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EDN: ALTQUO УДК 517.9
Solution of Convection Problem in a Rotating Tube by the Fourier Method
Igor V. Vakhrameev*
Institute of Computational Modelling SB RAS Krasnoyarsk, Russian Federation Siberian Federal University Krasnoyarsk, Russian Federation
Evgeniy P. Magdenko^
Siberian Federal University Krasnoyarsk, Russian Federation
Received 10.08.2022, received in revised form 18.09.2022, accepted 07.11.2022 Abstract. The non-stationary boundary value problem on the motion of a fluid in a rotating cylindrical pipe is studied in this paper,. The Oberbeck-Boussinesq equations are used to describe the motion of a fluid. From a mathematical point of view, the problem is inverse with respect to pressure gradient along the axis of the cylinder. The solution is found with the use of the method of separation of variables in the form of special Fourier series. Sufficient conditions are given for the solution of a non-stationary problem to reach a stationary regime with increasing time.
Keywords: convection, inverse problem, asymptotic behaviour, method of separation of variables , Bessel functions.
Citation: I.V. Vakhrameev, E.P. Magdenko, Solution of Convection Problem in a Rotating Tube by the Fourier Method, J. Sib. Fed. Univ. Math. Phys., 2023, 16(1), 17-25. EDN: ALTQUO.
Introduction
Thermal convection in rotating systems has been studied for various applications such as modelling heat exchangers, various large-scale circulations in the Earth atmosphere, and other phenomena [1-5]. However, convective flows were assumed to be stationary in these works, and the rotation angular velocity was constant.
In the present work, the angular velocity of a round tube depends on time, and the motion is rotationally symmetric and unsteady. In addition, the fluid rate through the pipe cross section is also the function of time. There are no mass forces. This takes place at a sufficiently large angular velocity of pipe rotation (centrifugal acceleration can be 106 times greater than gravity acceleration in practical vortex tubes) or in conditions close to weightlessness. The constant temperature gradient is applied along the pipe surface. From a mathematical point of view, an inverse initial-boundary value problem arises because it is also required to determine the non-stationary pressure gradient along the pipe axis, and the over determination condition for the fluid flow rate is set.
* vahrameevi@mail.ru
tmagdenko_evgeniy@icm.krasn.ru https://orcid.org/0000-0003-2354-1152 © Siberian Federal University. All rights reserved
1. Problem statement and formulation of basic equations
The fluid thermal convection problem is considered in a rotating cylindrical tube. Oberbeck-Boussinesq equations are used as a mathematical model. Here r, p, z are cylindrical coordinates, t is time, a is the cylinder radius and u(t) is the angular velocity of cylinder rotation around z axis. Further, u and w are radial and axial components of the velocity vector; v is the difference between the azimuthal velocity and the fluid rotation velocity u(t)r as a solid body; p is the deviation of true pressure from the equilibrium state pu2(t)r2/2; Q is the deviation of temperature from its average value Q = const. The fluid is characterized by the following physical parameters: density p at temperature Q, kinematic viscosity v, thermal diffusion x and the coefficient of thermal expansion ¡3. These parameters are assumed to be constant and positive. There is no gravity acceleration. The last assumption is satisfied if the angular velocity u(t) is sufficiently large.
Under given above assumptions the thermal convection equations have the form [6]
v r>/\ v 1 ( t 2 u\ h \n ut + uur--uv + w(t)uv + wuz — 2u(t)v--=--pr + v I Lu--jvv--j ) — w (t)ßrO,
uv 1 ( 2 v \
Vt + Ut (t)r + uvr + u(t)vv + wvv + 2u(t)u+ — = — ~Pv + v\Lv — rj uv — -j] — u2(t)ßrQ,
)r + uvr + u(t)vv + wvv + 2u(t)u+ — = -~Pv + v\Lv - uv - r2 1 - w2'
v 1
wt + uwr +— wv + w(t)wv + wwz =--pz + vLw,
rp u1
ur +---+ -vv + wz =0,
v r r
Qt + uQr + r Qv + u(t)Qv + wQz = xLQ,
where L = d2/dr2 + r-1d/dr + r-2d2/dp + d2/dz is the Laplace operator. We seek a solution of system (1) in the form
(1)
u = u(r,t), v = v(r,t), w = w(r,t), r2 ]
z + q(r,t), (2)
p = p ApJ2(t) J + f(t)
& = -Az + T (r,t), A = const.
According to the classification given in [7], it is an invariant solution of system (1) with respect to the infinite Lie subgroup defined by operators
A + (f(t) + A!3u2(t)pr2\ d_
dz AdO+ y (t)+ 2 ) dp.
