Lamb Oseen Vortex

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COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2007; 00 :10 Prepared using cnmauth.cls [Version: 2002/09/18 v1.02]

A Stochastic Lamb-Oseen Vortex Solution of the 2DNavier-Stokes Equations

J. L. Sereno, J. M. C. Pereira and J. C. F. Pereira

SUMMARY

The exact solution of the Lamb-Oseen vortices are reported for a random viscosity characterized bya Gamma probability density function.This benchmark solution allowed to quantify the analitic errorof the polynomial chaos expansion as a function of the number of stochastic modes considered andcompare it with its numerical counterpart. The last was obtained in the framework of the polynomialchaos expansion method together with a nite difference numerical discretization of the resultingsystem of Navier-Stokes equations for the expansion modes considered. The obtained solution may beused to test other numerical approaches for the solution of the Navier-Stokes equations with randominputs. Copyright c 2007 John Wiley & Sons, Ltd.

key words: Vortex,Lamb-Oseen, Stochastic, Polynomial Chaos

1. INTRODUCTION

Vortex dynamics is a dominant phenomenon in many science elds and engineeringapplications. Among them and with aeronautical interest, the aircraft trailing vortices havereceived particular attention, see e.g. Saffman [8], Gertz [10]. For such real wake vortex, andalso for other systems, there are a great deal of uncertainty sources, that may affect theirformation and development.Uncertainty quantication allows an insight into the scope of a physical system responseproviding an ensemble of solutions associated to a certain probability of occurrence. Amethod that has been recently applied for its effectiveness for short time integration forthe calculation of stochastic PDEs (Partial Differential Equations) is based on the spectralrepresentation of random variables and has been called the Polynomial Chaos ExpansionMethod. Norbert Wiener [1] proposed a spectral representation of general random variablesbased on Hermite orthogonal polynomials of Gaussian random variables. The span of theseorthogonal polynomials forms a complete basis for the L2 space and has been called aPolynomial Chaos (PC). The Polynomial Chaos representation of general random variables isconvergent (in the L2 sense), for the Gaussian measure, provided that the random variable is of

second order, see Cameron & Martin [2]. This method was rst applied in the context of partial

Correspondence to: Instituto Superior Tcnico, Mech. Eng. Dept., LASEF , Av. Rovisco Pais, 1049-001, Lisbon,Portugal

Contract/grant sponsor: This work was partially suported by FAR-Wake project from the EC under ContractAST4-CT-2005-012238 and by FCT project POCTI/EME/47012/2002.

Received DateCopyright c 2007 John Wiley & Sons, Ltd. Revised Date

Accepted Date


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2 J. L. SERENO ET AL.

differential equations in Ghanem & Spanos [3]. Later development of chaos decompositions

based on non-Gaussian basic variables was made in Xiu & Karniadakis [4] which has beencalled the Generalized Polynomial Chaos (gPC) and uses the orthogonal polynomials from theAskey family with weighting functions similar to probability density functions (Beta, Gamma,Binomial,. . . ). A formal exposition and a generalization of the theory to arbitrary probabilitymeasures was accomplished in Soize & Ghanem [5]. Local PC expansions, suited for long-termintegration and discontinuities in stochastic differential equations, where studied in Le Matreet al. [6] and in Wan & Karniadakis [7].In this work we investigate the Lamb-Oseen vortices subjected to a random viscosity (Reynoldsnumber) characterized by known PDFs. We present analytic solution and compare thepredicted Polynomial Chaos expansion convergence error and the numerical approximationerror.The Lamb-Oseen vortices correspond to exact solutions of the Navier-Stokes equations (see,e.g., Prolo et al. [9]) and have been used extensively as benchmark solutions for developingnumerical methods and has initial or boundary conditions for various studies. Our interestis to predict their two-dimensional evolution subjected to a random viscosity (Reynoldsnumber) and to investigate the interplay of the spacial discretization error with the stochasticapproximation error. Furthermore, these solutions can be used to test new numerical methodsdesigned for the stochastic Navier-Stokes equations.

