2000 [e.magere] simulation of the taylor-couette flow in a finite geometry by spectral element...
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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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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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(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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246 E. Magre, M.O. Deville / Applied Numerical Mathematics 33 (2000) 241249
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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E. Magre, M.O. Deville / Applied Numerical Mathematics 33 (2000) 241249 247
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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248 E. Magre, M.O. Deville / Applied Numerical Mathematics 33 (2000) 241249
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
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(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,
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[5] R.E. Lynch, J.R. Rice, D.H. Thomas, Direct solution of partial difference equations by tensor product methods,
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[6] E.M. Rnquist, Spectral element methods for the unsteady NavierStokes equations, Von Karman Institute
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