Computation of Viscous Flow in 

Cwved Ducts and Comparison With 

r~dsiu~-a354a) COUPUTATIGY or VISCOVS PLOY 184- 13494 

IY CU Vi3D DUCTS BliD C(3LIPABlSOY d1TH b 

EXPER 1iblTAL U&TB (bibSA) 22 p HC A02/Br A01 

CSCL 20D Uac la s 

63/34 42739 

Prepared for the 

Twenty-second Aerospace Sciences Meeting 

sponsored by the American Institute of 

Reno, Nevada, January 9-12, 1984 

t Aeronautics and Astrocautics 
COMPUTATION OF VISCOUS FLOW IN CURVED DUCTS AND COMPARISON 

WITH EXPERIMENTAL DATA 

Charles E. Towne 

NatiGnal Aeronautics and Space Administration 

Lewis Researc i Center 

Cleveland, Unio 44135 

SUMMARY 

A three-dimensional analysis for fully viscous subsonic internal flow is 

evaluated. The andlysis, designated PEPSIG, solves an approximate form of the 

Navier-Stokes equations by an implicit spatial marching procedure. Results of 

calculations are presented for laminar flow through two different circular 

cross-sectioned 180 degree bends, and for laminar and turbulent flow through 

circular and square cross-sectioned 22.5-22.5 degree S-ducts. 

comparisons with experimen a1 data are shown for all cases. 

is placed on verifying the ability of the analysis to accurately predict the 

distorted flow fields resu ting from pressure-driven secondary flows. 

Quantitative 

Special emphasis 

INTRODUCTION 

The presence of three-dimensional subsonic internal flow is relatively 

comnon in many of today's airbreathinq propulsion systems. Curved center1 ines 

and changes in cross-sectional shape are often present, resulting in the generation of complex three-dimensional secondary flows and significant flow distortion. 

layer thickness often comparable to ?he duct radius. These phenomena have important effects on ovel-all propulsici system performance. 

In addition, viscous effects are usualiy important, with the boundary 

Conventional boundary layer methods will not work for these complex flows. 

A full Navier-Stokes solution could be used, but would require large amounts 

of computer time and storage. However, by making certain approximations to the 

Navier-Stokes equations, such as 'ieglecting streamwise diffusion, a set of 

equations can be derived for fully viscous internal flow that can be solved by 

forward marching in space. An analysis method using tnese equations to compute 

three-dimensional subsonic flow through curved ducts with super-elliptic crosssections was developed by Briley and McDonald (ref. 1) and by Levy, McDonald, 

Briley, and Kreskovsky (ref. 2), and was designated PEPSIG. A preliminary 

evaluation of the PEPSIG method was presented by Towne and Anderson (ref. 3). 

However, in that evaluation ocly qualitat've comparisons were made with data. 

In addition, ;ever81 changes have since been made to the analysis by Levy, 

Briley, and Mcr)onA.ld (ref. 4). 

The objective of the present study, therefore, was to evaluate and verify 

the most recant version of the PEPSIG analysis by making detailed quantitative 

comparisons between predicted results and benchmark experimental data. Special 

emphasis was clirced on testing the ability of the analysis to accurately wedict the distorted tlow field resulting from pressure-driven secondary fiJws. 
GOVERNING EQUATIONS 

In this analysis, the flow is computed by a single sweep spatial marching 

procedure which solves an approximate form of the Navier-Stokes equations. It 

is assumed that the flow is primarily in the direction of the duct centerline, 

with transverse secondary flow. This allows two basic assumptions to be made. 

The first is that second derivatives in the primary flow direction are negligible. 

equation can be represented by the sum of a known three-dimensional pressure 

field and a one-dimensional correction computed as part of the marching procedure to account for viscous blockage. When these two assumptions are applied 

to the Navier-Stokes equations, a set of equations can be derived that can be 

solved by forward marching in the primary flow direction. 

