1 Introduction

Burgers vortex describes the interplay between the intensification of vorticity due to the imposed straining flow and the diffusion of vorticity due to the action of viscosity. The straining simulates locally the stretching undergone by each vortex in the velocity field induced by other vortices. Intermittent structures that exhibit velocity profiles similar to that of Burgers vortex were observed in grid turbulence^[1]. It is well known that the vortex stretching process is governed by the vorticity conservation equations^[2-7],

(1)

with ω=∇ × v for the velocity field v=(v₁(x, t, y), v₂(x, t, y), v₃(x, t, y)), which leads to the transport of energy among various scales of motion in a turbulent flow. It plays an important role in the vortex reconnection process. Therefore, there are many researchers who have investigated the exact solutions to the system for different velocities v within the solvable cases and their applications. The closed form of steady Burgers vortex solutions for special parameter values was found by Robinson and Saffman^[2]. Unsteady 2D Burgers vortex solutions have been used to model the spanwise structure of turbulent mixing layers by Lin and Corcos^[3] and Neu^[4]. Unsteady axisymmetric Burgers vortex solutions were used in modeling the fine-scale structure of homogeneous incompressible turbulence by Townsend^[5] and Lundgren^[6].

Let us consider a modified Burgers vortex flow with the velocity field of the form,

which describes the convection of the vortex lines toward the plane of x=0 and the stretching in the y-direction with the imposed straining flow. This streamline pattern is the same in each plane parallel to the xy-plane. Furthermore, note that the flow associated with the vortex in the question is perpendicular to the plane of the uniform straining flow. This situation is well suited for modeling a mixing-layer flow or jet flow^[7]. The corresponding vorticity field is then given by

Hence, the vortex lines for this model are aligned along the y-axis, which happens to be the principal axis of the uniform plane straining flow v. With the form of v, the vorticity conservation equation (1) becomes

(2)

where ν is the kinematic viscosity, r(t) is the flow parameter (which is a measure of the stretching rate of the vortices), and u is the vorticity component .

The exact solutions to the 2D Burgers vortex equation have been considered in some special cases of r(t). Shivamoggi^[7] obtained the exact solutions in which the flow parameter r(t) is constant. Then, exact solutions were obtained for the flow parameter r(t) satisfying the first-order ordinary differential equation^[8], i.e.,

(3)

where c is constant. Shivamoggi^[7] and Rollins^[8] obtained the solutions of the special case by separation of variables in the unsteady case. Van Gorder^[9] considered the unsteady modified 2D Burgers vortex equation for arbitrary forms of the flow parameter r(t) with Fourier analysis.

However, to our knowledge, the Lie symmetry group properties of the system (2) have not been investigated yet, even in the above mentioned cases.

For partial differential equations (PDEs) of physical phenomena, admitting various symmetries is an innate property of PDEs^{[11, 13]}. Generally speaking, the symmetries admitted by PDEs yield some intrinsic qualities of the solutions to the PDEs, such as exact solutions, conservation laws, Hamiltonian structures, and asymptotic behavior. The theory of Lie symmetry groups of differential equations started during the study of the heat equation^[10] and was developed by modern mathematicians^[11-13]. The classical symmetry method was expanded by introducing the non-classical (conditional) symmetry method^[14]. This generalization of the classical symmetry method was widely applied in solving various PDEs^[15-18].

In this paper, we investigate the classical and non-classical (Lie) symmetries of the modified 2D Burgers vortex equation (2) and their applications to solve exact solutions to the equation, which are necessary complements of existing literatures on study of the system (1). We will present the complete classical and non-classical symmetries of the 2D Burgers vortex equation for arbitrary forms of the flow parameter r(t), and show how they depend on the parameter r(t) by symmetries of the equation. The obtained results show such equation admits richer classical and non-classical symmetries. In addition to rediscovering the existing solutions in the literature obtained by different methods, some exact solutions of the several practical cases to the equation are derived in the unified symmetry scheme. This shows that the symmetry method is powerful and more general to solve PDEs.

The following remark is used to simplify the computations in the article.

Remark 1 We notice that the inverse transformation, i.e.,

(4)

is an equivalent one of (2), where a=constant ≠ 0. It means that under the translation in t and the scaling in r, (2) is similar to

with R(T)=ar(T + b). Hence, we can use the equivalence map to simplify the computations involved in the article. In fact, choosing a=c in the transformation, we always suppose c ≡ 1 in (3).

