Acta Mechanica 95, 131-156 (1992)
ACTA MECHANICA 9 Springer-Verlag 1992
Nonlinear stability of a viscous axisymmetric jet J. Horwitz and S. Rosenblat, Chicago, Illinois (Received July 20, 1991)
Summary. The nonlinear temporal stability of a viscous axisymmetric jet is studied with respect to axisymmetric disturbances. The weakly nonlinear Stuart Watson perturbation expansion is used to study the self interaction of the fundamental in a neighborhood of the critical Reynolds number (Re). The Landau constant is positive indicating there is a supercritical Hopf bifurcation. In order to extend these results past R ~ Re, a severely truncated Fourier modal expansion using the Stuart Watson functions to represent the r dependence, exponentials in z and unknown amplitudes in time is substituted into the Navier-Stokes equation. A projection onto an appropriate subspace leads to a low dimensional (five) system of amplitude equations for the disturbance of the fundamental, harmonic and distortion to the mean flow. Numerical results show that the periodic solution is stable for 55.3 < R < 72.3. There is a secondary bifurcation at R = 72.3 to a quasiperiodic solution with 2 incommensurate frequenciesfl and f2. Each peak in the Fourier spectrum can be indexedaccording t o f = f l + nf2 for n = 0, _+1, _+2.... As R increases past R = 78 there is a transition through another periodic regime and then finally a transition to intermittency for 100 < R < 1000. No chaotic solutions were observed in this low dimensional model.
1 Introduction This work is concerned with the stability of a viscous axisymmetric jet. A flow is considered to be stable if it can sustain itself against perturbations to which all systems are subjected. Stability of a jet and subsequent transition to turbulence, as a control p a r a m e t e r is increased, are i m p o r t a n t problems. F r o m a practical standpoint there is great interest in knowing how a laminar jet develops into a fully turbulent flow which is complex in time and space, and has vastly different t r a n s p o r t characteristics which m a y or m a y not be desirable. Although there is no complete or even moderately complete theory to describe the transition to turbulence [1], L a n d a u [2] attempted to describe the temporal aspect of transition. L a n d a u postulated that turbulence evolves only after an infinite sequence of supercritical bifurcations has occurred. A second theory due to Newhouse, Ruelle and Takens [3] suggests that transition to turbulence can occur after a finite number of bifurcations. Thus, after the third incommensurate frequency appears in L a n d a u ' s scheme, chaos could follow according to this scheme. This theory is also compatible with the numerous examples of low dimensional systems of O.D.E.'s which exhibit chaotic solutions after only a few bifurcations. Other routes to chaos include the period doubling route of F e i g e n b a u m [4] and the intermittency route of P o m e a u and Manneville [5]. The w o r k that has stimulated considerable interest along the lines of low dimensional models exhibiting chaotic solutions is due to Lorenz [6]. In the study of two dimensional RayleighB6nard convection, Lorenz expanded the stream function and a deviation from the mean temperature into a sum of linear eigenfunctions with time dependent coefficients. Retaining only the fundamental for the two physical quantities and a mean flow distortion term for the temperature, and using a Galerkin projection, Lorenz arrived at a three dimensional system of
132
J. Horwitz and S. Rosenblat
nonlinear O.D.E.'s which depend on the two control parameters, Prandtl number (Pr) and Rayleigh number (Ra). The well known sequence of bifurcations, as Ra increases, of this purely deterministic system using Pr = 10, is that the conduction solution is replaced by the convection solution which in turn undergoes a subcritical Hopf bifurcation and is followed by chaos. The relative ease of integrating such a low order system of O.D.E.'s compared to a direct numerical solution of the Navier Stokes equations has prompted much research along these lines, inspite of the fact that Lorenz's original truncation is invalid. However, the results of a low order model should be treated with caution. It has been shown [7] that a generalized Lorenz system consisting of a 14 dimensional system of O.D.E.'s has the bifurcation sequence: Steady (conduction) ~ Steady (convection) ~ Periodic ~ Period Doubling --+ Quasiperiodic ~ Chaos. It is important to note that although chaos is still observed, it occurs at a much higher Ra than in the lower order model. This raises the question of whether chaos in a low dimensional system is merely an artifact of the truncation or indeed related to turbulence in the fluid. Addressing this issue are the results of Curry et al. [8] using spectral methods, which show that chaos is not observed for high Prandtl number flows. It should be noted that Rayleigh-B~nard convection is a closed flow system in contrast to the boundary layer and axisymmetric jet which are open flow systems. The use of amplitude equations has also been applied to other types of fluid flow. The turbulent boundary layer which was modeled by a ten dimensional system of O.D.E.'s [9] gave periodic, quasiperiodic and finally intermittent solutions. The stability analysis for the axisymmetric jet, however, has not proceeded beyond the linear regime for which nonlinear interactions which are required for chaos have been neglected. Thus, our interest has been to modify the Lorenz method in order to derive a low dimensional model which would include a minimal number of nonlinear interactions. A low order model would give a sequence of bifurcations as the control parameter, in this case the Reynolds number (R), is increased. If a model could be found, the question of the range of validity could at least be determined in a neighborhood of the critical Reynolds number. This is done by comparing the results of the low order model with the results of the weakly nonlinear theory which is known to be valid close to Re. Matkowsky [10] describes this theory for a simple model problem and it was first applied to an actual fluid flow problem by Stuart [11] and Watson [12]. A perturbation expansion and a solvability condition lead to a single O.D.E., called the Landau equation for the amplitude of the disturbance. The lengthy calculation of the Landau constant determines whether the bifurcation at the critical Reynolds number is subcritical or supercritical based solely on the sign of the Landau constant. The Landau constant has contributions from the mean flow distortion, the fundamental and the harmonic. Maslowe [13] neglected the mean flow distortion and found supercritical stability for the case of a two dimensional plane jet. The first step in the perturbation expansion procedure is to solve the Orr-Sommerfeld equation which has been done for the case of the axisymmetric jet. The assumption of parallel flow is commonly used to simplify the analysis even though its validity remains questionable. For velocity profiles such as hyperbolic tangent profiles, numerical methods are required. The critical Reynolds number is shown [14] to be less ( R e - 21.75) for the helical mode than for the axisymmetric mode (Re = 55.3). Other authors [15]-[18] have considered various velocity profiles. Kambe [19] used a less realistic parabolic velocity profile and was able to obtain analytical results. For the sake of completeness we mention the inviscid linear analysis first studied by Batchelor and Gill [20]. They derived a necessary condition for the existence of unstable solutions and pointed out that there is no analogue of Squire's theorem and thus three dimensional disturbances cannot be neglected. Furthermore, the experimental evidence of Drubka [21] and Shakib [22] showed that the dominating modes which alternate in a jet at high
Nonlinear stability of a viscous axisymmetric jet
133
Reynolds numbers are the axisymmetric (n = 0) and helical (n = 1) modes and therefore they are the ones we choose to study. The major objective of this work is to derive a low order analogue of the Lorenz system for the case of the viscous axisymmetric jet in order to determine whether such a model exhibits chaotic solutions. The criterion for validating the model in a neighborhood of the critical Reynolds number will be the results of the Stuart-Watson weakly nonlinear theory. The exact relationship between the existence or nonexistence of chaotic solutions in a low dimensional model to turbulence in a real jet are open questions and beyond the scope of this paper.
