Phys. Fluids 31, 013602 (2019); https://doi.org/10.1063/1.5067291 31, 013602

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Mathematical modeling of Stokes flow inpetal shaped pipes Cite as: Phys. Fluids 31, 013602 (2019); https://doi.org/10.1063/1.5067291Submitted: 16 October 2018 . Accepted: 31 December 2018 . Published Online: 25 January 2019

Zhimin Xu (徐志敏), Siyuan Song (宋思远), Fengxian Xin (辛锋先) , and Tian Jian Lu (卢天健)

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Mathematical modeling of Stokes flow in petalshaped pipes

Cite as: Phys. Fluids 31, 013602 (2019); doi: 10.1063/1.5067291Submitted: 16 October 2018 • Accepted: 31 December 2018 •Published Online: 25 January 2019

Zhimin Xu (徐志敏),1,2,3 Siyuan Song (宋思远),1,3 Fengxian Xin (辛锋先),1,3,a) and Tian Jian Lu (卢天健)1,2,a)

AFFILIATIONS1State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049,People’s Republic of China

2State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, People’s Republic of China

3MOE Key Laboratory for Multifunctional Materials and Structures, Xi’an Jiaotong University, Xi’an 710049,People’s Republic of China

a)fengxian.xin@gmail.com and tjlu@nuaa.edu.cn

ABSTRACTA theoretical model is developed to characterize fully developed laminar flow (Stokes flow) in idealized petal shaped pipes byregarding the pipe wall as a circular boundary having sinusoidal perturbation (or, equivalently, surface roughness). Built upon themethod of perturbation, the model quantifies the effects of the relative roughness and wave number of the pipe boundary onthe Stokes flow. Approximate solutions of the velocity field, pressure gradient, and static flow resistivity are obtained. The sameapproach together with the method of Fourier transform is used to deal with low Reynolds flow in pipes having other cross-sectional morphologies such as triangle and square. Results obtained from computational fluid dynamics simulations are used tovalidate the theoretical model predictions, with good agreement achieved. The presence of surface roughness causes periodicfluctuation and global offset of velocity distribution and enlarges both the pressure gradient and static flow resistivity in petalshaped pipes.

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I. INTRODUCTIONHow surface roughness affects fluid flow is a fundamen-

tal problem in fluid mechanics.1,2 Von Mises3 first introducedthe concept of relative roughness to characterize the influ-ence of surface roughness. Colebrook et al.4,5 and Nikuradse6

carried out detailed experimental and theoretical studies onthe effect of the relative roughness and Reynolds number,providing a classic way to analyze the problem. They foundthat this effect can be divided into three regions, the lami-nar region, the transition region, and the turbulence zone; inthe laminar region, the influence of the rough boundary canbe ignored. A series of empirical formulas were also devel-oped to describe the relationships between relevant param-eters. Based on experimental and theoretical results, Moody7

plotted the relation diagram between the relative rough-ness, Reynolds number, and Darcy friction coefficient, whichis called the Moody diagram and widely used in industry.

However, these results are not applicable to cases involvinglarge relative roughness for existing studies concerned mainlysmall values (<5%) of relative roughness. Besides, in existingstudies, the specific morphology of surface roughness was notfully discussed.

For simplicity in theoretical and numerical modeling, therough surface has been commonly assumed to exhibit a peri-odic structure.8–12 Kandlikar et al.13 pointed out that fluidconcentration caused by the rough structure is not negligiblewhen the relative roughness becomes relatively large: even inthe laminar region, the change in the friction coefficient dueto the change in the hydraulic radius affects the fluid field.

Topologically speaking, roughness on the inner surfaceof a circular pipe may be divided into two main categories.In the first category, the roughness is distributed along theaxial direction (i.e., direction of fluid flow) of the pipe, whichhas been discussed in many studies.13–15 In the other, the

Phys. Fluids 31, 013602 (2019); doi: 10.1063/1.5067291 31, 013602-1

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rough structure is distributed along the circumference of thepipe, implying that its cross-sectional morphology is no longercircular. The present study focuses upon the latter one forit is poorly investigated in the open literature. Specifically,low Reynolds number flows in pipes with petal shaped crosssections are investigated.

