analysis of pipe flow transition

8/13/2019 Analysis of Pipe Flow Transition

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Digital Object Identifier (DOI) 10.1007/s00162-004-0105-9

Theoret. Comput. Fluid Dynamics (2004) 17: 273292TheoreticalandComputationalFluidDynamics

Original article

Analysis of pipe flow transition.

Part I. Direct numerical simulation

Jorg Reuter1,, Dietmar Rempfer2

1 Institut fur Aerodynamik und Gasdynamik, Universitat Stuttgart, Germany2 Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, USA

Received March 26, 2003 / Accepted January 14, 2004Published online July 1, 2004 Springer-Verlag 2004

Communicated by R.D. Moser

Abstract. We have developed an accurate hybrid finite-difference code for the simulation

of unsteady incompressible pipe flow. The numerical scheme uses compact finite differences

of at least eighth-order accuracy for the axial coordinate, and Chebyshev and Fourier poly-

nomials for the radial and azimuthal coordinates, respectively. Boundary conditions for the

incompressible flow are enforced using an influence-matrix technique, and the Poisson equa-

tion for pressure is solved using a fast direct method. The code has been used to simulate and

analyze the spatial transition process in developed laminar pipe flow at a Reynolds number of

Re = 2350. Results of the simulation are compared to experimental data given by Han, Tuminand Wygnanski [18].

Key words:direct numerical simulation, pipe flow transition, incompressible NavierStokes

equation

PACS:47.11.+j, 47.20.Ft, 47.27.Cn

1 Introduction

The process of transition to turbulence in incompressible pipe flow is still not fully understood. One of the

problems peculiar to laminar-turbulent transition in such flows follows from the observation that developed

pipe flow is asymptotically stable at any Reynolds number, with stability becoming marginal only in the limit

ofRe . It is well understood that, because the stability operator for pipe Poiseuille flow is non-normaland thus permits non-orthogonal eigenmodes (see, e.g., Farrell [13]; Reddy et al. [40]; Trefethen et al. [46]),

this situation does not preclude the temporary amplification of disturbances. On the other hand, the tem-

porary amplification of disturbances known as algebraic instability can only give a partial explanation

of the process of transition to turbulence in pipe flow. In particular, it is not clear whether the kind of op-

timal disturbances that are described in, e.g. (Farrell [13]; Butler and Farrell [7]; Bergstrom [5]; Schmid

Correspondence to:D. Rempfer (e-mail: [email protected]) This work was supported by Grant-# Wa 424/17-13 of the DFG (Deutsche Forschungsgemeinschaft). The authors are in-

debted to Professor Anatoli Tumin for providing them with detailed experimental data. Present address: Voith Paper GmbH & Co. KG, Heidenheim, Germany


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274 J. Reuter, D. Rempfer

and Henningson [43]), and more recently by Reshotko and Tumin [41], are indeed present at sufficient am-

plitude to induce natural transition to turbulence. Apart from that it is obvious that, at some point during

this transition process, nonlinear effects which are not included in linear theories must play an important

role.

One of the main motivations for our work in this area was to obtain a better understanding of pipe flow

transition at low Reynolds numbers of the order ofRe 20003000, in other words close to the known lowerlimits for pipe flow turbulence. By positioning our simulations near that limit, we were aiming at getting

insight into the interplay of factors that are responsible for allowing the flow to, or preventing it from, be-

coming turbulent. This also meant that to be able to capture the relevant situation with a sufficient degree

of accuracy, we had to strive to design a simulation code such that both amplitude and phase errors would

be kept as small as possible. To meet that goal, we have developed a new hybrid finite-difference-spectral

code which solves the incompressible NavierStokes equations in primitive variables using compact finite

differences of very high order for the streamwise coordinate, and Chebyshev and Fourier polynomials for

the radial and azimuthal coordinates, respectively. Using an influence-matrix method formulation for the

pressure boundary conditions, we make sure that both the incompressibility constraint and the boundary con-

ditions are satisfied to machine accuracy. Based on comparisons both with predictions of linear theory and

with experimental results as described below, we are confident that our code can indeed accurately describe

the spatial transition process in pipe flows.Here, we are focusing mostly on a description of the original parts of the numerical method that we

have developed and its validation, as well as on a phenomenological description of the process of pipe

flow transition as observed in a numerical experiment that was designed to mirror the experiments done by

