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CFD Simulation of a Bluff Body Shallow Wake in an OpenChannel

G. Nasif1, R.M. Barron2, R. Balachandar3

1Department of Mechanical, Automotive & Materials Engineering,University of Windsor, ON, Canada N9B 3P4

2Department of Mathematics & StatisticsUniversity of Windsor, ON, Canada N9B 3P4

3Department of Civil & Environmental Engineering,University of Windsor, ON, Canada N9B 3P4

E-mail: nasifg@uwindsor.ca

ABSTRACT

Three-dimensional numerical modeling usingDetached Eddy Simulation (DES) based on anunsteady RANS k-ω SST model has been carried outto evaluate the characteristics of shallow wake flow.Unsteady implicit formulation and second orderdiscretization are employed for pressure, momentum,turbulent kinetic energy and specific dissipationequations. The features of shallow wake flows arehighly dependent on the bed characteristics, the freesurface and the a non-uniform approaching flowvelocity profile. An approaching flow with fullydeveloped boundary layer was employed to model theeffect of the flow shallowness. The shallow wake isgenerated by introducing a disturbance in the form ofa bluff body placed in an open channel. A sharp-edged bluff body is used to reduce the flowsensitivity to Reynolds number and maintain aconstant position for boundary layer separation. Themean and instantaneous flow field characteristics inthe wake are examined and compared at differentdownstream locations to evaluate the effect ofvertical variability from the bed. Availableexperimental data have been used to validate thecomputational results.

1. INTRODUCTION

Turbulent shallow water flows are often encounteredin nature, e.g., wide rivers, lakes, estuaries, shallowcoastal waters, stratified flow and atmospheric flow.A proper understanding of shallow wake flow and itstransport capacity would be beneficial in a number ofcases, e.g., the tendency for pollutants to be trapped

in the lee of islands or headlands, nutrientaccumulation, or biological activity of various aquaticspecious. It would also aid in weather modelling asthe waterbeds and marshlands play a crucial role incontrolling the local weather [1]. In a shallow orbounded flow, the length scales in both the horizontaland transverse directions are much greater than in thevertical direction [2]. The flow is a boundary layertype, the bed imparts vertical shear while the freesurface acts as a stress-free weak boundary. Theshallow wakes occur when such flow is destabilizedby a sudden change of topology, such as byintroducing a horizontal shear (bluff body) ordeceleration of the flow. The vertical shear helps todissipate the kinetic energy and stabilizes the wakeflow. By contrast, in the deep wake, the flow isalways inviscidly unstable due to the absence of thevertical shear.The bed friction effect is one of the important factorsthat identify the features of shallow wake flow. Thiseffect is modelled as a stability parameter S = f(Cf, D,H) which accounts for the influence of stabilizationarising from bed friction and the destabilization in thenear wake due to the horizontal shear flow. The bedeffect might either suppress the vortex shedding orcause it to become intermittent if the stability numberexceeds a critical value. The friction effect reducesthe entrainment of the surrounding fluid into thewake, arrests the growth of the wake, andsubsequently weakens the interaction between theshear layers and stabilizes the wake at near bedlocations [3]. The recirculation flow behind the bluffbody is completely suppressed at near bed locations.Streamwise directed positive velocities have alsobeen experimentally observed in the wake centrelinenear the bed [4].

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Three categories of shallow wake flow pattern havebeen identified in [5]; vortex street, steady andunsteady wake bubble flows, and the stability numberis central to defining the limits between these modes.If the perturbation is due to the alternate separation ofthe shear layer (S ≤ 0.2), the shallow wake resembles the von Karman vortex street. If the bed frictioneffect increases, the shear layers separatesimultaneously, a steady bubble region is formedimmediately downstream of the body, and the degreeof stability of this region will identify the steadinessor unsteadiness of the bubble wake flow.The formation of a large-scale vortex street involves

upward ejection of fluid through its centre, whicheventually lead to a horizontal vortex that inducessignificant distortion of the free surface [6]. Thestrength of this vortex is an order of magnitudesmaller than the large vortex street. The existence ofthe upward axial flow along the core of the vortex isalso observed by using a dye visualization technique[7]. A technique of high-image-density particle imagevelocimetry was used in [8] to investigate the space-time development of the instantaneous flow patternsfor shallow-water wake. The study provided evidencefor the presence of the quasi-two-dimensional andthree-dimensional vortical structures. The vorticity ofthe three-dimensional pattern can exceed that of thequasi-two-dimensional pattern by a factor of two.The interaction between the non-uniform

