FULLY DEVELOPED PULSATING TURBULENT FLOWS

Reda. R. Mankbadi Cairo Universitv

Cairo, Abstract

The effect of imposed pulsations in a fully developed turbulent channel flow is examined. The calculations are obtained with an unsteady finite-volume method employing the k-E turbulence model. Results show that the effect of pulsation on the mean velocity is negligible. The time-averaged turbulence intensity and the time-averaged Reynolds shear stress are slightly influenced by high level of pulsation.

Nomenclature

amplitude of the imposed oscillations height of the channel turbulence energy length of the channel pressure Reynolds number Strauhal number = wH/Ut axial velocity component time-averaged axial velocity component transversal velocity component axial coordinate transversal coordinate Y/H

frequency displacement thickness momentum thickness phase angle wall shear stress dissipation rate of the turbulence cycle angle

Superscript: - time-average " oscillating component

I turbulence

Subscript: i inlet < > phase-average

1. Introduction

The interest in unsteady flows has been increasing in recent years due to the relevance of such studies to many technological applications. A number of experiments have been performed in recent years to study the behaviour of turbulent flows with a periodic variation in the velocity. A full review of such experiments is given by Carr [ l l . A number of these studies, have been directed toward turbulent boundary layers developed on plane surfaces as a first step in dealing with more complex flows. While these studies have provided a significant amount of information on the effect of the imposed oscillations on the behaviour of turbulent shear flows, there are large number of areas in which information is either not available or is contraversal. This is particulary true for flows in which the

imposed periodicity is either at high frequency or has a large amplitude. There is a general feeling that the imposed periodicity has no effect on the time- averaged properties of the flow. There is no general concensus on the qualitative response of the wall shear stress and near-wall flow to imposed periodicity. Therefore, the present work is concerened with examining the effect of the frequency and the amplitude of the imposed pulsation on the developement of the fully developed turbulent flow in a channel.

The purpose of the present investigation is two-folds: first to examine the applicability of quasi-steady turbulence modelling for solving unsteady pulsating flows, and secondly, to examine the effect of the amplitude and frequency of the imposed pulsation on the two-dimensional fully developed flow.

2. The Mathematical M o d d

The problem under consideration is that of the unsteady turbulent flow between two- dimensional plates of infinite aspect ratio. Since the flow is time-dependent, the mean value here is taken as the phase-averaged quantity which is composed of a time-mean part and a periodic component. The mean flow momentum equations and the transport equations of the turbulence energy and its rate of dissipation are written in the general differential form:

where for the continuity equation:

for the x-momentum equation: - a p

a = U , rU = vc f and SU = ax +

for the y-momentum equation:

+ = V , I.., .- 11,,,f a n d Sv = - - a p + a Y

for the k-equation:

= k, rk = ueff/ Ok , and

"Copyright (c) 1988 by AIAA, Inc. All Rights Reserved. " 376

and for the E-equation:

x and y are the coordinates parallel to and perpendicular to the plate, respectively. u and v are the coressponding velocity components and /' , P 6 p are the density, pressure and viscosity, respectively. The Reynolds stress closure problem are handled through the eddy viscosity concept. The turbulent viscosity is obtained from the k-E model of Jones & Launder [21, where k is the turbulent kinetic energy and E is its rate of dissipation. k and E are obtained by solving their transport equations ( 566 simultaneously with the mean flow equations. G is the production of turbulent energy by working of the mean flow on the turbulent Reynolds stresses. p f is the effective viscosity which is the sum of the molecular and eddy viscosities. The constants appearing in the above equations ( C1, C,,u6, uk, C, , Cd) are given in Jones & Launder t 2 1 .

Following Patankar [ 3 1 , the finite-difference equations are formulated by integrating the above differential equations over rectangular control volumes. The source terms S. are linearized and split into two parts S, and S, 4 such that S, must always be less than or equal to zero. Discretization of the finite difference equations are written in the form:

where dt is the time-step and dv is the volume of the control volume considered. For the x-momentum equation we obtain:

a P a 2 u Sp = 0, Sc ' - //-- d~+//(~-(ll~ff ) + ax

for the y-momentum equation:

6 v t p- k0 (10)

A t and for the E-equation:

2.1 Boundary Conditions

Ui is the time-averaged inlet velocity and is taken to be unity in all the calculation performed here. A is the amplitude of the imposed oscillations and w is its frequency. The Strauhal number S is defined here as S =WH/UI , H is the channel's height taken here to be unity. The turbulent kinetic energy at the inlet is taken k = 0.00005 Usz. The energy dissipation at the inlet is given by:

where L, is a characteristic length scale taken as 0.01 H. At the exit plane, the flov is assumed to be parabolic, i .e., independent of the unknown downstream variables. For the computations of the fully developed flow in a channel, parabolic conditions are assumed after a distance along the channel ten times of the channnel's height.

