Hydraulics Lab., Department of Civil Eng., Univ. ABSTRACT · PDF fileHydraulics Lab.,...

Turbulent flow in an open channel with a

backward-step

P. Prinos, J. Rasoul

Hydraulics Lab., Department of Civil Eng., Univ.

of Thessaloniki, Thessaloniki, Greece

ABSTRACT

The backward-step flow in an open channel is studied numerically usinga finite volume method for solving the Reynolds averaged Navier-Stokesequations in conjuction with turbulence models of the k-s type. The standardk-s model of turbulence and a model of the same type, based on the RNGapproach of Yakhot and Orszag (22), are used and computed results arecompared against experimental measurements of Nakagawa and Nezu (13).

The latter model is found to perform better than the standard k-s modelespecially in the recirculation region behind the step. Reattachment length,velocity, turbulence kinetic energy and shear stress are predicted better by thismodel. The standard k-s model substantially underpredicts the experimentalreattachment length ( up to 19%) while the RNG k-s model is in closeagreement with the measured reattachment length (maximum error 8%).

INTRODUCTION

The study of a backward-step flow in an open channel is quite essentialin hydraulic engineering since such flow features are often observeddownstream of sand waves on river bed and also of man-made hydraulicstructures. Also turbulent flow over a backward-step is quite appropriate fortesting the predictive ability of the various turbulence models for identifying anydeficiencies and for improving their effectiveness.

The characteristics of the backward-facing step flow in a duct have beenstudied extensively by several investigators (1,3,4,5,6,7,21) experimentally andnumerically. However such studies in open channels are rather limited(12,13,14).

Detailed experimental measurements of mean and turbulencecharacteristics in a duct flow with a backward-step have been untertaken byKim et al. (6) and Driver and Seegmiller (3) among others. The lattermeasurements have been used for comparison purposes against numericalpredictions in the recent "Collaborative Testing of Turbulence Models"

Transactions on Modelling and Simulation vol 4, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

324 Computational Methods and Experimental Measurements

organized by Stanford Univ. (2) which is a follow-up of the Stanford 1980-1981Conference (8). Numerical predictions of such a flow have been reported (3)based on turbulence models of the k-s type and of algebraic type which wereshown to yield rather poor results and especially to underestimate therecirculation length.

Rodi (20) has reported the use of a two-layer model for simulating suchflow, based on the near wall one-equation model due to Morris and Reynolds(16), which was found to give improved results. Also, Kim (6) has employed amultiple-time scale turbulence model of the k-s type and found that the modelyields improved results in the region where the turbulence is in a strongly non-equilibrium state.

Recently Thangam and Speziale (21) have evaluated the performanceof two-equation models in such a flow and demonstrated that errors in previouspredictions of the k-s model have as main origins the inadequate resolutionand the use of an isotropic eddy viscosity. They indicated that the standard k-smodel, modified with an indepedently calibrated anisotropic eddy viscosity canyield surprising good predictions for the back-step problem.

In open channel flow with a backward step the only detailedmeasurements reported are those of Nakagawa and Nezu (13). Theyconcluded that the separated step flow and its recirculation in an open channelare similar to that in boundary layer and duct flows but the former is morecomplicated due to Froude number effects. It was found that the recirculationlength tends to increase gradually with increasing Froude number.

Also numerical studies of such a flow are rather limited. Nezu et al. (14)have computed such a flow based on the k-s model of turbulence and foundthe largest difference between measured and predicted reattachment length tobe 16%. In a similar study Mendoza and Shen (11) have calculated the flowover dunes using an algebraic Reynolds stress turbulence model.

In this study the flow in an open channel with a backward-step ismodelled using turbulence models of the k-s type. Computed results arecompared against the measurements of Nakagawa and Nezu (13). Initially thestandard k-s (10) is used for testing the effectiveness of such a model inrecirculating flows. Subsequently the RNG-based k-s model is used, asdescribed by Yakhot and Orszag (22), which is of the same form as thestandard k-s model. However the numerical values of the constants, computedby the RNG approach, are different. The latter model has been applied byThangam and Speziale (21) in the backward-step flow of (3) but the resultswere poor due to the wrong value of the coefficient C-,, appearing in the s-equation, as proposed initially by Yakhot and Orszag (22). However, Yakhotand Smith (23) have recently determined the correct value of c^ which is usedin this study.

