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0Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 3, pp. 423-439. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220307
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 76D05
Approximate Weak Solutions to the Vorticity Evolution Equation for a Viscous Incompressible Fluid in the Class of Vortex Filaments
O. S. Kotsur, G. A. Shcheglov, I. K. Marchevsky
This paper is concerned with the equation for the evolution of vorticity in a viscous incompressible fluid, for which approximate weak solutions are sought in the class of vortex filaments. In accordance with the Helmholtz theorem, a system of vortex filaments that is transferred by the flow of an ideal barotropic fluid is an exact solution to the Euler equation. At the same time, for viscous incompressible flows described by the system of Navier-Stokes equations, the search for such generalized solutions in the finite time interval is generally difficult. In this paper, we propose a method for transforming the diffusion term in the vorticity evolution equation that makes it possible to construct its approximate solution in the class of vortex filaments under the assumption that there is no helicity of vorticity. Such an approach is useful in constructing vortex methods of computational hydrodynamics to model viscous incompressible flows.
Keywords: weak solution, vortex filament, helicity of vorticity, diffusion velocity, viscosity
Introduction
In nature and engineering there exist a number of phenomena and problems concerning the motion of vortex structures such as vortex tubes and vortex rings in infinite space [1]. Such phenomena can be both of natural and anthropogenic nature [2].
Received June 20, 2022 Accepted August 05, 2022
This work was supported by the Russian Ministry of Science and Higher Education, project 0705-20200047.
Oleg S. Kotsur
oskotsur@gmail.com
Georgy A. Shcheglov
shcheglov_ga@bmstu. ru
Ilia K. Marchevsky
iliamarchevsky@bmstu.ru
Bauman Moscow State Technical University
ul. 2-ya Baumanskaya 5, str. 1, Moscow, 105005 Russia
In some approximation it is admissible to consider the evolution of vortex structures separately, in the absence of any bodies in their neighborhood, which allows one to simplify the mathematical model of fluid flow by considering the flow in infinite space.
The class of vortex methods of computational hydrodynamics in which vorticity is the primary estimated value holds much promise for the modeling of incompressible vortex flows. In particular, the most convenient method for modeling structures such as vortex tubes is that of vortex filaments, which belongs to the class of Lagrangian vortex methods. The main advantage of such methods is the absence of a mesh and the relatively low computational complexity while exactly satisfying the condition of nondivergence (solenoidity) of vorticity.
The method of vortex filaments, which was proposed as early as the 1970s and developed by researchers such as A. Leonard, G. Winckelmans, was mainly applied to modeling nonviscous flows without boundaries. The absence of the mechanism of filament generation on the surface of bodies, the difficulties involved in the reconnection of filaments, and the absence of an adequate and feasible model of viscosity did not lead, at that time, to widespread use of the method of vortex filaments in the practice of engineering calculations.
In [3], special attention is given to engineering applications of the method of vortex filaments to the modeling of the flow around bodies, but the problems involved in taking into account the influence of viscosity remain open. The absence of a mathematical model incorporating the effects of the viscous diffusion of vorticity does not make it possible to use the method of vortex filaments for the modeling of vortex structures such as tubes in the space without boundaries either.
Thus, the development of mathematical models of viscosity for the method of vortex filaments is an important direction of research.
The purpose of this paper is to develop a method for constructing weak solutions to the equation for the evolution of vorticity in a viscous incompressible fluid in the space without boundaries in a special class of generalized functions — vortex filaments possessing the property of solenoidity.
1. Defining relations
In the hydrodynamics of an incompressible fluid, by the vorticity evolution equation one means an equation of the form
where V is the velocity field of the flow, u = VxV is the vorticity field, and v is the coefficient of kinematic viscosity1. This equation is a consequence of the Navier-Stokes equation and is obtained from it by applying the curl's operation to the right-hand and left-hand sides.
In the simplest case of flow in infinite space the relation between V and u is expressed by the integral Biot-Savart formula [4]:
1 Here and below, symbols V^, V • a and Vxb denote, the gradient of field the divergence of field a and the curl of field b, respectively.
—- + V ■ Vu> = u> ■ VV + uAu),
dt
(1.1)
(1.2)
where V^ is the velocity of the free flow at infinity which can, without loss of generality, be taken to be zero.
