Journal of Applied Mechanics and Technical Physics, Vol. 54, No. 2, pp. 207–211, 2013. c R.I. Mullyadzhanov, N.I. Yavorskii. Original Russian Text
SOLUTION OF THE PROBLEM OF FLOW OF A NON-AXISYMMETRIC SWIRLING SUBMERGED JETS R. I. Mullyadzhanov and N. I. Yavorskii
UDC 532.5
Abstract: This paper considers the problem of a non-axisymmetric swirling jet of an incompressible viscous fluid flowing in a space flooded with the same fluid. The far field of the jet is studied under the assumption that the angular momentum vector corresponding to the swirling of the jet is not collinear to the momentum vector of the jet. It is shown that the main terms of the asymptotic expansion of the full solution for the velocity field are determined by the exact integrals of conservation of momentum, mass, and angular momentum. An analytical solution of the problem describing the axisymmetric swirling jet is obtained. Keywords: Navier–Stokes equations, submerged swirling jet. DOI: 10.1134/S0021894413020041
We consider steady flow of an incompressible viscous fluid caused by the presence of a source of motion that has limited geometric dimensions. As is known, away from the source, this flow is self-similar and is described by the accurate solution of the Navier–Stokes equations corresponding to a submerged jet whose flow is due to the presence of a point source of momentum [1]. This solution corresponds to the class of conical flows in the case where the velocity is inversely proportional to the distance from the source of motion and is uniquely determined by the intensity of the source of momentum (the vector of the force applied to a particular point in space). Since, in this case, one vector is specified, the solution is symmetric. The exact solution of this problem was obtained in [1–3] (Slezkin–Landau–Squire solution). In general, the source of motion is not only the momentum but also the source of mass and angular momentum; the vector of the angular momentum is not necessarily collinear to the momentum vector. This case is studied in this paper. The stationary Navier–Stokes equations can be written as ∂u ∂Πij ∂uj i ; (1) = 0, Πij = ρui uj + pδij − ν + ∂xj ∂xj ∂xi ∂uj = 0, ∂xj
(2)
where Πij are the components of the total momentum flux tensor, ui are the velocity components, p is the pressure, ρ and ν are the constant density and viscosity of the fluid, and δij is the Kronecker symbol. Equations (1) and (2) satisfy the laws of conservation of momentum and mass, respectively. The law of conservation of the angular momentum can be written as ∂Kij = 0, Kij = εikm xk Πmj , (3) ∂xj
Kutateladze Institute of Thermal Physics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia. Novosibirsk State University, Novosibirsk, 630090 Russia;
[email protected];
[email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 54, No. 2, pp. 46–51, March–April, 2013. Original article submitted September 24, 2012. c 2013 by Pleiades Publishing, Ltd. 0021-8944/13/5402-0207
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where Kij are the components of the angular momentum flux tensor and εikm are the components of the Levi-Civita tensor. The differential equations (1)–(3) correspond to the integral conservation laws Πij nj dS = Pi , uj nj dS = Q, Kij nj dS = Li , (4) S
S
S
where P is the intensity of the source of momentum (the force applied to the fluid), Q is the fluid flow rate, L is the angular momentum vector, S is the arbitrary surface covering the coordinate origin from which the jet issues, and n is the unit outward normal vector to the surface. In general, the vectors P and L are arbitrarily directed. Suppose that the surface of integration is a sphere whose center coincides with the origin. Relations (4) and the expressions for the flux densities of momentum (1), mass (2), and angular momentum (3) imply that at infinity the momentum conservation law is satisfied if the velocity is inversely proportional to the radius of the sphere r, and the laws of conservation of mass and momentum are satisfied if the velocity is proportional to r−2 . The solution in which the velocity is proportional