The stability of steady, helical vortex ﬁlaments in a tube

The nonlinear conditions for the development of helical vortex ﬁlaments in a circular tube are considered. The helical ﬂow is assumed to be irrotational, except in a vortex ﬁlament of inﬁnitesimal core area. By introducing an appropriate image for this helical vortex ﬁlament, the boundary condition on the material frontier is satisﬁed. By assuming an axisymmetric ﬂow upstream and imposing the conservation laws, a dependence between the helix pitch and the nonlinear amplitude of the helical vortex developed downstream is obtained. Our results show that only helical ﬂows with the pitch in a certain range of values are allowed. The dependence of this range on the swirl ratio and on the tube cross section is considered. We discuss the usefulness of the nonlinear analysis of the allowed ﬂows to explain experimental results and to complement the usual linear study of stability. © 1999 American Institute of Physics. (cid:64) S1070-6631 (cid:126) 99 (cid:33) 01904-2 (cid:35)


I. INTRODUCTION
In recent years, there has been an important number of theoretical and experimental results on the development of structural changes in vortex flows. These phenomena, which produce abrupt unsteady changes in the pressure of the fluid, are usually associated with a loss of axial symmetry. They are of interest in several technical applications ͑for example, in aerodynamics and in the design of hydraulic machineries, as hydroelectric turbines͒. Laboratory experiments in tubes, described in Refs. 1-3, show that the flow is essentially irrotational, with constant circulation, except in a helical rotational core. ͑When the phenomenon just appears, the rotational core is close to the tube axis.͒ In this work, we study the possible steady, helical vortex filaments with finite helix radius ͑i.e., with arbitrary departure from an axially symmetric flow͒. To investigate the mechanisms that govern the structural changes in the flow, we consider the initial ͑inflow͒ field as axially symmetric and confined to a tube. We assume that ͑everywhere along the tube͒ the flow is irrotational except in a rotational core filament, with infinitesimal area and constant circulation around it. In Sec. II, the basic equations are derived. In Sec. III, an expression for the velocity field of the helical vortex filament is obtained, taking into account the material boundary conditions. In Sec. IV, a given upstream flow is considered and, by imposing conservations laws, a rule is obtained for the possible helical vortex filaments which can develop downstream departing from the axially symmetric flow. In Sec. V, criteria for the vortex stability are considered. In Sec. VI, the results of our analysis are applied to concrete examples.

II. BASIC EQUATIONS
We assume that the flow is incompressible and irrotational everywhere except in a vortex filament, i.e., in an infinite concentrated rotational core.
In the case of an incompressible inviscid fluid, the governing equations are ".vϭ0 and the Euler equation. In terms of the vorticity wϭ"؋v, the Euler equation takes the form of an evolution equation Therefore, the circulation is governed by the Kelvin theorem ((‫ץ/ץ‬t)ϩv-")⌫ϭ0. That is to say, the vortex filament moves with constant circulation around it. When the vorticity is concentrated, in the unbounded flow case, the velocity is determined by the Biot-Savart law; However, if boundary walls are present, this expression is not valid, because ͑in general͒ it does not satisfy the boundary conditions. We can construct an adequate solution for the flow around the vortex filament, adding to Eq. ͑3͒ a field v b (r), with "؋v b ϭ0 and "-v b ϭ0, provided that the sum satisfies the condition n-v͉ s ϭn-͑v u ϩv b ͉͒ s ϭ0 ͑5͒ on the limiting surface, where n is the unit vector in the normal direction to the surface. We note that v b depends on v u in a nonlinear form. Using Eq. ͑1͒ it is possible to prove ͑see Batchelor 4 ͒ that the vortex filament core moves with the particles without changes in the core strength ͑vorticity-transport theorem͒.
That is to say, the vortex line displacement ␦l is where r c is the position of the vortex line.
In principle, by using the previous equations, we could determine the vortex motion. However, this presents a serious difficulty in practice, because the first term of Eq. ͑4͒ is divergent when it is calculated on the filament itself ͑Batchelor 4 ͒. Obviously this singularity is unphysical because the real vortex has a finite core where Eq. ͑4͒ is not valid. Thus to determine the vortex velocity at the core, the real finite vortex core properties 5-8 must be considered. As a consequence, the resulting vortex motion depends critically on the core characteristics, and so the method is not very useful when the core size is not precisely known. In our work, we use only globally conserved magnitudes ͑as the mass rate, the circulation, and the total angular momentum͒, which do not depend on the vortex core characteristics. As we show in Sec. IV, significant results can be obtained which are independent of the concrete value of the core size, provided it is small in relation with the tube radius.

