-
Theoret. Comput. Fluid Dynamics (1992) 3:219-229 Theoretical and
Computational Fluid Dynamics © Springer-Verlag 1992
Parallel Spectral-Element-Fourier Simulation of Turbulent Flow
over Riblet-Mounted Surfaces 1
Douglas Chu, Ron Henderson, and George Em Karniadakis
Department of Mechanical and Aerospace Engineering and Program
in Applied and Computational Mathematics,
Princeton University, Princeton, NJ 08544, U.S.A.
Communicated by M,Y. Hussaini
Received 24 July 1991 and accepted 29 August 1991
Abstract. The flow in a channel with its lower wall mounted with
streamwise V-shaped riblets is simulated using a highly efficient
spectral-element-Fourier method. The range of Reynolds numbers
investigated is 500 to 4000, which corresponds to laminar,
transitional, and turbulent flow states. Our results suggest that
in the laminar regime there is no drag reduction, while in the
transitional and turbulent regimes drag reduction up to 10% exists
for the riblet-mounted wall in comparison with the smooth wall of
the channel. For the first time, we present detailed turbulent
statistics in a complex geometry. These results are in good
agreement with available experimental data and provide a
quantitative picture of the drag-reduction mechanism of the
riblets.
1. Introduction
Over the past two decades the field of simulation sciences (and,
in particular, computational fluid dynamics) has matured rapidly;
today two- and three-dimensional simulations of unsteady flows are
ordinary tools in fluid flow analysis. Simultaneous advances both
in algorithms as well as hardware have made possible simulations
involving more than 1 million degrees of freedom, thus allowing
accurate solutions of several fundamental flow problems [1]. Most
of the problems considered in the past, however, still involve
significant simplifications regarding both geometry and parameter
range. Recently there has been an increasing trend toward
simulation of more complicated flow problems with a level of
complexity that is close to industrial needs, and at least
equivalent to experimental laboratory conditions. The consideration
of these complex-geometry, complex-physics flows has initi- ated
the development of more flexible discretization schemes that are
typically based on a hybrid construction of the most popular
diseretization algorithms, i.e., finite differences, finite
elements, and spectral methods.
A typical example of such a confluence of numerical algorithms
is the spectral-element method [2],
1 This work was supported by National Science Foundation Grants
CTS-8906432, CTS-8906911, and CTS-8914422, AFOSR Grant No.
AFOSR-90-0124, and DARPA Grant No. N00014-86-K-0759. The
computations were performed on the Cray Y/MP's of NAS at NASA Ames
and the Pittsburgh Supercomputing Center, and on the Intel 32-node
iPSC/860 hypercube at Princeton University.
219
-
220 D. Chu, R. Henderson, and G.E. Karniadakis
H = l . 9
h=I 0.2
~ 0 ~
Lx=2.0
Flow
II IIIII - - J "
Figure 1. Geometry definition and skeleton of the
spectral-element mesh.
I-3] which is based on two weighted-residual techniques: finite
elements and spectral methods. The combination of spectral-like
accuracy with the flexibility in handling complex geometries has
made this method quite successful in a number of fluid dynamics
applications, including flows in the transitional and turbulent
regimes I-4]. On the other hand, the application of the
spectral-element method in simulating other types of flows, such as
flows over rough walls, is not straightforward and requires the
construction of hybrid spectral-element/finite-difference schemes,
which has been demonstrated re- cently in [5]. The hybrid
spectral-element/finite-difference technique provides a very
efficient approach to handling a rough wall geometry involving
extreme disparities in length scales.
There are also other situations involving complex-geometry flows
where efficiency can be explored by exploiting certain symmetries
or homogeneities in the geometry. The model problem we consider in
this study is flow over a surface mounted with streamwise aligned
riblets (Figure 1); the geometry is homogeneous in the streamwise
direction. This computational domain suggests the use of a hybrid
spectral discretization where Fourier expansions are employed in
the streamwise direction to represent data and unknowns, and a
standard two-dimensional spectral-element discretization is used in
the x-y planes. This algorithm allows the use of fast Fourier
transforms and thus substantially reduces both solution time and
memory requirements. Efficiency can be further enhanced by
implementing the spectral-element-Fourier algorithm in parallel by
decoupling the linear part of the Navier-Stokes equations and
computing each (or a group) of the Fourier modes on a different
processor.