The existence of wall similarity in a constant stress layer has enabled the use of the wall function for the velocity vector to reduce the number of grid points near the wall. The modified wall function:

is used where: U+ = U/U+ , U+ = ['W/P and

and r, is the wall shear stress. C and E are constants of the modified wall function taken from Dean's [41 survey of available experimental data as C =0.41 and E=0.331.

At the nodes nearest to the solid walls, the velocity vector is assumed to lie on the plane parallel to the wall and the wall function is applied for the velocity vector. With the assumption of local equilibrium and the wall function, the following relationships are obtained at the node next to the wall:

For the k-equation near the wall, the production G is replaced by:

Assuming local equilibrium near the wall, the source term Sp in the k-equation is replaced by:

The above system of equations is solved for pulsating inlet conditions. The inlet velocity profile is uniform in y and is periodic in time:

For the E-equation, the value of at LA.- node nearest to the wall is taken as the local equilibrium value:

2.2 The Solution Procedure

The present model is implicit in time- discretization. For space discretization the QUICK scheme of Leonard (51 is applied in order to reduce the numerical diffusion. ~t each time-step, the two momentum equations are first solved with an assumed pressure field using the strongly implicit method of Stone 161. As the new velocity field does not satisfy the continuity equation, the pressure field is adjusted towards the solution of the continuity equation as given by patankar [31. The procedure continues untill the solution converges for this time- step. A new time-step begings and the process continues for several cycles of the imposed oscillations untill the cycle-to-cycle variations are negligible.

2.3 Code Validation

In order to test the code and the turbulence model, the calculations are compared with the experimental data of Ramaprian & Tu [71 for a fully developed pipe flow. Ramaprian & Tu [71 pulsated a pipe flow at Reynolds number of 50,000. Measurements were conducted at the exit section. Two frequencies were studied: 0.5 Hz and 3.6 Hz. The two frequencies are equivalent to Strauhal numbers based on diameter of 0.175 and 1.25, respectively. The amplitude of oscillations were taken 0.65 for the low frequency and 0.15 for the high frequency. The calculations are performed here at the same conditions as those of Ramaprian & Tu's [71 experiment. A harmonic analysis of phase-averaged quantities is performed here to examine the oscillating components. Assuming the oscillating part to be dominated by the component of the same frequency as the imposed oscillations, one can write

where g" is the amplitude of the oscillations, * is the phase-difference with respect to the imposed oscillations. A check was made to examine the generations of other harmonics. It was found that the oscillation was dominated by the fundamental. This has been confirmed in several observations (e.g. Tu & Ramaprian [el). Figure 1 shows a comparison between the calculated velocity oscillations and the measured ones. At the low frequency the observed overshoot in the amplitude of the oscillating axial velocity is farther away from the wall than the measured one. At the high frequency case the calculated overshoot is at the same distance from the wall as the measured one, but the calculated overshoot is underestimated. The phase angle is compared in figure 2b. The figure shows qualitative agreement between theory and experiment.

378

clql~rc 1 . Cornpar I -,D~I of t I calculated ~*rotlle of ttre osrlllatlons In the axlal velocl ty componrtlt vl t11 tile mcasurernents of R?aaprl;lrl b Tu (1983) for pu1s;rtlnq plpe flow dt Re=50,000. Throry; Exper lment : oooooo S r l . 2 a1111 A : 0.15; A k a A a a 910.175 and A-O.Sr,. a ) Amplltufle b ) Phase angle.

FIql~r~2. r I t~etveen tho calculated v.yclc-var lat lonn of the wall 3hear stress wlttt Tu .S R.lrn.7llr Ian ' 3 (1983) measurements I n a pulsat lnq pipe flov at Re-:50,000. 3 1 5 - 1 . 2 #at111 A - 0 . 1 5 h ) S=O.175 and A-0.65.

Figure 2 shows a comparison between the cycle-variations of the wall shear stress and the measurements of Tu & Ramaprian [81 at the two sets of frequency and amplitude of their pipe's experiment. The agreement is generally satisfactory. However, the theory cannot predict the kink observed in the measured wall shear stress at the high frequency case. This kink could not be explained by Tu & Ramaprian either.

