brown university - parallel spectral-element—fourier ......spectral-element-fourier algorithm in...

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 initiated 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 recently 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.


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


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 computational domain onto a processor network (hypercube). All processors have local memory and may perform computations and message passing concurrently.


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 calculations, 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 coordinates through the riblet valley, from the riblet midpoint, and from the riblet peak.


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)


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).


, , , , ( , , , ,

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.


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 / #


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.


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

[1] Frisch, U., and Orszag, S,A. Turbulence: Challenges for theory and experiment. Physics Today, p. 22, January 1990. [2] Patera, A.T. A spectral-element method for fluid dynamics: Laminar flow in a channel expansion. Journal of Computa-

tional Physics, 54:468, 1984. [3] Karniadakis. G.E., Bullister, E.T., and Patera, A.T. A spectral-element method for solution of two- and three-dimensional

time dependent Navier-Stokes equations. Finite-Element Methods for Nonlinear Problems, Springer-Verlag, New York, p. 803, 1985.

[4] Karniadakis, G.E. Spectral-element simulations of laminar and turbulent flows in comlex geometries. Applied Numerical Mathematics, 6: 85, 1989.

I-5] Henderson, R.D., and Karniadakis, G.E. A hybrid spectral-element-finite-difference method for parallel computers. Unstructured Scientific Computations on Scalable Multi-processors, M.I.T. Press, Cambridge, MA, to appear.

[6] Walsh, M.J. Riblets. In Viscous Drag Reduction in Boundary Layers, Progress in Astronautics and Aeronautics, vol. 123, eds. Bushnell, D., and Hefner, J., 1990.

[7] Vukoslavcevic, P., Wallace, J.M., and Balint, J.L. On the mechanism of viscous drag reduction using streamwise aligned riblets: A review with new results. Turbulent Dra 9 Reduction by Passive Means, 1987.

[8] Chu, D., and Karniadakis, G.E. Numerical Investigation of Drag Reduction in Flow over Surfaces with Streamwise Aligned Riblets. AIAA-91-0518, 1991.

I-9] Karniadakis, G.E., Israeli, M., and Orszag, S.A. High-order splitting methods for the incompressible Navier-Stokes equations. Journal of Computational Physics, to appear.

[10] Karniadakis, G.E., Spectral-element-Fourier methods for incompressible turbulent flows. Computer Methods in Applied Mechanics and Engineering , 80: 367, 1990.

[11] Tomboulides, A.G., Israeli, M., and Karniadakis G.E. Efficient removal of boundary-divergence errors in time-splitting methods. Journal of Scientific Computing, 4(3):291, 1989.

1,12] Wallace, J.M., and Balint, J.L. Viscous drag reduction using streamwise aligned riblets: Survey and new results. In Turbulence Management and Relaminarisation, eds. Liepmann, H.W., and Narasimha, R., p. 133, 1987.

[13] Zores, R. Numerische untersuchungen mit einem grobauflosenden simulationsmodell fur die turbulente kanalstromung. Technical Report IB 221-89 A 24, Institut fiir Theoretische Stromungsmechanik, DLR Gottingen, 1989.

1,14] Gupta, A.K., and Kaplan, R.E. Statistical characteristics of Reynolds stress in a turbulent boundary layer. Physics of Fluids, 15:981, 1972.

[15] Hooshmand, A. An Experimental Investigation of the Influence of a Dray Reducing, Longitudinally Alighed, Triangular Riblet Surface on the Velocity and Streamwise Vorticity Fields of a Zero-Pressure Gradient Turbulent Boundary Layer. Ph.D. thesis, University of Maryland, 1985.

1-16] Dean, R.B. Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow. Journal of Fluids Engineering, 100:215, 1978.

[17] Henderson, R.D. Ph.D. thesis, Princeton University, in progress.

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