The class of methods of computational hydrodynamics, called vortex methods, is based on the solution to Eq. (1.1). The essence of these methods is to represent a continuous vorticity field using the system of so-called vortex elements (particles, segments, filaments [5]) which are the simplest weak solutions to the equation for the evolution of vorticity in a nonviscous fluid. For example, within the framework of the classical method of vortex particles (vortons) [6], which can be applied to model the flows of an ideal incompressible fluid, one considers the following representation of the vorticity field:
n
u(x,t) ^Y^ ^k (x,t), (1.3)
k=l
(x,t) = ak(t)S(x - Xk), (1.4)
where the functions uV (index V from Engl. "vorton") are elementary particle vorticity fields having intensity ak.
For the vorticity field of a system of particles to be a solution to the vorticity evolution equation in a generalized sense (the more exact meaning of this term will be explained below), it is necessary and sufficient that the following conditions concerning the functions of the position xk and the intensity ak of vortons [4] be satisfied for each particle:
dXk=V(xk, t),
dt da
k = 1, n. (1.5)
dt
Note that the velocity field appearing in these equations is also a function of the required parameters xk and ak, which can be written in explicit form by substituting (1.3) into (1.2) [7].
Conditions (1.5) form a system of ordinary differential equations (ODEs) in the parameters of vortex particles and are in this sense a discrete analog of the nonviscous form of the vorticity evolution equation. It can be seen from these conditions that, within the framework of the model of a nonviscous fluid, the particles are transferred through the velocity field, and their intensity changes in accordance with the local gradient of velocity. The vortex methods based on such an approach are called Lagrangian vortex methods. Reviews of the history and the state-of-the-art of vortex methods can be found in [8-10].
In the case of a viscous fluid, it is difficult to construct "exact" generalized solutions. Elementary functions of the form (1.4) cannot be weak solutions to (1.1) in the general case of spatial viscous flows. In the case of plane-parallel incompressible flows for which the vorticity evolution equation (1.1) has the form2
dw ^ . - + V-(Vuj) = VAUJ,
in the region where the vorticity is not zero, the diffusion term vAw can be represented in the same form as the convective term; it is determined by the vector field
v Vw
Vd = -!
d w
called the diffusion velocity.
2Here w defines the only nonzero vorticity component directed along the normal to the plane of the
Using this field, the vorticity evolution equation can be represented in the form of the transfer equation
^ + V-((V + Vd)uj)= 0, (1.6)
which can be obtained from the equation for a nonviscous fluid by replacing V with (V + Vd) in the convective term. Therefore, the weak solutions to the vorticity evolution equation for a viscous fluid in the planar case can be regarded as known if Vd is expressed in some way in terms of the parameters of vortex particles. In the theory of vortex methods, such an approach to taking viscosity into account was called the diffusion velocity method, DVM, and it was used as a basis to construct a number of methods such as the method of viscous vortex domains [11] and software packages for modeling planar problems of flow around moving bodies in the flow [12].
When DVM is extended to the general case of spatial flows, there are two main problems.
1. In the general case there exists no simple, practical method for obtaining a spatial analog of the diffusion velocity using which the diffusion term vAu could be represented in the same form as the convective term, as in (1.6).
2. Elementary functions of vortex particles (1.4) have the disadvantage that these functions in themselves do not possess the solenoidity property, which is inherent in the vorticity field. This leads to the necessity of using a large number of particles in modeling real-world problems.
From the viewpoint of the solenoidity of elementary vorticity fields of vortex elements, it seems more correct to discretize the vorticity not in the form of vortex particles, but in the form of a system of thin vortex tubes possessing the solenoidity property from the very beginning [6, 13]. This property turns out to be important in the practice of applying vortex methods to calculate the stresses acting on the streamlined structure. As the experience of calculations shows, the use of nonsolenoidal vortex elements in the neighborhood of the body leads to significant errors of the stresses to be determined.
From the mathematical point of view, thin vortex tubes can be represented in the form of closed curves (loops, filaments) lk, to which the intensity (circulation) of the tubes gk is related. The vorticity field of one such tube can be described by a generalized function of the form (index F from Engl. "filament")3
(xt) = jgkeks(x - y) dly, (1.7)
lk
whose mathematical meaning will be explained below. In what follows, it will also be shown that the system of vortex filaments forms a weak solution to the vorticity evolution equation for a nonviscous fluid under conditions similar to (1.5).
