

JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

44, 2, pp. 405-420, Warsaw 2006

NUMERICAL INVESTIGATION OF TRANSITIONAL FLOW
IN CO- AND COUNTER-ROTATING ANNULAR CAVITY

Ewa Tuliszka-Sznitko

Artur Zieliński

Institute of Thermal Engineering, Technical University of Poznań

e-mail: sznitko@sol.put.poznan.pl

A3Ddirect numerical simulation is performed to study the transitional flow
in co- and counter-rotating annular cavity of the aspect ratio 5. The identifi-
cation and characterization of mechanisms related to the laminar-turbulent
process in rotating cavities should improve the predictionmethods and lead
tonew,more effectiveboundary layercontrol strategies.Numerical computa-
tions arebasedon thepseudo-spectralChebyshev-Fouriermethod for solving
the incompressible Navier-Stokes equation. The numerical computations al-
low one to describe the steady axisymmetric basic state and 3D instability
structureswhich appear in the transitional flow.TheDNS computations are
interpreted in the light of ourLSA results.Moreover, the absolute instability
regions are theoretically identified and the critical Reynolds numbers of the
convective/absolute transition in both layers are given.

Key words: laminar-turbulent transition, rotating cavity, direct method

1. Introduction

The flow in rotating disk systems is not only a subject of fundamental
interest but it is also a topic of practical importance. Typical configurations
include cavities between rotating compressor and turbine disks. Numerous
works have been recently devoted to the investigation of instabilities associa-
ted with flows past single and differentially rotating disks, Serre et al. (2001,
2004), Lingwood (1997), Tuliszka-Sznitko and Soong (2000), Tuliszka-Sznitko
et al. (2002), Itoh (1991). For high rotation rates, flow in a rotor/stator cavity
consists of two boundary layers of the Ekman type on the rotating disk and of
the Bödewadt type on the stationary disk, separated by an inviscid rotating
core. The transition process in both layers is related to type I and type II


406 E. Tuliszka-Sznitko, A. Zieliński

generic linear instabilities. Type I instability is the inviscid instability. The
mechanism of type II instability is related to combined effects of Coriolis and
viscous forces. The instability structures in the rotating cavity were investiga-
ted numerically and experimentally byDijkstra and vanHeijst (1983), Szeri et
al. (1983), Serre et al. (2001, 2004), Lopez et al. (2002), Gauthier et al. (2002),
Schouveiler (1999, 2002) and others. The stability of the counter-rotating disk
cavity of the aspect ratio L > 10 was studied by Szeri et al. (1983). The
authors reported the existence of cylindrical and spiral vortices in boundary
layers with the core itself remaining stable. Experimental investigations on in-
stability of the co- andcounter-rotating cylindrical cavity (L=20.9)were also
carried out by Gauthier et al. (2002). For the counter-rotating case, Gauthier
has found a new instability pattern, which he called negative spirals.

Non-isothermal flow conditions were also considered (Mochizuki et al.,
1983; Tuliszka-Sznitko and Soong, 2000; Soong 1996), showing that thermal
effects and the rotation-induced buoyancy become influential on stability cha-
racteristics and on critical conditions.
In the present paper, three-dimensional direct numerical calculations are

performed for the annular co- and counter-rotating cavity of the aspect ratio
L=5 and curvature parameters Rm=1.5 and 3.0.We study spatial structu-
res which appear in boundary layers of both disks as well as the time depen-
dence of these flows. Theoretical investigations (LSA) are also performed in
order to enlighten the DNS results with respect to I and II instabilities. The
geometrical and mathematical models are described in Section 2. The DNS
and LSA numerical solution techniques are described in Sections 3 and 4, re-
spectively. In Sections 5 and 6, the results obtained from DNS methods are
analyzed.The theoretically identified absolute instability regions are discussed
in Section 7.

