NASA Technical Memorandum 102142 

ICOMP-89- 16 

Calculation of Turbulence-Driven 

Secondary Motion in Ducts With 

Arbitrary Cross Section 

A.O. Demuren 

Institute for Computational Mechanics in Propulsion 

Lewis Research Center 

Cleveland, Ohio 

TURBULEICE-DRIVEN (HASA-TH-3021 UZ) SECONDARY CALCULITIOB HOTIOM OF I10 DUCTS M89-27115 

WITH ARBITRARY CEOS SECTION (NASA. ,, Lewis 

Resarch Centr) 23 p CSCL 20D Unclas 63/34 0224119 -_ 

July 1989 

runsn LEWIS RESEARCH CENTER 

ICOMP 

* 
ItCALUTION OF TURBLENC-DRIVEN SECONDARY MOTIN IN 

DUCTS WITH ARBITY CROS-SECTION 

A.O. Demuren* 

Institute for Computational Mechanics in Propulsion 

Lewis Research Center 

Cleveland, Ohio 44135 

Abstrac 

Calculation methods for turbulent duct flows are generaliied for ducts with arbitrary crosssections. The irregular physical geometry is transformed into a regular one in computational 

space, and the flow equations are solved with a finite-volume numerical procedure. The turbulent 

stresses are calculated with an algebraic stress model derived by simplifying model transport 

equations for the individual Reynolds stresses. Two variants of such a model are considered 

in the present study. These procedures enable the prediction of both the turbulence-driven 

secondary flow and the anisotropy of the Reynolds stresses, in contrast to some of the earlier 

calculation methods. Model predictions are compared to experimental data for developed flow in 

triangular duct, trapezoidal duct and a rod-bundle geometry. The correct trends are predicted, 

and the quantitative agreement ia mostly fair. The simpler variant of the algebraic stress model 

procured better agreement with the measured data. 

*On leave from University of Lagos, Lagos, Nigeria. Work funded under Space Act Agreement CWG. 

1 
NOMECLATUR 

B half-width of duct 

Superscripts 

1 

2,3 

constant in algebraic stress model 

k - E turbulence model constants 

determinant of the Jacobian of the transformation 

near-wall proximity function in algebraic stress model 

rate of production of turbulent kinetic energy 

half-height of duct 

Jacobian of the transformation yj = yj(zj) 

turbulent kinetic energy 

mixing length 

pressure 

stress or flux of variable 

distance from point in flow to a point on wall 

source term for dependent variable Qi 

bulk velocity 

maxiu velocity 

friction velocity 

Cartesian velocity components; streamwise, lateral, vertical 

Reynolds normal stresses in Cartesian directions 

Reynolds shear stresses (2 primary and 1 secondary) 

curvilinear coordinates 

Cartesian coordinates 

square of the average distance from a point to all walls 

constants in algebraic stress model 

metric coefficients of transformation 

Kronecker delta 

rate of dissipation of turbulent kinetic energy 

angle 

von Karman constant 

turbulent eddy viscosity 

density 

turbulent Prandtl/Schmidt number for 

wall shear stress 

General representation of dependent variable 

longitudinal direction 

vertical and lateral directions 

2 
1 INTRODUCIN 

Secondary motion of Prandtl’s first kind is encountered in curved passages, where centrifugal forces 

act at an angle to the streamwise direction. It is pressure-driven, with magnitudes ranging from 

10-40 % of the bulk streamwise velocity. It usually leads to a first-order perturbation of the main 

flow properties, and there is no real difficulty in calculating it. On the other hand, secondary 

motion of Prandtl’s second kind is turbulence-driven and is believed to result from the anisotropy 

of the turbulent stresses. It is associated with internal or external turbulent flows in the presence of 

streamwise corners. Its magnitude has been observed to be mostly between 1-4 % of the bulk mean 

velocity in most straight ducts with non-circular cross-section, but the effects on wall shear stress 

distribution and heat transfer rates are quite significant. Some form of Reynolds stress modelling 

is usually required in order to calculate this type of secondary motion. 

Probably the first qualitative data on turbulence-driven secondary motion in non-circular ducts 

were presented by Nikuradse[l], using flow visualisation techniques. Actual quantitative measurements were presented by Hoagland[2] aboout 30 years later. Several detailed measurements have 

subsequently been presented. A comprehensive review of experimental data for ducts with regular 

cross-section can be found in Demuren and Rodi [3]. Additional data for irregularly-shaped channels are reported by Rodet [4], Aly et a1 [5], and Seale [6], for trapezoidal, triangular and simlated 

rod-bundle-type cross-sections, respectively. 

