cfd validation of high reynolds number flow past a...

CFD Validation of high Reynolds Number flow past a circular cylinder

P.Arnold, Minerva Dynamics, Bath UK

Abstract

The objective of this work is to validate the Flow-3D CFD code for the case of flow past a circular cylinder at high

Reynolds numbers in terms of the pressure distribution around the cylinders surface and the resulting drag and lift

coefficients and Strouhal numbers. The scope includes both 2D and 3D simulations of a circular cylinder at Reynolds

numbers of between 5 X 104 and 107 which are of interest in the marine environment. We also consider the

implications of the unexpected 2D inviscid and laminar results from the numerical simulation, which converge towards

the potential flow solution with very low drag. Whilst this is consistent theoretically it is not observed in 3D high

Reynolds number simulations or experiments as embodied by the d,Alemberts paradox.

Keywords: circular cylinder, CFD, Flow-3D, validation

*Authors email address: [email protected]

Background

The numerical simulation of flow over a cylinder for a range of Reynolds numbers between 40 and 106 has been

systematically studied by Stringer et al [8] in 2D using the CFX and OpenFoam CFD codes with the same mesh and low

Reynolds number k-w turbulence model and ensuring that y+ values did not exceed 1.0 . The cylindrical meshes used

close to the cylinders surface appeared to have a circumferential spacing of approximately one degree.

If we consider the representation of a circle with such a mesh, the cylinder is effectively represented as a multi sided

polygon. Geometrical consideration then dictate that the maximum difference between a radial line following the

surface and that made from polygons is given by;

/ D = 2/16 Eqn 1.

Where is the difference between the circle and polygon (m)

D is cylinder diameter (m)

is the angle between adjacent radial grid lines (rads)

The boundary layer thickness on a cylinder in cross flow as calculated numerically by Stringer et al [7] for a range of

Reynolds numbers (Re) is given by;

u / D = 1.5 Re-0.625 Eqn 2.

Where

u is the boundary layer thickness (m)

Whilst the relationship between the normal distance from the cylinders surface, y+ and Re is given from flat plate

theory as;

y1 / D = 8.60 y+ ReL -13/14 Eqn 3.

Where

ReL is the Reynolds number using the length of the boundary layer L which we approximate here as being equal to Re.

Y1 is the normal distance from the cylinders boundary (m)

Y+ is the dimensionless wall distance given by

Y+ = u* y1/ Eqn 4.

Where u* is the friction velocity = [ Tw/ ] 0.5 (m/s)

and Tw is the wall shear stress kg/(m.s2)

and is the dynamic viscosity ( kg/( m.s) )

Hence to ensure that the apparent distance between the line defined by a true circle and that of the polygon as defined

by the mesh does not exceed the boundary layer thickness we require ( combining equation 1 and 2)

2/16 1.5 Re-0.625 Eqn 5.

Which gives;

Relationship between Re and maximum mesh angle in circumferential direction to for boundary layer

Re Boundary Layer Relative Thickness u / D

(rads) Min Circumferential lines

103 2 X 10-2 0.565 12

104 4.7 X 10-3 0.275 24

105 1.2 X 10-3 0.134 48

106 2.7 X 10-4 0.0653 96

107 6.3 X 10-5 0.0318 192

In order that the apparent distance between the line defined by a true circle and that of the polygon as defined by the

mesh does not exceed the first normal mesh cell distance corresponding to y+ =1.0 we require ( combining equation

1 and 3)

2/16 8.60 Re-13/14 Eqn 6.

Which then gives;

Relationship between Re and maximum mesh angle in circumferential direction for Y+ =1.0

Re First Mesh cell relative normal distance (y1 / D)

(rads) Min Circumferential lines

103 1.4 X 10-2 0.475 13

104 1.7 X 10-3 0.163 39

105 2.0 X 10-4 0.056 112

106 2.3 X 10-5 0.0192 327

107 2.7 X 10-6 0.0066 952

Hence we see that for Re value up to 106 the number of circumferential mesh points used by Stringer et al was such

that the polygon nature of the mesh was insignificant in terms of the boundary layer thickness but that in terms of the

first mesh point the distance becomes of similar magnitude at an Re of 106.