approaching velocity and the front of the bluff bodycreates a variation in stagnation pressure along thebody. Consequently, a horseshoe vortex tube isgenerated near the channel bed and stretcheddownstream along the sides of the body. Avisualization study of the wake structure behind acircular cylinder was conducted in [9]. They found avertical oscillation of the shear layer at the sides ofthe body, which was attributed to the presence of ahorseshoe vortex wrapping around the body.Large eddy simulations (LES) have been used toevaluate the flow features of the near wake behind acylinder for a wide range of Reynolds numbers [10].LES provides good agreement with experimental datain describing the Reynolds stresses and the meanflow. However, in the close proximity of the cylinderat low Reynolds numbers, all simulations converge tomean velocity profiles that are different from theexperimental results. A combined PIV measurementsand Direct Numerical Simulation (DNS) study wasconducted to investigate the effect of Reynoldsnumber on the flow characteristics of the cylinderwake at transitional Re = 3900 and at a higher Re =10,000 [10]. The flow statistics changed significantlywith the change of Reynolds number, the principlefeatures of all turbulence quantities moved upstreamwith increasing Reynolds number. The shear layer

instability was found to be significantly influenced byReynolds number.The objective of the present study is to obtain furtherinformation on the shallow wake flow structuresoccurring downstream of a sharp-edged bluff bodyimmersed in an open channel flow using three-dimensional numerical simulation. The computationalresults provide support to previous experimentalobservations. One of the advantages of thecomputational model is its ability to capture moreinformation of the flow features at critical locationsthat could not be acquired by earlier experiments[11].

2. COMPUTATIONAL SCHEME

A schematic illustration of the flow, along with theco-ordinate system adopted in this study, is shown inFigure 1. The bluff body width is 0.03 m. Theunperturbed turbulent boundary layer profile,extracted by conducting a separate simulation for an8.0 m open channel flow with water depth of 0.1 m, isemployed as the inlet boundary condition. As shownin Figure 2, this profile compares well withexperimental data as well as the near wall anduniversal log-law equations. The wall shear stress andfriction velocity are estimated to be 0.36 Pa and 0.019m/s, respectively. The boundary layer thickness is δ = 80 mm, which constitutes 80% of the total depth. Thevelocity near the free surface is U∞ = 0.45 m/s. TheReynolds number is Re ≈ 7500 based on D and U∞.The flow field has been discretized usingapproximately 3.7M structured elements, taking intoaccount cell clustering near the solid walls and in thewake region. Detached Eddy Simulation (DES) basedon Shear-Stress Transport k-ω SST model wasemployed as the turbulence model [12]. The DESmodel is often referred to as a hybrid LES/RANSmodel. In the DES approach, unsteady RANS modelsare employed in the near-wall regions. Away fromthe wall the large scales are resolved, while thesubgrid scales are modelled using the same version ofthe RANS model that is used in the near-wall region.The k-ω SST model in ANSYS Fluent software incorporates modifications for low Reynolds numberaffects, shears flow spreading and is applicable towall-bounded flow. Unsteady implicit formulationand second order discretization are used for pressure,momentum, turbulent kinetic energy and specificdissipation equations. The standard SIMPLEalgorithm was employed to solve the pressure-velocity coupling. For complicated flow involvingturbulence or additional physics models, SIMPLECand SIMPLE will give similar convergence rate [12].A standard wall function was used to modeling thenear wall region. The values of non-dimensional wall

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distance Y+ have been inspected at the near walllocations to ensure it falls within acceptable limits.

Figure 1 Schematic illustration of the flow

Figure 2 Experimental and computational meanstreamwise velocity distribution of the approaching

flow

3. NUMERICAL MODEL VALIDATION

For validation purposes, physical and statistical flowquantities, e.g., mean velocity, turbulencefluctuations, and velocity deficit from the simulationhave been compared with experimental data atdifferent vertical locations from the bed, as reportedin [11].Transverse distribution of streamwise velocity deficitis one of the parameters which has been employed tovalidate the current numerical model. The validationwill be presented here, while specific definitions andcalculations of the velocity deficit and half-width ofthe wake can be found in the next section (Results).As shown in Figure 3, available PIV experimentaldata have been used to validate the computationalresults at three different vertical locations from thebed, at Y/H = 0.1, 0.5 and 0.8. In all three planes thecomputational results are in good agreement withexperimental data for | Z/Z0.5 | < 1 (i.e., in the wakecore), with a maximum difference less than 10%. At |

Z/Z0.5 | > 1, the computational and experimental dataare still in good agreement in the vicinity of the bluffbody for Y/H = 0.5 and 0.8. However, the differencebetween computational and experimental resultsincreases downstream at the medium-wake region(i.e., 3 ˂ X/D ˂ 6) as one moves towards the edge of the wake.