Han et al. [18]. We note that the Reynolds number range of these experiments, and thus of our simulation, is

in a range where the so-called puffs have been observed by Wygnanski and Champagne [51] as the charac-

teristic structures of natural transition. One might thus conjecture that the simulations that we present in the

following could conceivably represent the early stages in the life cycle of such puffs, although the differ-

ences between our simulation (and the experiments by Han et al. [18]) highly structured, forced transition

with simple periodic disturbances in our case versus natural transition in Wygnanskis work are sufficient

to label our conjecture as speculative, at best. In a following paper we intend to provide results from a de-

tailed analysis of energy flows in transitioning pipe flow, which will reveal some of the mechanisms behind

the phenomenology described in the paper at hand.

The present paper is structured as follows: To put our work into perspective, we start with a brief review

of previous work below. In Sect. 3 we present the differential equations underlying our numerical algorithm,

and Sect. 4 describes the associated boundary conditions. The fourth section discusses our discretisation of

the differential equations, and Sect. 6 deals with our direct solution of the Poisson equation for pressure and

the way we implemented the pressure boundary condition via an influence matrix method. The validation of

the numerical method by comparison of simulation data to predictions of linear theory is described in Sect. 7,

Sect. 8 and Sect. 9 detail the results of our attempt to numerically reproduce the experiment described by

Han et al. [18]. We end with some concluding remarks. A few of the more intricate particulars of our method

are described in three short appendices describing details of our finite differences, the direct method we use

to solve our Poisson equation, and our influence matrix technique.

2 Numerical simulation of unsteady pipe flows

Dixon and Hellums [10] were among the first to numerically simulate pipe flow. They applied a finite dif-

ference scheme to a two dimensional stream function formulation. Crowder and Dalton [8] also developed

a code using only finite differences. Finite volume methods were applied by Eggels et al. [11] and by Ak-

selvoll and Moin [3]. Leonard and Wray [27] developed an elegant spectral method for temporally evolving,

three dimensional perturbations and applied it to the linear stability problem. Due to the lack of efficient

transforms, similar fully spectral methods by Nikitin [29] and Boberg and Brosa [6] were restricted to only

a small number of modes. Landman [25] developed a method applying Fourier decomposition in the azi-

muthal direction and finite differences in the radial direction for helically symmetric flows. Nikitin [30] used

a similar scheme and extended it to the general spatially periodic case using Fourier modes in the axial di-

rection (Nikitin [31]. For the spatially evolving problem, he applied finite differences in the downstream


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Analysis of pipe flow transition. Part I. Direct numerical simulation 275

direction (Nikitin [32]). Several authors (see, e.g., Priymak and Miyazaki [38]; Zhang et al. [52]; Priymak

and Miyazaki [39]; Ma et al. [28] have proposed codes based on Fourier modes in the azimuthal and axial

directions and orthogonal polynomials in the radial direction.

Many transitional flows are dominated by so-called-vortices. The first detailed studies of these struc-

tures were made in flat plate boundary layers. Hama et al. [16] compared their shapes to milk bottles.

Klebanoff et al. [20] realised that three-dimensionality was an essential aspect of instability. They showedthat in the so-called peak planes, there is a velocity defect leading to an inflection point in the profile.

Breakdown is marked by spikes on the oscilloscope trace in that plane, with the number of spikes increas-

ing further downstream. Kovasznay et al. [24] plotted the shear (U+u) /yin the peak plane. They foundthat the spikes are due to the passage of strong shear layers. Hama and Nutant [17] visualised the vortices by

creating hydrogen bubbles in the boundary layer. They gave a detailed description of the development of the

vortices including the successive shedding of vortices. Further investigations have been made by Acarlar

and Smith [1, 2], and Haidari and Smith [15].

