Double-diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow

During tunnel construction, groundwater inrush from completely weathered granite strata is a significant challenge to geotechnical engineers. Up to the present, prevention of water inrushing hazards is almost exclusively based on the experience of engineers. This paper presents a coupled seepage–erosion water inrush model to investigate the characteristics of seepage–erosion properties. The proposed model is based on classical theories of solute transport and fluid dynamics in porous media. In the model, changes of porosity link permeability with the accumulation of particle loss in the seepage–erosion process. The coupled seepage–erosion model was applied to examine the influence of curtain grouting thickness on the seepage–erosion process. The results showed that the seepage–erosion process was attenuated as thickness increased. The results also showed that the porosity and permeability visibly changed and the water inflow clearly exceeded the acceptable engineering criterion when the thickness was less than 6 m. However, with a further increase in thickness, the seepage–erosion process was suppressed and little changes of the relative parameters were showed. The numerical results demonstrated that a curtain grouting thickness of 6 m was suitable for curtain grouting in completely weathered granite. Field investigation of Cenxi tunnel verified the effectiveness of the thickness determined by the proposed model.

Groundwater control and curtain grouting for tunnel construction in completely weathered granite

In the present study, double-diffusive convection in an anisotropic porous layer with an internal heat source, heated and salted from below, has been investigated. The generalized Darcy model is employed for the momentum equation. The fluid and solid phases are considered to be in equilibrium. Linear and nonlinear stability analyses have been performed. For linear theory normal mode technique has been used, while nonlinear analysis is based on a minimal representation of truncated Fourier series. Heat and mass transfers across the porous layer have been obtained in terms of Nusselt number Nu and Sherwood number Sh, respectively. The effects of internal Rayleigh number, anisotropy parameters, concentration Rayleigh number, and Vadasz number on stationary, oscillatory, and weak nonlinear convection are shown graphically. The transient behaviors of Nusselt number and Sherwood number have been investigated by solving the finite amplitude equations using a numerical method. Streamlines, isotherms, and isohalines are drawn for both steady and unsteady (time-dependent) cases. The results obtained, during the above analyses, have been presented graphically, and the effects of various parameters on heat and mass transfers have been discussed.

Double-Diffusive Convection in a Saturated Anisotropic Porous Layer with Internal Heat Source

A multilevel decoupled method for a mixed Stokes/Darcy model

A classical linear stability theory is applied to emphasize the effect of inertia on the stability of buoyancy-driven parallel shear flow in a vertical layer of porous medium. The Lapwood–Brinkman model with fluid viscosity different from effective viscosity is used to describe the flow in a porous medium. The resulting eigenvalue problem is solved numerically using the Chebyshev collocation method. The critical Darcy–Rayleigh number $$R_\mathrm{Dc} $$
 , the critical wave number $$a_\mathrm{c}$$
 and the critical wave speed $$c_\mathrm{c}$$
 are computed over a wide range of values of the Darcy–Prandtl number $$Pr_\mathrm{D}$$
 and the Darcy number $${\tilde{D}}a$$
 . Depending on the choice of physical parameters, instability occurs due to the presence of inertia. The value of $$Pr_\mathrm{D}$$
 at which the transition from stationary to traveling-wave mode instability takes place increases with decreasing $${\tilde{D}}a$$
 . Besides, the effect of decreasing $${\tilde{D}}a$$
 shows destabilizing effect if the instability is via stationary mode, and on the contrary, it exhibits a dual behavior if the instability is through traveling-wave mode. The streamlines and isotherms presented herein demonstrate the development of complex dynamics at the critical state. In the energy spectrum, transition of instability from one type to another is found to take place as a function of $$Pr_\mathrm{D}$$
 . The disturbance kinetic energy due to surface drag and viscous force plays no significant role in the stability of flow throughout the domain of $$Pr_\mathrm{D}$$
 considered.

Stability of natural convection in a vertical layer of Brinkman porous medium

Poiseuille flow in a fluid overlying a highly porous material

Penetrative convection via internal heating in anisotropic porous media

The onset of penetrative convection in an anisotropic porous medium, for fluids with quadratic density law, is studied. In particular, the effects of anisotropic permeability and thermal diffusivity have been taken into account.

Penetrative convection in anisotropic porous media with variable permeability

Studies of the nonlinear stability of fluid/porous systems have been developed very recently. A two-domain modelling approach has been adopted in previous works, but was restricted to specific configurations. The extension to the more general case of a Navier–Stokes modelled fluid over a porous material was not achieved for the two-domain approach owing to the difficulties associated with handling the interfacial boundary conditions. This paper addresses this issue by adopting a one-domain approach, where the governing equations for both regions are combined into a unique set of equations that are valid for the entire domain. It is shown that the nonlinear stability bound, in the one-domain approach, is very sharp and hence excludes the possibility of subcritical instabilities. Moreover, the one-domain approach is compared with an equivalent two-domain approach, and excellent agreement is found between the two.

/pdf/nonlinear-stability-of-the-one-domain-approach-to-modelling-2di2a8epal.pdf

Antony A. Hill

Papers

Double-diffusive penetrative convection simulated via internal heating in an anisotropic porous layer with throughflow

Poiseuille flow in a fluid overlying a highly porous material

Penetrative convection via internal heating in anisotropic porous media

Penetrative convection in anisotropic porous media with variable permeability

Nonlinear stability of the one-domain approach to modelling convection in superposed fluid and porous layers