Fluid Dynamics Research 37 (2005) 387–398

Reverse flow in channel with obstruction at entry

K. Senthil Kumara,1, E.G. Tulapurkaraa,∗, G. Biswasb, B.H.L. Gowdac

aDepartment of Aerospace Engineering, Indian Institute of Technology, Madras-600036, IndiabDepartment of Mechanical Engineering, Indian Institute of Technology, Kanpur-208016, India

cDepartment of Applied Mechanics, Indian Institute of Technology, Madras-600036, India

Received 17 January 2003; received in revised form 30 May 2005; accepted 19 July 2005

Communicated by S. Kida

Abstract

In this investigation the complex unsteady flow through and around a channel in the presence of an obstructionat the entry is studied by solving directly the unsteady Navier–Stokes equations. Depending on the location ofthe obstruction, the flow through the channel either reverses its direction, stagnates or continues in the samedirection as that of the flow outside but with reduced magnitude. The Reynolds number of 4000 is considered, asexperimental results are available for comparison. The computed results are in close agreement with experiments.The computations help with better understanding of the phenomenon of reverse flow and fluid pumping.© 2005 Published by The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.

Keywords: Reverse flow in channel; Unsteady laminar flow; Marker and Cell (MAC) method with third order upwinding

1. Introduction

When an obstruction is placed at the entry to a parallel walled channel (hence-forth called test channel)located within another wider channel (Fig. 1), it is observed that for certain relative positioning of theobstruction the flow within the test channel can stagnate or reverse its direction, i.e. in a direction oppositethe free stream flow direction (in the wider channel). When the distance between the obstruction and the

∗ Corresponding author. Fax: +91 44 22574002.E-mail address: egt@ae.iitm-ac.in (E.G. Tulapurkara).

1 Presently at Department of Engineering, University of Cambridge, UK

0169-5983/$30.00 © 2005 Published by The Japan Society of Fluid Mechanics and Elsevier B.V.All rights reserved.doi:10.1016/j.fluiddyn.2005.07.003

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Fig. 1. Flow situation.

test channel entry is sufficiently large, forward flow results, but its magnitude, even for very large distancesis less than the free stream velocity U∞. The flow velocity inside the test channel (Ui) is a measure offlow reversal.

Some of the well known applications where the reverse flow phenomenon occurs and find use are in:flow control, especially to obtain low velocities, interaction of shear layers at different distances, etc.

Gowda and Tulapurkara (1989) (henceforth referred to as GT) used a flat plate as obstruction andstudied the influence of various parameters like (I) gap g, (II) the length L of the test channel and (III)the Reynolds number Re, based on the width w of channel and the free-stream velocity U∞. In subse-quent studies, various facets of the phenomenon like the influence of the geometry of the obstruction(Gowda et al., 1993), the effect of obstructions placed both at the entry and the rear of the test channel(Tulapurkara et al., 1994), and the influence of splitter plates (Gowda et al., 1997) were investigated. Fur-ther investigations with flow visualization and pressure measurements (Gowda et al., 1998) attempted togain an understanding of the mechanism that triggers and sustains the reverse flow. The flow visualizationstudies by Kabir et al. (2003, 2004) were similar to those of GT, covering the range for the Reynoldsnumber between 1000 and 9000. They confirmed the results of GT that the reverse flow is maximum forthe ratio of g/w =1.5. Independently Bhuiyan et al. (2000) observed reverse flow also in non-Newtonianfluid flow.

The aim of the present investigation is to establish this complex flow, involving reverse flow, separa-tion, unsteadiness, vortex pairing and convection through computation. It would then improve the under-standing of the mechanism of pumping of fluid. In this study, two-dimensional unsteady Navier–Stokesequations are solved using Marker-and-Cell (MAC) method as modified by Hirt and Cook (1972) alongwith third order differencing scheme of Shirayama (1992). It is noted that experiments by GT madeuse of a channel of sufficient depth with free surface. Hence treating the flow to be two-dimensional,laminar but unsteady appears to be an acceptable approximation. This is confirmed by observing closecorrelation between the experiments and the computational results presented in the followingsections.

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2. Governing equations and boundary conditions

The computational domain is shown in Fig. 2. It is kept the same as the domain for the flow visualizationstudies used by GT. The entrance to the channel is at x/w = 14 and the exit at x/w = 38. The flowthrough and around the test channel is computed by solving the two-dimensional, unsteady Navier–Stokesequations for an incompressible fluid. The continuity and momentum equations in dimensionless formwith standard notation are

�u

�x+ �v

�y= 0, (1)

�u

�t+ u

�u

�x+ v

�u

�y= −�p

�x+ 1

Re

(�2u

�x2 + �2u

�y2

), (2)

�v

�t+ u

�v

�x+ v

�v

�y= −�p

�y+ 1

Re

(�2v

�x2 + �2v

�y2

). (3)

The reference velocity is U∞. The width w of the channel (same as the width of the obstruction in thepresent case) is the reference length. The reference time and the reference pressure are w/U∞ and �U2∞respectively; � being the fluid density. The boundary conditions are: (I) no-slip condition applies on thewalls of the outer channel, test channel and the surfaces of the obstruction; (II) velocity profile is uniformat the entrance to the domain (III) convective boundary condition applies at the exit to ensure smoothtransition through the outflow plane (Orlanski, 1976).

