MEMO No CFD/THERMO-39-2002Abstract: The relaminarization of a fully turbulent flat plate boundary layer subjected to a favorable pressure gradient inside a two-dimensional contraction has been carried out numerically. Two low-Reynolds number isotropic k-~ models are employed in the simulation. The simulated fluid flow and turbulence characteristics are compared with the experiment, demonstrating that the models have good agreement with the data as the flow approaches the laminar state

Rahman, M. M.

Siikonen, T.

Aaltodoc Publication Archive

1Turbulence Characteristics of Flow in a TwoDimensional ContractionM. M. Rahman 1 and T. Siikonen 2Helsinki University of Technology, Department of Mechanical Engineering,Laboratory of Applied Thermodynamics, Sa¨hko¨miehentie 4, FIN-02015 HUTFinlandAbstract — The relaminarization of a fully turbulent flat plate boundary layer subjected to afavorable pressure gradient inside a two-dimensional contraction has been carried out numer-ically. Two low-Reynolds number isotropic k-~ models are employed in the simulation. Thesimulated fluid flow and turbulence characteristics are compared with the experiment, demon-strating that the models have good agreement with the data as the flow approaches the laminarstate.Key Words: Contraction, turbulence, relaminarization, boundary layer.IntroductionPlane contraction is associated with the hydraulic head box utilized in the paper manufacturingmachine. It is characterized by the presence of a two-dimensional contracting part, mountedat the end of a step diffuser. In this device the flow is contracted in one direction and hencepragmatically called as the ”two-dimensional contraction”. The paper manufacturer presum-ably pays attention to the determination of turbulence characteristics of the flow through thiscontraction. The reasoning is that the mechanical properties of the paper may considerablydepend on the turbulence phenomena.Experiments show that the acceleration through the contraction suppresses the relative tur-bulence intensities, which have almost no effect on the mean velocity distributions in the con-traction. The streamwise component of the Reynolds stresses decays on passing through thecontraction, whereas the transverse components grow equally. A reduction in the contractionangle brings about an increased decay rate of the turbulence. Another phenomenon is the re-laminarization of turbulent boundary layer subjected to a favorable pressure gradient due toacceleration in the contraction. Herein the turbulent bursts near the wall disappear, the law ofthe wall breaks down and the turbulent intensity shows a tendency to decay. In principle, theshear stress distribution in the wall region rather than the Reynolds number is the most import-ant factor for the occurrence of boundary layer relaminarization. The ultimate penalty due tothe relaminarization is a significant reduction in the friction/heat transfer coefficient.Unambiguously, the flow structure is inherently connected with the industrial application.Therefore, it can be considered as a test case for the turbulence modeling. However, inadequateattempt has been made in computing the flow contraction due to the complexity in flow struc-1Research Scientist, Department of Mechanical Engineering2Professor, Department of Mechanical Engineering2ture and unavailability of suitable turbulent models. Unfortunately, the configuration lacks insufficient experimental data.In this study the flow inside a two-dimensional converging channel is simulated at a ReynoldsnumberRe  5:2104. Particular attention is paid to the relaminarization in the boundary layerof a flat plate, positioned at the centerline of the contraction. Two low-Reynolds number lineark-~ models, namely the original Chien (OCH) model [2] and the modified Chien (MCH)model [3] participate in the simulation. Computed velocity profiles and turbulent intensitiesare compared with the experimental data of Ref. [4].Turbulence ModelingThe two-dimensional Reynolds-averaged Navier-Stokes (RANS) equations, including the equa-tions for the kinetic energy k and dissipation ~, can be written in the following form:@U@t+@(F   Fv)@x+@(G Gv)@y= Q (1)where U = (; u; v; E; k; ~)T . The inviscid fluxes areF =0BBBBBBBB@uu2+ p+23kuvu(E + p+23k)uku~1CCCCCCCCA; G =0BBBBBBBB@vvuv2+ p +23kv(E + p+23k)vkv~1CCCCCCCCA(2)Here  is the density and p is the pressure. The total energy is defined asE = e +~V ~V2+ k (3)where e is the specific internal energy and ~V = u~i+ v~j is the velocity . The viscous fluxes areFv=0BBBBBBBB@0xx+23kxyuxx+ vxy  qxk(@k=@x)(@~=@x)1CCCCCCCCA; Gv=0BBBBBBBB@0xyyy+23kuxy+ vyy  qyk(@k=@y)(@~=@y)1CCCCCCCCA(4)and the viscous stress tensor can be given asij=  @uj@xi+@ui@xj 23(r ~V )Æij!  uiuj(5)where  is the laminar viscosity and uiujare the Reynolds stresses modeled using Boussinesqapproximation. The heat flux is calculated from~q =  (+ T)rT =  cpPr+ TcpPrTrT (6)3Figure 1: Computational grid for contracting channel.where Tis the coefficient of turbulent viscosity,  and Tare the laminar and turbulent thermalconductivity coefficients, Pr and PrTrepresent the laminar and turbulent Prandtl numbers, andT and cpimply the temperature and specific heat at constant pressure, respectively. Clearly, theturbulent part of the total heat-flux is estimated using Boussinesq approximation. The diffusionof turbulence is modeled askrk = (+Tk)rk; r~ = (+T)r~ (7)where kand are the appropriate empirical constants. The source term Q for the k and ~equations can be written asQ =0B@P    + EkC1P   C2f2~  f3DTt+ E1CA(8)where  = ~ +D and the turbulent production term P =  uiuj(@ui=@xj). The eddy viscosityand other variables are evaluated asT= CfkTt; Tt= maxk=~;q2=D = 2k=y2n; y+= uy=Ry=pkyn=; RT= k2=~(9)where uis the friction velocity, ynis the normal distance from the wall and  represents thekinematic viscosity. The turbulence time scale Ttprevents the singularity at yn= 0 in thedissipation equation.The MCH model defines the near-wall damping function fas a function of Rf= 1  exp( 0:01R  0:0068R3)R= yn=q Tt(10)wherep Ttis the Taylor microscale. The quantities Ekand Ein Eq. (8) are known ascross-diffusion terms evaluated as [Rahman and Siikonen, 2000]Ek= CkTmin"@(k=)@xj@~@xj; 0#E= CTT2t"@(k=)@xj@k@xj#(11)where the constants Ck= 0:5 and C=  2Ck. Table 1 summarizes functions and constants fordifferent turbulence models.4Table 1: Functions and constants.Model D ~w-B.C. C1C2PrTkCOCH 2 ky2n@~@yn= 0 1:44 1:92 0:9 1:0 1:3 0:09MCH 2 ky2n@~@yn= 0 1:44 1:92 0:9 1:0 1:3 0:09Model ff2f3EkEOCH 1:0  e 0:0115y+ 1:0  0:22e (RT =6)2 e 0:5y+ 0:0 0:0MCH Eq:(10) 1:0 e (Ry=80)2 Eq:(11) Eq:(11)Solution MethodA cell-centered finite-volume scheme together with an artificial compressibility approach isemployed to solve the flow equations [5]. In the artificial compressibility method, the artificialcompressibility is principally added to the derivative of density with respect to the pressure,influencing not only the continuity equation but also the other equations. The energy equationis not decoupled from the system of equations, facilitating a uniform treatment for both theprimitive and conservative variables. A fully upwinded second-order spatial differencing isapplied to approximate the convective terms. Roe’s damping term [6] is used to calculatethe flux on the cell face. A diagonally dominant alternating direction implicit (DDADI) timeintegration method [7] is applied for the iterative solution of the discretized equations. Amultigrid method is utilized for the acceleration of convergence [8]. The basic implementationof the artificial compressibility method and associated