The paper is arranged as follows. The complete classical symmetries of the modified 2D Burgers vortex equation are determined in Section 2. The complete non-classical symmetries of the equation are derived in Section 3. In Section 4, some exact solutions to the equation are constructed by the symmetries obtained in Sections 2 and 3. In Section 5, some concluding remarks are given.

2 Classical symmetries of modified 2D Burgers vortex equation

In this section, we determine the classical symmetries of (2) for complete cases of the parameter r(t).

According to the theory of the symmetry method^[13], the vortex equation (2) is invariant under the one-parameter Lie group of infinitesimal transformations in (t, x, u) given by

with an associated infinitesimal generator,

(5)

if and only if

(6)

Here, is a second-order prolongation of X, and ε is the group parameter. From (6) and algorithms given in Refs.[16]-[18], it is easy to derive the determining system satisfied by the infinitesimals ξ(t, x, u), τ(t, x, u), and η(t, x, u) as follows:

(7)

with the parameter r(t). By using the characteristic set method given in Ref. [17], we obtain the general solutions to the system as follows:

(8)

Here, c₀ is an arbitrary constant, and functions α(t), β(t), and r(t) are arbitrary solutions to the following ordinary differential equations (ODEs):

(9)

(10)

and ρ(t, x) is any solution to (2).

(8)-(10) give the general classical symmetry of (2). Consequently, the specific symmetries to solve exact solutions are obtained by the following two steps: find the kernel symmetry and extend the kernel symmetry by solving (9) and (10).

Step 1 Find the kernel symmetry. The kernel symmetry of (2) is the one corresponding to the arbitrary r (t). In this case, it is easily seen that α(t)=0 and β(t)=0 from (9) and (10). Therefore, it yields the kernel symmetry with the generator

(11)

which generates an infinite dimensional Lie algebra L^∞=L¹ ⊕ L⁰. Here, L¹ is spanned by X₁=u∂_u, and L⁰ is spanned by X⁰=ρ(t, x)∂_u.

Step 2 Extend the kernel symmetry. The solutions to (9) and (10) are various for different values of the parameter r(t), which yields different extensions of (11). However, it is hard to get the general solution to (9) and (10) without any restriction on the parameter r(t). Hence, in this paper, we consider the following two cases:

where d is a constant, and h (t) is a function of t. The two cases essentially extend the cases considered in Refs. [7]-[9]. In what follows we let m=.

Case Ⅰ  r′(t) -r²(t)=d=constant

In this case, the general solutions to the ordinary differential equation (ODE) are

with δ=±1.

Noticing the previous remark on the equivalent transformation (4), without loss of generality, let m=1, and b=0, and take d=1 when d > 0, and d=-1 when d < 0 in the expressions of the solutions r(t). Hence, the solutions r(t) are simplified to

(12)

By solving (9) and (10), one can correspondingly obtain

where c_i (i=1, 2, …, 5) are arbitrary constants. Now, from (8), one obtains the classical symmetries (5) with associated infinitesimals corresponding to the cases of r(t) as follows:

(13)

if r(t)=δ and d=-1 < 0;

(14)

if r(t)=-tanh t and d=-1 < 0;

(15)

if and d=0;

(16)

if r(t)=tan t and d=1 > 0.

Therefore, the corresponding infinitesimal generator (5) of the symmetries is obtained as follows:

(17)

if r(t)=δ and d=-1 < 0;

(18)

if r(t)=-tanh t and d=-1 < 0;

(19)

if r(t)= and d=0;

(20)

if r(t)=tan t and d=1 > 0.

Case 2 r′(t) -r²(t)=h(t)

The explicit solutions to the equation depend on the form of the right hand side function h(t). Therefore, for obtaining specific form symmetries, we consider the following three subcases: h(t)=a³t(2 -at), h(t)=a(at²ⁿ + nt^n-1), and h(t)=-at^n-2(atⁿ + n + 1).

Subcase 2.1 r′(t) -r²(t)=a³t(2 -at)

In this subcase, one can find the exact solutions to (10) (see Ref. [19]). Hence, one obtains the classical symmetries (5) with

(21)

Subcase 2.2 r′(t) -r²(t)=-a(at²ⁿ + nt^n-1)

In this subcase, we have and . Hence, one obtains the classical symmetries (5) with

(22)

Subcase 2.3 r′(t) -r²(t)=-at^n-2(atⁿ + n + 1)

In this subcase, a particular solution to (9) is , and a solution to (10) is α(t)=0. Hence, we obtain the classical symmetries (5) with

(23)

From the above procedure, we see that, once the parameter r(t) is given, the symmetries of (2) are determined by solving (9) and (10) and using (8). This also indicates that the dependence of the general symmetry of (2) on the parameter r(t) is completely determined by (9) and (10).