The first calculation of secondary flow in straight non-circular ducts was due to Launder and 

Ying [7] (hereafter denoted LY). Their method is based on approximating the Reynolds stresses with 

algebraic expressions by simplifying the corresponding transport equations proposed by Hanjalic 

and Launder [8] This model has subsequently been employed, sometimes with modifications in 

numerous studies [9-121 to calculate secondary flows in ducts with regular as well as gemetrical crosssections. In all these studies, it has been possible to predict the magnitudes of the secondary motion 

fairly well, but always at the expense of the Reynolds stress distributions. The anisotropy of the 

secondary normal stresses are usually underpredicted by an order of magnitude. This discrepancy 

has been discussed by various authors, including Demuren and Rodi [3],who showed that certain 

terms which are neglected in LY model could in fact be larger in magnitude than terms which 

are included. In order to avoid this shortcoming, Seale [6] in the calculation of developed flow in 

a rod-bundle-type configuration, utilises an algebraic model for the streamwise vorticity source, 

based on the correlation of measured turbulence data. Very good agreement between measured 

and computed velocity and wall shear stress distributions are reported. The prospects of such 

a method as a general simulation technique for arbitrary geometries is uncertain. The reported 

calculations are perfomed on a fairly coarse Cartesian grid, using a method which approximates the 

curved wall with steps. The influence of such a boundary treatment on the total flow is uncertain, 

but it doubtful that it can produce a true picture of the flow near the curved wall. A truly general 

computational method should employ a boundary-fitted coordinate system. 

Naot and Rodi [13] (hereafter denoted NR) calculated flows in rectangular ducts using an 

algebraic stress model developed from the Reynolds stress model of Launder Reece and Rodi [14]. 

The method was subsequently extended by Demuren anti Rodi [3] (hereafter denoted DR). To 

put the various models in perspective. LY model is derived from the Reynolds stress model of 

Hanjalic and Launder [8]. The latter is a precursor to the! more complete Reynolds stress model 

of Launder, Reece and Rodi [14]. The LY model neglects all terms involving secondary velocity 

gradients. NR model is based on the more complete Reynolds stress model and approximates terms 

3 
involving secondary velocity gradients by eddy viscosity type relations. DR model is also based on 

the same Reynolds stress model, but uses the full derivation. In all the models the algebraic stress 

relations are derived from the partial-differential equations governing the Reynolds stress transport 

by neglecting the convection and diffusion terms or relating them to those of the turbulent kinetic 

energy. This approximation is adequate in straight wall bounded flows, but would be inadequate 

for free shear flows or flows with high mean shear rates in which the convection and diffusion terms 

assume more significant roles. As may be expected, DR model is more difficult to implement than 

the other two. It is also much more sensitive to the solution procedure, because of the much stronger 

coupling between the Reynolds stresses. Both NR and DR model produce correct simulation of 

mean flow and Reynolds stresses in developing and fully developed flow in square and rectangular 

ducts [3,15]. In fully developed flow situations the secondary flow level predicted with the former 

agree better with experimental data. 

In the present study, NR and DR models are generalised for any curvilinear coordinate system 

and then applied to predict developed turbulent flow in straight ducts with arbitrary cross-sections. 

The governing equations are transformed into a general curvilinear coordinate system, but the 

Cartesian velocity components are retained as dependent variables, following the practice of Rhie 

and Chow [16]. 

2 MATHEICAL MODEL 

The time-averaged, three-dimensional, steady-state equations governing turbulent flows may be 

expressed in curvilinear tensor, for a general property CP as : 

where xj is a general curvilinear coordinate system related to the Cartesian coordinate system 

D is the determinant of the Jacobian of the transformation; Le., D = IJ/, with 

yj by the transformation yj = yj(xj). 

J= 

oi, also called the metrics, are the elements of the Jacobian J-' of the inverse transformation, 

i - ax' xi = d(yi), i.e., aj - -. 

aY3 

Urn represents the Cartesian velocity components. 

qgm and Sg are respectively fluxes/stresses and source terms for the corresponding Q. They 

may be expressed as : 

4 
Conservation Equation 

Continuity 

Q Qd s4 

1 0 0 

Momentum Components 

where, 6,; is the Kronecker delta 

and, G is the rate of production of turbulent kinetic energy by the interaction of turbulent 

stresses with mean shear, and is given by : 

0 a U. i3U U' -P6,i - pu,ui + p +a; + -g$af 

The turbulent eddy viscosity pt is given by the standard k - e model as 

Turbulent kinetic energy 

Dissipation rate of k 

k2 

fit = CrPy 

k bk dk a; G - PE 

6 +lG - C,2PE) 

(3) 

The empirical constants are: 

In the above, we have used the Einstein summation convention for repeated indices to write in 

compact form. Thus, for the 3-dimensional flow under consideration, 

The equations can be simplified if we consider only straight channels with constant cross-section 

in the streamwise direction (yl = 2'). Then, we can use parabolic assumptions to decouple the 

pressure gradient in the streamwise direction from those in the cross-plane, and also neglect the 

derivatives with respect to z1 or y' , in all but the convection terms. They will of course be zero 

throughout in fully developed flow, and the equations become 2-dimensional. 