If we now consider the results of Achenbach [2] for flow over rough cylinders, we see that the drag crisis is triggered

at lower Re values with increasing relative roughness Ks/D. Also that the drag coefficient is considerably higher

throughout and after the drag crisis than for a smooth cylinder. Assuming Ks corresponds in the geometrical height

of the roughness, then Ks/D values are in the range of 1.1X10-3 to 9X10-3.

The value of geometrical perturbation / D created by the polygon mesh corresponding to a one degree

circumferential spacing from equation 1 is then, 1.9 X10-5, which is less than 2% of the smallest relative roughness

height so would not seem significant. Relative to the minimum boundary layer thickness at an Re of 106 the geometrical

perturbation is approximately 7% , whilst the smallest relative roughness used in the Achenbachs [2] experiments is

almost four times the minimum boundary layer height yet. Hence the geometrical perturbation created by the polygon

mesh would seem to be insignificant.

In any event the results reported by Stringer et al indicate that at Re values higher than 104, there is significant

deviation in the results of the two CFD codes from each other and from the experimental results. This indicates that

the physical flow is very sensitive to surface roughness, inlet turbulence intensity and other geometrical details as seen

in the range of the experimental results and that the numerical solutions are also sensitive to small differences in the

solution methods, discretization, time advancements etc. It would therefore perhaps be more useful to perform a

numerical sensitivity study at each Reynolds number to provide a guide for engineers involved in the design of

cylindrical structures at high Reynolds numbers. An experiment to test various multisided polygons cylinders and

circular cylinders at various Re numbers may shed more light on the influence of the mesh discretization.

Simulation Setup

Domain and Mesh and boundary conditions

The computational domain and mesh for the 2D simulations is as shown in Figures 1, 2 and 3 where a 1m diameter

cylinder is placed 5 cylinder diameters (D) from the prescribed velocity inlet boundary and 15 D from the constant

pressure outlet boundary and 5 D from the top and bottom symmetry wall boundaries. The walls in the span wise

direction were set as symmetry planes. The mesh consisted of aspect ratio one cells arranged in a series of nested

blocks decreasing in size by a factor of two with each mesh block boundary so that the cells closest to the cylinder

were either 1/100th D , 1/200th D or 1/400th D. The number of nested mesh blocks was also varied between six on

the finest mesh to four blocks on the coarsest mesh, resulting in a mesh cell count of 75,000, 162,000, and 336,000

cells respectively. All mesh blocks were one cell in the span wise direction.

Figure 1 Coarse 2D Mesh Block Outline

Figure 1 Coarse 2D Mesh

Figure 1a Coarse 2D Innermost Mesh Block

Figure 2 Medium 2D Mesh Block Outline

Figure 2a Medium 2D Innermost Mesh Block

Figure 3 Fine 2D Mesh Block Outline

Figure 3a Fine 2D Innermost Mesh Block

The 3D meshes are shown in Figures 4, 5, and 6 where the domain is now extended 2D in the Z direction. Due to the

additional computational cost, the cell size closest to the cylinder was varied between 1/100th D, 1/50th D or 1/25th D.

The number of nested mesh blocks varied between four on the finest mesh, three on the medium mesh and two on

the coarsest mesh, with the finest blocks omitted on each of the progressively coarser meshes, resulting in a total cell

count of 6 million, 3 million or 1.5 million cells respectively. The top, bottom and span wise walls were all set as

symmetry planes.