Photographs of shear layers in pipe flow have been taken by Weske and Plantholt [50]. Extensive quanti-

tative results have been obtained by Eliahou et al. [12]. They found that the excitation of two counter-rotating

helical waves was a very effective way of triggering transition. This was also the case for a combination

of stationary streamwise rolls and periodic blowing and suction. These findings are consistent with similar

observations in boundary layers (Schmid and Henningson [42]), where the effect is referred to as obliquetransition. Eliahou et al. [12] also showed that for small disturbance amplitudes, the perturbations decay ex-

ponentially as predicted by linear theory. For intermediate amplitudes, there is transient growth eventually

followed by exponential decay. For even higher disturbances, transition takes place. Focusing on the latter

regime, Han et al. [18] made extensive measurements which have been used to validate the present numerical

method, see below.

3 Differential equations

The differential equations to be solved are the NavierStokes equations in cylindrical coordinates for in-

compressible flow. The dimensional quantities ( ) used to non-dimensionalise the differential equations are

the pipe radius a, the kinematic viscosity , and the mean velocity Vz in the axial direction. The onlycharacteristic parameter of the flow is the Reynolds number defined as

Re =2Vza

.

Velocityv+ and pressure p+ are split up into their laminar parts

V(r) = Vzez=

1r2

ez P(z) =P(z0) (4/Re)(zz0) (1)

and the disturbances v(t, r, ,z) and p(t, r, ,z), respectively. As shown by Patera and Orszag [37], the

momentum equations in the radial and azimuthal directions can be decoupled in the linear terms by defining

u= vr iv, (2)

thus simplifying the time integration. The resulting equations are

u

t+vr

u

r+

v

r

u

iu

+ (Vz +vz )

u

z=

p

r

i

r

p

+ 1

Re

1

r

r

r

u

r

+

1

r2

2u

2 2i

u

u

+

2u

z2

. (3)

The third velocity component vz is computed from continuity

1

r

rvr

r+

1

r

v

+

vz

z= 0, (4)


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ensuring a solenoidal velocity field. Taking the divergence of the momentum equations yields a Poisson

equation for pressure

1

r

r

r

p

r

+

1

r22p

2+

2p

z2 =

1

r

rr

r

1

r

z

z, (5)

where the non-linear terms are

r= vrvr

r+

v

r

vr

v

+ (Vz +vz )

vr

z, (6a)

= vrv

r+

v

r

v

+vr

+ (Vz +vz )

v

z, (6b)

z= vr(Vz +vz)

r+

v

r

vz

+ (Vz +vz )

vz

z. (6c)

4 Computational domain and boundary conditions

The pipe section considered is shown in Fig. 1. At the inflow boundary z = zi, fully developed laminar flowis assumed,

v|z=zi = p|z=zi = 0,

and the no-slip condition at the wall is

v|r=1= vw(t, ,z) =

vwr , v

w , v

wz

T. (7)

vw takes on non-zero values only at the disturbance strip modelling blowing and suction. Combining (7) and

(4) yields

vr

r

r=1 =vw

r

vw

vwz

z . (8)

A damping zone at the end of the domain gradually suppresses the perturbations. It resembles the one

described by Kloker et al. [23]. The velocity componentsvr andv are gradually reduced to zero by multi-

plying them, at each time step, by a function similar to the one shown in Fig. 2. Hence all quantities become

independent of the axial coordinate z giving

z

z=zo

= 2

z2

z=zo

= 0.

Fig. 1. Computational domain


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Fig. 2. Damping function

5 Discretisation

5.1 Time integration

The 1/rterms in (3) become large asr 0. If a purely explicit integration scheme was used, the maximumadmissible time step would be severely restricted. Hence a semi-implicit, third-order accurate, four-step

RungeKutta scheme as proposed by Ascher et al. [4] is used. The implicitly integrated terms are the rand

derivatives of the viscous terms in (3), i.e.

1Re

1r

r

ru

r

+ 1

r2

2

u2

2i u

u

.

