CC BY
13
MSC 35Q30, 35P20, 35B35
DOI: 10.14529/ mmp190303
ON THE STABILITY OF TWO-DIMENSIONAL FLOWS CLOSE TO THE SHEAR
O.V. Kirichenko1, S.V. Revina1'2
1 Southern Federal University, Rostov-on-Don, Russian Federation 2Southern Mathematical Institute of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences, Russian Federation E-mails: okirichenko@sfedu.ru, svrevina@sfedu.ru
We consider the stability problem for two-dimensional spatially periodic flows of general form, close to the shear, assuming that the ratio of the periods tends to zero, and the average of the velocity component corresponding to the "long" period is non-zero. The first terms of the long-wavelength asymptotics are found. The coefficients of the asymptotic expansions are explicitly expressed in terms of some Wronskians and integral operators of Volterra type, as in the case of shear basic flow. The structure of eigenvalues and eigenfunctions for the first terms of asymptotics is identified, a comparison with the case of shear flow is made. We study subclasses of the considered class of flows in which the general properties of the qualitative behavior of eigenvalues and eigenfunctions are found. Plots of neutral curves are constructed. The most dangerous disturbances are numerically found. Fluid particle trajectories in the self-oscillatory regime in the linear approximation are given.
Keywords: long-wave asymptotics; stability of two-dimensional viscous flows; neutral stability curves.
Introduction
Mathematical models that describe two-dimensional or spatially periodic fluid flows are widely used to study various natural phenomena in the atmosphere and the ocean [1]. In [2], a model of two-dimensional creeping motion of viscous liquid in a flat channel is considered. Based on a priori estimates, the solution of the problem is constructed and its properties are investigated. The Kolmogorov problem for a two-dimensional viscous fluid under the influence of spatially periodic external forces is considered in [3]. Using the Galerkin method, stationary and spatially periodic solutions are found numerically. In [4] nonstationary time-periodic structures are obtained using long-wave perturbations of the Kolmogorov flow.
In this paper we consider the two-dimensional (x = (x1,x2) £ R2) viscous incompressible flow driven by an external forces field F(x,t) that is periodic in x1 and x2 with periods l1 and l2, respectively. The flow is described by the Navier-Stokes equations
d v
— + (v, V)v - uAv = -Vp + F(x,t), divv = 0,
where v = 1 /Re is the kinematic viscosity and Re is the Reynolds number. The period l1 = 2n, and the ratio of the periods is characterized by the wave number a: l2 = 2n/a, a ^ 0. Let (/) denote the average with respect to x1, ((/)) denote the average over the period rectangle Q = [0,l1] x [0,l2]:
ii
(/> = ¿ / fMdxu «/»(i) = p/ f(x,t)dxidx2.
0 n
The spatial average velocity is assumed to be given: ((v)) = q. The velocity field is assumed to be periodic in x1, x2 with the same periods l1, l2 as the field of external forces.
A longwave asymptotics (a ^ 0) is constructed for the stability problem of the steady flow close to the shear, which will be called the basic flow:
V = (aVi(x2),V2(xi)), (V2) = 0. (1)
The class of flows under consideration generalizes the Kolmogorov flow with a sinusoidal velocity profile
V = (0, y sin(x1)). (2)
The problem of investigating the stability of a two-dimensional flow under the action of spatially periodic force was proposed by A.N. Kolmogorov in his seminar. The instability of the Kolmogorov flow with respect to long-wave perturbations in the linear approximation was proved in [5]. The long-wave asymptotic behavior of the stability problem for two-dimensional parallel flows of general form
V = (0,V2(xi)), (V2) =0
was considered in [6]. Research [7] is devoted to the study of self-oscillations arising in the loss of stability of parallel flows of a viscous fluid affected by long wavelength perturbations. In [8], the main terms of the asymptotics of the secondary self-oscillatory regimes in the case of the basic flow close to parallel were found, but general rules in coefficient expressions were not obtained.
