Both are characterized by the rapid growth of the core in a small portion of the vortex. Keeping only terms linear in the perturbation quantities gives the equations for the linearized stability of the compressed vortex. The growth of the solutions for the small compression can be compared with the values of Section 1.6.
The effect of the compression on these stable waves is examined in the next three sections. E > 0 on these waves can be examined by noting that the shape of .. the wave changes only slightly during a period of the oscillation. The behavior of the waves is examined numerically in the next section to determine the relationship.
The core boundary is a vortex jump due to the discontinuity in the derivative of the undeformed azimuthal velocity. As in the stationary vortex problem, the axial wavenumber is time dependent for the separation and is of the form The asymptotic behavior of the core displacement determines neither the stability nor the validity of the model.
Part of the difficulty is that the unknown constant coefficient in the asymptotic expression can be large.
TIME OF LINKING
LOG SCALE)
CHAPTER 2
Saffman (1977) discusses the effect of considering realistic core profiles on the critical \'/ave numbers and the corresponding 1st vorticity rates for straight vorticity in a stress field. The product of the wave number times the radius of the ring can be compared to the number of wave crests that appear in the observed rings. The relative growth rates for the first critical wave numbers can be examined to predict the number of waves in the ring.
The purpose of this chapter is to demonstrate a method for calculating the growth rate for a general axisymmetric distribution of vorticity in a straight eddy core of finite radius surrounded by potential flow. The fluid is assumed to be uniform, inviscid and incompressible with constant density taken as unity. In a cylindrical coordinate system (r,e,z) the undeformed vortex filament is aligned along the z axis \· velocity S'fiirl V .
The system is scaled by dividing the velocities by V0(a) and the radial coordinates by a. 2.2.5) The symbol t:, is used to denote vorticity to eliminate confusion with the symbol for frequency, which was introduced later. The parameter E is the ratio between the strain rate, e, and the dimensional speed of rotation at the edge of the core.
In order to evaluate the flux field corrections for a given axisymmetric core vorticity distribution, it is convenient to introduce flux functions, \jJ.
To investigate the stability of the steady flow field defined in the last section, the perturbation velocities (ij,~,0) and the pressure p are added to the steady solutions. The solution of the system (2.3.4) must satisfy the conditions at the origin and infinity, as well as the condition of continuity of velocities at the boundary of the core. The solutions of the perturbations can be written as an expansion in the small parameter t: defining.
Substituting this form into equation (2.3.4) and equating the coefficients to powers of t: gives the hierarchy. and replacing 8/8z in the matrix M by ik. To the lowest order at small strain, these are the stable deflections corresponding to w. By substituting the above into equation (2.4.2), the system can be reduced to two first-order differential equations.
In the region outside the core, the perturbation velocity field can be written in the form of a velocity potential. Using the form for the separation of axial, azimuth and time dependence given in equations (2.3.9) and (2.4.1), the velocity potential for mode n = 1 has the form. 2.4.12). The velocity with the addition of the disturbances must also be continuous over the disturbed boundary. 2.4.14), i.e. the parentheses indicate the jump in quantity over some curve.
The parameter £ is set to zero at order l OY.test in tension, giving the conditions. 0 and the values of the functions u and p at x=l depend on the shape of n(x) in the vortex core. The solution to the lowest-order problem must be determined in order to implement the Fredholm alternative.
The form of the matrix operators t·1~(+l) and M~(-l) yields the relationship given above bet>r/one U*(+l) and U*{-l) when they are appropriate. By replacing the components in equation (2.6.2), the sister can be written in terms of two ordinary differential equations. In the region outside the core the vorticity is zero and the solution to equations (2.6.5) and (2.6.6) can be found in closed form.
CHAPTER 3
In this case it is easier to calculate the corrections to the vortex velocity field due to the shear flow based on the governing equations. The tangential velocities must be continuous over the surface r = R(e), which provides an additional constraint. 3.2.9) The right side of this equation follows from the shape of the current. In the last chapter it was shown that stable spiral waves exist as superpositions of the n=+l and n=-1 modes.
The airfoil of length 2 lies along the x-axis from -1 to 1. The unit velocity flow is at an angle of attack a. The Kutta condition at the trailing edge and the t~·Jo equations for the equilibrium of the free vortex yield three equations for the four unknowns K, r, p,. This last equation determines the possible positions of the free vortex for a given angle of incidence a.
At the trailing edge of the body, the current should pass smoothly along the cusp. To devise a general scheme to determine the locus, equation (4.6.7) can be solved for the result of rand and then replaced by equation (4.6.6). From the results for the flat plate, the leading edge of the airfoil should also be examined for a second locus of solutions.
Note that the values of the maximum lift at the trailing loci are decreased compared to the values for the flat plate, even though the values of the lift without the free vortex are larger. The speed V does not approach the value of the tracking speed for the airfoil in the absence of the free vortex, since K tends to zero. For the motion") of the vortex of force K in the ~-plane, the trajectory function is given by.
5.2.1) Routh's result is that the trajectory function for the vortex motion in the z plane is. The Schwarz-Christoffel transformation gives a mapping from the s plane. and the image of vortex trajectories in the mapped plane s is described by . 12 I I I I I I I I I I I Figure 5.2 The dashed curve is the boundary trajectory for the eddies traveling through the channel in the geometry of Case I. Some other possible trajectories are plotted along with the time stamps.
Dots placed along the trajectory represent the time increment of L2jK. The curves are the trajectories of one of the pair's vortices. for the constant C or solutions. The distances from the walls to the asymptotes are the same as in case I. The separating trajectory crosses the x-axis at.
Structure of a linear array of hollow vortices of finite cross-section
In steady flow, constant pressure inside the cores requires the fluid velocity to have a constant value, q0 say, at the boundary of each vortex. We must compute a unique stable solution for 0 < R < 1, where each vortex has front-and-back symmetry, i.e. R = 1 gives a vortex sheet where each vortex is pulled out longitudinally and squeezed laterally to lie along a length L of the x-axis.
Core deformation is conveniently measured by PfA!::, which has a minimum value of 27Tl for a circle and becomes large with eccentricity. Mappings of the ABODE contour in the physical plane to the potential (
The physical plane is shown in Figure 1. Due to the symmetry, it is sufficient to calculate the flow within the contour .ABODE. The direction or magnitude of the velocity is known on the contour, and the methods of free-stream line theory can therefore be applied by mapping the potential plane onto the hodograph plane. For the vortex with the largest area for given L, the length of the major axis is 0.71£ and that of the minor axis is 0.25£.
Saffman (1975), we can estimate the critical value of A~JL for an array from the result for a single vortex by placing e == 7T r f6L2 in the critical value for the single vortex. The exact value of PJA~ for the critical vortex is 4.2 for the array and 4.5 for the single vortex. The existence of two possible configurations of the array suggests that at least one of them is unstable, and this one should be the most distorted.
The effect of the finite size of the core may have a bearing on the fact that Brown & Roshko (1974) do not seem to have found the Lamb-type instability.) It suffices to consider the strip - !L < x < !L, y > 0 and to use the intact velocity potential and stream function as independent coordinates. The critical value of fJ at which the group becomes unstable to disturbances of the type considered here gives the same value of R, 0·805, as that at which A!JL is a maximum, thus demonstrating that, when there are two contradictions of possible - figurations, the more deformed is unstable to perturbations for which the less deformed is stable (cf. Moore & Saffman 1971). Consider a member of the linear array of void or stagnant vortices with the geometry as shown in Figure 4.