The flow over streamwise riblets we consider here has important
technological applications. The effect of riblets is to reduce drag
(skin friction on the surface) by as much as 8~o in the turbulent
regime, as has been found experimentally in a number of
investigations [6], [7]. A more quantitative analysis of the
effects of riblets and some preliminary results obtained using
three-dimensional spectral-element simulations were presented in
[8]. In this work we extend these results to higher Reynolds
numbers and a wider computational domain using a highly efficient
spectral-element-Fourier code. We present for the first time
turbulent statistics of the flow over riblets that verify the drag
reduction observed experimentally, and which justify the use of
riblets as drag-reducing devices.
This paper is organized as follows: in Section 2 we present the
spectral-element-Fourier method and demonstrate its fast
(exponential) convergence for an exact three-dimensional solution
of the Navier-Stokes equations. Also included in this section is a
discussion on the parallel implementation of the hybrid algorithm
on the Intel iPSC/80 hypercube. In Section 3 we present results of
the simulation of turbulent flow over triangular riblets; finally,
we conclude in Section 4 with a brief discussion.
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Parallel Spectral-Element-Fourier Simulation of Turbulent Flow
over Riblet-Mounted Surfaces 221
2. Mathematical Formulation
We consider the flow of incompressible Newtonian fluids governed
by the Navier-Stokes equations of motion,
V.v = 0 in f~, (1)
Dv Vp + Re-lV2v in fl, (2)
Dt p
where v(x, t) is the velocity field, p is the static pressure, p
is the density, v is the kinematic viscosity, Re = W H / v is the
Reynolds number based on the channel height H = 1.9 measured from
the midpoint of the riblet to the upper wall (see Figure 1), and W
is the mean streamwise velocity. Note that D here denotes the total
derivative.
In order to sustain the flow, the momentum equation (2) should
include a nonzero pressure gradient in the prevailing direction of
motion. In practice, however, the pressure drop is an unknown
quantity, especially in complex-geometry flows or turbulent flows.
It is preferable, therefore, to sustain the motion by imposing a
volume (or mass) flow rate Q(t). This can be done efficiently by
solving in a preprocessing stage for a Green's function v* which
satisfies the equations of' motion for an equivalent Stokes flow
driven by a unit pressure drop and solving at subsequent time steps
the homogeneous Navier-Stokes equations to obtain an intermediate
solution v h. The requisite nondimensional forcing term Ap can now
be found by requiring that the mass flow rate remain at a
prescribed level, yielding the final velocity field as
v = v h + v*Ap. (3)
A numerical solution of the above system of equations will be
obtained in the domain fl shown in Figure 1. The riblets have a
height h = 0.2 and base s = 0.2 units in length. At Re = 3500,
these dimensions correspond to approximately 17.1 viscous wall
units; please see Table 1 for details. The streamwise length is Lz
= 5 and the spanwise length is L x = 2. This computational domain
is an extension of the one simulated in [8].
w(x, y, z, t) - ~=o I Wm(X, y, t) eilJzz" (4)
p(x, y, z, t) I p=(x, y, t)
Here, fl = 2n/L2 is the wave number associated with the
homogeneous direction z, and Lz is the periodicity length.
If we now substitute the expressions (4) into the
time-discretized equations (see [8] for details) and follow a
Galerkin approach in the z-direction we obtain the equations for
each Fourier mode m. The nonlinear equation is handled by
evaluating all products in physical space, while all z-derivatives
are computed in Fourier space [10]. The pressure and viscous
equations, however, are elliptic and linear with respect to z, so
they can be decomposed into M separate equations. A typical
equation for the velocity component v~ ÷1 (ruth mode at time level
(n + 1).At) is
n+l~ra [ 0 2 3 2 ] Vm -- __ Re-1 + __ m2 f12 ¥n+l in tim,
(5)
At b•
where ~m is the mth two-dimensional computational domain and {%
is a known forcing term computed at an earlier substep. This
equation can be expressed as a standard Helmholtz equation and
solved using a two-dimensional spectral-element method [10].
