Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 4, pp. 1217-1232, Warsaw 2011

NUMERICAL SIMULATION OF TURBULENT HEAT AND

MOMENTUM TRANSPORT IN ROTATING CAVITY1

Ewa Tuliszka-Sznitko

Wojciech Majchrowski

Kamil Kiełczewski

Pozna University of Technology, Institute of Thermal Engineering, Poznań, Poland

e-mail: ewa.tuliszka-sznitko@put.poznan.pl

The paper gives the results of the Direct Numerical Simulation (DNS )
and Large Eddy simulation (LES) which were performed to investigate
the 3D transitional non-isothermal flows within a rotor/stator cavity. A
Lagrangean version of the dynamic Smagorinsky eddy viscosity model
was used. Computations were performed for the cavity of the aspect
ratio L=3.0-5.0, curvature parameters R

m
=1.8-5.0, for the Reynolds

number Re = (1.0-2.5) · 105 and for different Prandtl numbers. The
results were obtained for coupled momentum and thermal transport in
the rotor/stator cavity flows.The obtaineddistributions of the turbulent
heat flux tensor components, theReynolds stress tensor components, the
turbulentPrandtl number andother structural parameters coincidewith
the experimental data (published in the literature).

Key words: laminar-turbulent transition, rotating cavities, LES, DNS

1. Introduction

The instability structures of the flow in the rotor/stator and rotor/rotor ca-
vity have been investigated since the sixties of the last century, mostly with
the reference to applications in turbomachinery. It is also a very interesting
fundamental problem: the flow between rotating disks is one of the simplest
3D flows, highly suitable for investigating the influence of mean flow para-
meters on transitional and turbulence structures. The flow in rotor/stator
cavity was investigated experimentally and numerically among others by An-
dersson and Lygren (2006), Gauthier et al. (2002), Moisy et al. (2004), Serre
andPulicani (2001), Serre et al. (2004), Lygren andAnderson (2004), Séverac

1
Paper presented at the XIXPolish National Fluid Dynamics Conference



1218 E. Tuliszka-Sznitko, W. Majchrowski

and Serre (2007), Tuliszka-Sznitko et al. (2002, 2008, 2009a,b, 2010). Lygren
and Anderson (2004) performed the LES of the flow in an open rotor/stator
cavity using three different models, and compared the results with those ob-
tained by DNS. Séverac and Serre (2007) performed numerical computations
for the enclosed cavity for the Reynolds number up to Re = ΩR21/ν = 10

6

(L=(R1−R0)/2h=5.0, Rm=(R1+R0)/(R1−R0)= 1.8) using the Spec-
tral Vanishing Viscosity (SVV) method, and compared the results with their
LDV experimental data. The non-isothermal flow conditions were also consi-
dered in some investigations (Tuliszka-Sznitko et al. 2002, 2008, 2009a, 2010;
Randriamampianina et al., 1987), which showed that the thermal effects and
the rotation-induced buoyancy influence the stability characteristics and the
critical conditions. Tuliszka-Sznitko et al. (2009a) performed the LES of the
non-isothermal flow in the rotor/stator cavity, delivering distributions of the
local Nusselt numbers along the stator and rotor for different configurations
and Reynolds numbers. Pellé and Harmand (2007) performed measurements
over the rotor (in the rotor/stator configuration), using a technique based on
infrared thermography. A very detailed experimental investigation of the tur-
bulent flow around a single heated rotating disk was performed by Elkins and
Eaton (2000).

The flow in the cavity between two disks heated from below (the Rayleigh
Bénard convection) with superimposed moderate rotation is used as a mo-
del problem for predicting geophysical phenomena (solar and giant planetary
convection, deep oceanic convection). The flow with moderate rotation un-
dergoes a series of consecutive bifurcations starting with unstable convection
rolls at moderate Rayleigh numbers. The transition culminates at a state do-
minated by coherent plume structures. The Rayeligh-Bénard convection with
superimposed rotation has been studied in rotor/stator cavity among others
by Kunnen et al. (2005).

