CC BY
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Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 2(18). C. 41-52. ISSN 2079-6641
DOI: 10.18454/2079-6641-2017-18-2-41-52
MSC 35R01, 76M60, 17B66.
SYMMETRY CLASSIFICATION OF NEWTONIAN INCOMPRESSIBLE
FLUID'S EQUATIONS FLOW IN TURBULENT BOUNDARY LAYERS
M. Nadjafikhah1, S. R. Hejazi2
1 Iran University of Science and Technology, Tehran, Iran.
2 Shahrood University of Technology, Shahrood, Semnan, Iran.
E-mail: [email protected], [email protected]
Lie group method is applicable to both linear and non-linear partial differential equations, which leads to find new solutions for partial differential equations. Lie symmetry group method is applied to study Newtonian incompressible fluid's equations flow in turbulent boundary layers. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra such as Levi decomposition, radical subalgebra, solvability and simplicity of symmetries is given.
Keywords: Fluid mechanics, Lie symmetry, Partial differential equation, Shear stress, Optimal system
© Nadjafikhah M., Hejazi S.R., 2017
УДК 548.71
КЛАССИФИКАЦИЯ СИММЕТРИИ УРАВНЕНИЙ НЬЮТОНОВСКОЙ НЕСЖИМАЕМОЙ ЖИДКОСТИ В ТУРБУЛЕНТНЫХ ПОГРАНИЧНЫХ
СЛОЯХ
М. Наджафихаx1, С. Р. Хеязи2
1 Иранский университет науки и техники, Тегеран, Иран
2 Шахрудский технологический университет, г. Шахруд, провинция Семнан, Иран
E-mail: [email protected], [email protected]
Метод группы Ли применим как к линейным, так и к нелинейным уравнениям в частных производных, что приводит к поиску новых решений для уравнений с частными производными. Метод группы симметрии Ли применяется для изучения уравнений потока ньютоновской несжимаемой жидкости в турбулентных пограничных слоях. Дана группа симметрии и ее оптимальная система, получены групповые инвариантные решения, связанные с симметриями. Наконец, дана структура алгебры Ли, такая как разложение Леви, радикальная подалгебра, разрешимость и простота симметрий.
Ключевые слова: механика жидкости, симметрия Ли, уравнение с частичным разложением, напряжение сдвига, оптимальная система
© Наджафихаx М. Хеязи С. Р., 2017
Introduction
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface. On an aircraft wing the boundary layer is the part of the flow close to the wing. The boundary layer effect occurs at the field region in which all changes occur in the flow pattern. The boundary layer distorts surrounding non-viscous flow. It is a phenomenon of viscous forces. This effect is related to the Reynolds number (In fluid mechanics and heat transfer, the Reynold's number is a dimensionless number that gives a measure of the ratio of inertial forces to viscous and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions). Laminar boundary layers come in various forms and can be loosely classified according to their structure and the circumstances under which they are created. The thin shear layer which develops on an oscillating body is an example of a Stokes boundary layer, whilst the Blasius boundary layer refers to the well-known similarity solution for the steady boundary layer attached to a flat plate held in an oncoming unidirectional flow. When a fluid rotates, viscous forces may be balanced by the Coriolis effect, rather than convective inertia, leading to the formation of an Ekman layer. Thermal boundary layers also exist in heat transfer. Multiple types of boundary layers can coexist near a surface simultaneously. The deduction of the boundary layer equations was perhaps one of the most important advances in fluid dynamics. Using an order of magnitude analysis, the well-known governing Navier—Stokes equations of viscous fluid flow can be greatly simplified within the boundary layer. Notably, the characteristic of the partial differential equations (PDE) becomes parabolic, rather than the elliptical form of the full Navier—Stokes equations. This greatly simplifies the solution of the equations. By making the boundary layer approximation, the flow is divided into an inviscid portion (which is easy to solve by a number of methods) and the boundary layer, which is governed by an easier to solve PDE [1].