Figure 3 Computational (solid lines) andexperimental (symbols) velocity deficit comparison at

three horizontal planes; Y/H = 0.1 (red), Y/H = 0.5(blue) and Y/H =0.8 (black).

4. RESULTS

In this section the averaged and instantaneousstatistical quantities for the wake flow field from theCFD simulation are presented and discussed.

4.1 Mean Velocity

The mean velocity distribution provides informationon bulk motion of fluid, i.e., convection of fluidparticles and relative mixing between differentregions in the flow field. Patterns of time-averagedstream traces at different vertical elevations from thebed are given in Figure 4. In this figure, the meanstreamwise velocity (U) contour has been superposedon the time-averaged stream traces in the X-Z plane.The effect of bed is obvious on the stream tracestopology as shown in Figure 4a, where therecirculation flow is completely suppressed andpositive directed velocity dominates the wake flow inthis plane. The existence of this particular distributionof streamwise velocity is also evident in the time-averaged streamline plot of a shallow wake close tothe bed in the investigation of [4] and [6]. Thecontour of high velocity is spread over a largertransverse Z/D as one moves from the near bedtowards the free surface location. The onset of thereverse flow occurs at vertical location Y/H ≈ 0.1, the formation of a pair of spiralling foci is observed in

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the near wake region as shown in Figure 4b. Theunstable foci are displaced downstream, coupled withexpansion of the recirculation region towards the freesurface as shown in Figures 4c and 4d.

(a) Y/H=0.01 (b) Y/H=0.1

(c) Y/H=0.4 (d) Y/H=0.8

Figure 4 Stream traces of the mean velocity fieldsuperposed with mean streamwise velocity (U) for

horizontal planes: (a) Y/H = 0.01; (b) Y/H = 0.1; (c)Y/H = 0.4; (d) Y/H = 0.8

4.2 Velocity Deficit

More insight into the streamwise velocitydevelopment can be realized by examining thestreamwise velocity deficit (Us-U(x,0)) distributionalong the wake centreline at different locations fromthe bed. The wake centreline velocity deficitdistribution at different elevations from the bed,normalized with the corresponding upstream velocity(Us), is given in Figure 5. The kinetic energy istransferred from the outer accelerating flow to thewake where the gradient is negative, and vice-versawhere the gradient is positive. A recirculation region(a negative streamwise velocity) in the wake isnormally expected where the normalized velocitydeficit is greater than unity. Therefore, in the nearwake, the energy usually transfers to the wake atlocations close to the bed, and from the wake at upperlocations. As shown in Figure 5, the downstreamdistance, where the reverse flow diminishes, increasesas the wake moves towards the free surface.Figure 6 shows the transverse distribution of thenormalized streamwise velocity deficit (U(x,z) - Us) /(U(x,0) - Us) at three different horizontal planes Y/H =0.1, 0.4 and 0.8, for each plane, the distribution isgiven at two downstream positions; X/D = 3 and 5.The transverse distance is normalized by the localhalf width (Z0.5), the half-width of wake is defined asthe transverse location (Z) where the velocity deficit

equals half of the maximum velocity deficit i.e., thevelocity deficit in the wake centreline.

Figure 5 Distribution of the wake centreline velocitydeficit at vertical elevations; Y/H =0.01, 0.1, 0.4 and

0.8

Figure 6 Transverse distribution of the streamwisevelocity deficit at three different vertical locations

4.3 Mean Vorticity

The contours of the time-averaged vorticity and itsstreamwise and transverse components are shown inFigure 7, where horizontal and vertical slices wereextracted at Y/H = 0.01, 0.4 and 0.8 and X/D = -0.5,0.5, 2.0 and 3.5 respectively. As shown in Figures 7aand 7d, the vorticity contours of the vortex streetextend over the whole depth with a dominant Y-vorticity component. These figures indicate that thesize of the contours tend to be larger as one movestowards the free surface. This demonstrates the effectof the bed, which suppresses the growth of vortices atlocations close to the bed.At the near bed location Y/H=0.01, the counter-rotating streamwise vorticity (red and blue contours)in Figure 7b immediately next to the body andemanating from the edges refer to the separatingshear layers. Adjacent to the shear layers are regionsof high vorticity generated by the flow negotiating thepresence of the body. The vorticity in these regionshas a magnitude opposite to that in the adjacent shear

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layers. The magnitude of these structures (wx)diminishes as one moves to the upper layers as shownin Figures 7b and 7d.

