COMPUTATIONAL ANALYSIS OF THE FLOW AROUND A CYLINDER AND OF THE DRAG FORCE

TUDOR BARACU - PhD Student, University POLITEHNICA of Bucharest, Department of Producing and Utilisation of the Energy, e-mail: tbaracu@yahoo.com.uk

STEFAN GRIGORAS-BENESCU - PhD Student, Technical University of Civil Engineering Bucharest, Hydraulic and Environmental Protection Department, e-mail: stefan.grigoras.benescu@gmail.com

Abstract: The flow around cylinder open the path for studying more complex shape bodies that still keep in their external flow properties the combinations of the flow properties of simpler bodies like flat plates, cylinders, ellipses. The aim of this study is to describe flow around a cylinder based on simulations with Comsol and Ansys Fluent and to make comparisons with the experimental and analytical results. Drag force that appear at flow around cylinders have an important practical impact in applications. Cylinder is a case where because of its simplicity the flow results could be checked broadly in all three ways – analytical, computational and experimental. Describing in exhaustive way the flow around cylinder with all of its flow properties opens the way of studying further more complex shapes like hulls of the ships and submarines.

Keywords: pressure drag, friction drag, external flow, Comsol, Ansys Fluent

1. Introduction

Drag force resulted from external flows express a permanent challenge for design engineers activating in diverse industries (aerospace, automotive, ship construction, etc). In most cases the designed shapes are very complex and appear in the streamlines interferences, influences (walls at different angles, edges, appendices, curvature variation). The best way in solving those issues is just to relate them to simpler shapes that could be better defined and solved by analytical and experimental methods. Mainly, in this study the interest is to obtain by numerical integration the drag force distributed around cylinder for different Reynolds numbers. The interest zone is just the interface cylinder - fluid, that part require a special attention to the shaping of the elements of mesh as relative proportions and degree of refinement. The fluid domain was modeled with length of ten times the diameter of cylinder in the direction of flow and five diameters in the other direction; so an enough distance in which imposed boundary conditions of the external domain (the fluid) does not influence significantly the flow near the cylinder wall.

2. Reviews

Wieselberger (1921) came with a lot of experimental results for the flow around cylinder at low and high Reynolds number and made some checkings of Lamb’s empirical formula. Roshko, A (1960) pointed the fact of very closed results of the drag coefficient shared by different researches, but in the region of supercritical values of the Re number the results are with varies discrepancies. In his researches and experiments implemented wall interference corrections in order to guarantee accurate results. Zdrakovich (1979) colected various empirical formulas for the drag coefficient recommended by researchers for low Reynolds number.

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Spalart (1983) created a code KPD2 capable of computing the boundary layer around cylinder based on a integral method, and succeeded to eliminate some distorsions in vortex simulation that was giving the previous versions of the code. Mital R. (1995) did an analysis of the flow around the cylinder based on LES model of turbulence a innitiative that wanted to use just other method of analysis than the conventional in this kind of flow. He met some difficulties in simulation of the three-dimensionality in the wake of the cylinder and linked them by the mesh density, a constraine imposed by the computational resources used.

3. Considerations on the theoretical and experimental results

On an axis-symmetrical body where the fluid flow around its surface with the direction parallel with its axis appear a distribution of pressure on the surface that after summation gives an overall external load D. This load called drag force is characterized mainly by the drag coefficient CD that has the form: (1)

This coefficient is very difficult to be exactly determined analytically, because it depend on many parameters: (2) The drag itself of a body is composed by two components, friction drag Df and pressure drag, Dp

(3) Also, drag coefficient CD could be put in the form of a sum between coefficient of friction drag and of pressure drag respectively: (4) where (5)

(6)

Aproximations from upper relations are as per [1]. For the flow around the cylinder, depending on Reynolds number, appear the separation of the flow from the wall of cylinder. We have the following situations:

• Re<1: is a creeping flow [2] where viscous forces dominates - friction drag prevail - and in this viscous flow the streamlines behind the cylinder are parallel and symmetric almost like in the front of it [ 3] .