Substituting solution (2) into (1) and assuming that there are no sources or sinks on the axis z, we obtain u = 0 and the system of linear equations
wt = v (^w„ + ^w^ — 1 Aßu2(t)r2 — f(t)
Tt = x(t„ + + Aw, (3)
11 \ pv2
( 1 M
Vt = v I vrr + -vr--2v - Utr, qr
r r2
The initial and boundary conditions for this system are as follows
w(r, 0) = wo(r), T(r, 0) = To(r), v(r, 0) = vo(r), 0 ^ r ^ a;
w(a,t)=0, T(a,t)=0, v(a,t)=0, 0 < t < t1; (4)
| w(0,t) \< œ, | T(0,t) \< œ, | v(0,t) \< œ.
In addition, the fluid volumetric flow rate through the tube cross section Q(t) is given as
fa Q(t)
rw(r,t)dr =-. (5)
J 0
In equations (1)-(3), v is the fluid kinematic viscosity; (—A) is the temperature gradient along the tube axis; ¡3 is the coefficient of thermal expansion; x is the thermal diffusivity; a is the tube radius; u(t) is rotation angular velocity of the tube around its axis; ut\t) is angular acceleration.
Given functions u(t), Q(t), one can give the following interpretation of problem (3)-(5). The fluid fills a cylinder of radius a rotating with an angular velocity u(t), and fluid is pumped with a flow rate Q(t) through the cross section of the cylinder. The temperature gradient —A along the axis is set on the cylinder surface. It is required to determine the resulting rotationally symmetric motion. If Q(t) = 0 then one can expect that the problem solution describes convection far from solid wall perpendicular to the axis of the cylinder, that is, motion in the core.
From a mathematical point of view, problem (3)-(5) is an inverse problem. Along with w(r,t),v(r,t),T(r,t), it is necessary to find an additional pressure gradient along z axis, that is, function f (t). Let us also note that problem for v(r,t) is separated. The main problem is for w(r,t)andf (t). For known w(r,t) function T(r,t) is found as a solution of the classical first initial-boundary value problem, and q(r, t) is found by integrating the last equation of system (3).
2. Stationary problem solution
In the stationary case, all variables in (3)-(5) do not depend on time
w = ws(r), f = fs = const, T = Ts(r), v = Vs (r),
and the angular velocity us and flow rate Qs are constant.
Simple mathematical treatment shows that solution of this problem is
(6)
ws(€j = x
a
Ra № - +i)+22Q (i - a
T "(£) = -Aa
R2 (^ - &+?>e -i) + - +3) fs = -VvX + ,
a3 \ n 6 /
(7)
where Ra = (Afius2a5)/(vx) is the analogue of the Rayleigh number, £ = r/a, Q = Qs/ax, 0 <
a < i.
It should be noted that in the vicinity of the solid wall, zones of the fluid return flow can arise. This is due to two types of flow mechanism that are present in the problem. The first one is related to the fluid rate, and the second one is related to thermal effects and fluid rotation. It is easy to find that the reverse flow occurs when Q < nRa/96, in particular, this always takes place when Qs = 0.
3. Solution of the unsteady problem by the method of separation of variables
The method of separation of variables, or the Fourier method, is one of the widely used methods for solving linear partial differential equations. Its application leads to a solution in the series form.
s
0
v
In our case, the main problem is to find w(r,t) and f (t) from the first equation of system (3). In order to use the method of separation of variables it is necessary to reduce the problem to a homogeneous one with respect to the boundary conditions. This can be achieved by doing the following change of variables
w(r,t) = w(r,t) + ^^Q(t)(a - r)r2 + d(t) ( r3 - 6ar2 + 1 a3 ) ,
\ 5 5 J
t (8)
3na5 12a
5 5
d(t) = [Q(t) - Q(0)] + ^ I J(t)dT.
25 , 5Aß I 2
/0
Since function f (t) is unknown the problem is inverse. Then we differentiate the equation with respect to r and introduce wr (r,t) = W(r,t). Thus the pressure gradient function is excluded. As a result, we obtain the following direct problem with integral (non-classical) condition on function W(r,t)
Wt = v(Wrr + 1 Wr - 1W^ + U,
U(r,t) = -v?0Q(t) + 9vd(t) + Qt(t) - 3dt(t)) r2, na° \na° J
W(r, 0) = W0(r) = w0r (r) - Q(0) (2ar - 3r2) , 0 < r <
na5
(9)
\W(0,t)\ < œ, 0 < t < t1,
i r2W(r,t)dr = 0, r G [0, a], t G [0,t0]. (10)
Jo
Integral condition (10) follows from (5).
