Aerodynamic 1

Aerosp. Sci. Technol. 5 (2001) 85–94 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS1270-9638(00)01082-8/FLA

Two-dimensional viscous vortex flow around a circular cylinder

Sébastien Rouvreaua,1, Laurent Peraultb

a Laboratoire de Combustion et de Détonique (UPR CNRS 9028), ENSMA, BP 40109, 86961 Futuroscope-Chasseneuil cedex,France

b Laboratoire d’Etudes Aérodynamiques (UMR CNRS 6609), ENSMA, BP 40109, 86961 Futuroscope-Chasseneuil cedex, France

Received 27 March 2000; revised 7 September 2000; accepted 25 October 2000

Abstract A code based on a vortex method for simulating viscous vortex flow around a circular cylinder has beendeveloped. The original method to obtain the no-through and no-slip condition introduced here is describedin detail. The diffusive part of the vorticity transport equation is treated using a deterministic method basedon the solution of the heat equation. Results have been obtained and are presented concerning the vorticityfield, the velocity field, the evolution of drag and lift coefficients and the Strouhal number. 2001 Éditionsscientifiques et médicales Elsevier SAS

vortex method / viscous / cyclinder / no-slip

Résumé Un code de calcul d’écoulement tourbillonnaire de fluide visqueux autour d’un cylindre, fondé sur uneméthode particulaire a été développé. La méthode originale d’obtention de la condition d’adhérence mise enplace ici est décrite en détail. Le traitement de la partie diffusive de l’équation de transport de la vorticitéutilise, lui, une méthode déterministe fondée sur la solution de l’équation de la chaleur. Les résultats sontprésentés pour l’évolution du champ de vorticité, du champ de vitesse, des efforts sur le profil et le nombre deStrouhal. 2001 Éditions scientifiques et médicales Elsevier SAS

vorticité / visqueux / cylindre / adhérence

Nomenclature

Cx drag coefficientCz lift coefficient→ez unit normal vector perpendicular to the planar

domainL reference length (diameter of the cylinder)n vortices shedding frequency

p dimensionless pressure such thatp = p∗ρU2

p∗ pressure of the fluid

Re Reynolds number defined asRe = ULν

1 Correspondence and reprints. Present address: University of Maryland, Fire Protection Engineering Department, College Park, MD 20742-0001,USA. Email address: [email protected]

St Strouhal number,St = nLU

t dimensionless time

t∗ characteristic time such thatt∗ = LU

u component of the velocity vector inx direction

U reference velocity (upwind flow velocity)

v component of the velocity vector iny direction

x longitudinal direction→x position vector

y transversal direction

ν kinematic viscosity of the fluid

86 S. Rouvreau, L. Perault / Aerosp. Sci. Technol. 5 (2001) 85–94

ρ density of the fluid→ω vorticity vector→Ω (t) solution for the diffusive part of the transport

equation

1. Introduction

Many incompressible flows are characterised by rota-tional regions in a mainly irrotational flow. Vortex meth-ods simulate such flows by discretising these concen-trated rotational zones into vortex elements. These par-ticle methods, where particles are advected in a La-grangian way, and which are being developed for three-dimensional studies, are particularly well adapted for in-compressible two-dimensional instationnary flows domi-nated by convective effects.

The main variable is vorticity→ω = →

× →u , defined

as the rotational of the velocity vector. For a two-

dimensional study, vorticity becomes a scalar (→ω =

(0,0,ω)), and the only non-zero component of the vectorrotational can be written

ω = ∂v

∂x− ∂u

∂y.

These methods, first used by Chorin [3], have been sig-nificantly developed during the last few years [7,9,12]and now constitute a family of methods with different ap-proaches. The most used approach remains close to theChorin’s exclusively Lagrangian method, where particlesare displaced by convection and viscous diffusion in time(Random Walk Method). In 1986, Cottet and Gallic [4]and Choquin and Huberson [2] concurrently developeda method to deal with the diffusion term in a determin-istic manner. For this method, when calculating the dif-fusion, it is not the particles’ position that is changedbut their weight. Other widely used approaches are theEuler–Lagrange approach and the semi-Lagrangian ap-proach (in the Vortex-In-Cell methods). Both of these ap-proaches require a grid.

