2000 [E.magere] Simulation of the Taylor-Couette Flow in a Finite Geometry by Spectral Element...

7/26/2019 2000 [E.magere] Simulation of the Taylor-Couette Flow in a Finite Geometry by Spectral Element Method

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Applied Numerical Mathematics 33 (2000) 241249

Simulation of the TaylorCouette flow in a finite geometry byspectral element method

E. Magre , M.O. Deville 1

EPFL, LMF/IMHEF/DGM, CH-1015 Lausanne, Switzerland

Abstract

The purpose of the present numerical method is to simulate the flow in between two concentric cylinders of finite

axial length. The numerical integration of the transient three-dimensional incompressible NavierStokes equations

is performed by a Legendre spectral element method in space while the time marching scheme is fully explicit. Bytaking full advantage of the tensor product bases, it is possible to set up fast diagonalization techniques in order to

speed up the solver. Numerical results are compared favorably with experimental data. 2000 IMACS. Published

by Elsevier Science B.V. All rights reserved.

Keywords:Spectral element method; Direct numerical simulation; Flow transition

1. Introduction

The TaylorCouette flow occurring in between two coaxial differentially rotating cylinders is one of

the fundamental problem in fluid mechanics. When only the inner cylinder is rotating and the outer one

is fixed, a transition occurs from the laminar Couette flow to a periodic superposition of axisymmetricvortices, called the Taylor vortex flow, as the angular velocity is increased. Increasing further the velocity

of the inner cylinder, the Taylor vortices are subject to a second transition and produce wavy vortices.

However, when the cylinders are counter-rotating, the physical situation is less clear. Some experimentalresults are available [2], but further investigations need to be carried out. This paper aims to design a

spectral element algorithm in order to handle this case.

The paper is organized as follows. Section 2 describes the weak formulation of the NavierStokes (NS)

equations. Section 3 presents the spatial discretization, while Section 4 treats the time marching scheme.

In Section 5, the boundary conditions are set up to avoid the singularity between the inner cylinder andthe top and bottom plates. The results are given in Section 6.

Corresponding author.1 E-mail: [email protected]

0168-9274/00/$20.00 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.PII: S0168- 9274( 99)00 089- 6


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242 E. Magre, M.O. Deville / Applied Numerical Mathematics 33 (2000) 241249

2. Weak formulation of the basic equations

The governing equations are the NS equations. The momentum and continuity equations are madedimensionless by introducing characteristic variables, the gap separating the cylinders, d, the viscoustimed2/, where is the kinematic viscosity, the velocity of the outer cylinder of radiusro and angular

velocityo, V = roo, and the pressure V/d, being the dynamic viscosity. The radius of the inner

cylinder is ri and its angular velocity is i. Writing the basic equations in vector form, and taking theboundary and initial conditions into account, one is left with:

v

t+Re nl(v) = p + 2v (r, t) D [0, T],

v = 0 (r, t) D [0, T],

v = vb (r, t) D]0, T],

v = vi (r, t) D {0}.

(1)

The notation nl refers to the nonlinear term. The cylindrical coordinates, r =(r,,z), are defined in

D =]ri, ro[]0, 2 []0, L[. The symbolRe = dV/ denotes the Reynolds number.Introducing the scalar product of unit weight: f, g =

D f g, we resort to a weak formulation of

Eq. (1). We search the velocity solution of these equations in the Sobolev space [H1(D)]3 = {v L2(D), v/r L2(D)}3 and the pressure in the Lebesgue space with zero mean, L20(D) = {

p

L2(D),

D p = 0}.

v

t+ p s(v),v

= 0 v

H1(D)

3,

v, p

= 0 p L2(D),

(2)

where s(v) = Re nl(v) + 2v, and v andp are high order polynomial test functions.

3. Spatial discretization

The azimuthal direction being periodic, we first approximate the velocity v and the pressure p byFourier series:

v(r, t) =

N/21k=N/2

vk(r,z,t)eik, p(r, t) =

N/21k=N/2

pk (r,z,t)eik.

The test functions

v

and p

are expressed as products of Fourier bases by continuous functionsof[H1()]3 andL2(), respectively, where =]ri, ro[]0, L[. We obtain a two-dimensional problemfor each of the N Fourier modes. is further decomposed in E rectangular elements: =

Ee=1

e .