The second is that the pressure in the primary, or streamwise, mgmentum 

The details of the derivation of the governing equations have been presented elsewhere (refs. 1, 2, and 4). For completeness, however, a brief disclrssion is presented here. The equations are first written in a centerline-based 

orthogonal reference coordinate system denoted by xi, xz, and x3, with corresponding velocities vs, ws, and up. The x3 direction is the primary flow 

direction, and is aligned with the duct centerline at each marching step. The 

xi and x2 directions are normal to the duct centerline and define the transverse, or secondary, flow plane. The primary momentum dquation is then given 

by 

This equation, and the others to be presented in this section,.are taken 

directly from reference 4. Here h is the metric scale coefficient associated 

with the primary direction . (The metric scale coefficients for the transverse directions are both uni x$ y). Also, p- = pi(x1,xf,x3) is the known threedimensional pressure field and pv = pv(x3] is the un nown one-dimensional 

viscous blockage correction. In this study, pi was computed using a threedimensional potential flow analysis, thus bringing elliptic effects due to 

geometry into the solution. 

The secondary velocities vs and ws are spl 

and rotational components. The continuity equation 

t into irrotational 

can then be writter, as 

2 
Here ~p is a scalar potential corresponding to the irrotational component of 

the secondary velocity. 

form an equation for the transport of streamwise vorticity in the primary flow 

direction, given by 

The transverse momentum equations are cross-differentiated and combined to 

axl ax2 ax2 axl axl ax2 

S apwS aw 

--- ax2 axl -L[k% axl (hua)] -%[I: h ax (ha) = 0 

1 

where 

Here 9 

rotational component of the secondary velocity, and a is the vorticity in 

the x3 direction. 

is the stream function, or vector potential, corresponding to the 

The fourth equation is simply the definition of n written in terms of JI. 

The energy equation is eliminated by assuming constant total enthalpy. For turbulent flow, an algebraic mixing length turbulence model is used. 

ordinate system. They are then solved by forward marching in the streamwise 

direction from an initial station, where the flow profiles are known, by an implicit finite-difference technique. The boundary conditions that are applied 

result in no-slip at the walls, and symnetry at the symmetry plane. Details of 

the solution procedure can be found in references 1, 2, and 4. 

The above equations are transformed into a non-orthogonal body-f i tted coCOMPUTED RESULTS 

The results of the PEPSIG analysis have been compared with experimental 

data for a variety of geometric configurations and flow conditions. In this 

3 
section, results are presented and discussed for the following cases: (1) incompressible laminar flow through two different circular cross-sectioned 180 

degree bends tested by Agrawal, Talbot., and Gong (ref. 5); (2) incompressible 

laminar and turbulent flow through a circular cross-sectioned 22.5-22.5 S-duct 

tested by Taylor, Whitelaw, and Yianneskis (ref. 6); and (3) incompressible 

laminar and turbulent flow through a square cross-sectioned 22.5-22.5 S-duct 

also tested by Taylor, Whitelaw, and Yiannes-is (ref. 7). All of these data 

are of benchmark quality, with streamwise and cross-flow velocities measured 

using laser-doppler velocimetry. 

presented by Anderson (ref. 8) and by Vakili, et al. (ref. 9). 

PEPSIG results for other configurations are 

180 Degree Bend 

The two geometries tested in reference 5 are shown in figure 1. Both have 

a cross-section radius r of 1.905 cm (0.75 in.). They had centerline radii 

R of 13.335 cm (5.25 in.) and 38.1 cm (15.0 in.), corresponding to radius 

ratios R/r of 7 and 20, respectively. 

Rlr = 7. - The first results to be presented are for the R/r = 7 bend. 

The flow was laminar, with a Reynolds number Re of 242 based on the crosssection radius r arid the average velocity 5, corresponding to a Dean number 

De of 183. The Dean number is defined as De = 2 Re m. With a 20x20 mesh 

in haif the cross-section (taking advantage of symmetry), and 181 streamwise 

marching steps, 1.7 minutes of CPU time were needed on a Cray-1 computer. 

