scholarly journals Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents

2014 ◽  
Vol 747 ◽  
pp. 218-246 ◽  
Author(s):  
Zhong Zheng ◽  
Ivan C. Christov ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Numerical Solutions ◽  
Gravity Currents ◽  
Similar Solution ◽  
Phase Plane Analysis ◽  
Plane Analysis ◽  
Self Similar Solution ◽  
Theoretical Predictions ◽  
Self Similar ◽  
Similar Solutions

AbstractWe report experimental, theoretical and numerical results on the effects of horizontal heterogeneities on the propagation of viscous gravity currents. We use two geometries to highlight these effects: (a) a horizontal channel (or crack) whose gap thickness varies as a power-law function of the streamwise coordinate; (b) a heterogeneous porous medium whose permeability and porosity have power-law variations. We demonstrate that two types of self-similar behaviours emerge as a result of horizontal heterogeneity: (a) a first-kind self-similar solution is found using dimensional analysis (scaling) for viscous gravity currents that propagate away from the origin (a point of zero permeability); (b) a second-kind self-similar solution is found using a phase-plane analysis for viscous gravity currents that propagate toward the origin. These theoretical predictions, obtained using the ideas of self-similar intermediate asymptotics, are compared with experimental results and numerical solutions of the governing partial differential equation developed under the lubrication approximation. All three results are found to be in good agreement.

Freely draining gravity currents in porous media: Dipole self-similar solutions with and without capillary retention

2007 ◽  
Vol 18 (3) ◽  
pp. 337-362 ◽  
Author(s):  
JOCHONIA S. MATHUNJWA ◽  
ANDREW J. HOGG
Keyword(s):  
Unsaturated Flow ◽  
Multiple Scales ◽  
Initial Conditions ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Theoretical Predictions ◽  
Analytic Expressions ◽  
Self Similar ◽  
Similar Solutions

We analyse the two-dimensional, gravitationally-driven spreading of fluid through a porous medium overlying a horizontal impermeable boundary from which fluid can drain freely at one end. Under the assumption that none of the intruding fluid is retained within the pores in the trail of the current, the motion of the current is described by the dipole self-similar solution of the first kind derived by Barenblatt and Zel'dovich (1957). We show that small perturbations of arbitrary shape imposed on this solution decay in time, indicating that the self-similar solution is linearly stable. We use the connection between the perturbation eigenfunctions and symmetry transformations of the self-similar solution to demonstrate that variables can always be specified in terms of which the rate of decay of the perturbations is maximised. Unsaturated flow can be modelled by assuming that a constant fraction of the fluid is retained within the pores by capillary action in the trail of the current. It has been shown (Barenblatt and Zel'dovich, 1998; Ingerman and Shvets, 1999) that in this case, the motion of the current is described by a self-similar solution of the second kind characterised by an anomalous exponent. We derive leading-order analytic expressions for the anomalous exponent and the self-similar quantities valid for small values of the fraction of fluid retained using direct asymptotic analysis and by using a novel application of the method of multiple scales. The latter offers a number of advantages and permits the evolution of the current to be clearly connected with its initial conditions in a way not possible with conventional approaches. We demonstrate that the theoretical predictions provided by these expressions are in excellent agreement with results from the numerical integration of the governing equations.

On a self-similar solution for the decay of turbulent bursts

1992 ◽  
Vol 3 (4) ◽  
pp. 319-341 ◽  
Author(s):  
S. P. Hastings ◽  
L. A. Peletier
Keyword(s):  
Asymptotic Behaviour ◽  
Physical Parameter ◽  
Dimensional Analysis ◽  
Existence And Uniqueness ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Eigenvalue Parameter ◽  
Self Similar ◽  
Similar Solutions

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

Converging gravity currents over a permeable substrate

2015 ◽  
Vol 778 ◽  
pp. 669-690 ◽  
Author(s):  
Zhong Zheng ◽  
Sangwoo Shin ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Gravity Current ◽  
Gravity Currents ◽  
Numerical Predictions ◽  
Front Position ◽  
Dependent Position ◽  
Self Similar ◽  
Vertical Leakage ◽  
Similar Solutions ◽  
Lock Gate

We study the propagation of viscous gravity currents along a thin permeable substrate where slow vertical drainage is allowed from the boundary. In particular, we report the effect of this vertical fluid drainage on the second-kind self-similar solutions for the shape of the fluid–fluid interface in three contexts: (i) viscous axisymmetric gravity currents converging towards the centre of a cylindrical container; (ii) viscous gravity currents moving towards the origin in a horizontal Hele-Shaw channel with a power-law varying gap thickness in the horizontal direction; and (iii) viscous gravity currents propagating towards the origin of a porous medium with horizontal permeability and porosity gradients in power-law forms. For each of these cases with vertical leakage, we identify a regime diagram that characterizes whether the front reaches the origin or not; in particular, when the front does not reach the origin, we calculate the final location of the front. We have also conducted laboratory experiments with a cylindrical lock gate to generate a converging viscous gravity current where vertical fluid drainage is allowed from various perforated horizontal substrates. The time-dependent position of the propagating front is captured from the experiments, and the front position is found to agree well with the theoretical and numerical predictions when surface tension effects can be neglected.