2. Geometrical model, physical parameters and equations (DNS)

The geometrical model is a co- and counter-rotating cavity (Fig.1). The
outer cylinder of radius R1, is attached to the stationary or slower rotating
disk. The inner one of radius R0 is attached to the faster rotating disk. The
faster rotating disk rotates at a uniform angular velocity Ω1 =Ω1ez, where
ez is the unit vector. The slower rotating disk rotates at a angular velocity
Ω2 = sΩ1. Positive s means that both disks rotate in the same direction
and negative s means that disks rotate in opposite directions. The flow is
controlled by the following physical parameters: the Reynolds number based


Numerical investigation of transitional flow... 407

on the external radiusof thedisks andon theangular velocity of faster rotating
disk Ω1, Re=R

2
1Ω1/ν, the local Reynolds number Reδ = r

∗/δ =
√
r∗2Ω/ν,

the aspect ratio L = (R1 −R0)/2h and the curvature parameter Rm =
= (R1 +R0)/(R1 −R0). In the present study, calculations are performed for
L=5, Rm=3.0 and 1.5 and for s from −0.2 to 0.5.

Fig. 1. Schematic picture of the rotating cavity

The governing equations are 3D Navier-Stokes equations, written in a
velocity-pressure formulation together with the continuity equation

∂V

∂t
=
1

Re
∆V − (V ·∇)V −∇p ∇·V =0 (2.1)

The equations are written in a cylindrical polar coordinate system (r,z,ϕ)
with respect to a stationary frame of reference. In equation (2.1) t is time,
p is pressure, V is the velocity vector and u, w, v are velocity components
in the r, z, and ϕ directions, respectively. The time, space and velocity are
normalized as follows: Ω−11 , h and Ω1R1. The dimensionless axial co-ordinate
is z = z∗/h; z ∈ [−1,1] (asterisks denote dimensional values). The radius
co-ordinate is additionally normalized in order to obtain the domain [−1,1]
requested by the used spectral method, based on the Chebyshev polynomials:
r=(2r∗− (R1+R0))/(R1−R0).
No slip boundary condition is applied to all rigid walls, so u = w = 0.

For the azimuthal velocity component, the boundary conditions are
v = (Rm + r)/(Rm + 1) on the top faster rotating disk, and
v = s(Rm+ r)/(Rm+1) on the slower rotating disk. However, because of
the singularity of the azimuthal velocity component at the junction betwe-
en the slower rotating disk and the inner end-wall, and between the faster
rotating disk and the outer end-wall, these boundary conditions must bemo-
dified. The singularity express a physical situation inwhich there is a gap, for


408 E. Tuliszka-Sznitko, A. Zieliński

instance, between the edge of the rotating disk and stationary end-wall. To
eliminate these singularities, we used the following exponential profiles along
the end-walls r=±1:

a) end-wall attached to the stator
v= [(Rm+r)/(Rm+1)](1−s)exp[(z−1)/0.006+s]

b) end-wall attached to the rotor
v= [(Rm+r)/(Rm+1)](1−s){1− exp[(−z−1)/0.006]+s}

Computations start with a sufficiently low Reynolds number Re=2000 to
obtain a stable flow. The solution is then used as the initial condition for
computation at a higher Reynolds number, with the small increment 500. We
start the process of consecutively increased Re with the zero meridional flow
and a linear distribution of the azimuthal velocity component.

3. Numerical approach (DNS)

The numerical solution is based on a pseudo-spectral collocation
Chebyshev-Fourier-Galerkin approximation. The approximation of the flow
variables Ψ = (u,w,v,p) is given by an expansion into a truncated series
(Serre et al., 2001)

ΨNMK(r,z,ϕ,t) =

K/2−1∑

p=K/2

N∑

n=0

M∑

m=0

Ψ̂nmp(t)Tn(r)Tm(z)e
ipϕ

−1¬ r,z¬ 1

0¬ϕ¬ 2π

(3.1)
where N, M and K are the numbers of collocation points in the ra-
dial, axial and azimuthal directions, respectively. Tn(r) = cos(narccosr)
and Tm(z) = cos(marccosz) are Chebyshev polynomials, ϕk = 2πk/K,
k = 0,1,2, . . . ,K − 1. Computations were performed for N = 79, M = 49
and K = 95. The time scheme is semi-implicit and second-order accurate.
It corresponds to a combination of the second-order backward differentiation
formula for the viscous diffusion term, and the Adams-Bashforth scheme for
the non-linear terms. The method uses a projection scheme to maintain the
incompressibility constraint. More information the reader can find in papers
by Serre et al. (2001, 2004).

