Closed vortex lines in fluid and gas

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Introduction. The classical Helmholtz theorems on the motion of vortices in barotropic fluid were summarized in Zoravski’s criterion [1], which is also re- ferred to the Friedmann theorem [2]. This criterion deals with the possibility of considering the evolution of an arbitrary vector field (not only the vorticity field) as the movement of vector lines and vector tubes of this field with the fluid par- ticles. Later, other patterns of vortex movements were discovered. One group of laws is associated with the conservation of certain quantities (which depend on the flow parameters) along streamlines or vortex lines in the general 3D case in a non-barotropic gas [3-6]. These laws deal with “generalized” circulation or “gener- alized” velocity and vorticity fields (which makes it difficult to use them to verify numerical calculations). The regularities have a simpler form, especially for plane and axisymmetric flows with an additional assumption on the isoenergeticity of the vortex gas flow (which is valid, for example, in the flows behind the detached shock wave). These regularities, in the first place, include the result of Crocco [7]. It consists in the fact that, in the plane case, along the streamlines, the ratio of vorticity to pressure I 1 = /p is maintained, and in the axisymmetric case the invariant is I 2 = /(pr), where r is the distance from the axis of symmetry. Another group of regularities includes the results discussed in [8-11], where for- mulas are obtained for calculating vorticity at some points of the flow behind a detached shock wave through the free stream parameters and through parameters determined by the shape of the shock wave (slope, curvature). We can also men- tion studies dealing with the existence of vortex flows of incompressible fluid with certain properties [12-16] and the principles of maximum in vortex flows [17-21]. In this paper, we study flows with closed vortex tubes. In some cases, it is possible a priori (before making calculations) to state that vortex lines and tubes will be closed (for example, a flow behind a detached shock wave [22]) and checking the closure of vortex lines can serve to verify the calculations. In other cases, closed vortex tubes are detected as a result of a calculation (for example, [23-28]), and checking the regularities specific to vortex tubes would allow verification of the calculation. This article is devoted to the search for such regularities in the general 3D case for continuous flows of fluid and gas, hereinafter called as fluid flows. 1. Basic notation and Zorawski’s criterion. In what follows, we consider the fluid flows whose velocity field V = V(x, y, z, t) is described by the equation ( V 2 ) V + V + = F, t 2 (1) where V = |V|, = rot V is vorticity, F = F(x, y, z, t) is the ratio of the density of the resultant of all forces (applied to the fluid) to the density of the 408Closed vortex lines in fluid and gas fluid. Such flows include stationary and non-stationary flows of compressible and incompressible, viscous (various rheologies) and non-viscous fluids. Let the spatial domain G be located inside a fluid with a velocity field V = V(x, y, z, t) and inside this field is vortex for some open time interval ( = 0). In the domain G, we also consider the flow of an imaginary fluid whose particles move with a velocity q = q(x, y, z, t). The particles of an imaginary fluid do not interact with the particles of a real fluid and do not affect its movement. Suppose that, in the domain G, within the time interval (t 1 , t 2 ), the vorticity of a real fluid and the velocity of an imaginary fluid q are related by the equation + rot( q) = 0. t (2) In this case, Zoravski’s criterion [1] states that, in the interval (t 1 , t 2 ), the evolution of vorticity can be viewed as the movement of vortex lines (and vortex tubes with preservation of their intensity), together with the particles of an imaginary medium moving with a velocity q as long as these particles are inside G. This conclusion from Zoravski’s criterion will be used below. For brevity, the particles of an imaginary fluid will be termed q-particles. All the parameters of the flows discussed in this paper are considered suffi- ciently smooth. 2. The velocity q in a closed vortex tube. Consider the region of vortex motion of a fluid ( = 0) in which there is a fixed flat region (surface) such that during a nonzero time interval (t 1 , t 2 ) each vortex line intersects this surface at an acute angle to the normal, and only once. Denote by G a fragment of space, the points of which can be reached if we fix the time and move from the flat region along the vortex lines starting at (due to the closedness of the vortex lines you can move to either side of ). In other words, the fragment G is the union of all vortex lines passing through . Although the surface is fixed in space, but, due to the change in the shape of the vortex lines with time, the shape of the G fragment may change with time, i.e. G = G (t). The fragment G (t) consisting only of closed vortex lines at each moment of time is a closed vortex tube. We construct the function f (x, y, z, t) as follows. At first we define it at points of the surface . Let for each time t (t 1 , t 2 ) at the points of the surface the function f be equal to zero. The surface has two sides. Let one of the sides be called the first, the other being called the second. We continue the function f in the rest part of the vortex tube G (t) by integrating along the vortex lines (from the first side of to the second side of ), so that the following equality holds (the dot denotes the scalar product): ·f = · F. (3) It follows from the theory of ordinary differential equations [29] that the field f constructed in this way will be uniquely determined for each time in- stant t (t 1 , t 2 ) at all points of the vortex tube G (t). In this case, at points of the surface , the field f will be continuous from the first side of (from which 409 integration began), and it may prove to be discontinuous from the second side of . We use the well-known formula for a double vector product [ {F f }] = ( · {F f }) {F f } ( · ). The first term in the right-hand side is zero by virtue of (3). Therefore, F = f [ {F f }] . 