Lagrange interpolation on the GaussLobattoLegendre (GLL) points is applied to the Fourier modes:

vk (r,z,t)|e =

Nri=0

Nzj=0

veijk (t)

eij(r, z),

where eij(r(x),z(y))= hi (x)hj(y). (r(x),z(y)) represents the affine mapping from ]1, 1[]1, 1[

to e . The basis hi(respectivelyhj) is the Lagrange interpolant of degreeNr (respectivelyNz) associatedto thei th (respectively jth) GLL point of]1, 1[. In order to avoid spurious pressure modes we use the


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E. Magre, M.O. Deville / Applied Numerical Mathematics 33 (2000) 241249 243

(PN/PN2)formulation, but instead of taking the pressure at the GaussLegendre points and the velocityat the GLL points, we apply the modified version of Azaez et al. [3] which consists in using the inner

GLL points for the pressure. Each pressure Fourier mode is then expanded as follows:

pk (r,z,t)|e =

Nr 1i=1

Nz1j=1

peijk (t)eij(r, z),

where eij(r(x),z(y))=hi (x)hj(y) andhi (hj) is the Lagrange interpolant of degree Nr 2 (Nz 2)

associated to thei th (jth) inner GLL point.

We use a unique discrete inner product defined at the GLL points of each element:

f, gd=

E

e=1Nr

i=0Nz

j=0feijg

eijr

eii j,

wherer ei = r(xi ),r belonging toe andi andjrepresent the GLL weights. The pressure is therefore

extrapolated on the interfaces of the elements to obtain the pressure gradient on the velocity grid. Thediscrete form of Eq. (2) is

vk

t+ pk sk (v),

d

= 0 (Xd)3,

vk , d= 0 Yd.

(3)

In the previous equations, Xd=H1() PN,E (),Yd= L

2() PN2,E (), PN,E () = {, e {1, . . . , E}, |e PN(

e), |e = 0}. PN(e)is the space of polynomials of degree Nr in the radial

direction andNzin the axial direction defined on e . The trial functions are chosen to be the same as the

test functions.We introduce the following matrices locally to each e . The mass matrices are defined as

rBr

ij= jr(xj)ij

ge

2,

Br

r

ij

=1

r(xj)jij

ge

2 and Bzij= jij

he

2,

whereg e is the width of e andhe, its height. The first-order derivative matrices are

drij=hj

x(xi ), d

zij=

hj

y(yi ).

The second-order derivative matrices:

Arij= 2

ge

Nrk=0

r(xk )rk d

rk,i d

rk,j, A

zij=

2

he

Nzk=0

zk dzk,i d

zk,j.

The mass matrices incorporating the extrapolation from the pressure grid to the velocity grid arerBr

ij= i r(xi )hj(xi ) ge

2, Br ij= ihj(xi ) ge

2, Bzij= ihj(yi ) he

2 .

The first-order derivative matrices going from the pressure space to the velocity space are

Gr =

2

gedr T

r

Br

Br ,

Gz =

2

hedzT

Bz.


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We perform the direct stiffness [6] on these matrices. We use calligraphic letters for the resulting globalmatrices. The NS equations become:

(rBr ) Bz dukdt

+ Gr Bzpk (rBr ) Bzsrk = 0,(rBr ) Bz

dvk

dt+ ikBr Bzpk (rBr ) Bzsk = 0,

(rBr ) Bzdwk

dt+ (rBr ) Gzpk (rBr ) Bzszk = 0,

Gr T BzTuk+ ikBr T BzTvk (rBr )T Gz Twk = 0,(4)

wheredenotes the tensor product of matrices. The velocity components uk , vk andwk are defined on

the grid dmade of all the GLL points in each of the elements. The pressure field pk is defined on thegriddmade of the inner GLL of all the elements. We now defineG=

Gr BzikBr BzrBr Gz

and B= rBr BzrBr Bz

rBr Bz

.In compact matrix form, the discrete equations are nowB

dvk

dt+Gpk Bsk = 0 in d,GTVk = 0 ind. (5)

In (5), vk= (uk , vk , wk)are the three components of the velocity defined in

d, the finite set that contains

the grid points ofd, the boundary points excluded. The notation Vk represents the three components ofthe velocity defined ind.