In the present paper, we investigate the flow in the rotor/stator cavi-
ties of the aspect ratio from the range L = 3.0-5.0, curvature parameters
Rm = 1.8-5.0, using the DNS and LES. The objective of our investigations
has been to compute the turbulent heat flux tensor components, theReynolds
stress tensor components, the turbulent Prandtl number and other structu-
ral parameters. The obtained data can be useful for heat transfer modeling,
but more importantly, such computations can deliver information about the
influence of three dimensionality of the mean flow on the turbulence structu-
re. The results are compared to data obtained in experimental investigation
of the flow with heat transfer over the heated rotating disk performed by
Elkins and Eaton (2000). The present paper results are also compared to the



Numerical simulation of turbulent heat... 1219

publisheddata obtained for heated two-dimensional turbulentboundary layers
(Wroblewski and Eibeck, 1990; Blair and Bennett, 1987).

The selected Rayleigh-Bénard convection results (with superimposed rota-
tion) are presented in Section 7.

2. Mathematical formulation

We investigate non-isothermal flows in the cavity between stationary and ro-
tating disks of the inner and outer radius R0 and R1, respectively. The inter-
disks spacing is denoted by 2h (Fig.1). The rotor rotates at a uniformangular

Fig. 1. Schematic picture of computational domain, meridian section

velocity Ω=Ωez, ez being the unit vector on the axis. The flow is described
by theNavier-Stokes, continuity and energy equations, written in a cylindrical
coordinate system (R,ϕ,Z) with respect to the rotating frame of reference

∇·V =0

ρ
∂V

∂t
+ρ(V ·∇)V +ρΩ× (Ω×R)+2ρΩ×V =−∇P +µ∆V − qρZ

(2.1)
∂T

∂t
+(V ·∇)T = a∆T

where t is dimensional time, R – radius, P – pressure, ρ – density,
V –velocity vector, a– thermaldiffusivity and µ is thedynamicviscosity.The
flow is governed by the following dimensionless geometrical parameters: aspect
ratio L=(R1−R0)/2h and curvature parameter Rm=(R1+R0)/(R1−R0).
The dimensionless axial and radial coordinates are: z = Z/h, z ∈ [−1,1],



1220 E. Tuliszka-Sznitko, W. Majchrowski

r=(2R−(R1+R0))/(R1−R0), r∈ [−1,1]. To take into account the buoyan-
cy effects induced by the involved body forces, the Boussinesq approximation
is used, i.e. the density associated with the terms of centrifugal and Corio-
lis forces due to disk rotation, curvilinear motion of the fluid and the Earth
acceleration is considered to be variable.

In the paper we consider the flow cases dominated by the centrifugal
and Coriolis forces in which the Earth acceleration is negligibly small (the
Rayleigh-Bénard convection in which there is a fixed ratio between the rota-
tional and thermal buoyancy is considered only in Section 7). For the flow
cases dominated by the rotational buoyancy, the velocity components and ti-
me are normalized as follows: ΩR1, (Ω)

−1. The governing parameters are: the
Reynolds number Re = ΩR21/ν, thermal Rossby number B = β(T2 −T1),
where β = −1/ρr(∂ρ/∂T)p, T1 and T2 are two chosen reference tempe-
ratures. The dimensionless temperature is defined in the following manner:
Θ = (T −T1)/(T2 −T1). The dimensionless components of the velocity vec-
tor in the radial, azimuthal and axial directions are denoted by u, v, w and
dimensionless pressure is denoted by p. The no-slip boundary conditions are
used with respect to all rigid walls, u = w = 0. For the azimuthal velocity
component, the boundary conditions are as follows: v=0on the rotating disk
and v=−(Rm+r)/(Rm+1) on the stator. In the paper, we consider the fol-
lowing configuration: the rotating upper disk is attached to the inner cylinder
and the heated stator is attached to the outer cylinder. T1 is the temperature
of the upper disk and the inner cylinder, and T2 is the temperature of the
bottom disk and the outer cylinder. The thermal boundary conditions are as
follows: Θ=1 for z =−1.0 and for −1.0¬ r ¬ 1.0, Θ=0 for z =1.0 and
for −1.0 ¬ r ¬ 1.0, Θ = 1 for r = 1.0 and for −1.0 ¬ z ¬ 1.0, Θ = 0 for
r = −1.0 and for −1.0 ¬ z ¬ 1.0. This configuration was chosen because is
themost unstable. For higherReynolds numbers, computations are performed
only for a section of cavity (for example 0 ¬ ϕ ¬ π/4) with the periodicity
condition in the azimuthal direction (Fig.2).