Flow and heat transfer of an incompressible viscous fluid over a stretching sheet appear in several manufacturing processes of industry such as the extrusion of polymers, the cooling of metallic plates, the aerodynamic extrusion of plastic sheets, etc. In the glass industry, blowing, floating or spinning of fibres are processes, which involve the flow due to a stretching surface. Mahapatra and Gupta studied the steady two-dimensional stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet when the sheet is stretched in its own plane with a velocity proportional to the distance from the stagnation-point [7]. They concluded that, for a fluid of small kinematic viscosity, a boundary layer is formed when the stretching velocity is less than the free stream velocity and an inverted boundary layer is formed when the stretching velocity exceeds the free stream velocity. Temperature distribution in the boundary layer is determined when the surface is held at constant temperature giving the so called surface heat flux. In their analysis, they used the finite-differences scheme along with the Thomas algorithm to solve the resulting system of ordinary differential equations.
The treatment of turbulent boundary layers is far more difficult due to the time-dependent variation of the flow properties. One of the most widely used techniques in which turbulent flows are tackled is to apply Reynolds decomposition. Here the instantaneous flow properties are decomposed into a mean and fluctuating component. Applying this technique to the boundary layer equations gives the full turbulent boundary
layer equations
d u d v 0 dx dy
_d U _d U 1 d p / d 2U
d x d y p d x \d x2
_ dv + _dv 1 dp + v i d2v
d x d y p d y \d x2
d2^
d y2 d^
d y2
d d _
- dy^ - dx<u'2',
- dx(U¥ )- dy^
(1)
where p is the density, p is the pressure, v is the kinematic viscosity of the fluid at a point and u and v are average of the velocity components in Reynold decomposition. Here u! and V are the velocity fluctations such that; u = u + U and v = v + V. By using the scale analysis(a powerful tool used in the mathematical sciences for the simplification of equations with many terms), it can be shown that the system (1) reduce to the classical form
d u d v 0 dx dy
_d U _d v 1 d p d 2U d
u^r + v^- =---+ V^r <uV),
d x d y p d x d y2 d y
d4=0.
d y
(2)
The term u'v' in the system (2) called Reynolds shear stress, a tensor that conventionally written
Rij = p li'U.
(3)
The divergence of this stress is the force density on the fluid due to the turbulent fluctuations. Using Navier-Stokes equations for a fluid whose stress versus rate of strain curve is linear and passes through the origin (Newtonian fluid) the tensor (3) reduces to
d Hi Rij=^ ,
where / is the fluid viscosity.
This paper is concerned with the symmetry of Newtonian incompressible fluid's equations flow in turbulent boundary layers of the form (2). Lie group theory is applied to the equations of motion for determining symmetry reductions of partial differential equations [2, 3, 4, 5, 10, 11]. The solution of the turbulent boundary layer equations therefore necessitates the use of a turbulence model, which aims to express the Reynolds shear stress in terms of known flow variables or derivatives. The lack of accuracy and generality of such models is a major obstacle in the successful prediction of turbulent flow properties in modern fluid dynamics.
Lie Symmetries of the Equations
A PDE with p—independent and q—dependent variables has a Lie point transformations %i = xi + e%i(x, u) + O (e2), via = Ua + £Va (x, u) + O (e2),
+
f
where ^ = ^ ^ for i = 1,...,p and qa = dUa
for a = 1,...,q. The action of the
£=0
e=0 d£
Lie group can be considered by its associated infinitesimal generator
p d q d
v = £ &(x, u) — + £ <Pa (x, u) — (4)
i=1 dx« a=1 d ua
on the total space of PDE (the space containing independent and dependent variables). Furthermore, the characteristic of the vector field (4) is given by
P d ua Qa(x, u(1)) = <Pa(x, u) - £ &(x, u),
«=1 Ox«
and its n—th prolongation is determined by
p d q n d
v(n) = £&(x, u)dx + £ £ J(^u(-)) .da,
i=1 ax« a =1 /=-=0 au/
where ^ = D/Qa + £P=1 ^«ua«. (D/ is the total derivative operator describes in (3)).
