as described before. In Figs. 18 to 20 we present theinstantaneous snapshots of the contours for each velocity component.Several observations can be made from these component-wise velocity contours. First,

the streamwise velocity near the pipe wall are well structured with alternative streaks ofhigh and low speeds, which are discovered as typical flow structures for the wall-boundedturbulence. Such strongly anisotropic streak structures stretch to the buffer layer andbecome more isolated, isotropic eddies in the logarithmic region. Second, by definition theradial velocity near the wall is weak, no strong ejection and sweeping events are observed.In the buffer region, strong ejection and sweeping events happen alternatively. These

Page 25 of 44


S (u′

z)

r/R

S (u′

r)

r/R

(a) (b)

S (u′

θ)

r/R

(c)

Figure 16: The skewness profiles of velocity components: (a) streamwise velocity, (b)radial velocity, and (c) azimuthal velocity.

strong ejection and sweeping events also occur in the logarithmic region, but appearingto be less structured. Third, the structured radial and azimuthal velocity distributionnear the wall indicates that this region is dominated by the highly-structured anisotropicstreaks and quasi-streamwise vortices (Robinson 1991). Approaching to the center of thepipe, the fluctuation velocities become more homogeneous and isotropic.Similarly, the instantaneous snapshots of component vorticity contours are shown in

Fig. 21 to Fig. 23. These figures confirm that the existence of structured long vorticesnear the pipe wall. Those long vortices break into shorter vortices in the buffer region andeventually turn into homogeneous distribution of isolated and isotropic vortices in thelogarithmic region. The azimuthal vorticity near the wall (Fig. 23) shows the strongeststreaky structures resembling the streamwise velocity structure. These show that thevelocity fluctuations and vorticity fluctuations do not have obvious scale separation nearthe wall. Put it another way, velocity and vorticity fluctuations both contain large (mainlyin the streamwise direction) and small (in azimuthal and radial direction) scales. In thecenter region of the pipe, the vorticity structures have a smaller length scales comparedto the velocity fluctuations, showing that a scale separation begins to emerge as in theisotropic homogeneous turbulence. However, at the same time, the circumferential length

Page 26 of 44


F (u′

z)

r/R

F (u′

r)

r/R

(a) (b)

F (u′

θ)

r/R

(c)

Figure 17: The flatness profiles of velocity components: (a) streamwise velocity, (b) radialvelocity, and (c) azimuthal velocity.

δ+ =3.5

θ

L/L

x

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =22.0

θπ/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =122.0

θ

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

2

4

6

8

10

12

14

16

18

20

Figure 18: Contours of the instantaneous streamwise velocity at three radial locations.

Page 27 of 44


δ+ =3.5

θ

L/L

x

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =22.0

θπ/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =122.0

θ

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 19: Contours of the instantaneous radial velocity at three radial locations.

δ+ =3.5

θ

L/L

x

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =22.0

θπ/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =122.0

θ

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−3

−2

−1

0

1

2

3

Figure 20: Contours of the instantaneous azimuthal velocity at three radial locations.

shrinks as the pipe center is approached, this scale separation is very limited in this flowgeometry. This perhaps is a major difference between turbulent pipe flow and channelflow.

4.6. Statistics in spectral space and analyses of flow length scales

To better quantify the flow scales at those representative radial locations, we calculatethe one-dimensional auto-correlations functions and energy spectra for all velocity andvorticity components. The auto-correlation functions are calculated in physical space, as

Page 28 of 44


δ+ =3.5

θ

L/L

x

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =22.0

θπ/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =122.0

θ

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 21: Contours of the instantaneous streamwise vorticity at three radial locations.

δ+ =3.5

θ

L/L

x

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =22.0

θπ/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =122.0

θ

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Figure 22: Contours of the instantaneous radial vorticity at three radial locations.