In [9] recurrence formulas for finding the k-th term of the long wavelength asymptotic for the stability of steady shear flows were derived in the case of nonzero average. The coefficients of the expansions are explicitly expressed in terms of some Wronskians, as well as integral operators of Volterra type. It is shown that the eigenvalues of the linear spectral problem are odd functions of the parameter a, and the critical viscosity is an even function. In the particular case, when the deviation of the velocity from its mean value V2(x) — (V2) is an odd function of x, the coefficients of expansion of the eigenvalues in series in powers of a, starting from the third, are zero and the eigenvalues can be found exactly: a1;2 = ±im(V)a,m = 0. In [10], recurrence formulas for finding the kth term of the long-wave asymptotics for the stability of two-dimensional basic shear flows of a viscous incompressible fluid with zero average are derived.
The aim of this paper is to generalize the results [9] related to shear flows in the case of basic flows close to shear.
1. Equations up to ak
Let H denote the subspace of functions / G L2(0,li) that are orthogonal to unity: (/) = 0. The operator I : H ^ H is the inverse of the differentiation operator and is completely continuous: x x ^
I/ = J /(s)ds -П /(s)ds У (3)
0 4 '
Let W(f,g) denote the Wronskian of functions / and g, the auxiliary function 9 characterizes the deviation of the second component of velocity from its average value:
Looking for a solution (v, p) of linearized on the basic flow (1) perturbation equation in the form of normal oscillations, we obtain the linear eigenvalue problem (here and below, x = x1, z = ax2):
<7^1 + a <p2 — + aV^z)—- + aV2(x) — - u[ + a = - —, (4)
dz dx dz V dx2 dz2 J dx
dV2 , d^2 fd2if2 2d2dP
+ ^ + + + ~adl> (5) 2n
= ¡Mx,z)dz = 0, (<p2) = 0. (6)
0
The value of the parameter v at which one or several eigenvalues a lie on the imaginary axis is called critical.
The unknown perturbations of velocity p(x,z), the function P(x,z), the eigenvalues a and the critical viscosity v are sought in the form of series in powers of a:
X X
a(a) = °kak, v = v* + ^ Vkak, (7)
k=0 k=1
((
p = ^2 <pkak, P = ^2 Pkak. (8)
k=0 k=0
We substitute (7) - (8) in (4) - (6) and equate the coefficients with the same powers of a. Up to a0 from the continuity equation (6) we deduce that = (z). Then it follows
dP 0
from (4) that a0if^(z) = —-—. Whence <r0 = 0 and P° = P°(z). From (5) we find the
dx
function that has the same form as in the case of shear basic flow:
'A = — ^i(z)a0(x), a0(x) = (9)
v* dx
Averaging equation (4) and equating the coefficients up to a1, we obtain:
°i<P°i{z) + = (10)
From (10) we find a1 = im(V2), <£?(z) = e-imz, where m = 0 is the wave number. Up to ak, k ^ 1, from (4) - (6) we derive the following system of equations:
02 fc Qpk fc fc l g2 fc-j fc 2 p k-2-j
^ = ^ + Lam - S - Sv*—d#
j=1 j=1 dx j=0 dz (11)
,dpk-1 dV1 k-2 M-1 +V2{x)^— + — '/2 2 + Fi(z)
dz dz dx
= Е <w2 - Е ч—ггрг - £ vi dz2 +
j=i j=i j=o (12)
2n
M + /^- = 0, <rf> = 0. (13)
0
Here, the sum is assumed to extend over those values of j for which the upper boundary is not less than the lower one. The solvability condition for equations (11) - (12) is that the average of the right-hand side over period l1 is zero. Averaging (11) for k > 2, we get the equation for finding (ip\-1):
(У2)[гш(^) + Щр-) = - ¿^Vn - Wi(z) -
k-1 k-3
d2
j=3 j=0
We will call (14) the averaged equation up to ak. On the other hand, the solvability condition of equation (14) serves to find vk-2 and ak:
((Fk-i(z)eimz)) = 0. (15)
For k = 2, from the solvability condition (15) for equation (14) we find: v2 = (0'2), a2 = 0. For the values v2 and a2 found, the right-hand side of (14) is zero. To exclude trivial nonuniqueness, we put = 0, as in the case of shear flow [9].