The computed secondary flow pattern after 60 degrees of turning is shown 

This is the classic patThe higher speed flow in the core region 

in figure 2. 

of the secondary velocity at the tail of the arrow. 

tern for flow in a curved pipe. The low energy flow near the wall migrates 

circumferentially away from the outside, or pressure side, of the bend and 

toward the inside, or suction side. 

moves toward the outside of the bend due to centrifugal effects. A pair of 

counter-rotating vortices is thus established. This secondary flow pattern 

persists through the length of the duct. 

Each arrow in this figure represents the direction and magnitude 

The resulting flow distortion is shown in figure 3 in the form of contours 

of constant streamwise velocity. These results are presented at the same stations used in the experiment. The boundary layer initially thickens slightly 

on the outside of the bend due to the pressure increase there as the flow enters the bend. 

ary flow is apparent in the thickening of the boundary layer on the inside of 

the bend. By 60 degrees of turning the secondary flow has caused significant 

flow distortion. A pocket of low speed flow has formed on the inside of the 

$end, displacing the region of maximum velocity toward the outside of the bend. 

The results at the final two stations are similar. 

By the second station, at 30 degrees, the effect of the secondIn the experiment of reference 5, data were taken ct five streamwise stations. At each station, streamwise vzlocity profiles were measured along the 

five data lines shown in the inset to figure 4. 

locity profiles along these data lines are compared with the data in figure 4. 

Computed results are shown for both 20x20 and 40x40 cross-sectional meshes; 

however, the effect of greater mesh resolution for this case is negligible. 

The computed streamwise ve4 
For both meshes, the agreement between the PEPSIG results and the data is 

excel lent. 

The computed and experimental secondary velocity profiles are presented 

in figure 5. These results, both computed and experimental, are for slightly 

different flow conditions. The Reynolds and Dean numbers were 183 and 138, 

respectively. The distance between adjacent data lines in figure 5 corresponds 

to a v/c value of 0.36. The agreement between theory and experiment is generally very good. 

R/r = 20. - For the R/r = 20 case, the flow was laminar with a Reynolds 

number of 1263 and Dean number of 565. Several different meshes were used. 

For a 50x50~226 mesh, 13.1 minutes of CPU time were used on a Cray-1 computer. 

in figure 6. 

but the boundary layers are thinner, the secondary velocities are not as large, 

and the vortices are tighter. The computed streamwise velocity contours are 

shown in figure 7. 

the R/r = 7 case. However, the flow is more distorted, and becomes distorted 

after less turning. 

The computed secondary flow pattern after 84 degrees of turning is shown 

The basic pattern is similar tG that seen in the R/r = 7 case, 

Again, these results are qualitatively similar to those in 

In figures 8(a) and (b), the computed streamwise velocity profiles are 

compared with the data. (The results along data line 5 are not shown because 

of problems encountered in interpolating the computed results in the high velocity gradient region near the wall.) In figure 8(a), results for both 25x25 

and 50x50 cross-sectional meshes are shown. 

the streamwise direction, with be = 0.5 degrees for e between 0 and 45 degrees, and be = 1.0 degree for e greater than 45 degrees. The agreement is 

generally very good for both meshes. In fact, the 25x25 mesh yives better results than the 50x50 mesh at e = 108 degrees. However, the coarser me-; is 

unable to resolve the velocity peak near the inner wall at e = 35 degrees seen 

along data lines 2 and 3. 

mesh are shown along with results for a 50x50~181 mesh with & = 1.0 degree 

for the entire duct. With the larger streamwise step size, the analysis is 

again unable to resolve the inner wall velocity peak at e = 35 degrees. 

These results indicate that proper mesh resolution in both the transverse and 

streamwise directions is critical in correctly predicting the initial development of the secondary flows. 

A total of 226 points were used in 

In figure 8(b), the results for the same 50x50~226 

Finally, in figure 9 the computed and experimental streamwise velocity 

contours are compared after 84 degrees of turning. 

for the 50x50~226 mesh. The agreement between theory and experiment is 

excellent. 