The flow in pipe also does much meaning for investigat-ing the acoustic properties of porous medium as it consists ofmany microtunnels (pores). However, in most practical cases,the pore morphology in the porous medium is not smooth. Toaddress the issue, the acoustical properties of model porousmedia with simple pore shapes (e.g., square and triangular)have been theoretically studied.16 In order to quantify theinfluence of the cross-sectional shape of the pore (e.g., devi-ation from circular) on complex density and compressibility—two important parameters governing the acoustic propertiesof the porous medium—the concept of shape factor was intro-duced.17 When investigating the sound absorbing ability, asnoted by Attenborough,18 the pore shape with a larger shapefactor provides a greater resistance for sound to pass, thusa larger air flow resistivity. In other words, the shape factorsfor different pore shapes reflect the relative flow resistivitiesfor fluid to pass through these pores. Although the introduc-tion of shape factor makes it easy to understand flow insidepores, there are some limitations to this approach. In addi-tion to frequency correlation, every time a new pore shapeis of concern, a series of complex operations need to be car-ried out to obtain its shape factor. By contrast, the methodof perturbation starts with a simplified form of the originalproblem, which is simple enough to be solved exactly. Themethod is more like a mathematical technique for finding anapproximate solution to a problem, by starting from the exactsolution of a related, simpler problem. A critical feature of thetechnique is a middle step that breaks the problem into “solv-able” and “perturbation” parts.19 In fact, as one of the newmethods for analyzing turbulent boundary layers, it has beendemonstrated that the perturbation method provides moreaccurate, yet practical, means for estimating the skin-friction,heat transfer, and separation on aircraft components underflight conditions.20,21

In the current study, the modified perturbation methodis employed to investigate analytically the fully developedlaminar flow at small Reynolds numbers (Stokes flow) insideidealized petal shaped pipes. The pipe wall is taken as acircular boundary having sinusoidal perturbation (or, equiv-alently, surface roughness). Approximate solutions for veloc-ity distribution, pressure gradient, and static flow resistivityof the petal shaped pipes are obtained. Numerical simula-tions with computational fluid dynamics (CFD) are performedto validate the analytical model predictions. The influence ofkey geometrical parameters characterizing the petal shapedcross section on pipe flow is quantified. Furthermore, thesame approach together with the method of Fourier trans-form is used to deal with low Reynolds flow in pipes hav-ing other cross-sectional morphologies such as triangle andsquare.

II. ANALYSIS OF STOKES FLOWIN PETAL SHAPED PIPES

Figure 1 depicts schematically a wide range of pipes hav-ing idealized petal shaped cross sections. For convenience,relative to the case of a smooth circular pipe, the petal-shapedpipe may be taken as a roughened pipe in a way similar toour previous study on surface roughness effects on stokesflow in circular pipes.22 Let n be the number of waves in thecross-sectional plane of the petal shaped pipe, which rep-resents the density of the periodic rough structure on thepipe wall. According to the definition of relative roughness, ε= e/D is introduced to quantify the relative altitude of therough structure, where D is the nominal diameter of the pipe(Fig. 1) and 2e is the radical distance between the peak and thevalley of the rough structure. Such periodic microstructuresare evenly distributed over the circumference of the pipe sothat its boundary may be described by

r̄ = Γ(θ)D, with Γ(θ) = 0.5 − ε sin(nθ). (1)

Here, the cylindrical coordinate system has its origin at thecenter of the pipe, the z-axis is along its axial direction of the

FIG. 1. Sketch of petal shaped cross-sectional pipes withdifferent wave numbers.

Phys. Fluids 31, 013602 (2019); doi: 10.1063/1.5067291 31, 013602-2

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pipe, and ρ, θ are in the cross-sectional plane, as shown inFig. 1. A dimensionless coordinate system is defined by settingr = ρ/D. So, the geometric boundary line can be expressed asr = r̄/D = Γ(θ).

