Steady-state flows of inviscid incompressible fluid and related particle dynamics in rectangular channels

An extended and improved particle‐spectral method for analysis of unsteady inviscid incompressible flows through a channel of finite length

“…Before outlining numerical methods, we are to expound on the known properties of flows with Yudovich's boundary conditions using the results of previous investigations 12,14,19,20 . The flow of fluid in $$$ D $$$ is called unseparated if each particle of fluid leaves channel for a finite time.…”

“…Thus, the function $$$ R $$$(y) is correctly determined on the interval $$$ y\in \left[0,b\right] $$$: $$$$ R:{y}^{+}\in \left[0,b\right]\to {y}^{-}\in \left[0,b\right]. $$$$ Stationary solutions to the problem (1)–(7) have several specific properties 18,20,42 . Here, $$$ R\left({y}^{+}\right) $$$ is invertible.…”

“…The stream function and vorticity distribution that define steady‐state flows are as follows 19,20 $$$$ {\omega}_s\left(x,y\right)=K{\psi}_s\left(x,y\right),\kern1em {\psi}_s\left(x,y\right)={q}_1y+\sum \limits_{i,j=1}^{\infty}\kern.3em {S}_{i,j}\sin \frac{i\pi x}{a}\sin \frac{j\pi y}{b}, $$$$ with coefficients $$$$ {S}_{i,j}=\frac{4\kern0.3em K\kern0.3em {b}^3{a}^2{\left(-1\right)}^j\kern0.3em \left(-1+{\left(-1\right)}^i\right)}{\pi^2i\kern0.3em j\kern0.3em \left({\pi}^2\left({i}^2{b}^2+{j}^2{a}^2\right)-K{a}^2{b}^2\right)}\kern0.3em {q}_1. $$$$ As test calculations, we consider a channel $$$ D $$$ with sides $$$ a=3 $$$ and …”

“…They proved that two‐dimensional inviscid steady flow is “totally nonunique” in the following sense: there is a continuum of solutions involving recirculation zones that can be always inserted into such a flow satisfying the same boundary conditions. Really, for Yudovich's boundary conditions, various stationary regimes under the same boundary conditions coexist in the channel 19,20 . The scenarios of the formation of nonstationary regimes with stagnant zones remain unclear and we study it also here.…”

“…Really, for Yudovich's boundary conditions, various stationary regimes under the same boundary conditions coexist in the channel. 19,20 The scenarios of the formation of nonstationary regimes with stagnant zones remain unclear and we study it also here.…”