Let us consider a homogeneous version of (9) (U = 0) in order to find the Fourier expansion basis. To do this, it is necessary to separate the variables and solve a Sturm-Liouville type problem to find eigenfunctions and eigenvalues. After some mathematical treatment, we find that expansion basis contains the Bessel functions of the first kind and first order. Therefore, the solution of inhomogeneous problem is sought in the form
W (r,t) = J2 Ck(t)Ji( v^kr), U (r,t) = Y] Uk (t)Ji( v^kr), (11)
k=i k=i
where Xk = Si/a2, and £k are roots of equation J2(£k) = 0. They are £1 = 5.13562, £2 = 8.41724,
= 11.6198, £4 = 14.796, £5 = 17.9598. Remark 1. Series (11) includes the Bessel functions terms J1(£k r/a), where £k are roots of equation J2(£k) = 0. However, taking into account relation J1 (£) — £J1(£) = £J2(£), we obtain that £k — are roots of equation J1(£k) — £k J1 (£k) = 0. Therefore, expansion (11) makes sense [8]. In particular, the system of functions {J1(£kr/a)} is orthogonal and complete in L2[(0, a); r] and
'Sn^
PrjJ—).Ji(—dr = 0, k = n; Jo \ a J V a J
£ r.j2( dr = a2 {j m2 + {i - jKï* )} = a2 j2(a).
(12)
Substituting expressions (11) into (9), we obtain after simple transformations the system of ordinary differential equations, and find Ck (t):
Ck (t) = fuk (r)e^ ^T {T-t)dr + Coke-u( \ (13)
Jo
Here coefficients C0k is obtained from the expansion of initial conditions W0(r) from (9) into the Fourier-Bessel series.
Integrating W(r,t) from (11), we return to the variable w. Then, taking into account (8), we obtain the solution of the first equation (3)
w(r,t)= a g ^ uk (r)e<%Y (r-t)dr + Coke-<^T^ [j0 (f r) — Jo (k + -10 Q(t)(a — r)r2 [Q(t) — Q(0)] + A 0 u2(t)dt) (r3 — 6 ar2 + i a^ ,
+
(14)
where
Cok = T 2 - !a r (wor (r) + ^ Q(0)3r2) J J ^ r) dr, [aJi(£k)]2 Jo \ ™5 J \a J
2 ra ( ....... ( 30
uk(t) = [ T ni2 J r {—v^O Q(t) + 9vd(t) + ^Qt(t) — 3dt(t)^j r2^j J^ar) dr
[ rJi{a0 dr = fk F (2) ^ [Ji(tk)Ho(tk) — Jo(tk)H1(^k)],
I r3J1 (a0 dr = ak fiJi(tk)G2,o(tk) — Jo(tk)Cs,i(£k)],
[aJi(Ck)]2 Jo V na5 \na° J J \a
are obtained using equalities (12).
Remark 2. When finding coefficients in (14), the following integrals arise [8]
- r ( 3 *
10 \a J £k V2,
r3^
o a k
where H0(x), H\(x) are Struve functions, (x) are Lommel functions.
In addition, it is necessary to find the remaining unknown pressure gradient function f (t). Multiplying the first equation (3) by r, integrating it over r from 0 to a and taking into account (4), (5), boundary and initial conditions, we obtain
f (t) = — W(a, t) - 1 Apu2(t)a2 Qt(t), (15)
a 4 na2
where W(r,t) is defined in (11).
It is interesting to find the initial value f (0). We have from (15) that
f (0) = — wor (a) - 1 Afiw2(0)a2 --L Qt(0) (16)
a 4 na2
Functions T(r,t) and v(r,t) are found in the standard way. Considering second and third equations (3), we have [9]
w r/* t ]
T (r,t) = V gk (t )ex( a)2(T-t)dT + Bok e-x( ^ P jJ ^ r) ,
t^i .Jo \ a J
'o —r,
a
fa )
Bok = ~ ,]2 rTo(r)Jo(^r) dr, (17)
[aJ0(ßk)]2 Jo \ a J
gk(T )=JAwL rw(T>r)Jo(a r)dr,
^^ ft
,(r,t) —]T hkwWWdr + Noke-W* Ji(^r) k=l ^
hk (T ) = J a
2
No k (t )
(18)
rv o (r
(r)JiÇ—r^ dr,
[aJ[(ek)]2Jo
where /i,k are roots of equation J0(^k) = 0. They are = 2.40483, ¡i2 = 5.52008, = 8.65373, 1^4 = 11.7915, = 14.9309. Parameters ek are roots of equation Jl(ek) = 0. They are el = 3.83171, e2 = 7.01559, e3 = 10.1735, e4 = 13.3237, e5 = 16.4706.