Mathematically speaking, the dimensionless geometricboundary Γ(θ) of (1) can be treated as a sine-type turbu-lence on a circle with a radius of 0.5. Having such a com-plex boundary makes it too difficult to solve directly theflow in the pipe. However, the perturbation method, whichreplaces the disturbance of the geometric boundary with adisturbance of the fluid field, can provide approximate solu-tions to the problem in a sophisticated way, as demonstratedbelow.

A. Modified perturbation methodWith reference to Fig. 1, the case considered is pressure-

driven, steady-state flow (i.e., Poiseuille flow) in straightchannels with different petal shaped cross sections. Here,the nonlinear Navier-Stokes equation governing the flow ofNewtonian incompressible fluid at small Reynolds numberscan be simplified to the linear Stokes equation. As both thecharacteristic length and characteristic velocity in the prob-lem of Fig. 1 are taken as considerably small, neglecting theinertial term is considered acceptable.23,24 Due to the linear-ity of the Stokes equation, it is reasonable to adopt a rela-tively large range of relative roughness (e.g., [0, 0.1]) for theroughened pipe without a significant loss in accuracy.

For fully developed steady incompressible flow, thedimensionless Stokes equation is

0 = −∂p∂z

+(

1r∂

∂r

(r∂

∂r

)+

1r2

∂2

∂θ2

)u, (2)

where p and u are the pressure and velocity of the fluid. Theinner boundary of the petal shaped pipe is assumed to be no-slip, and at the center of the pipe, the flow field is symmetrical.The boundary conditions can thus be written as

u = 0 at r = 0.5 − ε sin(nθ)∂u∂r= 0 at r = 0

. (3)

Assuming that the relative roughness (ε) of the pipe issufficiently small, one can take the Taylor expansion of thevelocity u(r, θ) about ε and retain only its zero- and first-orderterms as

u(r, θ) = u0(r, θ) + εu1(r, θ) + O(ε2

), (4)

where O(εn) is defined as the infinitesimal of the n-th order.According to the perturbation method, fluid velocity at theboundary can be expressed as

u(r, θ)��r=Γ(θ ) = u(r, θ)��r=0.5 − ε sin(nθ)(∂u(r, θ)∂r

) �����r=0.5+ O

(ε2

).

(5)

In Eq. (5), values at the complex boundary Γ(r, θ) can beobtained through the properties at the simplified bound-ary r = 0.5. As a result, the influence of wall rough-ness can be reflected by the first-order perturbation term

ε sin(nθ)(∂u(r,θ )∂r

) ����r=0.5. Consequently, the following analysis

and results are all obtained in the equivalent domain: r ∈ (0,0.5), θ ∈ (0, 2π).

The solution obtained with the zero-order term of theperturbed equation is

u0 = 2 − 8r2. (6)

This is actually the classical velocity field of Poiseuille flowinside a smooth circular pipe.

The first-order term of the perturbed equation togetherwith the revised boundary condition is given by

0 = −∂p1

∂z+

(1r∂

∂r

(r∂

∂r

)+

1r2

∂2

∂θ2

)u1

u1 = −4 sin(nθ) + Q, at r =12

∂u∂r

= 0 at r = 0

, (7)

where Q is a first-order infinitesimal, i.e., Q = O(ε). If theboundary condition is not revised by introducing Q, a non-ignorable error will occur. In other words, to be consistentwith the physical fact that the cross section of the roughenedpipe is closely related to flow resistance, the perturbation ofthe pipe boundary inevitably affects the pressure drop alongthe z-axis of the pipe. In fact, the nonlinearity of the boundarycurve, serving as the disturbance source, magnifies the high-order terms in the system so that the Q term in Eq. (7) cannotbe neglected.