5.2 Azimuthal direction

The quantities w

vr, v, vz ,p

are approximated by finite Fourier series

w(t, r, ,z) =

Nn=N

wn(t, r,z)ein . (9)

The sums have to be purely real, hence wn= w+n, wheredenotes complex conjugate. The coefficients of

the analogous series of the complex quantities usatisfy the equationsu,n= u,+n . Due to this coupling,

the solution process of the equations below can be restricted ton 0.With (9), the continuity (4), momentum (3), and Poisson (5) equations become

1

r

rvr,n

r+

inv,n

r+

vz,n

z= 0,

u,n

t+ ,n=

pn

r

npn

r+

1

Re

1

r

r

r

u,n

r

(1n)2u,n

r2 +

2u,n

z2

, (10)

1

r

r

r

pn

r

n2 pn

r2 +

2 pn

z2 =

1

r

rr,n

r

in,n

r

z,n

z, (11)

with ,n= r,n i,n ,

r,n =

n+n=n

|n |,|n|N

vr,n

vr,n

r+ v,n

invr,n v,n

r+ v+

z,n v

r,n

z

,

,n=

n+n=n

|n |,|n|N

vr,n

v,n

r+ v,n

vr,n + inv,n

r+ v+

z,n

v,n

z

,

z,n=

n+n=n

|n |,|n|N

vr,n

v+z,n

r+ v,n

invz,n

r+ v+

z,n vz,n

z

,


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and

v+z,n =

Vz + vz,0 if n= 0

vz,n if n= 0.

The convolution sums are computed pseudo-spectrally (Orszag [35]).

The cylindrical coordinate system introduces a mathematical boundary atr= 0. The series expansions(see, for example, Orszag and Patera [34])

u+,n

u,n

vr,n

v,n

vz,n

pn

=

l=0

r|n+1| u+,n,l

r|n1| u,n,l

r||n|1| vr,n,l

r||n|1| v,n,l

r|n| vz,n,l

r|n|

pn,l

r2l (12)

yield

u,n

r=0 =0 if n=1, (13a)

u,n

r

r=0

=0 if n / {0,2} , (13b)

vz,n

r=0 = pn

r=0

=0 if n= 0, (13c)

vz,n

r r=0= pn

r r=0 =0 if |n| = 1. (13d)

5.3 Radial direction

The radial dependence of the quantities is expressed in terms of finite Chebyshev series

wn (t, r,z) =

Kk=0

ckwn,k(t,z) Tk(r) where ck=

12

if k= 0

1 if k>0.

The Chebyshev polynomialsTk(r) = cos(karccos r)are even forkeven and odd forkodd. Due to (12), theseries can be restricted to either the even or to the odd indices, as appropriate:

wn (t, r,z) =

Ll=0

c2l+wn,l(t,z) T2l+(r) {0, 1} . (14)

If= 0, the resulting even function automatically satisfies the homogeneous Neumann conditions (13b)and (13d). If, on the other hand, = 1, the homogeneous Dirichlet conditions (13a) and (13c) need not beimposed separately.

The integration and the differentiation of Chebyshev series can be implemented by making use of recur-

rence relations (Gottlieb and Orszag [14]). The resulting band matrices can be inverted efficiently.


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5.4 Axial direction

The discrete analogs of the derivatives /z and 2/z2 are expressed as compact finite differences

(Lele [26]). The formulae are listed in Appendix A. They are at least eighth-order accurate. Due to their sym-

metries, they do not introduce any phase errors. Most of them damp short wavelength components, which is

a property that is beneficial for the stability of the integration scheme.At the inflow and outflow boundaries, asymmetric formulae have to be used to complete the linear sys-

tems. When differentiating in the axial direction, the errors at the fringes tend to be much higher than those at

the inner grid points, even if the formulae there are formally of the same orders as those of the symmetric fi-

nite differences. The computational domain is therefore extended by a few grid points at both ends. Because

of/z= 0, the new values can be directly extrapolated from the values at z = zi andz = zo, respectively.The accuracy of the approximations of the derivatives at the fringes is then of little significance.

The integration scheme is stabilised by filteringuin the axial direction at each time step (Vichnevetsky

and Bowles [47]). Since we use filters with a symmetric transfer function, this procedure does not introduce

any phase errors of its own into the scheme.

6 Solution of the Poisson equation

Inserting the recurrence relation for the r and derivatives and the compact difference representing the z

derivatives into the Poisson equations (11) for each n leads to linear equations for the vectors of unknowns.

