The Influence of Boundary Layer State on the Wake Topology of a Surface Mounted Bluff Body

Zixiang Chen, Zahra Hosseini and Robert Martinuzzi

Department of Mechanical and Manufacturing Engineering, The University of Calgary, AB Canada T2N 1N4

Email: zichen@ucalgary.ca

ABSTRACT A numerical analysis is conducted to investigate the flow structure in the wake of a vertical, surface-mounted square cross-section cylinder of height-to-width (h/d) ratio 4, for a laminar and a turbulent boundary layer of comparable thickness, δ/h ≈ 0.2. The mean and the dynamic flow fields are analyzed and compared. For the mean flow, the obstacle wake generated by a laminar on-coming flow is of the quadrupole type, while the one generated by a turbulent case is of the dipole type. A dynamic phase-averaged representation of the flow field shows that the laminar case sheds quasi-periodic full-loop vertical structures, while the turbulent case forms half-loop structures. The interaction between the streamwise coherent structures arising at the obstacle wall junction and the periodically shed vortices appear to be responsible for the difference in wake structure.

1. INTRODUCTION Surface-mounted bluff body flow is a simplified model for many practical engineering applications such as vortex generators, axial compressor/turbine blades and the mixing of a jet in cross flow. Despite the simple geometry, the wake structure of the obstacle is highly three-dimensional and complex. These flows are fundamentally interesting, retaining sufficient complexity to study three-dimensional vortex interactions while remaining sufficiently simple to be tractable with statistical tools. For surface mounted square cylinders, a great amount of effort has been made to relate different wake structures to the height to width ratio (aspect ratio) of the geometry and the incoming flow condition. The vortical structures arising from the bluff body wake are believed to be closely related to the ratio between the boundary layer thickness and the obstacle height (δ/h). Experimental (cf Wang et al.1) and numerical

(cf Afgan et al.2) studies, have shown the importance of the relative boundary layer thickness, δ/h. However, the impact of the boundary layer state has generally not been considered. The present study numerically investigates the influence of boundary layer state and shows that its influence on the wake structure can also be significant.

2. SIMULATION SETUP The numerical analysis simulates the flow over a surface-mounted square cylinder with an aspect ratio (h/d) of 4. The obstacle extends through a zero-pressure gradient boundary of nominal thickness δ/h ≈ 0.2. In one case, the boundary layer is laminar and in the second case turbulent. Figure 1 shows the schematic of the geometry and nomenclatures.

A boundary layer profile of the form: 32𝜂 −

12𝜂! =

𝑈𝑈!, 𝜂 =

𝑧0.15ℎ

is imposed at the inlet. Difference in Reynolds number is achieved by changing the inlet velocity

. A laminar boundary layer (Re=500 based on obstacle width) and a turbulent boundary layer (Re=11500) are then allowed to develop. The undisturbed boundary layer at the obstacle location is

€

U∞

Figure 1: Geometry and nomenclature

estimated by the velocity profile at x/d=0 and a sufficiently large traverse distance (y/d=7). The lower Re case develops a boundary layer with a thickness of approximately δ/h=0.2 and negligible turbulent intensity; while the higher Re case, the boundary layer thickness measured at the same location is slightly lower (δ/h≈0.18). The free stream turbulence intensity is about 0.5% and reaches a maximum of 2.8% within the boundary layer (Figure 2). ANSYS CFX is used for the simulations. A laminar solver is used to model low-Re case. For the high-Re case, Large Eddy Simulation (LES) with the Smagorinsky subgrid-scale model is used instead. The Smagorinsky Constant is set to 0.1 and von Karman’s constant is set to κ = 0.4. A finite volume formulation with a Central Difference Advection

Scheme and Second Order Backward Euler Transient Scheme was used to discretize the governing equation. The transient simulations are initialized with steady state simulations using the Shear Stress Transport turbulence model. Three levels of grid refinement were carried out. The number of nodes was doubled between levels to ascertain grid-converged results. A multi-block strategy is employed for meshing. The computational domain consists of 1.4 million hexahedral elements, with about 30 nodes per obstacle diameter. Five hexahedral blocks are extruded from the obstacle free surface to allow enough grid resolution of the obstacle wall shear layers (Figure 3). Inlet and surrounding outlet boundaries are placed at least 2.5h away from the location of the obstacle, and an axial distance of 6h is allowed for the wake to develop.