III. HELICAL VORTEX FILAMENTS IN A TUBE
In this work, we restrict our attention to the steady helical flows allowed inside a tube of radius b which can be modeled as vortex filaments. We assume that the helical vortex line has constant circulation along its extension and is situated on a cylinder of radius RϽb, with pitch ͑see Fig.  1͒. As there are two possible geometrical senses for the helix, we consider that can be positive or negative so that the sign of determines the sense of the helix. The filament can be parameterized as a line with an angular parameter ␣, where ␣ varies between Ϫϱ and ϩϱ.
The helical flow depends only on the helical coordinates (r,), where ϭϪ(2/)z and (r,,z) are the usual cylindrical coordinates. From Eqs. ͑3͒ and ͑7͒, we obtain the following expression for v u :

͑8͒
We construct the function v b , considering it as the velocity field of an induced image vortex, characterized by a radius R*Ͼb, a circulation ⌫*, and the same pitch . In this case, the total velocity field is vϭF͑r,,R,,⌫ ͒ϩF͑ r,,R*,,⌫*͒ϩU 0 e z , ϵv h ͑ r,,R,,⌫ ͒ϩU 0 e z , ͑9͒ where U 0 e z is an arbitrary constant term.
In the case of straight vortices (→ϱ), the values of R* and ⌫* can be computed using the well known Milne-Thompson theorem. 9 This theorem states that ⌫*ϭϪ⌫ and R*ϭb 2 /R. Unfortunately, there does not exist a general result for the case with finite . In this work, numerical values of R* and ⌫* ͑for given ⌫, R, and ͒ are determined by imposing condition ͑5͒ on some selected points on the surface, as shown in Fig. 2. In points 1 and 4 this condition holds automatically. We determine R* and ⌫* by imposing condition ͑5͒ in points 2 and 3 ͑and solving the system of the two resulting equations͒. Then, because of the symmetry, it is also satisfied in the points 5 and 6. Therefore, condition ͑5͒ holds exactly in the points ( P i , iϭ1,6) on the surface ͑see the surface ͑we verified this numerically͒. It is important to note that we need to evaluate the velocity field only on some points at the wall. Then, the divergence at the core does not affect our calculation. Thus we do not require to use special techniques of desingularization to solve the system. ͑In doing so, we eliminate ⌫* and then use a numerical bisection method to solve the resulting equation for R*, evaluating the integrals for each value of R*.) In Fig. 3, the curves denote the values of ⌫* and R* in the function of R and , taking bϭ1 and ⌫ϭ1. In this figure, it can be seen that as →ϱ, the numerical values for R* and ⌫* approach those obtained from the Milne-Thompson theorem, as expected. As mentioned before, these values are calculated considering only a finite number of points on the limiting surface. However, on the material frontier surface, the normal component of the velocity, n-v͉ s , is always less than 0.5% of the parallel velocity component. Then, Eq. ͑9͒ represents a good approximation to the flow, for our purpose ͑see Fig. 4͒. So, using Eqs. ͑9͒, ͑8͒, and the results shown in Fig. 3, we can determine the helical vortex filament for arbitrary values of R and .

IV. ADMISSIBLE FLOWS
In this section, we particularize the previous analysis ͑valid for arbitrary values of R, , and ⌫͒ to the special case in which the flow comes from an upstream cylindrical region with radius b 1 , where the velocity field is The downstream region is also considered to be cylindrical, but with radius b 2 . We suppose that the transition between the tubes is smooth enough to neglect transient effects. We are interested in determining the possible values of R and for the helical vortex filament developed in the downstream region. In order to determine these parameters, we impose the conservation of global magnitudes. First, we note that the Kelvin theorem implies that the circulation must be the same in both the upstream and downstream regions, that is to say, ⌫ϭ⌫ 0 . On the other hand, if an helical flow is present, it must satisfy the mass and angular momentum conservation.

