asymptotic stability of unsteady inviscid stratified flows
SEPARATUM
ASYMPTOTIC STABILITY OF UNSTEADY INVISCID STRATIFIED FLOWS S. CARMI*, and S. C. SINHA** [Manuscript
received:
15 March,
1978]
The stability of modulated atmospheric flows is analyzed. The equations govern. ing the disturbance motion are solved by Galerkin expansions with time-dependent coefficients. Asymptotic stability bounds are then established by constructing Liapuno"\ functions for the resulting differential systems.
1. Introduction The stability
I
of inviscid stably stratified
\ ~ subject ~ls
of intense interest to researches of continuous velocity and temperature . -'-, --.:I , ~C h_~ ~~C;~~..~
flows in a gravitational
fielc
of atmospheric phenomena profiles in infinite or sern' lo"o~o ,,~:th "'finnI hu""
~ AKAD~MIAI KIADO, BUDAPEST JDUBLISHING HOUSE OF THE HUNGARIAN ACADEMY OF SCIENCES VERLAG DER UNGARISCHEN AKADEMIE DER WISSENSCHAFTEN MAISON D'EDITIONS DE L'ACADEMIE DES SCIENCES DE HONGRIE 113):{ATEJIbCTBO AKA):{EMI1I1HAYI{ BEHrpI111
Acta Technica
Academiae
Scientiarum
Hungaricae,
Tomus 89 (1-2),
pp. 61-67
(1979)
ASYMPTOTIC STABILITY OF UNSTEADY INVISCID STRATIFIED FLOWS S. CARMI*, and S. C. SINHA** [Manuscript
received:
15 March,
1978]
The stability of modulated atmospheric flows is analyzed. The equations governing the disturbance motion are solved by Galerkin expansions with time-dependent coefficients. Asymptotic stability bounds are then established by constructing Liapunov functions for the resulting differential systems.
I. Introduction The stability of inviscid stably stratified flows in a gravitational field is a subject of intense interest to researches of atmospheric phenomena. Models of continuous velocity and temperature profiles in infinite or semiinfinite domains and ones of two semi-infinite layers with constant but different velocities were analyzed by many authors (for reviews see for example DRAZIN and HOWARD [5], LINDZEN [9] and EINAUDI and LALAS [6]). In the above studies linear theory was used and the basic velocity and density were varying only with height. ' In the current work the asymptotic stability of modulated atmospheric flows will be investigated. The basic flow variables will be time dependent in addition to their spatial variation and the modulation will be generated through the boundary (YIH [12] and ROSENBLAT and TANAKA [10]). Nonlinear stability analyses of thermoconvective and other modulated viscous flows were recently presented by DAVIS [4], HOMSY [7] and CARMI [2], but no such investigations were attempted yet for inviscid stratified fluids.
2. Basic flow and perturbation The governing equations of motion continuously stratified fluid are
-de dt
+ eV. v
equations
for
= 0
an inviscid,
nonconducting,
(1)
'
dV
(2)
e & = Vp - egk, * S. CARMI, Ph. D., Mechanical Engineering Department, Michigan 48202, USA . ** S. C. SINHA, Ph. D.: Mechanical Engineering Department, Manhattan, Kansas 66506, USA
Wayne
State
University,
Detroit,
Acta
Technica
Academiae
Kansas State University,
Scientiarum
Hungaricae
89, 1979
62
CARMI,
S.-SINHA,
S, C,
with V = (U, V, W), e, p are the velocity, density and pressure, respectively, g
=
gravitational
constant,
k
=
~o dt
normal
unit
vector
and
where
= a;- + v. V.
To system (1), (2) we apply the Boussinesq approximation yielding the solenoidity and incompressibility conditions in addition to the momentum equation V'
V=
0,
dew - .!!..-es dt g dV es a;,
WIt
h
sound
2 ~
--
Ides g = -g (-es - dz + 2Co = and
W = 0,
(3)
Vp -gek,
B runt-
v alsa 1a ,,,
"
"
f requency,
Co = spee
d
0
f
where
+ Vw(x, t) , Ps(z) + Pw(x, t) ,
V = Vs(z)
p = e = es(z) + ew(x, t) .
The subscripts s, w designate the steady and modulated parts, In the Boussinesq approximation we used ew = 0 (0-) with
1 a = -max
respectively.
