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Civil Avition High TECHNOLOGIES
Vol. 19, № 02, 2016
ON SYMMETRY REDUCTIONS AND INVARIANT SOLUTIONS OF THE k-£ TURBULENCE MODEL
N.G. KHOR'KOVA
Methods of theoretical group analysis of differential equations are applied to the k — £ turbulence model. Symmetry reductions of the k — e turbulence model with respect to some three-dimensional symmetry subalgebras are considered. Families of exact solutions are obtained.
Keywords: nonlinear differential equations, local infinitesimal symmetries, invariant solutions, the k — e turbulence model.
INTRODUCTION
The comprehension of turbulence is of fundamental interest from various aspects of human activities [1-6]. Although the turbulence problem has still remain an unsolved problem of classical physics, there exist various approaches to turbulence modeling. The k — e and other two-equation models are most widely used for engineering applications in spite of these models have several serious limitations [7, 8].
All two-equation models include the Navier-Stokes equations, which in principle can be integrated numerically [9]. However in many cases the computational efforts became enormous (see, for example, [5]), so numerical methods are viable for restricted class of problems.
Theoretical group analysis (symmetry methods) is a well known methodology to derive exact solutions of nonlinear differential equations. However, it is desirable to employ symmetry approach more systematically for development, improvement or calibration of turbulence models [10-13].
In this paper methods of theoretical group analysis of differential equations are applied to the k — e turbulence model. The classical symmetries of the k — e turbulence model have been calculated in [14]. The paper [14] contains also the complete ready-to-use list of symmetry subalgebras, which is necessary to construct invariant solutions of the equations under considerations. The aim of this paper is to obtain symmetry reductions and exact solutions of the k — e turbulence model. Here we shall not discuss physical meaning of obtained solutions [15].
The paper is organized as follows. In Section 2 we present the classical k — e turbulence model [7, 8, 14] and give a summary of the geometrical theory of nonlinear differential equations [16, 17]. In Section 3 we study invariance conditions for main classical symmetries of k — e turbulence model. In Section 4 we consider some reductions of k — e turbulence model with respect to three-dimensional subalgebras. In Section 5 we discussed methods of constructing solutions for reduction equations and obtain families of exact solutions for k — e turbulence model.
1. PRELIMINARIES
1.1. The k — £ turbulence model
The k — e turbulence model describes the motion of high Reynolds number turbulence flows and is derived from averages of Navier-Stokes equations by introducing the k- and ^-equations in order to obtain a closed set of equations [7, 8]:
ÔÛj
dTr0' (1)
d(puj) djpûjûj) dp 5/ .
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dk dk д (f cßk2\dk) -dm
+ ^ = Г + (3)
dt 1 dxi dxi (д ak e ) öxj 1 J dxt
de _ de d (i c„ k2\ ) e-düi e2
+ d^ - ^ ¥' (4)
where is the mean velocity component in the xi direction, p is the mean pressure, k is the turbulence kinetic energy, e is the rate dissipation of turbulence kinetic energy, v = const is the viscosity, p = const is the density,
c„k2 (dûi dûjS
-u<'u>'= * ^VrI611"'
is the Reynolds stress tensor, 5jy beeing the Kronecker delta, f£y = pv +
The five empirical constants c^, cEl, c£z, ak, a£ that appear in the equations are assigned the values: c^ = 0.09, cEl = 1.44 - 1.59, c£z = 1.9 - 2.0, ak = 1.0, aE = 1.3 - 1.47. Throughout, for repeated indices the summation convention is used, the indices running from 1 to 3.
1.2. Symmetries and invariant solutions of partial differential equations
We expose here in a simplified, local coordinate form the basics of the geometrical approach to differential equations and its symmetries [16, 17].
Let £ = {F = 0} be a system of differential equations given by
dxn
Fs ( x,u,..., ,...) = 0, s = 1, ...,т,
G
where u = (u1, ...,um) is the unknown vector-function in the variables x = (x1,^,xn), F = (F1,...,Fr). In the framework of the geometrical theory any differential equation £ of order k is considered as a submanifold in the space of k-jets Jk(n) for some vector bundle w. En+m ^ Mn. For example, system (1) - (4) is a submanifold £^J2(n), where n:Dx\R6^D, D £ IR4(x1( x2,x3, t), is the trivial bundle, ul = ul, i = 1,2,3, u4 = p, us = k, u6 = e are the coordinates in IR6.