Figure 4. 3D Coarse Mesh Block Outline

Figure 5. 3D Medium Mesh Block Outline

Figure 6. 3D Fine Mesh Block Outline

Numerical Modelling Options

The momentum advection were discretized using a 2nd order monotonic scheme whilst the governing equations were

integrated in time using a first order automatic time stepping method, starting from a uniform velocity field for up to

100 non dimensional time periods defined by D/U where U is the inlet velocity. Turbulence was accounted for using

the Large Eddy simulations based on the Smagorinsky model, with an unknown near wall modelling method. The inlet

turbulence intensity was not specified and was assumed to be zero. The incompressible fluids viscosity was varied to

achieve a Reynolds number of between 5 X 104 and 1 X 107. Once a sufficiently pseudo steady state solution had

developed, the pressure field on the surface of the cylinder and the resulting lift and drag forces decomposed in to

shear and pressure based forces, were extracted in 5 deg intervals every 1/100th non dimensional time unit for the

final 20 non dimensional time units of each run.

Results

2D simulations:

Figures 7 to 10 show the variation of the drag coefficient, RMS drag and lift coefficients and Strouhal number as a

function of Reynolds number (Re) for the coarse , medium and fine meshes. Clearly the results for all three quantities

are not fully mesh independent and the 2D fine mesh and medium mesh solutions overestimate the drag for all Re

whilst the rms values also increase with increasing mesh resolution. We can also see that the drag crisis seen in the

experimental results between a Re of 105 and 106 is not present in the simulations.

According to Blevins [1] the Strouhal Number for Re less than 2X105 follows the relationship:

St = 0.198[ 1-19.7/Re] Eqn [1.]

Where St is defined as =f L/V

Where f is the frequency of the lift signal , L is the cylinders diameter and V is the inlet fluid velocity.

and for Re values above 2 X 105 vortex shedding does not occur at a single frequency but rather over a narrow band

of frequencies as shown in figure 10a. The fine mesh results have distinct St values right up to Re=107, which are

close to the predicted values, but the St values increases with mesh coarsening to almost double the frequency

predicted by Blevins [1].

The coarse mesh results in particular indicate that all parameters are converging to particular values with increasing

Reynolds number. However the values shown in Table 1 for the inviscid case (with effectively an infinite Re) suggest

that the inviscid solution is something completely different which is converging with mesh refinement towards a

steady state solution with zero drag, i.e. the potential flow solution. This then raises the issue of d,Alemberts paradox

where without viscosity and the assumption of zero vorticity the 3D or 2D Euler equations become the Laplace

equation and Bernoullis equations, the solution of which are the well known symmetrical steady state with zero

drag. This is a paradox because it is well known that real 3D bodies at high Reynolds Numbers do not have steady

state zero drag solutions and is unresolved as yet because it is not possible to experimentally test the infinite Reynolds

number case whilst inviscid numerical simulation contain inherent numerical diffusion (viscosity) due to the

discretisation of the velocity derivatives. Consequently it is surprising to find that the numerical solution of the 2D

Euler equations of the flow over a bluff body gives rise to potential flow solution despite the inherent numerical

viscosity.

Figure 7 Drag Coefficient versus Re for 2 D simulation and Experimental data

Figure 8 RMS Drag Coefficient versus Re for 2D simulation

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Medium Mesh

Fine Mesh

Achenbach, Expt [1]

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Medium Mesh

Fine Mesh

Figure 9 RMS Lift Coefficient versus Re for 2D simulation

Figure 10 Strouhal Number for 2D simulation and Experimental data

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Fine Mesh

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2D Strouhal Number versus Reynolds Number

Coarse Mesh

Medium Mesh

Fine Mesh

Figure 10a Strouhal Number variation with Reynolds Number, (from Blevins 1990 [1] )

Figure 10b Four ranges of flow past a circular cylinder (from Achenbach 1971 [2] )

Figure 11 Drag Time History, 2D simulations

Table 1 Inviscid , Infinite Re Case

Mesh CD CD rms CL rms St

Coarse 0.03 0.00 0.00 N/A

Medium 0.02 0.00 0.00 N/A

Fine 0.02 0.00 0.00 N/A

Hence our 2D results indicate that as the Reynolds number increases the lift and drag coefficients and their rms

values along with the vortex shedding period all asymptote to particular values which appear be to a primarily a

function of mesh resolution. However at infinitely high Reynolds number the drag coefficient tends to zero as do

the fluctuations on the lift and drag coefficients, giving effectively the potential flow solution. This is a rather

puzzling result as it implies there is a discontinuity in the nature of the solution as the viscosity reaches a value of

zero.