to r−1 is the above Slezkin–Landau–Squire solution. This solution corresponds to zero flow rate and zero angular momentum. To describe flow with nonzero flow rate and nonzero angular momentum, it is necessary to consider the next terms of the asymptotic expansion of the far field parameters of the submerged jet proportional to r−2 . However, the term proportional to r−2 is determined by two independent integrals of conservation, which can lead to a contradiction. The solution corresponding to an axisymmetric swirling submerged jet with zero flow rate was found in [4]. An axisymmetric solution for a submerged jet with nonzero flow but zero angular momentum is presented in [5], but this solution contains a logarithmic singularity of the velocity field on the axis of symmetry, which is a consequence of the above contradiction. This contradiction was resolved in [6], where it was found that the radial velocity field should also contain a term proportional to r−2 ln r. The appearance of this term is due to the existence of the nontrivial solution of the problem for the term r−2 corresponding to zero flow rate, which was found in [5]. Similar results were presented in [7]. A solution for axisymmetric submerged jets for which the vector L is collinear to the vector P was obtained in [8]. This paper presents the results of solution of the problem for the general case of non-collinear vectors P and L. We consider the flow caused by the presence of a source of motion and characterized by a source of momentum P directed along the z axis, a nonzero source of mass with intensity Q, and a source of angular momentum L which has three nonzero components Lx , Ly , and Lz in a Cartesian coordinate system. According to the results of [6], we seek the velocity and pressure in a spherical coordinate system (r, θ, ϕ) (the angle θ is measured from the z axis) by representing them in the form of expressions which take into account the nonaxisymmetric terms containing the azimuthal angle ϕ: r 2 r 2 r 2 ν r0 r 0 0 0 y (t) + v (t) ur = − ln + w (t) + f (t) cos (ϕ − ϕ0 ) + . . . ; (5) r0 r r r0 r r uθ = −
r 2 r 2 r 2 ν r0 0 0 0 √ + v(t) + w(t) + F (t) cos (ϕ − ϕ0 ) + . . . , y(t) r r r r r0 1 − t2 uϕ =
p=
r 2 r 2 ν 0 0 √ γ(t) + F (t) sin (ϕ − ϕ0 ) + . . . , r r r0 1 − t2
r 2 r 3 r 3 r 3 ρν 2 r 0 0 0 0 g(t) + h(t) ln + q(t) + s(t) cos (ϕ − ϕ0 ) + . . . , 2 r0 r r r0 r r
(6)
t = cos θ.
Here the functions y(t) and g(t) correspond to the Slezkin–Landau–Squire solution: y(t) = 2
1 − t2 , A−t
g(t) = 4
At − 1 . (A − t)2
(7)
The expression for v(t) is obtained in [5], and that for γ(t) in [4]: v(t) = b0 v0 (t), 208
v0 (t) = (1 − t2 )
1 − At , A(A − t)2
γ(t) = d0
1 − t2 . (A − t)2
(8)
The parameter A > 1 is uniquely related to the intensity of the source of the momentum Pz [1], the constant b0 to the flow rate Q, and d0 to the axial intensity component of the source of angular momentum Lz [6]: A A + 1 4 − ln ; (9) Pz = 16πρν 2 A 1 + 3(A2 − 1) 2 A−1 Q=
2πr0 ν 1 A + 1 2 2 2 7 − 3A b + 4A(A , − a A + − 1) ln 0 3(A2 − 1) 3 A−1
a=−
(5A2 − 3)[4 − 6A2 + 3A(A2 − 1) ln ((A + 1)/(A − 1))] ; A[10A − 6A3 + 3(A2 − 1)2 ln ((A + 1)/(A − 1))] Lz = 4πr0 ρν 2 d0
A + 1 4 − 2 + A ln . 3(A2 − 1) A−1
(10)
(11)
The quantity r0 is the characteristic size of the source of motion (for example, the radius of the hole from which the jet issues). We will assume that r0 r, which corresponds to the far field of the jet in which expansions (5) and (6) are valid. The function w(t) is defined by the equation ((1 − t2 )(A − t)2 w (t)) + 6(A2 − 1)w (t) = (A − t)2 f0 (t),
(12)
where f0 (t) is a function which appears due to the presence of the nonlinear terms of the Navier–Stokes equations consisting of the terms of the asymptotic solution for the velocity field proportional to r−1 and r−2 ln r: 3A2 − 1 A2 − 1 A2 − 1 (A2 − 1)2 . (13) +3 f0 (t) = b0 a − 2 − 8A + 6 A(A − t) (A − t)2 (A − t)3 (A − t)4 The solution for w(t) can be found in the form of a quadrature [6] indicated below by w0 (t), to which we add the nontrivial solution of the Rumer homogeneous equation containing an arbitrary constant c0 : w(t) = w0 (t) + c0 v0 (t).
