Direct numerical simulations (DNS) of 2 and 3D turbulent flows in a lid-driven cavity have been performed. DNS are numerical solutions of the unsteady (here: incompressible) Navier-Stokes equations that compute the evolution of all dynamically significant scales of motion. In view of the large computing resources needed for DNS cost-effective and accurate numerical methods are to be selected. Here, various-order accurate spatial discretization methods for DNS have been evaluated by applying them to the 2D driven cavity at Re = 22,000. To analyze the results of the DNS of the 2D flow in a driven cavity at Re = 22,000 the proper orthogonal decomposition (POD) technique has been applied. POD is an unbiased method to determine coherent structures. The Galerkin projection of the Navier-Stokes equations on the space spanned by the POD-basis-functions yields a relatively low-dimensional set of ordinary differential equations that mimics the dynamics of the Navier-Stokes equations. 3D DNS with no-slip conditions at all walls of the cavity have been performed at both Re = 3,200 and Re = 10,000. The results reproduce the experimentally observed Taylor-Görtler-like vortices.

Cazemier, W.,

Veldman, A.E.P.,

Verstappen, R.,

Wissink, J.G.,

English

Verstappen, R.

Wissink, J.G.

Cazemier, W.

Veldman, A.E.P.

University of Groningen Research Database

  
 University of Groningen
Direct Numerical Simulations of turbulent flow in a driven cavity
Verstappen, R.; Wissink, J.G.; Cazemier, W.; Veldman, A.E.P.
Published in:
Future generation computer systems
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
1994
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Verstappen, R., Wissink, J. G., Cazemier, W., & Veldman, A. E. P. (1994). Direct Numerical Simulations of
turbulent flow in a driven cavity. Future generation computer systems, 10(2-3), 345-350.
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
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Download date: 22-05-2019
F G C S 
g3"'S'FEM S 
I St.~VII t,~ Future Generation Computer Systems 10 (1994) 345-350 
Di[rect Numerical Simulations of turbulent flow in a driven cavity 
R. Verstappen *, J.G. Wissink, W. Cazemier, A.E.P. Veldman 
Department ofMathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands 
.~i)stract 
Direct numerical simulations (DNS) of 2 and 3D turbulent flows in a lid-driven cavity have been performed. DNS 
are numerical solutions of the unsteady (here: incompressible) Navier-Stokes equations that compute the evolution 
of all dynamically significant scales of motion. In view of the large computing resources needed for DNS 
cost-effective and accurate numerical methods are to be selected. Here, various-order accurate spatial discretization 
methods for DNS have been evaluated by applying them to the 2D driven cavity at Re = 22,000. To analyze the 
results of the DNS of the 2D flow in a driven cavity at Re = 22,000 the proper orthogonal decomposition (POD) 
technique has been applied. POD is an unbiased method to determine coherent structures. The Galerkin projection 
of the Navier-Stokes equations on the space spanned by the POD-basis-functions yields a relatively low-dimensional 
set of ordinary differential equations that mimics the dynamics of the Navier-Stokes equations. 3D DNS with no-slip 
conditions at all walls of the cavity have been performed at both Re = 3,200 and Re = 10,000. The results reproduce 
the experimentally observed Taylor-G6rtler-like vortices. 
Key words: Direct Numerical Simulation; Turbulence; Navier-Stokes equations; Driven cavity; Numerical methods; 
Proper orthogonal decomposition 
1. Introduction 
Direct Numerical Simulations (DNS) are nu- 
merical solutions of the unsteady (here: incom- 
pressible) Navier-Stokes equations that compute 
the evolution of all dynamically significant scales 
of motion. Unfortunately, DNS of engineering 
flows exhaust he largest available computing re- 
sources by requiring machines in the exa(1018) 
flops range with exabytes of memory. Thus, for 
engineering applications acceptable computa- 
tional effort can only be obtained by modelling 
* Corresponding author. 
the turbulence of the flow. However, with existing 
turbulence models the simulation accuracy re- 
quired by industry cannot be reached in many 
technological applications, making turbulence 
modelling the main pacing item in the develop- 
ment of applied CFD codes. It is generally ex- 
pected that DNS will play the key role in obtain- 
ing better turbulence models; DNS have now 
become the chief source of highly detailed data 
for turbulence modelers and researchers [1]. 
In this paper, we report on DNS of turbulent 
flows in two- and three-dimensional, lid-driven, 
cavities and on their mathematical nalysis. The 
two-dimensional simulations have been per- 
formed to evaluate numerical methods for DNS 
0167-739X/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved 
SSDI 0167-739X(94)00032-A 
346 R. Verstappen taL / Future Generation Computer Systems 10 (1994) 345-350 
and to study the properties of the proper orthog- 
onal decomposition (POD) technique. 
2. I )NS  of t~vo-d imens iona l  flov~ in  a dr iven  tacit> 
In two dimensions, the flow converges to a 
steady state for Reynolds numbers up to 10,000 
[2]. For larger Reynolds numbers, 2D driven cav- 
ity flows undergo a transition to unsteadiness. 
Our direct simulations exhibit that the flow con- 
verges to a purely periodically oscillating state at 
Re = 11,000. The amplitude of the periodic mo- 
tion is small: in terms of the total kinetic energy 
0.2% of the mean value (more details can be 
found in [3]). DNS of the motion of the vortical 
structures at Re = 22,000 reveals that the dynam- 
ics has become chaotic: the correlation dimension 
is approximately 2.8, the Kolmogorov entropy is 
approximately 3.0. As expected, the number of 
vortices has increased compared to the simula- 
tion at Re = 11,000. Furthermore, the motion of 
vortical structures is no longer confined to small 
regions near the corners of the cavity. To illus- 
trate this, three instantaneous vorticity fields are 
shown in Fig. 1. 
3. Evaluation of numerical methods fi~r DNS 
In view of the large computing resources 
needed for DNS a careful selection of cost-effec- 
tive and accurate numerical methods is impor- 
tant. Here, various-order accurate spatial dis- 
cretization methods for DNS have been evalu- 
ated by applying them to the driven cavity at 
Re = 22,000. The evaluation focusses on the accu- 
racy with which the mean kinetic energy is com- 
puted using a 2002 economically stretched, stag- 
gered grid, and a time-step that is so small that 
the error resulting from the time-integration is 
negligible. Central as well as upwind discretiza- 
tions have been considered. The central method 
consisting of a rnth order central discretization of 
the convective term (in conservative form) com- 
bined with a nth order interpolation, and a kth 
order central discretization of the diffusive term 
is denoted by CmlnCk. Likewise, an upwind 
method is denoted by UmlnCk if the conserva- 
tive term (now in non-conservative form) is dis- 
cretized using a mth order upwind-biased 
method. A Lagrange interpolation is used to ob- 
tain the coefficients of the stencils. Since stencils 
with more than one gridpoint on either side of 
the point at which the derivative is to be calcu- 
lated cannot be applied near the boundaries, 
high-order methods are gradually replaced by 
