A more restrictively general stability criterion of two-dimensional inviscid parallel flow is obtained analytically. First, a sufficient criterion for stability is found as either $-μ_1<\frac{U''}{U-U_s}<0$ or $0<\frac{U''}{U-U_s}$ in the flow, where $U_s$ is the velocity at inflection point, $μ_1$ is the eigenvalue of Poincaré's problem. Second, this criterion is generalized to barotropic geophysical flows in $β$ plane. Based on the criteria, the flows are are divided into different categories of stable flows, which may simplify the further investigations. And the connections between present criteria and Arnol'd's nonlinear criteria are discussed. These results extend the former criteria obtained by Rayleigh, Tollmien and Fjørtoft and would intrigue future research on the mechanism of hydrodynamic instability.
The proper scale decomposition in flows with significant density variations is not as straightforward as in incompressible flows, with many possible ways to define a `length-scale.' A choice can be made according to the so-called \emph{inviscid criterion} \cite{Aluie13}. It is a kinematic requirement that a scale decomposition yield negligible viscous effects at large enough `length-scales.' It has been proved \cite{Aluie13} recently that a Favre decomposition satisfies the inviscid criterion, which is necessary to unravel inertial-range dynamics and the cascade. Here, we present numerical dem…
In this paper, the stability of inviscid parallel flow between two parallel walls is studied. Firstly, it is obtained that the profile of the base flow for this classical problem is a uniform flow. Secondly, it is shown that the solution of the disturbance equation is cr=U and ci=0, i.e., the propagation speed of the disturbance equals the flow velocity and the disturbance in this flow is neutral. Finally, it is suggested that the classical Rayleigh Theorem on inflectional velocity instability is incorrect which states that the necessary condition for instability of inviscid parallel flow is t…
The general stability criteria of inviscid Taylor-Couette flows with angular velocity $Ω(r)$ are obtained analytically. First, a necessary instability criterion for centrifugal flows is derived as $ξ'(Ω-Ω_s)<0$ (or $ξ'/(Ω-Ω_s)<0$) somewhere in the flow field, where $ξ$ is the vorticitiy of profile and $Ω_s$ is the angular velocity at the inflection point $ξ'=0$. Second, a criterion for stability is found as $-(μ_1+1/r_2)<f(r)=\frac{ξ'}{Ω-Ω_s}<0$, where $μ_1$ is the smallest eigenvalue. The new criteria are the analogues of the criteria for parallel flows, which are special cases of Arnol'd's n…
A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by developing a novel variational principle, where the velocity profile is assumed to be monotonic and analytic. It is shown that unstable eigenvalues of Rayleigh's equation (which is a non-selfadjoint eigenvalue problem) can be associated with positive eigenvalues of a certain selfadjoint operator. The stability is therefore determined by maximizing a quadratic form, which is theoretically and numerically more tractable than directly solving Rayleigh's equation. This variational stability c…
We investigate inviscid instability in an electrically conducting fluid affected by a parallel magnetic field. The case of low magnetic Reynolds number in Poiseuille flow is considered. When the magnetic field is sufficiently strong, for a flow with low hydrodynamic Reynolds number, it is already known that the neutral disturbances are three-dimensional. Our investigation shows that at high hydrodynamic Reynolds number(inviscid flow), the effect of the strength of the magnetic field on the fastest growing perturbations is limited to a decrease of their oblique angle i.e. angle between the dire…
Bedrossian and Masmoudi (2015) showed nonlinear stability of 2D inviscid parallel shear flows in the infinite stripe $(\mathbb{R}/2\mathbb{Z})\times\mathbb{R}$. Inspired by their result, combining a diffeomorphism group result by Misiołek (1993), in this paper we give a lower bound of perturbations in 2D inviscid shear flow in a periodic box.