2 Basic state In this work we will be concerned with an axial symmetric jet which is formed when an incompressible fluid emanates from a circular orifice into an unbounded region containing a similar fluid. Due to the geometry, cylindrical coordinates (r, 0, z) are used to specify the radial, azimuthal and axial directions respectively. The corresponding unit vectors are ?, O and ~. The governing equations are made nondimensional by scaling with respect to a characteristic length Lc and a characteristic velocity Uc and are given by V. V = 0 0V
--
&
+ (V.
(2.1)
V) V =
-VP
1 V2
+ --
R
(2.2)
V
where V = (U, V, W) is the velocity vector, t is time and P is the pressure. R = UcLc/v is the Reynolds number, v is the kinematic viscosity and V = ~r ~ + r-1
0 + ~z ~"
The domain of these nonlinear partial differential equations is given by 0 < r < v% 0 _< 0 _< 2~ and 0 < z < or. The boundary conditions in r are given by U, V, W, P finite
at
r = 0
(2.3.1)
U, V,, W,,P--,O
as
r~ov.
(2.3.2)
Due to the nature of the coordinate system a necessary condition is that U, V, W,, and P are 2k~z periodic in 0 where k is an integer. Since periodic boundary conditions in z are assumed, we will not be considering the spatial stability problem for which disturbances are allowed to grow in space. Thus, in order to formulate a well posed mathematical problem, only temporal growth can be considered, even though in a real jet temporal and spatial growth are relevant. The actual jet velocity profiles are known to vary from a top-hat profile to a more rounded profile as the distance from the jet exit is increased, indicating that the flow is not strictly parallel. However, we have used the parallel flow assumption and thereby consider the mean flow to be given by Vo = (Uo, Vo, Wo) = (0, 0, .5(1 + tanh (1.5625(1/r - r)))) which has been verified experimentally [14] to be a good approximation to an actual jet.
(2.4)
134
J. Horwitz and S. Rosenblat
3 W e a k l y nonlinear theory The major objective of this work is to develop a low dimensional system of O.D.E.'s for the disturbance amplitudes arising in a jet which depend on the Reynolds number. Since the governing equations are nonlinear partial differential equations, an approximation to the solution is obtained by a severe truncation of an infinite series. There is no guarantee that the resulting system of O.D.E.'s is valid for all values of R. However, in the limiting case when R is very close to Rc (critical Reynolds number), the solutions of the low dimensional system should agree with the results using a perturbation expansion about Re. Thus, the aim of this Section is to study the behavior of the growth of the disturbances close to Rc using the multiple scales perturbation expansion. It will be shown that the amplitude of the disturbances is governed by a single nonlinear ordinary differential equation, called the Landau equation, which has an analytic solution. The solution is either supercritical or subcritical depending solely on the sign of the constant multiplying the single nonlinear term. In the case of axisymmetric disturbances we may define a stream function 7~ which satisfies (2.1) and is given by 1 0k~
1 07~
rJ . . . .
W=
r ~z'
- --.
(3.1)
r Or
Then the nondimensional form of the Navier Stokes equation (2.2) becomes
G-R~
GT=--r 02
where G 1 37j r 0r' 1 OT r 0r'
3r 2
3z Hp
GT
1 3 02 3 - 2 and Hp - ~r r Or + -3z
1 3tP --finite r 3z 1 c~7j
~r 0z
- - - -+ 0 r 3z
(3.2) 2 r . The boundary conditions (2.3) become
at
r = 0
(3.3.1)
as
r ---, %.
(3.3.2)
Next we use the Stuart-Watson weakly nonlinear theory to derive a Landau equation for the disturbance amplitude of the fundamental. To study the stability of the mean flow ~0o(r) we look for a solution of (3.2) and (3.3) given by 7~(r, z, t, z) = 0o(r) + 7J'(r, z, t, r)
(3.4)
where q>o(r) and Wo are related by rWo = dOo(r)/dr. 7J'(r,z,t,r) and R are given by the perturbation expansions
~e'(r, z, t, z) = F~ ~"%'(r, z, t, ~)
(3.5)
n=l
R = Rc + ~2R2.
(3.6)
If r is a small time scale given by r = ~2t
(3.7)
Nonlinear stability of a viscous axisymmetric jet
135
then c~/& is transformed according to
c~/& --, ~/& + ~2 c~/c~.
(3.8)
Substitution of (3.4)-(3.8) into (3.2), separation of equations at order e", and neglecting primes gives the disturbance equations
Mtr/,(r, z, t, "c) = F, for
n=1,2,3
(3.9)
which can be solved sequentially. The linear partial differential operator M and the inhomogeneous terms F, are given by M =
G - R~ ~
G - - - r \ ~r ~zz G - HpG~~ ~z
FI=O
F2
F3
Rc (c3~1 c3
= T \~#
~ G~q -
c~1 HpG~I) 0~
Rc ( c~TJl c~1 0 0~2 c~2 0 ) = T --~-z H'GTJ2 + ~ r ~z GTJ2 -- ~-z H,G~P1 + ~-r ~z GTjl 0 G~P1- R2 + Rc ~z R~ G2 illl "
The linear problem at order e 1 is solved by the standard procedure of normal modes. Since ~l(r, z, t, z) must be real we look for a solution of the form IPl(r, z, t, z) = A(z) ~l(r) e i~(z-") + c.c.
(3.10)
where c.c. denotes complex conjugate and A(z) is an arbitrary amplitude. The axial wavenumber is real and c is complex. ~l(r) is called the fundamental. Substitution of (3.10) into (3.9, n = 1) leads to the eigenvalue problem which is the analogue to the Orr-Sommerfeld equation in a cylindrical coordinate system L1~1 = 0.
(3.11)
The differential operator L, is given by
L, = (Wo - c) G, - rd
+~
where d i = di/dr i and G, = d 2 - r-ld
G, 2
-
-
(n~) 2. The boundary conditions on •1 become
d@a/r,
c@z/r finite
at
r = 0
(3.12.1)
d@,/r,
7@i/r ~ 0
as
r--* oc.
(3.12.2)
At order ez we look for a solution 7Jr(r, z, t, z) of the form V2(r, z, t, r) = A2(z) $22(r) e 2'~(~-") + c.c. + A(,) A(z) $20(r)
(3.13)
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J. Horwitz and S. Rosenblat
where - denotes complex conjugate. 022(r) and 02o(r) are called the harmonic and mean flow distortion. Substitution of (3.13) into (3.9, n = 2), separation of the coefficients of A(r) J(r) and AZ(r) e 2~(z-") leads to the linear inhomogeneous differential equations Go2020 = ~Rc(ir)-* ((O,H + dO,) G,~I) + c.c.