The major difficulty in solving the set of equations is to determine the distribution of the 

Reynolds stresses -pW. Following Demuren and Rodi [3,15] , we can write the algebraic equation 

for the Reynolds stresses as : 

Equation (6) shows very strong coupling between the Reynolds stresses, and in order to stabilise 

the solution process only the secondary stresses ug , ui and fix are obtained from it. The primary 

normal and shear stresses are calculated from the turbulent kinetic energy and standard eddy 

viscosity relations. The resulting set of algebraic stress relat' ,ions are:- 

-- 

5 
- ((1 - Q)G - /3q + 7k)BaT - ((1 - Q)G - Pq + rk}&~Y] dU (12) 

In equations (10 - 12) wall proximity effects are accounted for by making the empirical coefficients functions of a dimensional distance from the walls, thus : 

ct! = 0.7636 - 0.06 f; p = 0.1091 + 0.06 f; 7 = 0.182; ~1 = 1.5 - 0.50 f (13) 

where f is the wall-proximity function given by : 

L2 ; <y>=- 2 = <y2> 7r 212r $ (14) 

c 314 k3/2 

L=L- K. is the length scale of the turbulent motion at the grid node, and < y2 >lf2 is ~ 

the average distance from all walls. 

In the NR model, equations (10 - 12) are further simplified by setting the denominator to unity 

and approximating terms involving secondary velocity gradients by eddy-viscosity-type relations 

as: 

6 
This practice has a stabilising effect on the solution procedure, since they can now be treated 

implicitly as diffusion terms in the momentum equations for the secondary velocity components. 

An analysis of the implications of these approximations, performed by Demuren and Rodi [3] for 

a square duct flow, has shown that the major effect is on the secondary shear stress calculation, 

of which equation (17) is a dominant part. That term is essentially reduced by a factor of 2 - 3. 

However, the secondary shear stress acts a sink of the streamwise vorticity, so that a reduction 

in its magnitude would lead to an increase in the secondary motion. This explains why the NR 

model usually produces stronger secondary motion than the DR model [3]. The terms in equations 

(15) and (16) are small compared to the rest of the terms in equations (10) and (ll), so that the 

effects of approximating them is minimal. If the calculations are performed with the secondary 

velocity terms in equations (15) - (17) neglected completely, as is done in LY model and derivatives 

thereof, the secondary motion will grow indefinitely and no stable solution can be obtained. In this 

case, a stable solution can only be obtained through an ad.-hoc reduction in the anisotropy of the 

secondary normal stresses, which is the source of streamwise vorticity. 

3 SOLUTION PROCEDURE AND BOUNDARY CONDITONS 

The differential equations to be solved are parabolic in the streamwise direction so that influences 

only travel from upstream to downstream. An efficient once-through forward marching procedure 

can thus be used, starting from given inlet conditions. The procedure is essentially an incorporation of a non-staggered grid arrangement into the SIMPLE algorithm of Patankar and Spalding 

[17]. Where the grid is not orthogonal, cross-derivative terms lead to the formation of a Qpoint 

computational molecule for the 2-dimensional plane. In the present study, these terms are treated 

explicitly in the equations for the velocity components. This practice proved not to be so robust 

in the solution of the pressure correction equation, which is the most critical for stability and 

convergence. We therefore employ a %point pressure-correction equation which is an extension of 

Stone’s strongly implicit scheme as proposed by Peric [18]. A comparative study [19] of various 

solution schemes, including coupled and decoupled method:s, for two-dimensional flow problems on 

a non-orthogonal grid shows the present arrangement to be the most effective. 

Four types of boundaries are usually encountered, at which conditions must be specified for 

the dependent variables. At inlet planes, known (or simulated) conditions are specified for all 

variables; usually uniform values for the primary velocity arid the turbulent quantities, and zero for 

the secondary velocity components. No specific conditions are required at outflow boundaries, since 

the equations are parabolic. Along a symmetry plane, the velocity component normal to the surface 

is set to zero. Further, the gradient of all other variables is zero in the direction normal to the 

surface. At solid walls, the velocity component normal to the surface is also set to zero, but rather 

than carry out the solution all the way down to the wall, we employ the wall-function method [3] to 

specify the values of the velocity components parallel to the! surface, and the turbulence quantities, 

along the first line of grid nodes away from the wall. These grid nodes are usually arranged to be in 

the fully turbulent region, where the logarithmic law of the wall may be assumed to be applicable. 