Let us consider a one-parameter Lie group of infinitesimal transformations (x,y, u, v,p) given by
x = x + e^(x, y, up, v, p) + O (e2), y = y + e&(x, y, u, v, p) + O (e2),
up = up + en1(x, y, UP, iP,p) + O (e2), iP = iP + en2(x, y, U, iP, p) + O (e2),
p = p + e^3 (x, y, U, iP, p) + O (e2),
where e is the group parameter. Then one requires that this transformations leaves invariant the set of solutions of the system (2). This yields to the linear system of equations for the infinitesimals ^1(x,y,UP,iP,p),
£2(x,y,u,iP,p), n1(x,y,u,iP,p), n2(x,y,u,iP,p) and n3(x,y,u,iP,p). The Lie algebra of infinitesimal symmetries is the set of vector fields in the form of
„ d „ d d d d v=* dx+& dy+du+n2 di+n3 dp ■
This vector field has the second prolongation
(2) = v + x + m y + mxx d + ,.xy d + vy d
= v+^xäx+**+^++^ (5)
+ ^3^" + $3^" + ^3^" + <3^" + <3^" + ^ d
d Vx 2 d Vy d Vxx 2 d Vxy 2 d Vyy d p.
+0>3' £ + ff 4- + tf + »T ^
d py d pxx 3 d pxy 3 d pyy'
with the coefficients
= Dx(Çl - Il Üx - |2Üy) + |l Uxx + ,
Çy = Dy(Çl - |lüx - |2Üy) + IlUy + |2Üyy,
çf* = ^^(Çl - IlÜx - feüy) + IlÜxxx + |2Üxxy,
Ç^ = DxDy(Çl - ^lUx - + IlÜxy + ^2"xxy,
= DJ(ÇI - |lÜx - |2Üy) + IlÜxxy + ^2Ûxyy,
Çf = Dx(Ç2 - Ilîx - ^2Vy) + Ilîxx + ^xy,
Çj = Dy(Ç2 - Ilîx - ^2Vy) + Ilîxy + |2Ïyy,
Ç3 = DJ(Ç3 - <§lpx - &pyH &pxy + &
çfx
xy _
Ç3 =
- €1 Px - €2Py) + €1 P xxx + €2 T^xx^ DxDy(^3 - €1 px - €2py) + €1 pxxy + €2 pxxy, = D;(P3 - €1 Px - €2py) + €1 Pxxy + €2pxyy,
where the operators Dx and Dy denote the total derivative with respect to x and y:
d
px—i ■ ■ ■, d p
d_
Pydp ■■■ '
Dx = d d d
x + Ux^— du + Vx^-z v
Dy = d d d
dy + Üy~dü +vVydV
(6)
Using the invariance condition, i.e., applying the second prolongation v(2), to system (2), and by solving the linear system
f v(2)/ dû + dvX = 0,
V d x d y/
d ü d v
--+ v— ,
d x d y p d x
<2,( I ) = ^
uu uv l dp < v(2M Ü— + v—- + - V
v
d2U d m
y 2 y
(uV)) = 0, (mod (2))
the following system of 17 determining equations yields:
Ily = 0, Ilü = 0, 1ll = 0, Ilv = 0,
|lxx = 0, I2x = 0, I2ü = 0, |2v = 0,
12 p = 0, I2yy = 0, nlx = 0, nl y = 0,
nlu = 0, nlv = 0, nlp = 2|lx - 4|2y, n2 = = ü(|lx
n3 = |2y.
The solution of the above system gives the following coefficients of the vector field v:
ni = C4U — 2C5U,
Il = Cl + C4x, n2 = -C5V,
I2 = C2 + C5y,
^3 = C3 + 2C4p - 4C5p,
where Q for i = 1,...,5 are arbitrary constants, thus the Lie algebra G of the Newtonian incompressible fluid's equations flow in turbulent boundary layers is spanned by the five
y
Таблица 1. Commutation relations of G
[, ] V1 V2 V3 V4 V5
V1 0 0 0 V1 0
V2 0 0 0 0 V2
V3 0 0 0 2V3 -4v3
V4 -V1 0 -2v3 0 0
V5 0 -V2 4v3 0 0
vector fields
vi =
V2 =
V3 =
V4 = V5 =
The commutation relations between these vector fields is given by the table (1), where entry in row i and column j representing [vj, Vj].