(LIes) ~ 1 and em = Ave es.
em
Z
By subtracting two flows ("starred" and "unstarred") which satisfy the same equations and boundary conditions but vary in their initial conditions we get the system governing the disturbance flow
v. u = 0, de'
-+
U'
(~: +
U
dt
es
Ve'
+ U 'Vew--esw
. Vu + U . vp') =
'}{,2
= 0,
(4)
g
- Vp' - ge' k,
where u = (u, v, w) = V* - V, p' = p* - p, (/ = e* - e are the disturbance variables. System (4) can serve as a starting point for both a linear and nonlinear theory analysis. Acta
Technica
Academiae
Scientiarum
Hungaricae
89, 1979
ASYMPTOTIC
Here, attention obtaining
we will consider
STABILITY
infinitesimal
to two-dimensional from (4)
systems
63
OF FLOWS
disturbances
(assuming
and will restrict
that
Squire's
theorem
our holds)
ou + ow = 0 , ox OZ d '
X2
dt
g
_L_-
esw
= 0,
du oU es-+es-w+-=O dt OZ dw
-+ge1
In the above, the atmospheric of a boundary layer type
velocity
dt
oP/ ox
oP/
es-+
(5)
= O.
OZ
is a two dimensional
v = (U, 0, 0,) with U = U(z, t) = Uiz) where e is not necessarily it is obtained by solving Since the basic flow Fourier Transform of (5) eliminating u, p'
0
-+
where
ikU
(
ot
(
:t +
)(
02ip oz2
--
ikU) ffJ-
we introduced
where
(1 =
(6)
)
J1jJ =
OU
--(1-
OZ2
l
OZ )
1jJ =
0, (7)
0,
the new variables
-1
es
comes the Richardson
des
dt
=
w e~/2, ratio
Number
ffJ= ge' e-;1/2,
of velocity
z, t
velocity
and
density
with
in the usual sense when
max
characteristic
+ eUw(z, t) ,
02U rJ21jJ-k2cp-ik
k = horizontal wave number, rJ= k2 + 1/4 (12 and
The
flow
small. This model is dynamically admissible since (1) with periodic upper boundary conditions. is a function of z and t only, we can take the xyielding in dimensionless form after using (6) and
1jJ =
and
parallel
scale
heights,
J = X2h2/U~ which
be-
-oU = --U0 h
OZ
and length Acta
are Technica
U 0 and Academiae
h, respectively. Scientiarum
Hungaricae
For the 89, 1979
64
CARMI, S.-SINHA,
S. C.
steady case, equations (7) reduce to the Taylor-Goldstein system which was previously solved for various sets of boundary conditions. System (7) is completed by specifying initial and boundary conditions. Here we will take tp = 0 at z with a necessarily
similar
=0
requirement
and z
(8)
-~OO,
for f{! as seen from (7).
3. Stability problem formulation Since the coefficients in (7) Laplace Transform method as can bounds from system (7), (8) we assume that tp, f{! can 'be written tp
=
depend on both z and t we cannot apply the be done in the steady case. To obtain stability first utilize the Galerkin method. Thus we as
-
~
An(t) Pn(z) , n=1
(9)
-
f{!
= n=1 ~ Bn(t) @n(z) ,
where Pn (t), @n (z) are complete orthogonal sets in z : (0, (0) satisfying the boundary conditions in (8) and with An (t), Bn (t) unknown functions of time. Substituting (9) in (7) yields
j' An(t)(P~n=1
rfPn)
+ ikU n~1 j' An(t)(P~
=
-
ik
.i
n=1
~
n=1
- rf~)
-
= (U - O"U')An(t)~ - k2
Bn(t)@n
+ ikU
~ Bn(t) @n -
n=1
~
n=1
J
Bn(t) @n =
j'
n=1
An(t) Pn
0,
(10)
= O.