A. Mass flux of the helical vortex
If we assume that the cross section area of the core is very small, in relation to the tube cross section area, the flow rate at the core can be neglected. In order to calculate the mass flow rate of the helical vortex filament, it is useful to introduce the vector potential of the velocity field. The surface integral of the velocity along the cross surface can be computed using Stokes theorem, integrating the potential vector round a closed curve C situated on the tube surface. This procedure avoids the singularities at the core, as we have mentioned in Sec. II. The potential vector H͑r͒ of the partial field ͑3͒ is easily obtained from the expression ϪR sin ␣ d␣ ͱ ͑͑r cos ϪR sin ␣͒ 2 ϩ͑r sin ϪR cos ␣͒ 2 ϩ͑zϪ␣/2 ͒ 2 ͒ , the potential vector A of the total velocity field is A͑r, ͒ϭH͑ r,,R,,⌫ ͒ϩH͑ r,,R*,,⌫*͒ϩ 1 2 U 0 re 0 , ϵA h ͑ r, ͒ϩ 1 2 U 0 re , ͑13͒ Then the mass conservation leads to the constraint

Helical stream function
When an incompressible flow depends only on the helical coordinates (r,), it is possible to define a stream function ⌿(r,) such that where v ϭv Ϫ(2/)v z . Taking derivatives, it can be seen that ⌿ϭA z ϩ(2/)A , where A z , A are the components of the velocity vector potential. From this last equality and Eq. ͑13͒, we can also find immediately the expression for the helical flow stream function ⌿(r,).

B. Flux of angular momentum
Let us consider that the streamlines which form the surfaces S 1 ,S 2 in the upstream region, form corresponding surfaces S 1 Ј ,S 2 Ј in the downstream region. If there is a helical flow, the section of surfaces S 1 Ј ,S 2 Ј looks as it is shown in Fig. 5 ͑we take S 2 and S 2 Ј on the material frontier͒. The mass conservation and the fluid incompressibility imply that the volume rate between surfaces S 1 Ј ,S 2 Ј is the same that between S 1 ,S 2 . Furthermore, as a consequence of the axial symmetry of the duct, the flow does not experiment a net torque and it conserves its angular momentum in relation to the tube axis. These two conditions lead to the equalities where h 2 () is the radial separation between S 1 Ј and S 2 Ј . By combining these equations, we obtain the condition Ϫ⌿ 2 )/v , where ⌿ 1 Ϫ⌿ 2 is independent of . Substituting in Eq. ͑17͒ and making simplifications, we finally obtain the following constraint equation In order to calculate the average ͗v 2 ͘, the function is considered a ''distribution function'' and v ,h ,v z,h are the components of the vector v h defined in Eq. ͑9͒.

C. Summarized constraint equations
The equations are a set of conditions which must be satisfied by the helical vortex flow in the downstream region. They constitute a nonlinear algebraic equations system for the variable parameters ⌫, U 0 , and R. The first and second conditions ͑19͒ immediately give the ⌫ value and the function U 0 (R,), which in turn gives the value of U 0 as a function of R and . Moreover, the last condition ͑19͒ is an undetermined nonlinear equation for the other two variable parameters, and R, of the helical flow. For a given value of , the equation becomes determined. In this way, solving numerically this equation for each value of ͑by bisection, evaluating the integrals for each value of the parameters͒, we obtain the possible values for R. ͑For a given , we can obtain two, one or none values of R, as we see in Fig. 6.͒ The curves in Fig. 6 show the values of R as a function of obtained by solving the last equation ͑19͒ for different values of the swirl ratio ⍀ϭ⌫ 0 b 2 /2Q, where Qϭb 1 2 U 1 is the flow rate. In the following, we do not consider the trivial case Rϭ0. ͓Notice that in this case the flow is axially symmetric ͑these conditions are always satisfied for the trivial solution, corresponding to an axisymmetric flow, R ϭ0, v ϭ⌫/2r, v z ϭ(b 1 2 /b 2 2 )U 1 . However, this solution is not interesting here, because it is not a helical flow.͔ It is remarkable that the solutions ͑in ͒ of Eq. ͑19͒ always have the opposite sign than ⍀. This means that the geometrical sense of the helical filament is the opposite of the rotation sense of the upstream vortex. This result agrees with the experimental observations reported by Sarpkaya 10 and cited by Leibovich. 11 Moreover, the curves of Fig. 6 show that the solutions ͑in ͒ of Eq. ͑19͒, have a module ͉͉ confined to a range between (0,͉͉ max ). The upper bound ͉͉ max depends strongly on ⍀. Figure 7 shows the contour plot of ͉͉ max as function of ⍀ and b 2 /b 1 . Furthermore, from the curve of Fig. 6 for ⍀ϭ0.16, the solutions of the lower branch with ͉͉Ͻ6 have a very small R. If we are interested in considering only helical flows of significant amplitude, then the range of permissible ͉͉ can be reduced. For example, in the case ⍀ϭ0.16, the ͉͉ values are actually in the range ͑1, 4.4͒. In the next section, we consider some consequences and applications of these results.