It is conjectured in the literature [3] that the dAlermbert paradox is due to the fact that the potential flow solution is

unstable, but in 2D at least it appears to be stable. This was checked in two ways, firstly with a simulation using a

viscous solution as a starting point and then switching to an inviscid simulation where upon the drag rapidly became

almost zero, shortly afterwards followed by the rapid decrease in the fluctuation in the lift and drag forces and vortex

shedding. Secondly the inviscid solution was given a large perturbation in the middle of the run by moving the cylinder

up and down only to find that the solution settled back down to the near zero drag solution as shown in figure 11.

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2D Drag coefficient time series with Re variation, Medium Mesh

Re = 5 X 10^4

Re = 1 X 10^5

Re = 5 X 10^5

Re = 1 X 10^6

Re = 5 X 10^6

Re = 1 X 10^7

Re = Infinity

RE = Infinity with pertabation

Further evidence of this sudden change in solution is provided by the averaged surface pressure profiles shown in

figures 12 to 16. Several features are apparent, firstly the pressure coefficient is insensitive to Re on the upstream 60

deg of the cylinder after which the differences become apparent. In general as Re increase the magnitude of the

negative pressure coefficient on the rear of the cylinder between 120 and 180 deg decreases resulting in a decrease

in drag (Remember that the drag coefficient is dependent on the angle as well as the magnitude, so that a large

pressure coefficient at 90 deg has very little effect). This is particularly apparent on the coarse mesh where Re values

up to 109 have been considered and the pressure profiles have converged. Also we see that the inviscid solution is

significantly different from the other viscous solutions and changes very little with mesh resolution suggesting that

the mesh is sufficient to capture this relatively simple flow pattern. Conversely the viscous solutions exhibit

increasingly large negative pressure coefficients in the critical 120 to 180 deg region with increased mesh resolution,

resulting in increased drag and deviating further from the inviscid solution.

Figure 15 also shows two sets of experimental at supercritical Re values close to 106 (Warschauer and Leene [4] and

Flacsbart [5]) along with the simulations at Re of 106 and clearly indicates increasing deviation of the numerical

solution from the experimental values as the mesh is refined. Taking all these results into consideration, this suggests

that the mesh independent 2D solution is significantly different from the 3D experimental results. The reasons for this

are unclear but could be related to the applicability of the LES turbulence model to 2D flows, or that the exact 2D

numerical solution obtained, by say direct numerical simulation, is different to the 3D solution.

Figure 12 Coarse Mesh Surface Pressure Coefficient

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2D Coarse Mesh Pressure Coefficient Distribution on upper surface of Cylinder

Re = 5 X 10^4

Re = 1 X 10^5

Re = 5 X 10^5

Re = 1 X 10^6

Re = 5 X 10^6

Re = 1 X 10^7

Re = 1 X 10^8

Re = 1 X10^9

Re = Infinity

Figure 13 Medium Mesh Surface Pressure Coefficient

Figure 14 Fine Mesh Surface Pressure Coefficient

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2D Medium Mesh Pressure Coefficient Distribution on upper surface of Cylinder

Re = 5 X10^4

Re = 1 X10^5

Re = 5 X10^5

Re = 1 X10^6

Re = 5 X10^6

Re = 1 X10^7

Re = Infinity

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2D Fine Mesh Pressure Coefficient Distribution on upper surface of Cylinder

Re = 5X10^4

Re = 1X10^5

Re = 5 X10^5

Re = 1 X10^6

Re = 5 X10^6

Re = 1X10^7

Re = Infinity

Figure 15 2D Simulations, Effect of Mesh Refinement at Re =10^6

Figure 16 2D Simulations, Effect of Mesh Refinement, Inviscid Case

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2D Mesh Effect Pressure Coefficient Distribution on upper surface of Cylinder