3. Method of solution

Saha et al. (1999) modified the temporal and spatial discretization of MAC method ofHirt and Cook (1972) further and applied it for direct numerical simulation of bluff body flows. A similarnumerical scheme is adopted for the present study. The computational domain is divided into Cartesiancells. Staggered grid arrangement is used i.e. velocity components are defined at the midpoints of and nor-mal to the cell-faces and pressure is defined at the cell centers. Numerical solution is obtained by advancingthrough a series of small time increments �t . The solution is obtained in two steps. Initially an explicit cal-culation is performed which uses velocities and pressures at previous time step to obtain a set of provisionalvelocities resulting from the acceleration caused by convection, viscous stresses and pressure gradients.Continuity is not satisfied necessarily during this explicit calculation. In order to ensure mass conservation

Fig. 2. Computational domain along with the obstruction and the test channel.

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in each cell, an iterative scheme is employed during the second step to adjust the velocities. This iterativecorrection of explicitly advanced velocity fields through an implicit continuity equation is equivalentto solving Poisson equation for pressure (Viecelli, 1971). In this study, for iterative pressure–velocitycorrection process, the over relaxation factor is chosen to be 1.8. Iterations continue until a divergence-free velocity field is obtained. However, for this purpose the divergence D in each cell is brought downto a preassigned upper bound (∼ 10−3). Initial trials with D ∼ 10−6 and 10−3 are found to yield resultsthat are same upto fourth place of decimal. Therefore D ∼ 10−3 is adopted for savings in computingtime.

The diffusion term is discretized using central differencing. For the convective terms, third orderaccurate upwind scheme of Shirayama (1992) is used. The velocities are explicitly advanced to thenext time step by second-order accurate Adams–Bashforth differencing scheme. For additional detailsSenthil Kumar (2002), and Fletcher (1991, Chapter 17) may be referred to.

The numerical instabilities are overcome by a combination of CFL condition and grid Fourier numberrestriction. i.e.

�t <

{�x

|u| ,�y

|v|}

(4)

or

�t <Re

2

{(�x)2(�y)2

(�x)2 + (�y)2

}, (5)

where �t is the time step and �x and �y are the grid spacings in x and y directions, respectively.The final time step for each time advancement is the minimum of the values obtained from these two

inequalities. The time steps employed in this computation are of the order of 0.003 units.To prescribe the initial conditions, the velocity, u, at each cell is taken to equal unity. Consequently,

the transverse velocity component v at each cell is zero.

4. Results and discussion

4.1. Validation

For the purpose of validation, the flow in a lid-driven cavity is computed for Re = 1000 and 5000.The results are compared with bench mark results of Ghia et al. (1982). The location of the center of theprimary vortex is regarded as one of the indicators of correctness of computations. After establishing gridindependence (Senthil Kumar, 2002) the center of the vortex is seen to be at (0.53, 0.56) and (0.52, 0.54)

for the two chosen Reynolds numbers. These values tally well with (0.53, 0.56) and (0.51, 0.53) of Ghiaet al. (1982). The streamline patterns (not presented for the sake of brevity) from the computations arealso almost identical with those of Ghia et al. (1982).

As a further check, the flow past a normal flat plate is computed and again the grid independent results(Senthil Kumar, 2002) compare closely with the computational results of Najjar and Vanka (1995) andexperimental data of Knisely (1990).

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Fig. 3. Time evolution of velocity in the test channel at x/w = 26.

4.2. Computations of flow in a channel with obstruction at entry

Computations of flow through and around the test channel with obstruction at entry are carried out forthree cases viz. g/w = 1.5, 3.5 and 6.0 at Re = 4000. These cases are selected for examination based onthe experimental observations of GT that (I) at g/w = 1.5 the reverse flow is maximum, (II) at g/w of3.5 the flow is stagnant and (III) for g/w = 6.0 the flow is in the free stream direction.

The phenomenon is unsteady but becomes nearly periodic after the initial period of development. Fig.3 shows the time evolution of centerline velocity at x/w = 26 (Point A in Fig. 1). The values of velocityare non-dimensionalized with reference to U∞ and are obtained by averaging over a suitable time spanwhich is based on the time evolution of the event under consideration. For g/w = 1.5 and 3.5 the timespan is from 100 to 300 and for g/w = 6.0, it is from 150 to 300.