lower-order methods in the vicinity of the bound- 
aries. Reference data for the comparison is ob- 
tained using the U514C6 method on a 4002 
stretched grid. The main results of the compari- 
son are shown in Fig. 2. From this figure, it can 
be concluded that C414C6 and U716C8 perform 
the best of the methods considered here. 
....... ,. 
iii " . . . . . . . . . .  
09 . . . .  ' 
0g ~. " " . i. 
1~.7 ~: !  .: i 
o.4 : .:~ 
c~3 . ! . " "  
iii / i i  .......... ' ,  
 ,iii : ! 
Fig. 1. Snapshots  of  the vort ic ity in a 2D driven cavity (Re = 22,000). 
R. Verstappen et al. / Future Generation Computer Systems 10 (1994) 345-350 347 
To analyze the results of the DNS of the 
turbulent flow in a driven cavity at Re = 22,000, 
we make use of proper orthogonal decomposition 
(POD). That is (see e.g. [4]), from fluctuating flow 
fields a set of (say N) basis-functions i extracted, 
which is optimal in the sense that any other 
decomposition (e.g. in N Fourier-modes) cap- 
tures less kinetic energy. The Galerkin projection 
of the Navier-Stokes equations on the space 
spanned by the basis-functions yields a relatively 
low-dimensional set of ordinary differential equa- 
tions. This 'cheap' (compared to Navier-Stokes) 
set can be analyzed using tools from dynamical 
systems theory, and thus we can advance our 
understanding of turbulence, provided that this 
set mimics the dynamics of the Navier-Stokes 
% 
/.12' 
L~ 
E 
2 
J Ib ~ X 
Fig. 2. Errors in the mean kinetic energy for a number of spatial discretization methods (2D driven cavity, Re = 22,000). 
I (X) 
R. Verstappen et aL /Future Generation Computer Systems 10 (1994) 345-350 
> 
"o 
/ 
l )O l )  - -  
. . j - - "  l~ourier ___ 
so / 
70 / 
6O 
50 ,, / J " "  
40 
/ 30 / 
20 / 
l0 y/'" 
I ltl l l l~;C'l" {/1" I'~.~t"q.s-ltlllCli(~ll>; 
Fig. 3. Comparison of POD and Fourier-modes. 
equations. To investigate how well it does this, we 
have performed a direct simulation of the turbu- 
lent flow in a 2D driven cavity at Re = 22,000 
during more than 700 large-eddy turn-around 
times to obtain the data necessary to compute up 
to 700 basis-functions. These basis-functions are 
computed using the so-called snapshot method 
[5]. The energy captured by POD-modes has been 
348 
Fig. 5. Snapshot of the vorticity in a 3D driven cavity at 
Re = 10,000. 
compared to the energy captured by Fourier- 
modes (Fig. 3), and, as expected, POD is far 
superior to Fourier. Furthermore, as can be seen 
(18 
0.7 
0,6 
0.5 
1).4 
0.3 
{).2 
(1.1 
0.9 
0.8 
0.7 
0.6 
>0.5 
0.4 
0.3 
0.2 
0.1 
~ ~ fir':l basis luncl ion 
I)? (14 0.6 (I.8 
x 
,i.q~ 
li.g F 
0-6 F 
>, (L5 i 
0"11 
i )?  
!,I i 
r 
st_',ctmd basis hmclion 
0.? () 4 II 6 (18 
lhird basis ltmclion 
{)2 (~.4 0.6 0.8 
X 
(D) 
lout01 basis hmclion 
0.8 
0.7 ¢ 
0.6 
(1.4 } 
0.2 
() 2 ( } 4 D.6 0.8 
x 
Fig. 4. The first four POD basis-functions. 
R. Verstappen et al. / Future Generation Computer Systems I0 (1994) 345-350 349 
in Fig. 4, POD is an unbiased method to deter- 
mine the coherent structures (eigenflows) in the 
driven cavity flow. 
It has been found that the evaluation obtained 
by numerical integration of a 80-dimensional dy- 
namical system, which is obtained by projecting 
the Navier-Stokes equations on the first 80 POD 
basis-functions, agrees well with direct numerical 
simulation data at Re = 22,000. Also, when the 
Reynolds number in the 80-dimensional dynami- 
cal system is taken equal to 11,000 the integration 
of the dynamical system reproduces the periodic 
solution as obtained by DNS at Re = 11,000. 
Acknowledgements  
Both the Stichting Nationale Computerfa- 
ciliteiten (National Computing Facilities Founda- 
tion, NCF) with financial support from the Ne- 
derlandse Organisatie voor Wetenschappelijk 
Onderzoek (Netherlands Organization for Scien- 
tific Research, NWO) and the National 
Aerospace Laboratory NLR are gratefully ac- 
knowledged for the use of supercomputer facili- 
ties. 
5. I~,NS of lh ree-d imens iona l  tu rbu len l  t]ow in ~ 
dri~ t,n t':l~il3 
Reliable experimental results (see e.g. [6]) have 
pointed out that the flow in a three-dimensional 
driven cavity is already unsteady for Reynolds 
numbers much smaller than 10,000. The experi- 
ments reveal that the flow becomes turbulent at 
Reynolds numbers between 6,000 to 8,000, and 
that the first instability in the 3D flow is certainly 
not two-dimensional: the experimentally observed 
instability is associated with 3D Taylor-G6rtler- 
like vortices. Thus, fluid flows in the direction 
perpendicular to the lid-motion are significant. 
For this reason, full 3D DNS (with no-slip condi- 
tions at all walls of the cavity) have been per- 
formed at Re = 3,200 and Re = 10,000. A 1003 
economically stretched, staggered grid has been 
used to solve the incompressible Navier-Stokes 
equations numerically. The CPU-time on one 
processor of NLR's NEC SX-3 is 0.6 /~s per 
gridpoint and time-step. The results reproduce 
the experimentally observed Taylor-G6rtler-like 
vortices, and the computed mean velocities agree 
well with those measured by Koseff and Street 
[6]. As an illustration of the structures in the flow, 
Fig. 5 shows an instantaneous vorticity field in 
the 3D cavity at Re = 10,000. In this figure, the 
orientation of the cavity is such that the upper- 
plane is driven from the left/back to the 
right/front. 
References  
[1] W.C. Reynolds, in: Wither Turbulence? Turbulence at the 
crossroads, J.L. Lumley (ed.) (Springer-Verlag, Berlin, 
1990), 313. 
[2] U. Ghia, K.N. Ghia and C.T. Shin, J. Comp. Phys. 48 
(1982) 387. 
[3] R. Verstappen, J.G. Wissink and A.E.P. Veldman, Appl. 
Sci. Research 51 (1993) 377. 
[4] N. Aubry, P. Holmes, J.L. Lumley and E. Stone, J. Fluid 
Mech. 192 (1988) 115. 
[5] L. Sirovich and M. Kirby, Physica D 37 (1987) 126. 
[6] J.R. Koseff and R.L. Street, ASME Z Fluids. Eng. 106 
(1984) 21. 
Rocl ~, erslappen was born in Venray 
(The Netherlands) on 22 June 1962. 
He received his education in applied 
mathematics at the University of 
Twente in the Netherlands. In 1989 
he received his Ph.D. in engineering 
from this university. After that he 
moved to the University of Groningen 
where he works on the field of com- 
putational f uid dynamics. 
After a working period of several 
years as an analyst at Delft Hy- 
draulics, Jan $~,issink started to study 
mathematics at the University of 
Groningen. Since his graduation he is 
a Ph.D. student, making researches 
into numerical methods (especially fi- 
nite-volume methods) for direct nu- 
merical simulation of turbulence. 
350 R. Verstappen et aL ~Future Generation Computer Systems 10 (1994) 345-350 
~,%illcm t a/t:mict studied mathemat . . . .  
ics at the University of Groningen. 
The last year of his study he spent at 
the National Aerospace Laboratory 
(NLR) in Amsterdam. After his grad- 
uation in 1992, he became a Ph.D. 
student in Groningen, where he is 
doing research on the Proper Orthog- 
onal Decomposition to study turbu- 
lence. 
Arthur L.l'. ~cldman obtained a 
Ph.D. in Applied Mathematics from 
the University of Groningen. In 1977 
he joined the National Aerospace 
Laboratory NLR in Amsterdam, 
where he was involved in various pro- 
jects in the area of computational 
aero- and hydrodynamics. In addition, 
between 1984 and 1990 he was part- 
time professor of CFD at Delft Uni- 
versity of Technology. In 1990 he re- 
turned to Groningen, where he now 
occupies the chair in Computational 
Mechanics. 


