Lower-branch travelling waves and transition to turbulence in pipe flow
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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276 J. Reuter, D. Rempfer
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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Analysis of pipe flow transition. Part I. Direct numerical simulation 277
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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278 J. Reuter, D. Rempfer
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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Analysis of pipe flow transition. Part I. Direct numerical simulation 279
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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280 J. Reuter, D. Rempfer
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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Analysis of pipe flow transition. Part I. Direct numerical simulation 281
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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282 J. Reuter, D. Rempfer
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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Analysis of pipe flow transition. Part I. Direct numerical simulation 285
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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Analysis of pipe flow transition. Part I. Direct numerical simulation 287
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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290 J. Reuter, D. Rempfer
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
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Analysis of pipe flow transition. Part I. Direct numerical simulation 291
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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