The computed results are 

Circular S-Duct 

The circular cross-sectioned S-duct geometry tested in reference 6 consisted of two 22.5 degree circular arc bends, as shown in figure 10. The crosssection diameter D was 48 mm and the centerline radius R was 336 mm, for a 

ratio R/D of 7. 

5 
Laminar flow. - For the laminar flow case, the Reynolds number based on 

cross-section diameter D and average velocity was 790, corresponding to 

a Dean number of 211. 

CPU time on a Cray-1 computer. 

shown in figure 11. 

tern typical of curved pipes is set up. 

pressure gradients change sign, tending to attenuate the secondary flows set up 

in the first bend. In fact, by the time the exit plane is reached, a vortextype motion has been set up in th2 lower half of the duct (i.e.? near the inner 

wall of the second bend) that is in the opposite sense as that in the first 

bend. However, as noted in references 6, 7, and 10, the distortion of the 

streamwise flow in the first bend causes the generation of streamwise vorticity 

in the top half of the second bend that is in the - same sense as that already 

present, in accordance with the Squire and Winter formula for the development 

of streamwise vorticity. The vortices set up in the first bend are thus sustained, or even intensified, in the second bend and become concentrated near 

the top of the duct. 

streamwise velocity contows in figure 12. 

of the duct initially thickens due to the adverse pressure gradient there as 

the flow enters the duct. 

ahead of the inflection plane, the secondary flows have caused the boundary 

layer in the top half of the duct to thicken significantly. Through the inflection region an adverse pressure gradient exists along the top of the duct, 

and the analysis predicts a small region af separated flow. This separation 

region has no appreciable effect on the rest of the flow, and the "FLARE" approximation (ref. 11) is used to allow the streamwise marching analysis to continue. In the second half of the duct the concentration of the vortices near 

the top causes a severely distorted flow field. In addition, the secondary velocities near the wall in the bottom half of the duct cause the boundary layer 

there to thicken. The flow at the duct exit is as distorted, or mare distorted, than it would be in a unidirectional bend with the same total amount of 

turning. 

With a 20x20~80 mesh, this case required 0.8 minutes of 

The computed secondary velocities near the inflection and exit planes are 

In the first bend the pressure-driven secondary flow patIn the second bend, the cross-flow 

The effect of the secondary flows on the streamwise flow is shown by the 

The boundary layer along the bottom 

By the time the second station is reached, just 

In figure 13, the computed streamwise velocity profiles in the symmetry 

plane are compared with the data. 

20x20~80 and 40x40~80 meshes. 

and the effect of greater mesh resolution is negligible. 

S-duct was 48,000, corresponding to a Dean number of 12,828. With a 50x50~80 

mesh, this case required 4.8 minutes of CPU time on a Cray-1 computer. 

Analytical results are shown for both 

Both give very good agreement with the data, 

Turbulent flow. - The Reynolds number for turbulent flow in the circular 

The computed secondary velocities near the inflection and exit planes are 

shown in figure 14. The physical phenomena here are the same as in the laminar 

flow case. However, the boundary layers are much thinner, resulting in generally lower secondary velocities. The vortices are also somewhat more concentrated and nearer the top of the duct. 

shown in figure 15. Again, except for the thinner boundary layers, these results are qualitatively similar to the laminar results. 

The streamwise velocity contours are 

6 
In figure 16, the computed and experimental streamwise velocity profiles 

in the symmetry plane are compared with .he data. !r: contrast to the laminar 

case, mesh resolution for the turbulent rase has a significant effect on the 

results. With the 50x50~80 mesh, the PEPSIG results agree very well with the 

data. With the 25x25~80 mesh, however, the thin boundary layers and secondary 

flow vortices are not resolved well enough for the analysis to predict the 

distorted flow region at the top of the duct. 