The behavior of the dependent variables is monitored at 15 points, in five
different positions in the radial direction N (1/6, 1/3, 1/2, 2/3, 5/6) and in
three positions in the axial direction M (9/10, 1/2, 1/10); Fig.1.


Numerical investigation of transitional flow... 409

4. Linear stability analysis (LSA)

The LSA computations were performed for the cylindrical rotor/rotor and
rotor/stator cavity of an infinite radius. The disturbance equations were ob-
tained by expressing the velocity and pressure fields as superposition of the
basic state and perturbation field. A similarity model of the flowwas used for
generation of axially symmetric solutions of the basic state (Tuliszka-Sznitko
and Soong, 2000; Tuliszka-Sznitko et al., 2002).

We assume that the perturbation quantities have the following normal-
mode form

[u′,v′,w′,p′]> = [û, v̂, ŵ, p̂]> exp(α∗r∗+mϕ−ω∗t∗)+ cc (4.1)

where û, ŵ, v̂ are the dimensional amplitudes of the three components of ve-
locity in the radial, axial and azimuthal directions, respectively, p̂ is pressure,
α∗ and β∗ = m/r∗ are the dimensional components of the wave number k∗

in the radial and azimuthal directions, respectively, ω∗ is the dimensional fre-
quency and t∗ is time. The co-ordinate system is located on the disk under
consideration. The linear stability equations plus the homogeneous bounda-
ry conditions (û(z∗) = v̂(z∗) = ŵ(z∗) = τ̂(z∗) = 0 at z∗ = 0 and z∗ = 2h)
constitute an eigenvalue problemwhich is solved in a globalmanner (Tuliszka-
Sznitko and Soong, 2000; Tuliszka-Sznitko et al., 2002). As in the DNS com-
putations, a spectral collocation method based on Chebyshev polynomials is
used for discretization of the LSA equations. The LSA is used to determine
the absolute/convective character of the boundary layer in the rotating cavity.

5. Basic state

In this Section we analyze the basic state obtained from the DNS com-
putations for L = 5, Rm = 3.0, 1,5 and for s ranging from −0.2 to 0.5.
The basic state solution is steady and axisymmetric. Figures 2a,b,c show the
velocity field in the meridional section (r∗/h,z∗/h,0.0), obtained for s=0.5
(Re = 94000), s = 0.0 (Re = 26000) and s = −0.2 for (Re = 2250). From
Fig.2a andFig.2b,we can see that the fluid is pumped radially outward along
the faster rotating disk (the upper one) and radially inward along the slower
rotating disk. For an arbitrary positive and small negative rotational rate s,
the flow is roughly the same as in the rotor-stator case: the flow consists of
two disjoint boundary layers on each disk and of the central core flow. For


410 E. Tuliszka-Sznitko, A. Zieliński

larger negative value of s (for instance s=−0.2), the flow is organized in a
two-cell structure. The centrifugal flow, induced by rotation of the upper disk,
recirculate along the bottomdisk towards the inner end-wall. This inward flow
meets the outward radial flow, induced by the rotation of the bottom disk, le-
ading to creation of a stagnation area where the radial component of velocity
vanishes.

Fig. 2. The velocity field in the plane (r∗/h,z∗/h,0.0) obtained for: (a) s=0.5
(Re=94000), (b) s=0.0, (Re=26000), (c) s=−0.2 for (Re=22500); L=5,

Rm=3.0; DNS

From Fig.2, we can see that the thickness of the bottom boundary layer
for s = −0.2 and s = 0.0 clearly depends on the radius. The bottom disk
boundary layer becomes more horizontal for the co-rotating case, s = 0.5.
The dimensionless thickness of the bottom boundary layer δ∗/(ν/Ω) in terms
of r∗/h, obtained for different s and Re, is illustrated in Fig.3. For s=−0.2,
the bottom disk boundary layer only exists for a larger radius; the thickness
of the boundary layer increases towards the inner cylinder rapidly and finally
reaches the upper disk boundary layer (at r∗/h ∼ 12, Fig.2c). At a slightly
smaller r∗/h∼ 11.6, the stagnation area appears.