Computed flow characteristics using the above mentioned turbulencemodels are compared against the experimental measurements of Nezu andNakagawa (13) for the ST-1 case. Computed recirculation lengths, frictioncoefficient c, and pressure coefficient Cp are compared against the measuredones in the ST-1 case which has a characteristic Fr number equal to 0.19 andRe number equal to 8200.Also, detailed computational and experimental meanvelocity and turbulence characteristics are compared at selected stationsdowstream of the backward step (x/Hg = 3.0, 5.0, 9.0 and 12.0, H = step


Computational Methods and Experimental Measurements 325

height).

GOVERNING EQUATIONS

The equations describing the steady, incompressible two-dimensionalflow over a backward-step can be written in the following form assuming thatthe eddy viscosity concept is valid for estimating the Reynolds stresses:Continuity equ:

-o d)ax,

Momentum equ:

ax, pax, ax, dxj

where Uj = component of mean velocity in the i-direction, p = fluid density, p= effective pressure, which is the difference between the actual and thehydrostatic pressure, v^f = effective viscosity (v , = v + v^, v = fluid viscosity,v, = eddy viscosity), f, = component of the external forces in the i-direction.

The eddy viscosity v, is calculated from the following relationship:

where c^ = constant (= 0.09), k = turbulence kinetic energy and s =dissipation of turbulence kinetic energy.

When the standard k-s model of turbulence is used, the characteristicsof turbulence k and s are calculated through the following transport equations:k-equ:

'ax, ax, o*. ax, *

s-equ:

'ax, ax, Og ax, k * k

where o^, o^, c^, = constants (= 1.0, 1.3, 1.44, 1.92 respectively) and P^= production of turbulence kinetic energy due to shear.When using the RNG-based k-s model of (22) the numerical values of theconstants, appearing in equations (3),(4) and (5), are computed by the RNGapproach and take the following values:C^ = 0.0837 0|< = 0.7179 O^ = 0.7179 C-, = 1.063 Cg = 1.7215Also the vonKarman constant K, appearing in the log-law of the wall, takes thevalue K = 0.372.



The model with the above constants has found to be too dissipative (21)due to the low value of the c-, constant as opposed to the more standard valuec., = 1.44. Hence the reattachment length is underpredicted considerably (21).

Recently Yakhot and Smith (23) have determined that there was anerror in the original calculations (21) and obtained an alternate value of c-, =1.42 which is very close to the standard value used. Hence the model with thenewly derived coefficient is tested in this study.

NUMERICAL METHOD

A finite volume method is used for solving equations (1), (2), (4) and (5)as described extensively by Patankar (17). The above equations can be put ina common form as:

A(pwi))_A(r )_A(r ) = (6)ay dx dx ay ay

where 0 = the dependent variable (= 1 for the continuity, U, V for momentumequations and k, s for the k-s model); f^ = diffusion coefficient and S^ =source term.

The main characteristics of the numerical method used are:(a) Equation (6) is transformed to an algebraic equation by integration over

a control volume.(b) A staggered grid is used for storing the dependent variables and hence

avoiding pressure oscillation.(c) The continuity equation is transformed to a Poisson Equation and a

pressure correction equation is solved, which gives the pressurecorrection needed for the velocities to satirfy the continuity equation(SIMPLE method (17)).

(d) The hybrid scheme, described by Patankar (17), is used for discretizingthe advection terms.

(e) The tri-diagonal matrix algorithm (TDMA) is applied for solving thealgebraic equations.

The finite volume method is applied with appropriate boundary conditionswhich are described briefly in the following paragraphs.(a) Inlet: At the inlet, located at x/Hg = -1.0, the experimental

measurements of Nakagawa and Nezu (13) for the mean velocity Uwere used while the k was estimated from measurements of the normalstresses. Finally the inlet s was calculated assuming that the productionof turbulence kinetic energy k is equal to dissipation s.

(b) Outlet: At the outlet, located at x/H^ = 100, all gradients with respectto x were set to zero, while the value of V was zero. However, thevelocity U was corrected to satisfy the overall continutity.

(c) Walls: When the standard k-s is applied, the wall function approach ofLaunder and Spalding (9) is used for the calculation of the velocitycomponents parallel to walls at the first grid point near the wall.

(d) Free-surface. The free-surface was treated as a rigid lid since theFroude numbers examined are relatively low. Hence, a symmetry



condition applies for the velocity component U and the turbulencekinetic energy k. The vertical velocity component V is set to zero whilez is specified according to the approach proposed by Lau andKrishnappan (8). Based on this approach, s at the free surface iscalculated from the following relationship:

k Y'S*f

fa)(7)

where k, = the turbulent kinetic energy at the free-surface, y, = thedistance between the nearest grid point and the surface and Cf =empirical constant (0.164).

