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ABSTRACT
This work presents numerical investigations for turbulent flow and heat transfer in a backward-facing step with and without porous inserts. Two classes of the model were employed, namely linear and nonlinear turbulence closures. The entire set of transport equations was discretized by means of the control volume method and the system of algebraic equations obtained was relaxed using the SIMPLE (Semi Implicit Pressure-Linked Equations) method. Results were first validated against the experimental data and the simulations follow experimental values and trends. Computations further indicated that when using the porous insert, the size, shape, and length of the recirculating region were drastically reduced in addition to being pushed toward the channel exit, leading eventually to a complete bubble suppression for thicker inserts. A more permeable medium gave better results in quickly suppressing the circulatory motions. By including porous inserts in the channel, turbulence generated due to the shear inside the recirculating region was damped, whereas high levels of k were concentrated within the permeable structure. Large variations for the skin friction factor along the bottom wall were also smoothed out by placing inserts, spanning from a typical distribution for an unobstructed back-step flow to a standard parallel channel flow distribution as the inserts got ticker. On the other hand, at the upper wall, flow pushed toward the top surface gave rise to a sudden increase of the skin friction factor, which was later stabilized downstream the flow. Heat transfer analysis followed showing damping for Nu at the bottom wall as the thickness of the porous substrate was increased. Overall, the thickness of the insert played a dominant role in changing the final flow and heat transfer characteristics rather than the porosity or permeability of the porous material. Finally, this work indicated that the sudden increase of Nu around the reattachment point, known to be undesirable in many practical situations for causing additional thermomechanical loads on the surface, may by avoided by the use of a porous obstacle past the back-step.
Nomenclature
| Latin characters | = | |
| A | = | area (m2) |
| a | = | porous Insert thickness (m) |
| cF | = | Forchheimer coefficient |
| Cf | = | skin friction coefficient: |
| c1NL | = | coefficient of the nonlinear model |
| c2NL | = | coefficient of the nonlinear model |
| c3NL | = | coefficient of the nonlinear model |
| cμ | = | turbulence model constant |
| c1 | = | turbulence model constant |
| c2 | = | turbulence model constant |
| ck | = | turbulence macroscopic model constant |
| cp | = | specific heat capacity (J/kg K) |
| D | = | inlet channel height (m) |
| D | = | deformation rate tensor: D = [∇u + [∇u]T]/2 |
| Dij | = | deformation rate tensor in index notation: |
| Dp | = | porous particle length (m) |
| Da | = | Darcy number: Da = K/H2 |
| E | = | wall log-law constant |
| ER | = | expansion ratio: ER = (H + D)/D |
| fg | = | dimensionless global head loss coefficient |
| fgo | = | dimensionless global head loss coefficient for the unobstructed channel |
| H | = | step height (m) |
| Hg | = | global head loss (m) |
| I | = | unit tensor |
| k | = | turbulence kinetic energy (m2/s2) |
| K | = | porous media permeability (m2) |
| Kdisp | = | conductivity tensor due to thermal dispersion (W/m K) |
| Kdisp, t | = | conductivity tensor due to turbulent thermal dispersion (W/m K) |
| Keff | = | effective conductivity tensor (W/m K) |
| Kt | = | conductivity tensor due to turbulent heat flux (W/m K) |
| Ktor | = | conductivity tensor due to tortuosity (W/m K) |
| n | = | coordinate normal to the interface (m) |
| nw | = | coordinate normal to the wall (m) |
| y+ | = | dimensionless normal interface/wall distance |
| Nu | = | Nusselt number: Nu = q″H/λf(Tw − Tin) |
| p | = | pressure (kg/m s2) |
| Pr | = | Prandtl number: Pr = cpμ/λf |
| Prt | = | turbulent Prandtl number constant |
| q″ | = | heat flux (W/m2) |
| = | wall heat flux (W/m2) | |
| ReH | = | Reynolds number: |
| St | = | Stanton number: St = Nu/(RePr) |
| T | = | time average temperature (K) |
| Tin | = | channel inlet temperature (K) |
| u | = | fluid velocity vector (m/s) |
| Uin | = | channel inlet velocity (m/s) |
| = | Darcy or superficial velocity vector (volume average of | |
| U | = | fluid x component velocity (m/s) |
| = | absolute average velocity (m/s) | |
| UD// | = | average surface velocity component parallel to the interface (m/s) |
| uτ | = | wall-friction velocity |
| x | = | horizontal Cartesian coordinate (m) |
| xu | = | upstream step length (m) |
| xd | = | downstream length after the step (m) |
| xR | = | boundary layer reattachment length (m) |
| Greek characters | = | |
| α | = | kinetic energy coefficient |
| β | = | interface stress jump coefficient |
| δ | = | boundary-layer thickness (m) |
