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Register a new userTranscritical flow of a stratified fluid over topography: analysis of the forced Gardner equation
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Transcritical flow of a stratified fluid past a broad localised topographic obstacle is studied analytically in the framework of the forced extended Korteweg--de Vries (eKdV), or Gardner, equation. We consider both possible signs for the cubic nonlinear term in the Gardner equation corresponding to different fluid density stratification profiles. We identify the range of the input parameters: the oncoming flow speed (the Froude number) and the topographic amplitude, for which the obstacle supports a stationary localised hydraulic transition from the subcritical flow upstream to the supercritical flow downstream. Such a localised transcritical flow is resolved back into the equilibrium flow state away from the obstacle with the aid of unsteady coherent nonlinear wave structures propagating upstream and downstream. Along with the regular, cnoidal undular bores occurring in the analogous problem for the single-layer flow modeled by the forced KdV equation, the transcritical internal wave flows support a diverse family of upstream and downstream wave structures, including solibores, rarefaction waves, reversed and trigonometric undular bores, which we describe using the recent development of the nonlinear modulation theory for the (unforced) Gardner equation. The predictions of the developed analytic construction are confirmed by direct numerical simulations of the forced Gardner equation for a broad range of input parameters.
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We develop modulation theory for undular bores (dispersive shock waves) in the framework of the Gardner, or extended Korteweg--de Vries, equation, which is a generic mathematical model for weakly nonlinear and weakly dispersive wave propagation, when effects of higher order nonlinearity become important. Using a reduced version of the finite-gap integration method we derive the Gardner-Whitham modulation system in a Riemann invariant form and show that it can be mapped onto the well-known modulation system for the Korteweg--de Vries equation. The transformation between the two counterpart modulation systems is, however, not invertible. As a result, the study of the resolution of an initial discontinuity for the Gardner equation reveals a rich phenomenology of solutions which, along with the KdV type simple undular bores, include nonlinear trigonometric bores, solibores, rarefaction waves and composite solutions representing various combinations of the above structures. We construct full parametric maps of such solutions for both signs of the cubic nonlinear term in the Gardner equation. Our classification is supported by numerical simulations.
We consider the stochastic convection-diffusion equation [ partial_t u(t,,{bf x}) = uDelta u(t,,{bf x}) + V(t,,x_1)partial_{x_2}u(t,,{bf x}), ] for $t>0$ and ${bf x}=(x_1,,x_2)inmathbb{R}^2$, subject to $theta_0$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure [ text{Cov}[V(t,,a),,V(s,,b)]= delta_0(t-s)rho(a-b)qquadtext{for all $s,tge0$ and $a,binmathbb{R}$}, ] where $rho$ is a continuous and bounded positive-definite function on $mathbb{R}$. We prove a quite general existence/uniqueness/regularity theorem, together with a probabilistic representation of the solution that represents $u$ as an expectation functional of an exogenous infinite-dimensional Brownian motion. We use that probabilistic representation in order to study the It^o/Walsh solution, when it exists, and relate it to the Stratonovich solution which is shown to exist for all $ u>0$. Our a priori estimates imply the physically-natural fact that, quite generally, the solution dissipates. In fact, very often, begin{equation} Pleft{sup_{|x_1|leq m}sup_{x_2inmathbb{R}} |u(t,,{bf x})| = Oleft(frac{1}{sqrt t}right)qquadtext{as $ttoinfty$} right}=1qquadtext{for all $m>0$}, end{equation} and the $O(1/sqrt t)$ rate is shown to be unimproveable. Our probabilistic representation is malleable enough to allow us to analyze the solution in two physically-relevant regimes: As $ttoinfty$ and as $ uto 0$. Among other things, our analysis leads to a macroscopic multifractal analysis of the rate of decay in the above equation in terms of the reciprocal of the Prandtl (or Schmidt) number, valid in a number of simple though still physically-relevant cases.
In this letter, we provide fundamental insights into the dynamic transcritical transition process using molecular dynamics simulations. A transcritical region, which covers three different fluid states, was discovered as a substitute for the traditional interface. The physical properties, such as temperature and density, exhibited a highly non-linear distribution in the transcritical region. Meanwhile, the surface tension was found to exist throughout the transcritical region, and the magnitude was directly proportional to $ - rho cdot { abla ^2}rho $
Imbibition, the displacement of a nonwetting fluid by a wetting fluid, plays a central role in diverse energy, environmental, and industrial processes. While this process is typically studied in homogeneous porous media with uniform permeabilities, in many cases, the media have multiple parallel strata of different permeabilities. How such stratification impacts the fluid dynamics of imbibition, as well as the fluid saturation after the wetting fluid breaks through to the end of a given medium, is poorly understood. We address this gap in knowledge by developing an analytical model of imbibition in a porous medium with two parallel strata, combined with a pore network model that explicitly describes fluid crossflow between the strata. By numerically solving these models, we examine the fluid dynamics and fluid saturation left after breakthrough. We find that the breakthrough saturation of nonwetting fluid is minimized when the imposed capillary number Ca is tuned to a value Ca$^*$ that depends on both the structure of the medium and the viscosity ratio between the two fluids. Our results thus provide quantitative guidelines for predicting and controlling flow in stratified porous media, with implications for water remediation, oil/gas recovery, and applications requiring moisture management in diverse materials.
Imbibition plays a central role in diverse energy, environmental, and industrial processes. In many cases, the medium has multiple parallel strata of different permeabilities; however, how this stratification impacts imbibition is poorly understood. We address this gap in knowledge by directly visualizing forced imbibition in three-dimensional (3D) porous media with two parallel strata. We find that imbibition is spatially heterogeneous: for small capillary number Ca, the wetting fluid preferentially invades the fine stratum, while for Ca above a threshold value, the fluid instead preferentially invades the coarse stratum. This threshold value depends on the medium geometry, the fluid properties, and the presence of residual wetting films in the pore space. These findings are well described by a linear stability analysis that incorporates crossflow between the strata. Thus, our work provides quantitative guidelines for predicting and controlling flow in stratified porous media.
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