Since the set Pn (z), @n(z) are known the z-dependence can be eliminated by multiplying them into (lOa, b) and integrating over the z-range. After performing the integration, truncating the series at N and rearranging we get (for details see CARMI and SHULZE [3]) dA ~
dt
N
=
-
ik
dB ~
where 4ela
dt
lXimn" i = 1 Technica
4eademiae
~ n=1
N
An(t)lXlmn
N
= - ik
~
n~1
+ k2 n=1 ~ Bn(t)1X2mn N
Bn(t)lXsmn
+ J n=1 ~ An(t)1X4mn ,
4 are known complex functions of t. Scientiarum
,
Hungaricae
89. 1979
(11)
ASYMPTOTIC
We can reduce
The vectors
(xi (t)
OF FLOWS
(11) to a single real vector Xi
using summation tions
STABILITY
= Gij(t)
65
equation
(12)
Xj,
convention with j = 1, . . . 4N and suhject to initial condi-
Xi (t) incorporate
.
X~n)(o) =
~in
the real
and imaginary
(13) parts
of Am (t), Bm (t)
= Re (AI (t)), Xz (t) = Re (BI (t)), etc.) and the matrix GiJ(t) is T-periodic. 4. Stability of periodically modulated flows
We will now ohtain general stahility hounds for periodically modulated flows hy applying lTAPUNOV'S second method and making use of FLNQUET'S theory and SYLVESTER'S test as descrihed in [11]. In periodic modulation of a steady atmospheric flow the velocity field (6) can he assumed in the form
U(z, t)
= a(z) + 8f(z)g(t) ,
(14)
where g (t) is T-periodic. The hasic velocity (14) renders (12) in the form Xi with
Cij a constant Taking 8 =
matrix
= (Cij + 8Biit))Xj,
(15)
and Bij (t) T-periodic.
0 we get from (15) the steady case (using direct notation) X= Cx
(16)
which is stahle assuming all the eigenvalues In this case, there exists a positive definite coefficients such that
of C lie in the left half-plane. quadratic form with constant
v = (Hx, x); (HT = H> 0) , whose
derivative
along the trajectory
. dV V = - dt = (HCx, x)
+ (Hx,
(17)
of the system
Cx) = ([HC
+
T
C H]x, x)
=
-
(Go x, x),
(18)
is negative definite. The matrix H can he found from (18) where Go is any positive matrix (e.g. Go = {JI, (J > 0). 5
Acta Technica
Academiae
ScientiaTum
HungaTicae 89, 1979
66
CARMI, S.-SINHA,
For the modulated SKU [11])
case (15) we have (see YAKUBOVICH and STARZHIN-
v= where
S. C.
G1 (t) is a T-periodic
+ sG1(t)]x,x),
([Go
-
matrix
function
(19)
given by
+ B!(t)H) .
G1(t) = - (HB(t)
+
We now let LJi (t, s) denote the principal minors of the matrix Go sG1 (t). As Go is positive definite, it follows from SYLVESTER'S text (see [11, p, 30]) that LJi (t, 0) > 0 (i =1, 2, . . . n). Since minors LJi (t, s) are continuous of both t and s we have for sufficiently small So 0
>
LJi(t,s)
>0
(i = 1,2,
. . . n)
for
0
< t :s;;: T ,
(20)
Isl:S;;: BO'
I
From (20) it follows that Go + sG1 (t) > 0 for 0 < t < T, s I< Bo hence by continuity there exists b > 0 such that all the principal minors of Go + BG1 (t) - bI are positive for 0 < t < T, B I< So' Therefore V in (19)
I
satisfies the inequality tion of (15) for B
I I
O
(24)
ASYMPTOTIC
Since the steady
state
STABILITY
67
OF FLOWS
case S = 0 is asymptotically
stable, it follows that
there exists a positive constant a such that II x II < C2e-atfor t > if II sEe-6t II < Cl, one can easily show that [1] equation (24) yields -+ 0 as t -+ CX) if ClC2 < a.
O.Then II
x
II -+
Actual stability bounds like So in (20) are now being established for model atmospheric flows (e.g. boundary layer type) and results will be reported in a subsequent paper.
Acknowledgements This work was supported
by the U.S. Army Research
Office.