features can be obtained [5, 9].Results and DiscussionThe reference velocity is Uref= 3:0m=s with an upstream turbulence intensity Tu = 4:5%,defined as Tu =q23k=Uref. The contraction inlet profiles for all dependent variables aregenerated by solving the models, invoking fully developed flow with Re = UrefLref= 5:2  104, where Lrefis the reference length at the contraction inlet. The contraction ratioC = Uout=Urefwith a half angle of 9:16 become approximately 13. A 288  96 nonuniformgrid having heavily clustered near the solid wall is used for the computations. The representativestructured grid is displayed in Fig. 1. Two low-Reynolds number linear k-~ models, namely theOCH [2] and MCH [3] models are used in the simulations. Computed velocity profiles andturbulent intensities are compared with the experimental data [4]. All quantities shown beloware normalized by the local maximum velocity corresponding to the position concerned.Figure exhibits the mean velocity profiles at six representative positions. As is observed, thevelocity defect in the outer region decreases in the presence of favorable pressure gradients. Inaddition, the boundary layer thickness continues to decrease as the flow develops downstream.The thickness of the last profile is approximately 1=20 of that of the initial profile, indicatingthat the boundary layer approaches a laminar state.Figure deals with the turbulence intensities at different downstream stations. It is evidentupon investigation that predictions of the models are somewhat on a lower level than the datashow at some positions. The agreement is better in the laminar region rather than in the turbulentregion.5Figure 2: Velocity profiles at selected locations.ConclusionsThe characteristic of a initially turbulent boundary layer inside a two-dimensional contractionis numerically studied. The boundary layer thickness decreases with increasing distance fromthe contraction inlet and reaches about 1=20 of that of the initial profile at the laminar region.Comprehensive comparisons conclude that the turbulent models maintain good correspondencewith the experiments as the flow approaches the laminar state.6Figure 3: Kinetic energy profiles at selected locations.REFERENCES1. B. K. Yoon and M. K. Chung, ”Computation of Compression Ramp Flow with a Cross-Diffusion Modified k- Model”, AIAA Journal, Vol. 33, No. 8, 1995, pp. 1518-1521.2. K-Y. Chien, ”Predictions of Channel and Boundary Layer Flows with a Low-ReynoldsNumber Turbulence Model”, AIAA Journal, Vol. 20, No. 1, 1982, pp. 33-38.3. M.M. Rahman and T. Siikonen, ”Improved Low-Reynolds Number k-~ Model”, AIAA J.,38, No. 7, 2000, pp. 1298-1300.4. M. Parsheh, ”Flow in Contractions with Application to Headboxes”, A Doctoral7Thesis,2001, Royal Institute of Technology, Department of Mechanics, Faxenlaborator-iet, SE-100 44 Stockholm, Sweden.5. M.M. Rahman, P. Rautaheimo and T. Siikonen, ”Numerical Study of Turbulent HeatTransfer from a Confined Impinging Jet Using a Pseudo-Compressibility Method”, 2nd In-ternational Symposium on Turbulence, Heat and Mass Transfer, Delft Univ. Press, Delft,The Netherlands, 1997, pp. 511-520.6. P.L. Roe, ”Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes”,J. Comput. Physics, 43, 1981, pp. 357-372.7. C. Lombard, J. Bardina, E. Venkatapathy and J. Oliger, ”Multi-Dimensional Formulationof CSCM-an Upwind Flux Difference Eigenvector Split Method for the CompressibleNavier-Stokes Equations”, 6th AIAA Computational Fluid Dynamics Conference , AIAAPaper 83-1895-CP, 1983, pp. 649-664.8. A. Jameson and S. Yoon, ”Multigrid Solution of the Euler Equations Using ImplicitSchemes”, AIAA J., 24, 1986, pp. 1737-1743.9. T. Siikonen, ”An Application of Roe’s Flux-Difference Splitting for k    TurbulenceModel”, Int. J. Numer. Meth. Fluids, 21, 1995, pp. 1017-1039.