(3.14)
L2022
(3.15)
=
(2r) -1 ( ( 0 , H - dO,) G,01)
where H = d - 2r -~ and boundary conditions
dO2o/r finite
at
r= 0
(3.16.1)
dOio/r ~ 0
as
r + o9
(3.16.2)
dOz2/r, o@22/rfinite at r = 0
(3.17.1)
dOaz/r,
(3.17.2)
~022/r ~ 0
as
r ~ oo.
At order e 3 we look for a solution 7J3(r, z, t, 7) of the form ~3(r, z, t, 7:) = (A3('c) 033(r) e 3ia(z-ct) q- A2('c) A(-c) 031(r) e '~(~-")) + c.c.
(3.18)
Substitution of (3.18) into (3.9, n = 3) and considering only the coefficients ofe ~(z-a) leads to the inhomogeneous differential equation involving A(r) A2('c)/l('c) L1031(r)=f3z
(3.19)
where the inhomogeneous term f3, is given by fa, = Az(T) A(z) (Q0 + Q2) + ~ \ ~ -
G~ + A(v)
G* 2 0a
(3.20)
and Q0 and Q2 are defined by rQo = (01/JCo - 6 , 0 1 d ) 0~o
rQ2 = (20z2H + d022) G I ~ , - ( ~ , H + 2 d ~ ) G2022. Since the condition Fredholm a solution
homogeneous equation (3.11) has a nontrivial solution at criticality, a solvability on (3.19) is required. However, since L , is a singular operator, the conditions of theory do not strictly apply [23] and thus we do not have a condition that guarantees to (3.19). However, if the adjoint 0 " satisfies
(r0*, L,0~> = ( L * 0 * , rOl>
=
0,
(3.21)
then taking the inner product of (3.19) with r0* leads to
(rO*,f3,) = S r0"*fa, dr = 0. o
(3.22)
The adjoint 0* is the solution to L * 0 * = 0 where L* is given by
L* = (Wo -- c) (G1 + 4r-~d) + 2dWo(d + 2r-*) + ~
i
(G, + 4r l d ) 2
Nonlinear stability of a viscous axisymmetric jet
137
and is subject to the boundary conditions ~*,
d~* finite
at
r = 0
1~*,
dO*~0
as
r+oo.
Substitution of (3.20) into (3.22) leads to the Landau equation dA(r) _ dr
R 2 ( r ~ * , G1201) A(r) + ic~(r~]*, (Qo + O2)) AZ(r)/~(~). Rc 2 (r~*, G1~1> (r~/*, G1~1}
(3.23)
Multiplication of (3.23) by A(r), adding its complex conjugate, using (3.6) and (3.7) and scaling of t leads to the real form of the Landau equation da d t = I i ( R - Re) a - (lo + 12) a2
(3.24)
where a and the linear growth rate l~ are given by a = IAI2 (c~ 2) > 0, l1 = - ~
( ( r o * ' G1201)~ (Rc2) -1 > O.
The two contributions to the Landau constant are due to the fundamental-mean flow distortion interaction (lo) and the fundamental-harmonic interaction (/2) and are given by
(
e.>
/
for
Here we have used the notation 9l and 3 to denote the real and imaginary parts respectively. The exact solution of (3.24) is given by
a = ao \ l l ( R -- Re) ao +
where
1
l l ( R - Re) ao
e l~(R-RJt
(3.25)
ao = a (t = 0) is the
initial condition. If R > Rc then according to (3.25), if lo + 12 > 0. If R < Rc then a ~ l l ( R - Rc)/(lo + /2) as t ~ - oo if lo + 12 < 0. Thus, the sign of lo + 12 determines whether there is a solution for R < Rc (subcritical bifurcation) or whether there is a solution for R > Rc (supercritical bifurcation). The lengthy calculation of lo + 12 requires solving two homogeneous boundary value problems and two linear inhomogeneous equations all of which are on infinite intervals and cannot be solved by simple numerical shooting procedures. We have used the method of matching [24] to solve the eigenvalue and adjoint problems. To insure that we found the least stable eigenvalue triple we used the contour integration method [25] and verified the results (cqc, R c ) = (1.028, .8033, 55.3977) at criticality. The method of c o m p o u n d matrices [26] was used to solve the inhomogeneous equations at order ~2. Simpson's rule was used to show that tb, Landau constant lo + 12 > 0 and in addition a ~ l l ( R - Rc)/(lo + /2) as t ~ ~
lo, 11, 12 > O,
12/lo = 8.8.
(3.26)
138
J. Horwitz and S. Rosenblat
Thus, the linear growth rate 11 is positive as required. The dominant contribution to the Landau constant (lo + 12) is due to the fundamental-harmonic interaction (12). The contribution due to the fundamental-mean flow distortion (lo) is also positive. Since lo + 12 is positive we have the result that the mean flow is supercritically stable. In the next Section we consider a system of O.D.E.'s which can be shown to be a generalization of the Landau equation. Therefore, in order for it to be consistent with the Landau equation it must be required to also yield a supercritical bifurcation at Re.
4 Nonlinear theory One method which has been used to approximate a solution to a class of nonlinear RD.E.'s (3.2), (3.3) is to assume the solution can be represented by an infinite series of known spatial functions whose unknown coefficients depend on time. One of the difficulties of this method is in choosing the functions which depend on space. For example, Lorenz [6] used the trigonometric eigenfunctions of the linear stability problem and Aubry [9] used the method of proper orthogonal decomposition. Our approach has been to incorporate the Stuart-Watson functions in a generalization of the expansion given in (3.5). In particular there will be no small parameter e and the temporal amplitudes are not predetermined.
4.1 Fourier modal expansion To study the stability of Oo(r) we first look for solutions to (3.2) and (3.3) of the form
~(r, z, t) = ~,o(r) + ~'(r, z, t).
(4.1)
The perturbation ~'(r, z, t) is given by the expansion N
tP'(r, z, t) = ~ A,(t) B,(z) S,(r) + c.c.
(4.2)
n=0
where B~(z) and the shape functions S,(r) are taken to be appropriately chosen sets of basis functions in the two spatial directions z and r. An(t) are the unknown coefficients which need to be determined. Comparison of (4.2) with (3.5) shows that there is no small parameter e in (4.2) which justifies a truncation. However, if we choose the basis functions B,(z) and S,(r) to include those functions used in (3.5) then the resulting system of amplitude equations will simply be a generalization of the Landau equation (3.24) and the range of validity of (4.2) could at least be assumed to be correct for R close to Re. Of course there is not a unique set of basis functions which can generalize the Landau equation. Thus, the objective of this Section is to derive the system of O.D.E.'s for the disturbance amplitudes A~(t) by choosing to retain only those terms in (4.2) which correspond to the leading order terms in (3.5). A second objective is to determine whether this system has chaotic solutions. The perturbation equations are obtained by substitution of (4.1) and (4.2) into (3.2), neglecting primes, and subtracting out the equations for the mean 00 which leads to the disturbance equations ~ - ~
6~, -
Wo ~
6 - Q ~
~' = -r
~
~z
~z t t
G~'
(4.3)
Nonlinear stability of a viscous axisymmetric jet
139
where Q = rd(r-ldWo). The basis functions B,(z) are chosen to be B.(z) = e "i=z.