In order to separate the two variables r and θ, takingFourier transform of the first-order term of velocity u1 in theθ (circumferential) direction, we find that only three termsremain, namely,

u1 = ϕ−1(r)e−(nθ−π/2)i + ϕ0(r) + ϕ1(r)e(nθ−π/2)i, (8)

where nθ − π/2 is adopted to make the real part agree with theboundary condition. In fact, ϕ−1(r) and ϕ1(r) terms representthe periodic fluctuation of flow velocity along the circumfer-ential direction, which is not directly related to the growth ofthe pressure gradient, while ϕ0(r) represents the overall offsetof velocity along the radial direction. According to Eq. (2), theϕ0(r) term is actually responsible for the increased pressuredrop due to surface roughness. Substituting Eq. (8) into thegoverning equations, we arrive at two independent equationsfor ϕ1 and ϕ0 as

0 = r2 ∂2ϕ1(r)∂r2

+ r∂ϕ1(r)∂r

− n2ϕ1(r)

ϕ1 = −2, at r = 0.5∂ϕ1

∂r= 0, at r = 0

, (9)

∂p1

∂z=∂2ϕ0

∂r2+

1r∂ϕ0

∂rϕ0 = Q, at r = 0.5∂ϕ0

∂r= 0, at r = 0

. (10)

Phys. Fluids 31, 013602 (2019); doi: 10.1063/1.5067291 31, 013602-3

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Due to symmetry, ϕ1 and ϕ−1 are complex conjugates. Thesolution of (9) is given by

ϕ1(r) = −2n+1rn. (11)

As discussed above, this solution does not cause any increasein the pressure gradient.

To get a better understanding of the influence of the petalshaped cross section on the pressure drop, the ϕ0 term isdetermined next. Solution to (10) can be expressed as

ϕ0(r) = E + Fr2, (12)

where E(ε, n) and F(ε, n) are two functions directly relatedto Q. Since incompressible fluid is assumed, the velocity fluxincrement in the whole domain should be zero: that is, theconservation of velocity holds,∫ 2π

0

∫ 1/2

0u1rdr

dθ = 0. (13)

It has been established that the terms involving ϕ1, ϕ−1have no contribution to the final result because of the inte-gration property of the trigonometric function. Inserting (12)into (13) yields

F = −8E. (14)

Equation (14) is an approximate expression, helpful for esti-mating the values of F and E. The precise expressions of F(ε, n)and E(ε, n) are determined in Sec. II B.

B. SolutionGiven that the petal shaped cross section blocks fluid flow

relative to its smooth counterpart, the blocking effect may bequantified based on the flow fields in two extreme cases. Onesuch limit is when the wave number is so large that the roughstructure is extremely dense, i.e., n → ∞. The other limit cor-responds simply to the case of smooth pipe, i.e., n = 0. Onemay consider these two extreme cases as asymptotic limits,which means that the pressure drops of the two cases repre-sent the upper and lower bounds, respectively. As the wavenumber n increases, it becomes more difficult for the fluidto flow through the rough pipe. Correspondingly, the pres-sure drop increases continuously, approaching the upper limitasymptotically as n→ ∞. To describe the two asymptotic lim-its, for the problem considered in the current study, the logis-tic function25 may be adopted. The logistic function is used toapproximate the dependence of E and F on n as

E = *,A

2e−1

12.5 n

1 + e−1

12.5 n+ B+

-, (15)

F = *,C

2e−1

12.5 n

1 + e−1

12.5 n+ D+

-, (16)

where A, B, C, and D are four coefficients that are dependenton relative roughness.

When n → ∞, the rough pipe is analogous to a smoothtube with diameter (1 − 2ε)D. Correspondingly, the Poiseuilleflow has velocity

un=∞ = 8(1 − 2ε)−4 [(0.5 − ε)2 − r2

], (17)

which yields

B =[2(1 − 2ε)−2

− 2] /ε

D =[−8(1 − 2ε)−4 + 8

] /ε

. (18)

When n → 0, the rough pipe is analogous to a smooth tubewith diameter D. The velocity is

un=0 = 2 − 8r2, (19)

from which

A =[2 − 2(1 − 2ε)−2] /

ε

C =[8(1 − 2ε)−4

− 8] /ε

. (20)