Direct Numerical Simulations of turbulent flow in a driven cavity

Direct numerical simulations (DNS) of 2 and 3D turbulent flows in a lid-driven cavity have been performed. DNS are numerical solutions of the unsteady (here: incompressible) Navier-Stokes equations that compute the evolution of all dynamically significant scales of motion. In view of the large computing resources needed for DNS cost-effective and accurate numerical methods are to be selected. Here, various-order accurate spatial discretization methods for DNS have been evaluated by applying them to the 2D driven cavity at Re = 22,000. To analyze the results of the DNS of the 2D flow in a driven cavity at Re = 22,000 the proper orthogonal decomposition (POD) technique has been applied. POD is an unbiased method to determine coherent structures. The Galerkin projection of the Navier-Stokes equations on the space spanned by the POD-basis-functions yields a relatively low-dimensional set of ordinary differential equations that mimics the dynamics of the Navier-Stokes equations. 3D DNS with no-slip conditions at all walls of the cavity have been performed at both Re = 3,200 and Re = 10,000. The results reproduce the experimentally observed Taylor-Görtler-like vortices

University of Groningen

   University of GroningenDirect Numerical Simulations of turbulent flow in a driven cavityVerstappen, R.; Wissink, J.G.; Cazemier, W.; Veldman, A.E.P.Published in:Future generation computer systemsIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1994Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Verstappen, R., Wissink, J. G., Cazemier, W., & Veldman, A. E. P. (1994). Direct Numerical Simulations ofturbulent flow in a driven cavity. Future generation computer systems, 10(2-3), 345-350.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019F G C S g3"'S'FEM S I St.~VII t,~ Future Generation Computer Systems 10 (1994) 345-350 Di[rect Numerical Simulations of turbulent flow in a driven cavity R. Verstappen *, J.G. Wissink, W. Cazemier, A.E.P. Veldman Department ofMathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands .~i)stract Direct numerical simulations (DNS) of 2 and 3D turbulent flows in a lid-driven cavity have been performed. DNS are numerical solutions of the unsteady (here: incompressible) Navier-Stokes equations that compute the evolution of all dynamically significant scales of motion. In view of the large computing resources needed for DNS cost-effective and accurate numerical methods are to be selected. Here, various-order accurate spatial discretization methods for DNS have been evaluated by applying them to the 2D driven cavity at Re = 22,000. To analyze the results of the DNS of the 2D flow in a driven cavity at Re = 22,000 the proper orthogonal decomposition (POD) technique has been applied. POD is an unbiased method to determine coherent structures. The Galerkin projection of the Navier-Stokes equations on the space spanned by the POD-basis-functions yields a relatively low-dimensional set of ordinary differential equations that mimics the dynamics of the Navier-Stokes equations. 3D DNS with no-slip conditions at all walls of the cavity have been performed at both Re = 3,200 and Re = 10,000. The results reproduce the experimentally observed Taylor-G6rtler-like vortices. Key words: Direct Numerical Simulation; Turbulence; Navier-Stokes equations; Driven cavity; Numerical methods; Proper orthogonal decomposition 1. Introduction Direct Numerical Simulations (DNS) are nu- merical solutions of the unsteady (here: incom- pressible) Navier-Stokes equations that compute the evolution of all dynamically significant scales of motion. Unfortunately, DNS of engineering flows exhaust he largest available computing re- sources by requiring machines in the exa(1018) flops range with exabytes of memory. Thus, for engineering applications acceptable computa- tional effort can only be obtained by modelling * Corresponding author. the turbulence of the flow. However, with existing turbulence models the simulation accuracy re- quired by industry cannot be reached in many technological applications, making turbulence modelling the main pacing item in the develop- ment of applied CFD codes. It is generally ex- pected that DNS will play the key role in obtain- ing better turbulence models; DNS have now become the chief source of highly detailed data for turbulence modelers and researchers [1]. In this paper, we report on DNS of turbulent flows in two- and three-dimensional, lid-driven, cavities and on their mathematical nalysis. The two-dimensional simulations have been per- formed to evaluate numerical methods for DNS 0167-739X/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0167-739X(94)00032-A 346 R. Verstappen taL / Future Generation Computer Systems 10 (1994) 345-350 and to study the properties of the proper orthog- onal decomposition (POD) technique. 2. I )NS  of t~vo-d imens iona l  flov~ in  a dr iven  tacit> In two dimensions, the flow converges to a steady state for Reynolds numbers up to 10,000 [2]. For larger Reynolds numbers, 2D driven cav- ity flows undergo a transition to unsteadiness. Our direct simulations exhibit that the flow con- verges to a purely periodically oscillating state at Re = 11,000. The amplitude of the periodic mo- tion is small: in terms of the total kinetic energy 0.2% of the mean value (more details can be found in [3]). DNS of the motion of the vortical structures at Re = 22,000 reveals that the dynam- ics has become chaotic: the correlation dimension is approximately 2.8, the Kolmogorov entropy is approximately 3.0. As expected, the number of vortices has increased compared to the simula- tion at Re = 11,000. Furthermore, the motion of vortical structures is no longer confined to small regions near the corners of the cavity. To illus- trate this, three instantaneous vorticity fields are shown in Fig. 1. 3. Evaluation of numerical methods fi~r DNS In view of the large computing resources needed for DNS a careful selection of cost-effec- tive and accurate numerical methods is impor- tant. Here, various-order accurate spatial dis- cretization methods for DNS have been evalu- ated by applying them to the driven cavity at Re = 22,000. The evaluation focusses on the accu- racy with which the mean kinetic energy is com- puted using a 2002 economically stretched, stag- gered grid, and a time-step that is so small that the error resulting from the time-integration is negligible. Central as well as upwind discretiza- tions have been considered. The central method consisting of a rnth order central discretization of the convective term (in conservative form) com- bined with a nth order interpolation, and a kth order central discretization of the diffusive term is denoted by CmlnCk. Likewise, an upwind method is denoted by UmlnCk if the conserva- tive term (now in non-conservative form) is dis- cretized using a mth order upwind-biased method. A Lagrange interpolation is used to ob- tain the coefficients of the stencils. Since stencils with more than one gridpoint on either side of the point at which the derivative is to be calcu- lated cannot be applied near the boundaries, high-order methods are gradually replaced by lower-order methods in the vicinity of the bound- aries. Reference data for the comparison is ob- tained using the U514C6 method on a 4002 stretched grid. The main results of the compari- son are shown in Fig. 2. From this figure, it can be concluded that C414C6 and U716C8 perform the best of the methods considered here. ....... ,. iii " . . . . . . . . . .  