Consider inviscid fluids in a channel {-1<y<1}. For the Couette flow v_0=(y,0), the vertical velocity of solutions to the linearized Euler equation at v_0 decays in time. At the nonlinear level, such inviscid damping has not been proved. First, we show that in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal period. This implies that nonlinear inviscid damping is not true in any (vorticity) H^{s}(s<(3/2)) neighborhood of Couette flow and for any horizontal period. Indeed, the long time behavior in such neighbor…
A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by a novel variational method, where the velocity profile is assumed to be monotonic and analytic. Unstable eigenvalues of the Rayleigh equation are shown to be associated with positive eigenvalues of a certain selfadjoint operator. The stability is therefore simply determined by maximizing a quadratic form, which is theoretically and numerically more tractable than directly solving the Rayleigh equation. This variational approach is based on the Hamiltonian nature of the inviscid fluid and …
We investigate the linear stability of shears near the Couette flow for a class of 2D incompressible stably stratified fluids. Our main result consists of nearly optimal decay rates for perturbations of stationary states whose velocities are monotone shear flows $(U(y),0)$ and have an exponential density profile. In the case of the Couette flow $U(y)=y$, we recover the rates predicted by Hartman in 1975, by adopting an explicit point-wise approach in frequency space. As a by-product, this implies optimal decay rates as well as Lyapunov instability in $L^2$ for the vorticity. For the previously…
In this paper, we study the Mach reflection phenomenon in inviscid flows using a higher order discontinuous Galerkin method and overset grids. We use the shock capturing procedure proposed in Siva Prasad Kochi et al. using overset grids to capture the discontinuities occurring in the supersonic flow over a wedge accurately. In this procedure, we obtain a coarse grid solution first and using the troubled cell data, we construct an overset grid which is approximately aligned to all the discontinuities. We rerun the solver with the coarse grid solution as the initial condition while using the tro…
Existence of a conjugate point in the incompressible Euler flow on a sphere and an ellipsoid is considered. Misiolek (1996) formulated a differential-geometric criterion (we call M-criterion) for the existence of a conjugate point in a fluid flow. In this paper, it is shown that no zonal flow (stationary Euler flow) satisfies M-criterion if the background manifold is a sphere, on the other hand, there are zonal flows satisfy M-criterion if the background manifold is an ellipsoid (even it is sufficiently close to the sphere). The conjugate point is created by the fully nonlinear effect of the i…
In the first part, the stability of two-dimensional parallel flow is discussed. A more restrictively general stability criterion for inviscid parallel flow is obtained analytically. In the second part, we report the numerical simulations of the partial-penetrating flow in horizontal convection within a squire cavity tank at high Rayleigh numbers $10^7<Ra<10^{10}$. The fast estab but slowly steadied flow is simulated, where a shallow and closed circulation cell is obtained numerically as partial-penetrating flow for the first time, which is consistent with the experiment.
Simulations of a supersonic, premixed, reacting flow over a wedge were performed to investigate the effect of a boundary layer on the wedge surface on ignition and stability of oblique detonation waves (ODWs). Two computational domains were used: one containing a wedge of a single angle with a straight after-body, and the other containing a double-angle wedge geometry. Both domains were channels with a supersonic inflow of stoichiometric hydrogen-air and a nonreflecting outflow, and the wedge was modeled using an immersed boundary method. The compressible reactive Navier-Stokes equations were …
Linear stability of inviscid, parallel, and stably stratified shear flow is studied under the assumption of smooth strictly monotonic profiles of shear flow and density, so that the local Richardson number is positive everywhere. The marginally unstable modes are systematically found by solving a one-parameter family of regular Sturm-Liouville problems, which can determine the stability boundaries more efficiently than solving the Taylor-Goldstein equation directly. By arguing for the non-existence of a marginally unstable mode, we derive new sufficient conditions for stability, which generali…
We sandwich a colloidal gel between two parallel plates and induce a radial flow by lifting the upper plate at a constant velocity. Two distinct scenarios result from such a tensile test: ($i$) stable flows during which the gel undergoes a tensile deformation without yielding, and ($ii$) unstable flows characterized by the radial growth of air fingers into the gel. We show that the unstable regime occurs beyond a critical energy input, independent of the gel's macroscopic yield stress. This implies a local fluidization of the gel at the tip of the growing fingers and results in the most unstab…