Qαα (δz, r) =u′

α (r, z, θ)u′

α (r, z + δz, θ)

u2α,rms(r)

, (4.6a)

Qαα (δθ, r) =u′

α (r, z, θ)u′

α (r, z, θ + δθ)

u2α,rms(r)

, (4.6b)

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δ+ =3.5

θ

L/L

x

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =22.0

θπ/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

δ+ =122.0

θ

π/2 π 3π/2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

0.2

0.4

0.6

0.8

1

1.2

Figure 23: Contours of the instantaneous azimuthal vorticity at three radial locations.

where (...) denotes averaging over the azimuthal (or streamwise) direction and over time.The streamwise and azimuthal energy spectra are given as

Eα (kz; r) =1

2

∑

kθ

[uα (r, kθ, kz) u∗

α (r, kθ, kz) + uα (r, kθ,−kz) u∗

α (r, kθ,−kz)] (4.7a)

Eα (kθ; r) =1

2

∑

kz

[uα (r, kθ, kz) u∗

α (r, kθ, kz) + uα (r,−kθ, kz) u∗

α (r,−kθ, kz)] (4.7b)

where uα (r, kθ, kz) is the velocity or vorticity component in the Fourier space, which isdefined as

uα (r, kθ, kz) =1

Nz

1

Nθ

∑

j

∑

k

uα (zk, θj , r) exp

(

−i2πkz · zk

Lz− ikθ · θj

)

(4.8)

and u∗

α (r, kθ, kz) is its complex conjugate.As shown in Fig. 25, the LBM results of two-point auto-correlation functions of

azimuthal and radial velocity are in good agreement with corresponding PS results, forthe separation of δz < D. For large streamwise separations, i.e., δz > D, the PS auto-correlation functions for three velocity components are slightly negative, even at thelargest separation distance δz = 2.5D. Such behavior makes the PS results questionablesince at the largest separation, the velocity components should be uncorrelated if the pipelength is sufficient to eliminate the effect of periodic boundary condition in the streamwisedirection. The same issue also exists in the FV correlation function of streamwise velocitycomponent. On the other hand, the correlation functions from LBM all decay to zero atδz = 3D. It is interesting to note that even at δz = 2.5D, the LBM correlation functionsare almost zero for all velocity components at all locations. This may imply that thereason for the PS and FV correlation functions to deviate from zero at large separationsis perhaps the insufficient number of grid points in the streamwise direction (192 for bothPS and FV compared to the much larger resolution of 1799 in LBM), rather than thelimited pipe length. For the streamwise velocity, the PS correlation functions drop faster

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−ε/(u∗4/ν)

δ+

Figure 24: Averaged dissipation rate profile.

than those of LBM and FV, indicating the streamwise elongated structures in the formerare shorter than in the latter.To confirm that, the averaged dissipation rate profile is presented in Fig. 24. The

minor oscillation for δ+ < 3 in the LBM result is mainly due to the pressure noiseand binning error near the wall. The LBM result is in very good agreement with thePS result of Loulou et al. (1997), and the FV result of Wagner et al. (2001) basedon a non-uniform grid. Since the flow near the pipe center is nearly isotropic, we couldapply the criterion for homogeneous isotropic turbulence (HIT) to estimate the resolutionrequirement. In HIT, the resolution requirement for the spectral simulation is kmaxη > 1,where kmax is the maximum wavenumber a spectral simulation can resolve and η is thelocal Kolmogorov length. Also, the previous studies have revealed that for the secondorder FV or LBM simulations (Peng et al. 2010; Wang et al. 2016b), the grid resolutionmust be at least doubled in order to achieve the same accuracy as the PS simulation.For the spectral simulation in Loulou et al. (1997), assuming no dealiasing truncation isapplied, the largest wavenumber can be resolved is half of the grid numbers, i.e., πnz/Lz.The dimensionless resolution parameter kmaxη can then be expressed as

kmaxη =1

∆z+

(

1

ε+

)1/4

, (4.9)

where ε+ = ε/(u∗4/ν). Using the dissipation rate data at the pipe center in Fig. 24 andthe grid resolution information listed in Table 4, we obtain kmaxη ≈ 0.32 for the PSsimulation of Loulou et al. (1997), approximately 0.45 and 0.85 for the FV simulationsof Eggels et al. (1994) and Wagner et al. (2001), respectively, and about 2.61 for ourLBM simulation. Based on these results, it appears that all the previous DNS studiesdid not have adequate grid resolution in the streamwise direction, while only our LBMsimulation has satisfied the resolution requirement.For the vorticity components, the spatial correlations are shown in Fig. 26. Although

all results are in good qualitative agreements, significant quantitative differences can beobserved between the PS correlation functions and their LBM counterparts, especially in

Page 31 of 44


Table 6: The integral length scales L0 of velocity and vorticity correlation functions atdifferent radial locations, normalized by D (or ν/u∗). The top table is for the streamwisecorrelations and the bottom refers to the azimuthal correlations.