The scheme for finding the k-th term of the asymptotics for k > 1 is as follows. Suppose ^T1, ^k-i, Pk-i, 0"k+i, vk-i, (^k) are known. Applying the integral operator I defined by (3), from the continuity equation (13) we find
^ = +<¥>?>• (16)
Substituting the found in (11), we define Pk up to an average values. After this, from (12) we find . The values found in the k-th step allow writing the solvability condition for the (k + 2)-th term of the asymptotics. From the averaged equations up to ak+2 we find ak+2, vk, Then the process is repeated.
For a non-parallel basic flow (1), the coefficients of expansion of the critical viscosity v and the eigenvalues a have the following structure:
Vk = [vk ] + Vk, ak+2 = [ak+2] + ak+2, (17)
where the square brackets are used to denote the coefficients of viscosity and eigenvalues in the case of basic shear flow, and the wave is used to denote additional term. If V1(z) = 0 then that additional term is equal to zero. It will be shown later that components of
v
eigenfunctions and pressure ^, Pk for k = 1, 2, 3 and ^ for k = 1, 2 have the same structure:
^ = bk ]+f+>, (18)
= ^k ] + & (19)
Pk = [Pk ] + Pk + (P k>, (20)
while for large values of k, additional terms appear in the expressions of the eigenfunctions ^ and the pressure P. These terms depend on the mean values of previous coefficients.
2. Asymptotic Terms up to a1
Putting k = 1 in (16), taking into account the expressions of ^2 in (9) and = 0, we find
2\ 1 d^O
I(ao),
v* dz
and P1, which coincides with the one found in the case of shear basic flow:
Represent ^2 as a sum (19), where [<^2] and ^2 satisfy the equations:
(Here and below the deviation of a periodic function from its period-average value is denoted by curly brackets: {F} = F(x) — (F)). Coefficient [^2] coincides with the coefficient ^2 for the shear basic flow except for the term containing v1, which has not
been determined yet. Suppose a 1 (0) = I{W(0',0)} [9], then
v* dz v* v* v*
For k = 3, we transform the right-hand side of (14), taking the known values [v1] = 0
and [<73J =--—{O ai) into account:
2
K
d2
W = 2 v^-JL^-a^z).
From the solvability condition (15), taking into account the orthogonality of 0' and ^2, we obtain vj = 0, (73 = 0. As in the case of shear basic flow, v1 = 0, 73 = [(3]. Hence the right-hand side of (14) is zero. To exclude trivial non-uniqueness, we put = 0. In the particular case when 0(x) is an odd function, 73 = 0, as in the case of shear basic flow.
3. Asymptotic Terms up to a2
We derive ^ from the continuity equation (16), representing y>1 as a sum (18), where
From (11) for k = 2, we conclude that [P2] coincides with the one found in the case of shear flow:
1 <j2 ir>0
[p2] = --¡¿rQ2, Q2 = -ai + I{d"I(a0)},
and P2 satisfies the equation
dP2 д2< 0dVi лг, ,д<
dx * dx2 dz 1W dx
from which, applying the integral operator I, we find
Let consider the equation (12) for k = 2. We represent ^ as a sum (19), where and ^2 satisfy the equations
Taking into account already known expressions of and ^, we find
v* dz* v*
O2{0) = I2[{W(9", Ia\)} + v2(qi(9) - ao(9))]. (22)
The coefficient a2 (9) (22) is the same as in the case of shear basic flow; ] matches structure with the corresponding expression of the shear basic flow. As it will be shown below, the coefficient v2 differs from the one found in the case of shear flow. Applying the integral operator I twice, from (21) we deduce
= -/2{w%2, e")} + + -УМЫ)-
v* v* dz v*
2/ 1 .. и IVх-;-1 \Y2) 2
m
From (14) for k = 4 considering [<74] = 0, [v2] = —^—^(9'a2) we get
d2 < 0 d2
To verify the solvability condition (15), we need the expression for ((^2eimz)):
Me™)) = —{{V^toix) + -AiV^Mx), (23)
V3
gl = /00, g2(x) = 12{W (/gi,0")} - Ifli, f2(x) = Igi(x) = 120o.