3. VALIDATION OF SIMULATIONS Validation of the simulation is performed for the high Reynolds Number case, for which experimental data is available. PIV measurements on a bluff body with the same geometry were previously performed in a wind tunnel and reported by Bourgeois et al 3. The Reynolds number in the experiment was 12000. The velocity profiles and surface pressure fluctuations (at the obstacle wall) recorded in this experiment are used to validate the numerical simulations.

The numerical results and experimental data were compared and found to be in satisfactory agreement. Sample profiles of the averaged and fluctuating velocity streamwise component U at 1 diameter downstream from the origin and ¼ obstacle height are shown in Figure 4. This location is a critical point for the wake feature since it is located at a height close to the boundary layer thickness. At this location, the discrepancy between the LES prediction

z=0

y=0

Figure 3: Grid resolution around the obstacle

Figure 4: Averaged and fluctuating velocity component U at z/h=1/4, x/d=1

Figure 2: Boundary layer profiles at x/d=0, y/d=7

and the experimental measurement is of the order of the experimental uncertainty.

Pressure measurements at the obstacle wall indicate that the vortex shedding frequency is similar to that measured and corresponding to a Strouhal number close to 0.1. The spectrum of the LES case also shows the expected 5/3 decay of fluctuation energy, which agrees with the experimental observation (Figure 5).

4. RESULTS AND ANALYSIS To reveal the effects of the boundary layer state on the wake structure, both the mean flow field and the dynamics of the vortical structures are compared for the two simulations.

To analyze the three-dimensional instantaneous velocity field, the method of Proper Orthogonal Decomposition (POD) is used to distinguish the energetic coherent motions from small-scale fluctuations in the obstacle wake. A comprehensive description of the POD method for fluid dynamic applications can be found in Lumley 4. Typically, the underlying assumption is that the velocity field of a fluid flow can be described separately as a function of space and time:

𝑢 𝑥, 𝑡 = 𝑢 𝑥 + 𝑎!(𝑡)𝑢!!(𝑥)!

!!!

Where 𝑎! is the temporal coefficient of the POD mode and 𝑢!! is the POD spatial mode. The term,  𝑎!(𝑡)𝑢!!(𝑥), gives the rank k approximation of the fluctuating velocity field. The columns, i.e. the order of k, are ranked by level of approximation, also called the energy of the POD mode.

Figure 6: a (top) POD temporal coefficient of the fist two modes; b (bottom) Phase of the shedding cycle obtained from the first two POD modes.

To obtain the temporal coefficient and POD spatial mode, the POD method (Snapshot POD) used is equivalent of solving the eigenvalue problem:

𝐴!𝑀𝐴 𝑉 = 𝜆𝑉

Where A is the temporal and spatial distribution of the fluctuating velocity in discrete form, M is the mass matrix. Since the size of the cells in CFD simulations is usually not uniform, the rows of M are collections of the fluid mass represented by each of the grid cell at a given time. The rows of M in this case are uniform since there is not sink/source in the computational domain. The POD modes are normalized with the number of snapshots and the associated eigenvalue, λ, so that the spatial modes are orthonormal in the column space. The magnitude of the eigenvalues in this approach is directly related to the amount of fluctuating kinetic energy carried by a particular POD mode.

The first two most energetic POD modes are mainly responsible for the von Karman-type vortex shedding motion in the recirculation region of the obstacle. As a result, the POD temporal coefficients of these two modes are not only orthogonal but also with an approximately 90o phase shift in spatial domain (Figure 6a). Assuming the dominating coherent motion is mainly represented by this (first) harmonic pair, the phase of the vortex shedding motion could

LES

Exp.