δ+ ur uθ uz ωr ωθ ωz

3.5 0.192(69.22) 0.214(76.90) 0.605(217.81) 0.372(133.88) 0.585(210.60) 0.267(96.25)22.0 0.237(85.16) 0.160(57.52) 0.544(195.90) 0.265(95.24) 0.140(50.43) 0.124(44.78)122.0 0.075(27.02) 0.055(19.82) 0.300(107.87) 0.082(29.51) 0.077(27.86) 0.071(25.38)

δ+ ur uθ uz ωr ωθ ωz

3.5 0.011(3.79) 0.015(5.45) 0.029(10.58) < 10−5(0.0011) 0.025(8.88) 0.028(10.21)22.0 0.012(4.46) 0.058(20.85) 0.019(6.87) 0.0015(0.533) 0.042(15.17) 0.035(12.61)122.0 0.038(13.62) 0.075(26.99) 0.059(21.24) 0.016(5.79) 0.063(22.83) 0.033(11.69)

the near-wall region. Again, some PS correlation functions do not decay to zero, makingit questionable for their reliability. The results together show that the vortices are moreelongated near the wall than in the pipe center, particularly for the azimuthal component.This phenomenon can also be observed in the vorticity visualization in Fig. 21 to Fig. 23.Away from the wall, the vortex structures are more isotropic, as the correlation functionsof three vorticity components essentially overlap.The azimuthal correlation functions from LBM and PS are shown in Fig. 27 and

Fig. 28 for velocity and vorticity, respectively. In these cases, the FV data are notavailable, and the PS data are only available for the near-wall and buffer regions. Here,the PS correlations for velocity vorticity components do decay to zero as the azimuthalseparation is increased. In general, the LBM results and PS results are in good agreement.As indicated by the location of the minimum correlations in Fig. 27(a), the typical widthin the azimuthal direction of the near wall alternating streaks is about 0.3R ≈ 50ν/u∗, orabout 5% of the circumferential length. Near the wall, the locations of minimum vorticitycorrelation can be roughly used to quantify the typical vortex size in the azimuthaldirection for each vorticity component, which indicates the radial vorticity component hasthe smallest size, and the streamwise vorticity component has the largest size. Towardsthe pipe center, the vortex size for each component increases slightly, as shown in Fig. 28.The integral length scales of each velocity and vorticity component at different radial

locations are computed and listed in Table. 6. Several observations can be made. Theintegral length scale in the streamwise direction decreases from the near-wall region tothe pipe center, for all velocity and vorticity components. The streamwise velocity hasthe largest streamwise length scale, compared to the other two velocity components. Thevelocity field is highly anisotropic, judged by both the discrepancies in the streamwisecorrelation lengths for different velocity components and the differences in the two spatialdirections, regardless whether it is in the near-wall region or the pipe center region. On theother hand, the vorticity field is highly anisotropic in the near-wall region, and becomesnearly isotropic in the pipe center, as judged by the streawise integral length.In order to compare with the benchmark results, we rescaled the PS benchmark dataset

by multiplying a factor∑

kz or kθEα,LBM/

∑

kz or kθEα,PS such that the areas under

the pair are the same. In general, both streamwise and azimuthal energy spectra obtainedfrom the LBM simulations are in excellent agreement with their PS benchmark curvesfor all velocity and vorticity components at the three different locations. The deviationsin the streamwise spectra in the logarithmic region may be related to the smaller numberof data points near the pipe center, while the deviations in the azimuthal spectra at largewavenumbers are likely due to the insufficient number of bins in the azimuthal direction

Page 32 of 44


(a)

Qαα (δz, r)

(b)

δz/D

(c)

Figure 25: The streamwise two-point auto-correlation functions for velocity componentsat different locations, (a): near wall (δ+ = 3.5), (b): buffer region (δ+ = 22.0), (c):logarithmic region (δ+ = 122.0). FV results (Eggels et al. 1994) are taken at slightdifferent representative radial locations, as δ+ ≈ 4, δ+ ≈ 17 and δ+ ≈ 91 in (a), (b) and(c), respectively.