09 . . . .  ' 0g ~. " " . i. 1~.7 ~: !  .: i o.4 : .:~ c~3 . ! . " "  iii / i i  .......... ' ,   ,iii : ! Fig. 1. Snapshots  of  the vort ic ity in a 2D driven cavity (Re = 22,000). R. Verstappen et al. / Future Generation Computer Systems 10 (1994) 345-350 347 To analyze the results of the DNS of the turbulent flow in a driven cavity at Re = 22,000, we make use of proper orthogonal decomposition (POD). That is (see e.g. [4]), from fluctuating flow fields a set of (say N) basis-functions i extracted, which is optimal in the sense that any other decomposition (e.g. in N Fourier-modes) cap- tures less kinetic energy. The Galerkin projection of the Navier-Stokes equations on the space spanned by the basis-functions yields a relatively low-dimensional set of ordinary differential equa- tions. This 'cheap' (compared to Navier-Stokes) set can be analyzed using tools from dynamical systems theory, and thus we can advance our understanding of turbulence, provided that this set mimics the dynamics of the Navier-Stokes % /.12' L~ E 2 J Ib ~ X Fig. 2. Errors in the mean kinetic energy for a number of spatial discretization methods (2D driven cavity, Re = 22,000). I (X) R. Verstappen et aL /Future Generation Computer Systems 10 (1994) 345-350 > "o / l )O l )  - -  . . j - - "  l~ourier ___ so / 70 / 6O 50 ,, / J " "  40 / 30 / 20 / l0 y/'" I ltl l l l~;C'l" {/1" I'~.~t"q.s-ltlllCli(~ll>; Fig. 3. Comparison of POD and Fourier-modes. equations. To investigate how well it does this, we have performed a direct simulation of the turbu- lent flow in a 2D driven cavity at Re = 22,000 during more than 700 large-eddy turn-around times to obtain the data necessary to compute up to 700 basis-functions. These basis-functions are computed using the so-called snapshot method [5]. The energy captured by POD-modes has been 348 Fig. 5. Snapshot of the vorticity in a 3D driven cavity at Re = 10,000. compared to the energy captured by Fourier- modes (Fig. 3), and, as expected, POD is far superior to Fourier. Furthermore, as can be seen (18 0.7 0,6 0.5 1).4 0.3 {).2 (1.1 0.9 0.8 0.7 0.6 >0.5 0.4 0.3 0.2 0.1 ~ ~ fir':l basis luncl ion I)? (14 0.6 (I.8 x ,i.q~ li.g F 0-6 F >, (L5 i 0"11 i )?  !,I i r st_',ctmd basis hmclion 0.? () 4 II 6 (18 lhird basis ltmclion {)2 (~.4 0.6 0.8 X (D) lout01 basis hmclion 0.8 0.7 ¢ 0.6 (1.4 } 0.2 () 2 ( } 4 D.6 0.8 x Fig. 4. The first four POD basis-functions. R. Verstappen et al. / Future Generation Computer Systems I0 (1994) 345-350 349 in Fig. 4, POD is an unbiased method to deter- mine the coherent structures (eigenflows) in the driven cavity flow. It has been found that the evaluation obtained by numerical integration of a 80-dimensional dy- namical system, which is obtained by projecting the Navier-Stokes equations on the first 80 POD basis-functions, agrees well with direct numerical simulation data at Re = 22,000. Also, when the Reynolds number in the 80-dimensional dynami- cal system is taken equal to 11,000 the integration of the dynamical system reproduces the periodic solution as obtained by DNS at Re = 11,000. Acknowledgements  Both the Stichting Nationale Computerfa- ciliteiten (National Computing Facilities Founda- tion, NCF) with financial support from the Ne- derlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for Scien- tific Research, NWO) and the National Aerospace Laboratory NLR are gratefully ac- knowledged for the use of supercomputer facili- ties. 5. I~,NS of lh ree-d imens iona l  tu rbu len l  t]ow in ~ dri~ t,n t':l~il3 Reliable experimental results (see e.g. [6]) have pointed out that the flow in a three-dimensional driven cavity is already unsteady for Reynolds numbers much smaller than 10,000. The experi- ments reveal that the flow becomes turbulent at Reynolds numbers between 6,000 to 8,000, and that the first instability in the 3D flow is certainly not two-dimensional: the experimentally observed instability is associated with 3D Taylor-G6rtler- like vortices. Thus, fluid flows in the direction perpendicular to the lid-motion are significant. For this reason, full 3D DNS (with no-slip condi- tions at all walls of the cavity) have been per- formed at Re = 3,200 and Re = 10,000. A 1003 economically stretched, staggered grid has been used to solve the incompressible Navier-Stokes equations numerically. The CPU-time on one processor of NLR's NEC SX-3 is 0.6 /~s per gridpoint and time-step. The results reproduce the experimentally observed Taylor-G6rtler-like vortices, and the computed mean velocities agree well with those measured by Koseff and Street [6]. As an illustration of the structures in the flow, Fig. 5 shows an instantaneous vorticity field in the 3D cavity at Re = 10,000. In this figure, the orientation of the cavity is such that the upper- plane is driven from the left/back to the right/front. References  [1] W.C. Reynolds, in: Wither Turbulence? Turbulence at the crossroads, J.L. Lumley (ed.) (Springer-Verlag, Berlin, 1990), 313. [2] U. Ghia, K.N. Ghia and C.T. Shin, J. Comp. Phys. 48 (1982) 387. [3] R. Verstappen, J.G. Wissink and A.E.P. Veldman, Appl. Sci. Research 51 (1993) 377. [4] N. Aubry, P. Holmes, J.L. Lumley and E. Stone, J. Fluid Mech. 192 (1988) 115. [5] L. Sirovich and M. Kirby, Physica D 37 (1987) 126. [6] J.R. Koseff and R.L. Street, ASME Z Fluids. Eng. 106 (1984) 21. Rocl ~, erslappen was born in Venray (The Netherlands) on 22 June 1962. He received his education in applied mathematics at the University of Twente in the Netherlands. In 1989 he received his Ph.D. in engineering from this university. After that he moved to the University of Groningen where he works on the field of com- putational f uid dynamics. After a working period of several years as an analyst at Delft Hy- draulics, Jan $~,issink started to study mathematics at the University of Groningen. Since his graduation he is a Ph.D. student, making researches into numerical methods (especially fi- nite-volume methods) for direct nu- merical simulation of turbulence. 350 R. Verstappen et aL ~Future Generation Computer Systems 10 (1994) 345-350 ~,%illcm t a/t:mict studied mathemat . . . .  ics at the University of Groningen. The last year of his study he spent at the National Aerospace Laboratory (NLR) in Amsterdam. After his grad- uation in 1992, he became a Ph.D. student in Groningen, where he is doing research on the Proper Orthog- onal Decomposition to study turbu- lence. Arthur L.l'. ~cldman obtained a Ph.D. in Applied Mathematics from the University of Groningen. In 1977 he joined the National Aerospace Laboratory NLR in Amsterdam, where he was involved in various pro- jects in the area of computational aero- and hydrodynamics. In addition, between 1984 and 1990 he was part- time professor of CFD at Delft Uni- versity of Technology. In 1990 he re- turned to Groningen, where he now occupies the chair in Computational Mechanics. 