4. Time evolution of the non-stationary solution
For simplicity, here we consider the practically important case Q(t) =0, Qs = 0. Let us show that solution of the non-stationary problem tends to the stationary regime as time increases, that is,
lim w(r,t) = ws(r), lim T(r,t) = Ts(r), lim v(r,t) = vs(r), lim f (t) = fs, (19) where ws(r), Ts(r), vs(r), fs are defined in (7) at Qs = 0. Let us introduce new functions
H(r,t)= w(r,t) - ws(r), Q(t)= u2(t) - (us)2, F(t) = f (t) - fs. (20)
Due to linearity of the main inverse problem function H(r,t) satisfies the same equation as w(r,t) but with modified initial data: H(0,r) = H0(r) = w0(r) — ws(r), u2(t) is replaced with tt(t), f (t) is replaced with F(t).
Substituting (20) into previously obtained solution (14), we find
H (r, t) — D(t)( r3 - 5 ar2 + ! a^ + a g Wk(t) J (f r)
5 5
D(t)=5~ÊÊ Î0 Ü(t)dT
Jo -r - Jo (&)
(21)
where
f2 ft £2 Wk(t) — e-va2tCok + / Dk(t)e-va2(t-T»,
Dk (t)
Cok — r , ^
[aJi(0]
2 [
o
2 ra
0k = [ . JQ rWo(r)Ji ^a^ dr, (22)
[aJi (£)]2
9vD(t) ^ rJl ^ar) dr - 3Dt(t) ^ r3Jl[^^^ dr
and Cok are the Fourier series coefficients of H0r (r). Further, we have
1 2
F (t) = —T,Wk (t)Ji(îk) - 7 Ap^(t)a2 + - a2 D(t). (23)
a 4 5
k=l
Let us assume that Q(t) behaves as follows
Q(t) ^ 0, i Q,(r)dr ^ 0, t ^œ. Jo
Let us take two functions that satisfy (24):
1) n(t) = C1e-at (1 — at), a > 0, i ü(t)dT = C1te-at;
Jo
ft e-at
2) ü(t) = C1e-at cos at, ü(t)dT = C1-T [at cos at + (at — 1)sin at] ^ 0.
o 2a2
In both cases we obtain
\il(t) \ < C2te-
ü(t )di
< C2te-
(25)
(26)
with positive constants C2 and a.
Let us estimate \H(r,t)\ and \F(t)\. To do this, we first estimate \Wk(t)|. From (22) we obtain
\Dk\ < A£k (\D(t)\ + \Dt(t)\) <
A1
[aJ1(0Y
^ ^ ü(t)dT +ü(t)]j
+ Q(t) < A2te-
(27)
because \J1\ < 1, and m1/£k ^ f rJ2 (£kr/a) dr < m2/£k [10], where A> 0, A\y2 > 0, m1y2 > 0
0
are constants.
Let us estimate the integral in the second relation (22)
A2 kt
ft c'2 A c'2 ft
Dk(t)e-vak(t-T)dT < A2Skte-v!»t / e(va-2£-a)Tdr =
Jo m1 Jo
_ e-(va-2Ç2k-a)t
m1 va 2 S2 — a
e-at — e-v att
A
2s ,t—t ) va-2 &-a
< -Skte
m1
(28)
t, a = v
The latter is true if for some k = k0 we have a = va 2Ck . For given v, a there is a constant
k\, and at k > k\ we have that a < va because £k increases with k (recall that £k ~ kn for k > 1).
Thus, the estimate for \Wk(t) \ is
\Wk(t)\ < \Cok\e-va te-va -2)t + — Skte-at I \va-2^-a
m1
1 _ e-(va-2il-a)t
t, a = v
2
(29)
Since < 6,..., \ J2() < 1, \Cok\ <C3,k > 1 it follows from (21), (29) that
19 c,-va-2at
\H(r,t)\ < \D(t)\a3 + C3ae—-^e-va -i2)t+
k=1
5
1
+ A1at J (e-at + e-va-2& m1
)
k=1
1
\va-2 k2 — a\
+ e-
: E
k=k i + 1
1
\va-2 Sk— a\
(2)
It is known that the "large" roots J2(i) are approximately equal to £k = « 7n/4 + kn. Then, in curly brackets in the numerator the term t appears for k = k0. Series (30) converges uniformly for t ^ e > 0 by the Dalamber criterion. Therefore, taking into account (26) \H(r,t)\ tends to zero as te-at when t ^ to uniformly for r G [0, a].