For viscous flows of general form, the search for approximate solutions to the vorticity evolution equation based on generalized functions of vortex filaments is of scientific and practical interest when it comes to applying vortex methods to model flows in the neighborhood of streamlined bodies and to calculate the hydrodynamical stresses acting on bodies which can arbitrarily move in the flow.
The above-mentioned conception of diffusion velocity cannot be applied directly for three-dimensional flows of general form, but allows a reasonable assumption such that an a posteriori
3Here and below, the subscript of the differential denotes the variable with respect to which the integration is performed.
estimate can be made to verify its adequacy and the system of filaments is, in the sense defined below, a weak solution to Eq. (1.1).
2. Weak formulation of the vorticity evolution equation
Consider the nonviscous variant of equation (1.1) in the spatial case
o
+ V • Vw = u; • VV. (2.1)
We will assume that all functions are given in the time interval (0, T) in the entire space R3 without any boundaries.
To describe various types of singular vortex elements applied in vortex methods, it is convenient to make use of the well-known mathematical tools of the theory of generalized functions. The role of the generalized (weak) formulation of linear differential equations consists in the possibility of searching for such a system of its generalized (weak) solutions, which under certain conditions converge to classical continuously differentiable solutions to the initial equation. The vorticity evolution equation is a nonlinear equation, it contains an operation of multiplying the derived fields V and u that has no definition in terms of generalized functions; therefore, in the strict sense, the generalized solutions to this equation are not defined. However, assuming the velocity field V to be a classical infinitely differentiable function, the vorticity evolution equation becomes linear in u, which makes it possible to search for a vorticity field satisfying it in the class of generalized functions, namely, in the class of vortex filaments (1.7). In this case, the dependence V(u) in the form of the integral relation (1.2) allows one to express V in terms of the parameters of vortex filaments in a natural way.
Set Q = R3 x (0, T). Introduce the space D(Q) of infinitely differentiable (on Q) functions 0(x, t) G (Q) with a compact carrier. The convergence in this space is defined in accordance with the monograph [14]. We will call the functions 0(x) test functions.
By the generalized function on the space of primary functions D(Q) we will mean any linear continuous functional T, the set of which we denote by D'(Q). The action of the functional T on the primary function 0 G D will be denoted by (T, 0).
For further discussion, we give several definitions, following [14].
The partial derivative of the generalized function T G D'(Q) with respect to the variable s,
Sj G x2, x3, t} is a generalized function ^ such that.
dT A /t W
This definition can be extended to the derivative of any order. In particular, the derivative of T with respect to the multiindex a G N4 is defined as
(daT, 0) = (—1)'"'(T, da0).
Multiplication of the infinitely differentiable function a G C^(Q) by the generalized function T is defined by the rule
(aT, 0) = (T,
In accordance with these definitions, to any function f (x, t) locally integrable in Q one can associate its generalized analog Tf using the following rule:
T
(Tf, 0) = j j f (x, t)0(x, t) dvdt, (2.2)
0 R3
from which it is seen that the class of generalized functions is wider than the class of classical functions.
Equation (2.1), which is formulated in the sense of generalized functions with the known field V, will be called the weak formulation of the vorticity evolution equation for a nonviscous flow, and the generalized solutions themselves, which turn it into a valid equality on any primary function 0 £ D(Q), will be called its weak solutions. By direct substitution into (2.1) one can verify that all classical solutions in the form of differentiable functions w(x, t) are also its weak solutions constructed by the rule (2.2).
3. Discrete analog of the vorticity evolution equation for an ideal fluid
In the classical method of point vortons, the mathematical tools are to a great extent based on the three-dimensional Dirac delta-functions S(x) (see (1.4)), which have there the meaning of vortex particles in which vorticity is concentrated. In our work, we consider the vorticity field from the viewpoint of vortex tubes for which the most convenient tool is the integral of the ¿-function along some smooth closed curve l to which the vortex tube is connected:
J S(x - y) dly.
i
The action of this generalized function on the primary function 0 £ D(Q) is performed by the rule
/ \ t
J 5(x - y) dly , 0j =J J 0(x, t) dlx dt.
'l '0 l The linearity and continuity of this functional is obvious.
Suppose we are given a system of vortex tubes defined by a set of smooth closed curves lk and intensities gk, k = 1, m. The curves are parameterized using the variable s £ [s0, s1] and, since the tubes are capable of moving in space, the curves are also functions of time: lk = lk(s, t), and lk(s0, t) = lk(s1, t), Vt £ [0, T] by virtue of closure.