Re = 1 X10^6, Coarse Mesh

Re =1 X 10^6 MediumMesh

Re = 1 X10^6 Fine Mesh

Warschauer and Leene Re= 1.26 X10^6

Flachsbart Re =6.7 X 10^5

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2D Mesh Effect Pressure Coefficient Distribution on upper surface of Cylinder, Inviscid Case

Inviscid, Coarse Mesh

Inviscid Medium Mesh

Inviscid Fine Mesh

Warschauer and Leene Re =1.26 X10^6


Finally we show results at Re = 106 comparing the LES and k-w turbulence models in Table 1a. The inlet turbulence

intensity for the LES model could not be specified in the code and so is assumed to be zero, however values of 0% ,1%

and 2% were used for the k-w model on the coarse mesh, with 0% used on the medium and fine meshes. The

turbulence length scale was specified as 0.07m. The results indicates that the drag coefficient, level of fluctuation and

vortex shedding period all increase with mesh refinement for both turbulence models.. However the drag coefficients

are smaller for the k-w model and closer to the experimental value, but are by no means mesh independent. Figures

16b of the surface strain profiles, indicate that the shear strain minimum point, corresponding to where flow reversal

and hence flow separation, occurs at lower angles as the mesh is refined varying from 135 deg on the fine mesh to

155 on the coarse mesh. The wider wake associated with the earlier separation point is then presumably the reason

for the increase in drag coefficients with mesh refinement. The pressure coefficient profiles of the two turbulence

models, shown in figure 16a, show the same trend between 120 and 180 degrees.

The level of inlet turbulence intensity is seen to have almost no effect, however this was only applied on the coarse

mesh which tended to damp out oscillations in the drag and lift presumably due to the higher level of numerical

viscosity, we might expect this parameter to have more effect on the finer meshes.

Table 1a comparison of LES and k-w turbulence model at Re = 106

Turbulence Intensity

CD CD rms CL rms St

Coarse Mesh LES 0 0.66 0.08 0.60 0.34

Coarse Mesh k-w 0 0.13 0.00 0.01 0.52

Coarse Mesh k-w 1 0.13 0.00 0.01 0.53

Coarse Mesh k-w 2 0.13 0.00 0.01 0.53

Medium Mesh LES

0 1.28 0.25 1.18 0.27

Medium Mesh k-w

0 0.39 0.04 0.38 0.38

Fine Mesh LES 0 1.75 0.33 1.49 0.24

Fine Mesh k-w 0 0.96 0.18 0.97 0.30

Figure 16a 2D Simulations, Effect of Mesh Refinement and turbulence model

Figure 16a 2D Simulations, Effect of Mesh Refinement and turbulence model on Y+

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2D Mesh and turbulence model Effect Pressure Coefficient Distribution on upper surface of Cylinder, Re =10^6

Coarse Mesh LES model

Medium Mesh LES model

Fine Mesh LES model

Coarse Mesh k-w model

Medium Mesh k-w model

Fine Mesh k-w model


Warschauer and Leene Re = 1.26 X10^6

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Y Plus Values versus angle with mesh and turbulence model variation

LES Coarse Mesh

k-w Coarse Mesh

k-w Medium Mesh

k-w Fine Mesh

Figure 16b 2D Simulations, Effect of Mesh Refinement on surface shear strain showing separation point

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Shear Strain values versus angle with mesh and turbulence model variation

k-w Coarse Mesh

k-w Medium Mesh

k-w Fine Mesh

3D Simulations:

Due to the additional computational cost, 3D simulations were only performed on the 1/25th D mesh for all Re

numbers. Several features are apparent, Figure 17 shows that the drag coefficient varies very little on the 1/25th D

mesh over the Re range . The rms lift and drag coefficients, shown in figures 18 and 19, are now an order of magnitude

lower than those on the 2D mesh, and the Strouhal number, shown on figure 20, is now double that given by the

Blevins equations [1] and is more consistent with the upper line of figure 10a, but in line with the coarse mesh 2D

simulations. The reduction is rms drag and lift values is likely to be due to the further reduced mesh resolution, as

found is the 2D simulation, and the spanwise averaging.