V. VORTEX STABILITY
A review of vortex stability is given by Leibovich. 11 At the present moment, a general criterion for the stability, to helical perturbations, of vortex flows in an inviscid incompressible fluid, is unknown. However, particular criteria are obtained for several simple cases. Thus Howard and Gupta 12 found a sufficient condition for the stability of an axial symmetric flow between concentric cylinders with axial velocity component v z (r) and swirl component v (r), to axisymmetric disturbances. This condition is where ⌽ϭr Ϫ3 (d/dr)(r 2 v 2 ). These authors also showed, for helical perturbations of the form ͕␦v r ,␦v ,␦v z ͖ϭ͕ f ͑ r ͒,g͑ r ͒,h͑ r ͖͒exp͓inϩik͑zϪct͔͒, ͑21͒ that the flow is stable when the condition is satisfied everywhere inside the fluid. This condition is always violated for sufficiently small values of ͉k͉, so no general stability criterion is obtained in this way. However, this condition permits us to determine a range of values of k corresponding to stable waves. We note that the perturbation ͑21͒ corresponds to a helix with pitch ϭ2/k. On the other hand, in some cases it is possible, making a normal mode analysis, to obtain the dispersion relation, and then to determine the flow stability. For example, Lessen et al. 13 have studied in this way the linear stability of the unbounded rotating top-hat jet. In this paper we are concerned with bounded flows, thus we now briefly study the linear stability of this flow bounded by a cylindrical surface rϭb, with velocity components If we assume that the disturbances are of the form ͕␦v r ,␦v ,␦v z ,␦p͖ ϭ͕ f ͑ r ͒,g͑ r ͒,h͑ r ͒,͑ r ͖͒exp͓inϩik͑zϪct͔͒, ͑24͒ the linearized disturbance equations for an inviscid fluid with velocity profile ͑23͒ are After eliminating the other variables, we get, in each region, a Bessel equation for the z-component of the disturbance, h. Solving this equation, we obtain an expression for h ͑in both regions͒, hϭC 1 I n ͑ r͒ϩD 1 K n ͑ r͒, 0рrрa, Boundary condition ͑5͒ implies that f (b)ϭ0, and then D 2 ϭϪ͓I n Ј͑kb͒/K n Ј͑kb͔͒C 2 .

͑28͒
Now, we suppose that the radial displacement of the vortex sheet due to the disturbance is ⌬ϭA exp(inϩik(zϪct)). The fact that the surface rϭaϩ⌬ moves with the fluid gives the kinematical condition which must be satisfied in both sides of the surface, being Uϭ(v r ,v ,v z ) the unperturbed velocity. The dynamical condition on the pressure states that it must be continuous across the surface rϭaϩ⌬ and gives, after linearizing, where the suffixes 1, 2 denote the inner and outer regions, respectively, and P 0 is the pressure of the unperturbed state. Equations ͑28͒, ͑30͒, ͑29͒, together with ͑26͒, ͑27͒, determine the dispersion relation where, by definition, H n (kr)ϵI n (kr)Ϫ͓I n Ј(kb)/ K n Ј(kb)͔K n (kr).
As b/a tends to infinity, this equation approaches the one obtained by Lessen et al. 13 Solving Eq. ͑31͒ we can obtain the value of the complex phase velocity cϭc r ϩic i , and so determine the stability of the basic flow Eq. ͑23͒. When the imaginary part of the phase velocity c i is positive, the flow is unstable. We have solved the dispersion relation ͑31͒ numerically. The values of c i /U 1 as a function of ka are shown in Fig. 8 for the quotient value b/aϭ10. It is remarkable that the values of c i are very close to those obtained by Lessen et al., in the limit of infinite b. This result means that the boundary has small influence on the linear stability of the vortex, when the core size is small in relation with the material frontier radius. Only for values of a of the same order of b, do we obtain a significant difference between the values of the phase velocity for the bounded and the unbounded vortex.