In general, such systems cannot be inverted directly and methods like multi-grid have to be used which are

either slow or inaccurate. In the present case, the special structure of the matrices can be exploited to ob-

tain the fast and accurate algorithm described in Appendix B. Similar methods have been presented, among

others, by Swarztrauber [44]. The present algorithm is superior to those methods because it is not restricted

to tridiagonal systems.

A boundary condition for pressure at the wall is derived from requiring that the velocity obtained from the

momentum Eq. (10) satisfy not only the Dirichlet (no-slip) condition (7) but also the Neumann (continuity)

condition (8). The method is based on an idea by Kleiser and Schumann [22]. The details can be found in

Appendix C.

7 Validation

7.1 Eigenvalue formulation

If the products of perturbation quantities are ignored and periodicity is assumed, (3)(5) can be transformed

to an eigenvalue problem providing an alternative approach to solving the equation set. Comparing the

two solutions gives an idea of the accuracy of the numerical scheme outlined above. More precisely, the

quantities w

vr, v, vz , p

are assumed to be of the form

wn= 12

wn +w

n

, wn = w(r)r

n exp [i(z+nt)] , (15)

where C,n Z, and R.n are the exponents from (12). The frequency being real, the amplitudesdo not vary with time. The imaginary part i of wave number determines the axial envelope curve eiz.

For the validation, the solutions (15) were prescribed as initial conditions and at the inflow boundary.

The integration was run for three periods T, where T= 2/. The parameters are shown in Table 1. Theeigenfunctions used are the least damped for the given parameter set.

Table 1. Parameters for validation. L is the highest index of the Chebyshev series (14)

Re (n= 0) (n= 2) t/T L z

2280 0.96 1.01791+0.06206 i 1.09129+0.07697 i 1/1500 40 0.2


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Fig. 3a,b. Perturbation velocity vz,2(r, = 0,z) (a) and error (b)

7.2 Comparison

Figures 3a and b show the solution vz forn = 2 in a plane = 0 and the corresponding error, respectively.The errors for the other velocity components and for other values ofn are very similar to Fig. 3b. An ex-

ception, however, is the pressure for n = 0. The error shown in Fig. 4b is considerably larger in spite of themodification forn = 0 to the influence matrix technique described in Appendix C. Clearly, p0 is computedless accurately than the other quantities. However, it is important to note that, since p/z does not appear

in our formulation, it is only the error in the radial direction which is relevant for the momentum Eq. (3).Figure 4c shows the quantityp0p0|r=0 which, again, is quite small.

The graphs from Fig. 3 exclude the region upstream of the damping zone and the damping zone itself. In

that area, by definition, the error is of the same order of magnitude as the perturbation itself shown in Fig. 3a.

The graphs demonstrate that further upstream, the error due to the damping is negligible compared to the

discretisation error.

7.3 Influence matrix

As has been pointed out by Werne [49], subtle errors in the formulation of an influence matrix technique

may lead to a considerable deterioration of the accuracy of the computer code (see, however, Kleisers reply

to Wernes paper, Kleiser et al. [21]. In the present case, testing the performance of the method is straight-

forward. The technique is to ensure that vrsatisfies not only the Dirichlet but also the Neumann boundarycondition at the pipe wall. Any error in vr/r|r=0 would lead to an error in vz|r=0, the latter being inte-grated from (4). As can be seen from Fig. 3b, vz perfectly meets the no-slip condition at the wall. In fact, the

error is of the same order as the roundoff error of the computer.