The surface pressure profiles on the 1/25th mesh shown in figure 21, demonstrate very little difference between the

various Re numbers, in line with the drag results. A range of experimental profiles are shown in figure 22 which

indicate that we should expect the derivative of the pressure coefficient to change from negative to positive at larger

angles with increasing Re. Instead the numerical results show the pressure gradient changing sign at approximately 85

deg regardless of Re.

The results for Re = 106 on the 1/25th, 1/50th and 1/100th D meshes are given in figure 23 and Table 2 along with some

of the experimental results available form [6,7] . The wide range in Drag coefficient and Strouhal numbers is clear and

is widely believed to be due to the different surface roughness and inlet turbulence intensities used in the experiments.

Our numerical simulations are within the experimental range both for Drag coefficient and Strouhal number and the

surface pressure profiles indicate that the results are almost mesh independent and close to the experimental values.

We also note that the 3D invisicid drag coefficient is now an order of magnitude higher than the 2D values and in the

range of the experimental results along with similar rms values to the viscous solutions. Also the surface pressure

coefficient is much closer to the viscous results, indicating that the 3D inviscid numerical solution is unsteady and is

very different to the 2D solution. However this is not a strictly fair comparison as the mesh cells in the 3D case were

four times as large.

Figure 17 Drag Coefficient versus Re for 3D simulation and Experimental data

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3D Drag versus Reynolds Number

V.V Coarse Mesh

V. Coarse Mesh

Coarse Mesh

Achenbach, Expt [1]

Figure 18 RMS Drag Coefficient versus Re for 3D simulations

Figure 19 RMS Lift Coefficient versus Re for 3D simulations

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Figure 20 Strouhal Number versus Re for 3D simulations

Figure 21 Surface Pressure Coefficient versus Re for 3D simulations on 1/25th D Mesh

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3D Strouhal Number versus Reynolds Number

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Re = 5X10^4 Very Very Coarse Mesh

Re =1X10^5 Very VeryCoarse Mesh

Re = 5X10^5 Very Very Coarse Mesh

Re = 1 X10^6 Very Very Coarse Mesh

Re = 5X10^6 , Very Very Coarse Mesh

RE = 1X10^7 , Very Very Coarse Mesh

Re = Infinity Very Very Coarse Mesh

Achenbach Re = 8.5 X10^5


Warschauer and Leene Re = 1.26X10^6Falcsbart Re = 6.7 X10^5

Figure 22 Experimental Surface Pressure Coefficients

Figure 23 Surface Pressure Coefficient for Re 106, effect of mesh refinement

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 20 40 60 80 100 120 140 160 180 200