To test the accuracy of the proposed spectral-element-Fourier
spatial discretization method, we consider here the following exact
solution of the three-dimensional incompressible Navier-Stokes
equation:
V ~ e ax cos #y cos mz
e ax sin #y cos mz
' co spys inmz
-
222 D. Chu, R. Henderson, and G.E. Karniadakis
where #, 2, and m are given real numbers which determine the
form of the solution (flow pattern). The associated pressure field
is computed so as to satisfy the momentum equation in the
y-direction with zero forcing (fy = 0):
[ m2 2 2 ] 2~c°s2/~Y [ ~ ] p = # R e - l e ~x cos # y cos mz 1 +
p2 ~2 + e ~ 1 + .
Using this solution for the pressure, we can determine the
applied force necessary to satisfy the x- and z-momentum equations.
These functions are given by
I )3 ~,m 2 f=(x, y, z) = R e - l e ~x cos #y cos mz - 2 2 + m E
- - - - + - -
# #
+ #e2~X[cos2 #y sin 2 mz - cos 2 mz sin 2 #y],
I,~ 3 ,~2 m m 3
f z (x , y, z) = R e - l e ~x cos #y sin mz ~ - 2m + + # #
A e 2 ~ X sin 2mz[2# + #2]. + 2m
] [ 'f] + 2 # + # 2 +e 2z=cos 2/.ty 2),+
2 2 P - 2m/z m 2/t2 m ~-]
This solution is periodic in the z-direction and thus Fourier
expansions are appropriate. In Figure 2 we demonstrate the
exponential convergence of the method in a simple geometry as the
polynomial order N is increased for a fixed number of elements K
and Fourier modes M (for this test case the values of the
parameters were Re = 0.1, # = 2re, 2 = 0.15, and m = 2.0). For the
particular case K = 16, M = 4, N = 12 the error in the computed
solution is reduced to machine zero. We note that the preceding
case is a test for the accuracy of the spatial discretization
scheme, since it involves only a steady-state solution. The
temporal discretization scheme of our method has been discussed at
length in [9] and [11]; time splitting errors have been eliminated
by the use of a new high-order pressure boundary condition and a
small time step (see [9] and [11] for details).
As mentioned above, applying a Fourier decomposition in the
z-direction yields separate equations for each Fourier mode m with
regard to the linear pressure and viscous equations. The
computational domain may then be maped as in Figure 3 onto a
network of processors in three separate phases: during the first
phase the domain is mapped in sheets of y - z planes, within which
FFTs are performed and nonlinear products computed with the
processors utilized as a simple array network in the second phase
the symmetries of the hypercube allow an efficient global transpose
(complete
0.01
0.001
0.0001
lO-S
10-8
-3 10 -~
10_a
10-~o
10-1~ 10-13
Bxt0 =t~
I
2 4 6 8 10 12 No,a~,
Figure 2. Error convergence of the spectral-element-Fourier
method.
I SHEETS: y - z planes 1 -- FFT in z -- Compute N(v) -- IFFT in
z
1 I I - - - - T - - - O - - - O - - - O . . . . . O - - -
I I I I FRAMES: x - y planes 1
- - Time Advance SEM solvers Compute derivatives
Processor Network
Global Exchange
(D, ,.©
Figure 3. Schematic for the 3-phase mapping of the computa-
tional domain onto a processor network (hypercube). All pro-
cessors have local memory and may perform computations and message
passing concurrently.