Thus, upon considering the asymptotic limits, the four coef-ficients A, B, C, and D are determined. Substituting them intoEq. (8), we obtain the velocity field as

u =[2 − 2(1 − 2ε)−2 +

(8(1 − 2ε)−4

− 8)r2

]*,

2e−1

12.5 n

1 + e−1

12.5 n− 1+

-− ε2n+2rn sin(nθ) + 2 − 8r2. (21)

The Stokes equation in the z-direction is written as

∂p∂z=

(1r∂

∂r

(r∂

∂r

)+

1r2

∂2

∂θ2

)u. (22)

Making use of (21), we calculate the ratio of the average pres-sure drop (i.e., area-weighted average of ∂p/∂z) across a petalshaped pipe to that across a smooth pipe, resulting in

kf (ε,n) =(∂p/∂z)rough(∂p/∂z)smooth

=1

(1 − 2ε)4+ *

,1 −

1

(1 − 2ε)4+-

2e−1

12.5 n

1 + e−1

12.5 n.

(23)

To quantify the influence of surface roughness on fluidflow, the concept of static flow resistivity is employed, whichis a fluidic parameter reflecting the viscous resistance of thepipe. By definition, the static flow resistivity is the quotient ofthe air pressure difference across a pipe divided by the volumevelocity of airflow through it.26 So, the static flow resistivitycan be expressed as σ = −∆P/LU, with ∆P being the fluid pres-sure drop across the pipe, L being the length of the pipe, andU being the average volume velocity of fluid in the pipe. Therelative static flow resistivity of the present petal shaped tubeis hence

σrough

σsmooth=

(∆P/LU)rough(∆P/LU)smooth

= kf , (24)

where σsmooth = 8µ/R2 is the static flow resistivity of asmooth tube with radius R. When ε = 0, the relative staticflow resistivity is equal to 1. When the wave number n → 0,the relative static flow resistivity takes the minimum 1; whenn → ∞, it takes the maximum 1/(1 − 2ε)4. That is, when n isincreased, the relative static flow resistivity increases from1 to 1/(1 − 2ε)4.

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C. Triangular and square cross-sectional pipesIn addition to sinusoidal (petal shaped) cross sections

considered above, several common cross-sectional mor-phologies, such as triangle and square, can be modeled bydecomposing their boundary curve functions into series ofsinusoidal functions via Fourier transform. This is demon-strated next.

To select a moderate length to apply the proposed mod-eling approach, the length corresponding to the unit circle isadopted. Thus, the areas of triangle and square are set equal tothat of the unit circle, as shown in Fig. 2. According to Eq. (7),the effect of the cross-sectional shape on flow is characterizedusing the revised boundary condition. For any given shape ofthe boundary curve, Fourier transform can be expressed bythe general form as

rshape(θ) − 0.5 = −∞∑n=1

sgnn · εn sin(nθ), (25)

where sgnn represents the sign of the n-th component. Asthe model present above, εn is the relative roughness, whosevalue is positive. To assure the completeness of the expres-sion, sgnn is introduced so that when the n-th effect is nega-tive, it should be “-.” In other words, the equilateral triangleand square are considered as a circle with numerous petalshaped disturbances.

The left side of (25) represents the disturbed part of theboundary rshape on a unit circle (r(θ) = 0.5), which can be trans-formed into the summation of a series of sinusoidal functions.The negative sign to the right side of (25) is set to be consis-tent with the shape function of (1). For triangular and squareshapes, the Fourier transform expressions are presented in theAppendix. By regarding the cross-sectional shape as the sum-mation of a unit circle and a series of petal shapes, it followsfrom (23) and (24) that the ratio of static flow resistivity of agiven shape to that of the unit circle is

kf_shape(s) = 1 +∞∑n=1

sgnn ·[kf (εn,n) − 1

]. (26)

Note that in each item of the series, the effect is subtractedby that of the unit circle, i.e., kf (εn, n) − 1. Inserting Fouriertransforms of triangle and square, i.e., Eqs. (A2) and (A4), intoEq. (26), we can obtain their relative static flow resistivities.

