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Turbulent Flow Around Fluid–Porous Interfaces Computed with a Diffusion-Jump Model for k and ε Transport Equations

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Abstract

Flow over vegetation and bottom of rivers can be characterized by some sort of porous structure of irregular surface through which a fluid permeates. Also, in engineering systems, one can have components that make use of a working fluid flowing over irregular layers of porous material. This article presents numerical solutions for such hybrid medium, considering here a channel partially filled with a flat porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both the regions. A diffusion-jump model for both the turbulent kinetic energy and its dissipation rate, across the interface, is presented and discussed upon. The discretization steps taken for numerically accommodating such model in the system of algebraic equations are presented. Numerical results show the effects of Reynolds number, porosity, and permeability on mean and turbulence fields. Results indicate that when negative values for the stress jump coefficient are applied, the peak of the turbulent kinetic energy distribution occurs at the macroscopic interface.

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Abbreviations

c F :

Forchheimer coefficient

c 1, c 2, c k , c μ :

Constant in turbulence model

D :

Deformation rate tensor, \({{\bf D}={[\nabla {\bf u}+(\nabla {\bf u})^{\rm T}]}/2}\)

G i :

Production rate of k due to the porous matrix, \({G^{\rm i}={c_k \rho \phi \langle k\rangle ^{\rm i}{\kern 1pt}|{\kern 1pt}{\bf \overline{u}}_{\rm D} |}/{\sqrt{K}}}\)

H :

Distance between the channel walls

I :

Unit tensor

k :

Turbulent kinetic energy per unit mass, \({k={\overline {{\bf {u}'}{\kern 1pt}\cdot {\bf {u}'}}}/2}\)

\({\langle k\rangle ^{\rm v}}\) :

Volume (fluid + solid) average of k

\({\langle k\rangle ^{\rm i}}\) :

Intrinsic (fluid) average of k

K :

Permeability

L :

Axial length of periodic section of channel

p :

Thermodynamic pressure

\({\langle p\rangle ^{\rm i}}\) :

Intrinsic (fluid) average of pressure p

P i :

Production rate of k due to mean gradients of \({{\bf \overline {\bf u}}_{\rm D}}\), \({P^{\rm i}=-\rho \langle \overline {{\bf {u}'}{\bf {u}'}} \rangle ^{\rm i}:\nabla {\bf \overline{{u}}}_{\rm D} }\)

\({{\bf \overline{{R}}}}\) :

Time average of total drag per unit volume

Re H :

Reynolds number based on the channel height, \({Re_{\rm H} ={\rho \left| {{\bf \overline{{u}}}_{\rm D}}\right|H}/\mu }\)

s :

Clearance for unobstructed flow

\({S_\varphi}\) :

Source term

\({{\bf \overline{{u}}}}\) :

Microscopic time-averaged velocity vector

\({\langle {\bf \overline{{u}}}\rangle ^{\rm i}}\) :

Intrinsic (fluid) average of \({\overline {\bf u} }\)

\({{{\bf \overline{{u}}}_{\rm D}} }\) :

Darcy velocity vector, \({{{\bf \overline{{u}}}_{\rm D}} = \phi \left\langle {\overline {\bf u}}\right\rangle ^{\rm i}}\)

\({{{\bf \overline{u}}_{\rm D}}_{\rm i}}\) :

Darcy velocity vector at the interface

\({{{\bf \overline{{u}}}_{\rm D}}_{\rm p} }\) :

Darcy velocity vector parallel to the interface

u D n, u D p :

Components of Darcy velocity at interface along η (normal) and ξ (parallel) directions, respectively.

u D i, v D i :

Components of Darcy velocity at interface along x and y, respectively

x, y :

Cartesian coordinates

β, β t :

Interface stress jump coefficient for mean and turbulent flow fields, respectively

μ :

Fluid dynamic viscosity

μ eff :

Effective viscosity for a porous medium

\({{\mu _{\rm t}}_\phi }\) :

Macroscopic turbulent viscosity

ε :

Dissipation rate of \({k, \varepsilon={\mu \overline {\nabla {\bf {u}'} :(\nabla {\bf {u}'})^{\rm T}}}/\rho }\)

\({\langle \varepsilon \rangle ^{\rm i}}\) :

Intrinsic (fluid) average of ε

ρ :

Density

\({\phi}\) :

Porosity

\({{\varphi}}\) :

General dependent variable

η, ξ :

Generalized coordinates

σ k , σ ε :

Turbulent Prandtl numbers for k and ε, respectively.

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  1. Departamento de Energia - IEME, Instituto Tecnológico de Aeronáutica - ITA, 12228-900, São José dos Campos, SP, Brazil

    Marcelo J. S. De Lemos

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Correspondence to Marcelo J. S. De Lemos.

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De Lemos, M.J.S. Turbulent Flow Around Fluid–Porous Interfaces Computed with a Diffusion-Jump Model for k and ε Transport Equations. Transp Porous Med 78, 331–346 (2009). https://doi.org/10.1007/s11242-009-9379-0

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