Lecture Notes on Fluid Dynamics (1.63J/2.21J) (original) (raw)

Kelvin-Helmholtz instability in a shallow-water flow with a finite width

Journal of Mathematical Physics, 2019

We examine an effect of side walls on the linear stability of an interface of tangential-velocity discontinuity in shallow-water flow. The flow is pure horizontal in the plane xy, and the fluid is bounded in a finite width 2d in the y− direction. In region 0 < y < d, the fluid is moving with uniform velocity U but is at rest for −d < y < 0. Without side walls, the flow is unstable for a velocity difference U<8c, with c being the velocity of gravity waves. In this work, we show that if the velocity difference U is smaller than 2c, the interface is always destabilized, also known as the flow is unstable. The unstable region of an infinite width model is shrunken by the effects of side walls in the case of narrow width, while there is no range for the Froude number for stabilization in the case of large width. These results play an important role in predicting the wave propagations and have a wide application in the fields of industry. As a result of the interaction of w...

Kelvin–Helmholtz Instability as a Boundary-Value Problem

Environmental Fluid Mechanics, 2005

The Kelvin-Helmholtz (KH) instability is traditionally viewed as an initial-value problem, wherein wave perturbations of a two-layer shear flow grow over time into billows and eventually generate vertical mixing. Yet, the instability can also be viewed as a boundary-value problem. In such a framework, there exists an upstream condition where a lighter fluid flows over a denser fluid, wave perturbations grow downstream to eventually overturn some distance away from the point of origin. As the reverse of the traditional problem, this flow is periodic in time and exhibits instability in space. A natural application is the mixing of a warmer river emptying into a colder lake or reservoir, or the salt-wedge estuary. This study of the KH instability from the perspective of a boundary-value problem is divided into two parts. Firstly, the instability theory is conducted with a real frequency and complex horizontal wavenumber, and the main result is that the critical wavelength at the instability threshold is longer in the boundary-value than in the initial-value situation. Secondly, mass, momentum and energy budgets are performed between the upstream, unmixed state on one side, and the downstream, mixed state on the other, to determine under which condition mixing is energetically possible. Cases with a rigid lid and free surface are treated separately. And, although the algebra is somewhat complicated, both end results are identical to the criterion for complete mixing in the initial-value problem.

Elastically driven Kelvin–Helmholtz-like instability in straight channel flow

Proceedings of the National Academy of Sciences, 2021

Significance Kelvin–Helmholtz instability (KHI) describes the growth of perturbations of the interface separating counterpropagating streams of Newtonian fluids with different velocities and densities with heavier fluid at the bottom. It is one of the most studied shear flow instabilities widespread in nature and industrial processes. The KHI mechanism is based on competition between destabilizing velocity difference and stabilizing stratification. We study KHI in channel shear flow of viscoelastic fluid, considered stable due to elastic stress generated by polymers stretched by shear strain. Contrary to generally accepted opinion, we discover the elastically driven KH-like instability with temporal dynamics similar to Newtonian KHI but driven by strikingly different elastically driven destabilizing mechanism of elastic wave interacting with wall-normal vorticity amplifying interface perturbations. Originally, Kelvin–Helmholtz instability (KHI) describes the growth of perturbations ...

Kelvin–Helmholtz Creeping Flow at the Interface Between Two Viscous Fluids

The ANZIAM Journal, 2015

The Kelvin–Helmholtz flow is a shearing instability that occurs at the interface between two fluids moving with different speeds. Here, the two fluids are each of finite depth, but are highly viscous. Consequently, their motion is caused by the horizontal speeds of the two walls above and below each fluid layer. The motion of the fluids is assumed to be governed by the Stokes approximation for slow viscous flow, and the fluid motion is thus responsible for movement of the interface between them. A linearized solution is presented, from which the decay rate and the group speed of the wave system may be obtained. The nonlinear equations are solved using a novel spectral representation for the streamfunctions in each of the two fluid layers, and the exact boundary conditions are applied at the unknown interface location. Results are presented for the wave profiles, and the behaviour of the curvature of the interface is discussed. These results are compared to the Boussinesq–Stokes appr...

