The Eighth Asia-Pacific Conference on Wind Engineering,December 1014, 2013, Chennai, India

UNIFORM FLOW AROUND A CIRCULAR CYLINDER IN THE SUBCRITICAL RANGE - USING THE SELF-INDUCED ANGULAR

MOMENT METHOD TURBULENCE MODEL

Jens Johansson1, M.P. Nielsen2, Leif O. Nielsen3 1 Assistant Professor, Ph.D., Department of Technology and Innovation, University of Southern Denmark,

jensj@iti.sdu.dk 2 dr.techn., Professor Emeritus, Department of Civil Engineering, Technical University of Denmark. Permanent

consultant at Alectia A/S, Virum, Denmark. 3 Associate Professor Emeritus, Department of Civil Engineering, Technical University of Denmark.

ABSTRACT The uniform flow around a circular cylinder at Reynolds number 1e5 is simulated in a three dimensional domain by means of the newly developed Self-induced angular Moment Method, SMoM, turbulence model. The global force coefficients, Strouhal number, pressure distributions and wall shear stress distributions are compared to experimental findings reported in literature. The SMoM turbulence model is found to provide maximum, minimum and time-mean pressure coefficient distributions in very good agreement with experimental findings.

Keywords: circular cylinder, SMoM turbulence model, sub critical

Introduction

The aerodynamic forces produced by flow around circular cylinders, with the longitudinal axis perpendicular to the flow, have been of interest since the ancient invention of the Aeolian harp (Jones et al. (1969)). The flow is of relevance for many practical applications e.g. offshore risers, bridge piers, chimneys, towers, masts, cables, antenna and wires (Norberg (2003)).The circular cylinder investigated in the present study is outlined in Figure 1. The goal of our investigation is the evaluation of the newly developed SMoM turbulence model in a civil engineering context.

Figure 1. The principal layout of the computational domain.

For bluff bodies with curved surfaces such as a circular cylinder, the positions of the

separation of the local surface boundary layers are dependent on viscous forces. This leads to a variation of drag forces with Reynolds number, in which is the diameter of

Proc. of the 8th Asia-Pacific Conference on Wind Engineering Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds)Copyright c 2013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: 978-981-07-8011-1doi:10.3850/978-981-07-8012-8 171 592

Proc. of the 8th Asia-Pacic Conference on Wind Engineering (APCWE-VIII)

the cylinder, is the freestream velocity and is the kinematic viscosity. Holmes (2007) mentions results from Scruton, and shows that a critical Reynolds number, i.e. the Reynolds number at which a sharp drop in drag can be observed, occurs at relatively low corner radii for square cylinders. Hence, the ability to predict such behavior might be of relevance, even though a circular cross section is not directly present.

The variation of the parameters with Reynolds number may be described by different regimes of the flow, with a regime being a range of Reynolds numbers. See e.g. Roshko (1961) for a definition of the regimes. The SMoM turbulence model has been used to simulate the flow around a circular cylinder at a range of Reynolds numbers spanning across the subcritical, critical, supercritical, uppertransitionel regimes. In the present study we shall limit ourselves to subcritical regime. The results presented in the present represent findings made during the preparation of Johansson (2011). In Johansson (2011) simulations three different grids were used due to grid convergence considerations. The results in the following correspond to the finest grid utilized at the time. However, for the specific Reynolds number presented here different model parameters were used with each of the three grids. Hence, more simulations are being prepared to cover grid and time-step convergence considerations.

The SMoM turbulence model

The SMoM turbulence model has been suggested by Nielsen et al. (2006). The model was further calibrated and boundary conditions were developed in Johansson (2011). A short introduction to the model is offered here.

If we let the axis , , and define a right-hand coordinate system, then the Navier-

Stokes equation in continuity form, with zero body forces, may be written in index notation as in Eq. (1).

(1)

where is time, is density, is the velocity vector, and is the Cauchy stress tensor.

The SMoM turbulence model is a zero equation turbulence model in the sense that no additional transport equations are introduced. The model is merely a modification of the constitutive equation of the Newtonian fluid. For the Newtonian fluid the classical terms takes into account the transfer of linear momentum between parts of the fluid. In the SMoM turbulence model a term is added which takes into account the transfer for angular momentum between parts of the fluid. Thus the constitutive equation of the fluid reads

(2) where is the strain rate tensor, is Kronecker delta, is the dynamic viscosity, and

is the pressure. The first two terms completes the usual constitutive equation for the Newtonian fluid. The term is thus the added term, where is the rate-of-rotation tensor, is the local value of SMoM models scale parameter and is the absolute velocity to the power of , which is a model parameter. The local value of is given by boundary conditions and the free stream value of the scale parameter, , which in the case of smooth walls may be estimated from Eq. (3).

(3)

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The value of the model parameters and was suggested in

Johansson (2011). The constant is labeled the scale factor, which is not a fluid property, but rather a flow property. The scale factor represents a characteristic length-scale in the simulated flow. In the case external flow around a circular cylinder we set the length equivalent to the cylinder diameter .

Simulations

The simulations have been performed by using a Finite Volume Method implementation of the SMoM turbulence model in the open source software package OpenFOAM. The size of the computational domain was chosen to be with a total length of , and the cylinder placed a distance of from the inlet plane. The length of the cylinder in the present case was , and a total of cells were used in the grid. The grid was constructed from the principal layout illustrated in Figure 2 using , , , , , and . A grid cell growth rate of was

used in the grid refinement zone. The resulting grid size near the surface, here expressed with respect to the cylinder diameter, was in the surface-normal direction, in the tangential direction, and in the span-wise direction.