The one-parameter groups Gi generated by the base of G are given in the following table.
G1 : (x + e,y,U,v,p), G2 : (x,y + e,U,v,p),
G3 : (x,y,U,v,p + e), G4 : (xee,y,Uee,v,pe2e)
G5 : (x, yee, Ue-2e, ve-e, pe-4e)
Since each group Gi is a symmetry group and if U = f (x,y),v = g(x,y) and p = h(x,y) are solutions of the system (2), so are the functions
Ui = f (x + e, y), vi = g(x + e, y), pi = h(x + e, y),
Ui = f (x, y + e), vi = g(x, y + e), pi = h(x, y + e),
Ui = f (x, y), vi = g(x, y), pi = h(x,y) + e,
Ui = e-e f (xee, y), vi = g(xee, y), pi = e-2e h(xee, y),
Ui = e2ef (x, yee), vi = eeg(x, yee), pi = e4e h(x, yee).
where e is a real number. Here we can find the general group of the symmetries by
considering a general linear combination civi +-----+ civ5 of the given vector fields.
In particular if g is the action of the symmetry group near the identity, it can be represented in the form g = exp(ev5)o---oexp(evi). Consequently, if U = f(x,y),v = g(x,y) and p = h(x,y) be a solution for system (2), so is
U = ee f ((x + e )ee, (y + e )ee),
v = g( (x + e )ee, (y + e )ee), (7)
p = e2eh( (x + e )ee, (y + e)ee) + ee-e,
д
д x'
д
dy '
д
д p'
д д x д „ д + + 2 p—, д u д p
д Уз" d y 2-d - д — 2u---v--- д u d v
д p'
Optimal system of steady two-dimensional boundary-layer stagnation-point flow equations
Let a system of differential equation A admitting the symmetry Lie group G,be given. Now G operates on the set of solutions S of A. Let s ■ G be the orbit of s, and H be a subgroup of G. Invariant H—solutions s g S are characterized by equality s ■ S = {s}. If h g G is a transformation and s g S,then h ■ (s ■ H) = (h ■ s) ■ (hHh--1). Consequently,every invariant H—solution s transforms into an invariant hHh-1—solution [4, 8, 9].
Therefore, different invariant solutions are found from similar subgroups of G. Thus, classification of invariant H—solutions is reduced to the problem of classification of subgroups of G,up to similarity. An optimal system of s—dimensional subgroups of G is a list of conjugacy inequivalent s—dimensional subgroups of G with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s—dimensional subalgebras forms an optimal system if every s—dimensional subalgebra of G is equivalent to a unique member of the list under some element of the adjoint representation: H = Ad(g) ■ H.
Let H and H be connected, s—dimensional Lie subgroups of the Lie group G with corresponding Lie subalgebras H and H of the Lie algebra G of G. Then H = gHg-1 are conjugate subgroups if and only H = Ad(g) ■ H are conjugate subalgebras (Proposition 3.7 of [9]). Thus,the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras, and so we concentrate on it.
One-dimensional optimal system
For one-dimensional subalgebras, the classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation, since each one-dimensional subalgebra is determined by a nonzero vector in Lie algebra symmetries of steady two-dimensional boundary-layer stagnation-point flow equations and so to "simplify"it as much as possible. The adjoint action is given by the Lie series
e 2
Ad(exp(evf)vj) = vj - e[vf, vj] + — [vf, [vf, vj]] - ■ ■ ■, (8)
where [v;,vj] is the commutator for the Lie algebra, e is a parameter, and i,j = i,•••,5. Let Fie : G m G defined by v m Ad(exp(evi)v) is a linear map, for i = i,•••,5. The
matrices Mf of Fi£, i = 1, (
,5, with respect to basis {v1,•••,v5} are
Mf =
Mf =
Mf =
1
0 0 —£
0 1
0 0
0 0
1
0 0
0 0
0 0 1 0 0 0 0 1
—2f 4f 0
0
4f
/
e
0 0
0
0 0 0
1
0
0 0
0 1
0
0 0
0 1
M22 =
M44 =
1
0 0
0 0
( e£ 0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
—£ 0 0 1 /
0 0 0 0
1 0 0 0
0 e2£ 1 0
0 0 1 0
0 0 0 1
(9)
e
e
by acting these matrices on a vector field v alternatively we can show that a one-dimensional optimal system of G is given by
Xi = v3, X2 = aivi + «2V2, (10)
X3 = ai V2 + «2V3, X4 = aiVi + «2V2 + OÎ3V3, ( )
where a/s are real constants.
