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The spectra of a plane Poiseuille flow and the problem of solving the Navier-Stokes equations for large Reynolds numbers
Short communications
This implies the inequality
.
us G aoexp
LJ
(min x) as’
-
0
247
To
I( r,
-‘I
1
’
which in the case of a finite temperature variation in the range considered imposes a constraint on the growth of the solutions of the homogeneous system (2) as a function of s. Here us is the value of u at the point s. For a monotonically decreasing function T of s the estimate is obtained similarly. In the case of an arbitrary function T(s) this treatment can be carried out on each segment of monotonicity. Translated by J. Berry. REFERENCES
Zh. @hid. Mat. mat.
1.
SHMYGLEVSKII, Yu. D. Calculation of radiative transfer by Gal&kin’s method Fiz., 13, 2, 398-407, 1973.
2.
KRIVTSOV, V. M., NAUMOVA, I. N., SHMYGLEVSKII, Yu. D. and SHULISHNINA, N. P. Check of two methods of calculating radiative transfer. Zh. ujkhisl. Mat. mat. Fiz., 15, 1, 163-171, 1975.
U.SS.R Comput. Maths Math. Phys. Vol. 18,pp. 247-253 0 Pergamon Press Ltd. 1978. Printed in Great Britain.
0041-5553/78/0101-0247$07
50/O
THE SPECTRA OF A PLANE POISEUILLE FLOW AND THE PROBLEM OF SOLVING THE NAVIER-STOKES EQUATIONS FOR LARGE REYNOLDS NUMBERS* Yu. L. LEVITAN, B. D. MOISEENKO, B. L. ROZHDESTVENSKII and V. K. SIDOROVA Moscow (Received 22 April 1976)
OVER a wide range of wave numbers (Ythe eigenvalues A,(o) and the eigen-functions of the boundary value problem for the Orr-Sommerfeld equation are calculated for supercritical Reynolds numbers Re = l/v. The observed features of the spectra demonstrate the use of these eigen-functions as a basis in the Bubnov-Cal&kin method of solving the non-linear problem of the non-stationary flow of a viscous incompressible fhrid between two parallel plates at supercritical Reynolds numbers. 1. In the interval [0,60] of wave numbers 01for the Reynolds numbers Re=i/v=i(T, we have calculated the eigenvalues h=h,(a) (Be h,,+i (cc)GRe X,(a)) and the of the boundary value problem for the Orr-Sommerfeld equation eigen-functions Ip=gn(a, y) 2.104, 5.5.102
lp(M)
=9’(d)
(1)
=o.
We have calculated the 20-20 leading symmetric and the same number of antisymmetric modes of problem (1). The calculations were performed by Galerkin’s method using the basis I (n/2)yl (p,,(y)=i-_(-I)” cos nny, n=l, 2, . . . , in the symmetric case, and %-sin functions -(--1)” sin [ (2n+l) (n/2) g), n=l, 2, . . . , in the antisymmetric case. *Zh. vj%hisl.Mat. mat. Fiz., 18,1,252-258,
1978.
Yu. L. Levitan et al.
248
The calculations carried out demonstrated the high accuracy, reliability and efficiency of the method over a wide range of wave numbers a, unhke some other methods of solving the boundary value problem (1) for example, that explained in [l] , which is very inefficient for values of aM. 2. The solution solution -
A,(a), +,(a, y) $(4x,
of the boundary value problem (1) corresponds to the
~)=~exp
IL(a)t+~l44a,
U)
(2)
of the Navier-Stokes equations linearized for Poiseuille flow. Thus, the quantity &,(a) describes the behaviour in time of small perturbations of a Poiseuille flow of a certain type. Figures l-4 show the functions Re k,(a), describing the decay in time (for A&(a) ~0 ) of perturbations of the type (2). Note that, in attempting to represent as much information about Re L(a), as possible, we have used variable scales, which explains the kinks in the curves in these graphs.