2 Substitute this expression for the vector F into equation (1): ( ( V [ {F f }] ) V 2 ) + V + = f . t 2 2 (4) Let n denotes a unit vector normal to the flat surface on the side from which integration began when constructing the function f ; (f ) 1 and (f ) 2 denote the limiting values of the gradient of the function f from the first side and from the second side , respectively. The surface is chosen so that · n = 0 everywhere on this surface. There- fore, there are an infinite number of functions g(x, y, z, t) defined (and smooth) at all the points of the tube G (t), except for the points on the surface , such that the limiting values of g 1 and g 2 on different sides of the surface satisfy the conditions ) ( [ {F (f ) i } ] + g · n = 0, i = 1, 2. V + i 2 (These conditions mean that on both the first and second sides of the surface , the vectors in round brackets are parallel to .) Let g(x, y, z, t) be one of such functions. This function is defined in the tube G (t) = G (t) (the tube G (t) is obtained by cutting out the surface from the tube G (t)). Inside the tube G (t), we consider the vector field q = V + [ {F f }] + g. 2 Since = 0, equation (4) is equivalent to ( V V 2 ) + q = f . t 2 Applying the rot operator to both sides of the latter equality leads to the equation + rot [ q] = 0. t Comparing this equation with (2) and using Zoravski’s criterion, we arrive at the following conclusion. 410Closed vortex lines in fluid and gas The evolution of the vorticity in the tube G (t) can be considered as moving the vortex lines (and vortex tubes with preserving their intensity) together with q-particles moving with velocity q. 3. The theorem on closed vortex tubes. At time t 1 (t 1 , t 2 ), we choose any closed vortex line in the tube G (t 1 ). Consider a set of q-particles that make up a part of this line lying in G (t 1 ), i.e. the part of the closed line without a point on the surface (where the velocity q is not defined and where there are no q-particles). The function g is chosen in such a way that the limit value of the velocity q on both sides of surface is parallel to this surface. Therefore, the q-particles cannot “get out” of the tube G (t) through the surface , and new q-particles cannot “get in”. Consequently, the considered set of q-particles inside G (t) will constitute a closed vortex line (except for one point on the surface ) for some non-zero time interval (t 1 , t 2 ) (t 1 , t 2 ). Since the vortex lines are closed, this means that, during the time interval (t 1 , t 2 ) (t 1 , t 2 ), the velocity limits q on different sides of the surface must coincide, V + [ {F (f ) 1 }] [ {F (f ) 2 }] + g 1 = V + + g 2 . 2 2 After rearrangements of the terms, we obtain {(f ) 2 (f ) 1 } = (g 2 g 1 ). (5) By the properties of the vector product, the left and right sides of equality (5) are perpendicular to each other. This is only possible if both parts are zero. Therefore, {(f ) 2 (f ) 1 } = 0, i.e. the vectors {(f ) 2 (f ) 1 } and = 0 are parallel. On the other hand, according to (3), · {(f ) 2 (f ) 1 } = 0, i.e. these vectors are orthogonal. This is only possible if (f ) 2 = (f ) 1 . By construction, the projection of (f ) 1 onto the surface is zero. Therefore, the projection of (f ) 2 onto the surface is also zero. The selected line and time t 1 (t 1 , t 2 ) were chosen arbitrarily (see the beginning of this section). Conse- quently, the projection (f ) 2 on the surface is zero at all points during the entire time interval (t 1 , t 2 ). This means that, at each instant of time, the limiting values of f at the ends of all lines are equal to one constant. This constant may be different for different points in time. According to equation (3), the aforemen- tioned constant is the circulation of the vector F along a closed vortex line, i.e. the circulation of the right-hand side of equation (1) over a closed vortex line is the same for different vortex lines crossing the surface . Thus, the following theorem is proved. A necessary condition for the existence in time of vortex tubes, consisting of closed vortex lines. Let the vortex flow of a fluid ( = 0) be described by equation (1). And let, during the nonzero period of time (t 1 , t 2 ), the entire (stationary) flat surface intersect with the closed vortex lines of this flow 411 at an acute angle to the normal and, at the same time, each vortex line intersect with only once. Then, for any fixed moment of time t (t 1 , t 2 ), the circulation of the right side of equation (1) along a closed vortex line is the same for different vortex lines crossing the surface . Note that the theorem leaves open the question of the time dependence on the circulation of the vector F. The classical Thomson (Kelvin) theorem [2] is also true for all types of fluids. It claims that the time derivative of the velocity circulation along a contour mov- ing together with the fluid particles is equal to the circulation of the right-hand side (1). In this theorem, no mention is made of the relation between the values of the circulation on different contours. In the theorem proved above, such a con- nection is found (equality of circulations) for contours of a special form (closed vortex lines). These contours are different from the contours referred to in the Thomson theorem since it cannot be stated in the general case that the vortex lines move with the particles of the fluid. Therefore, the Thomson theorem and the theorem proved above are incomparable and complementary. 4. Equivalent formulation of the theorem and some special cases. The circulation of the second term in the left-hand side of equation (1) is zero, since the vector V is orthogonal to the vortex line. The circulation of the third term is zero due to the closedness of the vortex line. Therefore, the circulation of F coincides with the circulation of V t . Consequently, a) under the conditions of the theorem proved above, for any fixed moment of time t (t 1 , t 2 ), the circulation of V t along a closed vortex line is the same for different vortex lines intersecting the surface ; b) for stationary flows, the circulation of the right-hand side of equation (1) along any closed vortex line is zero. For a non-stationary flow, it is also possible to specify a situation where the circulation of the right-hand side of equation (1) along any closed vortex line is zero. In particular, if there is a sequence of closed vortex lines in the vortex tube, the lengths of which tend to zero, then this circulation will be equal to zero. Conclusion. For all types of fluids (from an ideal fluid to a viscous gas), a previously unknown property of closed vortex lines has been obtained. It is formulated as a theorem, and it can be used both for qualitative analysis and for verification of numerical calculations of flows in which there are closed vortex tubes.

About the authors

Grigory Borisovich Sizykh

Moscow Aviation Institute (State Technical University)

Email: o1o2o3@yandex.ru
Candidate of physico-mathematical sciences, Associate professor 4, Volokolamskoe shosse, Moscow, 125993, Russian Federatio

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