Any infinitesimal symmetry of a system of differential equations has the form of evolutionary derivation
a,j a
d
where ^ = {cp1,...,cpm), e is the generating function of the symmetry, Di = — +
2igUii —T are the total derivative operators, Da = Dt o ... o Dj for a = (L, ...,ir), summation is tak-
J' at duJa 1 r
en over the internal coordinates on £, the bar means the restriction to £. We identify fields with their generating functions.
To find infinitesimal symmetries of the system £\F = 0, F = (F1(...,Fr), Ft G Cco(Jk(n)) one must solve the equation
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Vol. 19, № 02, 2016
h(.<P) = o.
(5)
where lF is the universal linearization operator
Let Sym £ denote the vector space of all infinitesimal symmetries of an equation £. We identify Sym £ with the solution space of equation (5). The vector space Sym £ is a Lie algebra with respect to the Jacobi bracket = —
Let g be a subalgebra of the Lie algebra Sym £. Then ^-invariant solutions of £:F = 0 are solutions of the system
where ... is a basis of g. System (6) is overdetermined and the fact that are symmetries means that the system is compatible. Under some regularity conditions, the problem of solving system (6) is equivalent to that of solving system with n — s independent variables. We shall say that system (6) is the reduction of equation £ with respect to the symmetry subalgebra g =< ~> (or with
respect to the symmetries ... ). Note that in equation (6) one can use also generating functions of nonlocal symmetries [16, 18]. It is reasonable to use Reduce package [19] to solve equations (5), (6).
2. INVARIANCE CONDITIONS
In this section we solve the part = 0, = 0 of system (6) for main classical symmetries of k — e turbulence model.
2.1. Generalized space translations and Galilean boost
Consider the following symmetries:
F = 0, (рг = 0, ...,q)s = 0,
(6)
^i(Z) = (/W - f. fu-l, fui, fp! + pfx±, fk±, fe±),
x2(g) = Cgub 9^1 - g, gub gpi + pg*2> gk2. g^)>
X3(K) = (hu\, hu3, hu3 ~ h, hp3 + phx3, hk3, h£3),
where f, g, h are arbitrary functions of t,
{X1(f),X2Cg)} = {X2(g), X3m = {X1(f),X3(K)} = 0,
while
{xmxim = P(fg-fg),
where Р(Л) = (0,0,0, h, 0,0) [14].
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When the functions /, g, h are being constant or linear, symmetries Xx(/), X2(g), X3(h) are assigned to space translations or Galilean boost respectively. For example, the system X3(K) = 0 is of the form
du1 du2 du3 h —— = 0, h —— = 0, h ---h = 0,
OX3 OX3 OX3
dp dk da
h -— + phx3 = 0, h -— = 0, h -— = 0,
0x3 0x3 0x3
and its solutions are
h
U —IX (t, X^, X2\ U —XI (t, X^, X2\ U = ~X3 "I- U (t, X^, X2\
h
ph 2 ~ ~ p = — ——x3 + p(t,Xi,X2), k = k(t,xi,x2), e = i(t,xi,x2). z a
In the same way one can solve system X^f) = X2(g) = X3(K) = 0 [20]:
,i-L f
u1 ^ij + a(t), u2 = — x2 + ß(t), w3 = т*з +
h h'
P(f
9
h
p = -2[7xl+gx2+hxl)+p(t)' к = k(t), e = £(t).
(7)
2.2. Rotations
The symmetry algebra of the k — £ turbulence model possesses three-dimensional rotation subalgebra < R12, R23, R13 > [14], where
Rtj =
/XjU} — XiUj + S-LjU1 — 0ци]'\ Xjuf — Xiuf + S2jUl — 02iUJ' Xjuf — xtu3 + S3jUl — S3iu]'
XjPi-XiPj Xjki Xjfcy Xy£j — Xj£2
\
/
Consider, for example, the system R12 = 0:
дй1 X2 я ' дй1 xl л " дх2 -и2 дй2 = 0,х2 дхг -хг дй2 дх2 + U1
дй3 я Х1 дй3 дх2 др = 0, х2 дхг -хх др дх2 = 0,
*2 дк я--Х1 дк дх2 д£ = 0, х2 — дхг -хг д£ дх2 = 0.
(8) (9) (10)
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Equations (9)-(10) are of the form yux — xuy = 0, u = u(x,y). The latter equation possesses the solutions u = f(x2 + y2).
System (8) (for a while we shall use the notations u = û1, v = u2, x = x1, y = x2)
{yUx — XUy = V, yvx — XVy = —u
Is reduced to the heat equation
y ^xx 2xyuxy + X Wyy XUx - yuy + u = 0,
which has the solution u(x,y) = y/(x2 + y2) + xg(x2 + y2), hence v(x,y) = yg(x2 + y2) — x/(x2 + y2).