Sqaare S-Duct 

The square cross-sectioned S-duct geometry of reference 7 also consisted 

Since the PEPSIG comof two 22.5 degree circular arc bends, as shown in figure 10. The cross-section 

width D was 40 mm and the centerline radius R was 280 nun, for a ratio R/D 

of 7, the same as the circular cross-sectioned S-bend. 

puter code is presently limited to super-ell iptic cross-sections, the square 

cross-section was represented by a super-ellipse with an exponent of 10. 

cross-section width D and average velocity u was 790, corresponding to a 

Dean number of 211, the same as in the circular cross-sectioned laminar flow 

Cray-1 computer. 

Laminar flow. - For the laminar flor; case, the Reynolds number based on 

. case. With a 20x20~111 mesh, this case required 1.2 minutes of CPU time on a 

The computed secondary velocities near the inflection and exit planes are 

shown in figure 17. 

same as for the circular cross-sectioned case, and the secondary flow patterns 

are very similar. 

bend. 

tne duct, and the near-wall secondary velocities in the lower half of the duct 

change direct ion. 

The important physical phenomena for this case are the 

A pair of counter-rotating vortices is set up in the first 

In the second bend, these vortices become concentrated near the top of 

The computed streamwise velocity contours are shown in figure 18. These 

results are, again, very similar to those in the circular cross-sectioned 

S-duct. In the first bend, the boundary layer in the top half of the duct 

thickens because of the secondary flows. In the second bend, the concentration 

of the vortices near the top of the duct again causes a severely distorted flow 

field. 

top of the duct near the inflection plane and along the bottom of the duct near 

the exit plane, where adverse pressure gradients exist. 

Small regions of separated flow were predicted for this case along the 

The covputed and experimental streamwise velocity profiles in the symmetry 

plane are compared in figure 19. Both the 20x20~111 and 40x40~111 mesh results 

agree very well with the data. 

gles, however, at the last two stations in the middle of the duct where the 

mesh resolution is poor. 

Turbulent flow. - The Reynolds and Dean numbers for the turbulent flow 

case were 40,000 and 10,690, respectively. With a 50x50~111 mesh, this case 

required 7.7 minutes of CPU time on a Cray-1 computer. 

The 20x20~111 case shows some undesirable wigThe computed secondary velocities ear the inflection and exit planes are 

shown in figure 20. As in the circular cross-sectioned case, these results are 

7 
basically similar to the corresponding laminar results, but with thinner boundary layers and generally lower secondary velocities. However, the vortices at 

the top of the duct have essentially disappeared by the time the exit plane is 

reached. In addition, in the second bend the streamwise velocity contours for 

turbulent flow, shown in figure 21, are shaped differently than for laminar 

flow. 

speed flow extending furthest into the core region a short distance away from 

the symmetry plane. 

extended furthest into the core region exactly in the symnetry plane. 

shape of these contour lines is in agreement with that found experimentally. 

In figure 22 the computed and experimental streamwise velocity prof i les 

in the symnetry plane are compared with the data. Computed results are shown 

for both 25x25~111 and 50x50~111 meshes. The 50x50~111 mesh gives slightly 

better agreement with the data than the 25x25~111 mesh, although the differerces are small. 

Near the top of the duct they have a gentle "W" shape, with the low 

The 

In the laminar flow case, in contrast, the low speed flow 

CONCLUDING REMARKS 

For the configurations studied, the PEPSIG analysis gave quantitatively 

accurate results for both laminar and turbulent flow when sufficient mesh resolution was used. In particular, the analysis correctly predicted the complex 

physical phenomena and distorted flow fields that occur in S-shaped ducts. For 

the cases with thick inlet boundary layers, 20x20 cross-sectional meshes were 

sufficient. However, for the thin boundary layer cases a 50x50 cross-sectional 

mesh was generally needed for accurate results. Proper mesh resolution in both 

the transverse acd streamwise directions was especially important in regions 

where the secondary flows were initially developing. 