The cycle-variations of the Reynolds shear stress is shown in figure 3 in comparison with Ramaprian 6 Tu's [71 data of the pulsating pipe flov. At the low frequency the agreement between theory and observations is quite satisfactory. However, at the high frequency the calculated oscillations in the Reynolds shear stress are much smaller than the measured ones. This indicates a complete failure of the quasi-steady turbulence model used here at high frequencies. In the turbulence modelling, several relations were deduced based on assumina 1 nral enuilibrium of the turbulence structure in the wall- region. Because of this assumption, quasi- steady turbulence modelling may fail in predicting unsteady effects.

The effect of the frequency of the imposed oscillations is studied here for low and high levels of pulsations. At the low level of pulsation the amplitude of the imposed oscillation is taken 0.10. At the high level of pulsation the amplitude is taken 0.50. The Strauhal number is varied from 0.20 to 5.0. The results presented here pertain to the section immediately before the exit of the channel where fully developed conditions are assumed.

3.1~he~hase-averaaed velocity The time-averaged velocity profile is shown in figure 4 for low and high level of oscillations. The figure shows that the profile under pulsation conditions is practically indistinguishable from that of the steady flow case. Thus, even at the large amplitude the time-mean velocity is not affected by pulsations. This has been observed in a flat plate by Karlsson [91 over a wide range of frequencies and amplitudes. For a flat plate at adverse pressure gradient Paraikh, Reynolds & Jayaraman [I01 and Cousteix, Houdeville & Javelle ill1 have reached the same conclusion. However, the measurements of Tu & Ramaprian [El have indicated that if the frequency is higher than the burst frequency, the mean flow increases by as much as 5% under high level of pulsations. Their calculations for the pipe flow did not show such an effect and lead them to believe that the failure of the calculations to predict a change in the mean flow profile under pulsations is due to using a quasi-steady turbulence model for predicting an essentially unsteady flow. However, Binder, Tardu, Blackwelder & Kueny's I121 observations for the channel flow showed that even when the frequency is higher than

the bursting frequency, high level of pulsations did not alter the mean velocity profile. 'Thus, this bulk of experimental evidence as well as the present theoretical finding leads one to conclude that the time- averaged velocity profile is not influenced by the imposed pulsation.

The cycle-variations of the displacement thickness under low and high levels of pulsations are shown in figure 5. The figure shows that there is a small change in the displacement thickness when the Strauhal number is increased from 0.2 to 0.5, but further increase in the Strauhal number has no effect on the displacement thickness. Thus, although the time-mean velocity profile does not change under high level of pulsation, the instantenous displacement thickness varies by as much as 50% of its mean value during the cycle. This is consistent with Carr's t11 conclusion that although the time-averaged profile is not affected by pulsation, a strong unsteady effects may be present.

The profile of the amplitude of the oscillation in the axial velocity component at several Strauhal numbers is shown in figure 6 for the low and the high pulsations' levels. The figure shows that the overshoot

Flgure 3. Tl~c calculat~d cycle-variatlons In t h e R~yr~olda shear streas at Re=50,000 In compbrlson vlth Tu 6 Rarnaprlan's (1983) d a t a for a pipe flov at y/R =0.035. a ) 8 - 1 . 2 and A .0 .15 b ) 5.0.175 R I I ~ A=0.65. Thoor y; oooooo Exper lment .

j 4 . ' I c f f p ( . l of pulsations on the proflle of the tlme-averaged axlal veloclty.

Flgure 5 . dlsplacement f requencles o A l O . 10 b

Flgurc G. rrofll~n of the amplltude of 03clllatl0ns In thr axial veloclty at several I r ~ q u ~ n c l c s of the Imposrd pulsatlon. a 1 h=O.lO b ) A ~ 0 . 5 0 .

is maximum around Strauhal number 0.5 and then decreases with increasing the frequency. The location of this overshoot moves closer to the wall with increasing the frequency. Thus, at high frequencies, the amplitude is constant for the bulk of the flow indicating that the flow oscillates almost like a solid body. This has been observed by Ramaprian h Tu [ 7 1 for the pipe flow.

The profile of the phase angle at both levels of pulsation at several Strauhal

C~clc-'Jarlatlot\n of the numbers is shown in figure 7. The phase angle thickness lJnder is always positive near the wall and

the imposed pulsation. a ) decreases to negative values before it A z 0 . 5 0 . reaches zero with increasing y . The phase-lag

is maximum at Strauhal number 0.5. The thickness of the region where the phase angle

Figure 7 . Prof llcs of the phase angle of the oscillations in the axlal veloclty component at several frequencles of the lmposed pulsatlon. a) A 0 0 . 1 0 b) A = 0 . 5 0 .