University of Groningen Digital Archive

F G C S 
g3"'S'FEM S 
I St.~VII  t,~  Future Generation Computer Systems 10 (1994) 345-350 
Di[rect Numerical  Simulations of turbulent flow in a driven cavity 
R. Verstappen *, J.G. Wissink, W. Cazemier, A.E.P. Veldman 
Department of Mathematics, University of Groningen, P.O. Box 800, 9700  AV Groningen, The Netherlands 
.~i)stract 
Direct numerical simulations (DNS) of 2 and 3D turbulent flows in a lid-driven cavity have been performed. DNS 
are numerical solutions of the unsteady (here: incompressible) Navier-Stokes equations that compute the evolution 
of  all  dynamically  significant  scales  of  motion.  In  view  of  the  large  computing  resources  needed  for  DNS 
cost-effective and accurate numerical methods are to be selected. Here, various-order accurate spatial discretization 
methods for DNS have been evaluated by applying them to the 2D  driven cavity at  Re = 22,000.  To analyze the 
results of the DNS of the 2D flow in a  driven cavity at  Re = 22,000  the proper orthogonal decomposition (POD) 
technique has been applied. POD is an unbiased method to determine coherent structures. The Galerkin projection 
of the Navier-Stokes equations on the space spanned by the POD-basis-functions yields a relatively low-dimensional 
set of ordinary differential equations that mimics the dynamics of the Navier-Stokes equations. 3D DNS with no-slip 
conditions at all walls of the cavity have been performed at both Re = 3,200 and Re = 10,000. The results reproduce 
the experimentally observed Taylor-G6rtler-like vortices. 
Key words:  Direct Numerical Simulation; Turbulence; Navier-Stokes equations; Driven cavity;  Numerical methods; 
Proper orthogonal decomposition 
1.  Introduction 
Direct  Numerical  Simulations  (DNS)  are  nu- 
merical  solutions  of the  unsteady  (here:  incom- 
pressible)  Navier-Stokes equations  that  compute 
the evolution of all dynamically significant scales 
of  motion.  Unfortunately,  DNS  of  engineering 
flows exhaust the largest available computing re- 
sources  by  requiring  machines  in  the  exa(1018) 
flops  range with  exabytes  of memory. Thus,  for 
engineering  applications  acceptable  computa- 
tional  effort can  only be  obtained  by modelling 
* Corresponding author. 
the turbulence of the flow. However, with existing 
turbulence  models  the  simulation  accuracy  re- 
quired  by  industry  cannot  be  reached  in  many 
technological  applications,  making  turbulence 
modelling  the  main  pacing  item  in  the  develop- 
ment  of  applied  CFD  codes.  It  is  generally  ex- 
pected that DNS will play the key role in obtain- 
ing  better  turbulence  models;  DNS  have  now 
become the  chief source  of highly  detailed  data 
for turbulence modelers and researchers [1]. 
In this paper, we report on DNS of turbulent 
flows  in  two-  and  three-dimensional,  lid-driven, 
cavities and  on their mathematical  analysis.  The 
two-dimensional  simulations  have  been  per- 
formed to  evaluate  numerical  methods  for  DNS 
0167-739X/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved 
SSDI 0167-739X(94)00032-A 346  R. Verstappen et aL / Future Generation Computer Systems 10 (1994) 345-350 
and to study the properties of the proper orthog- 
onal decomposition (POD) technique. 
2.  I)NS  of  t~vo-dimensional  flov~  in  a  driven  tacit> 
In  two  dimensions,  the  flow  converges  to  a 
steady state  for Reynolds numbers  up  to  10,000 
[2]. For larger Reynolds numbers, 2D driven cav- 
ity  flows  undergo  a  transition  to  unsteadiness. 
Our direct simulations exhibit that the flow con- 
verges to a purely periodically oscillating state at 
Re =  11,000.  The  amplitude  of the  periodic mo- 
tion is small:  in terms of the total kinetic energy 
0.2%  of  the  mean  value  (more  details  can  be 
found in [3]).  DNS  of the motion of the vortical 
structures at Re =  22,000 reveals that the dynam- 
ics has become chaotic: the correlation dimension 
is  approximately 2.8,  the  Kolmogorov entropy is 
approximately  3.0.  As  expected,  the  number  of 
vortices  has  increased  compared  to  the  simula- 
tion at  Re =  11,000.  Furthermore, the motion of 
vortical structures is no longer confined to small 
regions  near  the  corners of the  cavity. To illus- 
trate this, three instantaneous vorticity fields are 
shown in Fig. 1. 
3. Evaluation of numerical methods fi~r DNS 
In  view  of  the  large  computing  resources 
needed for DNS a careful selection of cost-effec- 
tive  and  accurate  numerical  methods  is  impor- 
tant.  Here,  various-order  accurate  spatial  dis- 
cretization  methods  for  DNS  have  been  evalu- 
ated  by  applying  them  to  the  driven  cavity  at 
Re =  22,000. The evaluation focusses on the accu- 
racy with which the mean kinetic energy is com- 
puted using a  2002  economically stretched, stag- 
gered grid,  and  a  time-step that  is so small that 
the  error  resulting  from  the  time-integration  is 
negligible.  Central  as well  as  upwind  discretiza- 
tions have been considered. The central method 
consisting of a  rnth order central discretization of 
the  convective term (in  conservative form) com- 
bined with  a  nth order interpolation, and  a  kth 
order central discretization of the diffusive term 
is  denoted  by  CmlnCk.  Likewise,  an  upwind 
method  is  denoted by  UmlnCk  if the  conserva- 
tive term (now  in  non-conservative form) is  dis- 
cretized  using  a  mth  order  upwind-biased 
method. A  Lagrange interpolation is used to ob- 
tain the coefficients of the stencils. Since stencils 
with  more  than  one  gridpoint  on  either  side  of 
the point  at which the  derivative is  to be calcu- 
lated  cannot  be  applied  near  the  boundaries, 
high-order  methods  are  gradually  replaced  by 
lower-order methods in the vicinity of the bound- 
aries.  Reference data for the  comparison  is  ob- 
tained  using  the  U514C6  method  on  a  4002 
stretched grid. The main results of the compari- 
son are shown in Fig. 2.  From this figure, it can 
be concluded that  C414C6  and  U716C8  perform 
the best of the methods considered here. 
.......  ,. 
iii  " .......... 
09  ....  ' 
0g  ~.  " "  .  i. 
1~.7  ~:!  .:  i 
o.4  :  .:~ 
c~3  .  !  ."" 
iii  /ii  ..........  ', 
 ,iii  :  ! 
Fig. 1. Snapshots of the vorticity in a  2D driven cavity (Re =  22,000). R. Verstappen et al. / Future Generation Computer Systems 10 (1994) 345-350  347 
To  analyze  the  results  of  the  DNS  of  the 
turbulent flow in a  driven cavity  at  Re =  22,000, 
we make use of proper orthogonal decomposition 
(POD). That is (see e.g. [4]), from fluctuating flow 
fields a set of (say N) basis-functions is extracted, 
which  is  optimal  in  the  sense  that  any  other 