Let us introduce a system of generalized functions of vortex filaments:
Wfe (», t) = J gk{y, t)6(x — y) dly, k = 1, m, (3.1)
h®
where gk(y, t)\y=l , t) = gkek(s, t) is the vector of intensity of the filament k and ek(s, t) is the unit vector of the tangent to lk(s, t). The superposition of the filaments
n
(x,t) = J2 "k (x,t) (3-2)
k=1
defines some generalized vorticity.
Let us define the conditions under which the system of vortex filaments (3.2) is a generalized solution to the vorticity evolution equation for an ideal fluid. Substitution of the "filament" representation of the vorticity field uF (x, t) into the weak formulation of the vorticity evolution equation makes it possible to obtain necessary conditions under which uF (x, t) will be a weak solution to Eq. (2.1). Let us substitute uF into each term (2.1) and consider its application to some primary function ф G D(Q).
д F
Nonstationary term :
д и
~dt
F
— { и
F
дф ~dt
T
"Ё / / 9k(V, t)dl
k=ln I
0
T
k=l
I / 7fc(M)T^(ifc(M),
dt
0 so
dl
where jk = gk-gj- and the derivative of the primary function is taken only with respect to the second (underlined) argument.
Introduce a material derivative jh which defines the rate of change of the function 0 along the trajectory of the point on the vortex line lk. It is defined as a partial derivative of the function 0 (lk(s, t), t) with respect to the variable t, which appears here both in the first argument 0 and in the second argument:
D<l>(lk(s, t), t) дф(1к(з, t), i_) + dlk(s, t)
Dt
dt
dt
Уф (lk(s, t), t),
where V0 [lk(s> ¿J denotes the gradient of the function 0(x, t) with respect to the first argument, taken at the point lk(s, t) at time t. Then the nonstationary term can be written as
/duF \ dt '
T si
£
k=i
. ,Dф (lk (s, t), t) . Blk (s, t) . .
7k(s, t) - 7k(s, t) kKJ ■ V0 lk(s, t), t
Dt
0 so
T si
E
k=i
dYk(s, t)^ f„ .Л .л , f„ ndlk(s, t)
dt
-ф(1к(з, t), t)+jk(s, t)-
dt
dt
•Уф lk(s,t),t
dsdt. (3.3)
0 So
Here, in the first term of the integrand, integration is performed by parts, taking into account the fact that the function 0 is finite on the set Q.
Convective term V ■ VuF. The convective term is applied to the primary function 0 as follows:
(V ■ VuF, 0) = - (uF, V ■ (V0)) = - (uF, V ■ V0). (3.4)
This follows from the fact that the same term is written in coordinate form (and from the fact that the fluid is incompressible, V^V = 0):
d_u4 dx■
F
F
d
-(coi, g^(V30)) = - К , V • (Уф)) = -(cjf,V. Уф), i G {1, 2, 3}.
m
Then, substituting into (3.4), we obtain
T «1
'V-Vu
F
IIV n n
t) = J Yk (s, t)(V ■ Vt) (lk (s, t), t) ds dt. (3.5)
k=10 s0
Deformation term uF
/ vdV
VV.
uF ■VV,
w
dx
w
F
3
dV
dx■
T s1
A / Yk (s, t) ■ (VV t) (lk (s, t), t) dsdt.
' k=10 s0
(3.6)
Substituting (3.3), (3.5) and (3.6) into (2.1) and regrouping the terms, we obtain the following expression for the vorticity evolution equation (the arguments s and t are omitted for brevity) :
E
k=i
T s1
0 so
dYk dt
- Yk ■VV (lk)
T s1
t (lk) dsdt + J J Yk
0 So
dh dt
- V (lk)
Vt (lk ) ds dt
0.
Since the primary function t S D(U) is chosen arbitrarily, this leads to necessary conditions for the superposition of vortex filaments to be a weak solution to the vorticity evolution equation:
dYk dt
k = 1, Til.
(3.7)
Yk ■VV (lk ),
di
Since 7k = i/fc-57, the left-hand side of the second of equations (3.7) can be rewritten as
dlk =
dt ds dt V ds
d9k dl
' +gk§-s(v(ik)) = %% ■ 7, • VVM/,).
dt dt ds dt \ds J dt ds ds dt ds
which, taking the same equation into account, leads to the equivalent system
d l
dt ˣk = 0
m '
V (lk ),
k = 1, Til.