7 
4 RESULT AND DISCUON 

4.1 Test Case 

Three test cases are considered in the present study. These coincide with flow in ducts with 

cross-sections requiring non-rectangular grids, for which experimental data could be found in the 

literature. In each case, the channel cross-section is constant in the streamwise direction, and the 

flow is nominally fully-developed. 

1) studied experimentally by Aly 

et a1 [5] at Reynolds number of 53,000, based on the bulk velocity and the hydraulic diameter. 

Measurements were made in one-sixth of the duct cross-section at a plane 133 diameters from the 

inlet. Case I1 is the flow, at a Reynolds number of 240,000, in a trapezoidal channel with two sides 

parallel and two slanting sides at angles 75' and 105' (fig. 2). Rodet [4] presents measurements 

of the velocity distribution and the Reynolds stresses at a cross-stream plane 50 diameters from 

the inlet. Judging by the review of duct flow data, this would barely be long enough for full flow 

development. The cross-section of the channel for case 111 is shown in fig. 3. It was studied 

experimentally by Seale [6] to simulate approximately flows encountered in rod-bundle geometries. 

Measurements were made in a, symmetric quadrant at a Reynolds number of 200,000 and 120 

diameters. 

Case I is the flow in an equilateral triangular duct (fig. 

4.2 Computainl Details 

In the calculations, advantage is taken of the inherent symmetry of the flow to reduce the size 

of the computational domain. Thus, we compute only a third of the triangular duct half of the 

trapezxoidal duct, and a quarter of the rod-bundle-type geometry. In each case, the grid is generated 

algebraically using transfinite mapping of the boundary points. Where necessary, a third-order 

smoother based on Newton's divided difference formula is applied. Although orthogonality is 

desirable, it is not a requirement of the present solution procedure. The sole constraint applied is 

that the normal distance of the first set of grid nodes, nearest to the wall should lie in the range 

y+ = 30 - 200. 

Since all the present test cases apply to fully-developed flow, the grid distribution in the streamwise direction is unimportant. The practice adopted here is to take an infinitely large step size, and 

then use a relaxation method to obtain a converged solution. An underrelaxation method with a 

maximum value of 0.5 is applied to all variables. A typical computation on a 40 x 40 grid required 

about 600 iterations to converge to a normalised residual norm of when the NR model is used. 

DR model calculatjons require 200-300 iterations more. Grid dependency tests are performed by 

doubling the total number of grid nodes, i.e., an increase by a factor of fi in each direction, and 

comparing the results. These tests show that a 26 x 26 grid is sufficient for case I. Case I1 requires 

a 18 x 36 grid, and case II a 40 x 40 grid. 

4.3 Comparison with Experimntal Dat 

4.3.1 Triangulr Duct 

The measured wall shear stress distribution is compared to present predictions with the k - E ) NR 

and DR models in fig. 4. Both the NR and DR models capture the trend of the data towards more 

uniformity, though the former has the better agreement. The k - E model predicts much wider 

8 
variation in wall shear stress. These can be explained with the results of the predicted secondary 

flow presented in figs. 5 and 6. This flow moves towards the corners along the corner bisector 

and returns along the walls towards the wall bisectors. Whilst the NR and DR models predict the 

correct trend, the k - E model produces no secondary motion. This motion leads to a redistribution 

of the wall shear stress by moving high velocity fluid from the central part of the duct to the 

corners thereby increasing the shear. In contrast, the shear stress near the wall bisector is reduced 

by the slower moving fluid convected along the wall. The influence of this secondary motion on the 

streamwise velocity distribution is shown in fig. 7. The experiments show strong bulging of the 

velocity contours towards the corners. The predictions mirror this effect to varying degrees. The 

k - E model predicts nearly circular contours (not shown). Fig. 8 shows similar comparisons for 

the turbulent kinetic energy. The normal stresses are compared in fig. 9. Both the NR and DR 

models predict the correct degree of anisotropy. 

4.3.2 Trapezoidl Duct 

The present predictions with both NR and DR models show that there are two eddies in each 

corner. The one stradling the longer side is much larger, as may be expected. Fig. 10 compares 

the predicted contours of the streamwise velocity component to the measurements of Rodet [4]. 

NR model predicts the bulging of the contours towards the corners to the correct degree, though 

there is some deviation between the actual levels along the centreline. DR model predictions agree 

better with the magnitudes along the centreline, but produces too little bulging of the contours 

towards the corners, due to its characteristic of producing weaker secondary motion. Predictions 

of the turbulent kinetic energy, k, are compared in fig. 11. The effect of the secondary motion 

in distorting the distribution of k is much stronger. Corrlespondingly, the differences between the 

predictions are magnified. Again NR model shows the correct trends overall, but DR model has 

better agreement with centreline values. 

The source of streamwise vorticity is the anisotropy of the secondary normal turbulent stresses. 