Numerical investigation of transitional flow... 411

The two cell recirculating structure was investigated by Dijkstra and van
Heijst (1983), Gauthier et al. (2002) and others.

Fig. 3. Boundary layer thickness δ/
√
ν/Ω1 of the slower rotating disk in terms of

the radial position r∗/R1 obtained for s=0.5 (Re=94000), s=0.3 (Re=44000),
s=0.15 (Re=34000), s=0.0 (Re=26000) and s=−0.2 for (Re=22400);

L=5,Rm=3.0, DNS

6. Instability structures

For all considered cases, we have found the same instability patterns in
the stationary or slower rotating disk’s boundary layer. With a fixed s, we
increased Re, and over a certain Reynolds number we observed propagating
cylindrical vortices in the slower rotating disk. Cylindrical vortices (2D), in-
terpreted as type II instability, appear in the middle area of the boundary
layer of the slower rotating disk and propagate towards the inner end-wall.
Above the second critical value of Re, 3D spiral structures appear in the area
near the outer end-wall. The spiral vortices, interpreted as type I instability,
are observed for higher r∗/h. Gauthier et al. (2002) called these spirals ”po-
sitive vortices” since they roll up to the centre in the direction of the faster
rotating disk. These spiral vortices were studied extensively in the rotor-stator
configuration among others by Schouveiler et al. (1999, 2002).
In Fig.4 the isolines of the azimuthal velocity component disturbances, in

the azimuthal section (r∗/h,z∗/h = −0.95,ϕ) and in the meridional section
(r∗/h,z∗/h,ϕ = 0) obtained for Re = 27000, Re = 30000, Re = 50000, re-
spectively, are presented (s = 0.0,L = 5,Rm = 3.0). We can see that for
Re=27000 the cylindrical vortices co-exist with the spiral vortices. We obse-
rve two pairs of cylindrical waves and 19 spiral vortices. For higher Reynolds


412 E. Tuliszka-Sznitko, A. Zieliński

Fig. 4. Isolines of azimuthal velocity component disturbances in the azimuthal
section (r∗/h,z∗/h=−0.95,ϕ) and in the meridional section (r∗/h,z∗/h,ϕ=0)
obtained for Re=27000, Re=30000, Re=50000, s=0,L=5,Rm=3.0; DNS


Numerical investigation of transitional flow... 413

numbers (Fig.4b), type I instability is fully dominant. For consequently in-
creasing Re, the flow becomesmore andmore disordered.We can notice that
the turbulence starts from the inner end wall (Fig.4c).

An exemplary instability characteristic of the azimuthal velocity compo-
nent, obtained in the monitored point M in the stator boundary layer for
Re=30000 is presented in Fig.5. The obtained solution is oscillatory and has
the angular frequency σ = 2π/T = 1.9, where T is the period of oscillation.
For L=5, Rm=3.0, s=0.0, the transition to unsteadiness is supercritical.

Fig. 5. The instability characteristic of the azimuthal velocity component obtained
in the monitored point M in the stator boundary layer. L=5,Rm=3.0, s=0,

Re=30000; DNS

For the co-rotating cases, the critical Reynolds number of the transition to
unsteadiness increased from Re=27000 for s=0up to 98000 for s=0.5. For
the counter-rotating case, the flow is significantly more unstable than for the
rotor-stator case. For s=−0.2, the critical Reynolds number of transition to
unsteadiness equals 23500. The influence of s on the critical Reynolds number
of transition to unsteadiness is presented in Fig.6.