Figure 5: Pressure fluctuations at obstacle side wall, z/d=1

be approximated based on the first two POD modes by taking (Figure 6b):

𝜙(𝑡) = tan!! 𝑎!(𝑡)/𝑎!(𝑡)

The phase averaged velocity field is given by the sum of coherent and incoherent motion:

𝑢 𝑥, 𝑡 = 𝑢 𝑥,𝜙 + 𝑢!!(𝑥, 𝑡)

4.1 Comparison of the Mean Flow Mean sectional streamlines in orthogonal planes are shown in Figure 7. Generally, the recirculation zone is longer for the Re=500 case than the Re=11500 case. In the median plane y = 0, the body-wall junction vortex is much larger for the low-Re case. This difference is believed to be significant and indicative of an interaction with the streamwise vortices originating upstream of the lee region. In the plane x/d = 7, for the high-Re case, two strong circulations are observed near the free-end region. The weaker, counter rotating vortices near the wall are associated with the streamwise extensions of the horseshoe vortex originating from the windward wall separation. For the low-Re case, however, the tip and near wall vortex pairs are of similar intensity. As will be addressed next, the near wall vortices are not extensions of the horseshoe vortex. In fact, the horseshoe vortex for the latter case is difficult to distinguish past the object trailing edge.

A representation of the principal mean three-dimensional vortical structure provides further insight to the mean field. The vertical structures are identified employing the λ2 criterion proposed by J. Jeong and F. Hussain6. Figure 8 shows λ2 = -0.02 isosurfaces. For the case Re = 11500, the mean wake

vortex structure is of the dipole type. The trajectory of the horseshoe vortex can be unambiguously followed from the windward origin, extending downstream along the streamwise extensions. The horseshoe vortex structure remains distinct from vortices arising in the wake region. For the case Re = 500, the four streamwise vortex extensions are seen to originate from the base wake region, consistent with a quadrupole structure.

Re=5

00

Re=1

1500

Figure 8: Mean vortical structures, λ2 criterion

In earlier studies, for obstacles of aspect ratio h/d > 3 and 0.06 < δ/h < 0.5, the mean wake structures are generally reported to be of the dipole type over a very large range of Reynolds numbers 1,2. The present simulations show that, even though the aspect ratio and relative boundary layer thickness are similar, the wake structures are fundamentally different. Thus the state of the boundary layer also plays an important role in determining the wake structure.

4.2 POD Analysis Due to the nature of different boundary layer states, the amount of kinetic energy carried by the coherent motions is relatively smaller for the turbulent boundary layer case than the laminar boundary layer one. As a result, the energy contents of the POD modes are not directly comparable for the two cases. Nevertheless, since the POD modes are ranked by their corresponding energy contents, the first three modes, for example, would still give the best rank 3 approximation of the fluctuating flow field. In fact, for both cases, the first three modes also best describe the coherent motion of the obstacle wake. Recall that

Re=500 Re=11500

y=0

z/h=

1/2

x/d=

7

Figure 7: Streamlines of mean flow on various planes

the POD spatial modes are normalized to be orthonormal in the column space, so the comparison between the two different Reynolds number cases can be made directly by comparing the spatial distribution of the fluctuation energy. The three most energetic modes are shown in Figure 9. Isosurfaces with both positive and negative isovalues are extricated from the POD spatial mode for the comparison. For both cases, the first two POD modes represent the coherent motion of the von Karman-type vortex shedding, at approximately the same Strouhal number (0.1).