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Parallel Spectral-Element-Fourier Simulation of Turbulent Flow
over Riblet-Mounted Surfaces 223
exchange) of data across the network so that it arrives as x - y
f r a m e s within processors, each frame representing a single
Fourier mode; during this third phase, the spectral element solvers
are applied and the solution is advanced to the next time step. On
the Intel iPSC/860 parallel supercomputer the message-passing
system operates independently of the microprocessors, allowing an
exchange to be initiated while continuing calculations in phases
one and three. Tests on the 32-node iPCS/860 at Princeton
University indicate that for large simulations (like the riblet
problem considered in this paper) an effective rate of 8 megaflops
per processor may be achieved for double precision calcula- tions,
giving performance superior to that of a single processor on the
Cray Y/MP. Details of the parallel implementation are described in
[17].
3. Results
In [8] full three-dimensional spectral-element results were
presented for the riblet problem in the laminar and transitional
regimes. We have validated those figures in a wider computational
domain (see Figure 1) and extended them into the turbulent regime
using the spectral-element-Fourier formulation discussed in Section
2. After verifying that the bulk flow properties were consistent
with those reported in [8], turbulent statistics were computed and
are presented below. Unless noted otherwise, the results discussed
in this section correspond to R e = 3500 (see definition in Section
2) and a spectral-element discretization of K = 100 elements,
polynomial order N = 8, and M = 16 Fourier modes.
3.1. Mean Flow Properties
Our simulations were performed with a constant flow rate Q ( t )
= 3.8, thus fixing the bulk velocity W = 1. The initial flow was
perturbed with a body force; exponential amplification occurred and
the simulation was carried out for a large number of convective
time units ( H / W ) , until a stationary state was reached. In
Figure 4 the time history of the nondimensional pressure gradient
(Ap from (3)) is shown soon after the initial perturbation. A
stationary state has been reached by approximately t = 550. We note
that this nondimensional forcing term oscillates with an amplitude
of roughly 8%, and that it contains numerous high- and
low-frequency fluctuations.
In Figure 5 profiles of the mean streamwise velocity at R e =
3500 (in global coordinates) are shown through the riblet valley,
from the midpoint, and from the riblet tip. As with the laminar and
transitional results of [8], the profile through the valley is
inflectional.
In Figure 6 we examine the mean velocity profile
(nondimensionalized in wall coordinates) above both the smooth and
riblet walls. The lines correspond to
Linear region: w + = y+, (6)
Spalding's Law of the Wall: y+ = w + + e-*P[e ~w÷ - 1 - x w + -
½0cw+) 2 - ~(xw+)3]. (7)
The solid line depicts (7) using Coles' values of (x,/~) =
(0.41, 5.0) and the dotted line represents Nikurade's values (x,
13)= (0.40, 5.5). The computed values above the smooth wall
(squares) are in excellent agreement with the two suggested
correlations, lying directly between the lines. The corre-
0,008 I i 0.006 # 0,004
0.002 400
i r i I i i i I i I i [ i i 600 800 1000 ~ i r n e
Figure 4. Instantaneous pressure drop history.
I
0.5
0
' v M i e y 7i m i d p o i n t
71 . . . . . p e a k
, b i I i i r 0 0.5 1 1.5 2 Y Figure 5. Mean streamwise velocity
profiles in global coordi- nates through the riblet valley, from
the riblet midpoint, and from the riblet peak.
-
224 D. Chu, R. Henderson, and G.E. Karniadakis
t , , l ~ .
C o l e s ' v a l u e s . . . . . - . . . . . . . . - . . . . .
. . . . . N i k u r a d s e ' s v a l u e s . . - . . . . -
" " l o g r e g i o n "
n n e r r g n . a " J b u f f e r r e g i ° n
r I r i r , , , 110
l o g ( y + )
(a)
3 0
20
~0
0 410 L , ,
l o g ( y + )
(b)
Figure 6. Mean streamwise velocity profiles (a) above the smooth
wall and (b) above the riblet wall.
sponding mean velocity profile above the riblet wall is shown in
Figure 6(b); the valley midpoint has been chosen for the origin y+
= 0.0 and a span-averaged value of the local shear velocity is used
for normalization. The riblets seem to thicken the viscous sublayer
effectively; the upward shift in the logarithmic region reported by
[12] is also evident here.