8 Setup

Han et al. [18] have made extensive measurements of transitional pipe flow. The following comparison is

based on the case of a perturbation with azimuthal modesn = 3.In their experiment, Han et al. have used a custom-built disturbance generator that excites oscillations in

a set of 24 settling chambers around the circumference of the pipe, which are then transmitted to a set of


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Fig. 4ac. Perturbation pressure p0(r, = 0,z) (a), error (b), and errorinr(c)

Fig. 5. Disturbance slot of Han et al. [18]. Lengths in [mm]

narrow exit slots where unsteady jets at an angle of 45 to the pipe axis are created (see Han et al. [18] for

more details). The excitation of our simulation differs from the experiment for two reasons. First, the velocity

amplitudes at the exit slots were not measured and hence could not be reproduced, and second,the slot shown


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Fig. 6. Solid: envelope curve f(x) of the simulated perturbation (16), dashed: assumed parabolic profile of the experiment. The

reference length and hence the width of the parabola is d/2 [cf. (16)]

Table 2.Parameters of the simulation.L is the highest index of the Chebyshev series (14). Because of the symmetry, only Fourier

modes n = 0, 3, 6, . . . , 117(= 339) are considered

Re t L N z2350 0.96 0.018 120 339 0.03

in Fig. 5 was too small to be resolved on the computational grid. In our simulation, the velocity components

w

vr, v, vz

atr= 1 were modelled as

w|r=1(t, ,z) =

W0+

Nn=1

Wnsin(n+n)

sin(t) f(z) ,

whereis the frequency used in the experiment. The amplitudes vanish with the exception ofW3. f(z) takes

on non-zero values only at the perturbation slot. As the discretisation assumes infinitely differentiable func-tions, f should be very smooth at the fringes of the slot, i.e. there should be high-order zeros in order to avoid

numerical difficulties. The function chosen and shown in Fig. 6 is

f[x(z)] =

(1x

2)6 , |x| 1

0 , |x| 1x=

zzs

d/2, (16)

wherezsis the position of the slot and dis its width. The value ofdused is 1.0 (about 33 grid points) which

gives an effective width of about 0.5 (cf. dashed curve in Fig. 6) as opposed to 2.8 mm/16.5 mm0.17 inthe experiment. The perturbation of the simulation was

vr|r=1(t, ,z) =+0.233 cos(3) sin(t) f(z)

vz|r=1(t, ,z) =0.233 cos(3) sin(t) f(z) (17)

v

r=1(t, ,z) = 0.

The amplitude of 0.233 was established to fit the initial structures (see below). The component vrat the wall

is displayed in Fig. 7.

The inherent symmetries of the setup were taken advantage of in order to economise computer mem-

ory and CPU time. The symmetry with respect to = 0 leads to purely real u, vr, vz, and p. v is purelyimaginary. Furthermore, only Fourier modes n= 0,3,6, . . . were excited. Tables 2 and 3 collect theparameters.


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Fig. 9ah.Gradient(Vz +vz )/rat z = 3.04 (a),(b), z= 4.84 (c), (d), z= 6.66 (e), (f), and z=10.30 (g), (h). Experiment (Han et al. [18]) (a, c,

e, g) and simulation (b, d,f, h). The four contour

levels are {4,3,2,1}

Fig. 10. Sense of rotation of a -vortex

Fig. 11a,b. Temporal average of vorticity z(a) and perturbation velocity vz (b) at z= 10.3.The arrows represent the sense of rotation


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Table 4. Factors for components (n, m) for plots of modal energies

m= 0 m> 0

n= 0 1 2n> 0 2 4

e

z

Fig. 15. Spatial development of

selected modes. Curves (n, m)

include all modes (n,m)

The energy of mode (0,0) (Fig. 14) includes the base flow (1). Upstream of the disturbance slot at z = 0,its value is

2

10

12

V2z r dr= 2

10

12

1r2

2r dr= 1

6 0.167 .

The gradual transition to turbulence is accompanied by a steady decline of this energy. The asymptotic limit

is given by the energy of a fully turbulent profile. An approximation at a comparable Reynolds number is

(Nikuradse [33])

Vz 91144

(1r)16 ,

the energy of which is 11839216

0.128.The evolution of the other modes (Fig. 15) is initially marked by the disturbance (3,1) which is directly

sustained by the blowing and suction (17). The generation of streaks [see Fig. 11b] soon gives rise to (6,0)

which dominates the spectrum downstream.

10 Conclusions

We have developed an accurate method for the direct numerical simulation of transition to turbulence in in-

compressible pipe flows. Using an appropriate combination of carefully selected compact finite differencerepresentations of streamwise derivatives with an explicit filter, in conjunction with spectral representation

of radial and azimuthal derivatives, we believe that we have managed to strike a good compromise between

stability requirements and a desire to minimize both amplitude and phase errors in the integration of the mo-

mentum equation. We are integrating the transport equation with zero phase error, and introduce amplitude

errors only as needed to prevent a build-up of energy at the small-scale end of our range of resolved scales.