Tlms, if we choose the truncation N = 2 in (4.2), then tP(r, z, t) is given by ~(r, z, t) = Ao(t) So(r) + (Al(t) Sa(r) e i~" + Az(t) S2(r) e 2'~z) + c.c.
(4.4)
where we have used the choice of shape functions {So(r), S~(r), S2(r)} = {~'2o, ~1, ~'z2}. Substitution of(4.4) into (4.3) and separation of the coefficients ofe "i~ (for n = 0, 1, 2) leads to G.Sn dt A . =
G.2S~ + nice(Q - WoG.) S.
A~ + icer-iF,,
(4.5)
where the nonlinear terms F. are given by Fo = (Jo,3A1A1 + Jo,4A2A2) Fa = (Ji,3A1Ao + J1,4-41A2)
F2 = (J2,3A12 + J2,4AoA2) and the J's are given by Jo,3 = (dS~G~S1 + S ~ H ~ S ~ ) -- c.c. Jo,4 = 2((dS1G2S1 + S1HGaS1) - c.c.) J1,3 = S1HGoSo - dSoGiS1 J~,4 = dSaG~ff, + 2S~HGa~q, - (S1HG2S1 ~- 2dff~G2S1) J2,3 = S1HGaS1 - dS1G1S1 J2,4 = 2(S1HGoSo - dSoG2S1).
Next, taking the projection of (4.5) with the projection functions S.(r) for n = 0, 1, 2 respectively leads to the system of O.D.E.'s dAo/dt = (ao,1/R) Ao + ao,aAl,Y.1 + ao,4A2-42
(4.6.1)
dA1/dt = (al,1/R + al,2) A1 + al,3AoA1 -[- al,4A1A2
(4.6.2)
dAz/dt = (aa,1/R + a2,2) A2 + a2,3A12 + aa,4AoA2.
(4.6.3)
The coefficients a,,,, are defined by the inner products
<~., G,?s~ a.,1 -
(~., G.S.> nicc(S.(Q - WoG.) S . )
a~ =
< ~2., G.S.>
a.,m - (~., G.S.>
for
n = 0, 1,2
for
n = 1,2
for
n = 0, 1,2,
m = 3,4.
140
J. Horwitz and S. Rosenblat
Converting (4.6) to a real valued system by the transformation an,m = rn,m -}- iSn,m, Ao = x~,
A s = x2 + ix3,
A2 = x4 + ix5
leads to the system of O.D.E.'s (4.7)
dx/dt = A x + f
where x = (x~, x2,..., xs) r, the 5 x 5 matrix A is given by to,, R
0
0
0
r l ' l q- rl, 2
SI,1 --m -- $1, 2
R
0
0
0
0
0
0
R
0
S1'1 -~ $1, 2
0
0
R
/~1,1 Av
R-
0
rl'2
.r2,1 . q-r2,2 . .
R
0
0
0
$2'~1 -}- $2, 2
R
$2,1 $2, 2 R r2,, R - q- r2'2
and f is given by
t ro,3(x22 + x32)+ 2+ x521 \ x t ( r t , 3 x 2 -- S1,3x3) -}- X4(rl,4X 2 -}- S1,4x3) q- x 5 ( - - s l , 4 x 2 q- F 1 , 4 X 3 ) / /
!
XI(S1,3X 2 q- rt,3x3) -I- X4(S1,4X 2 -- rl,4x3) -}- xs(rl,4x2 q- $1,4x3)
/
XI(F2'r
I / /
-- $2'4X5) -}- F2'3(X22 -- X32) -- 2S2"3X2X3
Xl(S2,gX 4 -}- r2,4Xs) q- S2,3(X22 -- X32) -~- 2r2,3x2x 3
"
Note that if (4.7) is to be meaningful, it must at least give consistent results with those results obtained by the linear and weakly nonlinear theories. The linear results give us the first condition I, which is that the null solution is stable for R < Rc and unstable for R > Re. The results of the Landau constants Io and 12 which are both positive give us the second condition II, which is that the first bifurcation from the null solution at Rc is supercritical. Thus, it must be shown that the choice of shape functions and projection functions satisfy these two conditions. It can be shown that the real part of the eigenvalues of A are given by ro,1/R, ra,i/R + r,,2 and r2,1/R + r2,2. Since there is a second bifurcation from the null solution, condition I is equivalent to I:
roA < 0,
rl, , < 0,
rl, 2 > 0,
r2A < 0,
r2, 2 > 0
where - r,, ~/r ~,2 = Rc and - r2, t/r 2,2 > Rc. In the next Section we examine the requirements that condition II places on some of the remaining coefficients in (4.7).
4.2 Reduction o f (4.7) to a Landau equation
Next we consider the limiting case of(4.7) as R --+ Re. Comparison of 7J'(r, z, t, ~) given by (3.5) and 7J'(r, z, t) given in (4.2) is shown below (e, = e"i~z): ~Jt(F, Z, t, T) = gA(z) (~a(-c) 9 1~/2o + e-i:ct~tlel -~ gA("~) a("~) e-2'~ctO22e2) - c.c. ~'(r, z, t) = (Ao(t) So + A,(t) S l e l + A2(t) $2e2) + c.c.
Nonlinear stability of a viscous axisymmetric jet
141
Thus for R close to Rc we have
dAo/dt = O,
dA 2/dt = - 2ic~cA2
(4.8)
which results from equating the time (t) dependence in the first and last terms in the two expansions for 7~'. Further comparison of (4.6) with (3.23) shows that ao,4
=
a2,~ = 0
(4.9)
which is the condition that the higher order interactions can be neglected in the limit as R ~ Re. Substitution of (4.8) and (4.9) into (4.6), solving for dA1/dt and converting to a real valued system leads to
dd
(r~,~
d~ = r1'2
g -{-
~} rl,2//
a -- ([0 ~- [2) a2
where d = IAll 2 R 2, { =
[o=~jt(al,3ao,3) \ ao,1 / '
(4.10)
2t/R and
[2=N
a,,4a2,s
(
)
a2,1 4- a2,2Rc + 2i7cRc "
Thus, we have reduced a system of O.D.E.'s to a single O.D.E. which has the same structure as the Landau equation (3.24). It is easy to see that condition II is equivalent to
-:
[o,[2 > 0.
If e{ = ('$1, J1,3)(So, in the form
Jo,a)/(So, GoZSo), then it can be shown that [o and lo can be written
[0 = 9t(A/(S,, G,S,)),
10 = .3((r~*, Qo)/(rt~*, G?p~>).
Comparison of the denominators (Sa, G,Sa) and (r~*,Gl~> suggests that a suitable projection function is given by S, = rO* since G,S, = G,~,. We also make the choice $2 = rO* by comparison of [2 and 12. A reasonable projection function So is obtained by noting that the linear growth rate of Ao must be negative. If we let So -- r-*So, then
r-*SoGo2So dr r0,i =
0
~ r(d(r-'dSo)) 2 dr __
oo
--
r-lSoGoSo dr o
0 --
< 0
~ r-~(dSo) 2 dr 0
for any choice So. Using these shape functions and projection functions
= {r-102o, rOx, r l}, Simpson's rule can be used to verify that both conditions I and II are satisfied. The complex coefficients aj,k appearing in (4.6) are given in Table 1. The terms in (4.10) are given by
--rl,1/rl,2
[o, [2 > O,
=
Rc = 55.32
(4.11.1)
[2/[0= 10.72
(4.11.2)
which verifies qualitative agreement with (3.26).