FIG. 1 The symmetric case The quantity Im h,,(a) is characteristic of the ripple frequency of a perturbation of the type (2), and if we write Im L(a) --a(l-sr,2(a)), then y,(a) described the place in the channel -IQ&~, where the vortex of the perturbation ~,,(a, v)- a21p,,(a, v) -$,,“(a, I/) is mainly concentrated. This is confirmed by graphs of the functions Re *l(a, Y) (curves l), -4OIm $t(a, Y) or 41m 9, (a, v) (curves 2) and 10-210i(a, rm)1 or iO-sl o, (a, g) ) (curves 3) for Re’= 10W4, and correspondingly, for a! = 1 (Fig. 5) and CY = 20 (Fig. 6)
Short communications
249
On some of the curves of these graphs figures are written indicating the value of at the given point 01. 7
2
4
6
8
70
40
20
FIG. 2 The antisymmetric case We will discuss the results of the calculations only for the case Re = lo4 (Figs. 1,2). The results will be mainly the same for the case Re = 2010~ (Fig. 3) and must be modified a little for Re = 5.S.104 (Fig. 4). An arbitrary solution of the linearized Navier-Stokes equations describing the development in time of small perturbations, periodic in X with period 2L, is given by a linear combination of the functions (2):
q (4 2, Y)= y.
where
rc c&o=-, L
c,,(t)exp(iaoms)~,,(aom,
Y),
(3)
Gnn(t) = Gnn(O)expth,(aom)rl.
It is obvious from this that it is mainly the modes $,,(aom, y), corresponding to the greatest values of Re h,(a0m) which are responsible for the development of the perturbations. It is obvious from Figs. 1,2 that on the segment CC=10, 21 these are the modes corresponding to
250
Yu. L. Levitan et al.
Re hi (a) >O (the source of instability of the Poiseuille flow), which is mainly concentrated along the wall of the channel (1-y,2~0.23), and about three tenths of the other modes competing with one another (within the limits of the decay rate Re A>-0.2). within The position changes when we consider more short-wave perturbations. For G-2 the limits of the given decay rate there remain only 5 symmetric modes and 5 antisymmetric modes, soon (for cz>5) there remain only 5 of them altogether, and for a>10 only 3-4 modes. It is important to note that for a>2 alI the principal modes are perturbations concentrated at the centre of the channel, since for them yn (a) is close to 0. Therefore, the long-wave perturbations (k=&z/a>3) are concentrated both at the wall and also throughout the entire width of the channel. As for the short-wave perturbations, for they are concentrated in a narrow band in the region of the centre of the channel, and a>2 it is apparently possible to speak of a short-wave vortex strip in this region of the channel. We note that these results are found to be in agreement with the calculations of some eigenvalues of problem (1) for Re = 104, published in [2] . It was mentioned there that for large a the principal modes are concentrated at the centre of the channel, However, in this paper the correct hierarchy of eigenfunctions for large a was not established, this being due to too few modes being considered. Therefore, a large number of diverse long-wave modes exist, which must be taken into account in the process of development of the perturbations, and conversely, the number of short-wave.modes determining the picture of the development of small perturbations is extremely small.
FIG. 3 Symmetric case
Short communications
251
FIG. 4 Symmetric case
10
08
06
?4
06 04 02
0
0
0.4
FIG. 5
FIG. 6
4. We will consider the problem of the two-dimensional non-stationary flow of a viscous fluid between two parallel platesy = kl with a given flow rate
s
u(g) dy = Q =
const;
and let Re be the Reynolds number corresponding to this flow rate. Defining the velocity vector of the flow by the equation
252
Yu.L. Levitan et al.