3. Numerical approach

In theLESwe use a version of the dynamic Smagorinsky eddy viscositymodel
proposed byMeneveau et al. (1996), inwhich the required averaging is perfor-
medover thefluidparticles pathlines, instead of averaging over thedirection of
statistical homogeneity. The Smagorinsky coefficient is determined byminimi-
zing themodeling error over the pathlines of the fluid particles. Thenumerical



Numerical simulation of turbulent heat... 1221

Fig. 2. Section of computational domain

algorithm used for the LES of the non-isothermal flow in the annular cavity,
proposed in the papers Tuliszka-Sznitko et al. (2008, 2009a,b), is an extended
version of the DNS algorithm developed by Serre and Pulicani (2001). The
numerical solution is based on a pseudo-spectral Chebyshev-Fourier-Galerkin
collocation approximation. In the time approximation we use a second-order
semi-implicit scheme, which combines an implicit treatment of the diffusive
terms and an explicit Adams-Bashforth extrapolation for the non-linear co-
nvective terms. In the non-homogeneous radial and azimuthal directions, Che-
byshev polynomials are used with the Gauss-Lobatto distributions to ensure
high accuracy of the solution inside the very narrow boundary layers at the
disks.

4. Mean flow

For all considered Reynolds numbers 105 ¬ Re ¬ 2.5 · 105 the flow exhibits
typical Batchelor behavior, whichmeans that the flow consists of two disjoint
boundary layers on each disk and of a central inviscid core flow. The flow
is pumped radially outward along the rotor and recirculates along the sta-
tor. Positive thermal Rossby number B> 0 means that the buoyancy-driven
secondary flow enforces the basic rotation-driven flow. In the transitional bo-
undary layers, theaxisymmetric propagatingvortices interpretedas the type II
instability andpositive spiral vortices interpreted as the type I instabilitywere
observed. For higher Re (considered in the present paper) the spiral vortices
evolve to more annular vortices.

Figure 3presents the iso-lines of dimensionless temperature in themeridian
section (Rm = 5, L = 5, Re = 105, B = 0.1). From Fig.3 we can see that



1222 E. Tuliszka-Sznitko, W. Majchrowski

the flow is propagated radially outward along the cooled rotor, then it is
transported down to the stator along the heated stationary outer cylinder.
The flow recirculates along the heated stator and finally is lifted up along the
cooled rotating inner cylinder. We observe the largest temperature gradients
in the area near to the inner and outer cylinders.

Fig. 3. Iso-lines of temperature; Rm=5,L=5, Re=105,B=0.1

Figure 4 presents the profiles of the mean tangential velocity (computed
with respect to the stationary frame of reference) normalized by friction velo-
city uσ = [ν

2((∂u/∂z)2+(∂v/∂z)2)]0.25 in terms of z+ =uσz/ν. The profiles
were obtained in the middle section of the stator boundary layers (cavities of
different L,Rm and Reynolds numbers were considered). In Fig.4, the mean
tangential velocity profiles are compared to the conventional logarithmic law
of the wall with constants κ=0.41 and C =5.0.

Fig. 4. Mean tangential velocity profiles v/u
σ
in terms of z+. Comparison to the

traditional logarithmic law. Profiles obtained in the middle sections of the stator
boundary layers

Figures 5a and5b showmean temperatureprofiles, (T2−T)/Tσ in terms of
z+, obtained in the middle section of the stator boundary layers, where Tσ is
the friction temperature; Tσ =−λ(∂T/∂z)/ρcpuσ. The results are compared
to the traditional thermal lawof thewallwith κ=0.46and C =3.6 (Kaysand
Crawford,1980). InFig.5b, the influenceof thePrandtlnumberon (T2−T)/Tσ
is presented.