A measure of this anisotropy is shown in fig. 12. The present prediction (NR model) compares 

quite well with the data. The predictions by Nakayama et a1 [12], which is characteristic of those 

based on the LY model show magnitudes which are at least an order of magnitude too low. This 

observation has, of course, ben widely discued in the literau without isolatng the true cause. 

4.3.3 Rod-Bundle Geomtry 

Case I11 represents the most complicated geometry of all the test cases computed in this study. 

There are 5 corners, all different from each other. We have combinations of intersections of straight 

walls, curved walls and symmetry planes. The geometry certainly poses a severe test on the predictive ability of the present mathematical procedure and turbulence model. There are, however 

significant differences from the usual rod-bundle geometries, which contain no wall/wall intersections. Figure 13 compares the wall shear stress distributions predicted with the three turbulence 

models to measured data of Seale [6]. The data shows nearly uniform wall shear stresses, away 

from the corners. This effect is, of course, due to the equalizing action of the the secondary motion. 

The k - E model cannot simulate this effect since it does not produce any secondary motion. DR 

model does produce a more uniform distribution, but the secondary motion still appears to be 

too weak. NR model predicts the correct level of shear stress, but this has a wavy distribution, 

not apparent in the measurements. It is not certain if the waviness is physical, but the results 

9 
have been reproduced with the NR model, using a different numerical procedure, so it does not 

appear to be numerical in origin. The measurements do not appear to be sufficiently detailed to 

provide definitive answer. Examination of the lateral velocity data, presented in fig. 14, shows 

very strong periodic variations along the verticals. The predictions do simulate this behaviour. As 

usual, the NR model performs better. The peaks in the predicted wall shear stress profiles are 

seen to coincide with the maxima in the velocity component directed towards the vertical wall. 

Thus the data suggests the action of the secondary motion near the vertical wall could produce 

miltiple peaks in the shear stress distribution as predicted. A similar effect would be expected on 

the turbulent kinetic energy. These measurements represent the most detailed set presented by 

Seale [6], and the good agreement with the predictions provides the confidence that the correct 

behaviour is being simulated. Fig. 15 compares measured and predicted vertical velocity profiles. 

Again we see a periodic variation in the predictions, but the measured data points are too scanty 

to discern any trend. Considering the measured data, Seale claims large continuity errors ( 0.19 

U,) along horizontal lines. Close to the horizontal wall, the imbalance was as high as 1.78 U,, 

which is considerably larger than the maximum secondary velocity measured. The presented data 

certainly do not provide an accurate representation of the flow, and have been included here only 

for completeness. Since the lateral velocity field is well predicted, continuity would ensure that the 

vertical velocity field is too. Figure 16 shows the velocity vectors and the stream function contours 

predicted with the NR model.The eddy formation appears to be quite complex. There are at least 

two eddies at each wall/wall juction, and the wall/symmetry plane junctions each contain one eddy. 

The two large central eddies appear to be in the process of breaking up, as has been observed in 

channels with large aspect ratios. The predictions with the DR model are similar, but with reduced 

strengths. 

Predicted contours of the streamwise velocity component and the turbulent kinetic energy are 

compared to measured data in figures 17 and 18, respectively. The distortion of the contours due to 

the secondary motion is well predicted, but the quantitative agreement is only fair. The predicted 

components of the turbulent intensities along the vertical centreline are compared to correlations 

of the measured data in fig. 19. The agreement is fairly good, considering that there could be 

uncertainties of about f 29 % in the correlation coefficients [SI. 

It is pertinent to mention that the wall-function method used here precludes adequate resolution 

of the secondary velocity profiles very close to the wall. However, there is as yet no consistent 

Reynolds stress model for this near-wall region. Mansour et a1 [20], have discussed the inadequacy 

of the standard Launder, Reece and Rodi I141 rnodel in this region. Rodi [21] presents a review of 

current attempts to bridge this gap. Preliminary calculations by the present author for a developed 

square duct fiow, using such a near-wall model, shows that the computed mean flow and turbulence 

quantities are essentially the same as those obtained with the present model in the region more 

than 1% of the duct width, away from the wall. However, in consonance with other attempts [Zl], 

the predicted Components of the Reynolds stresses do not agree with the limiting two-dimensional 

behaviour of turbulence in the region very close to the wall. It appears that major changes to the 

pressure-strain modelling would be required to overcome this problem. 

5 CONCLUDING REMAKS 

Calculation of turbulent flow in complex geometries have been presented. The performance of 

three turbulence models are compared to measured data for three different channel configurations. 