142
J. Horwitz and S. Rosenblat
Table 1. Coefficients a~,9
j j j j
= = = =
1 2 3 4
i=0
i=1
i=2
-9.6
-6.97 - 0.52i 0.126 - .817i - 1.095 + 15.02i -6.41 - 6.08i
-11.17 + 0.87i 0.122 - 1.45i 0.07 - 0.216i -1.77 + 22.73i
0.0926 4.23
4.3 Mathematical analysis o f the O.D.E.'s Before we consider the full five dimensional system (4.7) for all three of the modes (fundamental, harmonic and mean flow distortion) we first consider subsystems corresponding to the interactions of only two of the three modes. We examine the local and global stability of the equilibrium solutions. In the case of the fundamental-mean flow distortion we neglect the harmonic by substitution of X4 ~
X5 ~
0
into (4.7), transform t, xl, x2 and x3 according to R t=--t', rl,2
rl, 2
xl = - - Z , R
x2=
rl. 2
R
X,
F1'2
x3 = - -
y,
R
make use of (4.11.1) and neglect primes so that d X / d t = (R - Re) X - b Y + r l , 3 X Z - s1,3YZ
(4.12.1)
dY/dt = b X + (R - Re) Y + s l , 3 X Z + r l , a Y Z
(4.12.2)
dZ/dt = ro,1/rl,2Z + ro,3X 2 + ro,3Y z.
(4.12.3)
For the sake of comparison we note that (4.12) is the direct analogue of the 3-D Lorenz system which only included the fundamental and the mean flow distortion. The stability of the null solution X=Y=Z=O of (4.12) is obtained by substitution of X=0+x,
Y=0+y,
Z=0+z
into (4.12) and linearizing which gives dx/dt = Js x
(4.13)
where x = (x,y,z) r, b = (sl,1 + Sl,aR)/rl,z and Js is given by
J~=
b
R-Rc
0
0
0
ro,1/rl,2
.
(4.14)
Nonlinear stability of a viscous axisymmetric jet
143
Substitution of x = (CI, C2, c3)Te "~'t i n t o (4.13) and looking for nontrivial solutions gives the eigenvalues 2 of J~ to be
2 = (R - Rc • ib, ro,1/rl,2). Since ro,i/r~,2 < 0 and b is real, the stability of the null solution is governed by a single pair of complex conjugates whose real part crosses the imaginary axis at R = R~. Thus, R = Rc is a H o p f bifurcation point and bifurcating from the null solution at R = R~ is a unique periodic solution whose period T is given by T = lim [T(R)] = lira [2~/b]. R~Rr
(4.15)
R~R~
It can be shown that (4.12) has a periodic solution given by X = A cos (cot),
Y = - A sin (cot),
Z = zo
(4.16)
where Zo, co and A are given by 1
Zo -
/'1,3
(R - Re)
co = - ( b + sl,3Zo) A = (
to,1
(R
\rl,zrl,3ro,3
\1/2
- Rc))
.
Since ro,1/rl,2 < 0 by condition I, the second condition II imposes the inequality
rl,3ro,3 < 0
(4.17)
so that A is real when R > Re. The stability of (4.16) is obtained by substitution of X = A cos (cot) + U,
Y = - A sin (cot) + V,
Z = Zo + W
into (4.12) and linearizing which gives (4.18)
dV/dt = 4 ( 0 V where U = (U, V, W) T and Jr(t) = J;(t + 2~/co) is periodic and is given by
(0 --CO
co
A(rl,3 cos (cot) + sl,3 sin (coO))
0
A($1,3 cos (cot) -- rl,3 sin (cot)) .
2ro,3A cos (cot) - 2 r o , 3 A sin (cot) ro,1/rl,2 Next we define 7+ according to
2y+ = -ro,1/rl,2 +_ ]/(ro,z/rl,2) 2 + 8(R - Re) ro,1/rl,z and note that 9t(7+ ) > 0 for R > Rc since ro,1/rz,2 < 0. Then, it can be shown that the fundamental matrix solution U is given by U = P(t)
e Bt
144
J. Horwitz and S. Rosenblat
where P(t) = P(t + 2~/co) is given by t P(O =
sin (cot)
- c o s (cot)- sl,3 sin (cot) rl,3
cos (cot)
sl,3 cos (cot) + sin (cot) rl,3 2ro,3A ro,1/rl,2 + 7+
0
- c o s (cot)
-
sl,3 sin (cot) \ rl,3
\
_ sl,3 cos (cot) + sin (cot) ] F1,3 2ro,aA
ro,1/rli2 + 7-
/
and e ~ is given by
e Bt =
e-y-
(i ~ 0
t
:)
.
e -~-t
The Floquet multipliers Q and the corresponding Floquet exponents # are given by Q = 1, e -r-v, e -~-~,
# = 0, - 7 + , - 7 - .
Thus, according to Floquet theory the periodic solution (4.16) is orbitally stable since the two nonzero Floquet exponents # have negative real parts. The important result is that the sign of ro,1/r1,2 governs the stability of the periodic solution (4.16). We have already shown that ro,1/rl,2 < 0 is the condition I which was imposed on our system by the linear theory. Therefore, the periodic solution remains orbitally stable for R > Rc and this is a result that must be true for any choice of shape functions, and not just the Stuart-Watson functions we have chosen to use in this work. From this we can conclude that for any reasonable choice of shape functions, projection functions, and mean velocity profile, the supercritical solution which bifurcates from the null solution at R = Rc will remain orbitally stable for all R > R~. It is interesting to compare this result with that obtained from the Lorenz system where the bifurcation parameter is the Rayleigh number r. In the Lorenz system the null, or conduction solution is stable for r < 1 and loses stability to a pair of convection solutions. The interesting case is when the Prandtl number is high enough so that these convection solutions lose stability through a subcritical Hopf bifurcation point r , > 1. In addition, the divergence of the flow is negative (eliminating the possibility of quasiperiodic solutions) and a Liapunov function exists (solutions are bounded). This implies that for R > R~, the only possible solutions are periodic and chaotic ones which have both been found to exist by numerical methods. However, the previous results show that the Lorenz system and our model differ in a fundamental way. In particular, in the jet, there is no case analogous to the high Prandtl number case which yields a subcritical Hopf bifurcation and chaotic solutions. Next we verify the boundedness of the solutions of (4.12) which proves that we have not introduced any singularities into our system. We consider the Liapunov function
1 (X2+y2--rl'~3(Z-h(R))2)>O g= ~
(4.19)
to,3
where h(R) is a continuous function of R and is given by
h = h(R) =
3
-- 2r1,~3 0,
(R--Re),
for
R>Rc;
for
R < Re.