Numerical simulation of turbulent heat... 1223

Fig. 5. Distributions of the mean dimensionless temperature (T2−T)/Tσ in terms
of z+. Stator boundary layers.Middle section of different cavities. (a) Pr= 0.71,

(b) Pr= 0.71 and 2.79

5. Turbulent velocity and temperature characteristics

The three main Reynolds stress tensor components obtained in the heated
stator and cooled rotor boundary layers, normalized by friction velocity√
u′u′/uσ,

√
v′v′/uσ,

√
w′w′/uσ are presented in Fig.6. We can see strong

anisotropy of turbulence in both boundary layers. The turbulence is mostly
confined in the stator boundary layer withmaximum at the junction between
the stator and the stationary outer cylinder.We have found that for the same
aspect ratio L and Reynolds number, the Reynolds stress tensor components
increase with the decreasing curvature parameter Rm (Fig.6). Areas of the



1224 E. Tuliszka-Sznitko, W. Majchrowski

most intensive turbulence are also visible in Fig.7, where iso-lines of the axial
velocity component are presented in the section of the whole cavities.

Fig. 6. Reynolds stress tensor components profiles normalized by wall frictions
velocity. Results obtained for different L and Rm; B=0.1, Re=105

Fig. 7. Iso-lines of the axial velocity component obtained for Rm=4, Re=105,
B=0.1 and for different aspect ratios: (a) L=3, (b) L=4



Numerical simulation of turbulent heat... 1225

Figure 8 shows the turbulent temperature fluctuations Θ′/Tσ normalized
by friction temperature in terms of the axial coordinate normalized by the
thickness of the boundary layer z/δ (Re=105). In Fig.8, the present results
are compared to the experimental data obtained by Elkins and Eaton (2000)
for the single heated rotating disk (byBC1andBC2Elkins andEaton denoted
different thermal adiabatic boundary conditions) and to the results obtained
for different two-dimensional boundary layers (Wroblewski and Eibeck, 1990;
Blair and Bennett, 1987). For all our computations we obtained the maxi-
mum value of Θ′/Tσ at z/δ = 0.35-0.45 whereas Elkins and Eaton (2000)
obtained maximum of temperature fluctuations at z/δ ∼ 0.35. In the case
of two-dimensional boundary layers, Θ′/Tσ equals about 1.7 near the wall
and decreases gradually to very small values near the edge of the bounda-
ry layer. In Fig.8, the influence of the Prandtl number on Θ′/Tσ distribu-
tion is presented. Numerical simulations have showed that the temperatu-
re fluctuations reach maximum approximately at z+ ∼ 15 in all considered
cases.

Fig. 8. Distributions of temperature fluctuations Θ′/T
σ
normalized by friction

temperature in terms of z/δ. Re= 105,B=0.1.Middle section of the stator
boundary layers. Comparison to the results obtained by different authors for 2D and

3D boundary layers

Figure 9 shows three components of the turbulent heat flux tensor (norma-
lized by the product of friction velocity and friction temperature) versus z/δ.
We can see that the largest value was obtained for the component v′Θ′/uσTσ
with the maximum at z/δ∼ 0.1.



1226 E. Tuliszka-Sznitko, W. Majchrowski

Fig. 9. Turbulent heat flux tensor components profiles obtained in the middle
section of the cavity. Stator boundary layers. Re=105, Pr= 0.71

6. Structure parameters

Correlation coefficients and structure parameters are very useful for mode-

ling purposes. The (u′2 + v′2)/w′2 parameter is a measure of the coherence
of the turbulent structures. For all analyzed cases, in the stator boundary
layer, this parameter reaches a peak near the disk and then decreases rapi-
dly to the value of about 2 near the edge of the boundary layer, showing
that the vertical motion is very weak close to the disk. Similarly to other 3D
TBLs, the parameter a1 (defined as the ratio of shear stress vector magni-
tude to twice the turbulent kinetic energy) obtained in our investigations is
reduced significantly below the limit of 0.15 typical for 2D TBL, showing
that the boundary layers in rotating cavities are less effective in creating
shear stresses from the turbulent motion. The turbulent Prandtl number is
defined as the ratio of the eddy diffusivity for momentum to the eddy dif-
fusivity for heat Prt = (−w′v′/∂v̄/∂z)/(w′Θ′/∂Θ/∂z). This is not a strict
definition for strongly 3D TBLs, however, we used it to compare our re-
sults to the data published in the literature (Elkins and Eaton, 2000). In
many 2D TBLs, the turbulent Prandtl number equals 1 in the area near the
wall and decreases to 0.8 with increasing z. In Fig.10, we present Prt in
terms of z+. We can see that Prt obtained in the present investigations re-
aches the value of 0.9-1.2 near the wall, then decreases to reach theminimum
at z+ ∼ 15-20.