2. STOCHASTIC NAVIER-STOKES EQUATIONS

A random variable (r.v.) can be represented using the polynomial chaos (PC) expansion

X () =

n =0

an n ( ()) (1)

where the functions n ( ) are orthogonal polynomials of the basic r.v. dened in the inner-product space ( L2w ) with

u, v = uvwd (2)w is the weighting function (dened in the domain ) and is often similar to certain probabilitydensity functions (Gaussian, Beta, Binomial, . . . ). The parameter represents a random eventand will be dropped to simplify the notation. Orthogonal polynomials which have weightingfunctions that closely resemble PDFs can be found in the Askey family of hypergeometricpolynomials (Xiu and Karniadakis [4]). A general differential equation containing randomvariables can be represented by

(x , t ,, u ) = f (x , t , ) (3)

where x , t and represent the space coordinates, time and a random event. Substituting theexpansion (1) in the differential equation we obtain the following

(x , t ,,

n =0

u n n ) = f (x , t , )

Copyright c 2007 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2007; 00 :10Prepared using cnmauth.cls


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A STOCHASTIC LAMB-OSEEN VORTEX SOLUTION 3

The equation above must be projected into the space spanned by the polynomial basis dened

earlier in order to absorb the random variable and to minimize the representation error. Thisprocedure leads to

(x , t,P

n =0

u n n ), k = f (x , t ), k

where the rst P + 1 terms were retained. This means that instead of solving a system of differential equations with random variables, one has to solve a larger system of coupleddeterministic equations. The solution of equation (3) is obtained usually by a samplingprocedure, whereas the employed method leads to non-statistical method.Assuming the PC expansion of the primary variables for the Navier-Stokes equations (for anincompressible uid with constant properties)

u (x , t , ) =P

n =0

u n (x , t )n ( ()) (4)

p(x , t , ) =P

n =0

pn (x , t )n ( ()) (5)

and performing the projection into each element of the polynomial basis leads to the followingsystem of deterministic partial differential equations

u k = 0 (6)

u kt

+ 1k 2

P

i =0

P

j =0

eijk [u i] u j = 1 pk + 1k 2P

i =0

P

j =0

eijk i2

u j (7)

where

eijk = i j k w( )d (8)and it was assumed that the viscosity is a r.v. The term eijk in equation (7) is a sparse,constant and symmetric tensor and can be calculated a priori .Equation (6) corresponds to the continuity equation and forms a set of independent differentialequations, which means that all modes are divergence free. On the other hand, equations (7)form a coupled set. These equations are coupled by the convective term and the diffusive term.An important aspect of these equations is that they are equivalent for all the stochasticmodes, which means that all modes are subjected to convection, diffusion and dissipationphenomenons.The solution of the system (6) and (7) provides the elds u k (x , t ) and pk (x , t ) which may beused to calculate the approximate statistics (statistical moments, correlations and PDFs) of the ow.The variance can be calculated using the expansion (1) and the orthogonality property of the



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polynomials. This is equal to the following inner product

u(x,t, ) u0 (x, t ), u(x,t, ) u0 (x, t ) ==

P

i =1

u i (x,t, ) i ( )P

j =1

u j (x, t , ) j ( )w( )d

=P

k =1

u2k (x, t ) k2 (9)

where terms higher than P were neglected.The Monte Carlo method is obtained by choosing the weighting function w( ) = (i ), where is the Kronecker delta function and i represents an outcome from a sampling procedure.A detailed description of the Monte Carlo methods can be found, e.g., in Hammersley andHandscomb [11].