Fig. 1 Flow around cylinder. Re=0.16, Van Dyke, M [9]

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• 1≤Re<30: behind the cylinder the boundary layer begins to separate, and in the zone between the two symmetrical points of separation is appearing two twin eddies of opposed spins – Von Karman street of shed vortices; behind the two eddies the streamlines of the lateral sides of cylinder get closed together parallel and symmetric [3]. Cd depends comparable on both form drag and friction drag. Cd decreases when increasing Re.

a) b) c)

Fig. 2 Flow around cylinder a) Re=1.54; b) Re=9.6; c) Re=26, Van Dyke, M [9]

• 30≤Re<90: periodic oscillations inside the wake of the flow behind cylinder begin to be detected [3]. Cd depends comparable on both form drag and friction drag. Cd decreases when increasing Re.

Fig. 3 Flow around cylinder. Re=30.2, Van Dyke, M [9]

• Re≥90: the eddies alternatively spill in the backward wake of the flow and continuously

regenerate on both sides of the axis parallel with the flow. CD depends comparable on both form drag and friction drag. CD decreases when increasing Re.

• 102<Re<103: begins the separation on a point at about 800 from the front edge. In the range of values. Cd depends comparable on both form drag and friction drag. CD decreases when increasing Re.

Fig.4 Flow around cylinder. Separation point at Re=100

• 103<Re<2x105 (Fig. 5 a, b) is seen that drag coefficient is almost constant, CD=1…1.2. This is

because in this range pressure drag (form drag) is dominant in the total drag. The same situation is for Strouhal number that remain almost constant around the value . This region practically is independent of Re number.

a) b)

Fig. 5 Flow around cylinder. a) Re=2000, b) Re=10.000, Van Dyke, M [9]

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• 2x105<Re<5x105: Critical flow. At Re=2x105 attached boundary layer begin to transit to a turbulent state because most of the drag is based on shear force that is greater for turbulent flow [1], and it is manifest by a significant drop of drag coefficient. This effect is called drag crisis. This is justified by the fact that turbulent boundary layer is more capable to resist to the adverse pressure gradient around the cylinder so it will separate later and go further more in the back of the cylinder – and on this surface turbulent boundary layer is capable to maintain a pressure that reduce overall the drag. The wake will be thinner, it creates smaller vortices and determine a smaller a smaller zone of low pressure on the downstream surface, contributing to a smaller drag [2]. At Re=3.8x105 boundary layer becomes turbulent and the point of separation moves behind up to 1200 from the front edge, drag coefficient have a sudden decrease to CD=0.3.

Fig. 6 Flow around cylinder.

Separation point at Re=3.8x105 where θ=1200 and drag coefficient CD=0.3 is minimum

• 5x105<Re<2x106: supercritical flow with CD increasing slowly. • Re>2x106: post-supercritical flow, where drag coefficient close to a steady values of about 0.67

[4]. There are a lot of empirical formulas that describe the drag coefficient of the flow around cylinder, the most known are presented below:

• Wieselsberger-Lamb (1921) [9]: (7)

• Clift (1978) [10]: (8)

• Munson [1]: (9)

In the graph from below are made some comparisons between those empirical formulas related to the experimental values taken from F. M. White [7].

Fig. 7 – Comparison of experimental results with empirical formulas of prediction

for the drag coefficient of flow around cylinder

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As can be seen, no one of formulas is capable to represent with enough acuracy the entire range of variation of Re number. Wieselberg-Lamb formula approximate the experiments only in a short range of the Reynolds number 1<Re<10; Clift formula only in the range 1<Re<100; Munson’s formula looks to cover a large range 1<Re<2x105 with good approximation – it fails in prediction of drag coefficient value only at the beginning of the drag crisis. From the comparisons of empirical formulas that try to predict the drag coefficient, the proposed by Munson’s looks to be the most complete solution and in the widest range of the Reynolds number. It can be used with accuracy up to the drag crisis region. The bigger part of the total drag is due to the pressure drag. As per results obtained by method of conformal mapping, for the cylinder is calculated the coefficient of pressure at the flow around

(10)

Fig. 8 Theoretical variation of the pressure coefficient around the cylinder

Strouhal number characterize the vortex shedding in the wake of cylinder. The generation of vortexes encounter with a specific frequency depending on the Reynolds number. The formula of Strouhal number is: (11)