The instability pattern obtained for the considered negative values of s
in the bottom disk’s boundary layer is the same as for the rotor-stator and
co-rotating flow. Figure 7a shows isolines of azimuthal velocity component
disturbances in the azimuthal plane (r∗/h,z∗/h = −0.95,ϕ) obtained at
Re = 24000, Rm = 3.0, L = 5 and s = −0.2. As in the rotor-stator cavi-
ty (s=0.0), also for s=−0.2 we obtained 19 spiral vortices and two pairs of
cylindrical vortices. The isolines of axial velocity component disturbances in
themeridional section (r∗/h,z∗/h,ϕ=0) are presented inFig.7b.We can see
from Fig.7b that in the inner end-wall area, the structures fill the whole cell.
The instability structures in the boundary layers of the slower rotating disk
are observed for r∗/h > 12.5. The instability characteristic of the azimuthal


414 E. Tuliszka-Sznitko, A. Zieliński

Fig. 6. The critical Reynolds number of transition to unsteadiness in terms of s;
L=5,Rm=3; DNS

Fig. 7. Isolines of azimuthal velocity component disturbances in the
(r∗/h,z∗/h=−0.95,ϕ) plane (a), isolines of axial velocity component disturbances
in the meridional plane (r∗/h,z∗/h,ϕ=0.0) (b), instability characteristics of the
azimuthal velocity component obtained in the monitored point M (c), Re=24000,

L=5,Rm=3.0, s=−0.2; DNS


Numerical investigation of transitional flow... 415

velocity component, obtained in the monitored point M in the stator boun-
dary layer (L= 5,Rm = 3.0,s =−0.2), is presented in Fig.7c. The angular
frequency of oscillations is equal to σ=2π/T =2.
For Rm = 1.5, L = 5, we obtained similar instability structures as for

Rm = 3.0, L = 5. We observed two cylindrical and twelve spiral vorti-
ces. Exemplary isolines of azimuthal velocity component disturbances in the
(r∗/h,z∗/h=−0.95,ϕ) plane and isolines of axial velocity component distur-
bances in the (r∗/h,z∗/h,ϕ=0.0) plane, obtained for s=0,Re=26000, are
presented in Fig.8a and Fig.8b, respectively.

Fig. 8. Isolines of fluctuations of the axial velocity component at ReR =26000,
Rm=1.5,L=5, s=0.0, (a) azimuthal plane (r∗/h,z∗/h=−0.95,ϕ),

(b) meridional plane (r∗/h,z∗/h,ϕ=0.0); DNS

We want to emphasize that the results were obtained for the inner end-
wall attached to the faster rotating disk and the outer one attached to the
slower rotating disk. This configuration was chosen because it was the most
unstable. Our computations showed that the end-wall boundary conditions
strongly influence the instability of boundary layers.

7. Absolute instability

The LSA results (Tuliszka-Sznitko and Soong, 2000) showed the existen-
ce of absolutely unstable areas in both stationary and rotating disk boundary
layers of cylindrical rotor/stator cavities, with the critical localReynolds num-
bers equal to Reδca = r

∗/δ=
√
r∗2Ω/ν =48.5 and Reδca =562, respectively.

In Tuliszka-Sznitko and Soong (2000), the Briggs (1964) criterion was used to
determine the regions of absolute instability. The critical Reynolds numbers of
type II instability in the rotor and stator boundary layers were established at


416 E. Tuliszka-Sznitko, A. Zieliński

ReδcII =90.23 and 35.5, respectively. The critical Reynolds number of type I
instability in the boundary layer of the rotating disk equaled ReδcI = 278.6.
The critical Reynolds number of absolutely unstable area Reδca =48.5, in the
boundary layer of the stationary disk, was very close to the critical Reynolds
number of the convectively unstable area ReδcI =47 (type I).TheLSAresults
(Tuliszka-Sznitko and Soong, 2000) showed that almost the whole convective-
ly unstable area in the boundary layer of the stationary disk, was absolutely
unstable.