(3.8)
The systems of ODEs (3.7) and (3.8) are a discrete analog of the vorticity evolution equation for an ideal incompressible fluid when the vorticity field is discretized using vortex filaments. We note that the intensities (circulations) of the filaments gk remain constant in the process of motion in a nonviscous fluid. This fact is in complete agreement with the well-known Helmholtz theorem, which states that, in an ideal barotropic fluid, vortex tubes move together with the material particles carrying them and retain their intensity [15].
In the case of a viscous fluid, the presence of the diffusion term vAu prevents one from obtaining in a similar way necessary conditions for the existence of weak solutions to the vorticity evolution equation, which does not contradict the Helmholtz theorem either.
4. Calculation of the velocity field
The problem that still remains open is that of calculating the velocity field V(x, t), which was assumed to be known in the calculations of the preceding section. In fact, the velocity field and the vorticity field are related to each other by the integral Biot-Savart law (1.2). And if the vorticity field is replaced with vortex tubes (filaments), it is necessary to find a velocity field that is "induced" by this system of tubes. Direct substitution of the elementary vorticity field of the filament k (3.1) into the Biot-Savart formula (1.2) leads to the integral expression of the velocity field of the filament in terms of its parameters lk and gk,
9k(У, t) x (x - y)
Vk
F
(x,t) = J
h №
4n|x — y\3
dy.
(4.1)
Analysis of the integral (4.1) shows [16] that infinitely thin vortex tubes have an infinite velocity of self-induction at the location where they have nonzero curvature. To overcome this singularity, one usually uses two approaches: the LIA-approximation of the filaments [17] or the method of smoothed filaments. In the former case, in the calculation of the velocity at some point of the filament, the influence of the part of the filament that is the nearest to this point is excluded. In the case of the method of smoothed filaments, when one substitutes the expression (3.2) into the integral relation (1.2), one uses, instead of the singular ¿-function, delta-shaped functions with some characteristic size e [6].
For an approximate calculation of the integral (1.2), one can use the method of vortex segments in accordance with which the filament lk is split into segments of length 2hk possessing the intensity Yk = gkhk (in this case, index k runs along all segments of all m filaments). The velocity field of the segment k has the following form (index S from Engl. "segment") [3]:
Vk(x> = ¥^(cosak + cos/ifc),
4nd2k
where dk = ek x sk, in Fig. 1.
= i^j, d,k = I dk I, and the vector sk and the angles ak and ßk are defined
Fig. 1. Calculation of the velocity field of a vortex segment
e
k
To limit the velocity when approaching the axis of the segment in some e-neighborhood, the velocity profile is taken to be linearly increasing from zero (on the axis of the segment) to some value (at distance e to the axis) according to the Rankine vortex principle [18], see Fig. 2. To calculate the velocity at the point x lying in the e-neighborhood of segment k, it is convenient to introduce an additional vector
, / N , dk X ek
x = xk + (sk ■ ek)ek + e—^-
\dkxeky
so that the resultant velocity field is defined as
Vks<€(x,t) = { w ^ ^ , Vk = l,n
VkS(x, t), dk > e,
~Vk(x, t), dk < e,
The superposition Vk,<: for each segment defines the velocity field corresponding to the vorticity field of all vortex segments:
n
VS't(x,t) = Y, VkS'e(x,t). (4.2)
k=l
It should be noted that the resulting approximate vector velocity field VS'e is not a differentiate function on the boundary of the e-neighborhood of the segments. Formally, this is a violation of the condition in the case of a weak formulation of the vorticity evolution equation according to which the velocity field V must be infinitely differentiate in the entire space. This inaccuracy can be eliminated by applying a smoothing function of higher order for the velocity field in the e-neighborhood of vortex segments, for example, using the Gauss function or smooth rational functions [6].