decomposition  (e.g.  in  N  Fourier-modes)  cap- 
tures less kinetic energy. The Galerkin projection 
of  the  Navier-Stokes  equations  on  the  space 
spanned by the basis-functions yields a relatively 
low-dimensional set of ordinary differential equa- 
tions.  This  'cheap'  (compared  to  Navier-Stokes) 
set  can  be  analyzed  using  tools  from dynamical 
systems  theory,  and  thus  we  can  advance  our 
understanding  of turbulence,  provided  that  this 
set  mimics  the  dynamics  of  the  Navier-Stokes 
% 
/.12' 
L~ 
E 
2 
J  Ib  ~  X 
Fig. 2. Errors in the mean kinetic energy for a number of spatial discretization methods (2D driven cavity, Re =  22,000). I (X) 
R. Verstappen et aL /Future Generation  Computer Systems 10 (1994) 345-350 
> 
"o 
/ 
l)Ol)  -- 
..j--"  l~ourier  ___ 
so  / 
70  / 
6O 
50  ,,  /J"" 
40 
/  30  / 
20  / 
l0  y/'" 
Iltllll~;C'l"  {/1" I'~.~t"q.s-ltlllCli(~ll>; 
Fig. 3. Comparison of POD and Fourier-modes. 
equations. To investigate how well it does this, we 
have performed a direct simulation of the turbu- 
lent  flow  in  a  2D  driven  cavity  at  Re = 22,000 
during  more  than  700  large-eddy  turn-around 
times to obtain the data necessary to compute up 
to 700  basis-functions.  These basis-functions  are 
computed  using  the  so-called  snapshot  method 
[5]. The energy captured by POD-modes has been 
348 
Fig.  5.  Snapshot  of  the  vorticity  in  a  3D  driven  cavity  at 
Re =  10,000. 
compared  to  the  energy  captured  by  Fourier- 
modes  (Fig.  3),  and,  as  expected,  POD  is  far 
superior to Fourier. Furthermore, as can be seen 
(18 
0.7 
0,6 
0.5 
1).4 
0.3 
{).2 
(1.1 
0.9 
0.8 
0.7 
0.6 
>0.5 
0.4 
0.3 
0.2 
0.1 
~ 
~  fir':l  basis  lunclion 
I)?  (14  0.6  (I.8 
x 
,i.q~ 
li.g F 
0-6 F 
>, (L5 i 
0"11 
i)? 
!,I i 
r 
st_',ctmd basis hmclion 
0.?  () 4  II 6  (18 
lhird basis ltmclion 
{)2  (~.4  0.6  0.8 
X 
(D) 
lout01 basis hmclion 
0.8 
0.7  ¢ 
0.6 
(1.4  } 
0.2 
() 2  (  } 4  D.6  0.8 
x 
Fig. 4. The first four POD basis-functions. R. Verstappen et al. / Future Generation  Computer Systems  I0 (1994) 345-350  349 
in Fig. 4,  POD  is  an unbiased method to deter- 
mine the coherent structures (eigenflows) in the 
driven cavity flow. 
It has been found that the evaluation obtained 
by numerical integration of a  80-dimensional dy- 
namical system, which is  obtained by projecting 
the Navier-Stokes equations on the first 80 POD 
basis-functions, agrees well with direct numerical 
simulation data  at  Re = 22,000.  Also, when the 
Reynolds number in the 80-dimensional dynami- 
cal system is taken equal to 11,000 the integration 
of the dynamical system reproduces the periodic 
solution as obtained by DNS at Re =  11,000. 
Acknowledgements 
Both  the  Stichting  Nationale  Computerfa- 
ciliteiten (National Computing Facilities Founda- 
tion, NCF) with financial support from the Ne- 
derlandse  Organisatie  voor  Wetenschappelijk 
Onderzoek (Netherlands Organization for Scien- 
tific  Research,  NWO)  and  the  National 
Aerospace  Laboratory  NLR  are  gratefully  ac- 
knowledged for the use of supercomputer facili- 
ties. 
5.  I~,NS  of  lhree-dimensional  turbulenl  t]ow  in  ~ 
dri~ t,n  t':l~il3 
Reliable experimental results (see e.g. [6]) have 
pointed out that the flow in a three-dimensional 
driven  cavity  is  already unsteady for  Reynolds 
numbers much smaller than  10,000. The experi- 
ments reveal that the flow becomes turbulent at 
Reynolds numbers between  6,000  to  8,000, and 
that the first instability in the 3D flow is certainly 
not two-dimensional: the experimentally observed 
instability is  associated with 3D  Taylor-G6rtler- 
like  vortices.  Thus,  fluid  flows  in  the  direction 
perpendicular  to  the  lid-motion  are  significant. 
For this reason, full 3D DNS (with no-slip condi- 
tions  at  all  walls of the  cavity) have  been  per- 
formed  at  Re = 3,200  and  Re = 10,000.  A  1003 
economically stretched, staggered grid has been 
used  to  solve  the  incompressible  Navier-Stokes 
equations  numerically.  The  CPU-time  on  one 
processor  of  NLR's  NEC  SX-3  is  0.6  /~s  per 
gridpoint  and  time-step.  The  results  reproduce 
the  experimentally observed  Taylor-G6rtler-like 
vortices, and the computed mean velocities agree 
well with those  measured by Koseff and  Street 
[6]. As an illustration of the structures in the flow, 
Fig.  5  shows  an  instantaneous vorticity field  in 
the 3D  cavity at  Re =  10,000. In this figure, the 
orientation of the cavity is  such that the upper- 
plane  is  driven  from  the  left/back  to  the 
right/front. 
References 
[1]  W.C.  Reynolds,  in:  Wither  Turbulence?  Turbulence  at  the 
crossroads,  J.L.  Lumley  (ed.)  (Springer-Verlag,  Berlin, 
1990), 313. 
[2]  U.  Ghia,  K.N.  Ghia  and  C.T.  Shin,  J.  Comp.  Phys.  48 
(1982) 387. 
[3]  R. Verstappen, J.G. Wissink and A.E.P.  Veldman,  Appl. 
Sci. Research  51 (1993) 377. 
[4]  N. Aubry, P.  Holmes, J.L. Lumley and E. Stone,  J.  Fluid 
Mech.  192 (1988) 115. 
[5]  L. Sirovich and M. Kirby, Physica D  37 (1987) 126. 
[6] J.R.  Koseff and  R.L.  Street,  ASME Z  Fluids. Eng.  106 
(1984) 21. 
Rocl  ~, erslappen was born in Venray 
(The  Netherlands) on 22 June  1962. 
He received his education in applied 
mathematics  at  the  University  of 
Twente  in  the  Netherlands.  In  1989 
he received his Ph.D.  in engineering 
from  this  university.  After  that  he 
moved to the University of Groningen 
where he works on the field of com- 
putational fluid dynamics. 
After  a  working  period  of  several 
years  as  an  analyst  at  Delft  Hy- 
draulics, Jan $~,issink started to study 
mathematics  at  the  University  of 
Groningen. Since his graduation he is 
a  Ph.D.  student,  making  researches 
into numerical methods (especially fi- 
nite-volume  methods)  for  direct  nu- 
merical simulation of turbulence. 350  R. Verstappen et aL ~Future Generation  Computer  Systems 10 (1994) 345-350 
~,%illcm  t a/t:mict  studied mathemat  .... 
ics  at  the  University of  Groningen. 
The last year of his study he spent at 
the  National  Aerospace  Laboratory 
(NLR) in Amsterdam. After his grad- 
uation  in  1992, he  became  a  Ph.D. 
student  in  Groningen,  where  he  is 
doing research on the Proper Orthog- 
onal  Decomposition to  study  turbu- 
lence. 
Arthur  L.l'.  ~cldman  obtained  a 
Ph.D.  in Applied Mathematics from 
the University of Groningen. In 1977 
he  joined  the  National  Aerospace 
Laboratory  NLR  in  Amsterdam, 
where he was involved in various pro- 
jects  in  the  area  of  computational 
aero- and hydrodynamics. In addition, 
between 1984 and  1990 he was part- 
time professor of CFD at Delft Uni- 
versity of Technology. In 1990 he re- 
turned to Groningen, where he now 
occupies the chair in  Computational 
Mechanics. 