Existence of Leray's Self-Similar Solutions of the Navier-Stokes Equations In 𝒟 ⊂ ℝ3

2004 ◽  
Vol 47 (1) ◽  
pp. 30-37
Author(s):  
Xinyu He
Keyword(s):  
Boundary Condition ◽  
Stokes Equations ◽  
Local Existence ◽  
Similar Solution ◽  
Navier Stokes ◽  
Smooth Bounded Domain ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar ◽  
Similar Solutions

AbstractLeray's self-similar solution of the Navier-Stokes equations is defined bywhere . Consider the equation for U(y) in a smooth bounded domain D of with non-zero boundary condition:We prove an existence theorem for the Dirichlet problem in Sobolev space W1,2(D). This implies the local existence of a self-similar solution of the Navier-Stokes equations which blows up at t = t* with t* < +∞, provided the function is permissible.

The self-similar solutions to full compressible Navier–Stokes equations without heat conductivity

2019 ◽  
Vol 29 (12) ◽  
pp. 2271-2320
Author(s):  
Xin Liu ◽  
Yuan Yuan
Keyword(s):  
Small Perturbation ◽  
Heat Conductivity ◽  
Stokes Equations ◽  
The Self ◽  
Similar Solution ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Self Similar Solution ◽  
Self Similar ◽  
Similar Solutions

In this work, we establish a class of globally defined large solutions to the free boundary problem of compressible full Navier–Stokes equations with constant shear viscosity, vanishing bulk viscosity and heat conductivity. We establish such solutions with initial data perturbed around the self-similar solutions when [Formula: see text]. In the case when [Formula: see text], solutions with bounded entropy can be constructed. It should be pointed out that the solutions we obtain in this fashion do not in general keep being a small perturbation of the self-similar solution due to the second law of thermodynamics, i.e. the growth of entropy. If, in addition, in the case when [Formula: see text], we can construct a solution as a global-in-time small perturbation of the self-similar solution and the entropy is uniformly bounded in time. Our result extends the one of Hadžić and Jang [Expanding large global solutions of the equations of compressible fluid mechanics, J. Invent. Math. 214 (2018) 1205.] from the isentropic inviscid case to the non-isentropic viscous case.

Self-similar solution in the Leith model of turbulence: anomalous power law and asymptotic analysis

2013 ◽  
Vol 47 (2) ◽  
pp. 025501 ◽  
Author(s):  
V N Grebenev ◽  
S V Nazarenko ◽  
S B Medvedev ◽  
I V Schwab ◽  
Yu A Chirkunov
Keyword(s):  
Asymptotic Analysis ◽  
Power Law ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Self Similar

scholarly journals Propagation of a Plane Strain Hydraulic Fracture With a Fluid Lag in Permeable Rock

2018 ◽  
Vol 85 (9) ◽  
Author(s):  
B. Chen ◽  
Andrew R. Barron ◽  
D. R. J. Owen ◽  
Chen-Feng Li
Keyword(s):  
Hydraulic Fracture ◽  
Dimensionless Parameter ◽  
Numerical Solutions ◽  
Similar Solution ◽  
Confining Stress ◽  
Self Similar Solution ◽  
Permeable Rock ◽  
Self Similar ◽  
Fluid Lag ◽  
Limiting Case

Based on the KGD scheme, this paper investigates, with both analytical and numerical approaches, the propagation of a hydraulic fracture with a fluid lag in permeable rock. On the analytical aspect, the general form of normalized governing equations is first formulated to take into account both fluid lag and leak-off during the process of hydraulic fracturing. Then a new self-similar solution corresponding to the limiting case of zero dimensionless confining stress (T=0) and infinite dimensionless leak-off coefficient (L=∞) is obtained. A dimensionless parameter R is proposed to indicate the propagation regimes of hydraulic fracture in more general cases, where R is defined as the ratio of the two time-scales related to the dimensionless confining stress T and the dimensionless leak-off coefficient L. In addition, a robust finite element-based KGD model has been developed to simulate the transient process from L=0 to L=∞ under T=0, and the numerical solutions converge and agree well with the self-similar solution at T=0 and L=∞. More general processes from T=0 and L=0 to T=∞ and L=∞ for three different values of R are also simulated, which proves the effectiveness of the proposed dimensionless parameter R for indicating fracture regimes.

Self-similarity in coupled Brinkman/Navier–Stokes flows

2012 ◽  
Vol 699 ◽  
pp. 94-114 ◽  
Author(s):  
Ilenia Battiato
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Flexural Rigidity ◽  
Similar Solution ◽  
Navier Stokes ◽  
Self Similar Solution ◽  
Hydrodynamic Shear ◽  
Self Similar ◽  
Definition Of ◽  
Similar Solutions

AbstractIn this paper we derive self-similar solutions of flows through both a porous medium and a pure fluid. Self-similar filtration velocity and hydrodynamic shear profiles are obtained by means of asymptotic analysis in the limit of infinitely small permeability, and for both laminar and turbulent regimes over the porous medium. We show that a spatial length scale, related to the porous layer thickness, naturally emerges from the limiting process and suggests a more formal definition of thick and thin porous media. We finally specialize the analysis to porous media constituted of patterned cylindrical obstacles, which can freely deflect under the aerodynamic shear exerted by the fluid flowing through and over the forest. A self-similar solution for the bending profile of the elastic cylindrical obstacles is obtained as intermediate asymptotics, and applied to carbon nanotube (CNT) forests’ response to aerodynamic stresses. This self-similar solution is successfully used to estimate flexural rigidity of CNTs by linear fit of appropriately rescaled maximum deflection and average velocity measurements.

Sign in / Sign up

Export Citation Format

Share Document