The method of segments can also be used in finding an approximate numerical solution to the system (3.8). In this case, the system of ODEs in the parameters of the filaments (3.8) can
be approximately replaced with a system of ODEs in the parameters of segments of the form [5]:
V S,i(xk),
hk-VVs^(xk), Vfc=I7n. (4.3)
Yk ■VVS'<(xk),
The gradient of the approximate velocity field VVS and its smoothened analog VVS'e can also be obtained analytically. Since their explicit form is cumbersome, it is not presented in this paper, but can be found in [5]. However, in reality, when the algorithm of numerical solution (4.3) is implemented, no explicit calculation of the velocity gradient is usually required because vortex segments that make up a vortex filament move in a constrained way. The algorithm consists in the parallel transfer of vortex segments through the velocity field by moving the markers xk, followed by combining the ends of adjacent segments. In this case, the deformation of material segments is performed immediately, instead of the cost-intensive calculation of the velocity gradient VVF'e.
5. On the existence of the velocity of transfer of vortex tubes in a viscous fluid
dx
k
dt dh
k
dt dlk dt
The existence of the diffusion velocity Vd in two-dimensional flows using which the vortic-ity evolution equation can be represented in the convective form (1.6) leads to two important questions concerning the possibility of searching for such a velocity in the case of spatial viscous flows [19]:
1. Can one choose for any viscous flow a vector field U (which may be regarded as the velocity field of some imaginary continuous medium) such that, when moving through it, the vortex tubes would remain vortex tubes and retain their intensity?
2. If one can, then do there exist practical methods of searching for such a field?
According to the Fridman theorem [20], such a velocity U (called in [21] the Fridman velocity) must satisfy the equation
d ^
— + U ■ Vu> — up ■ VU + u>V ■ U = 0 (5.1)
dt
or, which is the same,
d &
- + Vx(ux[/)=0.
For nonviscous incompressible flows, the Helmholtz equation holds, which can be obtained from (5.1) by replacing U with the velocity field of the flow V. Therefore, in the case of such flows the velocity field V itself is an example of the Fridman velocity.
In [22], a positive answer is given to the question of the existence of velocity U for an elementary vortex fragment which is a segment of the spatial vortex tube with nonzero vorticity. Moreover, in the same paper it is proved that there exist infinitely many Fridman velocities, and for a viscous incompressible fluid all of them are described by the following expression:
TT u x (Vxu) u x (-Vf + VW)
U = V-u--^ +--5.2
\u\2 \u\2
where W is some scalar field that is constant along vortex lines, 7 is an arbitrary scalar field, and f is some scalar function satisfying the condition
u- Vf = -vS, (5.3)
where
S = u • (Vx u)
is called the helicity of the vorticity field [19] by analogy with the concept of helicity of the velocity field [16], which is defined as the scalar product of the velocity by its rotor.
Thus, the question of the existence of the Fridman velocity U for the flow region is linked to the question of the existence of a potential f that would satisfy condition (5.3). The authors of [22] conclude that such a function f exists for any elementary vortex fragment and can be obtained by integrating Eq. (5.3) along vortex lines.
Also, it is clear that in the case S = 0, which corresponds, for example, to any plane-parallel or axisymmetric flow of the incompressible fluid without twisting, one can choose identical zeroes as f, W and 7 in (5.2). In this case, the Fridman velocity takes the form
u = V + Vd, (5.4)
where the velocity
u x (V x u) .
= (5.5)
was called the diffusion velocity [11, 23, 24] as an additive to the principal velocity of the flow which describes the transfer of vorticity related to the viscous diffusion.
In the more general case where S = 0, in the expression for the Fridman velocity it is necessary to take into account the correction term Vs:
U = V + Vd + Vs, (5.6)
where
n = .x(-v/ + vM-)+7u,. (57)
In this case, the diffusion velocity is defined by the sum Vd + Vs and is multivalued by virtue of the arbitrariness of the fields W and 7, which may be taken to be zero for definiteness.
The second question posed above concerning the possibility of practical search for the Frid-man velocity is closely related to the question of the existence of a local method of calculating this velocity for the flow under consideration. By locality one means a method such that the Fridman velocity at a given point is calculated in terms of the parameters of the flow in an arbitrarily small neighborhood of this point (i.e., in terms of the parameters of the flow and their space and time derivatives at this point) [22]. The existence of flows for which there exist no local methods for calculating the Fridman velocity greatly complicates the development of numerical algorithms based on the motion of vortex objects, as is implemented in the planar case in the method of viscous vortex domains.
It is well known that the Fridman velocity always has a local formula as a minimum for two types of three-dimensional vortex flows: stationary (in this case, one can always choose an identical zero as U) and nonstationary flows with zero helicity of vorticity S. For the latter the Fridman velocity is obtained from the formulae (5.4) and (5.5), which are obviously local.