So, we get the solution of the system (8)-(10):
= x2/^x2 "i- x2, x3, t) "i" x^ ^(x2 "i- x2, x3, t), W2 = X2 ^(x2 "I"" X2, X3, t) XX/(X2 "I"" X2, X3, t), W — U (x^ "I- X2 , X3f t), p = p(xl + xf,x3,t),
k = k(xt
In the same way one can solve system P12 = P23 = ^13 = 0 and get
ul = xj$(x2 + xf + xf, t), p = p(x2 + xf + xf, t), k = £(x2 + xf + xf, t), £ = £(x2 + xf + xf, t).
(11) (12) (13)
2.3. Scale symmetry
Consider the generating function of scale symmetry [14]:
/XjUj + X2U2 + X3U3 + 2tUt + v}\ xxu2 + x2uf + x3u| + 2tu2 + u2 ^ _ + x2u| + x3u| + 2tuf + u3
*iPi + x2p2 + x3p3 + 2tpt + 2p X^k^ "I- X2&2 "I" "I-
\ Xj^ + X2£2 + X3£3 + 2t£t + 4£ /
All equations of the system 5 = 0 are of the form
x1ux1 + x2ux2 + X3UX3 + 2tut + au = 0,
(14)
where и = u1, p, к or e, a = 1,2,4.
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Using the characteristic equation for (14)
dx2 dx3 dt du
X-i x2 x3 2t Q,U we get the solution of the system S = 0 in the form
-i 1 ~i(X 1 X2 *f\ - 1 *2 *3
u =Ttu{~T-)'p=Tp{T-
i /222
/ ■v* "V* 'V"
± ~ / A3
3. REDUCTIONS OF THE k-£ TURBULENCE MODEL
In this section we consider three reductions of the k — e turbulence model with respect to three-dimensional subalgebras. Let us remark that the complete list of three-dimensional symmetry subalge-bras, which is presented in the paper [14], contains more than 20 items. Calculations show that reductions under considerations involves ordinary or partial differential equations only for functions k and £, while for other four function expression containing arbitrary constants (and function of t in case of p ) are obtained. This means that regularity conditions mentioned at the end of Section 2.2 in case of the k — £ turbulence model are broken.
1. The reduction with respect to subalgebra < Xx (/), X2 (g), X3 (h) >.
Combining (7) and (1)-(4), we obtain (see [20])
k2
k = L--e, (15)
E2
è = cEiLk-cE—, (16)
where
i = 2c„iin
while
fx1 + a1 gx2 + a2 hx3 + a3
u1 =---=-,u6 =---,fgh = a, (18)
f g h
P = ¡)+РЮ> k = KV, e = e(t), (19)
where a, at are constants.
2. The reduction with respect to subalgebra < Xx(/), X2(f), R12 >. Combining equations X1(f) = X2(f) = R12 = 0 and (1), we get
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и1 = jxlt и2 = ^x2, и3 = -2^x3 + a(t),
Pf ? ? P = -2f(xi+ 4) + P(x3, *),
к = k(x3, t), а = г(х3, t).
(20)
(21) (22)
Substituting (20), (21) in (2), we obtain
^P = - 3 (j^ ) x3 + ^a - 2a0 x3 - - ^fc + p(t).
Finally, using (3), (4) and (22), we get the reduction
cß к2
лч 2
fct+u3fc3 + jfc3 1 + 12^]--e.
\f) e
3. The reduction with respect to subalgebra < R12, R23, R13 >. Substituting (11) in (1), we obtain
_3
ul = A(t)xtR 2, R=xl + xl+ xf. Using (2) and (12), one get
-p =-A2R~2 --АЛ'1'2 — — k — Aca p 3 3 t 3 M
Finally, combining (3), (4) and (13), we obtain the reduction system for functions k = k(t, R) and £ = £(t, R):
cß к2 e
kt + 2AR~1/2kR = 2R[[v + ) ) + 6 ( v + ^^ ) kR + 12смЛ2Д-3 — - e,
c^ fc2
fc2
£
cu к
Си к2
+ 2AR-1/2£j> = 2R\\v + I e„ I +6\v + -t— ) £r + 12 cßcEiA2R~3k - cP„ —
'£2
4. INVARIANT SOLUTIONS
Reductions 2 and 3 in Section 4 are complicated systems of nonlinear partial differential equations for functions k and £. To find solutions of these system one can use symmetry method again or try to solve the systems numerically (cf. [12], [13]). This will be discussed elsewhere.