(a) Vorticity magnitude (b) Streamwise vorticity

(c) Transverse vorticity (d) Iso-surfaces vorticity

Figure 7 Time-averaged; (a) vorticity magnitudecontours; (b) streamwise vorticity component

contours; (c) transverse vorticity component contoursand (d) iso-surfaces of vorticity components (wx, wy

and wz)

The transverse vorticity (wz) distributions in the Y-Zplanes are given in Figure 7c. These patterns clearlyindicate the presence of the horseshoe vortex thatdevelops in front of the body at X/D = - 0.5 andextends downstream. The horseshoe vortex envelopsthe vortices that form part of the vortex street. Moreinsight into the horseshoe vortex can be gained fromFigure 7d. In this figure, the iso-surfaces of vorticitycomponents are plotted. Part of the horseshoe vortex(wz) is captured in front of the body (dark bluesurface). The contour of wz is merged with the iso-surface vorticity (wx) which extends downstream.The counter-rotating streamwise vorticity at the nearbed location (Figure 7b) involves upward ejection offluid through the wake centre as shown in Figure 8.This ejection (Region A in Figure 8) provides a three-dimensional imprint for the shallow wake. This flowdoes not travel upwards perpendicular to the bed, butat an inclined angle towards the body and transfersthe turbulent kinetic energy to the transverse andstreamwise components at upper locations from thebed (see also Figure 9a). This observation has beenconfirmed experimentally in [4]. Comparing thevelocity field at three different horizontal locationsfrom the bed reveals that the ejection event is robustin the horizontal plane Y/H = 0.1, and gets weakerwith increasing Y/H. At Y/H = 0.9, the velocityvectors exhibit a planar distribution due to thepresence of the free surface.

Figure 8 Time-averaged velocity vectorssuperimposed on contours of time-averaged velocity

magnitude at horizontal planes Y/H = 0.1, 0.4 and 0.9

4.4 Turbulent Kinetic Energy

The time-averaged contours of turbulent kineticenergy normalized with the free surface velocity (U∞)at three vertical locations Y/H = 0.1, 0.4 and 0.8 aregiven in Figure 9a. As shown in this figure, the highmagnitude kinetic energy contours spread over largerX/D and the wake flow becomes more chaotic as thefree surface is approached. Near the bed, the presenceof the boundary damps out and alleviates the disorderin the fluid.

(a) Turbulent KE (b) Reynolds stress

Figure 9 Time-averaged contours of (a) turbulentkinetic energy and (b) Reynolds stress <uw>,

extracted at horizontal planes Y/H = 0.1, 0.4 and 0.8

The contours of normalized Reynolds stress atdifferent locations from the bed are shown in Figure9b. The Reynolds stress gives a positive contributionand represents a production term for the turbulentkinetic energy. The Reynolds stress distribution isanti-symmetric with respect to the body, the highervalue on each plane stretches along the shoulder ofthe body. This observation indicates that themomentum exchange by Reynolds stress between the

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accelerating flow and wake region increases as oneapproaches the free surface.