3. NUMERICAL METHOD

The 4 th Runge-Kutta order (RK4) explicit time integration method together with sixthorder spacial central differences was used to solve the system of Navier-Stokes equations forthe stochastic modes (see equations 6 and 7). The continuity equations (6) were used in aprojection method to calculate each mode pressure eld and to ensure that each velocitymode is divergence free. The Poisson equations for each mode were solved using the conjugategradient method and the implicit Laplace operator was approximated recurring to a secondorder scheme.The computational cost of the numerical solution of the coupled set of equations is, with this

method, approximately ( P + 1) times higher than the cost of the numerical solution of thecorresponding deterministic problem because the Poisson equations for the pressure correctionsare uncoupled.The approximation error of this system has two origins, one is due to the truncation of seriesand the second is the numerical discretization. If 2P is the variance approximation error, then

2P =

k =1

a2k (x, t ) k2

P

k =1

a2k (x, t ) k2 + x

=P

k =1

(a2k (x, t ) a2k (x, t )) k

2

I

+

k = P +1

a2k (x, t ) k2

II

+ x

II I (10)

In equation (10) the approximation error sources correspond to: term I is the error due to thenite mode approximation where the computed modes ak (x, t ) are different from the exactsolution modes ak (x, t ); term II is the error due to not considering stochastic modes higherthan P and term III is the numerical discretization error. These terms are not independentand it can occur that the magnitude of the terms higher than P is small but the term I islarge due to the coupling of the equations. These error sources will be refered as I , II andII I .



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4. LAMB-OSEEN VORTICES UNDER A RANDOM VISCOSITY

The Lamb-Oseen vortex under a random viscosity characterized by a Gamma PDF wasconsidered. This vortex is dened by the following equation in cylindrical polar coordinates

v = c2r

1 exp r 2

4t (11)

where c is the circulation and r is the radius. A non-dimensional velocity, time and lengthwill be used and an expected viscosity of 10 2 m 2 /s is considered so that a Reynolds numberof approximatly 100 was obtained. The reference velocity and time scale chosen are

(v )ref = c2r 0

(12)

t ref = r0(v )ref =

2r 20c (13)

where r0 was considered to be equal to 1. A random viscosity ( ) was considered and fromequation (11) it is possible to conclude that for r the solutions will become increasinglycloser to the potential ow solution, independently of the probability distribution considered.As could be expected, only the viscous vortex core is affected by the viscosity randomness.Four different PDFs of the Gamma family were considered for the parameters = 0 , 1, 2 and3. For = 0 the exponential PDF is recovered. The parameter is related with the relativeimportance of events near = 0 and the parameter inuences the probability decay farfrom the mean value. The value of the parameter was chosen so that the expected value of the viscosity corresponds to 0 = 102 m 2 /s . The coefficient of variation ( cv) for the GammaPDF, considering an integer , is given by:

cv( ) = 11 +

(14)

where the cv is the coefficient of variation, the quotient between the standard deviation andthe mean value. The decrease of the parameter will make the mean velocity solution departfrom the deterministic solution ( Fig. 1 and Fig. 3) due to the cv increase. It can be seen inthese gures that the mean velocity solution varies considerably with the shape of the viscosityPDF. In average, the vortices will have a longer lifetime than the predicted by the deterministicsolution.The variance is represented in Fig. 2 where it can be seen that the uncertainty is restrictedto the vortex core. For = 0 the variance peak is achieved at the vortex center, whereas inthe other cases this maximum is achieved near the mean velocity peak, inside the vortex core.The = 0 solution departs more from the deterministic case because in this case we havecv = 100%.The exact mean velocity solution for a general Gamma probability density function is givenby the following equation,

E v (r,,, )

(v )ref =

r0r

1 2 r

2

t

1+ 2

J k 1 , r t(1 + )

(15)



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0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

r *

v *

=0 =1 =2 =3 Determ.

4 6 8 10 12 14 16 18 20 22 240.2

0.3

0.4

0.5

0.6

0.7

0.8

t *

v *

Determ.