cldman obtained a Ph.D. in Applied Mathematics from the University of Groningen. In

erslappen was born in Venray (The Netherlands)

NARCIS 

ARTS repository - University of Groningen

   University of GroningenDirect Numerical Simulations of turbulent flow in a driven cavityVerstappen, R.; Wissink, J.G.; Cazemier, W.; Veldman, A.E.P.Published in:Future generation computer systemsIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1994Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Verstappen, R., Wissink, J. G., Cazemier, W., & Veldman, A. E. P. (1994). Direct Numerical Simulations ofturbulent flow in a driven cavity. Future generation computer systems, 10(2-3), 345-350.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 20-06-2022F G C S g3"'S'FEM S I St.~VII t,~ Future Generation Computer Systems 10 (1994) 345-350 Di[rect Numerical Simulations of turbulent flow in a driven cavity R. Verstappen *, J.G. Wissink, W. Cazemier, A.E.P. Veldman Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands .~i)stract Direct numerical simulations (DNS) of 2 and 3D turbulent flows in a lid-driven cavity have been performed. DNS are numerical solutions of the unsteady (here: incompressible) Navier-Stokes equations that compute the evolution of all dynamically significant scales of motion. In view of the large computing resources needed for DNS cost-effective and accurate numerical methods are to be selected. Here, various-order accurate spatial discretization methods for DNS have been evaluated by applying them to the 2D driven cavity at Re = 22,000. To analyze the results of the DNS of the 2D flow in a driven cavity at Re = 22,000 the proper orthogonal decomposition (POD) technique has been applied. POD is an unbiased method to determine coherent structures. The Galerkin projection of the Navier-Stokes equations on the space spanned by the POD-basis-functions yields a relatively low-dimensional set of ordinary differential equations that mimics the dynamics of the Navier-Stokes equations. 3D DNS with no-slip conditions at all walls of the cavity have been performed at both Re = 3,200 and Re = 10,000. The results reproduce the experimentally observed Taylor-G6rtler-like vortices. Key words: Direct Numerical Simulation; Turbulence; Navier-Stokes equations; Driven cavity; Numerical methods; Proper orthogonal decomposition 1. Introduction Direct Numerical  Simulations (DNS) are nu- merical solutions of  the unsteady (here: incom- pressible) Navier-Stokes equations that compute  the evolution of all dynamically significant scales of  motion. Unfortunately, DNS of engineering flows exhaust the largest available computing re- sources by requiring machines in the exa(1018) flops range with exabytes of memory.  Thus, for engineering applications acceptable computa-  tional effort can only be obtained by modelling * Corresponding author. the turbulence of the flow. However, with existing turbulence models the simulation accuracy re- quired by industry cannot be reached in many technological applications, making turbulence modelling the main pacing item in the develop- ment  of applied CFD codes. It  is generally ex- pected that DNS will play the key role in obtain- ing bet ter  turbulence models; DNS have now become the chief source of highly detailed data for turbulence modelers and researchers [1]. In this paper,  we report  on DNS of turbulent flows in two- and three-dimensional,  lid-driven, cavities and on their mathematical  analysis. The two-dimensional simulations have been per- formed to evaluate numerical methods for DNS 0167-739X/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0167-739X(94)00032-A 346 R. Verstappen et aL / Future Generation Computer Systems 10 (1994) 345-350 and to study the properties of the proper  orthog- onal decomposition (POD) technique. 2.  I ) N S  of t ~ v o - d i m e n s i o n a l  f lov~ i n  a d r i v e n  tacit> In two dimensions, the flow converges to a steady state for Reynolds numbers up to 10,000 [2]. For larger Reynolds numbers, 2D driven cav- ity flows undergo a transition to unsteadiness. Our direct simulations exhibit that the flow con- verges to a purely periodically oscillating state at Re = 11,000. The amplitude of the periodic mo- tion is small: in terms of the total kinetic energy 0.2% of the mean value (more details can be found in [3]). DNS of the motion of the vortical structures at Re = 22,000 reveals that the dynam- ics has become chaotic: the correlation dimension is approximately 2.8, the Kolmogorov entropy is approximately 3.0. As expected, the number of vortices has increased compared to the simula- tion at Re = 11,000. Furthermore,  the motion of vortical structures is no longer confined to small regions near the corners of the cavity. To illus- trate this, three instantaneous vorticity fields are shown in Fig. 1. 3. Evaluation of numerical methods fi~r DNS In view of the large computing resources needed for DNS a careful selection of cost-effec- tive and accurate numerical methods is impor- tant. Here,  various-order accurate spatial dis- cretization methods for DNS have been evalu- ated by applying them to the driven cavity at Re = 22,000. The evaluation focusses on the accu- racy with which the mean kinetic energy is com- puted using a 2002 economically stretched, stag- gered grid, and a time-step that is so small that the error resulting from the time-integration is negligible. Central as well as upwind discretiza- tions have been considered. The central method consisting of a rnth order central discretization of the convective term (in conservative form) com- bined with a nth order interpolation, and a k th  order central discretization of the diffusive term is denoted by CmlnCk. Likewise, an upwind method is denoted by UmlnCk if the conserva- tive term (now in non-conservative form) is dis- cretized using a mth order  upwind-biased method. A Lagrange interpolation is used to ob- tain the coefficients of the stencils. Since stencils with more than one gridpoint on either side of the point at which the derivative is to be calcu- lated cannot be applied near the boundaries, high-order methods are gradually replaced by lower-order methods in the vicinity of the bound- aries. Reference data for the comparison is ob- tained using the U514C6 method on a 4002 stretched grid. The main results of the compari- son are shown in Fig. 2. From this figure, it can be concluded that C414C6 and U716C8 perform the best of the methods considered here. ....... ,. iii " . . . . . . . . . .  0 9  . . . .  ' 0g ~. " " . i. 1~.7 ~ : !  . :  i o.4 : .:~ c~3 . ! . " "  iii / i i  .......... ' ,   ,iii : ! Fig.  1. S n a p s h o t s  o f  t he  vor t ic i ty  in a 2 D  dr iven  cavi ty  (Re = 22,000).  R. Verstappen et al. / Future Generation Computer Systems 10 (1994) 345-350 347 To analyze the results of the DNS of the turbulent flow in a driven cavity at Re = 22,000, we make use of proper orthogonal decomposition (POD). That is (see e.g. [4]), from fluctuating flow fields a set of (say N)  basis-functions is extracted, which is optimal in the sense that any other decomposition (e.g. in N Fourier-modes) cap- tures less kinetic energy. The Galerkin projection of the Navier-Stokes equations on the space spanned by the basis-functions yields a relatively low-dimensional set of ordinary differential equa- tions. This 'cheap' (compared to Navier-Stokes) set can be analyzed using tools from dynamical systems theory, and thus we can advance our understanding of turbulence, provided that this set mimics the dynamics of the Navier-Stokes % /.12' L~ E 2 J Ib ~ X Fig. 2. Errors in the mean kinetic energy for a number of spatial discretization methods (2D driven cavity, Re = 22,000). I (X) R. Verstappen et aL /Future Generation Computer Systems 10 (1994) 345-350 > "o / l ) O l )  - -  . . j - - "  l~ourier ___ so / 70 / 6O 50 ,, / J " "  40 / 30 / 20 / l0 y/'" I l t l l l l~;C'l"  {/1" I'~.~t"q.s-ltlllCli(~ll>; Fig. 3. Comparison of POD and Fourier-modes. equations. To investigate how well it does this, we have performed a direct simulation of the turbu- lent flow in a 2D driven cavity at Re = 22,000 during more than 700 large-eddy turn-around times to obtain the data necessary to compute up to 700 basis-functions. These basis-functions are computed using the so-called snapshot method [5]. The energy captured by POD-modes has been 348 Fig. 5. Snapshot of the vorticity in a 3D driven cavity at Re = 10,000. compared to the energy captured by Fourier- modes (Fig. 3), and, as expected, POD is far superior to Fourier. Furthermore, as can be seen (18 0.7 0,6 0.5 1).4 0.3 {).2 (1.1 0.9 0.8 0.7 0.6 >0.5 0.4 0.3 0.2 0.1 ~ ~ fir':l basis lunc l ion I)? (14 0.6 (I.8 x ,i.q~ li.g F 0-6 F >, (L5 i 0"11 i )?  !,I i r st_',ctmd basis hmclion 0.? () 4 II 6 (18 lhird basis ltmclion {)2 (~.4 0.6 0.8 X (D) lout01 basis hmclion 0.8 0.7 ¢ 0.6 (1.4 } 0.2 () 2 ( } 4 D.6 0.8 x Fig. 4. The first four POD basis-functions. R. Verstappen et al. / Future Generation Computer Systems I0 (1994) 345-350 349 in Fig. 4, POD is an unbiased method to deter- mine the coherent structures (eigenflows) in the driven cavity flow. It has been found that the evaluation obtained by numerical integration of a 80-dimensional dy- namical system, which is obtained by projecting the Navier-Stokes equations on the first 80 POD basis-functions, agrees well with direct numerical simulation data at Re = 22,000. Also, when the Reynolds number in the 80-dimensional dynami- cal system is taken equal to 11,000 the integration of the dynamical system reproduces the periodic solution as obtained by DNS at Re = 11,000. A c k n o w l e d g e m e n t s  Both the Stichting Nationale Computerfa- ciliteiten (National Computing Facilities Founda- tion, NCF) with financial support from the Ne- derlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for Scien- tific Research, NWO) and the National Aerospace Laboratory NLR are gratefully ac- knowledged for the use of supercomputer facili- ties. 5. I~,NS of  l h r e e - d i m e n s i o n a l  t u r b u l e n l  t]ow in ~ dri~ t,n t':l~il3 Reliable experimental results (see e.g. [6]) have pointed out that the flow in a three-dimensional driven cavity is already unsteady for Reynolds numbers much smaller than 10,000. The experi- ments reveal that the flow becomes turbulent at Reynolds numbers between 6,000 to 8,000, and that the first instability in the 3D flow is certainly not two-dimensional: the experimentally observed instability is associated with 3D Taylor-G6rtler- like vortices. Thus, fluid flows in the direction perpendicular to the lid-motion are significant. For this reason, full 3D DNS (with no-slip condi- tions at all walls of the cavity) have been per- formed at Re = 3,200 and Re = 10,000. A 1003 economically stretched, staggered grid has been used to solve the incompressible Navier-Stokes equations numerically. The CPU-time on one processor of NLR's NEC SX-3 is 0.6 /~s per gridpoint and time-step. The results reproduce the experimentally observed Taylor-G6rtler-like vortices, and the computed mean velocities agree well with those measured by Koseff and Street [6]. As an illustration of the structures in the flow, Fig. 5 shows an instantaneous vorticity field in the 3D cavity at Re = 10,000. In this figure, the orientation of the cavity is such that the upper- plane is driven from the left/back to the right/front. R e f e r e n c e s  [1] W.C. Reynolds, in: Wither Turbulence? Turbulence at the crossroads, J.L. Lumley (ed.) (Springer-Verlag, Berlin, 1990), 313. [2] U. Ghia, K.N. Ghia and C.T. Shin, J. Comp. Phys. 48 (1982) 387. [3] R. Verstappen, J.G. Wissink and A.E.P. Veldman, Appl. Sci. Research 51 (1993) 377. [4] N. Aubry, P. Holmes, J.L. Lumley and E. Stone, J. Fluid Mech. 192 (1988) 115. [5] L. Sirovich and M. Kirby, Physica D 37 (1987) 126. [6] J.R. Koseff and R.L. Street, ASME Z Fluids. Eng. 106 (1984) 21. Rocl ~, erslappen was born in Venray (The Netherlands) on 22 June 1962. He received his education in applied mathematics at the University of Twente in the Netherlands. In 1989 he received his Ph.D. in engineering from this university. After that he moved to the University of Groningen where he works on the field of com- putational fluid dynamics. After a working period of several years as an analyst at Delft Hy- draulics, Jan $~,issink started to study mathematics at the University of Groningen. Since his graduation he is a Ph.D. student, making researches into numerical methods (especially fi- nite-volume methods) for direct nu- merical simulation of turbulence. 350 R. Verstappen et aL ~Future Generation Computer Systems 10 (1994) 345-350 ~,%illcm t a/t:mict studied mathemat . . . .  ics at the University of Groningen. The last year of his study he spent at the National Aerospace Laboratory (NLR) in Amsterdam. After his grad- uation in 1992, he became a Ph.D. student in Groningen, where he is doing research on the Proper Orthog- onal Decomposition to study turbu- lence. Arthur L.l'. ~cldman obtained a Ph.D. in Applied Mathematics from the University of Groningen. In 1977 he joined the National Aerospace Laboratory NLR in Amsterdam, where he was involved in various pro- jects in the area of computational aero- and hydrodynamics. In addition, between 1984 and 1990 he was part- time professor of CFD at Delft Uni- versity of Technology. In 1990 he re- turned to Groningen, where he now occupies the chair in Computational Mechanics. 