10 
The standard k - e turbulence model does not predict any secondary motion of Prandtl’s second 

kind, and thus fails to predict important flow features. Although the NR model can be seen 

as an approximation of the DR model, it procures better quantitative agreement with measured 

secondary flow data. It thus enables the correct degree of redistribution of the primary velocity 

and the turbulence fields that takes place in complex channels to be predicted. Computations with 

the DR model also show the correct trends, but the magnitudes of the secondary flow is mostly 

underpredicted. Previous analysis had shown that differences in the treatment of the secondary 

shear stress are responsible for the different strengths of secondary motion predicted. Unfortunately, 

this stress component is the most difficult to measure, andl it has received the least attention in 

turbulence model studies. More attention will have to be given to this component if there is to 

be significant progress in turbulence modelling of flow in channels with complex configuration. 

Coupled with this is a need for better near-wall modelling. The widely used wall-function approach 

now appears inadequate, for accurate simulation of flow in such complex geometries. Work is 

progressing on such a model. It must ensure that the various components of the Reynolds stresses 

exhibit the correct limiting behaviour as the wall is approached. Since detailed experimental data 

is limited, we have to make use of data generated by direct simulation of channel flows, such as 

those by Kim et a1 [22], and analysed by Mansour et a1 [20] in the light of popular Reynolds stress 

modelling concepts. 

6 REFNCS 

1. Nikuradse, J. : Untersuchungen Uber Turbulente Stromungen in Nicht-Kreisformigen Rohren. Zng. 

Arch, vol. 1, 1930, pp. 306-332. 

2. Hoagland, L.C. : Turbulent Flow in Straight Rectangular Ducts-Secondary Flow, Its Cause and 

Effect on the Primary Flow. PhD thesis, MIT, 1961. 

3. Demuren, A.O. ; and Rodi, W. : Calculation of Turbulence-Driven Secondary Motion in Non-Circular 

Ducts. J. Fluid Mech., vol. 140, Mar. 1984, pp. 189-222. 

4. Rodet, E.: Etude de 1’Ecoulement d’un Fluide Dans un Tunel Prismatique de Section Trapezoidal. 

Publ. Sci. et Tech. du Min. de l’Air, No. 369, 1960. 

5. Aly , A. A. M . ; Trupp, A. C . ; and Gerrard, A. D . : Measure:ments and Prediction of Fully Developed 

Turbulent Flow in an Equilateral Triangular Duct. J. Fluid Mech., vol. 85, pt. 1, Mar. 7, 1978, 

pp. 57-83. 

6. Seale, W.J.: Measurements and Predictions of Fully Developed Turbulent Flow in a Simulated Rod 

Bundle. J. Fluid Mech., vol. 123, Oct. 1982, pp. 399-423. 

7. Launder, B.E.; and Ying, W.M.: Prediction of Flow and Heat Transfer in Ducts of Square CrossSection. Proc. Znst. Mech. Eng., vol. 187, no. 37, 1973, pp. 455-461. 

8. Hanjalic, K.; and Launder, B.E.: A Reynolds Stress Moldel of Turbulence and its Application to 

Thin Shear Flows. J. Fluid Mech., vol. 52, pt. 4, Apr. 25, 1972, pp. 609-638. 

11 
9. Gessner, F.B.; and Emery, A.F.: The Numerical Prediction of Developing Turbulent Flow in 

Rectangular Ducts. J. Fluids Eng., vol. 103, no. 3, Sept. 1981, pp. 445-455. 

10. Trupp, A.C.; and Aly, A.A.M.: Predicted Secondary Flows in Triangular Array Rod Bundles. 

J. Fluids Eng., vol. 101, no. 3, Sept. 1979, pp. 354-363. 

11. Gosman, A.D.; and Rapley, C.W.: Fully Developed Flow in Passages of Arbitrary Cross-Section. 

Recent Advances in Numerical Methods in Fluids, Vol. 1, C. Taylor and K. Morgan, eds., 1980, 

pp. 335-399. 

12. Nakayama, A.; Chow, W.L.; and Sharma, D.: Calculation of Fully Developed Turbulent Flows 

in Ducts of Arbitrary Cross-section. J. Fluid Mech., vol. 128, Mar. 1983, pp. 199-217. 

13. Naot, D.; and Rodi, W.: Calculation of Secondary Currents in Channel Flow. AXE J. Hydraulic 

Div., vol. 108, no. 8, Aug. 1982, pp. 948-968. 

14. Launder, B.E.; Reece, G.J.; and Rodi, W.: Progress in the Development of a Reynolds Stress 

Turbulence Closure. J. Fluid Mech., vol. 68, pt. 3, Apr. 15, 1975, pp. 537-566. 

15. Demuren, A.O.; and Rodi, W.: Three-Dimensional Calculation of Turbulent Duct Flow with NonUniform Inlet Conditions. Computers Fluids, vol. 15, no. 1, 1987, pp. 47-57. 