0

1 2

3

Figure 1. Stochastic solution for the Gamma viscosity. Left: Mean velocity solutions for t = 25 / 2 .Right: Mean velocity decay.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

r *

( v

* )

=0 =1 =2 =3

Figure 2. Variance of the stochastic solution for the Gamma viscosity for t = 25 / 2 .

and the non-centered velocity variance will be

E v (r,,, )

(v )ref

2

= 2

r 2 (1 + )2 (1 + )

2J k

1

,

r

tr 2

t

1+ 2

+21+

2 J k 1 , 2r t

r 2

t

1+ 2

(16)

where ( x) is the Euler gamma function an J k (, x ) is the modied Bessel function of thesecond kind.



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4 6 8 10 12 14 16 18 20 22 240.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c

/

0

t *

Determ.

0

12

3

5 10 15 20 25 30

0.5

1

1.5

2

2.5

3

t *

r * c

Determ.

0

1 2

3

Figure 3. Stochastic solution for the Gamma viscosity. Left: Vortex core circulation. Right: Vortexcore growth.

4.1. NUMERICAL RESULTS

The computational domain was chosen so that the viscous effects and its uncertainty werenegligible at the boundaries location. Therefore, all Polynomial Chaos expansion modes canbe set to zero at the boundaries except the rst ( P = 0), the mean solution is equal to thedeterministic solution because the ow is irrotational in that region.The mean velocity prole using several computational meshes and a Polynomial Chaosexpansion with P = 1 is represented in Fig. 4. Accurate results were obtained even for theminimal approximation level for the mean velocity solution, provided that a sufficient number

of grid points were used. On the other hand, a good aproximation of the velocity variance wasmore dicult to obtain as can also be seen in Fig. 4 were the results for Polynomial Chaosexpansions up to P = 5 are represented. The difference to the exact solution represented in thisgure is related to the errors II and II I discribed in section 3. The variance approximationerror components are clearly evidenced in Fig. 5. The velocity variance computation can beimproved by considering ner grids but this is not sufficient because the overall numericalerror reduction is also affected by the approximation level in the random variable. The exactsolution for the U 1 expansion coefficient was calculated and is also represented in Fig. 5. It canbe seen that despite of the good numerical resolution considered the solution obtained witha nite mode approximation is far from the exact rst mode U 1 solution. This large varianceapproximation error is related to the II error source.The predicted and numerical errors are represented in Fig. 5. The numerical error obtainedpresented a similar behaviour to the predicted Polynomial Chaos expansion error. Nevertheless,a saturation effect was found due to the stagnation of the convergence related to the interplayof the I and II with the numerical discretization error component II I . When coarse grids areused the II I error component becomes predominant causing the convergence rate to decrease.In Fig. 6 the Polynomial Chaos expansion method and the Monte Carlo method are comparedusing the assumption that the computational cost of the solution of the system of PDEs forthe stochastic modes is proportional to the corresponding deterministic problem, as discussedin section 2. The Monte Carlo solutions were obtained by sampling equation 11 and the



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5 4 3 2 1 0 1 2 3 4 5

0.25

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

0.25

x/r 0

U 0

Grid 8x8

Grid 16x16

Grid 32x32

Exact Solution

5 4 3 2 1 0 1 2 3 4 50

0.002

0.004

0.006

0.008

0.01

0.012

V a r i a n c e

x/r 0

P=1

P=2 P=3 P=4 P=5 Exact solution

Figure 4. Stochastic numerical solution for a Gamma viscosity with = 3. Left: Mean velocity proleobtined with different spacial discretizations. Right: Velocity variance obtained using a grid with

64 64 points.

same, error of numerical variance, numerical error measure was used. The Polynomial Chaosexpansion method was by far the most effective in capturing both the velocity mean andvariance solution, being 10 3 faster than its counterpart.For problems with low or moderate dimension very large improvements in the computationalcost can be made relative to the cost of Monte Carlo. Adding a new equation (sample), whenusing the Monte Carlo method, has little inuence on the statistics of the solution becausethe number of samples is usually large. If we consider the mean solution, for example, theefect of the new sample will be scaled by 1 /N , being N the overall number of samples. Onthe other hand, when using the Polynomial Chaos method, an additional sotchastic mode willmake it possible to represent a larger set of stochastic processes. For example, in the case of the Gaussian PDF and Hermite polynomials, a two mode approximation can only representfairly a symmetric PDF with similar tails to the Gaussian PDF. Adding another stochasticmode will make it possible to represent PDFs that are not symmetric, which corresponds toa qualitative improvement to the approximation of the random process.