2. Mathematical formulation

Viscous incompressible flows are governed by dimen-sionless Navier–Stokes equations (mass and momentumconservation) that can be written in 2D:

∂u

∂x+ ∂v

∂y= 0,

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −∂p

∂x+ 1

Re

(∂2u

∂x2 + ∂2u

∂y2

),

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y= −∂p

∂y+ 1

Re

(∂2v

∂x2 + ∂2v

∂y2

).

The boundary conditions associated with these equa-tions include the no-slip condition on solid walls, whichis the physical constraint that generates vorticity, and theno-penetration condition (i.e. zero normal speed on solidwalls). Since it is impossible to get an analytical solutionfor this non-linear system, the use of a numerical schemeis necessary.

Taking the rotational of the Navier–Stokes equations(divergence of the velocity vector being zero) and intro-

ducing vorticity→ω leads to the two-dimensional vorticity

transport equation:

∂→ω

∂ t+ (

→u · →∇)

→ω = 1

Re

→ω .

The main difficulty concerning time discretisation is to

treat the temporal derivative of→ω . The method used here

splits the equation and is known as the fractionated-step-method.

The equation is split into two steps: one of advectionand one of diffusion, solved simultaneously:

∂→ω

∂ t= −(

→u · →∇)

→ω,

∂→ω

∂ t= 1

Re

→ω .

Proves of the convergence of these equations to thesolution of the transport equation do exist [1,13,14].

The vorticity field is discretised into vortex elementsthat will be advected and diffused throughout time. Ad-vective transport for an element is governed by a firstorder differential equation that uses the position of theelement and its velocity induced by the vorticity field.The way this velocity is obtained depends on the methodand an advective displacement is associated to it. Diffu-sive transport with the Random Walk method, is statisti-cally simulated with Gaussian random displacements togive a diffusive displacement. Concerning the determin-istic method used here, diffusive transport is simulatedby varying particles weight, in other words their vorticalintensity.

3. The deterministic diffusion model

This method, developed firstly by Cottet and Gallic [4]and Choquin and Huberson [2] treats the diffusion equa-tion in a deterministic way by using the exact solution ofthe heat equation. The conservative algorithm proposedby Choquin and Huberson, on which mathematical analy-sis has been made by Cottet and Gallic, takes into accountthe vorticity diffused from each particle. In this method,concerning the diffusion simulation, the vortex intensityvaries, not the particle position. The solution for diffusive

S. Rouvreau, L. Perault / Aerosp. Sci. Technol. 5 (2001) 85–94 87

part of the transport equation of the vorticity is then givenby:

→Ω (t) = Re

4πt

∫

R2

exp

(−|→x − →

x′ |2

4Re−1t

)→ω 0 (

→x

′)d

→x

′.

Figure 1. Schematic of the method used to obtain the no slipcondition on solid walls for one control point.

With this equation, the vortex contained in a particle‘P ’ can be calculated, and then, taking into account thevortex diffused out of each particle leads to:

→Ω (δt)= →

Ω0 + Re

4πδt

∫P

∫

R2−P

exp

(−|→x − →

x′ |2

4Re−1δt

)

× →ω 0 (

→x

′)d

→x

′d

→x

· · · − Re

4πδt

∫

R2−P

∫P

exp

(−|→x − →

x′ |2

4Re−1δt

)

× →ω 0 (

→x

′)d

→x

′d

→x .

The second term of the right hand side of this equationrepresents vorticity entering the particle and the third onerepresents the vorticity leaving the particle.

4. Vorticity generation: no-slip condition

4.1. Conventional methods

On solid walls, the velocity vector must be zero, whichgives the no-slip condition. However, during a perfect-fluid type calculation, this condition is not obtained,giving a tangential velocity called the slip-velocityug .Thus, for each time step, an appropriate quantity ofcirculation must be generated on solid walls to obtain

(a)

Figure 2. Vorticity field, Re = 1000, black: clockwise, grey: counterclockwise: (a)t∗ = 2.0; (b) t∗ = 6.0; (c) t∗ = 10.0; (d) t∗ = 12.0;(e) t∗ = 20.0.


(b)

(c)

Figure 2. (Continued)

a zero slip-velocity, which then simulates the no-slipphenomenon due to fluid viscosity [5].