I z 3 L 5 5

Figure 8 . Calculated osclllntlons ln the w a l l shear stress. a ) Amplitude b) Phase a11qlf-.

varies decreases with increasing the Strauhal number. This trend of behaviour has been observed by Parikh et al. [lo1 for a pulsating turbulent boundary layer under adverse pressure gradient. However, because of the adverse pressure gradient present in their experiment, the observed overshoot in the amplitude and the observed variation in the phase angle are generally greater than the corresponding values obtained here.

Shemer, Wygnanski & Kit [131 examined laminar and turbulent pulsating pipe flows at Reynolds number of 4000. Their observations have indicated that the dependency of the phase angle on the level of the imposed pulsation is weak. This is consistent with the present calculations. The behaviour of the phase angle under low level of pulsations (figure 7a) is almost the same as its behaviour under high level of pulsations (figure 7b). Figure 7 also shows that the phase-lag or -lead tends to diminish with increasing the frequency as Shemer et al. have observed.

3.2. The Wall Shear S t r a The calculated time-averaged wall shear stress is found to be the same like the calculated one for the steady flow case. Thus, the pulsation has no effect on the time-averaged wall shear stress as observed by Binder et al. [121. The effect of pulsation on the wall shear stress at high and low levels of pulsations is shown in figure 8. The figure shows that the oscillations in the wall shear stress increase with increasing the Stauhal nummber. This is in accordance with Cousteix et al.'s 1111 data for the pulsating boundary layer case. The figure also shows that the relative oscillations in the wall shear stress normalized by the imposed oscillations decreases with increasing the pulsation level. Thus, the process is nonlinear. The phase anglle of the oscillations in the wall shear stress also increases with increasing the Strauhal number as figure 8b indicates. Unlike, the magnitude of the wall shear stress oscillations, the phase angle of these oscillation is weakly dependent on the level of pulsation.

At low levels of pulsations, the time- averaged Reynolds shear stress is the same as in the steady flow case. The effect of high level of pulsation on the time-averaged Reynolds shear stress is shown in figure 9. The figure shows that only at Strauhal number of 5, there is some noticable difference between the pulsated case and the steady flow case. This is consistent with Ramaprian & Tu's [71 conclusion that only sever pulsation at high frequencies can affect the time- averaged values. The amplitude 0 f oscillations in the Reynolds shear stress is shown in figure 10. The figure shows that the oscillations are restricted to a small layer next to the wall. For both the low and the hiqh level of pulsation, the fiaure indicates

F l g u r e 9 . The e f f e c t of high l e v c l of pulsation on t h e t l m e - a v e r a g e d R e y n o l d s 3 t 1 r j r s t r e s s . Re = 2 0 0 , 0 0 0 and A = 0 . 5 0 .

F l g u r e 11. The e f f e c t o f h l g h l e v e l o f p u l s a t l o n on t h r t l m e - a v r r n g c d t u r b u l e n c e e n e r g y . Re = 2 0 0 , 0 0 0 a n d A 7 0 . 5 0 .

I , f F l q u r ~ 1 2 . P r o f 1 l r n o f ttlr nmpl l t u d e of F l q u r r 1 0 . P r o f l l e s n f t h e nmnl l t i ~ ~ l * o s c l l l a t l o n s I n t h e R e y n o l d s s t w a r - s t r e a r i t t o s c ~ l l a t l o r ~ n I n t h e t u r l , u l c r ~ c r e n e r g y a t s e v e r a l f r e q u e n c i e s o f t h e imposed pulsations s e v e r a l f r - q u - n r l e s o f t h e lmposed a ) A=0.10 b ) A=0.50 o s c l l l a t l o n s . a ) A-0.10 b ) A=0.50 .

that the oscillations decrease with increasing the frequency. Thus, at S=5.0 the oscillations in the Reynolds shear stress is almost zero for most of the cross-section. This is consistent with Parikh et al.'s [lo1 conclusion that at high frequencies, the Reynolds shear stress is frozen over the oscillation cycle.