The PEPSIG analysis is very fast compzred to the alternative, a full 

Navier-Stokes calculation. For the results presented in this paper, the current FEPSIG computer code processed 600-700 grid points per second on the 

Cray-1 computer. (The analysis is approximately 10 times slower on an IBM 370/ 

3033). In reference 12, a time of 0.028 seconds per grid povnt per time step 

is reported for a three-dimensional problem solved by an efficient full NavierStokes procedure on an IBM 370/3033. Assuming 100 time steps for a converged 

solution, and a speed-up factor of 10 between the IBM and Cray, the NavierStokes procedure would process 3.6 grid points per second. The PEPSIG analysis 

is thus approximateiy 150-200 times faster, for coinparable accuracy, than an 

efficient Navier-Stokes procedure. 

REFERENCES 

1. 

2. 

Briley, W, R., anlj McDonald, H., "Analysis and Computation Df Viscous Subsonic Primary and Secondary Flows," AIAA Computational Fluid Dynamics Conference, Williamsburg, VA, July 1979, pp. 74-88. 

Levy, R., McDonald, H., Briley, W. R., and Kreskovsky, J. P., "A ThreeDim2nsional Turbulent Compressible Subsonic Duct Flow A.ialysis for Use with 

Constructed Coordinate Systems," AIAA Paper 80-1398, July 1980. 

8 
3. Towne, Charles E., and Anderson, Bernhard H., "Numerical Simulation of 

Flows in Curved Diffusers with Cross-Sectional Transitioning Using a ThreeDimensional Viscous Analysis," AIAA Paper 81-0003, Jan. 1981. 

4. Levy, R., Briley, W. R., and McDonald, H., "Viscous Primary/Secondary Flow 

Analysis for Use with Nonorthogonal Coordinate Systems," AIAA Paper 

83-0556, Jan. 1983. 

5. Agrawal, Y., Talbot, L., and Gong, K., "Laser Anemometer Study of Flow Development in Curved Circular Pipes," Journal of Fluid Mechanics, Vol. 85, 

Pt. 3, Apr. 13, 1978, pp. 497-518. 

6. Taylor, A. M. K. P., Whitelaw, J. H., and Yianneskis, M., "Developing Flow 

in S-Shaped Ducts. 

be published. 

11 - Circular Cross-Section Duct," NASA CR-3759. To 

7. Taylor, A. M. K. P., Whitelaw, J. H., and Yianneskis, M., "Developing Flow 

in S-Shaped Ducts. 

1982. 

I - Square Cross-Section Duct," NASA CR-3550, May 

8. Anderson, 6. H.. "Three-Dimensional Viscous Design Methodology for Advanced 

Technology Aircraft Supersonic Inlet Systems,o1 AIAA Paper 84-0194, Jan. 

1984. 

9. 'Iakili, A., Wu, J. M., Bhat, M. K., Liver, P., Hingst, W. R., and Towne, 

C. E., "Comparison of Experimental and Computational Compressible Flow in 

an S-Duct," AIAA Paper 84-0033, Jan. 1984. 

10. Rowe, M., "Measurements and Computations of Flow in Pipe Bends," -- Journal 

of Fluid Mechanics, Vol. 43, Pt. 4, Oct. 2, 1970, pp. 771-783. 

11. Reyhner, T. A., and Flugge-Lotz, I., "The Interaction of a Shock Wave with 

a Laminar Boundary Layer," International Journal of Non-Linear Mechanics, Vol. 3, No. 2, June 1968, pp. Im. - 

12. Liu, N.-S., Shamroth, S. J., and McDonald, H, 81Nunterical Solution of 

Navier-Stokes Equations for Compressible Turbulent Two/Three Dimensional 

Flows in the Terminal Shock Region of an Inlet/Diffuser," NASA CR-3723, 

Aug. 1983. 

9 
Figure 1. - 180 degree bend configurations. Fipre 2 -Computed secondary flow at 0 = 600 

in Rlr - 1 duct 

INSID 

Figure 3. - Computed streamwise velocity contours in Rli * 7 duct 
0 EXPERIMENT 

ANALYSIS, 4€&4(k181 ------ ANALYSIS, am181 

Figure 4 - Compiited and experimental streamwise velocity profiles in Rlr = 7 duct 

WPERIMENT 
Figure 6. - CoaputeC secondary flow at 0 = 88 

in Rlr = 20duct. 