we will seek a solution $ (t, x, y), periodic in x with period 2L, of the Navier-Stokes equations in the exact formulation (taking into account the non-linear terms). Using the representation $(r, x1 Y) in the form (3) in the Bubnov-Gal&kin method, we arrive at an infinite system of ordinary differential equations dC?M -=
dt
hmrt(cm) cm,+
c
mtl
r ij,klCijcklr
(4)
f,l,k.l
where the quadratic terms in the subscripts i, j, k, 1 are summed between the corresponding limits. The quantities rc,$ are constants and are calculated in terms of the eigen-functions of the boundary value problem (1) and the boundary value problem conjugate to it. (We solved the boundary value problem conjugate to (1) by Galerkin’s method also). The practical realization of the Bubnov-Gal&in method consists of considering the finite system (4) obtained by discarding terms cmn with large numbers m, n. The difficulty of the realization of the calculation of the non-stationary flow by Galdrkin’s method is due mainly to the large number of non-linear terms in system (4), which have to be considered and the number of which increases in proportion to @fl in each equation of (4) where M and N are the limits of summation with respect to m and n, respectively, in the representation (3) for 4 (t, I, y). We note that attempts to calculate systems of the type (4) were undertaken previously, in particular in [3]. However, there in the expansion (3) the antisymmetric modes were disregarded, and also the number of terms in m in (3) was taken as 3, which is, of course, insufficient and cannot lead to qualitatively correct results. Another attempt to investigate systems of the type (4) (with a different choice of the basis in Galerkin’s method was made in [4]. Finally, we also mention that general and important properties of systems of the type (4) have been studied in papers by A. M. Obukhov and others (see, for example, [5] ). 5. It seems plausible that at least for weak perturbations (for low supercritical Reynolds numbers) the flow picture in the initial stage in the non-linear case will essentially depend only on those modes in the expansion (3), which decay weakly in the linear approximation, that is, which correspond to large Re A,,(aom). Specifying the limit Re h, (aon) 2 -0.2, we arrive at the result that in a considerable part of the interval OGa<60 it is sufficient in the first calculations to take into account a small number of modes. A calculation of the number of functions c,, which need to be considered in (4) for the simulation of flows for Re-104, leads to the figures 1000-2000. Therefore, it appears possible to realize qualitatively the correct simulation of the unstable flow in a channel with Re N10” by means of a system of ordinary differential equations with 1000-2000 unknowns. The study of such a system appears to be feasible on modern computers. A further simplification of the problem resulting from a detailed study of the matrices is possible. rg,;,
253
Short communications
6. In [6,7] turbulent
we attempted
to estimate the complexity
flow by means of the traditional
schemes. The estimates obtained
of the simulation
methods of numerical
showed the great difficulties
of a two-dimensional
mathematics
- difference
of this approach.
The spectra of the Poiseuille flow obtained by us illustrate these difficulties: calculation
of the principal modes it is necessary to represent perturbations Such a requirement
concentrated
in
and in even narrower zones in the
narrow zones close to the walls of the channel (for a(2) middle of the channel (for ~02).
for a correct
leads to a fine step of the spatial
mesh, which entails a tine time step, and all this makes the problem difficult to realize on modern computers. On the other hand, the calculation enables us to describe, economically Navier-Stokes
equations
of these features of the spectrum and spectral functions
and sufficiently
well, the solution
of the non-linear
in the form (3).
The authors thank A. A. Samarskii for valuable discussions. Translated by J. Berry. REFERENCES stability equation. J. FZuid Mech., 50,4,
1.
ORSZAG, St. A. Accurate solution of the Orr-Sommerfeld 689-703.1971.
2.
GOL’DSHTIK, M. A., SAPOZHNIKOV, V. A. and SHTERN, V. N. Local properties of the problem of hydrodynamic stability. Prikl. matem. i. tekhn. fiz., No. 2, 56-61, 1970.
3.
PERKERIS, C. L. and SHKOLLER, B. Stability of plane Poiseuille flow to periodic disturbances of finite amplitude. J. Fluid Mech.. 39, 3,611-628, 1969; The neutral curves for periodic perturbations of finite amplitude of plane Poiseuille flow. 629-639.
4.
DOWELL, E. H. Non-linear theory of unstable plane Poiseuille flow. J. FIuid Me&., 38,2,401-414,
5.
DOLZHANSKII, F. V., KLYATSKIN, V. I., OBUKHOV, A. M. and CHUSOV, M. A. Non-linear systems ofhydrodynamic type (Nelineinye sistemy gidrodinamicheskogo tipa), “Nauka”, Moscow, 1974.
6.
ROZHDESTVENSKII, B. L. On the applicability of difference methods of solving the Navier-Stokes equations for large Reynolds numbers. Dokl. Akad. Nauk SSSR, 211,2, 308-311, 1973.
7.
MOISEENKO, B. D., ROZHDESTVENSKII, B. L. and SIDOROVA, V. K. The spectral characteristics of difference schemes and conditions for the numerical simulation of limiting flow modes of a viscous fluid. Zh. vj%hisl. Mat. mat. Fir., 14,6, 1499-1515, 1974.