VI. APPLICATIONS OF NONLINEAR ANALYSIS OF ALLOWED FLOWS
Of course, the linear analysis is not valid when the helical flow has a large ͑finite͒ departure from an axially symmetric flow. In this case, the nonlinear analysis of allowed flows can be useful as a complement to the linear stability analysis. Knowledge of the range of allowed values of , for the full developed helical flow, can be used to check if a linear unstable mode can actually grow and develop into a helical flow. In order to clarify this point, let us study two concrete examples.

A. Example 1
First, we consider again, as in the previous section, the development of helical flows departing from the columnar vortex Eq. ͑23͒. The results of the linear analysis, for U 1 0, are showed in Fig. 8. We see that, for helical perturbations ͑21͒ with wave number modulus ͉k͉ϭ2/͉͉ sufficiently large, the flow is linearly unstable. In particular, when ⌫*ϭ⌫/aU 1 ϭ0.6, the flow is unstable for all positive values of k, i.e., of the pitch . However, nonlinear analysis shows that the of waves with finite amplitude are contained in a range (Ϫ͉͉ max ,0), and then the unstable modes with Ͼ0 or with ϽϪ͉͉ max are not able to grow and develop helical flows. Of course, in these cases the instabilities can lead to another type of change in the flow, but not to helical flows. In particular, the inexistence of helical waves with Ͼ0 is in agreement with the experiments. 10,11 It is not possible to obtain this result from the linear analysis. 14,15

B. Example 2
Now we consider the basic flow ͑in appropriate units͒ inside a tube of radius b, with continuity in the velocity ͓J 0 (x),J 1 (x) are the Bessel functions͔. For the values A ϭϪ0.04, aϭ0.2, bϭ1, ␥ϭ6 the linear criterion ͑22͒ guarantees stability under perturbations ͑21͒ with in the range ͑Ϫ54, 1.04͒. The system can be unstable under perturbations with out of this interval, which violate the criterion. For these values of the parameters, the swirl ratio ⍀ϭ0.038. Nonlinear analysis of Sec. IV implies ͑for bϭ1, ⍀ϭ0.038) that all possible helical flows have included in the interval ͑Ϫ0.62, 0͒. We note that the pitch of the waves which violates the stability criterion is out of this last range. Then, no helical flow will develop departing from the basic flow Eq. ͑32͒ for the given values of the parameters. On the other hand, when ␥ϭ20, with the same values of a and b, the range of which satisfies the stability condition ͑22͒ is ͑Ϫ1.35, 0.026͒, which does not contain the allowed interval ͑Ϫ2.54, 0͒ obtained from our nonlinear analysis. Therefore, in this case, there are unstable modes which can develop helical flows.

VII. CONCLUSIONS
In this paper a study of the allowed helical flows inside a circular tube is presented. The flow is supposed irrotational except in a core of infinitesimal section. An expression for the flow around this helical filament is obtained. The boundary conditions are satisfied, calculating numerically the image for the helical vortex. Conservation laws are imposed to the helical flow, supposing a given upstream flow. Using this procedure, a dependence between pitch and amplitude R of the helix is obtained. A range for admissible , for helical vortex flows with finite amplitude developed from an axially symmetric basic flow, is also found. The results are independent of the cutoff distance d, which is a fortunate fact, because the determination of d constitutes a serious problem. 4,8 Experimental observations ͑sense of the helix͒ are explained by this theory. The present analysis, which considers the effects of nonlinear terms in the balancing of conserved quantities, may be used to complement linear studies of stability, as it was shown before in two simple examples.