Case 1: (g/w = 1.5)

This corresponds to the maximum reverse flow at Re = 4000. Grid independence is determined byusing three different grades of meshes namely, 395 × 157, 498 × 177 and 597 × 217. Fig. 4 shows thetime averaged velocity profiles at x/w = 26 for the three grids. The differences in values of velocity for

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Fig. 4. Time averaged velocity profile at x/w = 26 for different grids. — 498 × 177, 395 × 157, 597 × 217.

grid sizes of 498×177 and 597×217 are seen to be negligible and hence subsequent results are obtainedfor 597 × 217 grid. The velocity in the test channel for this grid is found to be −0.21, which compareswell with the experimental value of −0.2.

Case 2: (g/w = 3.5)

This corresponds to the stagnant flow at Re = 4000. Typical grid used in this study is 546 × 177. Thevelocity in the test channel is found to be −0.02.

Case 3: (g/w = 6.0)

This corresponds to the forward flow at Re = 4000. Typical grid used in this study is 591 × 177. Thevelocity in the test channel is found to be 0.58, which compares well with the experimental value of 0.57.

Fig. 5 shows the experimental streakline patterns taken from GT and the streak-line patterns obtainedfrom computed results for the three cases. To plot the streaklines, the computational time step is so selectedthat the computed streaklines can be compared meaningfully with the experimental flow pattern. Startingfrom the selected time step, seed particles are released for a period of one eighth of a second, whichcorresponds to the speed of the camera used in the experiments to photograph the flow patterns. Thestreaklines are then obtained by joining the locations of all the particles that are released from the chosenlocation. It is seen (Fig. 5) that the experimental and computed streak-line patterns are in close agreementwith one another. It is pointed out that the most important features of the flow are found to occur in theregions near the entry and exit of the test channel. To show the details clearly in photographs/figures ofmanageable size, only the forward and the rear portions of the test channel are shown. The gaps introducedbetween the left and right parts of the patterns are meant to indicate that the central portion of the testchannel is not shown.

4.3. Establishment of reverse flow

Hereafter we consider the g/w=1.5 case. As seen in Fig. 3, at t =0 the flow in the test channel is in theforward direction (Ui = 1). Subsequently the flow decelerates inside the test channel, becomes stagnantand attains finally nearly constant reverse flow velocity. To highlight the establishment of reverse flowwe begin with the description from non-dimensional time t = 30 and continue up to t = 55.

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Fig. 5. Comparison of experimental (top) and computed (bottom) streak lines. (a) g/w = 1.5, (b) 3.5, (c) 6.0. Note: The left andthe right panels show respectively the regions near the entry and the exit of the test channel.

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Fig. 6(a)–(g) show two features at selected values of the non-dimensional time t , viz. the instantaneousstreamline pattern in the flow field (upper part of each figure) and vorticity plots in the region near therear end of the test channel (lower part of each figure). At time t = 30 (Fig. 6a), the fluid inside the testchannel flows in the forward direction, but at a velocity lower than U∞. This can be inferred from thefollowing flow features. At the rear end of the test channel the streams inside and outside the test channel,having different velocities, cause formation of shear layers (lower part of Fig. 6a). The two shear layersstarting from the two edges of the channel are seen to be distinct indicating only a small difference invelocities of the two streams. At t = 35 (Fig. 6b), the flow in the test channel has decelerated further andthe vorticity plot shows that the shear layers from the two edges of the channel roll up into vortices whichare larger in size than those at t = 30. At t = 38 the vortices at the entry and exit of the test-channel seemto block the flow at the two ends and the flow is temporarily stagnant. This is evident from the vorticityplot in Fig. 6c. It is also seen that the shear layers from the two sides of the channel begin to interactwith each other near the exit of the channel. Shedding of vortices also appears to begin. At a subsequenttime t = 40 (Fig. 6d, lower part), the vortices are shed and the fluid enters the channel from the rear end(Fig. 6d, upper part). Fig. 6e–g, corresponding to t = 45, 50 and 55, show alternate shedding of vorticesand the fluid entering the channel from the two sides of the channel.

GT argued that the triggering of the reverse flow is due to the reduced pressure created at the front endof the test channel by the obstruction. To verify this, the pressure distribution along the centerline of thechannel is obtained from the computed results. Fig. 7 shows the variation, along the centerline, of thetime-averaged pressure coefficient Cp defined as

Cp = (p∗ − p∗∞)/12 �U2∞,

where p∗ and p∗∞ are the local and free stream pressures respectively. The values of Cp are obtained byaveraging over t = 100–300. The variation of Cp at Re = 4000 is qualitatively similar to that obtainedexperimentally by Gowda et al. (1998) for Re = 26 000. It is observed that Cp is −0.818 at the front end(x/w = 14). This low pressure at the front end continues, with a slight increase, till the rear end of thechannel where a sharp dip (Cp = −1.29) and then an increase (Cp = −0.446) are observed along thevortical region between x/w = 38–41.