The nonlinear behavior of a sheared immiscible fluid interface

Physics of Fluids, 2002

The two-dimensional Kelvin-Helmholtz instability of a sheared fluid interface separating immiscible fluids is studied by numerical simulations. The evolution is determined by the density ratio of the fluids, the Reynolds number in each fluid, and the Weber number. Unlike the Kelvin-Helmholtz instability of miscible fluids, where the sheared interface evolves into well-defined concentrated vortices if the Reynolds number is high enough, the presence of surface tension leads to the generation of fingers of interpenetrating fluids. In the limit of a small density ratio the evolution is symmetric, but for a finite density difference the large amplitude stage consists of narrow fingers of the denser fluid penetrating into the lighter fluid. The initial growth rate is well predicted by inviscid theory when the Reynolds numbers are sufficiently high, but the large amplitude behavior is strongly affected by viscosity and the mode that eventually leads to fingers is longer than the inviscidly most unstable one.

Nonlinear Stability of two-layer flows

Communications in Mathematical Sciences, 2004

We study the dynamics of two-layer, stratified shallow water flows. This is a model in which two scenarios for eventual mixing of stratified flows (shear-instability and internal breaking waves) are, in principle, possible. We find that unforced flows cannot reach the threshold of shearinstability, at least without breaking first. This is a fully nonlinear stability result for a model of stratified, sheared flow. Mathematically, for 2X2 autonomous systems of mixed type, a criterium is found deciding whether the elliptic domain is reachable -smoothly-from hyperbolic initial conditions. If the characteristic fields depend smoothly on the system's Riemann invariants, then the elliptic domain is unattainable. Otherwise, there are hyperbolic initial conditions that will lead to incursions into the elliptic domain, and the development of the associated instability.

Pressure Corrections for the Potential Flow Analysis of Kelvin-Helmholtz Instability

Applied Mechanics and …, 2012

Pressure corrections for the viscous potential flow analysis of Kelvin-Helmholtz instability at the interface of two viscous fluids have been carried out when there is heat and mass transfer across the interface. Both fluids are taken as incompressible and viscous with different kinematic viscosities. In viscous potential flow theory, viscosity enters through normal stress balance and effect of shearing stresses is completely neglected. We include the viscous pressure in the normal stress balance along with irrotational pressure and it is assumed that this viscous pressure will resolve the discontinuity of the tangential stresses at the interface for two fluids. It has been observed that heat and mass transfer has destabilizing effect on the stability of the system. A comparison between viscous potential flow (VPF) solution and viscous contribution to the pressure for potential flow (VCVPF) solution has been made and it is found that the effect of irrotational shearing stresses stabilizes the system.

Elastically driven Kelvin-Helmholtz-like instability in planar channel flow

2020

Kelvin-Helmholtz instability (KHI) is widely spread in nature on scales from micrometer up to Galactic one. This instability refers to the growth of perturbation of an interface between two parallel streams of Newtonian fluids with different velocities and densities, destabilized by shear strain and stabilized by density stratification with the heavier fluid at the bottom. Here, we report the discovery of the purely elastic KH-like instability in planar straight channel flow of viscoelastic fluid, which is theoretically considered to be stable. However, despite the remarkable similarity to the Newtonian KHI temporal interface dynamics, the elastic KHI reveals qualitatively different instability mechanism. Indeed, the velocity difference across the interface strongly fluctuates and non-monotonically varies in time due to energy pumping by elastic waves, detected in the flow. A correlation of the elastic wave intensity and efficiency of the elastic KHI in different regimes suggests th...

Nonlinear instability in an ideal fluid

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 1997

Linearized instability implies nonlinear instability under certain rather general conditions. This abstract theorem is applied to the Euler equations governing the motion of an inviscid fluid. In particular this theorem applies to all 20 space periodic flows without stagnation points as well as 20 space-periodic shear flows.