REFERENCES 1. BELLMAN, R.: Stability Theory of Differential Equations, Dover Publications, Inc., N. Y., 1969 2. CARMI, S.: Energy Stability of Modulated Flows, Phys. Fluids, 17 (1974), 1951-1955 3. CARMI, S.-SCHULZE, R. S.: Stability of Modulated Flows in the Atmosphere, Proc. Sixth Can. Congo Appl. Mech., (1977), 617 -618 4. DAVIS, S. H.: Finite Amplitude Instability of Time-Dependent Flows, J. Fluid Mech., 45 (1970), 38-48 5. DRAZIN, P. G.-HOWARD, L. N.: Hydrodynamic Stability of Parallel Flow of Inviscid Flnids, Adv. Appl. Mech., 9 (1966),1-89 6. EINAUDI, F.-LALAS, D. P.: Some New Properties of Kelvin-Helmholtz Waves in an Atmosphere with and without Condensation Effects,J. Atm. Sci., 31 (1974), 1995-2007 7. HOMSY, G. M.: Global Stability of Time-Dependent Flows: Impulsively Heated or Cooled Fluid Layers, J. Fluid Mech., 60 (1973), 129-139 8. LAKSHMIKANTHAM,V.-LEELA, S.: Differential and Integral Inequalities, Vol. 1, Academic Press, N. Y., 1969 9. LINDZEN, R. S.: Stability of a Helmholtz Velocity Profile in a Continuously Stratified Infinite Boussinesq Fluid. Application to Clear Air Turbulence, J. Atm. Sci., 31 (1974), 1507-1514 10. ROSENBLAT, S.-TANAKA, G. A.: Modulation of Thermal Convection Instability, Phys. Fluids, 14 (1971), 1319-1322 11. YAKUBOVICH, V. A.-STARZHINSKII, V. M.: Linear Differential Equations with Periodic Coefficients, Vols. I and II, John Wiley & Sons, N. Y., 1975 12. YIH, C. 5.: Instability of Unsteady Flow Configurations, Part 1. Instability of a Horizontal Liquid Layer on an Oscillating Plane, J. Fluid Mech., 31 (1968),737-752
Asymptotische Stabilitat der nichtviskosen laminaren Stromung. - Die Stabilitat der modulierten atmospherischen Stromung wird untersucht. Die Grundgleichungen der Stromungsbewegungen werden mit Hilfe der Galerkinschen Reihenentwicklung mit zeitabhangigen Koeffizienten gelOst. Sodann werden durch Herstellung der Liapunovschen Funktionen die Stabilitatsgrenzen fiir die resultierenden Differentialsysteme ermittelt.
5*
Acta
Technica
Academia.
Sci.ntiarum
Hungarica.
89, 1979
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ASYMPTOTIC STABILITY OF UNSTEADY INVISCID STRATIFIED FLOWS S. CARMI*, and S. C. SINHA** [Manuscript
received:
15 March,
1978]
The stability of modulated atmospheric flows is analyzed. The equations govern. ing the disturbance motion are solved by Galerkin expansions with time-dependent coefficients. Asymptotic stability bounds are then established by constructing Liapuno"\ functions for the resulting differential systems.
1. Introduction The stability
I
of inviscid stably stratified
\ ~ subject ~ls
of intense interest to researches of continuous velocity and temperature . -'-, --.:I , ~C h_~ ~~C;~~..~
flows in a gravitational
fielc
of atmospheric phenomena profiles in infinite or sern' lo"o~o ,,~:th "'finnI hu""
~ AKAD~MIAI KIADO, BUDAPEST JDUBLISHING HOUSE OF THE HUNGARIAN ACADEMY OF SCIENCES VERLAG DER UNGARISCHEN AKADEMIE DER WISSENSCHAFTEN MAISON D'EDITIONS DE L'ACADEMIE DES SCIENCES DE HONGRIE 113):{ATEJIbCTBO AKA):{EMI1I1HAYI{ BEHrpI111
Acta Technica
Academiae
Scientiarum
Hungaricae,
Tomus 89 (1-2),
pp. 61-67
(1979)
ASYMPTOTIC STABILITY OF UNSTEADY INVISCID STRATIFIED FLOWS S. CARMI*, and S. C. SINHA** [Manuscript
received:
15 March,
1978]
The stability of modulated atmospheric flows is analyzed. The equations governing the disturbance motion are solved by Galerkin expansions with time-dependent coefficients. Asymptotic stability bounds are then established by constructing Liapunov functions for the resulting differential systems.