As a consequence of vortexes generation and shedding over the cylinder appear a vibration with a frequency that is described by Strouhal number. This unsteady phenomenon determine fluctuations of the lift and drag coefficients. For St≈1 begin to dominate the viscosity in the fluid flow, and for St<10-4 dominates the oscilations in the wake. An empirical formula (Roshko) gives St number function of Re number for the range Re<2x105 (12)

4. Aspects of the flow modeling

The turbulence method used for analysis is that is associated to the equations of Navier-Stokes in RANS form of the conservation of momentum (13) and equation of continuity (14)

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The is the method of two equations, of turbulent kinetic energy k and of dissipation rate . The method is iterative and contain a predictor k-L (L is the turbulence eddy scale length) and a corector ε. The equations involved are

• for k (15)

• for ε (16) where turbulent kinetic energy k is (17) dissipation rate ε is

(18) the production of turbulent kinetic energy is , (19) the turbulent viscosity of eddy is (20) the turbulence eddy scale length L is

(21)

The default values of empirical parameters used in equations are (22) As initial condition in this method, the field of velocity is considered zero; then is considered a transition time after which already is formed an steady turbulent flow. Near the wall the effect of viscosity prevail, the velocity is very small but its gradient is very high and get a big amplification closing to the wall. To avoid solving the equations for huge gradients of velocity, are used wall functions that are applied at a distance y from the wall.

(23), (24), (25)

where is the von Karman’s constant. So all the upper equations are applied on a surface envelope at distance y from the wall, and not directly on the wall. The tangential speed u to the wall is used to calculate the frictional speed from the equation (26)

that is available in the logharitmic sublayer of the fluid near the wall, where Reynolds number is (27)

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The domain is computed using k-epsilon turbulence model because gives quality results especially near wall where have to be obtained the drag coeficient along the surface. Flow results around a cylinder is a start in understanding the flow around more complex shapes of the applications from engineering, where the main issues involve for example the design of a hydrofoil or a hull of the ship where it is desired that the separation of boundary layer to happen as backward as possible, in idea to have the resistance more from the viscous drag than from pressure drag. The domain is meshed in 2 parts, such as near the wall is used structured mesh by quad elements, and the rest of the domain of the fluid is meshed by triangular elements. The Delaunay triangulation have specific maximization of the minimum angle inside the triangular element, avoiding skinny triangles that distorts the results. This triangulation method is commonly used in CFD for meshing the domains. On the surface of the cylinder are chosen very small elements at the scale of 10-4 in order to be comparable with the von Karman sublayer. Entire domain is a 16x8 m water fluid and the cylinder is 1 m diameter.

Fig.9 The mesh of the domain of flow around cylinder

5. Results The main issue remain to compare experimental results with the numerical ones obtained by CFD analysis in COMSOL. Fig. 10 a), b), c) shows the field of velocities around the cylinder and can be distinguish that at about 300 - 400 all curves from that zone are gathered in that point, where practically the pressure begins to take a negative gradient (fig 10 b). Fig. 10 c) shows the streamlines around cylinder delimiting at the backward the turbulent flowing mass of water.

a) b)

c)

Fig. 10 Representation of the field of velocities and streamlines. a) velocity magnitude field; b) velocity magnitude contour; c) velocity streamlines

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On the surface of the cylinder, the pressure has a maximum on front edge, then at about 300 - 500 gradient becomes negative for any Re number of the fluid flow. Then, beginning with 800 and up to 1200, function of Re number, pressure becomes zero and as consequence appear the separation of the boundary layer (fig.11 ).

Fig. 11 Pressure contours around the cylinder

Pressure coefficient describes how the external loads distribute on the surface of the cylinder. But for an overall action of the loads on cylinder is necessary to determine drag coefficient.

Fig. 12 Pressure coefficient of the flow around cylinder

An important issue is to try to obtain drag coefficient in a computational model and to compare it with experimental values and empirical formulas. Following this path, for a list of Reynolds numbers (Re=102, 2x102, 103, 2x103, 104, 2x104, 105, 2x105, 106, 107) was obtained afferent drag coefficients that gave enough information to build the drag curve.