3.4 The Turbulence Eneray At low levels of pulsations, the time- averaged turbulence intensity is not affected by pulsation as in Cousteix, Desopper & Houdevillels [I41 experiment. At the high level of pulsation, figure 11 shows that the turbulence energy increases over that of the steady flow case and the effect is most pronounced at the high Strauhal number case of S =5. This effect has been observed by Mizushina, Maruyama & Shiozaki [I51 and by Ramaprian & Tu [71. The oscillations in the turbulence energy, shown in figure 12, decrease with increasing the frequency as observed by Ramaprian & Tu 171 for the case of pulsating pipe flow, and by Cousteix et al. [11,141 for the case of pulsating boundary layer. Binder et al.'s [I21 observations for the channel flow have also indicated that the higher the frequency the lower the oscillation in the turbulence energy. Comparing figure 12 to figure 10 shows that the oscillations in the Reynolds shear stress follows the same pattern as the oscillations in the turbulence energy under the various pulsation's conditions. This indicates the coupling between the phase- averaged Reynolds shear stress and the phase- averaged turbulence energy as discussed before. - The conclusions drawn from the present investigations of pulsating fully developed turbulent channel flow can be summarized as follows:

1) The quasi-steady turbulence model adopted here seems to produce satisfactory results. However, if the frequency is large, i.e. short length scale of time-variations, the predicted oscillations in the turbulence are less than the observed ones.

2) The time-averaged velocity profile is not affected by pulsation. However, The time- averaged turbulence energy and the time- averaged Reynolds shear stress are slightly affected by pulsation only if the pulsation level is high.

3) The unsteady effects are confined to a thin layer next to the wall. The thickness of this layer decreases with increasing the Strauhal number of the imposed pulsation.

4) The dependency of the phase angle of the osscilations in the phase-averaged velocity flow on the level of the imposed pulsations is weak.

5) The oscillations in the turbulence energy and Reynolds stresses decrease with increasing the frequency of the imposed pulsation.

References [11 Carr, L.W., "A compilation of existing unsteady turbulent boundary layers experimental data," AGARDograph AG-265, 1981.

[21 Jones, W.P. & Launder, B.E., ItThe prediction of laminarization with two- equation model of turbulenceIn Int. J. Heat m-, Vo1.15, 1972, pp. 301-314.

t 31 Patankar, V.S. Numerical Methods in Heat fer and Fluid Flow. McGraw Hill, New

York, 1980.

[ 4 1 Dean, R.B., "Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional and rectangular duct flow," ASME Journal o f Fluids Enaineerinq, Vo1.100, 1978, pp.215-223.

[51 Leonard, B. P., "A stable and accuarate convective modelling procedure based on quadratic upstream interpolation," m ~ u t e r Hethods in A ~ ~ l i e d Mechanics and Enaineerinq, Vol. 19, 1979, pp.59-98.

[61 Stone, H. L., I1Iterative solution of implicit approximations of multidimensional partial differential equations,"- Numerical Analysis, Vol. 5, 1968, pp. 530- 560.

[71 Ramaprian, B.R. & Tu, S.W. ,*IFully developed periodic pipe flow. Part 2. The detailed structure of the flow," J.Fluid Mech. 137,1983, pp. 59-81.

I81 Tu, S.W. & Ramaprian, B.R., "Fully developed periodic turbulent pipe flow. Part 1. Main experimental results and comparison with p~edictions,~~ J , Fluid Mech. 137, 1983,pp. 31-58.

[91 Karlsson, S.F. , nAn unsteady turbulent boundary layer," J.Fluid Mech. 5, 1959, pp. 622-636.

[lo1 Parikh, P.G., Reynolds, W.C. & Jayaraman, R., "Behaviour of an unsteady turbulent boundary layerrt9 AIAAJ Vo1.20, No.6, 1982, pp.769-775.

[I11 Cousteix, J., Houdeville, R. & Javelle, J., "Response of a turbulent boundary layer to a pulsation of the external flow with and without adverse pressure gradient," In m t e a d v Turbulent Shear Flows,(ed. R. Michel, J. Cousteix 6. R. Houdeville). 1981, PP.120-144. Springer.

t121 Binder, G.,Tardu, S., Blackwelder, R.F. & Kueny, J.L., "Large amplitude periodic oscillations in the wall region of a turbulent channel flov," Fifth Symposium on Turbulent Shear Flows. Cornell University, Ithaca, New York, August 7 - 9, 1985.

[I31 Shemer, L., Wygnanski, I. & Kit, E., "Pulsating flow in a pipe," J.Fluid Me- 153, 1985, PP. 313-337.

I141 Cousteix, J., Desopper, A. & Houdeville, R., wStructure and development of a turbulent boundary layer in oscillating external flowrW In Turbulent Shear flows L, 1977, pp.154-170. Springer-Verlag.