145
Nonlinear stability of a viscous axisymmetric jet If 12 is defined by .
.
OV OX . . 8X dt
+
8V dY 8Y dt
+
OV dZ 8Z dt
(4.20)
then substitution of (4.12) into (4.20) and making use of (4.19) leads to l) = (R - R~) ( X 2 -~- g2) + r l , 3 h ( X 2 q_ y2)
ro,lrl,3 Z(Z -- h).
(4.21)
rl,2ro,3
For the case R < Rc, h = 0 and (4.21) simplifies to give 171= (R - Rc) (X 2 + y2)
r~ Z 2 ~ O. rl,2ro,3
Therefore, the null solution is globally stable for R < Re. When R > Rc it can be shown by setting Pin (4.21) = 0 and completing the square in Z that I? = 0 on the surface of an ellipsoid E given by rt,2ro,a(R2ro,ara,3-Re) (X 2 q- y2) +
(
Z +
3(R-~-Rc!~Z = (3(R- ~---Rc!~2 4ra,3 ] \ 4rl.3 /
(4.22)
and l? < 0 outside the ellipsoid. Thus, all solutions (4.12) eventually become trapped in E and are therefore bounded for all time. These results also follow from conditions I and II and not from the particular shape functions we have chosen. Analogous results follow for the second subsystem of (4.7) where we neglected the fundamental by substitution of X2 ~
X3 ~
0
into (4.7). For example, the null solution loses stability at a Hopf bifurcation point/~. There is a supercritical periodic solution when the symmetry condition (analogue of (4.17)) r2,4ro,4 < 0
(4.23)
is imposed, and this solution remains orbitally stable for R > / ~ . The null solution is globally stable for R ~c. For R > / ~ all solutions become trapped in an ellipsoid/~. Finally, we may consider a third subsystem of (4.7) in which dxl/dt = 0.
(4.24)
Using the Liapunov function 5
V - -ro,1 ~ xl 2 > 0 R
=
i=2
and the subsystem obtained by substitution of (4.24) into (4.7) leads to = (ro.3(x2 2 + x3 2) + ro,~(x4 2 + x,2)). (r.,3(~2 ~ + ~3 2) + ~z,4(x~ ~ + xs~))
+ lower order terms. Thus, a sufficient condition for 1?< 0 for sufficiently large values of xi (i = 2 ..... 5) is that r1,3, r2,~ < 0
and
ro,3, ro,a > 0.
(4.25)
146
J. Horwitz and S. Rosenblat
It is interesting to note that the global stability condition (4.25) on the third subsystem is consistent with the Symmetry condition (4.17) and (4.23) on the first two subsystems. Finally, we have also verified that every coefficient of the nonlinear terms in (4.6) has the correct sign which supports our choice of shape functions and projection functions. Thus, we have shown that the mechanism whereby chaos occurs in the Lorenz system cannot occur here. Furthermore, this result must be true for any set of shape functions, and not just the particular ones that we have chosen. Therefore, what remains to be answered is, does adding in the harmonic to obtain the full 5-D system (4.7) change this result or not? To answer this question we need to use numerical methods and the discussion of the results is the subject of the next Section.
4.4 Numerical results The numerical results were obtained using A U T O [27] to track the stability of the steady and periodic solutions. When the periodic solutions were shown to possess a torus, a Runge-Kutta variable step size time integration scheme with arbitrary initial conditions was used. Characterization of the resulting time series was accomplished by analyzing its Fourier spectrum using the I M S L program FTFPS. The stability of the null solution X 1 ~X
2 ~X
3 =X
4 ~X
(4.26)
5 ~ 0
of (4.7) is obtained by substitution of
xi=O+Xi
for
i = 1..... 5 i
i
i
i
",,!
x~
//
iI
I I
"
/I\
I
x
--
I /
I
'" I
I
1
i",
: /9 ",-L J" I
t
a)
X
2,5 '
!!i i
[i f l
[iNi N, bo 0
q ~.L~ii'i',iilp'" "l F q ' 4 -Io. o I -I
0.0
1
I I
~ ;
i,
i 1 [
!
I
frequency
"
IiI 'ql O. 5
Fig. 1. R = 58 a Phase plane projection of b) x2 vs. xs h Fourier spectrum of x 2
Nonlinear stability of a viscous axisymmetric jet
147
into (4.7), linearizing and solving in the usual manner. This leads to the eigenvalues 2 of J~ which are given by
ZR = (ro,,, rL2(R -- g<) • isl,2(i + s,,,/Sl,2), r2,2(R - R J +_ is2,2(R
-}- $2,1/$2,2) ).
The stability of the null solution is governed by two pairs of complex conjugate eigenvalues. The two corresponding H o p f bifurcation points on the null solution occur at R = R~, = 55.32 and at R = R,~ = 91.56. The frequency f~ of the periodic solution occuring at R = R< is given by f~ = (sl,2 + s~a/R~,)/(2n) = 0.131 5.
(4.27)
This periodic solution has all nonzero components and so we designate it a 0 - 1 - 2 type solution to denote which amplitudes are nonzero in the original expansion (4.4). Figure l a shows the periodic phase plane projections of x2 vs. xs at R = 58. The corresponding F o u r i e r spectrum in Fig. l b shows one sharp peak at f = 0.1309 in agreement with (4.27). As R increases the fundamental frequency f l decreases. Figures 2 a and 2 b show the solution is still periodic at R = 70 but that f l has decreased to f~ = 0.1270. As R increases further the periodic solution loses stability at R = 72.3 as a pair of Floquet multipliers 01,2 cross the unit circle at 01,2
= --0.7403 + 0.6742i
i
1_I
.,
1
!
.
11
7!r
x8
i
I
i
~i
i
!/
....
-
i[
i
I
r!r
r\r/
t
!
i
[,
al
X2
2.5
! -4
4 'I "l o
I v
-
-i0.0
=,1 ,
0.0
is1
i
A /t
i
! f
.
~
~'
1
'~"
--_7. ~ = :
x x227-~.._._ ~ - - .
!
. . . . . . . I .... frequency
Fig. 2. R = 70 a Phase plane projection of b) x2 vs. xs b Fourier spectrum of x2
i
I ....
0.5
148
J. Horwitz and S. Rosenblat
,~,.-x-.~
z. 5
1
'~-.J. "~'~. ! .-~;/! A
t,,,,
k[ . . . . . . . . . . . .
.......
.~/N24--"7"
"%i.g+" ! ! N " ~ . : , ~ / ii i i ~ \ V \:%~ V \~%~
I
~.;z..2g,,~ ,
,~.,!. ~,
."
t",..t ~ . . - "
I
f
" I ~ '
"%J..:". \ !'t.-" /-~'q---k~:-i"- ", \
I
r'-,!L
",..\1
I
k~
t'1 '~-.
Y! "M ! ~' / ;'~//"-wl r162 I l l
, /
a)
z2 5.o
i
II
i(1, '
O)
Ilt
J
2
all'"
q 1
t,,
, .