1969.
This implies the inequality
.
us G aoexp
LJ
(min x) as’
-
0
247
To
I( r,
-‘I
1
’
which in the case of a finite temperature variation in the range considered imposes a constraint on the growth of the solutions of the homogeneous system (2) as a function of s. Here us is the value of u at the point s. For a monotonically decreasing function T of s the estimate is obtained similarly. In the case of an arbitrary function T(s) this treatment can be carried out on each segment of monotonicity. Translated by J. Berry. REFERENCES
Zh. @hid. Mat. mat.
1.
SHMYGLEVSKII, Yu. D. Calculation of radiative transfer by Gal&kin’s method Fiz., 13, 2, 398-407, 1973.
2.
KRIVTSOV, V. M., NAUMOVA, I. N., SHMYGLEVSKII, Yu. D. and SHULISHNINA, N. P. Check of two methods of calculating radiative transfer. Zh. ujkhisl. Mat. mat. Fiz., 15, 1, 163-171, 1975.
U.SS.R Comput. Maths Math. Phys. Vol. 18,pp. 247-253 0 Pergamon Press Ltd. 1978. Printed in Great Britain.
0041-5553/78/0101-0247$07
50/O
THE SPECTRA OF A PLANE POISEUILLE FLOW AND THE PROBLEM OF SOLVING THE NAVIER-STOKES EQUATIONS FOR LARGE REYNOLDS NUMBERS* Yu. L. LEVITAN, B. D. MOISEENKO, B. L. ROZHDESTVENSKII and V. K. SIDOROVA Moscow (Received 22 April 1976)
OVER a wide range of wave numbers (Ythe eigenvalues A,(o) and the eigen-functions of the boundary value problem for the Orr-Sommerfeld equation are calculated for supercritical Reynolds numbers Re = l/v. The observed features of the spectra demonstrate the use of these eigen-functions as a basis in the Bubnov-Cal&kin method of solving the non-linear problem of the non-stationary flow of a viscous incompressible fhrid between two parallel plates at supercritical Reynolds numbers. 1. In the interval [0,60] of wave numbers 01for the Reynolds numbers Re=i/v=i(T, we have calculated the eigenvalues h=h,(a) (Be h,,+i (cc)GRe X,(a)) and the of the boundary value problem for the Orr-Sommerfeld equation eigen-functions Ip=gn(a, y) 2.104, 5.5.102
lp(M)
=9’(d)
(1)
=o.
We have calculated the 20-20 leading symmetric and the same number of antisymmetric modes of problem (1). The calculations were performed by Galerkin’s method using the basis I (n/2)yl (p,,(y)=i-_(-I)” cos nny, n=l, 2, . . . , in the symmetric case, and %-sin functions -(--1)” sin [ (2n+l) (n/2) g), n=l, 2, . . . , in the antisymmetric case. *Zh. vj%hisl.Mat. mat. Fiz., 18,1,252-258,
1978.
Yu. L. Levitan et al.
248
The calculations carried out demonstrated the high accuracy, reliability and efficiency of the method over a wide range of wave numbers a, unhke some other methods of solving the boundary value problem (1) for example, that explained in [l] , which is very inefficient for values of aM. 2. The solution solution -
A,(a), +,(a, y) $(4x,
of the boundary value problem (1) corresponds to the
~)=~exp
IL(a)t+~l44a,
U)
(2)
of the Navier-Stokes equations linearized for Poiseuille flow. Thus, the quantity &,(a) describes the behaviour in time of small perturbations of a Poiseuille flow of a certain type. Figures l-4 show the functions Re k,(a), describing the decay in time (for A&(a) ~0 ) of perturbations of the type (2). Note that, in attempting to represent as much information about Re L(a), as possible, we have used variable scales, which explains the kinks in the curves in these graphs.