Two-dimensional optimal system
Next step is to construct two-dimensional optimal system, i.e., classification of two-dimensional subalgebras of G. The process is by selecting one of the vector fields in (10), say, any vector field of (10). Let us consider X1 (or X^i = 2,3,4,5). Corresponding
to it, a vector field X = a1v1 +-----+ a5v5, where ai's are smooth functions of (x,y,u,v,p)
is chosen, so we must have
[X1, X ]= A X1 + ^ X, (11)
the equation (4) leads us to the system
Cjkajflk = A a,- + ^aj (i = 1,2,3,4,5). (12)
The solutions of the system (5), give one of the two-dimensional generator and the second generator is X1 or, X,, i = 2,3,4,5 if selected. After the construction of all two-dimensional subalgebras, for every vector fields of (10), they need to be simplified by the action of (6) in the manner analogous to the way of one-dimensional optimal system.
Consequently the two-dimensional optimal system of G has three classes of G's members combinations such as
V1, V2, V1, V3, V2, V3, V3, V4,
v1 , v4, v1 , v5, v2, v4 v2, v5,
V3, V5, V4, V5, V1, Li=2 Av,',
^1V2 + &V3, V1 + 2&(V4 + V5), V1 + &V2, V3 + ¿3 (V4 + V5)'
Three-dimensional optimal system
This system can be developed by the method of expansion of two-dimensional optimal system. For this take any two-dimensional subalgebras of (7), let us consider the first two vector fields of (7), and call them Y1 and Y2, thus, we have a subalgebra with basis
{Y1,Y2}, find a vector field Y = a1V1 +-----+ a5V5, where a/s are smooth functions of
(x,y, u, v,p), such the triple {Y1, Y2, Y} generates a basis of a three-dimensional algebra. For that it is necessary an sufficient that the vector field Y satisfies the equations
[Y1, Y] = A1Y + M1Y1 + V1Y2,
[Y2, Y ] = A2Y + M2Y1 + V2Y2, (13)
and following from (13), we obtain the system
jS/afc = A1ai + + vX a = 1,2,3,4,5 (14)
jS/afc = A2ai + + V2^Si', a = 1,2,3,4,5.
The solutions of system (14) is linearly independent of {Yi, Y2} and give a three-dimensional subalgebra. This process is used for the another two couple vector fields of (7).
Consequently the three-dimensional optimal system of G is given by
vi, v2, v3, vi, v2, v4, vi, v2, v5, vi, v3, v4, vi, v3, v5, (^
vi, v4, v5, v2, v3, v4, v2, v3, v5, v2, v4, v5, v3, v4, v5.
Four-dimensional optimal system
Four-dimensional optimal system can be developed by the method of expansion of three-dimensional optimal system analogous to three-dimensional. First step is selecting a three-dimensional subalgebra of (15), and call them Zi,Z2 and Z3, then find a vector
field Z = aivi +-----+ a5v5, where a/s are smooth functions of (x,y,U,v,p), such the
foursome {Zi,Z2,Z3,Z} generates a basis of a four-dimensional algebra. It is necessary an sufficient that the vector field Z satisfies the equations
[Zi, Z] = AiZ + MiZ + ViZ2 + YiZ3,
[Zi, Z ] = A2Z + M2Zi + V2Z2 + 72Z3, (16)
[Zi, Z] = A3Z + M3Zi + V3Z2 + Y3Z3,
following from (16), we obtain the system
Cjkak = Aia,- + Mi + Vi + Yi, a = i, 2,3,4,5,
Cjkak = ^ + M2 + V2 + Y2, a = i, 2,3,4,5, (17)
C^k = ^ + M3 + V3 + Y3, a = i, 2,3,4,5.