(14)
The above solution (5)–(13) provides the presence of all three nonzero components of the angular momentum. In the case of axisymmetric fluid flow, the angular momentum vector has only one component Lz , which defines the constant d0 . The solution v0 (t) corresponds to flow with nonzero flow rate, so that this term of the expansion satisfies the law of conservation of mass. In the axisymmetric case, this solution has no effect on the law of conservation of angular momentum. Thus, the constant c0 in (14) is not defined by the conservation integrals listed above. According to [6, 8], the problem should also contain other expansion terms of higher order in r−1 inversely proportional to the radius of the sphere raised to non-integer powers that are functions of the Reynolds number determined from the momentum flux of the jet: Re = Pz /(ρν 2 ). In this case, if the corresponding term of the expansion is a definite conservation integral, the exponent is independent of the Reynolds number. Such terms of the expansion are the terms proportional to r−1 and r−2 . The constant c0 depends on the hidden conservation integral which occurs due to the additional symmetry of the problem since in the axisymmetric case, Lx = Ly = 0. Let us consider in more detail the nonaxisymmetric flow of the submerged jet. It should be noted that the principal term of the asymptotic expansion of the velocity field at infinity is proportional to r−1 , is defined by the law of conservation of momentum, is given by the value and direction of one vector P , and is therefore axisymmetric. The difference between the solution considered and the solution in the case of axial symmetry can manifest itself only in the next terms of the expansion [see (6) and (7)]. In these expansions, only the term with the cosine of the azimuth angle is added. This is sufficient to describe the flow because it is completely defined by two vectors P and L and the scalar Q. This assumption is confirmed by the fact that in the case of higher harmonics in ϕ for the terms proportional to r−2 , there is only the trivial solution. Higher harmonics occur in the following orders of the expansion of the general solution for r−1 . The constant angle ϕ0 characterizes the orientation of the vectors P and L. We obtain equations that define the nonaxisymmetric terms proportional to r−2 in the asymptotic expansion. These terms are described by the dimensionless function f (t), F (t), and s(t). We substitute the expressions for the velocity field and the pressure (6), (7) into the Navier–Stokes stationary equations (1). In the leading order in the inverse powers of the radius of the sphere, we obtain the following system of linear ordinary differential equations: 209
(1 − t2 )f − 2tf − yf − 3y f −
f + 2yF − y F + 3s = 0, 1 − t2
(1 − t2 )(F + s − 2f ) − 2tF + 2F − yF − 2y F −
F + 2tyF = 0, 1 − t2
(1 − t2 )2 F − (1 − t2 )(4tF + y F + yF ) + 2tyF − F − This system of equations has an analytical solution 1 A2 − 1 2(A2 − 1) , f (t) = λ0 1 − t2 + − A − t (A − t)2 (A − t)3 s(t) = 2λ0
1 − t2
2tF − 2f + s = 0. 1 − t2
√ 1 − t2 F (t) = λ0 , A(A − t)
1 A 2(A2 − 1) , + − A − t (A − t)2 (A − t)3
(15)
where λ0 is a constant that characterizes the non-axisymmetric nature of the jet. We find all components of the angular momentum flux vector L in Cartesian coordinates. The component Lz is given by expression (10), and Lx and Ly are given by the expressions Lx = −r
3
2π 1
Πrθ sin ϕ + tΠrϕ cos ϕ dt dϕ,
0 −1
2π 1 Ly = r Πrθ cos ϕ − tΠrϕ sin ϕ dt dϕ. 3
0 −1
Substituting the values of the momentum flux tensor components in spherical coordinates into these expressions and using (4), (6), (7), and (15), we obtain Lx = −Λ sin ϕ0 , where
Λ = 8πρν 2 λ0 1 +
Ly = Λ cos ϕ0 ,
(16)
4 A A + 1 − ln . − 1) 2 A−1
3(A2
We introduce the unit vector τ = (− sin ϕ0 , cos ϕ0 , 0) in a plane orthogonal to the vector P . In this case, the expression for the total angular momentum flux is given by L = Λτ + Lz ez . By virtue of (10) and (16), the parameters ϕ0 , λ0 , and d0 fully determine the magnitude and direction of the vector L. We note that the constant c0 , which depends on the hidden conservation integral is absent in the expression of the vector L. Consequently, the hidden conservation law is not related to the law of conservation of angular momentum, or to the law of conservation of mass, which contradicts the statement given in [6]. Thus, we obtained a solution of the problem of flow of a non-self-similar submerged jet in a general formulation where the total momentum flux and the total angular momentum flux vectors are given arbitrarily. The solution is a correct asymptotic of the exact solutions of the Navier–Stokes equation, which is valid in a neighborhood of infinity. The solution is obtained in explicit analytical form and the parameters of the solution are determined by the values of the exact integrals of conservation of momentum, mass, and angular momentum. It is shown that the principal term of the asymptotic expansion describes the flow of an axisymmetric submerged jet. The difference from the case of axial symmetry are manifested in the next terms of the asymptotic expansion, proportional to r−2 , which are determined by the magnitude and direction of the total angular momentum flux. Given the results of [8], we can assume that the solution of the problem also contains other higher-order expansion terms proportional to r−α(Re) ; in this case, the exponents for certain values of the Reynolds numbers can be complex-valued. This work was supported by the Federal Target Program Scientific and Scientific-Pedagogical Personnel of Innovative Russia for 2009–2013 (State contract No. 8609). 210
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