FIG. 2. (a) Circular, (b) equilateral triangular, and (c) square cross sections withidentical area S.

III. RESULTS AND DISCUSSIONA. Validation against numerical simulation results

Numerical simulations with Computational Fluid Dynam-ics (CFD) software FLUENT are carried out on petal shapedpipes. Figure 3 displays the simulation settings. The left sur-face is set as the velocity inlet, and the right surface is setas the outflow. The boundary between the velocity inlet andoutflow is set as the wall, and the fluid region inside is setas FLUID. Quadrilateral mesh is selected to sweep the wholeregion, and the maximum size of the grid is sufficiently smallto ensure convergence of numerical simulation results: gen-erally, if the grid number is more than 100 000, it can meetthe requirement of mesh independence. In accordance withthe assumption of small Reynolds numbers, the fluid velocityis set as 0.001 m/s. The CFD results are used to validate thetheoretical model predictions, as illustrated below.

B. Velocity distributionWith the wave number fixed at 6 (i.e., n = 6), Fig. 4(a) dis-

plays the remarkable influence of relative roughness on thevelocity field in a petal shaped pipe (D = 1 mm), as obtainedseparately from the present theoretical model and CFD sim-ulation. Corresponding results for the case of fixed relativeroughness (ε = 0.1) but varying wave number are presentedin Fig. 4(b). Overall, the comparison between model predic-tion and numerical simulation is reasonably well. As the sameconclusion holds for other combinations of the wave numberand relative roughness, the results are not presented here forbrevity.

The results of Fig. 4 show that, due to the no-slip bound-ary condition, the velocity at the boundary is zero and thecloser to the pipe center, the greater the fluid velocity. Toexamine clearly the difference in velocity fields between thetheoretical and numerical results, line-scatter figures are pre-sented in Fig. 5. It shows that the theoretical predictionsagree well with the numerical simulations in a whole. Thepredicted velocities at the center are greater than the corre-sponding numerical ones, especially when the relative rough-ness or wave number is small. This occurs because when solv-ing the perturbation equation, the logistic function form isassumed to characterize the two asymptotic limits so thatthe two parameters E and F in Eq. (12) can be determined. It

FIG. 3. Sketch of numerical simulations with FLUENT.

Phys. Fluids 31, 013602 (2019); doi: 10.1063/1.5067291 31, 013602-5

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FIG. 4. Comparison between theoretically predicted andnumerically simulated velocity fields as contour maps inpetal shaped pipes: (a) n = 6 and (b) ε = 0.1.

FIG. 5. Comparison between theoretically predicted andnumerically simulated velocity fields as line-scatter figuresin petal shaped pipes: (a) n = 6, ε = 0.06 and (b) n = 4,ε = 0.1.

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FIG. 6. Influence of surface roughness on the pressuregradient in a petal shaped pipe: (a) n = 6 and (b) ε = 0.1.

should, however, be pointed that, as an approximate approach,this inevitably involves certain errors, thus causing the dis-crepancy between the analytically predicted and numericallycalculated velocities.

As for the influence of petal shapes, as shown in Fig. 4,upon changing the boundary morphology from a circle intoa petal shape, the velocity contours also exhibit a petal-likeshape, especially when the relative roughness or wave num-ber is increased. From Eq. (21), we find that the disturbance ofthe pipe boundary affects the velocity field in two ways. One isreflected by the term containing θ, representing the change inthe shape of the velocity contour. The other is reflected by theterm without θ, representing the global offset of the velocityfield. It should be pointed out that, in the presence of surfaceroughness, the fluid in the protruding portions of the petalshaped pipe is trapped (i.e., u = 0) due to the fluid viscosityand no-slip boundary condition. This feature is well capturedby the proposed theoretical model.