Animations of the spatio-temporal behavior of the disturbances, generated
by the obtained DNS results, showed that type II and I structures propa-
gate in the boundary layer of the stationary disk in the opposite directions.
Cylindrical disturbances (type II) propagate towards the inner end-wall in ac-
cordance with the direction of the basic state flow, whereas type I structures
propagate in the opposite direction towards the outer end-wall. This behavior
could suggest (in accordance with the LSA results) that the area of domi-
nance of type I instability in the stator boundary layer may be absolutely
unstable.

To complete our previous LSA computations obtained for the rotor/stator
cavity (Tuliszka-Sznitko and Soong, 2000), we performed LSA computations
for co-rotating rotor/rotor configurations. We found that the critical local
Reynoldsnumber of type I instability increases in the slower rotating disk from
ReδcI =47 for s=0.0 to ReδcI =770 for s=0.75 and in the faster rotating
disk from ReδcI = 278.6 to ReδcI = 930. The critical Reynolds number of
type II instability increases in the slower rotating disk from ReδcII = 35.5
for s=0.0 to ReδcII =405 for s=0.75 and in the faster rotating disk from
ReδcII =90.23 to ReδcII =419. These results are in good agreementwith Itoh
(1991).We also observed rapid increase in the critical Reynolds number of the
absolutely unstable area with increasing s in both boundary layers; for the
faster rotating disk it increases from Reδca =562 for s=0.0 to Reδca =1685
for s = 0.75. Exemplary neutral curves in the plane (σ,Reδ) obtained for
s = 0.0, 0.5, 0.75 and 0.9 in the boundary layer of the faster rotating disk
are presented in Fig.9. The critical local Reynolds numbers ReδcI, ReδcII and
Reδca in termsof s obtained for the faster rotatingdisk are presented inFig.10.
From Fig.9 and Fig.10, we can see that with increasing s the flow becomes
more and more stable, and the absolutely unstable flow vanishes. These LSA
results are in good agreement with our DNS results presented in Fig.6 where
we observe a rapid increase in the critical Reynolds number of transition to
unsteadiness with increasing s.


Numerical investigation of transitional flow... 417

Fig. 9. Exemplary neutral curves in the plane (σ,ReδI) obtained for s=0.0, 0.5,
0.75 and 0.9 in the faster rotating disk; LSA

8. Conclusions

An incompressible 3D fluid flow between co- and counter-rotating disks
with enclosing inner and outer cylinders attached to faster and slower rotating
disks, respectively, was investigated numerically using the DNS method. We


418 E. Tuliszka-Sznitko, A. Zieliński

Fig. 10. Critical local Reynolds numbers ReδcI, ReδcII and Reδca in terms of s
obtained for the boundary layer of the faster rotating disk; LSA

investigated an annular cavity of the aspect ratio L=5 and Rm=1.5, 3.0.
The computations showed complexity of the flow. In the stationary or, slower
rotating disk’s boundary layer two different patterns of instability structures
were observed and described: two-dimensional vortices propagating towards
the inner end-wall along the slower rotating disk and three-dimensional spi-
ral vortices propagating in the opposite direction. We analyzed the counter-
rotating computations in the light of Gauthier’s et al. (2002) experimental
results, obtained for a cylindrical cavity of the aspect ratio L=20.9. We re-
ceived the same structure of the basic state with a stagnation circle in which
the radial component of the velocity vanishes.

The animation of the flow, based on the obtained DNS results, showed
that type II instability structures in the boundary layer of the stationary
disk propagate downstream towards the inner end-wall, whereas type I pro-
pagates in the opposite direction (up-stream). It could possibly indicate (in
accordance with previous LSA results, see Tuliszka-Sznitko and Soong, 2000)
that partly the stator boundary layermay be absolutely unstable.Our present
LSA computation carried out for a co-rotating rotor/rotor cavity proved that
with increasing s the flow becomes more and more stable and the absolutely
unstable area slowly vanishes.

Acknowledgements

Thecomputationswerecarriedoutat theSupercomputingandNetworkingCenter

in Poznań within the framework of the KBN grant 4T07A04528 supported by the

State Committee for Scientific Research.