3 Non-classical symmetry of modified 2D Burgers vortex equation

The non-classical symmetry X of the modified 2D Burgers vortex equation is the classical symmetry of the system consisted by both (2) and the associated invariant surface equation, i.e.,

(24)

In what follows we consider two cases, i.e., τ ≠ 0 and τ ≡ 0.

Case 3 τ ≠ 0

Without loss of generality, let τ(t, x, u) ≡ 1. By the algorithm given in Ref. [17], the infinitesimals ξ(t, x, u) and η(t, x, u) satisfy a determining system,

(25)

The nonlinear system differs from the case of classical symmetry. It has general solutions as follows:

Here, β=β(t, x), γ=γ(t, x), and ρ=ρ(t, x) satisfy the following equations:

(26)

with the parameter r(t). The solutions to (26) can extend the classical symmetry and generate abundant non-classical symmetries of the modified 2D Burgers vortex equation. As a result, more new exact solutions or reduced forms of (2) will be obtained by symmetries, which cannot be derived from classical symmetries.

We consider Case 1 of r(t) in the classical symmetries, namely,

to see the efficient extension of the classical symmetry by the non-classical symmetry.

Subcase 3.1 r(t)=δ

By solving (26), one obtains the non-classical symmetries (5) with τ ≡ 1 and

where c_i (i=1, 2, …, 6) are arbitrary constants, ρ(t, x) is any solution to the modified Burgers vortex equation (2), and erf () is the error function. Obviously, the non-classical symmetry extends the classical symmetry when the flow parameter r(t)=δ.

The corresponding infinitesimal generators of the symmetries are obtained as follows:

(27)

Subcase 3.2 r(t)=-tanh t

By solving (26), one obtains the non-classical symmetries (5) with τ ≡ 1 and

where c_i (i=1, 2, …, 5) are arbitrary constants, and ρ(t, x) is any solution to the modified Burgers vortex equation (2).

The corresponding infinitesimal generators of the symmetries are obtained as follows:

(28)

Subcase 3.3

By solving (26), one obtains the non-classical symmetries (5) with τ ≡ 1 and

where c_i (i=1, 2, …, 6) are arbitrary constants, and ρ(t, x) is any solution to the modified Burgers vortex equation (2).

The corresponding infinitesimal generators of the symmetries are obtained as follows:

(29)

Subcase 3.4 r(t)=tan t

By solving (26), one obtains the non-classical symmetries (5) with τ ≡ 1 and

where c_i (i=1, 2, …, 5) are arbitrary constants, and ρ(t, x) is any solution to the modified 2D Burgers vortex equation. The corresponding infinitesimal generators of the symmetries are obtained as follows:

(30)

Case 4 τ ≡ 0

Without loss of generality, let ξ(t, x, u) ≡ 1. In this case, the determining equation for η(t, x, u) is

(31)

One is unable to find the general solution to (31). Hence, by an ansatz of the form, η(t, x, u)=f(t, x)+g(t, x)u +h(t, x)u², we further investigate the non-classical symmetry of Eq. (2) for the previous four cases (12) of the flow parameter r(t).

Subcase 4.1 r(t)=δ

By solving (31), one obtains the non-classical symmetries (5) with τ ≡ 0, ξ ≡ 1 and

where c_i (i=1, 2, and 3) are arbitrary constants, and p₀(x) is any nontrivial particular solution (p₀(x) ≠ 0) to the equation, i.e., νp"(x)+δxp′(x)+2δp(x)=0. The corresponding non-classical symmetries are

(32)

Subcase 4.2 r(t)=-tanh t

By solving (31), one obtains the infinitesimal η(t, x, u) as follows:

where c_i (i=1, 2, and 3) are arbitrary constants. The corresponding non-classical symmetry is

(33)

Subcase 4.3

By solving (31), one obtains the non-classical symmetries (5) with τ ≡ 0, ξ ≡ 1, and

where c_i (i=1, 2, and 3) are arbitrary constants. The corresponding non-classical symmetry is

(34)

Subcase 4.4 r(t)=tan t

By solving (31), one obtains the infinitesimal η(t, x, u) as follows:

where c_i (i=1, 2, and 3) are arbitrary constants. The corresponding non-classical symmetry is

(35)

Obviously, the above examples show that the non-classical symmetry effectively extends and enriches the classical symmetry, which will give many new ways to solve the given PDEs.