I
[
' 'i [ I l l i il'
IIIIq "t"
ml'~1 I ' n ' F l I l ' [ ~ q T t ' ~ m " I/ I "t1 ~[ i! I /t I II
I
lI - 1 5 . 0 - I0.0
, ,
I
n
~
[
i
I
. . . . . . . .
I
. . . .
~equency
I
Fig. 3. R = 73 a Phase plane projection
b)
of x2 vs. x5 b Fourier spectrum of x 2
0.5
Table 2. Fourier spectrum at R = 73 (f2/fl = 0.36) (n, m)
f(,,m)
nf l + mf2
(1, -- 2) (1, -- 1) (1, 0) (1, 1)
0.033 2 0.0791 0.1240 0.1699
0.034 2 0.0791 0.1240 0.1689
indicating the possibility of a torus bifurcation. F l o q u e t theory predicts that the ratio of the new frequency f2 to the original frequency f l is given by
f2/fl = ( t a n - 1 ( - 0 . 6 7 4 2 / 0 . 7 4 0 3 ) ) / 2 = = 0.38.
(4.28)
A typical phase plane p r o j e c t i o n of the solution at R = 73 a p p e a r s in Fig. 3 a. The corresp o n d i n g F o u r i e r s p e c t r u m is shown in Fig. 3 b. The m a j o r f u n d a m e n t a l p e a k f l still a p p e a r s at f = 0.1240. If we choose f l = 0.1240 a n d f z = 0.0449 then there is g o o d a g r e e m e n t between the frequencies of the r e m a i n i n g peaks f(,,,,) and the family of frequencies rm where rm is given by 1",, = nfl + m f2
for
m = 0, _+ 1, _+2 . . . . . n = 1,
Nonlinear stability of a viscous axisymmetric jet
q I
I
I
[
I
l
I
I
, ~- t ~
Ml~qw'.a~"
I
i
i
, _ I~-.-,",--
I
.,,~--r162
,~-a.-------
i
I ,
I
,
149 1
, +
t
~
~
I
x5 gl
~,C.-Te~.:"C'~-,?---J Y I / \\'~\ . ..e.W./. . . . /d/L]tql .
.21
I ~ "'-
%
~ ' \ . .\ .' 4. .
%~ %~k I
I.J ..,"
9
~
.
I
~%
I
I
] . . .
I
.
.
.
Ii
I
, .
, g,,
'
I 9
;,,'.r~:~'l
.~."LZ#N % "%kl II 9 /// -.4;~- ~--; k%k)\ .)111
'
"::,'K I
I 4
~,
- -
.~
'% / 2~
I
, . . . .
I
, . . . .
I
.
I
.
.
a)
X2
!0,o)
5'~i
(1, - 1 ) "-2-(1 , _
o
i i
hO
2),I 'I,,; ,;I"
|, ; 1 1 2)
J(:
( ,iT3)I / " 4 JV -,~
i
IPm,.
I ,,T'"' 't" .,?r[ I
-i0"0 0.~' '
'I
.
.
.
.
.
.
.
.
.
.
.
.
frequency
1 o.J
b)
Fig. 4. R = 75 a Phase plane projection of x2 vs. x5 b Fourier spectrum of X 2
Table 3. Fourier spectrum at R = 75 (f2/fl = 0.32)
(n, m)
f{,,m)
nfl + mf2
(1, -- 3) (1, -- 2) (1, -- 1) (1, O) (1, 1) (1, 2) (1, 3)
0.003 4 0.042 5 0.081 5 0.1201 0.1592 0.198 2 0.2368
0.004 3 0.042 9 0.081 5 0.1201 0.1587 0.197 3 0.2359
as s h o w n b y c o m p a r i n g the last two c o l u m n s in Table 2. N o t e that f2/fl = 0.36 which is in close a g r e e m e n t with (4.28). As R increases the ratio f2/fl decreases a n d f2/fl = 0.32 at R = 75. The phase plane projection a n d c o r r e s p o n d i n g F o u r i e r s p e c t r u m are given in Figs. 4 a a n d 4b. If f l = 0.1201 a n d f2 = 0.038 6 the r e m a i n i n g peaks f(m,,) m a y be indexed as s h o w n in Table 3, which include the first seven m e m b e r s of rm. At R = 78 the ratio fz/fa has decreased to 0.27. Figures 5 a a n d 5 b show the phase plane projection a n d F o u r i e r s p e c t r u m at R = 78. If f l = 0.116 2 a n d f2 = 0.030 8 t h e n the r e m a i n i n g peaks m a y be indexed as s h o w n in Table 4. F i g u r e 6 shows that the ratiof2/f~ decreases m o n o t o n i c a l l y o n the interval 72.3 < R < 78. Since the solutions in this interval c o n t a i n frequencies which can be indexed by a c o m b i n a t i o n of integral multiples of two i n c o m m e n s u r a t e frequencies f l a n d f2 a n d the ratiof2/fl decreases over
150
J. Horwitz and S. Rosenblat
i
I
3
x5
,. _.:~......
!
/~ll.
1 I
/
t
# # t "1 / / ~l
!U" I I
t
~
/
.KT," xl I
I
tl~
I
I\r,,
I
\
k
I\
"~
I
I
t/
',
VA
I
I
~./~
v
d
"--.
t
...~
%
i
i
I
#
.t
7-s / r ~ s ,' /
is.."
#'~" "~'
I
~" t
\
~._
;,
"-
i
"w'~.
/
il
~-
,
1' I'~.~
"?w
,-~. ,,"
..-"
~
"~
",-.
i
% ~" I
I
i}
, I
I
I 't 1 \
I"~
I
,.~
,
I
I
iN
1
~ } ~
~ [ 2
/'-f--~2+-~, t
-i
,
%
~
% I
I
I
I
i
%
,'
~.l
~ + l
I
i
"~
I
I
i
r~..'
I
-
I I I
I
I
i
I
t
I
z2
5.0.5q
t
i_(1
N
1,~
I1 (lil) !
-
,
' II t l l ~ i~'l~'ff,[hd'l.~tW~L~ m | ~
~ k L 1. s L
.L
Fig. 5. R = 78 a Phase plane projection of
-io.o "i
. . . . . .
i,
0.0
I
....
I
....
I
....
frequency
Table 4. Fourier spectrum at R = 78 (f2/fl
I b) x2 vs. x5 b Fourier spectrum of x2 O. 5
0.27)
(n, m)
f(,,,,)
nfl + mfz
(1, - 3) (1, - 2)
0.024 4 0.054 7
0.023 8 0.054 6
(1, - 1) (1, 0)
0.0854 0.1161
0.0854 0.1162
(1, 1)
0.1465
0.1470
(1, 2) (1, 3)
0.1772 0.2080
0.177 8 0.208 6
the interval, there is evidence that the solution is a Q2 (quasiperiodic solution with two incommensurate frequencies) solution. As R increases past 78, all of the frequencies move toward the major frequency f l and the solution becomes periodic again. Figures 7a and 7b show the phase plane projection and the Fourier spectrum at R = 80. There is only one sharp peak at f = 0 . 1 1 9 1 . As R increases past 100 there is a gradual transition to intermittency. Figure 8 a gives the time series at R = 200 which shows the fairly regular behavior is interrupted by chaotic bursts. Figure 8b shows the phase plane projection and Fig. 8c shows the Fourier spectrum with broad band behavior. As R increases further up toward R = 1000, the time interval between the chaotic bursts increases. Figure 9a shows the time series at R = 600 and Fig. 9 b shows that the Fourier spectrum has broad band behavior.