EXPERIMENTAL AND COMPUTATIONAL RESULTS

Computed and experimental results are presented at the locations x/H^= 3.0, 5.0, 9.0 and 12.0. The distributions of the computed mean velocity U,the turbulent kinetic energy k and the shear stress -uv are compared with therespective experimental measurements of Nakagawa and Nezu (13) for the ST-1 experiment using the two turbulence models. Hence the effect of theturbulence models (standard k-s, RNG model) on the flow characteristics couldbe studied. Also, computed and experimental reattachment length, pressurecoefficient Cp (Cp = (p - p f) / 0-5 p U /) and friction coefficient Cf arecompared for the various conditions.

Before studying the effects of the discretization schemes and theturbulence models on the flow characteristics a grid consisting of 180 x 50nodes with a total length x/H,, = 100 was selected after comparing the resultswith those obtained using coarser grids and shorter lengths. For the sake ofbrevity no results are presented from the grid indepedent tests.

Figure 1 shows the experimental and computed reattachment lengthsx/Hg for two test cases (ST-1 and ST-3) of (13) together with other data fromArmaly et al (1), indicating the Reynolds number effects on x/H . It is shownthat the RNG model produces x/H very close to the experimental ones whilethe standard k-s model underestimates the reattachment length . A higherunderestimation is observed for the ST-1 case for which the Re is lower. Tesuperiority of the RNG model is shown in fig. 2 where the variation of Cf (Cf =T / O.SpU ma* *i TW = wall shear stress, U^^ = maximum velocity at inlet)along the channel bed is plotted for both turbulence models. For the ST-1 case(fig. 2(a)) the reattachment length predicted by the RNG model isoverestimated by 8% with regard to the experimental one (x%/Hg = 5.25)while the standard k-s underpredicts it considerably (12.5%). In the ST-3 case(fig. 2(b)) the length of reattachment predicted by the RNG model is almostequal to the experimental one while the standard k-s model underpredicts it by12.5%. The performance of the standard k-s model and the errors producedby it are similar with presented in (14). Nezu et al (14) have calculated errorsin the predicted reattachment length up to 16%.



The better performance of the RNG model is also demonstrated in figs.(3) and (4) where computed and experimental friction coefficient c, (defined asCf = 2(11 * / U maxjoc) * where U. = shear velocity and U axjoc ~ maximumlocal velocity) and pressure coefficient Cp ( Cp = (p - p f) / d-SpU^x ) arecompared for the two test cases.

The accurate prediction of the reattachment length by the RNG modelresults in a better prediction of the flow characteristics in this region. Fig. 5shows the observed and predicted velocity distributions by both models atselected stations of the ST-1 case. At x/Hg = 5.0 the RNG model predictsnegative velocities near the bed due to the extent of the reattachment region,while the standard k-s model predicts very small negative velocities very closethe channel bed. Also, in the developing region (x/Hg = 9.0) the RNG modelpredicts velocities close to the measured ones.

Similar conclusions can be drawn from the comparison of computedand observed turbulence kinetic energy (fig. 6). At x/Hg = 3.0 the RNG modelpredicts k levels very close to the experimental ones while the standard k-smodel overpredicts them. In the developing region the two models have similarbehaviour with the RNG model slightly underestimating the turbulence levels.

Finally predicted and observed shear stress -uv indicate a betterperformance of the RNG model in the recirculating region and anunderestimation of shear stress by both models in the developing region (fig.7).

CONCLUSIONS

The flow characteristics in an open channel with a backward step havebeen studied numerically using the standard k-s model and a model of thesame type based on the RNG approach of Yakhot and Orszag (22). Theperformance of the models used has been assessed by comparing thecomputed results against extensive experimental measurements of Nezu andNakagawa (13) for their ST-1 case. The following conclusions can be derivedfrom the above study.(a) The RNG - based k-s model is found to be superior to the standard k-s

model for such flow conditions. The model predicts more accurately thereattachment length and the flow characteristics in the recirculatingregion. The standard k - s model underpredicts the reattachment length(up to 19%) while the new model is in close agreement withexperimental data ( Maximum Error 8%).

(b) The variation of the friction and pressure coefficients along the channelbed are predicted with the RNG model than with the standard k-smodel.

(c) Velocity profiles in the recirculation region are predicted by the RNGmodel in close aggreement with the experimental results.