Separating the real and imaginary parts in the solvability condition (15) for k = 4, we obtain
1 1 3 ¿2 = 2^«V?»<072> = - 2^«V?»<02>, = (24)
Note that the imaginary part of (23) contains the mean values of the odd degrees of V1, and the the real part contains the mean values of the even degrees of V1.
The solution of (14) for k = 4 will be sought in the form (^f) = c(z)e-imz, where c(z) is a 2n-periodic function. Let /z denote the operator / defined by (3) with variable upper limit z. Condition (15) uniquely determines the function c(z). Then c(z) = e-imz /z (Fa (z)eimz )/(V>).
Further, in each of the orders, two special cases will be considered. Case 1. 0(x) is an odd function (it means that the deviation V2 — (V2) is an odd function). Hence g1(x) is odd and from (23) it follows that g2(x) and f2(x) are even functions.
Case 2. V1(z) is an odd function of variable z. Then ((V1(z))) = 0 and <74 = 0.
4. Asymptotic Terms up to a3
From the continuity equation (13) for k = 3 we find i1, in the form of a sum (18), where
r 3n 1 dVl I/2 1
P3 can be represented as a sum (20) where
1 rfa^o
[■P3] = Qs(0) = -02 + I{9"I{ai)} - v2J2a0,
and P3 is determined by expression
As in [10] we use the following notation. If the function f is expressed in terms of a linear combination of the functions i1(z) and its derivatives with coefficients depending on x, then by f((^^)) we denote the expression that coincides with the expression f if i1(z) is replaced by (^)(z). Since = 0, it is convenient to represent i as a sum
i2 = b2 ] + i + i0 (((i1))), (25)
where [^3] has a form
г 3n 1 ¿Vi ( . /2, in о
a3(9) = I2[{W(9",Ia2)} + vl(q2 - <ц)] - ^4(0) - (9'al)I2(a0).
v 2
*
Then v3 is determined from the following equality:
^ = V/ + ¿'Ч W» + + f)
im
To find (</?4), v3 and <r5, we write F4 taking into account [//3] = 0, [<r5] = — (9'a3{9))\
v*
Separating the real and imaginary parts (15) for k = 6, we obtain
K = 05 = ^<<v?>><073> - —^
g3 = 12{W(Ig2, 9'')} + Ia2, /3 = I2{W(If2, 9'')} + (26)
Now, unlike the previous step (24), the expression for the viscosity expansion coefficients includes the averages of the odd powers of V1, and the eigenvalue coefficient contains the averages of even powers of Vi. The solution of (14) for k = 5 has the form (vi) = e-imzIz(F4(z)eimz)/(V2), similar to the order a3.
Case 1. 9(x) is an odd function. Then it follows from (26) that f3(x) and g3(x) are odd functions, hence v3 = 0 and, taking into account a3 = 0, we find a5 = [a5] = 0.
Case 2. V1(z) is an odd function of z. Then ((V1(z))) = 0 and v3 = 0, as in the case of shear basic flow.
5. Asymptotic Terms up to a4
From the continuity equation (13) for k = 4 and v2 defined in (25), we find v4:
vi = [v1] + it + vi ((v1)) + (Vi),
where [v4], V4 are defined by formulas:
1 t (a\ 0^2 r 21 ^з
, 2 a2 fdv-, 4 л 1 „ U! dvt a,
The pressure P4 is representable as the sum P4 = [P4] + P4 + P 1(((if))) + (P4), where
and q4 is found in [9].