Additional mean flow properties have been computed; those
corresponding to Re = 3500 have been listed in Table 1. Values of
the Reynolds number based on shear velocity and centerline
velocity, displacement thickness, momentum thickness, shape factor,
Clauser shape parameter, velocity defect ratio, and normalized
velocities are shown for both the smooth wall and the riblet wall.
Also listed are the nondimensionalized riblet dimensions h ÷, s ÷
(height, base). Measurements for the riblet wall were based on a
virtual origin located at the riblet valley midpoint (y = 0.i).
3.2. Flow Structure
We proceed to examine the instantaneous flow field further. In
Figure 7 contours of the instantaneous streamwise velocity are
plotted on an x-plane (x --- 0.2) through a valley. The spectral
element skeleton of the computational domain is pictured, and the
flow is from left to right. The existence of large-scale streaky
structures near both top and bottom walls can be seen. Further
insight into the three dimensionality of the flow field can be
gained by viewing contours on a different plane. Figure 8 also
shows W contours at the same instant in time, but on a z-plane (z =
0.0), with the flow direction being into the page. This view
reveals the spanwise and normal extents of the aforementioned
structures.
While from these figures it is apparent that the velocities
inside the riblet valleys are small, they are not completely
nonnegative. In I-8] we reported that strong burstin9 and sweeping
motions in the near-riblet regions caused flow reversal in some
locations even deep inside the riblet valleys. In Figure 9(a)
profiles of the instantaneous streamwise velocity are plotted along
the span of the domain at z = 0.0, y = 0.1 (within the valleys);
there are regions where negative velocities exist. In Figure 9(b)
we examine in detail one of these reverse flow regions; W is
plotted at x = 0.2 (at the valley through). We see that there is a
considerable region of negative velocity deep within the valley.
This region extends all the way up to the midpoint of the valley (y
= 0.1) at this particular time instant.
Table 1. Comparison of bulk flow properties.
Smooth wall Riblet wall
R e 3500 Rec 2250 Re~ 131 86 Uc/U 1.22 U/U~ 14.0 21.5 6" 0.140
0.205 0 0.79 0.89 H = 6*/0 1.77 2.30 G = (Uc/U,)((H -- 1)/H) 7.45
14.85 J = (Uc -- U)/U~ 3.08 4.74 h +, s + 17.1 (riblet
dimensions)
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Parallel Spectral-Element-Fourier Simulation of Turbulent Flow
over Riblet-Mounted Surfaces 225
Figure 7. Instantaneous streamwise velocity contours on an
x-plane (Re = 3500).
Figure 8. Instantaneous streamwise velocity contours on a
z-plane at the same time instant (as Figure 7).
-
226 D. Chu, R. Henderson, and G.E. Karniadakis
, , , , ( , , , ,
0.1
0.05
- I , i l l l J ~ , . . . . ~ , , ,
0 0.5 1.5 x
(a)
0,04
0.0~
3.02
0.02 0.04 0.00 0.08 0.1 0.12 Figure 9. Instantaneous W profiles
within the riblet valleys (a) along the span of the domain and (b)
detail of flow
(b) reversal region.
3.3. Turbulence Statistics
In [8] it was noted that the spanwise length (L~) used was most
likely too small. In this computation we have extended the domain
to L~ = 2.0 (see Figure 1) and we have verified its adequacy via
two-point velocity correlations. Figure 10 shows examples of
two-point correlations in the spanwise direction at Re = 3500,
measured at three y-locations: (a) near the smooth wall (y =
1.625), (b) at the centerling (y = 1.0), and (c) near the riblet
wall (y = 0.375). The three velocity correlations have been
overlaid in each plot. In all cases the correlations approach zero
for increasing separation distances, indicating that L~ is
sufficiently large. It is interesting to note that there is
actually a pronounced negative W velocity correlation near the
riblet wall (in Figure 10(c)). Current computational resources have
limited the streamwise extent of the computational domain to L~ =
5.0, or about 680 viscous wall units at Re = 3500. Profiles of
quantities in the streamwise direction are modulatory and consist
of up to five complete waves; this suggests at least marginal
resolution in the z-direction. Recent findings [13] demonstrate
that spectral methods can sustain turbulence in channel flow with
as few as four Fourier modes in the streamwise direction; our
present simulations employ a minimum of 16 modes in z. Future work
will use higher resolutions in the z-direction, and will expand the
streamwise length L~.