As for F(t), the first part of its representation (23) is estimated as \H(r,t)\ in (30), and the estimation of the last terms follows from (26).
Estimates for T 1(r,t) = T(r,t) - Ts(r) and V(r,t) = v(r,t) - vs(r) can be found in a similar way. Thus, if we additionally require \w'(t)\ < C4e-at then the following theorem is true.
.
(30)
o
2
a
1
Theorem 1. Let w0(r) £ C 1[0, a], f rw0(r)dr = 0, T0(r) G C[0, r], v0(r) £ v[0,r] and condi-
o
tions (26) are satisfied. Then non-stationary solution of the inverse problem tends to zero with increasing time to stationary solution (7). To be more specific, when t ^ m, r £ [0, a] we obtain that
\w(r,t) - ws(r)\ < C5te-a0t, \f (t) - fs\ < C6te-a0t,
t t (31)
\T(r,t) - Ts(r)\ < C7te-ait, \v(r,t)\ < C8te-a2t
with positive constants C5, C6, C7, C8, a0 = min(a,va-2^f), a1 = min (a,xa-2^i), a2 = = min (a,\a-2£i), where J2(^h) = 0, Ji(ek) = 0, Jo(^k) = 0.
Remark 3. When ws = 0, fs = 0, = 0, Qs = 0, Ts = 0, vs =0 inequalities (31) are the a priori estimates of the formulated problem under conditions (24), (26).
Thus, it follows from estimates (31) that solution of the non-stationary problem exponentially tends to the stationary solution with increasing time. It means the asymptotic stability of the stationary solution (7).
Remark 4. The resulting solution is classical. This is proved by estimating the Fourier series with respect to the Bessel functions [10].
Conclusion
The following results were obtained in the work:
1. A stationary solution of the inverse problem is found in terms of polynomials in r.
2. The non-stationary solution of the problem is obtained by the method of separation of variables. In this case, the problem is reduced to the direct problem with an integral (nonclassical) condition.
3. With the help of the a priori estimates, it is established that non-stationary solution tends to the stationary mode with increasing time if conditions of the theorem are satisfied.
The obtained results are of theoretical and practical importance. They can be used to simulate rotationally symmetric convective motions of a viscous heat-conducting fluid in rotating tubes.
This research was supported by the Russian Foundation for Basic Research (grant 20-0100234) and the Krasnoyarsk Mathematical Center financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement no. 075-02-2022-873).
References
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[2] G.M.Homsy, J. L.Hudson, Centrifugal covection and its effect on the asymtotic stability of a bounded rotating fluid heated from below, Jour. Fluid Mech., 48(1971), 605.
[3] J.EHart, On the influence of centrifugal buoyncy on rotating convection, Jour. Fluid Mech., 403(2000), 133. DOI: 10.1017/S0022112099006928
[4] F.H.Busse, Om the influence of centrifugal buoyncy on rotating convection, Jour. Fluid Mech. 44(1970), 441.
[5] J.Herrmann, F.H.Busse, Convection in a rotating cylindrical annulus. Part 4. Modulation and transition to chaos at low Prandtl numbers, Jour. Fluid Mech., 350(1997), 209.
[6] G. Z.Gershuni, E.M.Zhukhovitsky, Convective stability of an incompressible fluid, Moscow, Nauka, 1972.
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[8] G.Bejtmen, A.Erdane, Higher transcendental functions. Vol. 2. Bessel Functions, Parabolic Cylinder Functions, Orthogonal Polynomials, Moscow, Nauka, 1974.
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Решение задачи о конвекции во вращающейся трубе методом Фурье
Игорь В. Вахрамеев
Институт вычислительного моделирования СО РАН Красноярск, Российская Федерация Сибирский федеральный университет Красноярск, Российская Федерация
Евгений П. Магденко
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Исследована нестационарная краевая задача о движении жидкости во вращающейся цилиндрической трубе. Для описания движения жидкости используются уравнения Обербека-Буссинеска. С математической точки зрения задача является обратной относительно градиента давления вдоль оси цилиндра. Методом разделения переменных решение найдено в виде специальных рядов Фурье. Даны достаточные условия выхода решения нестационарной задачи с ростом времени на стационарный режим.
Ключевые слова: конвекция, обратная задача, асимптотическое поведение, метод разделения переменных, функции Бесселя.