As was shown in [19], in the class of incompressible flows with nonzero helicity S there exist both flows for which there exists a local method of calculating the velocity U and flows for which no such method exists.
In general, this result gives a negative answer to the question of the possibility of extending the classical method of the diffusion velocity to three-dimensional flows of general form.
6. Approximate weak solutions to the vorticity evolution equation under the assumption that the vorticity has zero helicity
Despite the complexity of practical construction of the transfer velocity of vortex tubes in spatial flows, the theoretical results of the preceding section provide a basis for constructing an approximate method for taking viscosity into account, based on the assumption that the vorticity has zero helicity.
Assuming S = 0, the component Vs in the expression (5.6) may be taken to be zero (by choosing W, y and f in (5.7) to be identically zero), and in this case the field
Ui = V + Vd
is the Fridman velocity and Vd is the diffusion velocity.
Consider the vorticity evolution equation (1.1) for a viscous incompressible fluid (VV = 0), written as
d u
-— + V ■ Vu> - u> ■ VV + u>V ■ V - vAu = 0. dt
Adding the field Vd (5.5) to the velocity field V and subtracting, we obtain the identity
^ + (V + Vd)-Vu>-u>-V(V + Vd)+u>V-(y + Vd)-B = 0, or, which is the same,
^+Vx(u>x(V + Vd))=B, (6.1) where B denotes the remainder term
B = vAu + Vd ■Vu - u ■ VVd + uV ■ Vd = vAu + Vx(u x Vd). (6.2) Substituting (5.5) into (6.2) yields the explicit form for B:
_ „ „-, , ^ , „ u x (u xVxu)' B = -uV x (V x u>) + V x (u; x Vd) = -uV x ( V x u> H------'
= -vV x(Vxu + ew x (eu xVx u)), (6.3)
where ew =-fa.
The second term in the brackets can be transformed by the rule of expansion of the triple vector product:
x (e„ xVxu)= eu, (e„ ■ (V x u)) - (Vx u)(e^ ■ eu,).
Since e^ ■ e^ = 1, substituting the last expression into (6.3) gives the final form B:
B = -vVx(e„ (ew ■ (Vx w))).
It follows from the formula for B that, in the case of vorticity having zero helicity (w ■ (Vx xw) = 0), the remainder term B is zeroed and the vorticity evolution equation (6.1) takes the form of the Fridman equation (5.1):
d w
— + Ul • Vu; = u> • VUl - uV • Uv (6.4)
We show that in this case the set of vortex filaments (3.2) is also a weak solution to the Fridman equation (6.4) when conditions similar to (3.8) are satisfied. We note that the vector field U1 describes the motion of some "medium" which is compressible in the general case. Therefore, the Fridman equation differs from the vorticity evolution equation describing the dynamics of a nonviscous incompressible fluid in that it has the term w(V ■ U1 ) responsible for compressibility.
Let us substitute the filament representation of the vorticity field (3.2) into each term of the Fridman equation for some primary function 0. The expressions obtained for the nonstationary term ^f- and for the first deformation term up ■ VU1 are similar to those in the nonviscous case and are obtained by replacing V with U1 in the expressions (3.3) and (3.6), respectively.
Consider the convective term U1 ■ VwF. Its application to the primary function 0 can be shown as follows:
<U1 ■ VwF, 0) = - (wF, V ■ (U10)> = - (wF, U1 ■ V0) - (wF, 0V ■ U1> =
= - <wF, Ul ■ V0) - <wF(V ■ Ul),0).
The underlined term is exactly equal to the second deformation term, applied to 0, on the right-hand side of Eq. (6.4) and, when it is substituted into (6.4), both terms cancel out. Therefore, the weak formulation of the Fridman equation (6.4) finally takes the form
Note that the resulting expression differs from the formulation for the nonviscous case only in that the last equation contains, instead of the velocity field of the flow V, the transfer velocity of vortex tubes U1, taken under the assumption that the vorticity has zero helicity. Therefore, the necessary set of conditions under which the vortex tubes will be a weak solution to the Fridman equation in the above-mentioned sense reduces to a system of ODEs similar to (3.7)
dh dt dlk dt
Also valid is the equivalent system
U1(Zfc ), Ik ■ VU1(lk),
k = 1, m.