4.2. CONCLUSIONS

The Lamb-Oseen vortex solution for the stochastic Navier-Stokes equations under a randomviscosity eld was presented and its approximation by the Polynomial Chaos expansion methodwas studied. The analytic solutions were compared with several numerical approximations andthe different error sources were described and quantied.The results show that good approximations of the mean velocity solution can be obtainedeven when the minimal stochastic approximation with P = 1 is considered if enough gridpoints are used. Accurate approximations for the velocity variance require additional stochasticmodes, increasing the computational effort. The variance error was divided into three differentdependent components: the I which is related to the nite mode approximation with P + 1expansion terms, the II error that accounts for the truncated terms of the Polynomial Chaos



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5 4 3 2 1 0 1 2 3 4 50

0.002

0.004

0.006

0.008

0.01

0.012

x/r 0

V a r i a n c e

Grid 8x8 Grid 16x16 Grid 256x256 Exact P=1 Aprox.Exact Solution

1 1.5 2 2.5 3 3.5 4

104

103

P

E r r o r

Predicted

16x16

32x32

64x64

128x128

Figure 5. Left: Comparison between different velocity variance computations. Right: Numericalapproximation || e|| error for the velocity variance.

100

101

102

103

104

106

10 5

104

103

102

101

Number of samples / Equations

E r r o r

Monte Carlo

Polynomial Chaos Expanstion

Figure 6. Comparison between the Polynomial Chaos expansion method and the Monte Carlo method.

expansion and the II I which is related to the numerical discretization error. If the grid iskept the same, incresing the number of Polynomial Chaos expansion terms rapidly led to asaturation of the error decrease.In general, the effect of incresing the number of stochasticmodes is more effective in reducing the approximation error than to double the number of grid

points.The set of solutions presented can be used as a benchmark for validation and error evaluationof stochastic Navier-Stokes equations solvers.



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REFERENCES

1. N. Wiener, The Homogeneous Chaos, Amer. J. Math. 60 (1938) 8979362. R. Cameron, W. Martin, The orthogonal development of non-linear functionals in series of Fourier-Hermite

functionals, Ann. of Math. 48 :2 (1947) 3853923. R. G. Ghanem and P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag. (1991)4. D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations,

SIAM, J. Sci. Comput. 24 :2 (2002) 6196445. C. Soize and R. G. Ghanem, Physical systems with random uncertainties: chaos representation with

arbitrary probability measure, SIAM, J. Sci. Comput. 26 :2 (2004) 3954106. O. P. Le Matre, H. N. Najm, R. G. Ghanem and O. M. Knio, Multi-resolution analysis of Wiener-type

uncertainty propagation schemes, J. Comput. Physics 197 (2004) 5025317. X. Wan and G. Karniadakis, An adaptive multi-element generalized polynomial chaos method for

stochastic differential equations, J. Comput. Physics 209 :2 (2005) 6176428. D.I. Pullin and P.G. Saffman, Vortex dynamics in turbulence, Ann. Rev. Fluid Mechanics , 30 (1998)

31519. G. Prolo, G. Soliani and C. Tebaldi, Some exact solutions of the two-dimensional Navier-Stokes equations,

Int. Jour. Engng. Sci. , 36 :4 (1998) 45947110. T. Gerz, F. Holzapfel and D. Darracq, Commercial aircraft wake vortices, Progress in Aerospace Sciences ,38 :3 (2002) 18120811. J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods, Methuens Monographs on Applied

Probability and Statistics, Fletcher & Son Ldt, Norwich , 1964


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Lamb Oseen Vortex