| δij | = | Kronecker delta |
| ϵ | = | turbulent dissipation (m2/s3) |
| ϕ | = | porosity |
| κ | = | von Kármán constant |
| μ | = | fluid dynamic viscosity (kg/m s) |
| = | macroscopic eddy dynamic viscosity (kg/m s) | |
| ν | = | fluid kinematic viscosity (m2/s): |
| = | macroscopic eddy kinematic viscosity (m2/s): | |
| Ωij | = | vorticity tensor in index notation: |
| λ | = | thermal conductivity (W/m·K) |
| ρ | = | density (kg/m3) |
| σk | = | turbulence model constant |
| σϵ | = | turbulence model constant |
| τw | = | wall shear stress (kg/m s2) |
| Special characters | = | |
| φ | = | general variable |
| = | time average | |
| φ′ | = | time fluctuation |
| ⟨φ⟩i | = | intrinsic average |
| ⟨φ⟩v | = | volume average |
| iφ | = | spatial deviation |
| |φ| | = | absolute value (Abs) |
| φ | = | general tensor variable |
| φeff | = | effective value of φ, φeff = ϕφf + (1 − ϕ)φs |
| φs, f | = | solid/fluid |
| φ∞ | = | freestream position |
| φin, out | = | inlet/outlet |
| φw | = | wall position |
Nomenclature
| Latin characters | = | |
| A | = | area (m2) |
| a | = | porous Insert thickness (m) |
| cF | = | Forchheimer coefficient |
| Cf | = | skin friction coefficient: |
| c1NL | = | coefficient of the nonlinear model |
| c2NL | = | coefficient of the nonlinear model |
| c3NL | = | coefficient of the nonlinear model |
| cμ | = | turbulence model constant |
| c1 | = | turbulence model constant |
| c2 | = | turbulence model constant |
| ck | = | turbulence macroscopic model constant |
| cp | = | specific heat capacity (J/kg K) |
| D | = | inlet channel height (m) |
| D | = | deformation rate tensor: D = [∇u + [∇u]T]/2 |
| Dij | = | deformation rate tensor in index notation: |
| Dp | = | porous particle length (m) |
| Da | = | Darcy number: Da = K/H2 |
| E | = | wall log-law constant |
| ER | = | expansion ratio: ER = (H + D)/D |
| fg | = | dimensionless global head loss coefficient |
| fgo | = | dimensionless global head loss coefficient for the unobstructed channel |
| H | = | step height (m) |
| Hg | = | global head loss (m) |
| I | = | unit tensor |
| k | = | turbulence kinetic energy (m2/s2) |
| K | = | porous media permeability (m2) |
| Kdisp | = | conductivity tensor due to thermal dispersion (W/m K) |
| Kdisp, t | = | conductivity tensor due to turbulent thermal dispersion (W/m K) |
| Keff | = | effective conductivity tensor (W/m K) |
| Kt | = | conductivity tensor due to turbulent heat flux (W/m K) |
| Ktor | = | conductivity tensor due to tortuosity (W/m K) |
| n | = | coordinate normal to the interface (m) |
| nw | = | coordinate normal to the wall (m) |
| y+ | = | dimensionless normal interface/wall distance |
| Nu | = | Nusselt number: Nu = q″H/λf(Tw − Tin) |
| p | = | pressure (kg/m s2) |
| Pr | = | Prandtl number: Pr = cpμ/λf |
| Prt | = | turbulent Prandtl number constant |
| q″ | = | heat flux (W/m2) |
| = | wall heat flux (W/m2) | |
| ReH | = | Reynolds number: |
| St | = | Stanton number: St = Nu/(RePr) |
| T | = | time average temperature (K) |
| Tin | = | channel inlet temperature (K) |
| u | = | fluid velocity vector (m/s) |
| Uin | = | channel inlet velocity (m/s) |
| = | Darcy or superficial velocity vector (volume average of | |
| U | = | fluid x component velocity (m/s) |
| = | absolute average velocity (m/s) | |
| UD// | = | average surface velocity component parallel to the interface (m/s) |
| uτ | = | wall-friction velocity |
| x | = | horizontal Cartesian coordinate (m) |
| xu | = | upstream step length (m) |
| xd | = | downstream length after the step (m) |
| xR | = | boundary layer reattachment length (m) |
| Greek characters | = | |
| α | = | kinetic energy coefficient |
| β | = | interface stress jump coefficient |
| δ | = | boundary-layer thickness (m) |
| δij | = | Kronecker delta |
| ϵ | = | turbulent dissipation (m2/s3) |
| ϕ | = | porosity |
| κ | = | von Kármán constant |
| μ | = | fluid dynamic viscosity (kg/m s) |
| = | macroscopic eddy dynamic viscosity (kg/m s) | |
| ν | = | fluid kinematic viscosity (m2/s): |
| = | macroscopic eddy kinematic viscosity (m2/s): | |
| Ωij | = | vorticity tensor in index notation: |
| λ | = | thermal conductivity (W/m·K) |
| ρ | = | density (kg/m3) |
| σk | = | turbulence model constant |
| σϵ | = | turbulence model constant |
| τw | = | wall shear stress (kg/m s2) |
| Special characters | = | |
| φ | = | general variable |
| = | time average | |
| φ′ | = | time fluctuation |
| ⟨φ⟩i | = | intrinsic average |
| ⟨φ⟩v | = | volume average |
| iφ | = | spatial deviation |
| |φ| | = | absolute value (Abs) |
| φ | = | general tensor variable |
| φeff | = | effective value of φ, φeff = ϕφf + (1 − ϕ)φs |
| φs, f | = | solid/fluid |
| φ∞ | = | freestream position |
| φin, out | = | inlet/outlet |
| φw | = | wall position |
Acknowledgments
The authors would like to thank FAPESP, CNPq, and CAPES, funding agencies in Brazil, for their financial support during the preparation of this work.