used in data conversion from the Cartesian to cylindrical coordinates. The energy spectrafor both velocity and vorticity components confirm the nearly isotropic flow in the pipecenter. It is particularly interesting to observe an unphysical peak in the azimuthalspectrum of the azimuthal velocity (Fig. 31(a)) at kθ = 4 in the near wall region. Thiscould be a result of the weak unphysical secondary flow pattern caused by the use ofD3Q19 lattice, as reported in (White & Chong 2011; Kang & Hassan 2013; Suga et al.

2015). The azimuthal spectra of radial and streamwise vorticity components (Fig. 32(a))also exhibit the similar weak unphysical behavior at kθ = 4, which corresponds to theunphysical pattern in the azimuthal spectrum of azimuthal velocity. However, comparedwith the significant secondary flow that reported in the previous LES study (Kang &

Page 33 of 44


(a)

Qαα (δz, r)

(b)

δz/D

(c)

Figure 26: The streamwise two-point auto-correlation functions for vorticity componentsat different locations, (a): near wall (δ+ = 3.5), (b): buffer region (δ+ = 22.0), (c):logarithmic region (δ+ = 122.0).

Hassan 2013), the secondary flow observed in the present simulation is much weaker.In Fig. 33 we show the cross-sectional contour of the averaged streamwise velocity overabout 44.6 large eddy turnover times. Although a weak secondary flow patterns can bestill observed, the contours generally have no obvious azimuthal dependence. The muchweaker secondary flow due to numerical artifact in our LBM is due to several factors:first, we shift the pipe center a little bit so the lattice arrangement is no longer perfectlysymmetric; second, compared with Kang & Hassan (2013) who used only 100 latticeunits for the pipe diameter, our resolution is three times better; finally, our improvedboundary implementation also likely reduces such artifact.

Page 34 of 44


(a)

Qαα (δθ, r)

(b)

δθr/R

(c)

Figure 27: The azimuthal two-point auto-correlation functions for velocity componentsat different locations, (a): near wall (δ+ = 3.5), (b): buffer region (δ+ = 22.0), (c):logarithmic region (δ+ = 122.0).

5. Summary and conclusions

In this work, we present a direct numerical simulation of a fully developed turbulentpipe flow at the bulk flow Reynolds number Reb ≈ 5200 (or Reτ = 180), using the latticeBoltzmann method on a D3Q19 lattice. A number of difficulties associated with accuracyand numerical instability of LBM involving a curved wall have been resolved to make thispossible for the first time. To handle this turbulent flow with reasonable computationalresources, an extended MRT LBM model has been introduced to improve the numericalstability of the simulation by relaxing the strict relationship between the viscosities andrelaxation parameters in standard LBM. To accelerate the transition from laminar toturbulence, a divergence free perturbation force field is introduced to excite the flow fieldat the early stages. The simulation set-up and other implementation details have alsobeen presented. Compared with the turbulent channel flow simulation, we discovered

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(a)

Qαα (δθ, r)

(b)

δθr/R

(c)

Figure 28: The azimuthal two-point auto-correlation functions for vorticity componentsat different locations, (a): near wall (δ+ = 3.5), (b): buffer region (δ+ = 22.0), (c):logarithmic region (δ+ = 122.0).

a more critical Ma constraint due to the presence of severe checkerboard instability.To overcome this problem, we added a negative velocity to the pipe wall to reduce thevelocity magnitude in the center region of the pipe. This novel treatment however, bringsanother issue related to the Galilean invariance on the pipe wall surface. This issue isresolved by an improved bounce back scheme based on coordinate transformation. Webelieve these implementation details in our work provide a useful guidance for others toconduct similar simulations in the future.