Numerical investigation of transitional flow... 419

References

1. Briggs R.J., 1964, Electron-Steam Interaction with Plasmas, Chap.2, MTT
Press

2. Dijkstra D., van Heijst G., 1983, The flow between finite rotating discs
enclosed by a cylinder, J. Fluid Mech., 128, 123, 123-154

3. Gauthier G., Gondret P., Moisy F., Rabaut M., 2002, Instabilities in
the flow between co- and counter-rotating disks, J. Fluid Mech., 473, 1-21

4. Itoh M., 1991, On the instability of the flow between coaxial rotating disks,
In: Boundary Layer Stability and Transition to Turbulence, ASME FED, 114,
83

5. Lingwood R., 1997, Absolute instability of the Ekman layer and related ro-
tating flows, J. Fluid. Mech., 331, 405, 405-428

6. Lopez J., Hart J., Marques F., Kittelman S., Shen J., 2002, Instabi-
lity and mode interactions in a differentially driven rotating cylinder, J. Fluid
Mech., 462, 383-409

7. Mochizuki et al., 1983,Heat transfermechanismsandperformance inmultiple
parallel disk assemblies,Trans. ASME, J. Heat Transfer, 105, 598-604

8. OwenJ.,RogersR., 1995,Heat transfer in rotatingdisc systems, In:Rotating
Cavity, Vol.2,W.D. Morris (edit.), Wiley

9. Schouveiler L., Le Gal P., Chauve P., 2002, Instabilities of the flow
between a rotating and stationary disk, J. Fluid Mech., 443, 329-350

10. Schouveiler L., Le Gal P., Chauve P., Takeda Y., 1999, Spiral and
circularwaves in the flowbetweena rotating anda stationarydisc,Experiments
in Fluids, 26, 179

11. Serre E., Crespo del Arco E., Bontoux J., 2001, Annular and spiral
patters in flowbetween rotating and stationary disks, J. Fluid. Mech., 434, 65,
65-100

12. Serre E., Tuliszka-Sznitko E., Bontoux P., 2004, Coupled numerical
and theoretical study of the flow transition between a rotating and stationary
disk, Phys. of Fluids, 16, 3

13. Soong C.Y., 1996, Theoretical analysis for axisymmetric mixed convention
between rotating coaxial disks, Int. J. Heat Mass Transfer, 39, 8, 1569-1583

14. Szeri A., Giron A., Schneider S., Kaufman H., 1983, Flow between ro-
tating disks, Part II. Stability, J. Fluid Mech., 443, 329-350

15. Tuliszka-Sznitko E, Soong C.Y., 2000, Instability of non-isothermal flow
between coaxial rotating disks, Proceedings of European Congress on Compu-
tational Methods in Applied Sciences and Engineering, Barcelona


420 E. Tuliszka-Sznitko, A. Zieliński

16. Tuliszka-Sznitko E., Serre E., Bontoux P., 2002, On the nature of the
boundary layers instabilities in a flowbetween a rotating and a stationary disc,
C.R. Mecanique, 30, 90-99

Badanie przejścia laminarno-turbulentnego pomiędzy dyskami
wirującymi w konfiguracji wirnik/wirnik

Streszczenie

W pracy badany jest metodą bezpośrednią przepływ nieściśliwy w przestrzeni
pomiędzy dwoma wirującymi tarczami i dwoma wirującymi pierścieniami. Do obli-
czeń numerycznych zastosowanometodę spektralnej kolokacji bazującą na szeregach
Czebyszewa i szereguFouriera.Obliczenia przeprowadzonodla współczynnika rozcią-
głości obszaruL=5 i współczynnika krzywizny Rm= 1.5 i 3.0. Badania wykazały
olbrzymiązłożonośćbadanegoprzepływu i różnorodność strukturniestabilnościowych
występujących w obszarze przejścia laminarno-turbulentnego. Badania bezpośrednie
(DNS) uzupełnione zostałybadaniami teoretycznymi (LSA), któreumożliwiłymiędzy
innymi wyznaczenia w przestrzeni pomiędzy wirującymi tarczami obszarów o niesta-
bilności absolutnej.

Manuscript received August 31, 2005; accepted for print December 7, 2005