4 Some exact solutions to (2)

Now we give some exact solutions to (2) by using some of the obtained symmetries.

Example 1 Use the classical symmetry with the infinitesimal X₁ in the case r(t)=δ and d=-1 < 0 in (17). For the generator X₁, one has the following similarity variables:

Therefore, the group-invariant solution to (2) has the form,

(36)

Substituting (36) into (2), one reduces (2) to the following ODE, m′(t)=0. Hence, the invariant solution to (2) under the generator X₁ is

(37)

This solution is the same as the result for the steady flow problem (i.e., =0) given by Shivamoggi^[7]. The steady-state solution is the shear-layer flow solution.

Example 2 (2) admits the classical symmetry with X₄ of the subcase r(t)=δ and d=-1 < 0 in (17). For the generator X₄, one has the following similarity variables:

Therefore, the group-invariant solution to (2) is

(38)

Substituting (28) into (2), one reduces (2) to the following ODE, m"(ζ)=0 with ζ=xe^-t, which leads to invariant solutions to (2) under the generator X₄,

(39)

This solution is obtained by Shivamoggi^[7] for the unsteady case.

Example 3 (2) admits the classical symmetry with X₅ of the subcase r(t)=tan t and d=1 > 0 in (20). For the generator X5, one has the following similarity variables:

Therefore, the corresponding group-invariant solution to (2) is

(40)

Substituting (40) into (2), one reduces (2) to the following ODE:

(41)

Solving (41), one obtains

(42)

Here, c₁ and c₂ are arbitrary constants, and . Therefore, one gets invariant solutions to (2) under the generator X₅,

(43)

This is a new solution to the unsteady case. The exact solution (43) with in the denominator develops some singularities for . Therefore, the solutions are defined for . When , the solutions represent a normal phenomenon of vortex. When , the solutions become complex. The complex solutions represent the vortex pair with different rotation directions (k=0, 1, 2, …). The solutions are different from those of Ref. [7] which exist for all time. Therefore, the exact solution (43) is a new solution to the modified 2D Burgers vortex equation for the unsteady case.

Example 4 (2) admits the non-classical symmetry with Y₈ of the subcase r(t)=δ in (27). For the generator Y₈, one has the following similarity variables:

Therefore, the corresponding group-invariant solution to (2) is

(44)

Substituting (44) into (2), one reduces (2) to the following ODE:

(45)

Solving (45), one obtains

Here, c₁ and c₂ are arbitrary constants. Therefore, one gets invariant solutions to (2) under the generator Y₈,

(46)

This solution is a new result for the unsteady case. It will grow without bound as t becomes large.

5 Conclusions

The new solutions given in this paper display that the vortex phenomenon is obvious for little time, and the vortex will break down for much time. This corresponds with an actual phenomenon. The vortex generates for little time and disappears slowly as t becomes large. Some of the obtained similarity solutions will result in solutions which exist only for little time. The new solutions given in our paper are only partial solutions to the modified 2D Burgers vortex equation. These solutions are constructed by selecting several infinitesimal generators of the obtained classical symmetries and non-classical symmetries. We can obtain more new solutions by using other infinitesimal generators of the present paper.

With the symmetry method, we give the complete classical and non-classical symmetries of the modified 2D Burgers vortex equation (2) for arbitrary forms of the flow parameter r (t). The obtained results show that the equation admits richer classical and non-classical symmetries which are useful for solving the equation. By the symmetries, exact solutions or reductions of the equation are obtained.

In particular, as specific examples drawing close attention of researchers in the literature, we consider several different cases of the flow parameter r(t) and obtain corresponding infinitesimal generators and some exact solutions. We recover the existing exact solutions to (2) obtained by different ways in the literature and produce new exact solutions to the equation. Moreover, comparing the system (26) with systems (9) and (10), we see that the non-classical symmetry admitted by (2) is richer. For each case of the parameters r(t)=δ, tanh t, -1/t, and tan t, both the classical and non-classical symmetries are admitted by the equation which shows that the non-classical symmetry nontrivially extends the classical one of the equation.

These show that the symmetry method is general and powerful to solve PDEs exactly. In the coming paper, as extension of the present article, we will further study the conservation laws of (2) corresponding to the obtained symmetries.