0 0.2 0.4 0.6 0.8 1 r (y=1.625)
(a)
0 0.2 0.4 0.6 0.8 :" (y=l.0)
(h)
= "', . . - - R ~
0 0.2 0,4 0.6 0.8 1 ,- (y=o.3"zs)
(c)
Figure 10. Spanwise two-point velocity correlations (a) near the
smooth wall, (b) at the centerline, and (c) near the riblet
wail.
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Parallel Spectral-Element-Fourier Simulation of Turbulent Flow
over Riblet-Mounted Surfaces 227
0.1
0.05
I . . . . I ' ' ' ' I ' ' ' ' I ' ' ' '
........ v ~ - - - u ~
/ / ',, d / ",
0 0 . 5 1 1 . 5 2
y (~hrough valley)
(a)
0.1
0.05
(
. . . . . . v ~
0 0.5 1 1.5
(b)
i
I :
Figure 11. The three components of the turbulence intensities
(a) through the riblet valley and (b) from a riblet tip. The riblet
wall is located on the left (at y = 0.0 to 0.2) and the smooth wall
is located on the right (at y = 2.0).
In Figure 11 we show all three components of the velocity
fluctuations normalized by the bulk streamwise velocity plotted
across the channel at spanwise locations (a) through the riblet
valley (y = 0.0) and (b) at the riblet peak (y = 0.2). All
statistical quantities have been additionally averaged in z (the
homogeneous direction) and also "horizontally averaged" in x (the
spanwise direction), thus collapsing the domain to one single
riblet.
This particular time sample corresponds to 315.5 nondimensional
time units. The profile shapes and peak values of the turbulence
intensities above both the riblets and the smooth wall (y = 2.0)
are in excellent agreement with the experimental results for smooth
and riblet-mounted flat plates [7-1, [6], [14]. We note that all
three root mean square velocity components are smaller in the
vicinity of the riblet wall, even at locations above the riblet
tips. To investigate this in more detail we examine the profile of
the streamwise turbulence intensity near the wall in Figure 12.
Profiles from the smooth wall, the riblet tip, and the riblet
valley have been overlaid and shifted such that the virtual origin
(y/6 = 0.0) corresponds to the wall surface of each profile. Here 6
is the boundary-layer thickness (i.e. channel half-height). We see
that the presence of the riblets reduces the peak velocity
fluctuations near the wall (compared with the smooth wall) and,
furthermore, the turbulence intensity is effectively suppressed
inside the valleys of the riblets (y/6 < 0.2). Near the bounding
surface the smooth wall intensity peaks at roughly 14% of the bulk
velocity, while the intensity inside the valley reaches only 7% at
y/6 = 0.2. It is interesting to note that although very low, the
streamwise turbulence intensity within the valley is not
negligible. The small bump in the profile at y/6 ~ 0.05 suggests
slight activity even in regions deep within the riblet valleys,
thus further confirming the need for full Navier-Stokes simulations
of riblet flows; this is consistent with the flow reversal findings
of 1-8] and Section 3.2.
Experimental results to date have shown contradicting
measurements of the Reynolds stress in flow over riblets; Walsh
1-6] found a reduction over the entire boundary layer, while in
[15] an increase for locations near the riblets is reported. In
Figure 13 we plot the -pv'w' component of the Reynolds stress
normalized by W 2 at a location (a) through a valley and (b) from a
tip. The peak values and profile shapes are in good agreement with
the available fiat-plate data 1-12], [14].