Figure 2. Principal structure of the computational grid. designates the number of cells in

the indicated directions. The Crank-Nicolson scheme was used for temporal discretization. For the spatial

discretization second order central differencing was used, while a Total Variation Diminishing scheme, based on the Sweby limiter, was used for the convective terms. At the outlet boundary an advective boundary condition was used, and the symmetry boundary condition was applied for the upper, lower, and cylinder-end-plane boundaries. At the cylinder surface the no-slip condition was enforced. The scale parameter of the SMoM turbulence model, , must be zero at the surface of the cylinder in the case of smooth surfaces and a transition to the free-stream value must be made. An analytical variation of can be derived which corresponds to enforcing the logarithmic law-of-the-wall. However, in Johansson (2011) it was suggested to use a simplified variation of the scale parameter, where is given a linear and proportional variation with the distance to the wall. This approach has been used in the present simulation where is varied from zero to the free stream value within a zone around the cylinder extending from the cylinder surface to the transformation point an absolute distance of away from the surface. Beyond this point the scale parameter value of was used together with the model parameter .

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A non-dimensional time-step of was used. Here is the

actual time-step. The simulations ran for a total non-dimensional simulation time of =, in which is the actual time. All quantities were sampled in every time-step. However, the results presented in the following are based on the non-dimensional time interval from to , as the simulations were allowed to stabilize.

Results

Figure 3 shows the time histories of the drag coefficient and lift coefficient . The time histories are shown from the beginning of the simulations, and the vertical dashed lines indicate the time from which the data was used in the further analysis.

Figure 3. The time histories of the drag (top) and lift coefficient (bottom).

The time-mean drag coefficient, was found together with a fluctuation drag

coefficient of . The power spectral density of the lift-force was found using the Welch estimator with a Hamming window by using four segment of equal length with overlap. The resulting spectrum is shown in Figure 4. A Strouhal number of was found from the peak in the spectrum.

The distribution of the time-mean surface pressure, , is shown together with the

distribution of maximum and minimum pressure coefficients in Figure 5. The location on the cylinder surface is given by the angle, , as defined in Figure 2. The results are compared to the time mean pressure distributions reported by Achenbach (1968) and James et al. (1980) and the maximum and minimum values reported by the latter. For both the time mean and min/max values a very good agreement is found.

The simulated max/min values lie just within the values reported from experiments. Even

the small reduction in minimum around , followed by the increase from is predicted.The largest deviation is found when comparing the simulated time-mean distribution with the one reported by James et al. (1980) for , where is found to be , while was reported. However, one must keep in mind that the experiment by James et al. (1980) was conducted at a slightly higher Reynolds number. Moreover the time mean pressure distribution is still in agreement with Achenbach (1968).

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Figure 4. Normalized power spectral density of the lift-force.

Figure 5. The calculated local time mean, minimum and maximum surface pressure

coefficient are compared to the experimental findings. James et al. (1980) : , ; (the shaded area), range between minimum and maximum . Achenbach (1968) : , . Numerical results : , ; , maximum

; , minimum . Two times the time-mean wall shear stress distribution, , is shown in Figure 6

compared with the distribution reported by Achenbach (1968). The simulated shear stress distribution is both qualitatively and quantitatively different from the distribution reported by Achenbach (1968). Though a slightly negative value indicates separation around it seems that the flow immediately reattaches. The deviation is likely to be caused the simple linear and proportional variation of the SMoM models scale parameter b near the surface. A more sophisticated boundary condition is currently under development by the authors.

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Figure 6. Wall shear stress distribution at . , Achenbach (1968) ; ,

Numerical results

Conclusions The SMoM turbulence model is found to provide maximum, minimum and time-mean

pressure coefficient distribution in very good agreement with experimental findings. The predicted wall shear stress distribution however, differs both qualitatively and quantitatively from reported experimental findings. This deviation is most likely caused by the simplified boundary conditions employed at the surface of the cylinder.

References Achenbach, E. (1968). Distribution of local pressure and skin friction around a circular cylinder in cross-flow up

to Re = 5x10e6. Journal of Fluid Mechanics, 34(04):625_639

Holmes, J. D. (2007). Wind loading of structures. Taylor and Francis, London, 2nd ed. edition.

James, W. D., Malcolm, G. N., and Paris, S. W. (1980). Study of viscous cross-flow effects on circular cylinders at high Reynolds numbers. AIAA Journal, 18:1066_1072

Johansson, J. (2011), Wind loads on structures, Ph.D. thesis, Institute of Technology and Innovation, University of Southern Denmark. (Pending publication)

Jones, J., Cincotta, J., and Walker, R. (1969). Aerodynamic forces on a stationary and oscillating circular cylinder at high Reynolds numbers. Technical report TR R-300, NASA.

Nielsen, M.P., Shui, Wan, Petersen, Ulrik E. (2006). Turbulence model. Unpublished. A draft is available with the authors.

Norberg, C. (2003). Fluctuating lift on a circular cylinder: review and new measurements. Journal of Fluids and Structures, 17(1):57-96.

Roshko, A. (1961). Experiments on the flow past a circular cylinder at very high Reynolds number. Journal of Fluid Mechanics, 10(03):345-356.

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