4.5 Instantaneous Characteristics

One of the disadvantages in using the time-averagedquantities is that it conceals much of the unsteadycharacteristics of the flow field. The time dependentquantities divulge more details of the flow fieldstructures. For the current flow. The vortex sheddingfrequency f ≈ 1.8 Hz was calculated using the FastFourier Transform (FFT). The Strouhal number basedon the width of the body (D) and free stream velocity(U∞) is St = 0.2.The vortical or coherent structures play an importantrole in understanding many of turbulent features ofthe flow, e.g., entrainment, mixing, drag. Therefore,there is a need to identify the large-scale vorticalregions in turbulent flows. Construction ofinstantaneous stream trace patterns allowsidentification of such structures in the flow field. Theevolution of instantaneous stream traces topology atdifferent elevations from the bed are given in Figure10 for two sequential shedding from the right and leftsides at t/T= 0 and 0.6 respectively. Each pattern inthis figure shows an identifiable focus, whichcorresponds to the centre of the spiraling streamtraces pattern. A further feature in this figure is theexistence of well-defined saddle points (S). Thecombination of saddle points and a spiral streamtraces pattern leads to limiting the cycle of the streamtraces and identifies the size of the vortical structuresas shown in Figures 10a-10f. The patterns ofinstantaneous stream traces topology slightly exhibita slight spatial expansion of the spiraling portiontowards the free surface. The size of the largestructures in Figure 10 occupy more than 150% of thewidth of the bluff body, and they appear to meanderlike a sine wave about the vertical central plane dueto interaction between instantaneous vortices andmean flow. Figure 11 shows other selected examplesof the large vortical structures extracted at differentvertical planes. From this figure, one can identifythree type of structures; (1) spiralling stream tracestructures as shown in Figure 11a and 11b, (2)structure with upward flow close to the bed (at X/D =2.2 and 1.5 in Figures 11a and 11b, respectively)which seems to be due to thrust by streetwisevorticity that resides near the bed and (3) structuresthat appears as a thick vertical line and span almostthe entire water depth as shown in Figure 11c. Thedirection of the flow at the sides of this elementindicates that it is a side-section of a vortical elementwhich resembles a hairpin leg. This shape of thevortical element may be a result of the interaction of ahairpin vortex with quasi-streamwise structures thatseem to be common in this kind of flow [13].

(a) Y/H = 0.1, t/T = 0 (d) Y/H = 0.1, t/T ≈ 0.6

(b) Y/H = 0.5, t/T = 0 (e) Y/H = 0.5, t/T ≈ 0.6

(c) Y/H = 0.8, t/T = 0 (f) Y/H = 0.8, t/T ≈ 0.6

Figure 10 Evolution of the stream traces ofinstantaneous velocity superposed with vorticity

magnitude contours at horizontal planes Y/H = 0.1,0.5 and 0.8

(a) Z/D = - 0.5, t/T ≈ 0.6 (c) Z/D = 0.5, t/T ≈ 0.6

(b) Z/D = 0.0, t/T ≈ 0.6 (d) Z/D = 1.0, t/T ≈ 0.6

Figure 11 Stream traces of instantaneous velocitysuperposed with vorticity magnitude contours at

vertical planes Z/D = - 0.5, 0.0, 0.5 and 1.0

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5. CONCLUSIONS

Three-dimensional numerical simulation wasconducted to study the flow characteristics of theshallow wake flow. The effect of the bed is apparenton the time-averaged topology of stream traces andvorticity at locations close to the wall. A stablenarrow wake region associated with positive directedvelocity has been found in the centre of the wake inplanes close to the bed. The wake region widens withincreased recirculation towards the free surface. Theformation of a large-scale street vortex involvesupward oriented axial flow through the centre of thevortex, this ejection provides a three-dimensionalimprint for the shallow wake. A signature of thehorseshoe vortex is found at the front of the bluffbody at the near bed location.Comparison of the stream traces topology at differentelevations above the bed show that a common saddlepoint can be identified at approximately the samespatial position in the near and medium wake for allelevations. The vertical and horizontal slicesextracted from the instantaneous flow field atdifferent locations reveal the existence of differenttypes of vortical structures.

NOMENCLATURE

Cf [-] Bed skin friction coefficientD [m] Width of the bluff bodyH [m] Depth of flowk [m2/s2] Turbulent kinetic energyDES [-] Detached Eddy SimulationRANS [-] Reynolds Averaged Navier-StokesS [-] Stability parameterSt [-] Strouhal numberU∞ [m/s] Free surface velocity magnitudeU [m/s] Velocity magnitudeU, V,W

[m/s] Velocity components

u, v, w [m/s] Velocity fluctuation componentsw [1/s] Vorticity magnitudeX [m] Cartesian axis directionY [m] Cartesian axis directionZ [m] Cartesian axis directionZ0.5 [m] Wake half width

Special charactersδ [-] Boundary layer thickness ω [1/s] Specific turbulent dissipation rate

Subscriptsc, deficit Centreline deficitdeficit Deficitrms Root mean squares Upstreamx, y, z Component directions

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