It is apparent that the riblets significantly reduce the
Reynolds stress compared with the smooth wall (the peak value is
reduced by roughly 24%) and thus result in decreased vertical
momentum
Figure 12. Streamwise turbulence intensity in the near-wall
region.
0.1
\
~ 0.05
' ' ' I ' ' , J , , ,
/ / / .......
i / / - - smoo th w a l l ! ~ ~ ~ i . . . . . . . . . . r i b l
e ~ t J p
0 0.2 0.4 0.6 y / #
-
228 D. Chu, R. Henderson, and G.E. Karniadakis
0 . 0 0 2
- 0 . 0 0 2
I ' ' ' ' I ' ' ' ' J ' , ' , I ' ' ' ' I
i i i i ~ r i i i i i i r i i i i
0 0 . 5 'f 1 . 5 2
y (~hrough val ley)
(a)
0 . 0 0 2
0
- 0 , 0 0 2
. . . . . , , - , , . . . . . , ,
0 0 . 5 1 1 . 5 2
y (from up)
(b)
Figure 13. - p v ' w ' component of the Reynolds stress across
the channel (riblet wall on the left, smooth wall on the
right).
transport; this is consistent with the drag reduction found in
[8]. In addition, Figure 13(a) shows that virtually zero vertical
momentum transport occurs in the valleys of the riblets.
Finally, exact drag measurements were made by computing directly
the viscous stress tensor on both walls and checking these figures
via a global momentum balance using the pressure drop from (3). The
narrow channel transitional results from [-8] at Re = 2750 and 3000
have been renormalized and plotted in Figure 14, along with our
turbulent results at Re = 3500 and 4000 and laminar flow results at
various Re's.
The solid lines in the figure correspond to the exact laminar
solution and an empirical data fit [16] for a channel with two
smooth walls. Compared with the smooth wall, the riblet wall has a
higher drag in the laminar regime, with the difference in drag
diminishing as the Reynolds number increases. This trend reverses
in the transitional and turbulent regimes, where it appears that a
10% drag reduction exists at Reynolds numbers 3000 and 3500. The
amount of drag reduction at Re = 4000 is lower, but this should be
viewed as a preliminary result, since it is possible that higher
resolution is needed above Re = 3500.
4. Discussion
We have presented an efficient spectral-element method for the
solution of the incompressible Navier-Stokes equations in complex
geometries. The spatial discretization is accomplished via a hybrid
spectral-element-Fourier scheme that is highly amenable to a
parallel implementation. Com- putations on the 32-node Intel
iPSC/860 Hypercube achieved performance superior to the Cray Y/MP.
The algorithm was validated for an exact solution of the
Navier-Stokes equations, achieving exponential error
convergence.
0 . 2
0 . 1 5
0 . 1
0 . 0 5
- - s m o o t h c h a n n e l r e s u l t • f la t wail
• • r ible£ wall
1000 2000 3000 4000 Reynolds Number
Figure 14. Drag on each wall versus Reynolds number for riblet
channel simulation.
-
Parallel Spectral-Element-Fourier Simulation of Turbulent Flow
over Riblet-Mounted Surfaces 229
The spectral-element-Fourier code was used to investigate drag
reduction in flows over surfaces with streamwise-aligned V-shaped
riblets. The results, consistent with our previous computations
I-8,1, showed that in the laminar regime there is no drag
reduction, while in the transitional and turbulent regimes drag
reduction up to 10~o exists for the riblet-mounted wall in
comparison with the smooth wall of the channel. For the first time
we presented detailed turbulent statistics in a complex geometry.
These results are in excellent agreement with available
experimental data [-12-1, [7,1, I-6], 1,14,1 and provide a
quantitative picture of the drag-reduction mechanism of the
riblets.
Future work will include higher-resolution computations at large
Reynolds number in the fully turbulent regime, performed with the
fully parallel version of the spectral-element-Fourier code on the
Intel iPSC/860 hypercube.
References
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