C. Pressure gradientBoundary disturbance also affects significantly the pres-

sure gradient in petal shaped pipes. As described in Eq. (2), thepressure gradient is the second derivative of the fluid veloc-ity. Substituting the velocity field [Eq. (21)] into the Stokesequation, i.e., Eq. (2), we find that the second derivative ofthe period term containing θ in the velocity expression iszero. This implies that there is no period term in the pres-sure gradient. It is thus interesting to observe that while thevelocity field inside a petal shaped pipe exhibits periodic dis-turbance, the pressure gradient is constant over a specifiedcross section. The latter actually ensures that the fluid field issymmetrical.

The theoretical predictions and numerical results of thepressure gradient inside a petal shaped pipe (normalized bythat inside a smooth circular pipe) are compared in Fig. 6 forselected values of wave number and relative roughness. Again,reasonably good agreement is achieved. Relative to a circu-lar pipe (either ε = 0 or n = 0), the pressure gradient in apetal shaped pipe is much greater. As the relative roughnessand/or wave number is increased, the perimeter of the pipecross section becomes longer, which implies more contactarea between the fluid and the solid wall. The wall imposesa drag force to the flowing fluid due to its viscosity, increas-ing as the surface roughness of the pipe is increased. In fullydeveloped flow, this drag force is balanced with the pressuredifference.

D. Static flow resistivityThe flow resistance depends largely on the parameters

characterizing the rough structure, as shown in Fig. 7. Theinfluence of the relative roughness and wave number onrelative static flow resistivity is quantified theoretically andnumerically. Theoretical expressions of the static flow resis-tivity for petal shaped pipes are given in Eqs. (23) and (24).Table I presents further explicit expressions of static flowresistivity by taking the Taylor expansion and retaining termsup to ε2 (which ensures the accuracy of calculation). Theagreement between the theory and numerical simulation isgood. As the relative roughness increases, the relative staticflow resistivity rises, increasing more sharply [Fig. 7(a)]. Bycontrast, the results of Fig. 7(b) show that as the wavenum-ber increases, the numerically calculated relative static flowresistivity first rises and then tends to an upper limit. In com-parison, the theoretical model predictions deviate from the

FIG. 7. Comparison of static flow resistivity between theo-retical predictions and numerical calculations: (a) n = 6 and(b) ε = 0.1.

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TABLE I. Theoretically predicted static flow resistivity (σ) of petal shaped pipes withvarying wave numbers and relative roughness.

Wave number n Shape σ

n = 1(1 + 0.32ε + 1.6ε2

) 8µa2

n = 2(1 + 0.64ε + 3.2ε2

) 8µa2

n = 3(1 + 0.96ε + 4.8ε2

) 8µa2

n = 4(1 + 1.28ε + 6.4ε2

) 8µa2

n = 5(1 + 1.6ε + 8ε2

) 8µa2

. . . . . . . . .

n = ∞ . . .(1 + 8ε + 40ε2

) 8µa2

numerical results when the wave number is small. When n issmall (e.g., n = 1, 2), the pipe cross section is yet to developinto a periodic petal-like structure, reducing thus the effec-tiveness of the calculation process. Furthermore, as differentmorphologies of the cross section require different grids forsimulation, the accuracy of numerical results varies among themorphologies considered.

From Table I, it is seen that as the wave number increases,the static flow resistivity also increases, which is consistentwith Fig. 7(b); when the wave number becomes infinitely large,the static flow resistivity approaches asymptotically its upperlimit. Increasing the wave number enables more fluid to betrapped due to its viscosity and the no-slip condition, andhence it becomes harder for the fluid to pass through. Theresults of Table I also reveal that when the wave number issmall, the static flow resistivity is approximately linearly pro-portional to the wave number, providing thus a quick wayto estimate the effect of the wave number on static flowresistivity.

E. Triangular and square pipesIn Secs. III B–III D, the flow field, pressure gradient, and

static flow resistivity of petal shaped pipes have been obtainedusing the modified perturbation method. It is interesting thatthe same method can be combined with Fourier transformto deal with triangular and square pipes, as demonstrated inSec. II C and the Appendix.