Representing i as the sum i = [i2] + + i2((i1)) + i°((i4)) and taking into account
4 1 <4 2 0 1/2 r 2l 2v3 r 1n 2v,2 2 v4 0
= -5+ -/V2 - - — - -2-i2{w([<p%e")} - -ip02,
V. dz V* V* V* V. V*
we find
(f))
- ^ - ^ - ^ (0)++
To find (if), <6 and v4, we write out the right-hand side of the averaged equation up to
a6:
Fs(z) = uS^ ~ ^A) ~ W\{z) - + ufMi. (27)
4
Taking into account [//4] = ^ {0'0,4(6)), [<r6] = 0, (27) yields
2v*
Separating the real and imaginary parts of (15) and considering the previous coefficients, we obtain
£ = - t^-M - - 0")}),
2//* 2J v,
2
<76 = m2lm(((0'tpf)eimz)) - ^-(92) - 2^. Introduce the notation:
h1 = —/2{W (f2,0")} + (V>)/f2, h2 = —/2 {W №,0")} + (V>)/2h1, f4 = /2{W(/fa, 0'')} + /ga, g4 = /2{W(/ga, 0'')} — /03.
Then the expression Re(((d')eimz)) has the following structure
V 5
dz
V 5
where Ai, A2 are determined by the formulas
Ai = -m2/4 - 3v2v,/2 + 2m212f2, A2 = 3v*212/2 - Ih2.
Similarly, Im(((9'^)eimz)) has a form:
m
m
/m(((0V!)e^)> = в'Вг) + -Ж))(9'В2),
where Bi, B2 are determined by the formulas:
Bi = 13[9''/2} + 12{W(/2,9")} + 12[W(12/2,9")} + 1292 -
V2
1
-^(2I2{W(Igue")} + g2)--9i,
V
v 2
B2 = I2 {W (I2 / ,0'')} + I/.
Note that in the order a4, as in the order a2, the expression for the viscosity expansion coefficients includes the averages of the even powers of V, and the eigenvalue coefficient contain the averages of odd powers of V. As in the previous order, for k = 6 the solution of the equation (14) has the form: ) = e-imzIz(F5(z)eimz)/(V2).
Case 1. 0(x) is an odd function. Then /4(x) and g4(x) are even functions.
Case 2. V (z) is an odd function of z. Then ((V (z))) = 0, ((V3(z))) = 0 and aa = 0. Using the obtained formulas, we construct graphs of curves of neutral stability y =
. For general flows, as well as for special cases considered above, there is a similar
1
v (a)
qualitative picture. The most dangerous are perturbations at m = 1 (Fig. 1).
m=3 m=2
m=l
*
*
*
0 0.1 0.2 0.3 0.4 0.5 0.6 «
Fig. 1. Neutral stability curves
The found asymptotics allows us to investigate the trajectories of the motion of passive impurity particles in the secondary self-oscillatory flow [8]. The trajectories of particles in the linear approximation satisfy the equation
x = V(x) + u(x, t), u(x, t) = + e-iWt.
The qualitative behavior of the trajectories is presented in Fig. 2.
y
10 50 -5 -10
Fig. 2. Trajectories of the motion of particles of a passive admixture Conclusion
The first terms of the long-wavelength asymptotics with respect to the parameter a of linear spectral problem for a class of non-parallel flows close to the shear are found. A comparison with the case of the shear basic flow is made. It is shown that the expansion coefficients in terms of a eigenvalues and the critical value of viscosity have the form (17), where the square brackets are used to denote the coefficients of viscosity and eigenvalues in the case of basic shear flow, and the wave is used to denote additional term. If V1 (z) = 0 then that additional term is equal to zero.
The first terms of the expansion in series in the parameter a of velocity tf0(z), tf2(x, z), tf1(x, z) and pressure P0, P1 coincide with the coefficients for the case of shear basic flow. Average values (<^1) = = 0, as in the case of shear basic flow, but ) = 0 for k ^ 3.
The coefficients of the decomposition in a series of eigenfunctions have the following structure:
tf = ]+?++ (tf), >4 = ] + ? + + tf ((^1)),
and the coefficients of decomposition in a series of pressure have the following structure
Pk = [Pk] + px + P 1((tf1-1)) + (Pk).