Turbulence Characteristics of Flow in a Two Dimensional Contraction

Aaltodoc

1Turbulence Characteristics of Flow in a Two
Dimensional Contraction
M. M. Rahman 1 and T. Siikonen 2
Helsinki University of Technology, Department of Mechanical Engineering,
Laboratory of Applied Thermodynamics, Sa¨hko¨miehentie 4, FIN-02015 HUT
Finland
Abstract — The relaminarization of a fully turbulent flat plate boundary layer subjected to a
favorable pressure gradient inside a two-dimensional contraction has been carried out numer-
ically. Two low-Reynolds number isotropic k-~ models are employed in the simulation. The
simulated fluid flow and turbulence characteristics are compared with the experiment, demon-
strating that the models have good agreement with the data as the flow approaches the laminar
state.
Key Words: Contraction, turbulence, relaminarization, boundary layer.
Introduction
Plane contraction is associated with the hydraulic head box utilized in the paper manufacturing
machine. It is characterized by the presence of a two-dimensional contracting part, mounted
at the end of a step diffuser. In this device the flow is contracted in one direction and hence
pragmatically called as the ”two-dimensional contraction”. The paper manufacturer presum-
ably pays attention to the determination of turbulence characteristics of the flow through this
contraction. The reasoning is that the mechanical properties of the paper may considerably
depend on the turbulence phenomena.
Experiments show that the acceleration through the contraction suppresses the relative tur-
bulence intensities, which have almost no effect on the mean velocity distributions in the con-
traction. The streamwise component of the Reynolds stresses decays on passing through the
contraction, whereas the transverse components grow equally. A reduction in the contraction
angle brings about an increased decay rate of the turbulence. Another phenomenon is the re-
laminarization of turbulent boundary layer subjected to a favorable pressure gradient due to
acceleration in the contraction. Herein the turbulent bursts near the wall disappear, the law of
the wall breaks down and the turbulent intensity shows a tendency to decay. In principle, the
shear stress distribution in the wall region rather than the Reynolds number is the most import-
ant factor for the occurrence of boundary layer relaminarization. The ultimate penalty due to
the relaminarization is a significant reduction in the friction/heat transfer coefficient.
Unambiguously, the flow structure is inherently connected with the industrial application.
Therefore, it can be considered as a test case for the turbulence modeling. However, inadequate
attempt has been made in computing the flow contraction due to the complexity in flow struc-
1Research Scientist, Department of Mechanical Engineering
2Professor, Department of Mechanical Engineering
2ture and unavailability of suitable turbulent models. Unfortunately, the configuration lacks in
sufficient experimental data.
In this study the flow inside a two-dimensional converging channel is simulated at a Reynolds
numberRe  5:2104. Particular attention is paid to the relaminarization in the boundary layer
of a flat plate, positioned at the centerline of the contraction. Two low-Reynolds number linear
k-~ models, namely the original Chien (OCH) model [2] and the modified Chien (MCH)
model [3] participate in the simulation. Computed velocity profiles and turbulent intensities
are compared with the experimental data of Ref. [4].
Turbulence Modeling
The two-dimensional Reynolds-averaged Navier-Stokes (RANS) equations, including the equa-
tions for the kinetic energy k and dissipation ~, can be written in the following form:
@U
@t
+
@(F   F
v
)
@x
+
@(G G
v
)
@y
= Q (1)
where U = (; u; v; E; k; ~)T . The inviscid fluxes are
F =
0
B
B
B
B
B
B
B
B
@
u
u
2
+ p+
2
3
k
uv
u(E + p+
2
3
k)
uk
u~
1
C
C
C
C
C
C
C
C
A
; G =
0
B
B
B
B
B
B
B
B
@
v
vu
v
2
+ p +
2
3
k
v(E + p+
2
3
k)
vk
v~
1
C
C
C
C
C
C
C
C
A
(2)
Here  is the density and p is the pressure. The total energy is defined as
E = e +

~
V 
~
V
2
+ k (3)
where e is the specific internal energy and ~V = u~i+ v~j is the velocity . The viscous fluxes are
F
v
=
0
B
B
B
B
B
B
B
B
@
0

xx
+
2
3
k

xy
u
xx
+ v
xy
  q
x

k
(@k=@x)


(@~=@x)
1
C
C
C
C
C
C
C
C
A
; G
v
=
0
B
B
B
B
B
B
B
B
@
0

xy

yy
+
2
3
k
u
xy
+ v
yy
  q
y

k
(@k=@y)