The solutions of system (17) is linearly independent of {Zi,Z2,Z3} and give a four-dimensional subalgebra. This process is used for the another triple vector fields of (15). All previous calculations lead to the table 2 and 3 for the optimal system of G.
Lie Algebra Structure
G has a no any non-trivial Levi decomposition in the form of G = R x Gi, because G has no any non-trivial radical, i.e., if R be the radical of G, then G = R.
If we want to integration an involuting distribution, the process decomposes into two steps:
• integration of the evolutive distribution with symmetry Lie algebra G/R, and
• integration on integral manifolds with symmetry algebra R.
First, applying this procedure to the radical R we decompose the integration problem into two parts: the integration of the distribution with semisimple algebra G/R, then the integration of the restriction of distribution to the integral manifold with the solvable symmetry algebra R.
The last step can be performed by quadratures. Moreover, every semisimple Lie algebra G/R is a direct sum of simple ones which are ideal in G/R. Thus, the Lie-Bianchi theorem reduces the integration problem to involutive distributions equipped
Таблица 2. Optimal system of subalgebras
dimension 1 2
Ы (V1, V2)
subalgebras (a v1 + a2v2 ) (V1, V3)
(«1V2 + «2V3 ) (V2, V3)
(aiVi + «2V2 + «3 V3) (V2, V4)
(V3, V4)
(V1, V4)
(V1, V5)
(V2, V5)
(V3, V5)
(V4, V5)
(V1, L5=2 0^)
(01V1 + 02 V2, V3 + 03 (V4 + V5))
(01V2 + 02 V3, V1 + 2 03 (V4 + V5))
Таблица 3. Optimal system of subalgebras
dimension 3 4 5
(V1, V2, V3) (V1, V2, V3, V4) (V1, V2, V3, V4, V5)
subalgebras (v1, v2, v4) (V1 , V2, V3, V5)
(V1, V2, V5) (V1 , V2, V4, V5)
(V1, V3, V4) (V1 , V3, V4, V5)
(V1, V3, V5) (V2, V3, V4, V5)
(V1, V4, V5)
(V2, V3, V4)
(V2, V3, V5)
(V2, V4, V5)
(V3, V4, V5)
with simple algebras of symmetries. Thus, integrating of system (2), become so much easy.
The Lie algebra G is solvable and non-semisimple. It is solvable because if G(1) = (vf, [vf,vj]) = [G,G], we have G(1) = [G,G] = (vb•••,V5), and G(2) = [G(1),G(1)] = (vi,v2,2vs), so, we have a chain of ideals G(1) d G(2) d {0}, and it is non-semisimple because its killing form
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 5 -8 \ 0 0 0 -8 17 У
is degenerate.
According to the table of the commutators, G has two abelian 2 and 3-dimensional subalgebras spanned by (v1, v2, v3) and (v4, v5) respectively such that the first one is an ideal in G, so, we can decompose G in the to semidirect sum of G = R3 к R2.
Conclusion
In this article group classification of Newtonian incompressible fluid's equations flow in turbulent boundary layers and the algebraic structure of the symmetry group was analyzed. Classification of r-dimensional subalgebra is determined by constructing r-dimensional optimal system. The optimal system of system of differential equations is usefull for reducing a system of differential equations. The method described in [4] widely, but it is suggested to use a computational software for calculations such as Maple or Mathematica.
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Для цитирования: Nadjafikhah M., Hejazi S.R. Symmetry classification of newtonian incompressible fluid's equations flow in turbulent boundary layers // Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 2(18). C. 41-52. DOI: 10.18454/2079-6641-2017-18-2-41-52
For citation: Nadjafikhah M. , Hejazi S.R. Symmetry classification of newtonian incompressible fluid's equations flow in turbulent boundary layers, Vestnik KRAUNC. Fiz.-mat. nauki. 2017, 18: 2, 41-52. DOI: 10.18454/2079-6641-2017-18-2-41-52
Поступила в редакцию / Original article submitted: 04.01.2017