A direct solution method for the pressure allows for a highly efficient treatment of the corresponding Poisson

equation. By using an influence matrix method, the incompressibility constraint can be satisfied to machine

accuracy. The numerical method has been validated both by comparisons with predictions of linear stability

theory and by matching the experimental results of Han et al. [18].

Formally, pipe flow transition appears quite different from the analogous process in boundary layers.

In boundary layers, an important path to turbulence proceeds from primary instabilities which lead to the


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As shown by Davis (1979), the eigenvectors ofCwritten as a matrix are identical to the ones of the operator

F= c

exp

2ijk

N+5

j,k

2 j, k N+2

of the discrete Fourier transform, wherec = 0 is an arbitrary constant. The inverse of that matrix is

F1 = 1

c(N+5)

exp

+

2ijk

N+5

k,j

.

Given its eigenvectors,Ccan be diagonalised:

C= F1C F= diag {k} ,

where

k=

2

j=2

tj exp2i j k

N+5

are the eigenvalues ofC. The solution of (22) is therefore given by

v = C1b =

FC F11

b =FC1 F1b, (23)

where C1 = diag {1/k}.The first step of the algorithm derived from (23) is an inverse transform

F1

of the right hand sideb.

Using the Fast Fourier Transform (see, for example, [45] this can be implemented in a highly efficient

manner. In the simplified one-dimensional example, the next stepC1

consists of weighting the result-

ing vector by 1/k. In the actual application, an LUdecomposition (Patankar [36]) is applied to solve the

decoupled systems resulting from the radial discretisation. Another Fourier Transform ( F) yieldsv.

Initially, the coefficientsbj, j S={2,1,N+1,N+2

}, are unknown. On the other hand, the solu-tion forvj , j Shas to meet the boundary conditions. In a first run, (22) is solved with arbitrary values b

j ,

j S. Multiplying an influence matrix by the error vj , j S, provides the correct values bj = bj + bj .

The solution obtained in a second run is the desired solution to (21).

C. Influence matrix method

In a first step, arbitrary values at the wall are imposed when computing a provisional pressure distribution

from (11). A solution of (10) based on this pressure and (7) does not, in general, satisfy (8). The pressure

terms in (10) being linear, there is a linear relation between the error in vr,n /rand pn

r=1. The inverse

of this dependence transforms the error into a correction pnr=1. Once the boundary condition has been

fixed, a second cycle gives the desired solution.

Due to the discretisation, there is a finite number of points in both vectors pn

r=1

j

and

vr,n /r

r=1

j

.

In an initialisation step, the square matrix Mn of the relationvr,n /r

r=1

i= {Mn}ij

pn

r=1

j

is computed for each n . The desired influence matrix is the inverse of Mn . The elements j of pn

r=1

j

are set to unity one at a time, keeping all of the others equal to zero. Solving the homogeneous counterpart

of (11)

1

r

r

r

pn

r

n2 pn

r2 +

2 pn

z2 = 0


19/20


isolates the impact of the specific boundary point on pressure. The effect on velocity is given by solving, with

homogeneous initial conditions, (10) in its reduced form

u,n

t=

pn

r

npn

r+

1

Re 1

r

rru,n

r (1n)2u,n

r2 ,retaining only the implicit and pressure terms. u,n with (2) gives vr,n and hence its derivative at the wall.The boundary points of the latter are the values of column j of Mn . The above procedure is repeated for

each j .

For n= 0, the method cannot determine the dependence of p0

r=1 on z, because only the derivative

p0/rappears in (10). As an additional condition, the finite difference form of p0/z

r=1,z=ziis required

to vanish. The condition on the grid point preceding the outflow boundary is dropped in favour of this con-

straint, because otherwise Mn would not be a square matrix and could not be inverted. That last grid point

being situated in the damping zone, this modification has no impact on the satisfaction of (8).

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