FIG. 1 The symmetric case The quantity Im h,,(a) is characteristic of the ripple frequency of a perturbation of the type (2), and if we write Im L(a) --a(l-sr,2(a)), then y,(a) described the place in the channel -IQ&~, where the vortex of the perturbation ~,,(a, v)- a21p,,(a, v) -$,,“(a, I/) is mainly concentrated. This is confirmed by graphs of the functions Re *l(a, Y) (curves l), -4OIm $t(a, Y) or 41m 9, (a, v) (curves 2) and 10-210i(a, rm)1 or iO-sl o, (a, g) ) (curves 3) for Re’= 10W4, and correspondingly, for a! = 1 (Fig. 5) and CY = 20 (Fig. 6)
Short communications
249
On some of the curves of these graphs figures are written indicating the value of at the given point 01. 7
2
4
6
8
70
40
20
FIG. 2 The antisymmetric case We will discuss the results of the calculations only for the case Re = lo4 (Figs. 1,2). The results will be mainly the same for the case Re = 2010~ (Fig. 3) and must be modified a little for Re = 5.S.104 (Fig. 4). An arbitrary solution of the linearized Navier-Stokes equations describing the development in time of small perturbations, periodic in X with period 2L, is given by a linear combination of the functions (2):
q (4 2, Y)= y.
where
rc c&o=-, L
c,,(t)exp(iaoms)~,,(aom,
Y),
(3)
Gnn(t) = Gnn(O)expth,(aom)rl.
It is obvious from this that it is mainly the modes $,,(aom, y), corresponding to the greatest values of Re h,(a0m) which are responsible for the development of the perturbations. It is obvious from Figs. 1,2 that on the segment CC=10, 21 these are the modes corresponding to
250
Yu. L. Levitan et al.
Re hi (a) >O (the source of instability of the Poiseuille flow), which is mainly concentrated along the wall of the channel (1-y,2~0.23), and about three tenths of the other modes competing with one another (within the limits of the decay rate Re A>-0.2). within The position changes when we consider more short-wave perturbations. For G-2 the limits of the given decay rate there remain only 5 symmetric modes and 5 antisymmetric modes, soon (for cz>5) there remain only 5 of them altogether, and for a>10 only 3-4 modes. It is important to note that for a>2 alI the principal modes are perturbations concentrated at the centre of the channel, since for them yn (a) is close to 0. Therefore, the long-wave perturbations (k=&z/a>3) are concentrated both at the wall and also throughout the entire width of the channel. As for the short-wave perturbations, for they are concentrated in a narrow band in the region of the centre of the channel, and a>2 it is apparently possible to speak of a short-wave vortex strip in this region of the channel. We note that these results are found to be in agreement with the calculations of some eigenvalues of problem (1) for Re = 104, published in [2] . It was mentioned there that for large a the principal modes are concentrated at the centre of the channel, However, in this paper the correct hierarchy of eigenfunctions for large a was not established, this being due to too few modes being considered. Therefore, a large number of diverse long-wave modes exist, which must be taken into account in the process of development of the perturbations, and conversely, the number of short-wave.modes determining the picture of the development of small perturbations is extremely small.
FIG. 3 Symmetric case
Short communications
251
FIG. 4 Symmetric case
10
08
06
?4
06 04 02
0
0
0.4
FIG. 5
FIG. 6
4. We will consider the problem of the two-dimensional non-stationary flow of a viscous fluid between two parallel platesy = kl with a given flow rate
s
u(g) dy = Q =
const;
and let Re be the Reynolds number corresponding to this flow rate. Defining the velocity vector of the flow by the equation
252
Yu.L. Levitan et al.