Numerical simulation of turbulent heat... 1227

Fig. 10. Turbulent Prandtl number in terms of z+ obtained in the middle section.
Stator boundary layer

7. Rayleigh-Bénerd convection with superimposed rotation

In this Section, we consider the flow between two disks heated from be-
low (from stator) with superimposed moderate rotation. The flow is descri-
bed by equations (2.1) but the velocity components are normalized with the
free-fall velocity

√

qβ∆T(2h), time is normalized by a convection time scale
2h/
√

qβ∆T(2h) and temperature by ∆T =T2−T1. Length is normalized as
in Section 2. The final system of equations received after normalization is pre-
sented inAppendixA.We consider the simplest case inwhich the direction of
rotation is alignedwith gravity. Thedynamics of the rotatingRayleigh-Bénard
convection is completely determined by specification of the boundary condi-
tions and by the following governing dimensionless parameters: the Rayleigh
number Ra= qβ∆T(2h)3/νa, theTaylor number Ta= (2Ω(2h)2/ν)2 and the
Prandtl number.Boundary conditions are identical as in Section 2 (no-slip bo-
undary conditions are used with respect to all rigid walls, u=w = 0, v = 0
on the rotating disk and the inner cylinder, and v=−(Rm+r)/(Rm+1) on
the stator and the outer cylinder, isothermal boundary conditionswith heated
stator and outer cylinder). This paper is not intended to analyze carefully the
rotating Bayleigh-Bénard flow properties and statistics. We only would like
to demonstrate how significantly different structure of this flow is in compa-
rison to the flow structure analyzed in previous Sections (when the flow was
fully dominated by centrifugal and Coriolis forces). With increasing Ra, the
flowundergoes a succession of bifurcations before reaching the turbulent state.
The first transition is from the static, conducting state, to a convecting flow
(in Fig.11 the iso-lines of the axial velocity component obtained for cavity
Ro = 0.75, Pr = 0.71, L = 5, Rm= 1.5 and different Ra = 5 ·104, 2.5 ·105
and 3 ·106 are presented). For the Rayleigh number higher than the critical
Ra of the first bifurcation, two-dimensional rolls are observed. In the next



1228 E. Tuliszka-Sznitko, W. Majchrowski

Fig. 11. Iso-lines of the axial velocity component obtained for L=5,Rm=1.5,
Ro=0.75, Pr= 1 and different Rayleigh numbers: (a) Ra=5 ·104,

(b) Ra=2.5 ·105, (c) Ra=3 ·106



Numerical simulation of turbulent heat... 1229

step the oblique rolls oriented at finite angles appear.As Ra increases further,
the periodic convection roll pattern gradually gives way to a state dominated
by the chaotic interaction between vertical vortices associated with convec-
tion cells. At about Ra=2.5 ·105, coherent structures called plumes appear.
We can see that the flow pattern in the rotating Rayleigh-Bénerd convection
is significantly different from the flow structures fully dominated by rotation
where we observe a one-cell structure.

8. Conclusions

In the paper we presented the Direct Numerical Simulation and Large Eddy
Simulation of the non-isothermal transitional and turbulent flow in enclosed
cavities of the aspect ratio L=3.0-5.0 and curvature parameters Rm=1.8-
5.0 with the heated stator and outer cylinder. The computations have been
performed for the thermal Rossby number B=0.1 and for different Reynolds
numbers.
The investigated flows belong to the Batchelor family, which means that

the flows are divided into two boundary layers separated by a central rota-
ting inviscid core.We have found that the fluid turbulence concentrates in the
stator boundary layer and its intensity increases towards the outer cylinder.
The instability structures and the level of turbulence depend on the curvatu-
re parameter. We focused on the analysis of the three Reynolds stress tensor
components and the turbulent heat flux tensor components, which are discus-
sed in the light of experimental results obtained by Elkins and Eaton (2000)
for a single rotating disk heated by uniform flux. The obtained distributions

of the structural parameter (u′2+v′2)/w′2 in the stator and rotor boundary
layers show that vertical movement near the disks is very weak. The turbu-
lent Prandtl number is analyzed in terms of z+; Prt equals 0.9-1.2 near the
wall and decreases almost linearly to the value from the range (0.8-0.9) at
z+ ∼ 15-20.