I. Introduction The stability of inviscid stably stratified flows in a gravitational field is a subject of intense interest to researches of atmospheric phenomena. Models of continuous velocity and temperature profiles in infinite or semiinfinite domains and ones of two semi-infinite layers with constant but different velocities were analyzed by many authors (for reviews see for example DRAZIN and HOWARD [5], LINDZEN [9] and EINAUDI and LALAS [6]). In the above studies linear theory was used and the basic velocity and density were varying only with height. ' In the current work the asymptotic stability of modulated atmospheric flows will be investigated. The basic flow variables will be time dependent in addition to their spatial variation and the modulation will be generated through the boundary (YIH [12] and ROSENBLAT and TANAKA [10]). Nonlinear stability analyses of thermoconvective and other modulated viscous flows were recently presented by DAVIS [4], HOMSY [7] and CARMI [2], but no such investigations were attempted yet for inviscid stratified fluids.
2. Basic flow and perturbation The governing equations of motion continuously stratified fluid are
-de dt
+ eV. v
equations
for
= 0
an inviscid,
nonconducting,
(1)
'
dV
(2)
e & = Vp - egk, * S. CARMI, Ph. D., Mechanical Engineering Department, Michigan 48202, USA . ** S. C. SINHA, Ph. D.: Mechanical Engineering Department, Manhattan, Kansas 66506, USA
Wayne
State
University,
Detroit,
Acta
Technica
Academiae
Kansas State University,
Scientiarum
Hungaricae
89, 1979
62
CARMI,
S.-SINHA,
S, C,
with V = (U, V, W), e, p are the velocity, density and pressure, respectively, g
=
gravitational
constant,
k
=
~o dt
normal
unit
vector
and
where
= a;- + v. V.
To system (1), (2) we apply the Boussinesq approximation yielding the solenoidity and incompressibility conditions in addition to the momentum equation V'
V=
0,
dew - .!!..-es dt g dV es a;,
WIt
h
sound
2 ~
--
Ides g = -g (-es - dz + 2Co = and
W = 0,
(3)
Vp -gek,
B runt-
v alsa 1a ,,,
"
"
f requency,
Co = spee
d
0
f
where
+ Vw(x, t) , Ps(z) + Pw(x, t) ,
V = Vs(z)
p = e = es(z) + ew(x, t) .
The subscripts s, w designate the steady and modulated parts, In the Boussinesq approximation we used ew = 0 (0-) with
1 a = -max
respectively.
(LIes) ~ 1 and em = Ave es.
em
Z
By subtracting two flows ("starred" and "unstarred") which satisfy the same equations and boundary conditions but vary in their initial conditions we get the system governing the disturbance flow
v. u = 0, de'
-+
U'
(~: +
U
dt
es
Ve'
+ U 'Vew--esw
. Vu + U . vp') =
'}{,2
= 0,
(4)
g
- Vp' - ge' k,
where u = (u, v, w) = V* - V, p' = p* - p, (/ = e* - e are the disturbance variables. System (4) can serve as a starting point for both a linear and nonlinear theory analysis. Acta
Technica
Academiae
Scientiarum
Hungaricae
89, 1979
ASYMPTOTIC
Here, attention obtaining
we will consider
STABILITY
infinitesimal
to two-dimensional from (4)
systems
63
OF FLOWS
disturbances
(assuming
and will restrict
that
Squire's
theorem
our holds)
ou + ow = 0 , ox OZ d '
X2
dt
g
_L_-
esw
= 0,
du oU es-+es-w+-=O dt OZ dw
-+ge1
In the above, the atmospheric of a boundary layer type
velocity
dt
oP/ ox
oP/
es-+
(5)
= O.
OZ
is a two dimensional
v = (U, 0, 0,) with U = U(z, t) = Uiz) where e is not necessarily it is obtained by solving Since the basic flow Fourier Transform of (5) eliminating u, p'
0
-+
where
ikU
(
ot
(
:t +
)(
02ip oz2
--
ikU) ffJ-
we introduced
where
(1 =
(6)
)
J1jJ =
OU
--(1-
OZ2
l
OZ )
1jJ =
0, (7)
0,
the new variables
-1
es
comes the Richardson
des
dt
=
w e~/2, ratio
Number
ffJ= ge' e-;1/2,
of velocity
z, t
velocity
and
density
with
in the usual sense when
max
characteristic
+ eUw(z, t) ,
02U rJ21jJ-k2cp-ik
k = horizontal wave number, rJ= k2 + 1/4 (12 and
The
flow
small. This model is dynamically admissible since (1) with periodic upper boundary conditions. is a function of z and t only, we can take the xyielding in dimensionless form after using (6) and