Nonlinear stability of a viscous axisymmetric jet
151
0.38
0.36
0.3/..
f2/fl 0.32
0.30
0.28
0.26
,,,,
72
73
q,ll
i~
ir
iiii
i
75
7&
i
i
i
i
77
78
r
~
I
78
Fig. 6. Ratio j)/fa vs. R
R
x~
~z
i_i
~
r
~
II i ;'
\
t
t,,
it i i
I i
It
\
"7
i 71~i.i ,!,~
F
9
9
,,(
,,
sl f i
%1s
I
X i
/
/
%
i
,s
I
i
i
i I ..i i , ~ , , ! ....
.......
1.
. . . . . .
....
i
r
~-
i
a)
i
i I . . . . . . ! .... ! . . . . . . . .
i
,
x2 5,0
""!
I i Ii mi !I
d
t
I /
N?
L~,
!
'~
--10,0
I
O. o
i
'~a.. ......
i
I
I
~
I
~
I
~
J
,
frequency
I
i
l
i
Fig. 7. R = 80 a Phase plane projection of b) Xz vs. x5 b Fourier spectrum of x2
o.'
T h e frequency of the periodic s o l u t i o n o c c u r i n g at R = Re2 is given by %
=
( $ 2 , 2 -}-
s2,t/Rc=)/(2=) =
0.2322.
This p e r i o d i c s o l u t i o n has c o m p o n e n t s x2 = x3 = 0 a n d is labeled a 2 - 0 type solution. It is supercritical and unstable. These results are s u m m a r i z e d in the bifurcation d i a g r a m in Fig. 10, where the vertical axis represents the E u c l i d e a n n o r m of the solution. T h e sequence of bifm'cations as R increases is I:
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152
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although the last transition may be more complex. Since the range of validity of (4.4) is still unknown we cannot say whether the Q2 and intermittency solutions are artifacts of the truncation or if they would persist in a higher order system. The best we can say is that the m a x i m u m amplitudes of Ao and A2 do remain smaller than the m a x i m u m amplitude of A1 for the interval in R that has been examined. The ratio of max JA21/max IAll which is shown in Fig. 11 is fairly constant and small ( ~ .1) in the interval 70 < R < 400. This can only be regarded as weak evidence that the series (4.4) captures the dominant behavior of a higher dimensional system using a particular set of basis functions
Sin(r). Unfortunately we still do not k n o w if our choice of shape functions is the proper one to use. For example it is possible to choose a different set of shape functions and still obtain
Nonlinear stability of a viscous axisymmetric jet
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results that are consistent with the Stuart-Watson results for R close to Re. Using the shape functions
{So(r), sl(r), s2(,-)} = {~,2o, 01, ~1} we found that the resulting sequence of bifurcations as R increases is II:
Steady --, Periodic ~ Quasiperiodic (Qa) ~ Homoclinic Orbit.
For large enough values of R the solutions eventually become unbounded. A possible explanation for the difference between sequence I and II is that the choice of 01 for S2(r) changes the sign of r2, 4 and therefore (4.23) and (4.25) are no longer satisfied. Thus, the periodic solution at Re1is supercritical and the periodic solution at Re2is subcritical. Therefore, the proper symmetry does not exist, which leads to singularities in the system. We also found sequence II if we neglected the mean flow distortion ~'z0. However, the c o m m o n feature in both sequences I and II is that the secondary bifurcation leads to a Q2 solution rather than to a chaotic regime. This is not surprising since we have already found that the 3-D analogue of the Lorenz system (4.12) does not have chaotic solutions either. Thus, we conclude by saying that the low dimensional system of amplitude equations leads to the sequence Steady -~ Periodic ~ Quasiperiodic (Q2) and whether intermittency or homoclinic orbits are observed for higher values of R depends on the choice of shape functions or whether or not the mean flow distortion term is incIuded.
5 Conclusions The nonlinear stability of a viscous axisymmetric jet having a pseudo-hyperbolic tangent mean profile has been investigated. A severely truncated form of a Fourier modal expansion has taken the form of an axisymmetric disturbance. The main assumptions leading to the low order model of amplitude equations are:
Nonlinear stability of a viscous axisymmetric jet
155
(i) parallel flow of mean velocity, (ii) decomposition of disturbance into separate modes in each of the two spatial directions z and r, (iii) severe truncation of an infinite series and (iv) Stuart-Watson type shape functions in the radial direction. In an effort to keep the dimension of the system of O.D.E.'s low, we have limited the number of terms in the expansion of the perturbation. Only those terms which would arise at first and second order in a Stuart-Watson perturbation expansion have been retained. In the case of axisymmetric disturbances we were able to obtain the 3-D analogue of the Lorenz equations which includes only the fundamental and mean flow distortion. Unlike the Lorenz system which has chaotic solutions when the Prandtl number is high enough, there are no chaotic solutions for the model of the jet, since there is only an orbitally stable periodic solution for R > Re. We have also shown that the absence of chaotic solutions is independent of the choice of shape functions. Adding in the harmonic to give a 5-D system of O.D.E.'s causes the periodic solution to lose stability to a quasiperiodic solution with 2 incommensurate frequencies. For one choice of shape functions the quasiperiodic branch eventually goes through a periodic regime and then becomes intermittent. However, a second choice of shape functions leads to a homoclinic orbit and unbounded solutions following the Q2 regime. The common feature that distinguishes these low order models, which represent an open flow system from the Lorenz system, which represents a closed flow system, is that the secondary bifurcation leads to a Q2 regime rather than to chaos. Of course the range of validity of these expansions remains untested. Thus, the main conclusion is that a very low order truncation does not lead to a model which has chaotic solutions and the model is therefore inadequate to study transition to turbulence. Higher order truncations could verify whether the results obtained in this work still persist or whether they are artifacts of the truncation used, and in theory it can be done. However, in view of the large amount of work required to perform even a severe truncation using only three terms it would not be practical to use the Fourier modal expansion to include higher order terms, especially in the case of nonaxisymmetric disturbances. A better approach would be to use an entirely different method such as finite differences or finite elements. This work represents the first attempt to proceed beyond the linear stability analysis for the axisymmetric jet to calculate the Landau constant which can then be used to derive a low dimensional system of O.D.E.'s which is valid for R close to Re, We have also incorporated three dimensional helical disturbances and derived a ten dimensional model using the Fourier modal expansion. These results will be reported in a forthcoming paper.
Acknowledgement This work was supported by AFOSR Grant 86-0165.
References
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J. Horwitz and S. Rosenblat: Nonlinear stability of a viscous axisymmetric jet
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Chicago, IL 60616, U.S.A.