(@~=@y)
1
C
C
C
C
C
C
C
C
A
(4)
and the viscous stress tensor can be given as

ij
= 
 
@u
j
@x
i
+
@u
i
@x
j
 
2
3
(r 
~
V )Æ
ij
!
  u
i
u
j
(5)
where  is the laminar viscosity and u
i
u
j
are the Reynolds stresses modeled using Boussinesq
approximation. The heat flux is calculated from
~q =  (+ 
T
)rT =  


c
p
Pr
+ 
T
c
p
Pr
T

rT (6)
3Figure 1: Computational grid for contracting channel.
where 
T
is the coefficient of turbulent viscosity,  and 
T
are the laminar and turbulent thermal
conductivity coefficients, Pr and Pr
T
represent the laminar and turbulent Prandtl numbers, and
T and c
p
imply the temperature and specific heat at constant pressure, respectively. Clearly, the
turbulent part of the total heat-flux is estimated using Boussinesq approximation. The diffusion
of turbulence is modeled as

k
rk = (+

T

k
)rk; 

r~ = (+

T


)r~ (7)
where 
k
and 

are the appropriate empirical constants. The source term Q for the k and ~
equations can be written as
Q =
0
B
@
P    + E
k
C
1
P   C
2
f
2
~  f
3
D
T
t
+ E

1
C
A
(8)
where  = ~ +D and the turbulent production term P =  u
i
u
j
(@u
i
=@x
j
). The eddy viscosity
and other variables are evaluated as

T
= C

f

kT
t
; T
t
= max

k=~;
q
2=

D = 2k=y
2
n
; y
+
= u

y=
R
y
=
p
ky
n
=; R
T
= k
2
=~
(9)
where u

is the friction velocity, y
n
is the normal distance from the wall and  represents the
kinematic viscosity. The turbulence time scale T
t
prevents the singularity at y
n
= 0 in the
dissipation equation.
The MCH model defines the near-wall damping function f

as a function of R

f

= 1  exp( 0:01R

  0:0068R
3

)
R

= y
n
=
q
 T
t
(10)
where
p
 T
t
is the Taylor microscale. The quantities E
k
and E

in Eq. (8) are known as
cross-diffusion terms evaluated as [Rahman and Siikonen, 2000]
E
k
= C
k

T
min
"
@(k=)
@x
j
@~
@x
j
; 0
#
E

= C


T
T
2
t
"
@(k=)
@x
j
@k
@x
j
#
(11)
where the constants C
k
= 0:5 and C

=  2C
k
. Table 1 summarizes functions and constants for
different turbulence models.
4Table 1: Functions and constants.
Model D ~
w
-B.C. C
1
C
2
Pr
T

k


C

OCH 2 k
y
2
n
@~
@y
n
= 0 1:44 1:92 0:9 1:0 1:3 0:09
MCH 2 k
y
2
n
@~
@y
n
= 0 1:44 1:92 0:9 1:0 1:3 0:09
Model f