1jJ =
and
parallel
scale
heights,
J = X2h2/U~ which
be-
-oU = --U0 h
OZ
and length Acta
are Technica
U 0 and Academiae
h, respectively. Scientiarum
Hungaricae
For the 89, 1979
64
CARMI, S.-SINHA,
S. C.
steady case, equations (7) reduce to the Taylor-Goldstein system which was previously solved for various sets of boundary conditions. System (7) is completed by specifying initial and boundary conditions. Here we will take tp = 0 at z with a necessarily
similar
=0
requirement
and z
(8)
-~OO,
for f{! as seen from (7).
3. Stability problem formulation Since the coefficients in (7) Laplace Transform method as can bounds from system (7), (8) we assume that tp, f{! can 'be written tp
=
depend on both z and t we cannot apply the be done in the steady case. To obtain stability first utilize the Galerkin method. Thus we as
-
~
An(t) Pn(z) , n=1
(9)
-
f{!
= n=1 ~ Bn(t) @n(z) ,
where Pn (t), @n (z) are complete orthogonal sets in z : (0, (0) satisfying the boundary conditions in (8) and with An (t), Bn (t) unknown functions of time. Substituting (9) in (7) yields
j' An(t)(P~n=1
rfPn)
+ ikU n~1 j' An(t)(P~
=
-
ik
.i
n=1
~
n=1
- rf~)
-
= (U - O"U')An(t)~ - k2
Bn(t)@n
+ ikU
~ Bn(t) @n -
n=1
~
n=1
J
Bn(t) @n =
j'
n=1
An(t) Pn
0,
(10)
= O.
Since the set Pn (z), @n(z) are known the z-dependence can be eliminated by multiplying them into (lOa, b) and integrating over the z-range. After performing the integration, truncating the series at N and rearranging we get (for details see CARMI and SHULZE [3]) dA ~
dt
N
=
-
ik
dB ~
where 4ela
dt
lXimn" i = 1 Technica
4eademiae
~ n=1
N
An(t)lXlmn
N
= - ik
~
n~1
+ k2 n=1 ~ Bn(t)1X2mn N
Bn(t)lXsmn
+ J n=1 ~ An(t)1X4mn ,
4 are known complex functions of t. Scientiarum
,
Hungaricae
89. 1979
(11)
ASYMPTOTIC
We can reduce
The vectors
(xi (t)
OF FLOWS
(11) to a single real vector Xi
using summation tions
STABILITY
= Gij(t)
65
equation
(12)
Xj,
convention with j = 1, . . . 4N and suhject to initial condi-
Xi (t) incorporate
.
X~n)(o) =
~in
the real
and imaginary
(13) parts
of Am (t), Bm (t)
= Re (AI (t)), Xz (t) = Re (BI (t)), etc.) and the matrix GiJ(t) is T-periodic. 4. Stability of periodically modulated flows
We will now ohtain general stahility hounds for periodically modulated flows hy applying lTAPUNOV'S second method and making use of FLNQUET'S theory and SYLVESTER'S test as descrihed in [11]. In periodic modulation of a steady atmospheric flow the velocity field (6) can he assumed in the form
U(z, t)
= a(z) + 8f(z)g(t) ,
(14)
where g (t) is T-periodic. The hasic velocity (14) renders (12) in the form Xi with
Cij a constant Taking 8 =
matrix
= (Cij + 8Biit))Xj,
(15)
and Bij (t) T-periodic.
0 we get from (15) the steady case (using direct notation) X= Cx
(16)
which is stahle assuming all the eigenvalues In this case, there exists a positive definite coefficients such that
of C lie in the left half-plane. quadratic form with constant
v = (Hx, x); (HT = H> 0) , whose
derivative
along the trajectory
. dV V = - dt = (HCx, x)
+ (Hx,
(17)
of the system
Cx) = ([HC
+
T
C H]x, x)
=
-
(Go x, x),
(18)
is negative definite. The matrix H can he found from (18) where Go is any positive matrix (e.g. Go = {JI, (J > 0). 5
Acta Technica
Academiae
ScientiaTum
HungaTicae 89, 1979
66
CARMI, S.-SINHA,
For the modulated SKU [11])
case (15) we have (see YAKUBOVICH and STARZHIN-
v= where
S. C.
G1 (t) is a T-periodic
+ sG1(t)]x,x),
([Go
-
matrix
function
(19)
given by
+ B!(t)H) .