16. Rhie, C.M.; and Chow, W.L.: A Numerical Study of Turbulent Flow Past an Isolated Airfoil With 

Trailing Edge Separation. AIAA Paper 82-0998, June 1982. 

17. Patankar, S.V.; and Spalding, D.B.: A Calculation Procedure for Heat, Mass and Momentum Transfer 

in Three-Dimensional Parabolic Flows. Znt. J. Heat Mass Transfer, vol. 15, Oct. 1972, pp. 1787-1806. 

18. Peric, M. : Efficient Semi-Implicit Solving Algorithm for Nine-Diagonal Coefficient Matrix. Nurner. 

Heat Trans., vol. 11, no. 3, 1987, pp. 251-259. 

19. Demuren, A.O.: Evaluation of Coupled and Decoupled Methods for Flows in Complex Geometries. 

in preparation. 

20. Mansour, N.N.; Kim, J.; and Moin, P.: Reynolds-Stress and Dissipation-Rate Budgets in a Turbulent 

Channel Flow. J. Fluid Mech., vol. 194, Sept. 1988, pp. 15-44. 

21. Rodi, W. : Recent Developments in Turbulence Modelling. Proceedings, 3rd. International Symposium 

on Refined Flow Modelling and Turbulence Measurements, Tokyo, 1988. 

22. Kim, J.; Moin, P.; and Moser, R.: Turbulence Statistics in Fully Developed Channel Flow at Low 

Reynolds Number. J. Fluid Mech., vol. 177, Apr. 1987, pp. 133-166. 

12 
tY2 

.2 

4 

L - DR ROKL 

I I I 

FIGURE 1. - FLOW CONFIGURATION AND GRID DISTRIBUTION FOR 

EQUILATERAL TRIANGULAR DUCT. 

4 P 

Y3 

FIGURE 3. - FLOW CONFIGURATION AND GRID DISTRIBUTION FOR 

SIMULATED ROD-BUNDLE GEORTRY. 

* Y3 

FIGURE 2. - FLOW CONFIGURATION AND GRID 

DISTRIBUTION FOR TRAPEZOIDAL DUCT. 

1.2 

1.0 

.4 

0 EXPERIMENTAL DATA OF ALY 

ET AL [51 

PREDICTIONS 

--- k - E MODEL 

-- NR MODEL 

y 3/B 

FIGURE 4. - WALL SHEAR STRESS DISTRIBUTION. 

13 
0 EXPERIMENTAL DATA OF 

ALY ET AL [51 

PREDICTIONS 

- 

- - NR MODEL 

DR MODEL 

.3 - 

.2 - 

> 

hl 

.6 

.5 

.4 

.3 

.2 

.1 

(A) y3/B = 0.18. (B) y3/B = 0.32. 

I 

n/U* 

(D) ?/B = 0.86. (C) J/B = 0.52. 

FIGURE 5. - LATERAL VELOCITY PROFILES. 

14 
*4 r 

.2 

0 

-.2 

0 EXPERIMENTAL DATA OF ALY 

ET AL 151 

PREDICTIONS 

-- 

- 

NR MODEL 

DR MODEL 

Ii\ 

\O \do I 

I 

0 .4 .8 1.0 

y3/B 

FIGURE 6. - VERTICAL VELOCITY PROFILES. 

DR MODEL 

EXPERIENT [SI 

DR MODEL 

FIGURE 7. - CONTOURS OF STREAHWISE VELOCITY U/Ub. FIGURE 8. - CONTOURS OF TURBULENT KINETIC ENERGY k/(U*)’. 

15 
0 U'/U* 

0 V'/U* 

a W'/U* 

PREDICTIONS 

3.0 1 

2.5 - 

- 

- NR MODEL 

DR MODEL I 

a > 2.0 

3 1.5 

3 

I 

> 

1 3 1.0 2 

0 .2 .4 .6 

$8 

FIGURE 9. - TURBULENT INTENSITIES ALONG CENTRELINE. 

- NR MODEL 

-__- EXPERIMENT 

141 

- DR MODEL 

FIGURE 11. - CONTOURS OF TURBULENT KINETIC ENERGY 

k/(U,)2. 

NR MODEL - DR MODEL 

---- EXPERIMENT 

(41 

FIGURE 10. - CONTOURS OF STREANWISE VELOCITY U/U,. 

- NR MODEL PREDICTIONS OF 

---- EXPERIMENT NAKAYANA ET AL [121 

[41 

FIGURE 12. - ANISOTROPY OF THE SECONDARY NORMAL -- 

STRESSES (U: - U;)/(U*)*. 

16 
I I I I 

.2 .4 .6 .8 

$/H 

0 

(C) VERTICAL WALL. 

I I I 

$0 .2 .4 .6 .8 

i y3/H 

(A) HORIZONTAL WALL. 