4. Time discretization

A fully explicit treatment of the source term, sk (V)= Re nlk(V) + B1Avk , is chosen instead of

the classical implicit linear viscous/explicit non linear decomposition. Here, A denotes the Laplacianoperator. In order to be able to carry out long term integration over large time scales, the skew-symmetric

form of the nonlinear term is chosen. In the general case of projection methods, a Helmholtz operator for

the velocity has to be inverted. In cylindrical coordinates, this operator cannot be expressed in a separateform. Hence, the fast diagonalization technique [4,5] cannot be applied to invert this matrix. An iterative

method is then needed. For a three-dimensional problem with one periodic direction, each matrix

vector multiplication involved in the algorithm requires 2N4 operations (1 addition and 1 multiplicationcounts for one operation), with N3, the total number of grid points. Each of the iteration steps can be

achieved with the order of 10 matrixvector multiplications. In the case of the totally explicit choice,the pseudo-Laplacian operator for the pressure can be inverted with the fast diagonalization method

and requires therefore 4N4 operations. The solver in itself is more efficient in the fully explicit case

than in the implicit/explicit one. However, the time step requirements are more stringent for the former.

Indeed, when an explicit treatment of the discrete NS equations is undertaken, two stability constraints


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have to be enforced on the time step t. The first condition is imposed on the diffusive part of theoperator

tCSDRer2min + (r)2min + z2min1, CSD < 0.4, (6)while the second condition comes from the nonlinear term (CFL condition)

tCSN

umax

rmin+

vmax

rmin+

wmax

zmin

1, CSN < 0.2. (7)

Usually, rmin = zmin and, in the case of the TaylorCouette geometry, rmin (r)min.Moreover, we have for a Legendre GLL grid: rmin 1/N

2. Hence, the viscous constraint (6) can bewritten as:tmax Re/N

4. Let us assume umax wmax 1, then we obtain from (7)tmax 1/N2. For

a polynomial degree, N 10, and a Reynolds number, Re 100, the diffusion constraint is of the sameorder as the CFL constraint. Consequently, an explicit treatment of the linear viscous term is possible,for the Reynolds range we are interested in: between 100 and 1000. In practice, at a Reynolds number

of 100, the viscous constraint imposes a time step about three times smaller.Dropping the indices k of the Fourier modes, the semi-discrete NS equations give rise to the set of

ordinary differential equations:dv

dt= B1 Gp + s(V) in d,GTV = 0 ind. (8)

We have chosen a second-order RungeKutta (RK2) scheme for the time discretization. It reads asfollows:

v

n+1 = vn + t

B1

Gpn+1/2 + sn+1/2

,

GTVn+1 = 0, (9)with

pn+1/2 = p

tn +

t

2

, Vn+1/2 = V

tn +

t

2

and sn+1/2 = s

V

n+1/2

.

Using the fractional step method, we first obtain Vn+1/2 from Vn and then Vn+1 from Vn+1/2. The firststep can be decomposed in three stages: a prediction, where we obtain v = vn + (t/2)sn which is not

divergence-free, a projection on the divergence-free velocity space, where we get the pressure

pn =GTB1

G1 2

t

GT

v

vn+1/2b

, (10)

and, finally, a correction step, where we find the divergence-free velocity

vn+1/2 = v

t

2 B1 Gpn.

So that,

Vn+1/2 =

v

n+1/2

vn+1/2b

.

The second and last step is also performed in three successive stages. We first obtain: v = vn +ts(Vn+1/2), then:

pn+1/2 =

GTB1

G

1

1

t GT

v

vn+1b

, (11)


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Fig. 1.i = 1,o= 2,ri= 7,d= 1, = 0.05d.

and, finally, we find: vn+1 = v tB1 Gpn+1/2. So that,V

n+1 =

v

n+1

vn+1b

.