Having validated the code, it is used now to gain further insight into the reverse flow phenomenon. e.g.pumping cycle, and to carry out parametric studies. These are described in the next section.

4.4. Pumping cycle

To examine the mechanism of fluid pumping we consider a time period nearly one cycle long afterthe flow becomes nearly periodic. The interval between t = 200 and 207 is chosen for this purpose.The instantaneous streamline plots are shown in Fig. 8(a)–(d). At time t = 200 (Fig. 8a), it is observedthat the fluid enters the test channel mostly from the upper side and a vortex is formed at the rear end.Senthil Kumar (2002) presents streamline plots at t = 200, 201, 202, . . . , 209. Here we present a fewtypical results. By t = 202 the vortex grows (Fig. 8b) almost blocking the channel exit. Subsequently,it is shed and convected downstream. This enables the fluid to enter from the lower side (Fig. 8c). Thisvortex also grows and is shed subsequently. Thereafter the fluid begins to enter from the top side again(Fig. 8d). At this stage the situation is similar to that shown in Fig. 8a and the pumping cycle continues.In this way the reverse flow is sustained. It may be remarked that in Fig. 8(a)–(d) the fluid moving inthe reverse direction inside the channel, leaves mainly from one side only. However, the computations

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Fig. 6. Instantaneous streamlines (upper part of each figure) and vorticity plots for region near the trailing edge (lower part ofeach figure) for various time steps. (a) t = 30, (b) 35, (c) 38, (d) 40, (e) 45, (f) 50, (g) 55.

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-1.5

-1

-0.5

0

0.5

1

0 5 10 15 20 25 30 35 40

x/w

Cp

Fig. 7. Variation of Cp along the centerline (g/w = 1.5).

Fig. 8. Instantaneous streamline plots at various instants during the pumping cycle. (a) t = 200, (b) 202, (c) 205, (d) 207.

at other instants of time do show that sometimes the reverse flow leaves the channel from the otherside and sometimes systematically from both sides. This behavior is observed also by GT and confirmsthat computations have been able to capture various features of the flow. Fig. 9(a) shows the variation,with time, of the normal velocity component v at a point on the centreline at x/w = 42 (Point B inFig. 1). The spectrum of the time signal is shown in Fig. 9(b). From this spectrum the frequency of vortexshedding at the rear end is estimated as 0.152 and hence a time period of 6.58 units. This is only slightlydifferent from the experimentally observed period of 6.66 units (1 s physical time). The signal is notexactly periodic as the flow at the rear end is affected by the vortices convected along the walls of the testchannel.

It may be mentioned that Senthil Kumar (2002) has further presented (I) time average vorticity plotsfor g/w=1.5 and 3.5, (II) autocorrelation and space–time correlation data for pressure on the walls of thetest channel which give the convective velocity of vortices and (III) effect of ratio of the width w of the

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Fig. 9. Variation of normal velocity at x/w = 42, (a) time signal (b) spectrum.

test channel to the width H of the main channel; which indicates that the reverse flow slightly decreaseswith increase in w/H .

5. Conclusions

Computations using unsteady, laminar, two-dimensional Navier–Stokes equations for the flow throughand around a test channel with an obstruction at entry are performed using MAC algorithm with higherorder upwind scheme. The flow patterns and the magnitudes of reverse flow for the three cases considered

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are in good agreement with the experiments. Computations provide further insight into the establishmentof the reverse flow, pumping mechanism and permit parametric studies. Three conclusions are drawnfrom the investigation. First the mechanism of reverse flow is due to the following. When an obstructionis introduced at the entrance to the channel, the flow initially moving in the forward direction deceleratesdue to the reduced pressure created by the obstruction, subsequently stagnates temporarily and thenreverses direction due to vortices at the end of the channel. Secondly as regards pumping of the fluid, thefollowing cycle is observed. (A) At the rear end of the test channel the flow enters from one side (topor bottom), (B) A vortex is formed that grows and is convected downstream, leaving way for the fluidto move in from the other side (bottom or top), (C). Thereafter another vortex is formed at the rear endof the channel which grows in size and convects downstream causing the fluid to enter from the otherside. This process repeats and sustains reverse flow.Thirdly the magnitude of the reverse flow decreasessomewhat with increase in the width of the test channel.

The computations were carried out at the Department of Science and Technology funded CFD Center,Indian Institute of Technology Madras. The support of the agency is gratefully acknowledged.

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