G1(t) = - (HB(t)
+
We now let LJi (t, s) denote the principal minors of the matrix Go sG1 (t). As Go is positive definite, it follows from SYLVESTER'S text (see [11, p, 30]) that LJi (t, 0) > 0 (i =1, 2, . . . n). Since minors LJi (t, s) are continuous of both t and s we have for sufficiently small So 0
>
LJi(t,s)
>0
(i = 1,2,
. . . n)
for
0
< t :s;;: T ,
(20)
Isl:S;;: BO'
I
From (20) it follows that Go + sG1 (t) > 0 for 0 < t < T, s I< Bo hence by continuity there exists b > 0 such that all the principal minors of Go + BG1 (t) - bI are positive for 0 < t < T, B I< So' Therefore V in (19)
I
satisfies the inequality tion of (15) for B
I I
O
(24)
ASYMPTOTIC
Since the steady
state
STABILITY
67
OF FLOWS
case S = 0 is asymptotically
stable, it follows that
there exists a positive constant a such that II x II < C2e-atfor t > if II sEe-6t II < Cl, one can easily show that [1] equation (24) yields -+ 0 as t -+ CX) if ClC2 < a.
O.Then II
x
II -+
Actual stability bounds like So in (20) are now being established for model atmospheric flows (e.g. boundary layer type) and results will be reported in a subsequent paper.
Acknowledgements This work was supported
by the U.S. Army Research
Office.
REFERENCES 1. BELLMAN, R.: Stability Theory of Differential Equations, Dover Publications, Inc., N. Y., 1969 2. CARMI, S.: Energy Stability of Modulated Flows, Phys. Fluids, 17 (1974), 1951-1955 3. CARMI, S.-SCHULZE, R. S.: Stability of Modulated Flows in the Atmosphere, Proc. Sixth Can. Congo Appl. Mech., (1977), 617 -618 4. DAVIS, S. H.: Finite Amplitude Instability of Time-Dependent Flows, J. Fluid Mech., 45 (1970), 38-48 5. DRAZIN, P. G.-HOWARD, L. N.: Hydrodynamic Stability of Parallel Flow of Inviscid Flnids, Adv. Appl. Mech., 9 (1966),1-89 6. EINAUDI, F.-LALAS, D. P.: Some New Properties of Kelvin-Helmholtz Waves in an Atmosphere with and without Condensation Effects,J. Atm. Sci., 31 (1974), 1995-2007 7. HOMSY, G. M.: Global Stability of Time-Dependent Flows: Impulsively Heated or Cooled Fluid Layers, J. Fluid Mech., 60 (1973), 129-139 8. LAKSHMIKANTHAM,V.-LEELA, S.: Differential and Integral Inequalities, Vol. 1, Academic Press, N. Y., 1969 9. LINDZEN, R. S.: Stability of a Helmholtz Velocity Profile in a Continuously Stratified Infinite Boussinesq Fluid. Application to Clear Air Turbulence, J. Atm. Sci., 31 (1974), 1507-1514 10. ROSENBLAT, S.-TANAKA, G. A.: Modulation of Thermal Convection Instability, Phys. Fluids, 14 (1971), 1319-1322 11. YAKUBOVICH, V. A.-STARZHINSKII, V. M.: Linear Differential Equations with Periodic Coefficients, Vols. I and II, John Wiley & Sons, N. Y., 1975 12. YIH, C. 5.: Instability of Unsteady Flow Configurations, Part 1. Instability of a Horizontal Liquid Layer on an Oscillating Plane, J. Fluid Mech., 31 (1968),737-752
Asymptotische Stabilitat der nichtviskosen laminaren Stromung. - Die Stabilitat der modulierten atmospherischen Stromung wird untersucht. Die Grundgleichungen der Stromungsbewegungen werden mit Hilfe der Galerkinschen Reihenentwicklung mit zeitabhangigen Koeffizienten gelOst. Sodann werden durch Herstellung der Liapunovschen Funktionen die Stabilitatsgrenzen fiir die resultierenden Differentialsysteme ermittelt.
5*
Acta
Technica
Academia.
Sci.ntiarum
Hungarica.
89, 1979
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