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For Reductions 1 in paper [20] families of exact solutions were obtained. For example, if X1 (/), X2{g), X3(h) are translations (f = g = h= 1, L = 0) we get invariant solution:
u1 = alt ù2 = a2, u3 = a3, p = p(t),
i c C£2 k = (C1t + Q)1"^, £ =-+ C2)1"c^2.
where at, Cx, C2 are arbitrary constants.
Also in [20] it was shown, that in the case L ^ 0 system (15)-(16) transforms to
è = (cEiLw — cE?w~1')e, k = we, (23)
where
w = -l(t)w2 + a, (24)
¿(0 = (c£l ~1)L>0, a = cE2-l> 0. (25)
1 V
The substitution w = y- transforms equation (24) to linear equation
lv — lv = al2v. (26)
Integrating (23), one get
r , c£i f lv f lv
In I e\ = l(cE Lw — cE w i)dt =- I I -——dt — cE I —dt =
J 1 2 cEi-l] lv 2 j v
Co f lv - lv
= —lnM-^j-âîrdt =
= \v\--^-(ln |v|-ln \l\) + ln\C\.
Ce1 ce2 ^
Therefore, we have
E = Cl^(iY^1, k = we = "^(L)& = c^(Lft=*. (27)
\W I v \v/ \v/
These results can be summarize as follows. To find invariant solutions in the case L ^ 0 we must choose functions /, g, h such that fgh = A = const and find , p by (18), (19), then calculate function Z(t) (see (17), (25)), solve equation (26) and find function v. Then, using (27), we get e and k. Thus, the main problem in this procedure is to solve equation (26). Below we consider two cases when this equation can be integrated.
In the case f = , g = b2e^t, h = , where X, p, bi are arbitrary constants,
I = const, v = C2e^"~lt + we get:
Научный Вестник МГТУ ГА_Том 19, № 02, 2016
Civil Avition High TECHNOLOGIES Vol. 19, № 02, 2016
û1 = Ахх + a1e_At, и2 = дх2 + a2e~ßt, û3 = —(А + д)хх + а3е(я+^, р = -|(А2х^ + д2х| + (А + д)2х|) + p(t),
1
£ = С5 с^. к = c5jj —
(aj, Cj are arbitrary constants).
In the case / = ta, a ^ 0, g = 1/f, h = const, equation (26) is the Euler equation
t2v + 2ti> - 4cAi(c£i - l)(c£z - l)a2v = 0 with the solution v = C6t^~ + C7t~^~, where
V = + 16cß(c£l - l)(c£z - l)«2.
So one get family of invariant solutions
и1 = aXi/t + a1t a,u2 = —ax2/t + a2ta,u3 = a3,
ap (a — l)x2 + (a + l)x2
P = ---p-+
1 1
k =
£ =
Cg2
(c6(M - l/2)t"+I - C7(M + l/2)t-"+l)
where ca = 4cAi(c£i — l)(c£z — !)a2, while at, Ct are arbitrary constants.
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4. Kalugin V.T., Strijhak S.V Physical and mathematical modeling of a detachable flow of the probe-device with disk stabilizers in the gas swirling flow. The Scientific Herald MSTUCA. Ser. Aeromechanics and Strength. Moscow. 2008. No. 125. Pp. 63-67. (in Russian).
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О РЕДУКЦИЯХ И ИНВАРИАНТНЫХ РЕШЕНИЯХ МОДЕЛИ к-Е ТУРБУЛЕНТНОСТИ
Хорькова Н.Г.
Методы группового анализа дифференциальных уравнений применяются к модели к - е турбулентности. Рассмотрены редукции модели к - е турбулентности по отношению к трехмерной подалгебре симметрий. Получены семейства точных решений.
Ключевые слова: нелинейные дифференциальные уравнения, локальные симметрии, инвариантные решения, модель к - е турбулентности.
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СВЕДЕНИЯ ОБ АВТОРЕ
Хорькова Нина Григорьевна, 1960 г.р., окончила МГУ им. М.В. Ломоносова (1983), кандидат физико-математических наук, доцент кафедры «Прикладная математика» МГТУ им. Н.Э. Баумана, автор более 20 научных работ, область научных интересов - алгебро-геометрическая теория дифференциальных уравнений, локальные и нелокальные симметрии, законы сохранения, инвариантные решения, электронный адрес: nkhorkova@diffiety.ac.ru, шпа-khorkova@yandex.ru.