we will seek a solution $ (t, x, y), periodic in x with period 2L, of the Navier-Stokes equations in the exact formulation (taking into account the non-linear terms). Using the representation $(r, x1 Y) in the form (3) in the Bubnov-Gal&kin method, we arrive at an infinite system of ordinary differential equations dC?M -=
dt
hmrt(cm) cm,+
c
mtl
r ij,klCijcklr
(4)
f,l,k.l
where the quadratic terms in the subscripts i, j, k, 1 are summed between the corresponding limits. The quantities rc,$ are constants and are calculated in terms of the eigen-functions of the boundary value problem (1) and the boundary value problem conjugate to it. (We solved the boundary value problem conjugate to (1) by Galerkin’s method also). The practical realization of the Bubnov-Gal&in method consists of considering the finite system (4) obtained by discarding terms cmn with large numbers m, n. The difficulty of the realization of the calculation of the non-stationary flow by Galdrkin’s method is due mainly to the large number of non-linear terms in system (4), which have to be considered and the number of which increases in proportion to @fl in each equation of (4) where M and N are the limits of summation with respect to m and n, respectively, in the representation (3) for 4 (t, I, y). We note that attempts to calculate systems of the type (4) were undertaken previously, in particular in [3]. However, there in the expansion (3) the antisymmetric modes were disregarded, and also the number of terms in m in (3) was taken as 3, which is, of course, insufficient and cannot lead to qualitatively correct results. Another attempt to investigate systems of the type (4) (with a different choice of the basis in Galerkin’s method was made in [4]. Finally, we also mention that general and important properties of systems of the type (4) have been studied in papers by A. M. Obukhov and others (see, for example, [5] ). 5. It seems plausible that at least for weak perturbations (for low supercritical Reynolds numbers) the flow picture in the initial stage in the non-linear case will essentially depend only on those modes in the expansion (3), which decay weakly in the linear approximation, that is, which correspond to large Re A,,(aom). Specifying the limit Re h, (aon) 2 -0.2, we arrive at the result that in a considerable part of the interval OGa<60 it is sufficient in the first calculations to take into account a small number of modes. A calculation of the number of functions c,, which need to be considered in (4) for the simulation of flows for Re-104, leads to the figures 1000-2000. Therefore, it appears possible to realize qualitatively the correct simulation of the unstable flow in a channel with Re N10” by means of a system of ordinary differential equations with 1000-2000 unknowns. The study of such a system appears to be feasible on modern computers. A further simplification of the problem resulting from a detailed study of the matrices is possible. rg,;,
253
Short communications
6. In [6,7] turbulent
we attempted
to estimate the complexity
flow by means of the traditional
schemes. The estimates obtained
of the simulation
methods of numerical
showed the great difficulties
of a two-dimensional
mathematics
- difference
of this approach.
The spectra of the Poiseuille flow obtained by us illustrate these difficulties: calculation
of the principal modes it is necessary to represent perturbations Such a requirement
concentrated
in
and in even narrower zones in the
narrow zones close to the walls of the channel (for a(2) middle of the channel (for ~02).
for a correct
leads to a fine step of the spatial
mesh, which entails a tine time step, and all this makes the problem difficult to realize on modern computers. On the other hand, the calculation enables us to describe, economically Navier-Stokes
equations
of these features of the spectrum and spectral functions
and sufficiently
well, the solution
of the non-linear
in the form (3).
The authors thank A. A. Samarskii for valuable discussions. Translated by J. Berry. REFERENCES stability equation. J. FZuid Mech., 50,4,
1.
ORSZAG, St. A. Accurate solution of the Orr-Sommerfeld 689-703.1971.
2.
GOL’DSHTIK, M. A., SAPOZHNIKOV, V. A. and SHTERN, V. N. Local properties of the problem of hydrodynamic stability. Prikl. matem. i. tekhn. fiz., No. 2, 56-61, 1970.
3.
PERKERIS, C. L. and SHKOLLER, B. Stability of plane Poiseuille flow to periodic disturbances of finite amplitude. J. Fluid Mech.. 39, 3,611-628, 1969; The neutral curves for periodic perturbations of finite amplitude of plane Poiseuille flow. 629-639.
4.
DOWELL, E. H. Non-linear theory of unstable plane Poiseuille flow. J. FIuid Me&., 38,2,401-414,
5.
DOLZHANSKII, F. V., KLYATSKIN, V. I., OBUKHOV, A. M. and CHUSOV, M. A. Non-linear systems ofhydrodynamic type (Nelineinye sistemy gidrodinamicheskogo tipa), “Nauka”, Moscow, 1974.
6.
ROZHDESTVENSKII, B. L. On the applicability of difference methods of solving the Navier-Stokes equations for large Reynolds numbers. Dokl. Akad. Nauk SSSR, 211,2, 308-311, 1973.
7.
MOISEENKO, B. D., ROZHDESTVENSKII, B. L. and SIDOROVA, V. K. The spectral characteristics of difference schemes and conditions for the numerical simulation of limiting flow modes of a viscous fluid. Zh. vj%hisl. Mat. mat. Fir., 14,6, 1499-1515, 1974.
1969.