   University of GroningenDirect Numerical Simulations of turbulent flow in a driven cavityVerstappen, R.; Wissink, J.G.; Cazemier, W.; Veldman, A.E.P.Published in:Future generation computer systemsIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1994Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Verstappen, R., Wissink, J. G., Cazemier, W., & Veldman, A. E. P. (1994). Direct Numerical Simulations ofturbulent flow in a driven cavity. Future generation computer systems, 10(2-3), 345-350.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 17-07-2023F G C S g3"'S'FEM S I St.~VII t,~ Future Generation Computer Systems 10 (1994) 345-350 Di[rect Numerical Simulations of turbulent flow in a driven cavity R. Verstappen *, J.G. Wissink, W. Cazemier, A.E.P. Veldman Department of Mathematics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands .~i)stract Direct numerical simulations (DNS) of 2 and 3D turbulent flows in a lid-driven cavity have been performed. DNS are numerical solutions of the unsteady (here: incompressible) Navier-Stokes equations that compute the evolution of all dynamically significant scales of motion. In view of the large computing resources needed for DNS cost-effective and accurate numerical methods are to be selected. Here, various-order accurate spatial discretization methods for DNS have been evaluated by applying them to the 2D driven cavity at Re = 22,000. To analyze the results of the DNS of the 2D flow in a driven cavity at Re = 22,000 the proper orthogonal decomposition (POD) technique has been applied. POD is an unbiased method to determine coherent structures. The Galerkin projection of the Navier-Stokes equations on the space spanned by the POD-basis-functions yields a relatively low-dimensional set of ordinary differential equations that mimics the dynamics of the Navier-Stokes equations. 3D DNS with no-slip conditions at all walls of the cavity have been performed at both Re = 3,200 and Re = 10,000. The results reproduce the experimentally observed Taylor-G6rtler-like vortices. Key words: Direct Numerical Simulation; Turbulence; Navier-Stokes equations; Driven cavity; Numerical methods; Proper orthogonal decomposition 1. Introduction Direct Numerical  Simulations (DNS) are nu- merical solutions of  the unsteady (here: incom- pressible) Navier-Stokes equations that compute  the evolution of all dynamically significant scales of  motion. Unfortunately, DNS of engineering flows exhaust the largest available computing re- sources by requiring machines in the exa(1018) flops range with exabytes of memory.  Thus, for engineering applications acceptable computa-  tional effort can only be obtained by modelling * Corresponding author. the turbulence of the flow. However, with existing turbulence models the simulation accuracy re- quired by industry cannot be reached in many technological applications, making turbulence modelling the main pacing item in the develop- ment  of applied CFD codes. It  is generally ex- pected that DNS will play the key role in obtain- ing bet ter  turbulence models; DNS have now become the chief source of highly detailed data for turbulence modelers and researchers [1]. In this paper,  we report  on DNS of turbulent flows in two- and three-dimensional,  lid-driven, cavities and on their mathematical  analysis. The two-dimensional simulations have been per- formed to evaluate numerical methods for DNS 0167-739X/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0167-739X(94)00032-A 346 R. Verstappen et aL / Future Generation Computer Systems 10 (1994) 345-350 and to study the properties of the proper  orthog- onal decomposition (POD) technique. 2.  I ) N S  of t ~ v o - d i m e n s i o n a l  f lov~ i n  a d r i v e n  tacit> In two dimensions, the flow converges to a steady state for Reynolds numbers up to 10,000 [2]. For larger Reynolds numbers, 2D driven cav- ity flows undergo a transition to unsteadiness. Our direct simulations exhibit that the flow con- verges to a purely periodically oscillating state at Re = 11,000. The amplitude of the periodic mo- tion is small: in terms of the total kinetic energy 0.2% of the mean value (more details can be found in [3]). DNS of the motion of the vortical structures at Re = 22,000 reveals that the dynam- ics has become chaotic: the correlation dimension is approximately 2.8, the Kolmogorov entropy is approximately 3.0. As expected, the number of vortices has increased compared to the simula- tion at Re = 11,000. Furthermore,  the motion of vortical structures is no longer confined to small regions near the corners of the cavity. To illus- trate this, three instantaneous vorticity fields are shown in Fig. 1. 3. Evaluation of numerical methods fi~r DNS In view of the large computing resources needed for DNS a careful selection of cost-effec- tive and accurate numerical methods is impor- tant. Here,  various-order accurate spatial dis- cretization methods for DNS have been evalu- ated by applying them to the driven cavity at Re = 22,000. The evaluation focusses on the accu- racy with which the mean kinetic energy is com- puted using a 2002 economically stretched, stag- gered grid, and a time-step that is so small that the error resulting from the time-integration is negligible. Central as well as upwind discretiza- tions have been considered. The central method consisting of a rnth order central discretization of the convective term (in conservative form) com- bined with a nth order interpolation, and a k th  order central discretization of the diffusive term is denoted by CmlnCk. Likewise, an upwind method is denoted by UmlnCk if the conserva- tive term (now in non-conservative form) is dis- cretized using a mth order  upwind-biased method. A Lagrange interpolation is used to ob- tain the coefficients of the stencils. Since stencils with more than one gridpoint on either side of the point at which the derivative is to be calcu- lated cannot be applied near the boundaries, high-order methods are gradually replaced by lower-order methods in the vicinity of the bound- aries. Reference data for the comparison is ob- tained using the U514C6 method on a 4002 stretched grid. The main results of the compari- son are shown in Fig. 2. From this figure, it can be concluded that C414C6 and U716C8 perform the best of the methods considered here. ....... ,. iii " . . . . . . . . . .  0 9  . . . .  ' 0g ~. " " . i. 1~.7 ~ : !  . :  i o.4 : .:~ c~3 . ! . " "  iii / i i  .......... ' ,   ,iii : ! Fig.  1. S n a p s h o t s  o f  t he  vor t ic i ty  in a 2 D  dr iven  cavi ty  (Re = 22,000).  R. Verstappen et al. / Future Generation Computer Systems 10 (1994) 345-350 347 To analyze the results of the DNS of the turbulent flow in a driven cavity at Re = 22,000, we make use of proper orthogonal decomposition (POD). That is (see e.g. [4]), from fluctuating flow fields a set of (say N)  basis-functions is extracted, which is optimal in the sense that any other decomposition (e.g. in N Fourier-modes) cap- tures less kinetic energy. The Galerkin projection of the Navier-Stokes equations on the space spanned by the basis-functions yields a relatively low-dimensional set of ordinary differential equa- tions. This 'cheap' (compared to Navier-Stokes) set can be analyzed using tools from dynamical systems theory, and thus we can advance our understanding of turbulence, provided that this set mimics the dynamics of the Navier-Stokes % /.12' L~ E 2 J Ib ~ X Fig. 2. Errors in the mean kinetic energy for a number of spatial discretization methods (2D driven cavity, Re = 22,000). I (X) R. Verstappen et aL /Future Generation Computer Systems 10 (1994) 345-350 > "o / l ) O l )  - -  . . j - - "  l~ourier ___ so / 70 / 6O 50 ,, / J " "  40 / 30 / 20 / l0 y/'" I l t l l l l~;C'l"  {/1" I'~.~t"q.s-ltlllCli(~ll>; Fig. 3. Comparison of POD and Fourier-modes. equations. To investigate how well it does this, we have performed a direct simulation of the turbu- lent flow in a 2D driven cavity at Re = 22,000 during more than 700 large-eddy turn-around times to obtain the data necessary to compute up to 700 basis-functions. These basis-functions are computed using the so-called snapshot method [5]. The energy captured by POD-modes has been 348 Fig. 5. Snapshot of the vorticity in a 3D driven cavity at Re = 10,000. compared to the energy captured by Fourier- modes (Fig. 3), and, as expected, POD is far superior to Fourier. Furthermore, as can be seen (18 0.7 0,6 0.5 1).4 0.3 {).2 (1.1 0.9 0.8 0.7 0.6 >0.5 0.4 0.3 0.2 0.1 ~ ~ fir':l basis lunc l ion I)? (14 0.6 (I.8 x ,i.q~ li.g F 0-6 F >, (L5 i 0"11 i )?  !,I i r st_',ctmd basis hmclion 0.? () 4 II 6 (18 lhird basis ltmclion {)2 (~.4 0.6 0.8 X (D) lout01 basis hmclion 0.8 0.7 ¢ 0.6 (1.4 } 0.2 () 2 ( } 4 D.6 0.8 x Fig. 4. The first four POD basis-functions. R. Verstappen et al. / Future Generation Computer Systems I0 (1994) 345-350 349 in Fig. 4, POD is an unbiased method to deter- mine the coherent structures (eigenflows) in the driven cavity flow. It has been found that the evaluation obtained by numerical integration of a 80-dimensional dy- namical system, which is obtained by projecting the Navier-Stokes equations on the first 80 POD basis-functions, agrees well with direct numerical simulation data at Re = 22,000. Also, when the Reynolds number in the 80-dimensional dynami- cal system is taken equal to 11,000 the integration of the dynamical system reproduces the periodic solution as obtained by DNS at Re = 11,000. A c k n o w l e d g e m e n t s  Both the Stichting Nationale Computerfa- ciliteiten (National Computing Facilities Founda- tion, NCF) with financial support from the Ne- derlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization for Scien- tific Research, NWO) and the National Aerospace Laboratory NLR are gratefully ac- knowledged for the use of supercomputer facili- ties. 5. I~,NS of  l h r e e - d i m e n s i o n a l  t u r b u l e n l  t]ow in ~ dri~ t,n t':l~il3 Reliable experimental results (see e.g. [6]) have pointed out that the flow in a three-dimensional driven cavity is already unsteady for Reynolds numbers much smaller than 10,000. The experi- ments reveal that the flow becomes turbulent at Reynolds numbers between 6,000 to 8,000, and that the first instability in the 3D flow is certainly not two-dimensional: the experimentally observed instability is associated with 3D Taylor-G6rtler- like vortices. Thus, fluid flows in the direction perpendicular to the lid-motion are significant. For this reason, full 3D DNS (with no-slip condi- tions at all walls of the cavity) have been per- formed at Re = 3,200 and Re = 10,000. A 1003 economically stretched, staggered grid has been used to solve the incompressible Navier-Stokes equations numerically. The CPU-time on one processor of NLR's NEC SX-3 is 0.6 /~s per gridpoint and time-step. The results reproduce the experimentally observed Taylor-G6rtler-like vortices, and the computed mean velocities agree well with those measured by Koseff and Street [6]. As an illustration of the structures in the flow, Fig. 5 shows an instantaneous vorticity field in the 3D cavity at Re = 10,000. In this figure, the orientation of the cavity is such that the upper- plane is driven from the left/back to the right/front. R e f e r e n c e s  [1] W.C. Reynolds, in: Wither Turbulence? Turbulence at the crossroads, J.L. Lumley (ed.) (Springer-Verlag, Berlin, 1990), 313. [2] U. Ghia, K.N. Ghia and C.T. Shin, J. Comp. Phys. 48 (1982) 387. [3] R. Verstappen, J.G. Wissink and A.E.P. Veldman, Appl. Sci. Research 51 (1993) 377. [4] N. Aubry, P. Holmes, J.L. Lumley and E. Stone, J. Fluid Mech. 192 (1988) 115. [5] L. Sirovich and M. Kirby, Physica D 37 (1987) 126. [6] J.R. Koseff and R.L. Street, ASME Z Fluids. Eng. 106 (1984) 21. Rocl ~, erslappen was born in Venray (The Netherlands) on 22 June 1962. He received his education in applied mathematics at the University of Twente in the Netherlands. In 1989 he received his Ph.D. in engineering from this university. After that he moved to the University of Groningen where he works on the field of com- putational fluid dynamics. After a working period of several years as an analyst at Delft Hy- draulics, Jan $~,issink started to study mathematics at the University of Groningen. Since his graduation he is a Ph.D. student, making researches into numerical methods (especially fi- nite-volume methods) for direct nu- merical simulation of turbulence. 350 R. Verstappen et aL ~Future Generation Computer Systems 10 (1994) 345-350 ~,%illcm t a/t:mict studied mathemat . . . .  ics at the University of Groningen. The last year of his study he spent at the National Aerospace Laboratory (NLR) in Amsterdam. After his grad- uation in 1992, he became a Ph.D. student in Groningen, where he is doing research on the Proper Orthog- onal Decomposition to study turbu- lence. Arthur L.l'. ~cldman obtained a Ph.D. in Applied Mathematics from the University of Groningen. In 1977 he joined the National Aerospace Laboratory NLR in Amsterdam, where he was involved in various pro- jects in the area of computational aero- and hydrodynamics. In addition, between 1984 and 1990 he was part- time professor of CFD at Delft Uni- versity of Technology. In 1990 he re- turned to Groningen, where he now occupies the chair in Computational Mechanics. 

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