In the commonly used method to generate circulation,known as the Vortex Sheet Method, segment-vortices areemitted by solid walls and need to be transformed intovortex blobs when going out of a user-defined boundarylayer called vortex sheet. The inverse transformation

also has to be performed when a blob enters the vortexsheet. This method relies on an arbitrary parameter (thevortex sheet thickness) which can be difficult to computeand increases the calculation cost, since at each timestep the transformation of vortex elements (segment orblob) passing through the vortex sheet frontier must becomputed.


(d)

(e)


In other methods, that do not use the vortex sheet, twosteps are commonly used to obtain the velocity boundarycondition at the solid surface. One set of blobs is emittedto obtain the zero-normal velocity at the surface andthen another one is created to obtain the zero-tangentialvelocity at the surface. It seems hazardous to use sucha method without an appropriate iterative process sincethe creation of the second set of blobs must modify the

velocity field and then a normal velocity may appear atthe solid surface.

4.2. A one-step method

To avoid an iterative process or the use of the vortexsheet method that both induce time cost it is then of


(a)

(b)

Figure 3. Velocity vectors of passive particles,Re = 1000: (a)t∗ = 2.0; (b) t∗ = 6.0; (c) t∗ = 10.0; (d) t∗ = 12.0.

interest to try to find a method to obtain this boundarycondition in one step.

In our method, solid walls are discretised into wall-segments with a control point in the middle of eachone. In order to obtain the no-slip condition we use theassociation of one vortex-blob and one source-segmentfor each control point. The source is placed insidethe profile close to the wall-segment and parallel to it

whereas the vortex blob is outside of the profile, straightabove the control-point (figure 1). In that position, thevortex blob and the source-segment induce tangentialand normal velocities respectively at the control-point.This association enables a one step process to obtainthe velocity boundary condition on a solid wall andavoid the use of an iterative process or the calculation ofvortex elements passing through a vortex sheet frontier.


(c)

(d)


Then, solving a single linear system of 2N equations ateach time step gives directly the no-slip condition for anobstacle discretised intoN wall-segments.

Blobs are then advected, due to the velocity field andthe new intensity of each blob due to the diffusion iscalculated. Then, considering the resulting new velocityfield, a new set of vortices is emitted and new intensitiesfor the sources are calculated to keep the no-slip condi-tion.

5. Drag and lift coefficients

The force applied on the body by the fluid (with unitdensity) can be easily calculated in the case of a flowaround a non-rotating circular cylinder [6]. It is given by:

→F b = − d

dt

∫fluid+body

→ω ∧ →

x dσ.


(a) (b)

(c) (d)

Figure 4. Experimental visualisation of a flow around a circular cylinder,Re = 1000: (a)t∗ = 2.0; (b) t∗ = 6.0; (c) t∗ = 10.0; (d)t∗ = 12.0.

It is then easy to get the lift coefficientCz and the dragcoefficientCx .

6. Results

6.1. Vorticity and velocity fields

Cylinder has been chosen for this study in order tocompare calculation results to some experiments con-ducted at Université de Poitiers in the last decade [8].Simulation has been performed for cylinders discretisedinto 50 segments and for a Reynolds number of 1000 forthe first twenty dimensionless seconds of the flow. TheReynolds number is defined asRe = UL

ν, whereU is the

velocity of the incoming flow,L is the diameter of thecylinder andν is the kinematic velocity of the fluid.

Concerning the velocity field and the vorticity field, agood accordance has been found between experimentaland numerical results. The formation of two symmetri-cal vortices is observed (figures 2(a) and 3(a) ) as in theexperiment (figure 4(a) ), creating a wake of about onediameter up to 3.0< t < 4.0. Then, dissymmetry gradu-

ally takes place (figures 2(b), 3(b) and4(b) ), leading tothe beginning of the periodic regime (figures 2(c), 2(d),3(c), 3(d), 4(c), 4(d) ).

Observation of the trailing edge of the cylinder alsoshows an evolution in accordance with the literature( figure 5 ). First, two primary vortex zones are createddue to the impulsive start . Then contra-rotating layersappear between these vortex zones and the surface ofthe body, develop and make primary zones to separatefrom the surface. At the same time a third set of vortexzones appear, develop and join the primary vortex zone,making it grow until separation. This process graduallyleads to the creation of the Von Kármán vortex street( figure 2(e) ).