f
2
f
3
E
k
E

OCH 1:0  e 0:0115y+ 1:0  0:22e (RT =6)2 e 0:5y+ 0:0 0:0
MCH Eq:(10) 1:0 e (Ry=80)2 Eq:(11) Eq:(11)
Solution Method
A cell-centered finite-volume scheme together with an artificial compressibility approach is
employed to solve the flow equations [5]. In the artificial compressibility method, the artificial
compressibility is principally added to the derivative of density with respect to the pressure,
influencing not only the continuity equation but also the other equations. The energy equation
is not decoupled from the system of equations, facilitating a uniform treatment for both the
primitive and conservative variables. A fully upwinded second-order spatial differencing is
applied to approximate the convective terms. Roe’s damping term [6] is used to calculate
the flux on the cell face. A diagonally dominant alternating direction implicit (DDADI) time
integration method [7] is applied for the iterative solution of the discretized equations. A
multigrid method is utilized for the acceleration of convergence [8]. The basic implementation
of the artificial compressibility method and associated features can be obtained [5, 9].
Results and Discussion
The reference velocity is U
ref
= 3:0m=s with an upstream turbulence intensity Tu = 4:5%,
defined as Tu =
q
2
3
k=U
ref
. The contraction inlet profiles for all dependent variables are
generated by solving the models, invoking fully developed flow with Re = U
ref
L
ref
= 
5:2  10
4
, where L
ref
is the reference length at the contraction inlet. The contraction ratio
C = U
out
=U
ref
with a half angle of 9:16 become approximately 13. A 288  96 nonuniform
grid having heavily clustered near the solid wall is used for the computations. The representative
structured grid is displayed in Fig. 1. Two low-Reynolds number linear k-~ models, namely the
OCH [2] and MCH [3] models are used in the simulations. Computed velocity profiles and
turbulent intensities are compared with the experimental data [4]. All quantities shown below
are normalized by the local maximum velocity corresponding to the position concerned.
Figure exhibits the mean velocity profiles at six representative positions. As is observed, the
velocity defect in the outer region decreases in the presence of favorable pressure gradients. In
addition, the boundary layer thickness continues to decrease as the flow develops downstream.
The thickness of the last profile is approximately 1=20 of that of the initial profile, indicating
that the boundary layer approaches a laminar state.
Figure deals with the turbulence intensities at different downstream stations. It is evident
upon investigation that predictions of the models are somewhat on a lower level than the data
show at some positions. The agreement is better in the laminar region rather than in the turbulent
region.
5Figure 2: Velocity profiles at selected locations.
Conclusions
The characteristic of a initially turbulent boundary layer inside a two-dimensional contraction
is numerically studied. The boundary layer thickness decreases with increasing distance from
the contraction inlet and reaches about 1=20 of that of the initial profile at the laminar region.
Comprehensive comparisons conclude that the turbulent models maintain good correspondence
with the experiments as the flow approaches the laminar state.
6Figure 3: Kinetic energy profiles at selected locations.
REFERENCES
1. B. K. Yoon and M. K. Chung, ”Computation of Compression Ramp Flow with a Cross-
Diffusion Modified k- Model”, AIAA Journal, Vol. 33, No. 8, 1995, pp. 1518-1521.
2. K-Y. Chien, ”Predictions of Channel and Boundary Layer Flows with a Low-Reynolds
Number Turbulence Model”, AIAA Journal, Vol. 20, No. 1, 1982, pp. 33-38.
3. M.M. Rahman and T. Siikonen, ”Improved Low-Reynolds Number k-~ Model”, AIAA J.,
38, No. 7, 2000, pp. 1298-1300.
4. M. Parsheh, ”Flow in Contractions with Application to Headboxes”, A Doctoral
7Thesis,2001, Royal Institute of Technology, Department of Mechanics, Faxenlaborator-
iet, SE-100 44 Stockholm, Sweden.
5. M.M. Rahman, P. Rautaheimo and T. Siikonen, ”Numerical Study of Turbulent Heat
Transfer from a Confined Impinging Jet Using a Pseudo-Compressibility Method”, 2nd In-
ternational Symposium on Turbulence, Heat and Mass Transfer, Delft Univ. Press, Delft,
The Netherlands, 1997, pp. 511-520.
6. P.L. Roe, ”Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes”,
J. Comput. Physics, 43, 1981, pp. 357-372.
7. C. Lombard, J. Bardina, E. Venkatapathy and J. Oliger, ”Multi-Dimensional Formulation
of CSCM-an Upwind Flux Difference Eigenvector Split Method for the Compressible
Navier-Stokes Equations”, 6th AIAA Computational Fluid Dynamics Conference , AIAA
Paper 83-1895-CP, 1983, pp. 649-664.
8. A. Jameson and S. Yoon, ”Multigrid Solution of the Euler Equations Using Implicit
Schemes”, AIAA J., 24, 1986, pp. 1737-1743.
9. T. Siikonen, ”An Application of Roe’s Flux-Difference Splitting for k    Turbulence
Model”, Int. J. Numer. Meth. Fluids, 21, 1995, pp. 1017-1039.


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Turbulence Characteristics of Flow in a Two Dimensional Contraction

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