Fig. 13 Comparisons of the drag coefficient:

Experimental vs Empirical formula vs Computational

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The computational analysis of the model for different Re numbers got relatively closed results to the experimental values, but have to be mentioned that in the zone of the drag crisis the results differ significantly. Practically the computation model delivered results in the crisis zone in a form that shows still a continuous variation of the drag coefficient, it does not detect that pit in the zone of the flow crisis; the empirical formula of Munson too is not able to cover the drag crisis. The Von Karman street is also analyzed to point on the transient processes that take place in the formation and evolution of the vortexes in the wake of the cylinder. Because of the transitory processes of the flow at Re>100 lift and drag coefficients will fluctuate. This transient study of the wake of the flow around the cylinder is made using Ansys Fluent, because for transient processes was more accurate after a preliminary comparison with Comsol. For present model are used time steps of 0.4 seonds in an entire interval of 320 seconds (800 time steps) for a Reynolds number of the flow Re=105.

a) b)

Fig. 14 Transient processes in the wake of the flow around the smooth cylinder a) Lift coefficient; b) Drag coefficient

Analyzing the graph of lift coefficient obtained by computation from Fig.14 a), results that the oscilation period is about 40 seconds. Lift coefficient fluctuate with an amplitude of value 1 around the value zero, an drag coefficient with an amplitude at about 0.20CD from the main value CD. (28)

On the other way from the empirical formula results that (29)

Making comparison between empirical and computational values results that the error is (30)

The obtained error could be reduced further if more densed mesh is used that also will require much more computational resources.

a) b)

Fig. 15 Unsteady flow in the wake of the fluid flow around cylinder a)Velocity magnitude; b) Vorticity magnitude

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6. Conclusions The applied computational analysis of the flow around cylinder with Comsol and Ansys Fluent shows that obtained results keep closed to experimental. In some special cases like the range of the Reynolds number in the drag crisis zone appear differences in the values of the computational drag coefficient obtained with Comsol related to the experimental results. The results for pressure coefficient present some differences from the analytical formula, but they are more closed to experimental results. Transient analysis for the wake of the cylinder reproduced the turbulence including the two vortexes that are permanently created at regular time steps. The accuracy of computational analysis could be improved further by meshing with higher density the entire domain of analysis and using more powerful hardware resources. Being given the details and results of the flow around cylinder is opened the path for further analysis of more complex shapes in 2D or 3D dimensions. In fact in any other shape with at least one axis of symmetry under external flow could be reviewed similarities and similitudes related to the cylinder.

The flow around cylinder is still a subject of research because in the complex processes of flow still is not imposed only one method or solution of determination the specific unknowns. Being the simpliest possible closed shape, at this point are meeting in a large enough way the analytical theories and experiments. More complex shapes still have much more degrees of freedom, their determination rise much more problems and interferences of phenomenas, boundaries influentions. But then they could be determined in the simpliest way by trying to define them by combinations of simpler shapes where the variations are approximatively linear.

7. Future work By establishing the results of the flow around cylinder is calibrated already a basic method of studying external flows around more complex bodies.

Generalizing the cylinder could be followed further by studies of ellipse in 2D space and next in 3D space establishing the flow properties of an ellipsoid.

The hull of a submarine is an application where could be established relations between its complex geometry and an equivalent elipsoid that can approximate it in all three dimensions. The idea is not only to relate complex shapes to shapes of simpler geometry; those simpler shapes is better to be just parametrized surfaces that can be exprimated directly by a matematical equation. In this way are kept in all stages of research all three main methods of research: experimental, computational and analytical.

Symbols

– dinamic viscosity, Ns/m2 – turbulent viscosity, Ns/m2 - transition time, s

– kinematic viscosity, m/s2 – density, m3/s – time, s; shear stress, N/m2

– drag coefficient – drag coefficient from friction

– drag coefficient from pressure

- production of turbulent kinetic energy, N/m2/s Fr – Froude number k – turbulent kinetic energy, J L – turbulence eddy scale length, m p, p∞ - pressure, inlet pressure, N/m2 u – speed in x direction of the model, m/s

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– friction velocity, m/s U∞ - inlet velocity, m/s y – distance from the wall in boundary layer, m

– surface of the cylinder wall, m2 – Reynolds number

– dissipation rate - von Karman’s constant

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