Axi

s Ti

tle

Theta

Experimental Pressure Coefficient Distribution on upper surface of Cylinder

Achenbach Re =10^5

Achenbach Re = 2.6 X 10^5

Achenbach Re = 8.5 X 10^5


Warschauer and Leene Re = 1.26X 10^6

Flacsbart Re = 6.7 X10^5

-3.50E+00

-3.00E+00

-2.50E+00

-2.00E+00

-1.50E+00

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

0 20 40 60 80 100 120 140 160 180 200

Axi

s Ti

tle

Theta

3D Mesh Effect Pressure Coefficient Distribution on upper surface of Cylinder

Re = Inviscid Solution, 1/25th CylinderMesh

Re =1X10^6 1/25th Cyclinder mesh

Re =1X10^6 1/50th Cylinder Mesh

Re =1X10^6 1/100th Cylinder Mesh

Warschauer and Leene Re = 1.26 X10^6


Table 2. 3D simulations and experimental results at Re=106

Mesh CD CD rms CL rms St

Very Very Coarse ( 1/25th D) Inviscid 0.20 0.01 0.05 N/A

Very Very Coarse (1/25th D) 0.31 0.04 0.06 0.41

Very Coarse (1/50th D) 0.42 0.01 0.05 0.39

Coarse ( 1/100th D) 0.35 0.01 0.03 N/A

Zdravkovich [6] 0.4 - 0.7

Zdravkovich [7] 0.17- 0.4 0.18-0.50

Achenbach [2] 0.48

Shih et al [8] 0.24 0.22

Stringer et al [9] 0.15 to 0.54 0.13 to 0.19 0 to 0.3

Norberg [10] 0.13 0.5

Figure 24 Surface Pressure Coefficient for Re 106, 2D and 3D solutions on 1/100th D Mesh

-3.50E+00

-3.00E+00

-2.50E+00

-2.00E+00

-1.50E+00

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

0 20 40 60 80 100 120 140 160 180 200

Cp

Theta

2D and 3D Pressure Coefficient Distribution on upper surface of Cylinder at Re=10^6

2D Coarse Mesh

3D Coarse Mesh

Figure 25 Near Wall Y+ values

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+03

1.00E+04

1.20E+04

1.40E+04

0.00E+00 2.00E+01 4.00E+01 6.00E+01 8.00E+01 1.00E+02 1.20E+02 1.40E+02 1.60E+02 1.80E+02 2.00E+02

Yplu

s

Theta

Y Plus Values versus angle

2D Coasse LES Re=10^6

3D V.V Coarse LES Re=5X10^5

3D V.V Coarse LES RE=10^7

Shear to pressure forces ratios:

Finally table 3 shows that the ratio of shear drag to pressure derived drag is at most 7% and decreases with

increasing Re. On the 2D mesh the ratio decreases with mesh cell size but this is not repeated on the 3D mesh for

some unknown reason.

Table 3. Shear Drag to Total Drag Force Ratios

Re Mesh 2D/3D Mean Shear Drag to Total Drag

5X104 1/100th D 2D 0.035

1X105 1/100th D 2D 0.033

5X105 1/100th D 2D 0.028

1X106 1/100th D 2D 0.026

5X106 1/100th D 2D 0.021

1X107 1/100th D 2D 0.019

1X106 1/200th D 2D 0.01

1X106 1/400th D 2D 0.005

1X106 1/100th 3D 0.046

1X106 1/50th D 3D 0.035

1X106 1/25th D 3D 0.042

5X104 1/25th D 3D 0.072

1X105 1/25th D 3D 0.047

5X105 1/25th D 3D 0.042

1X106 1/25th D 3D 0.046

5X106 1/25th D 3D 0.032

1X107 1/25th D 3D 0.030

Conclusions

The flow over a cylinder has been widely studied and reported in the literature both experimentally and numerically

probably because despite the geometry being simple the resulting experimental and numerical solutions are complex

and sensitive to small influences. Probably the most important aspect of this flow is the transition of the boundary

layers from laminar to turbulent, to which the separation point is particularly sensitive and as a result the drag

coefficient and vortex shedding frequency and magnitude. The transition process is known to be influenced by the

product of the magnitude of the initial disturbance and the amplification factor which is a function of the Reynolds

number. The former is dependent on the surface roughness of the cylinder and the inlet turbulence intensity in the

experimental setup. In the numerical simulations the cylinder is effectively a multi sided polygon due to the mesh

discretization, which could be considered a form of roughness.

Accurate modelling of the laminar and turbulent parts of the boundary layer itself and the transition process requires

at least a low Reynolds number model with Y+ values of the order of 1.0 to have any chance of modelling the transition

process and predicting the correct separation point. In our simulations a high Reynolds number model was used for

the near wall modelling with Y+ values of the order of 100 or even several thousand on the coarsest meshes which is

outside the log law region. The implications of using such inappropriate Y+ appears to be that the drag crisis cannot

be predicted, instead our results are insensitive to Reynolds number.