The third mode gives further insight to coherent structure interaction. For the case Re = 11500, the energy content for mode 3 is concentrated in the wake. Spectra of the modal coefficient indicate that this mode is associated with low frequency oscillations of the base flow. For the Re=500 case,

the third mode shows a large concentration of activity in the obstacle-wall junction region. Spectra of the modal coefficient show, in addition to the low frequency fluctuation, a distinct peak at a non-dimensional frequency of approximately 0.04 (or approximately half of the Strouhal number of 0.1). This spectral energy concentration is indicative of an interaction between coherent structures, most likely related to the interaction between the shed vortices and the streamwise vortices from the base region.

4.3 Phase Averaged Flow Field

The first two POD modes are used to reconstruct a phase-averaged representation of the flow field. The phase of the shedding cycle is obtained from the temporal coefficients of these two modes. Instantaneous velocity data are averaged in 20 evenly spaced bins within the shedding cycle. The λ2

Re=500 Re=11500

Mod

e 1

Mod

e 2

Mod

e 3

Figure 9: Frist 3 POD Modes. Isosurface of spatial modes and FFT of temporal coefficient; Isovalues: 5e-3 for red and -5e-3 for blue

criterion is again used to identify the evolution of the vortical structures during one vortex shedding cycle (Figure 10). Due to the different levels of influence from the vortical structures arising from the wall junction, the three-dimensional vortex shedding mechanism for the two cases are fundamentally different. The Re=500 case forms full-loop vortex tubes connecting the tip and the junction on alternating side of the obstacle, which shed alternatingly from the obstacle. Upon averaging, the repeated passage of these organized full-loop structures result in the quadrupole mean wake structure, as shown in Figure 8. For the Re=11500 case, the influence of the wall-junction vortices do not interact strongly with the periodically shed vortices, such that streamwise re-orientation of the

shed vortices occurs mainly towards the free-end, yielding half-loop vortex tubes. Hence, upon averaging, a dipole structure is observed.

5. CONCLUDING REMARKS The effect of the boundary layer state on the wake structure of an aspect ratio 4 square cylinder was investigated using CFD. The results indicate that for boundary layers with similar thicknesses but different intensities, the wake structures can be largely different, which shows the significant influence of boundary layer state on wake topology. For the low-Re case, the mean wake structure is of quadrupole type, resulting from full-loop shedding of vortical structures. For the high-Re case, the dipole structure is a result of half-loop shed vortices. The difference in structures appears related to the interaction with streamwise vortices generated at the obstacle-wall junction. The present results thus highlight that the type of wake structure observed for low-aspect ratio obstacles depends on the boundary layer characteristics, in addition to the aspect ratio and relative boundary layer thickness.

ACKNOWLEDGEMENTS The authors thank Dr. Doug Phillips from IT University of Calgary for providing technical support on the Westgrid high performance computing system.

REFERENCES [1] H.Wang, Y. Zhou, C. Chan, and T. Zhou,

“Effect of initial conditions on interaction between a boundary layer and a wall-mounted finite-length-cylinder wake,” Phys. Fluids, 18, 1–12 (2006).

[2] I.Afgan, C. Moulinec, R. Prosser, and D. Laurence, “Large eddy simulation of turbulent flow for wall mounted cantilever cylinders of aspect ratio 6 and 10,” Int. J. Heat and Fluid Flow, 28, 561–574 (2007).

[3] J.Bourgeois, P.Sattari, and R.Martinuzzi, “Alternating half-loop shedding in the wake of a finite surface-mounted square cylinder with a thin boundary layer,” Phys. Fluids 23, 095101 (2011)

[4] J.L. Lumley, “The structure of inhomogeneous turbulence”, in A.M. Yaglom and V.I. Tatarski, eds., Atmospheric Turbulence and Radio Wave Propagation, Nauka, Moscow, pp. 166-178. (1967)

[5] J. Jeong and F. Hussain, “On the identification of a vortex”, J. Fluid Mech., Vol. 285, pp. 69-94. (1995)

Re=500 Re=11500 Ph

ase

1

Phas

e 4

Phas

e 8

Phas

e 12

Figure 10 Phase averaged velocity field, λ2 criterion