A. Appendix

Equations (2.1) after the following normalization: velocity components
by the free-fall velocity

√

qβ∆T(2h), time by the convection time scale
2h/
√

qβ∆T(2h) and temperature by ∆T =T2−T1



1230 E. Tuliszka-Sznitko, W. Majchrowski

1

L

∂u

∂r
+

u

(Rm+r)L
+

1

(Rm+r)L

∂v

∂ϕ
+
∂w

∂z
=0 (A.1)

∂u

∂t
+
u

L

∂u

∂r
+

v

L(Rm+r)

∂u

∂ϕ
+w
∂u

∂z
−

v2

L(Rm+r)
+

√

PrTa

4Ra
v

+(Rm+r)L
TaPr

16Ra
=−
1

L

∂p

∂r
+2∂

√

Pr

Ra

[ 1

L2
∂2u

∂r2
+

1

(Rm+r)L2
∂u

∂r
(A.2)

+
1

(Rm+r)2L2
∂2u

∂ϕ2
+
∂2u

∂z2
−

u

L2(Rm+r)2
−

2

L2(Rm+r)2
∂v

∂ϕ

]

∂v

∂t
+
u

L

∂v

∂r
+

v

(Rm+r)L

∂v

∂ϕ
+w
∂v

∂z
+

uv

L(Rm+r)
+

√

TaPr

4Ra
u

=−
1

(Rm+r)L

∂P

∂ϕ
+2

√

Pr

Ra

[ 1

L2
∂2v

∂r2
+

1

(Rm+r)L2
∂v

∂r
(A.3)

+
1

(Rm+r)2L2
∂2v

∂ϕ2
+
∂2v

∂z2
−

v

(Rm+r)2L2
+

2

(Rm+r)2L2
∂u

∂ϕ

]

∂w

∂t
+
u

L

∂w

∂r
+

v

(Rm+r)L

∂w

∂ϕ
+w
∂w

∂z
=−
∂P

∂z
+
Θ

2
(A.4)

+2

√

Pr

Ra

[ 1

L2
∂2w

∂r2
+

1

(Rm+r)L2
∂w

∂r
+

1

(Rm+r)2L2
∂2w

∂ϕ2
+
∂2w

∂z2

]

∂Θ

∂t
+
u

L

∂Θ

∂r
+

v

(Rm+r)L

∂Θ

∂ϕ
+w
∂Θ

∂z
(A.5)

=

√

4

PrRa

[ 1

L2
∂2Θ

∂r2
+

1

(Rm+r)L2
∂Θ

∂r
+

1

(Rm+r)2L2
∂2Θ

∂ϕ2
+
∂2Θ

∂z2

]

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partment

Numeryczna symulacja transportu ciepła i pędu w konfiguracjach

wirujących

Streszczenie

W artykule przedstawiono wyniki symulacji przepływu (z wymianą ciepła) w ob-
szarze pomiędzy stojanem i wirnikiem oraz dwoma pierścieniami uzyskane z za-
stosowaniem metod DNS i LES. Badania przeprowadzono dla rozciągłości obszaru
L = 3.0-5.0 oraz dla współczynnika krzywizny R

m
= 1.8-5.0. Badano struktury

niestabilnościowe występujące w warstwie przyściennej wirnika i stojana oraz profile
osiowe naprężeń reynoldsowskich, fluktuacji temperatury, turbulentnej liczby Prand-
tla, profile parametrów strukturalnych i korelacyjnych. Obliczenia przeprowadzono
dla różnych liczb Reynoldsa i Prandtla. Uzyskane rozwiązania porównano z wynika-
mi badań eksperymentalnych Elkinsa i Eatona (2000) uzyskanymi podczes badania
przepływu wokół pojedynczego wirującego dysku podgrzewanego jednorodnym stru-
mieniem. Rezultaty badań porównywano również z wynikami uzyskanymi dla dwu-
wymiarowych turbulentnych warstw przyściennych.

Manuscript received December 15, 2010; accepted for print February 28, 2011