Figure 7. - Computed streamwise vekity contours in Klr - 20 duct 
f 

i 

i 

k 

VELOCITY 

5 o WPERIMM 

ANALYSIS. %5at226 

------ hkAL?SlS. 25~25x226 

Figure 8 

{ai tffect d cross-section;: mesh. 

ComputeC and experimental streamwise velocity profiles in Wr = 20 duct 

2 

VELOCITY 

EXPERIMEF'T 

ANALYSIS. WtWt226 

ANALYSIS, 5(k5(k181 

(b) Effect of streamwise mesh. 

Figure 8, - Concluded. 
CONTOUR 

A 

8 

C 

D 

E 

F 

C 

H 

I 

J 

K 

UlU 

a25 

.63 

.75 

-88 

LOO 

Lob 

L 13 

L 19 LB INSIDE 

L38 

1.44 

Figure 9. - Computed and experimental streamwise velocity contours at 0 - 840 in Rlr - P 

duct. 

Figure 10. - S-duct configuration. 
NEAR INFLECTION PLANE MAR EXIT PLANE 

Figure 11. - Computed secondary velocities for laminar flow in 22 5-22 5 circular S-duct 

/ 

Figure 12 - Computed streamwise velocity contours 

for laminar flow in 22.5 - 22.5 circular S-duct 
EXPERIMENT 

ANALYSIS, 40I4rki80 

ANALYSIS, akakaQ 

0 

1.0 

.6 

- .2 

----_ 

- L 

% 

Y 

z 5 -a2 

B 

-0.6 

-1.0 

0 .5 1.0 L5 20 2 2 2 

VELOCITY. Ulij 

Figure 13. -Computed and experimental streamwise velocity profiles in symmetry 

plane for laminar ftm in 22 5 - 22 5 circular S-duct 

NGIR iwcEcnon3 PLANE NEAR MI1 PIANE 

Figure 14 -Computed secondary velocities for turbulent flow in 22 5 - 22 5circular S-duct 
Figure 15. -Computed streamwise velaity contours 

for turbulent flow in 225 - 225cinular Sduct 

0 .5 1.0 1.5 1.5 1.5 1.5 

Figure 16. - Computed and experimental 

streamwise velocity prdiles in symmetry 

plane for turbulent flow in 22 5 - 22 5 

circular S-dud. 

VELOCITY, UlU 

EXPERIMENT 

ANALYSIS, sQtwt80 

ANALYSIS , 25~25x80 
NEAR I.WLECTION PLANE NEAR EXIT PLANE 

Figure 17. -Computed secondary vslocities for laminar flow in 22 5 - 2Z 5 square S-duct 

Figure 18. - Computed streamwise velocity contours for laminar flow in 

22 5 - 22 5 sauare S-duct. 
fXPERIMEM 

ANALYSIS, 4k4klll 

ANALYSIS. 2Ikablll 

0 .4 .8 L2 1.6 2 2 2 2 2 

VELOCITY. ulii 

Figure 19. - Computed and experimental streamwise velocity prdiles in symmetry plane for laminar flow in 

22 5 - 22 5 square S-duct 

NEAR iwcnm PLANE NEAR EXIT PLANE 

Flgure 20. -Computed secondaryvelmities for turbulent flow in 22 5 - 22 5 square S-duct 
Figure Zl. -Computed dreamwise velocity contours for turbulent flow in 

22 5 - 22 5 square S -duct 

1.0 

.6 

.2 

L - 

)r 

ui 

or 

4, -.2 s -. 6 

-1.0 

0 .5 1.0 1.5 1.5 1.5 1.5 1.5 

VELOCITY, ulti 

Figure 22 - Computed and experimental dreamwise velocity profiles in symmetry plane for 

turbulent flow in 22 5 - 22 5 square S-duct. 

tXPERlMENT 

ANALYSIS, %5(klll 

ANALYSIS. 25a2hrlll