The projection equation is obtained from the combination of the momentum equation in

dwith the

boundary conditions ond

d.v

n+1 = v tB1 Gpn+1/2 in d,v

n+1 = vn+1b ond

d.(12)

We then apply the divergence operator,GT, to this equation to recover (11).5. Boundary conditions

We impose Dirichlet boundary conditions on the velocity. The rotation velocity of the inner cylinder,i, is different from the rotation velocity common to the outer cylinder and the end plates, o, resultingin a singularity at(r,z) = (ri, 0)and (r, z) = (ri, L). We replace the rotation velocity of the end plates,o, by (r)on [ri, ri + ], d, where(r) = C (r)fi(r) + ofo(r), withfi and fotwo C

1 functionsdefined so that evolves smoothly from C to o on [ri, ri +] (see Fig. 1), C being the rotationvelocity of the circular Couette flow defined between the radii riandri + :

C(r) = a +br2

, a= (ri + )2o r2i i

(ri + )2 r2i

and b = (i o)r2i(ri + )2

(ri + )2 r2i

.

We impose:

(ri) = C(ri) = i,d

dr(ri) =

dC

dr(ri), (ri + ) = o and

d

dr(ri + ) = 0.

We choose:

fi,o = Ai,o sin

2

ri + r

+ Bi,o cos

2

ri + r

.

The four constraints determine the constants Ai,oandBi,o.


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Fig. 2. Stream traces in the meridian plane. Re = 165. = L/d= 1.281.

Fig. 3. experiments of Aitta et al. simulations of Streett et al. ours.

6. Results

To validate our code, we have reproduced the simulations of Streett et al. [7] in a short aspect ratio

TaylorCouette geometry (see Figs. 2 and 3). Our simulations compare well to theirs, and also to theexperiments of Aitta et al. [1]. We observe a sub-critical transition from a symmetric to an asymmetric

flow. They define an asymmetry parameter,

=

L0 w(r, z) dzL

0 |w(r, z)| dz,

to track this transition.

In Fig. 4, we also show the beginning of the instability in the counter-rotating case. The aspect ratio

is = 12, The inner Reynolds number, based on the velocity of the inner cylinder, is 400 and theouter Reynolds number, based on the outer cylinder velocity, is 612. The polynomial degree we use


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Fig. 4. Stream traces and azimuthal velocity contour in the meridian plane.

is Nr =Nz =12, the number of elements is E =3 26. The number of Fourier modes is 32. The

instability starts at mid-height and propagates towards the end plates. The region of instability, as givenby the non-viscous theory, is confined near the inner cylinder. The flow forms spirals after a sufficienttime.

7. Conclusions

In this paper, we have detailed a numerical algorithm based on Fourier-spectral element discretizationin space. The velocity nodes are the GLL nodes while the pressure nodes are the inner GLL points. This

formulation avoids the spurious pressure modes. The explicit time discretization relies on a two-stage

RungeKutta scheme. In order to speed up the pressure calculation, a fast diagonalization method isdevised.

The numerical method is well adapted to the problem at hand. The algorithm is efficient, the CPU time

spent on a CRAY J90 per time step, per node is 1.3 105 s. The physics is resolved correctly in spaceand time. A good agreement is found between numerical results and experimental data.

References

[1] A. Aitta, G. Ahlers, D.S. Cannel, Tricritical phenomena in rotating CouetteTaylor flow, Phys. Rev. Lett. 54

(1985) 673676.[2] C.D. Andereck, S.S. Liu, H.L. Swinney, Flow regimes in a circular Couette system with independently rotating

cylinders, J. Fluid Mech. 164 (1986) 155183.[3] M. Azaez, A. Fikri, G. Labrosse, A unique grid spectral solver of thenD Cartesian unsteady Stokes system.

Illustrative numerical results, Finite Elements in Analysis and Design 16 (1994) 247260.[4] D.B. Haidvogel, T. Zang, The accurate solution of Poissons equation by expansion in Chebyshev polynomials,

J. Comput. Phys. 30 (1979) 167180.


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[5] R.E. Lynch, J.R. Rice, D.H. Thomas, Direct solution of partial difference equations by tensor product methods,

Numer. Math. 6 (1964) 185199.

[6] E.M. Rnquist, Spectral element methods for the unsteady NavierStokes equations, Von Karman Institute

Lecture Series, 1991.[7] C.L. Streett, M.Y. Hussaini, A numerical simulation of finite-length TaylorCouette flow, Comput. Fluid

Dynamics (1988) 663675.

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