Ëik = o
dt '
k = 1, m, (6.5)
in accordance with which the vortex tubes move in the vector field Ui while retaining their intensity.
To close the system of ODEs (6.5), it is also necessary to relate the transfer velocity of the vortex tubes U1 = V + Vd to the parameters lk and gk. The velocity field of the flow V is expressed in terms of these parameters in the same way as in the nonviscous case (4.2). The expression for the analog of the diffusion velocity Vd depends on the field u and its derivatives. In this case, the general approach is to apply an integral approximation of differential operators (for a description of the general idea of this approximation, see [25]. Derivation of the corresponding calculation formulae goes beyond the scope of this paper.
7. Limitations of the model of vortex filaments in modeling a viscous fluid
Among the limitations of the proposed method based on the assumption that the vorticity has no helicity, one should point out the following.
1. The model of closed vortex filaments does not describe the processes of reconnections of vortex tubes, which play an essential role in the dynamics of a viscous fluid.
2. The method does not provide for an a priori estimate of the contribution of the remainder term B (the helicity of vorticity) to the diffusion term vAu as compared to the part of it that is modeled using an analog of the diffusion velocity Vd.
The first limitation can be overcome by introducing the heuristic models of reconnection of filaments which are used in the practice of numerical calculations [3]. The second limitation requires an additional analysis in solving specific problems. A. V. Setukha [26] makes a case for modeling the viscous diffusion of vorticity by a simple formula of the diffusion velocity (5.5) near the body's surface, since in the neighborhood of the body the helicity of the vortex lines is negligibly small. This conclusion provides a justification for the use of the proposed approach in modeling a viscous incompressible flow using the method of vortex filaments at least near streamlined boundaries. In a more general case of spatial flows, for example, those for which the helicity of vorticity is nonzero, the question of the limits to the applicability of the assumption about zero helicity of vorticity is the subject of future research.
However, the remainder term B that is discarded can be calculated using the formulae of integral approximation [25] to obtain an a posteriori estimate of the correctness of the assumption about the absence of helicity in a specific flow. If its value is smaller than the part of the diffusion term that is solved using an analog of the diffusion velocity, then its omission in this calculation is justified. But if B is comparable or larger than the other terms (6.1), then this method requires additional mechanisms for taking into account the effects due to the helicity of vorticity. One of such examples is the above-mentioned phenomenon of reconnection of vortex filaments, which cannot be modeled in principle by taking into account only the diffusion transfer of vortex tubes by means of some field.
In [27], an example is given of the modeling of evolution of an elliptic vortex ring (Re = 834) in a viscous fluid by the method of vortex filaments using an analog of the diffusion velocity for taking into account the viscous diffusion of vorticity. A special feature of the problem is the three-dimensional character of the flow of this vortex structure in which a quasi-periodic change of semiaxes of the ellipse occurs in the process of motion. The flow is also characterized by regions of nonzero helicity of vorticity, which allows an estimate of the contribution of the effect of twisting of vortex lines on the dynamics of the motion of the elliptic ring. Calculations for this
model are in qualitative and quantitative agreement with the results of the physical experiment and calculation by the grid method of control volume. At the same time, the modeling of this problem in a nonviscous setting has yielded results that differ greatly from those of the experiment conducted in a viscous medium, which demonstrates the adequacy of the proposed approach at least at similar Reynolds numbers.
8. Conclusion
This paper presents a method for constructing weak solutions to the equation for the evolution of vorticity in a viscous incompressible fluid. The solutions are based on a special class of generalized functions — vortex filaments possessing the solenoidity property. Vortex filaments, which can be understood as infinitely thin vortex tubes, are weak solutions to the vorticity evolution equation for an ideal fluid, and move together with the liquid particles carrying them while retaining their intensity.
In the case of a viscous spatial flow, approximate weak solutions based on vortex filaments can be constructed under the assumption that the vorticity has no helicity. In this case there exists a vector field that is a transfer field of vortex tubes in a viscous fluid with their intensity remaining unchanged. This makes it possible to construct a discrete analog of the vorticity evolution equation in the form of a system of ODEs that ensures necessary conditions for the system of vortex fields to be a weak solution to the vorticity evolution equation.
These theoretical results are a basis for the computational method of vortex filaments and extend the scope of its application to the class of viscous flows.
Conflict of interest
The authors declare that they have no conflicts of interest.
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