#e--- 

1.2 - 

0 10 20 30 40 50 60 70 80 90 

e 

(B) CURVED WALL. 

0 EXPERIMENTAL DATA OF SEALE [61 

PREDICTIONS 

k -E MODEL 

- 

NR MODEL 

DR MODEL 

-- 

FIGURE 13. - WALL SHEAR STRESS DISTRIBUTION. 

17 
w/u* 

(A) y3/H = 0.09. 

.* r 

0 EXPERIMENTAL DATA OF 

SEALE E61 

PREDICTIONS 

- 

- 

- NR MODEL 

DR MODEL 

.6 

.4 

0 

(B) ?/H = 0.23. (D) y3/H = 0.50. 'I .4 

.2 

0 .4 0 

-.4 

w/u* 

(C) y3/H = 0.36. (E) y3/H = 0.63. 

FIGURE 14. - LATERAL VELOCITY PROFILES. 

18 
PREDICTIONS - NR MODEL EXPERINENT [61 

FIGURE 18. - CONTOURS OF TURBULENT KINETIC ENERGY k/(U.)2. 

CORRELATION CURVES DUE 

TO SEALE [61 

- - w'/U, 

- - - V'/U* 1.6 1 

1.4 A v'/U, 

0 W'/U, 

A 

00 

PRESENT PREDICTIONS .' 0 

1.2 

1 .o 

.8 

a > .6 

I 

(A) NR MODEL. > 

1.2 

1.0 

.8 

.6 

1.2 1.6 2.0 2.4 

u'/U* 

(B) DR MODEL. 

FIGURE 19. - TURBULENT INTENSITIES ALONG CENTRELINE. 

20 
.4 

.2 

0 

* -.2 

3 \ 

0 EXPERIMENTAL DATA OF SEALE [61 

PREDICTIONS 

- 

- - NR MODEL 

DR MODEL 

IC 

r 

-.4 

(A) y2/H = 0.07. 

.2 r 

-.2 0 .2 .Ir .6 .8 - 

(B) y2/H = 0.18. 

- 

y '/H 

(C) y2/H = 0.31. (D) y2/H = 0.44. 

FIGURE 15. - VERTICAL VELOCITY PROFILES. 

I. 

VELOCITY VECTORS STREAM FUNCTION CONTOURS 

FIGURE 16. - PREDICTED FLOW PATTERNS IN ROD-BUNDLE GEOMETRY (NR 

MODEL). 

PREDICTIONS - NR MODEL EXPERIMENT [61 

FIGURE 17. - CONTOURS OF STREAMWISE VELOCITY U/U,. 

19 
National Aeronautics and %ace Administration 

2. Government Accession No. '' Report No' NASA TM-102142 

ICOMP-89-16 

4. Title and Subtitle 

Calculation of Turbulence-Driven Secondary Motion in Ducts With 

Arbitrary Cross Section 

Report Documentation Page 

3. Recipient's Catalog No. 

5. Report Date 

July 1989 

6. Performing Organization Code 

7. Author@) 

A.O. Demuren 

8. Performing Organization Report No. 

E-494 I 

10. Work Unit No. 

505-62-2 1 

9. Performing Organization Name and Address 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 44135-3 191 113. Type of Report and Period Covered 

2. Sponsoring Agency Name and Address 1 Technical Memorandum 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

14. Sponsoring Agency Code 

5. Supplementary Notes 

A.O. Demuren, Institute for Computational Mechanics in Propulsion (work funded under Space Act Agreement 

C99066G). Space Act Monitor, Louis A. Povinelli. The author is on leave from the University of Lagos, 

6. Abstract 

Calculation methods for turbulent duct flows are generalized for ducts with arbitrary cross-sections. The irregular 

physical geometry is transformed into a regular one in computational space, and the flow equations are solved 

with a finite-volume numerical procedure. The turbulent stresses are calculated with an algebraic stress model 

derived by simplifying model transport equations for the individual Reynolds stresses. Two variants of such a 

model are considered in the present study. These procedures enable the prediction of both the turbulence-driven 

secondary flow and the anisotropy of the Reynolds stresses, in contrast to some of the earlier calculation 

methods. Model predictions are compared to experimental data for developed flow in triangular duct, trapezoidal 

duct and a rod-bundle geometry. The correct trends are predicted, and the quantitative agreement is mostly fair. 

The simpler variant of the algebraic stress model procured better agreement with the measured data. 

Turbulence modeling Unc1assifie.d - Unlimited 

Reynolds stress modeling 

Complex channel flows 

Finite volume method 

Subject Category 34 

9. Security Classif. (of this report) 20. Security Classif. (of this page) 

Unclassified Unclassified 

1 21. No of;T 22. Price' 1 A03 

*For sale by the National Technical Information Service, Springfield, Virginia 221 61 NASA FORM 1626 OCT 86