A further complication is the numerical viscosity which is added due to the numerical scheme which is often large

compared to the laminar viscosity and can even exceed the turbulent viscosity. Hence it may be the case that varying

the dynamic viscosity to change the Reynolds number is ineffective and the Reynolds number would be better served

by the inlet velocity. This again will need to be checked separately to ensure that the non-dimensionalised equation

all give the same result regardless of the manner in which the Reynolds number has been changed.

However despite all of the above we have obtained drag coefficients and Strouhal numbers and surface pressure

coefficient distribution within the experimental range at supercritical Reynolds numbers.

Pending further investigations using finer meshes and more appropriate near wall modelling we make the following

comments on what is observed in the 2D and 3D results;

The drag crisis is not predicted by the 2D solution, instead increasing the Reynolds number gives a very slow

decrease in the drag coefficient and other parameters which appear to asymptote to constant values. This is

accompanied by an increase in the magnitude of the maximum negative pressure coefficient which occurs at

around 85 deg and a reduction in it between 120 and 180 deg. Both of which are consistent with a slow

decrease in the size of the wake to a constant value due to the separation point moving downstream with

increasing Reynolds number. The increase in drag coefficient which occurs in the supercritical region is not

seen.

The 2D solution with mesh refinement is converging to a different surface pressure distribution to the

experimental data and drag coefficients which are too large. This appears to be due to the separation point

converging to a location too far upstream, resulting in a wider wake with the accompanying greater scope

for vortex shedding and increasingly large fluctuation in the drag and lift coefficients and decrease in vortex

shedding frequency associated with the larger length scale of the wake.

In 2D the inviscid solution is significantly different to the viscous solutions and has a very small wake with no

detectable fluctuation and a very small drag coefficient. This solution appears to be stable and returns after

a large perturbation in the cylinders vertical position.

The 3D solutions which were obtained on the 1/25th D mesh also shows a slow decrease in drag coefficient

with Reynolds number.

The 3D solution follow the patterns of coarser mesh, later separation higher pressure recovery in the wake ,

narrower wake, lower fluctuations, higher fluctuation frequency and less drag.

The Re=106 solution using the same size mesh cells in 2D and 3D as shown in figure 24, shows that the

pressure coefficient is smaller in magnitude across the entire profile , the reason for this in not clear.

The 3D inviscid solution is closer to the viscous solutions but only the 1/25th mesh results are reported.

References

1. Blevins R. D. (1990) Flow Induced Vibrations, Van Nostrand Reinhold Co.

2. Achenbach, E. (1971). Influence of surface roughness on the cross-flow around a circular cylinder. J. Fluid

Mech., 46, 321-35

3. Computational Turbulent Incompressible Flow, Johan Hoffman and Claes Johnson, Applied Mathematics ,

Body and Soul, Volume 4, 2007, XIX, 401p, Springer

4. Warschauer, K. A. & Leene, J. A. 1971 Experiments on mean and fluctuating pressures of circular cylinders at cross flow at very high Reynolds numbers. Proc. Int. Conf. on Wind Effects on Buildings and Structures, Saikon, Tokyo, 305-315.

5. Falchsbart as given in Zdravkovich 1997 [2]

6. Zdravkovich,M.M.,1990, A Conceptual overview of laminar and turbulent flows past smooth and rough circular-cylinders. J.WindEng.Ind.Aerodyn.33,5362.

7. Zdravkovich, M. M. 1997 Flow Around Circular Cylinders. Vol. 1: Fundamentals. Oxford University Press, Chap. 6.

8. Shih, W. C. L., Wang, C., Coles, D. & Roshko, A. 1993 Experiments on flow past rough circular cylinders at

large Reynolds numbers. J. Wind Engg and Industrial Aerodynamics 49, 351-368.

9. Unsteady RANS simulations of the flow around a cylinder for a wide range of Reynolds numbers, R.M Stringer, J. Zang and A.J.Hillis, Ocean Engineering, 87 (2014) 1-9

10. Norberg,C.,2003.Fluctuating lift on a circular cylinder: reviewand new measure-ments. J. Fluids Struct.17, 5796.

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