Stability Of Inviscid Flows Research Articles

On the Hydrodynamic and Hydromagnetic Stability of Inviscid Flows between Coaxial Cylinders

On the Hydrodynamic and Hydromagnetic Stability of Inviscid Flows between Coaxial Cylinders

An energy criterion for the linear stability of conservative flows

We investigate the linear stability of inviscid flows which are subject to a conservative body force. This includes a broad range of familiar conservative systems, such as ideal MHD, natural convection, flows driven by electrostatic forces and axisymmetric, swirling, recirculating flow. We provide a simple, unified, linear stability criterion valid for any conservative system. In particular, we establish a principle of maximum action of the formformula herewhere η is the Lagrangian displacement,e is a measure of the disturbance energy, T and V are the kinetic and potential energies, and L is the Lagrangian. Here d represents a variation of the type normally associated with Hamilton's principle, in which the particle trajectories are perturbed in such a way that the time of flight for each particle remains the same. (In practice this may be achieved by advecting the streamlines of the base flow in a frozen-in manner.) A simple test for stability is that e is positive definite and this is achieved if L(η) is a maximum at equilibrium. This captures many familiar criteria, such as Rayleigh's circulation criterion, the Rayleigh–Taylor criterion for stratified fluids, Bernstein's principle for magnetostatics, Frieman & Rotenberg's stability test for ideal MHD equilibria, and Arnold's variational principle applied to Euler flows and to ideal MHD. There are three advantages to our test: (i) d2T(η) has a particularly simple quadratic form so the test is easy to apply; (ii) the test is universal and applies to any conservative system; and (iii) unlike other energy principles, such as the energy-Casimir method or the Kelvin–Arnold variational principle, there is no need to identify all of the integral invariants of the flow as a precursor to performing the stability analysis. We end by looking at the particular case of MHD equilibria. Here we note that when u and B are co-linear there exists a broad range of stable steady flows. Moreover, their stability may be assessed by examining the stability of an equivalent magnetostatic equilibrium. When u and B are non-parallel, however, the flow invariably violates the energy criterion and so could, but need not, be unstable. In such cases we identify one mode in which the Lagrangian displacement grows linearly in time. This is reminiscent of the short-wavelength instability of non-Beltrami Euler flows.

Stability of inviscid flow in a flexible tube

The equivalents of the classical theorems of hydrodynamic stability are derived for inviscid flow through a flexible tube. An important difference between flows in plane and cylindrical geometries is that the Squire transformation, which states that two-dimensional perturbations in plane parallel flows are always more unstable than three-dimensional perturbations, is not valid for tube flows. Therefore, it is necessary to analyse both axisymmetric and non-axisymmetric perturbations in flows in cylindrical geometries. Perturbations of the form exp[ik(χ –ct) + inϕ] are imposed on a steady axisymmetric mean flow V(r), and the stability of the mean velocity profiles and bounds for the phase velocity of the unstable modes are determined. Here r, ϕ and x are the radial, polar and axial directions, and k and c are the wavenumber and phase velocity. The flexible wall is represented by a standard constitutive equation which contains inertial, elastic and dissipative terms. Results for general velocity profiles are derived in two limiting cases: axisymmetric flows (n = 0) and highly non-axisymmetric flows (n [Gt ] k). The results indicate that axisymmetric perturbations are always stable for (V″ – r−1V′) V < 0 and could be unstable for (V″ – r−1V′) V < 0, while highly non-axisymmetric perturbations are always stable for (V″ + r−1V′) V [ges ] 0 and could be unstable for (V″ + r−1V′) V < 0. In addition, bounds on the real part (cr) and imaginary part (ci) of the phase velocity are also derived. For the practically important case of Hagen–Poiseuille flow, the present analysis indicates that axisymmetric perturbations are always stable, while highly non-axisymmetric perturbations could be unstable. This is in contrast to plane parallel flows where two-dimensional disturbances are always more unstable than three-dimensional ones.

Complex Mapped Matrix Methods in Hydrodynamic Stability Problems

The ordinary differential equations governing the linear stability of inviscid flows contain singularities at real or complex points called critical latitudes, which degrade the accuracy of standard numerical schemes. However, the use of a complex mapping prior to the numerical attack offers some respite. This mapping shifts the computational domain to a contour in the complex plane to avoid the critical latitudes. Both quadratic and cubic complex maps are considered in some detail. An analytic result for choosing the optimum quadratic complex map in the case of a single critical latitude is presented. Numerical results are given for two test problems and a barotropic vortex model. A comparison is made between methods with and without these mappings. The results show that the use of complex maps can lead to remarkably accurate solutions.

The stability of inviscid flows over passive compliant walls

The linear temporal stability of incompressible semi-bounded inviscid parallel flows over passive compliant walls is studied. It is shown that some of the well-known classical results for inviscid parallel flows with rigid boundaries can, in fact, be extended in modified form to passive compliant walls. These include a result of Rayleigh (1880) which shows that the real part of the phase velocity of a non-neutral disturbance must lie within the range of the velocity distribution; the semi-circle theorem of Howard (1961) and a result of Høiland (1953) which places a bound on the temporal amplification rates of unstable disturbances. The bounds on the phase velocity and the temporal amplification rates of unstable two-dimensional disturbances provide useful guides for numerical studies.The results are valid for a large class of passive compliant walls. This generality is achieved through a variational-Lagrangian formulation of the essential dynamics of wall motion. A general treatment of the marginal stability of thin shear flows over general passive compliant walls is given. It represents a generalization of the analysis given by Benjamin (1963) for membrane and plate surfaces. Sufficient conditions for the stability of thin shear flows over passive compliant walls are deduced. The applications of the stability criteria to simple cases of compliant wall are described to illustrate the use and the effectiveness of these criteria.