6.2. Evolution of lift and drag

Seefigure 6. This evolution can be divided into twoparts. The first is fromt = 0 to t = 10. Here thedrag obtained by calculation is characteristic of the flowunder study. Initially,Cx is large, due to the impulsivestart. Then the drag quickly decreases untilt = 1.0and then tends slowly towardsCx = 1.0, which is a


Figure 5. Velocity vectors of passive particles,t∗ = 2.0, 3.0, 4.0 and 5.0,Re = 1000. Zoom on the trailing edge of the cylinder.

Figure 6. Evolution of dragCx and liftCz for 0.0< t∗ < 20.0 for a flow around a circular cylinder,Re = 1000.


characteristic value for a flow past a circular cylinderat Re ≈ 1000 [10]. As it can be expected from a flowaround a circular cylinder, the mean lift is zero in thefirst period of this part. It then increases slowly and dropssuddenly aroundt = 10, when the Von Kármán vortexstreet takes place. Beyond this point the drag is no longersteady and the lift oscillates widely and regularly. Thefrequency of this oscillations leads toSt ≈ 0.22 which isa characteristic value of the Strouhal number for a flowaround a circular cylinder withRe = 1000 [11].

7. Conclusion

In this paper a particle method, based on a verysimple model for the no-slip condition, coupled with awell known deterministic diffusion model, is introducedto simulate a viscous flow past a circular cylinder.Results are in good correlation with both literature andexperiments. Qualitatively, the vorticity field and thevelocity field are in good accordance with experimentaldata. Moreover, concerning quantitative results, the bodyforces evolution and the Strouhal number arising fromit correlate well with literature. The one step methodused here to obtain the no-slip condition then provedcorrect. Since it is very simple and easily adaptable tomore complicated geometries, this method should beof significant interest for particle simulations of flowsaround obstacles such as wings with flaps.

Now that this method is proven to give qualitatively aswell as quantitatively good results for the first 20 secondsof the flow, future work will include an evaluation of theadvantage concerning time cost with using this method.Then, the following step will be to try to extend thismethod to 3d simulations.

Acknowledgements

The authors would like to thank M. Gérard Pineau forthe experimental data and for his precious contribution.

References

[1] Beale J.T., Majda A.J., Rates of convergence for viscoussplitting for the Navier–Stokes equations, Math. Comput.37 (1981) 243–259.

[2] Choquin J.P., Huberson S., Particles simulation of viscousflow, Comput. Fluids 17 (2) (1989) 97–410.

[3] Chorin A.J., Numerical study of slightly viscous flow,J. Fluid Mech. 57 (4) (1973) 785–796.

[4] Cottet G.H., Gallic S., Rapport interne 115, CMAP, ÉcolePolytechnique, 1986.

[5] Ghoniem A.F., Gagnon Y., Vortex simulation of laminarrecirculating flow, J. Comput. Phys. 68 (2) (1987).

[6] Koumoutsakos P., Leaonard A., High-resolution simula-tion of the flow around an impulsively started cylinder us-ing vortex methods, J. Fluid Mech. 26 (1995) 1–38.

[7] Leonard A., Vortex methods for flow simulation, J. Com-put. Phys. 37 (1980) 289–335.

[8] Pineau G., Mise en évidence et évaluation des effetstridimensionnels dans l’écoulement instationnaire autourd’un cylindre circulaire d’envergure finie, PhD thesisUniversity of Poitiers, 1992.

[9] Puckett E.G., Vortex methods: a survey of selected re-search topics, in: Nicolaides R.A., Gunzburger M.D.(Eds.), Incompressible Computational Fluid Dynamics –Trends and Advances, Cambridge University Press, 1991.

[10] Roberson J.A., Crowe C.T., Engineering Fluid Mechan-ics, 6th edition, Wiley, 1997, pp. 429–436.

[11] Roberson J.A., Crowe C.T., Engineering Fluid Mechan-ics, 6th edition, Wiley, 1997, pp. 436–437.

[12] Sarpkaya T., Computational methods with vortices – The1988 Freeman scholar lecture, J. Fluid. Eng.-T. ASME111 (1989).

[13] Strang G., Accurate partial difference method II. Nonlin-ear problems, Numer. Math. 6 (1964) 37–49.

[14] Strang G., On the construction and comparison of differ-ence schemes, SIAM J. Numer. Anal. 5 (1968) 506–517.

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Aerodynamic 1