Here k = 1, 2, 3, 4. The expressions in square brackets ], [Pk] are the solutions of the same equations, as in the case of shear basic flow and coincide with corresponding coefficients if Vj = 0, <jj- = 0.
If at least one of the conditions is met: 6(x) is odd or V1 (z) is odd, then critical value of viscosity V1(z) is an even function of a as in the case of shear basic flow (up to order k = 4 inclusively):
v(a) = v* + v2a2 + v4a4 + O(a5), a ^ 0.
If 0(x) is odd, then odd components of the eigenvalue decomposition, starting with the third, are zero:
a (a) = ai a + a4 a4 + a6a6 + O(a7), a ^ 0.
If V1(z) is odd, then even components of the eigenvalue decomposition are zero:
a(a) = aia + a3a3 + a5a5 + O(a6), a ^ 0.
Thus, under both conditions eigenvalues up to terms of order a5 have the form
a(a) = ±ima (V2) + O(a5), a ^ 0,
as in a shear basic flow case.
For a basic flow close to a shear of general form the inequality v2 < [v2] is fulfilled, therefore, the loss of stability of such flows occurs at higher Reynolds numbers, than for shear flow.
Neutral stability curves qualitatively coincide with the corresponding curves for the Kolmogorov-flow (2). The trajectories of the passive admixture, found numerically, consistent with Obukhov's hydrodynamics experiments [11].
References
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Received December 11, 2018
УДК 517.958 DOI: 10.14529/mmp190303
ОБ УСТОЙЧИВОСТИ ДВУМЕРНЫХ ТЕЧЕНИЙ, БЛИЗКИХ К СДВИГОВЫМ
О.В. Кириченко1, С.В. Ревина1'2
1 Южный федеральный университет, г. Ростов-на-Дону, Российская Федерация 2Южный математический институт Владикавказского научного центра РАН, г. Владикавказ, Российская Федерация
Рассматривается задача устойчивости двумерных пространственно-периодических течений общего вида, близких к сдвиговым, в предположении, что отношение периодов стремится к нулю, а среднее скорости вдоль «длинного» периода отлично от нуля. Найдены первые члены длинноволновой асимптотики. Коэффициенты асимптотических разложений явно выражаются через некоторые вронскианы и интегральные операторы типа Вольтерра, как и в сдвиговом случае. Выявлена структура собственных значений и собственных функций для первых членов асимптотики, произведено сравнение со сдвиговым случаем. Исследованы подклассы рассматриваемого класса течений, в которых обнаруживаются общие свойства качественного поведения собственных значений и собственных функций. Построены графики нейтральных кривых. Численно найдены наиболее опасные возмущения. Приведены траектории движения пассивной примеси во вторичном автоколебательном потоке в линейном приближении.
Ключевые слова: длинноволновая асимптотика; устойчивость двумерных течений вязкой жидкости; кривые нейтральной устойчивости.
Литература
1. Должанский, Ф.В. Лекции по геофизической гидродинамике / Ф.В. Должанский. -М.: ИВМ РАН, 2006.
2. Андреев, В.К. О решении одной обратной задачи, моделирующей двумерное движение вязкой жидкости / В.К. Андреев // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2016. - Т. 9, № 4. - С. 5-16.
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Ольга Валерьевна Кириченко, аспирант, кафедра вычислительной математики и математической физики, Южный федеральный университет (г. Ростов-на-Дону, Российская Федерация), okirichenko@sfedu.ru.
Светлана Васильевна Ревина, кандидат физико-математических наук, доцент, кафедра вычислительной математики и математической физики, Южный федеральный университет (г. Ростов-на-Дону, Российская Федерация); научный сотрудник, Южный математический институт Владикавказского научного центра РАН (г. Владикавказ, Российская Федерация), svrevina@sfedu.ru.
Поступила в редакцию 11 декабря 2018 г.