Table II compares the static flow resistivity predicted bythe present model with that obtained by Bruus et al.27 forcircular, triangular, and square pipes. Excellent agreement isachieved, validating again the modified perturbation methodproposed in the present study.

TABLE II. Static flow resistivity (σ) for circular, triangular, and square pipes:comparison between the present model and the existing study.

Cross-sectional morphology σ (present study) σ (Ref. 27)

8µa2

8µa2

76.95µa2

80µa2

28.46µa2

28.4µa2

IV. CONCLUSIONA theoretical model has been established to estimate the

fully developed flow at a small Reynolds number (Stokes flow)inside petal shaped pipes by regarding the pipe wall as acircular boundary with sinusoidal perturbation (or, equiva-lently, surface roughness). The effects of the relative rough-ness and wave number of the petal shaped pipe wall are quan-tified using the modified perturbation method. Approximatesolutions of the velocity field, pressure gradient, and staticflow resistivity are obtained, which are validated by resultsobtained from direct numerical simulations. The main findingsare summarized as follows:

(1) The presence of surface roughness leads to periodicfluctuation and global offset of velocity distribution.

(2) Increasing the relative roughness or wave number of thepipe wall enlarges both the pressure gradient and staticflow resistivity.

(3) For relatively small wave numbers of roughness, thestatic flow resistivity is linearly proportional to the wavenumber.

(4) With the triangle or square taken as the summation ofa series of sinusoidal perturbations and a unit circle, theflow resistivity inside triangular and square pipes can beobtained via Fourier transform.

ACKNOWLEDGMENTSThis work was supported by NSFC (Grant Nos. 11761131003,

U1737107, and 11772248), DFG (No. ZH15/32-1), and theFundamental Research Funds for the Central Universities(No. Z201811253).

APPENDIX: FOURIER TRANSFORM EXPRESSIONSFOR THE BOUNDARY CURVE FUNCTIONSOF TRIANGULAR AND SQUARE SHAPES

With reference to Fig. 2, because the equilateral triangleand square are assumed to have an identical cross-sectional

Phys. Fluids 31, 013602 (2019); doi: 10.1063/1.5067291 31, 013602-8

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Physics of Fluids ARTICLE scitation.org/journal/phf

area as the unit circle (with radius r = 0.5), the side length of

the triangle is√π/√

3, while that of the square is√π/2. Their

Fourier transforms are presented below so that the flow fieldis equivalent to that obtained from the summation of a seriesof petal shaped pipes.

The boundary curve function of the equilateral triangle isexpressed using a piecewise function as

rtriangle(θ) =

√π/√

3/2√

3

cos(θ − π/6), −

π

6< θ <

π

2√π/√

3/2√

3

cos(θ − 5π/6),π

2< θ <

7π6√

π/√

3/2√

3

cos(θ − 3π/2),7π6< θ <

11π6

. (A1)

Regarding the boundary curve as the summation of a seriesof petal shapes with increasing wave numbers, we can obtainFourier transform of the triangle. Here, to assure the accuracyof calculation, we obtain Fourier transform by retaining itemsup to 9 as

rtriangle(θ) − 0.5 = −0.010 86 sin(θ) − 0.132 19 sin(3θ)

+ 0.026 34 sin(5θ) − 0.038 15 sin(7θ)

+ 0.009 28 sin(9θ) + . . . . (A2)

For the case of a square pipe, the boundary curve functionis

rsquare(θ) =

√π/2/2

cos(θ), −

π

4< θ <

π

4√π/2/2

cos(θ − π/2),π

4< θ <

3π4

√π/2/2

cos(θ − π),

3π4< θ <

5π4

√π/2/2

cos(θ − 3π/2),

5π4< θ <

7π4

. (A3)

Similarly, to assure the accuracy of calculation, we obtainFourier transform by retaining items up to 9 as

rsquare(θ) − 0.5 = −0.00246 sin(θ) + 0.03363 sin(3θ)

−0.05339 sin(5θ) − 0.03185 sin(7θ)

+ 0.0032 sin(9θ) + .... (A4)

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