Next, the results from the LBM simulations are compared systematically with previousDNS and measurement data. We have demonstrated that the violation of Galileaninvariance error in the usual interpolated bounce back causes a significant error on themean velocity, although this seems to have a minor impact on the fluctuating velocitystatistics. The VGI error on the mean velocity can be removed by using the improved

Page 36 of 44


Eα (kz)

kz

(a) (b) (c)

Figure 29: The streamwise energy spectra for velocity components at different locations,(a): near wall (δ+ = 3.5), (b): buffer region (δ+ = 22.0), (c): logarithmic region (δ+ =122.0). FV results (Eggels et al. 1994) are taken at slight different representative radiallocations, as δ+ ≈ 4, δ+ ≈ 17 and δ+ ≈ 91 in (a), (b) and (c), respectively.

Eα (kz)

kz

(a) (b) (c)

Figure 30: The streamwise energy spectra for vorticity components at different locations,(a): near wall (δ+ = 3.5), (b): buffer region (δ+ = 22.0), (c): logarithmic region (δ+ =122.0).

bounce back based on coordinate transformation. The comparisons in the r.m.s. velocities,using the spectral simulation data (Loulou et al. 1997) as the benchmark, reveal thatthe LBM results are much more accurate than the second-order finite difference resultson a uniform grid. The deficiency of the FV results is likely due to the inadequate gridresolution near the pipe wall, implying that DNS of the turbulent pipe flow requiresa more demanding grid resolution requirement (say δr+ 6 1.25) than the case of theturbulent channel flow (δ 6 2.25) at similar friction Reynolds numbers. Additionally,LBM treats the advection in the mesoscopic space exactly (except when the interpolatedbounce-back is used near the curved wall) and preserves the exact local mass andmomentum conservations, implying that LBM could be physically more accurate than

Page 37 of 44


Eα (kθ)

kθ

(a) (b) (c)

Figure 31: The azimuthal energy spectra for velocity components at different locations,(a): near wall (δ+ = 3.5), (b): buffer region (δ+ = 22.0), (c): logarithmic region (δ+ =122.0). FV results (Eggels et al. 1994) are taken at slight different representative radiallocations, as δ+ ≈ 4, δ+ ≈ 17 in (a) and (b), respectively.

Eα (kθ)

kθ

(a) (b) (c)

Figure 32: The azimuthal energy spectra for vorticity components at different locations,(a): near wall (δ+ = 3.5), (b): buffer region (δ+ = 22.0), (c): logarithmic region (δ+ =122.0).

the second-order FV method based on solving the Navier-Stokes equations. The r.m.s.pressure and vorticity comparisons indicate that the LBM has a better accuracy thanthe second-order FV methods on both uniform and non-uniform grids for the velocitygradient calculations. The comparisons of higher-order statistics (skewness, flatness) ofvelocity fluctuations again confirm the better accuracy of the LBM than the second-orderFV on uniform grid. Therefore, we argue that the early DNS data by Eggels et al. (1994)are rather inaccurate and as such should not be used as a benchmark in the future.Previously, the DNS data by Eggels et al. have often been used as a benchmark, forexample, in Kang & Hassan (2013),Suga et al. (2015) and Wu et al. (2011).On the other hand, the general consistency of our LBM data with the spectral

Page 38 of 44

Page 39 of 44

Page 40 of 44


situations in the vorticity calculation, as shown in (a), (b), (c) and (d) in Fig. 34. Foreach case, the vorticity at node (i, j, k)is calculated as follows

(i) Case (a),

∂uy

∂x(i, j, k) =

uy (i+ 1, j, k)− uy (i− 1, j, k)

2∆x(6.1)

(ii) Case (b),

∂uy

∂x(i, j, k) =

q2uy (i+ 1, j, k)− uwy +(

1− q2)

uy (i, j, k)

(q2 + q)∆x(6.2)

(iii) Case (c),

∂uy

∂x(i, j, k) = −q2uy (i− 1, j, k)− uwy +

(

1− q2)

uy (i, j, k)

(q2 + q)∆x(6.3)

(iv) Case (d),

∂uy

∂x(i, j, k) = (q1 − q2)

uwy − uy (i, j, k)

q1q2(6.4)

7. acknowledgments

This work has been supported by the U.S. National Science Foundation (NSF) undergrants CNS1513031, CBET-1235974, and AGS-1139743 and by Air Force Office ofScientific Research under grant FA9550-13-1-0213. Computing resources are providedby National Center for Atmospheric Research through CISL-P35751014, and CISL-UDEL0001 and by University of Delaware through NSF CRI 0958512.

Page 41 of 44


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