(d) The prediction of turbulence kinetic energy and shear stresses by bothmodels has some deficiencies in the recirculation region due to highflow anisotropy inthere. However, the predictions improve in thedeveloping region.



REFERENCES

1. Armaly, B.F., Durst, F., Pereira, J.C.F., Schonung, B. (1983) 'Experimentaland theoretical investigation of backward-facing step flow', J. Fluid Mech. 127,473-496.2. Bradshaw, P., Launder, B.E., Lymley, J.L (1991) 'Collaborative testing ofturbulence models', AIAA paper, 91-0215.3. Driver, D. M., Seegmiller H.L (1985) 'Features of a reattaching turbulentshear layer in divergent channel flow ', J. of AIAA,vol. 23, 163-171.4. Durst, F., Tropea, C. (1983) 'Flows over two-dimensional backward-facingsteps', in IUTAM Symp. on Structure of Complex Turbulent Shear flow,Springer41-52.5. Etheridge, D.W., Kemp. P.M. (1978) 'Measurements of turbulent flowdownstream of a rearward-facing step', J. Fluid Mech. 86, 545-566.6. Kim, J., Kline, S.J., Johnston, J.P. (1978) 'Investigation of separation andreattachment of a turbulent shear layer: flow over a backward-facing step'Report MD-37, Dept. of Mech. Eng., Stanford University. .7. Kim, S.V. (1991) 'Calculation of divergent channel flows with a multiple-time-scale turbulence model', J. of AIAA, vol. 29, no. 4, 547-5548. Kline, S.J., Cantwell, B.J., Lilley, G.M. (1982) 'Proc. 1980-81 AFOSR-HTTM -Stanford Conf. on complex turbulent flows - comparison of computations andexperiments' Vols.1-3.9. Lau, Y.I., Krishnappan, B.G. (1981) 'Ice cover effects on stream flows andmixing', J. of Hydraulics Div., ASCE, vol. 107, no 10, 1225-1242.10. Launder, B.E., Spalding, D.B. (1974) ' The numerical computation ofturbulent flow',Comp. Methods in Appl. Mech. and Eng., 3, 269-289.11. Leonard, B.P. (1978) 'A stable and accurate convective modellingprocedure based on quadratic upstream interpolation', Comp. Meth. in AppliedMech. and Eng., vol. 19, no 1,pp 59-9812. Mendoza, C., Shen, H.W. (1990) 'Investigation of turbulent flow overdunes', J. Of Hydraulic Eng., vol. 116, no 4, 459-477.13. Nakagawa, H., Nezu, I. (1987) 'Experimental investigation on turbulentstructure of backward-facing step flow in an open channel', J. Of HydraulicResearch, vol. 25, no 1,pp 67-88.14. Nezu, I., Papritz-Wagner, B., Scheuerer, G. (1988) 'Numerical calculationsof turbulent open channel backward-facing step flows', 3rd Int. Symp. onRefined Flow Modelling and Turbulence, IAHR, Tokyo, 183-190.15. Nezu, I., Nakagawa, H. (1989) 'Turbulent structure of backward-facing stepflow and coherent vortex shedding from reattachment in open channel flows',Turbulent shear flows 6, 313-337.16. Norris, L.H., Reynolds, W.C. (1975) Turbulent channel flow with movingwavy boundary',Rept. No FL-10, Stanford Univ., Dept. Mech. Eng.17. Patankar, S.V. (1980) 'Numerical heat transfer and fluid flow', HemispherePublishing Co.18. Patel, V.C., Rodi, W., Scheuerer, G. (1985) Turbulence models for near walland low- Reynolds number flows. A review', J. AIAA, vol. 23. 1308-1329.19. Roache, P.J. (1972) 'Computational fluid dynamics' Hermosa, Albuquerque,NM, USA.



20. Rodi, W. (1990) 'Some current appraches in turbulence modelling', AGARDTR in Appraisal of the Suitability of Turbulence Models in Flow Calculations.21. Thangam, S., Speziale, C.G. (1992)' Turbulent flow past a backward-facingstep: A critical evaluation of two-equation models' J. of AIAA, vol. 30, no. 5, pp.1314-1320.22. Yakhot, V. , Orszag S.A. (1986) 'Renormalization Group Analysis ofturbulence. I. Basic Theory .' J. of Scientific Computing, vol. 1, no. 1, pp. 3-51.23. Yakhot, V. , Smith, L (1992) ' The renormalization group, the s-expansionand derivation of turbulence models.', J. of Scientific Computing.



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