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ABSTRACT
The effect of the angle of attack on an elliptical cylinder in the steady forced convection regime of Bingham plastic fluids is investigated numerically. The elliptical cylinder with an aspect ratio (≡a/b), E = 0.5 is subjected to either the constant wall temperature (CWT) or the constant heat flux (CHF) conditions. The parameters influencing the flow and heat transfer in Bingham plastic fluids are the Reynolds number (0.01 ≤ Re ≤ 30), the Prandtl number (1 ≤ Pr ≤ 100), the Bingham number (0.01 ≤ Bn ≤ 100), and the angle of inclination (0° ≤ λ ≤ 90°). The detailed structure of the flow and the heat transfer characteristics have been analyzed in terms of the distributions of pressure coefficient, vorticity, and the local Nusselt number on the surface of the cylinder; in addition, the flow domain is mapped in terms of the streamline and isotherms in the vicinity of the heated cylinder. In all cases, the orientation of the elliptical cylinder plays an important role in the onset of the asymmetry in the flow and the morphology of the yielded/unyielded regions under the combined effects of the yield stress and inertial forces. Finally, the present numerical values of the drag coefficient and average Nusselt number have been correlated in terms of the pertinent dimensionless parameters with acceptable levels of accuracy over the range of inclinations considered in the present study. The dependence of the Nusselt number on the angle of inclination is found to be nonmonotonous.
Nomenclature
| a | = | semiminor axis of the elliptical cylinder (m) |
| Ap | = | projected area of the elliptical cylinder per unit length (m2/m) |
| b | = | semimajor axis of the elliptical cylinder (m) |
| Bn | = | Bingham number [2bτo/ (μBV∞)] (–) |
| Bn* | = | modified Bingham number (–) |
| C | = | specific heat of fluid (J/kg K) |
| CD | = | total drag coefficient (–) |
| CL | = | lift coefficient (–) |
| CDF | = | frictional drag coefficient (–) |
| CDP | = | pressure drag coefficient (–) |
| CP | = | pressure coefficient (–) |
| D∞ | = | diameter of the computational domain (m) |
| E | = | aspect ratio of the elliptical cylinder, [=a/b] (–) |
| FD | = | drag force per unit length of the cylinder (N m−1) |
| FDF | = | frictional component of drag force per unit length of the cylinder (N m−1) |
| FDP | = | pressure component of drag force per unit length of the cylinder (N m−1) |
| FL | = | lift force per unit length of the cylinder (N m−1) |
| h | = | local heat transfer coefficient (W/m2 K) |
| k | = | thermal conductivity of fluid (W/m K) |
| m | = | growth rate parameter (s) |
| n | = | unit vector normal to the surface of the cylinder (–) |
| Nu | = | local Nusselt number (–) |
| Nuavg | = | average Nusselt number (–) |
| p | = | pressure (–) |
| ps | = | local pressure on the surface of the cylinder (Pa) |
| p∞ | = | reference pressure far away from the cylinder (Pa) |
| Pe | = | Peclet number [Re × Pr] (–) |
| Pr | = | Prandtl number [CμB/k] (–) |
| Pr* | = | modified Prandtl number (–) |
| qw | = | surface heat flux, W/m2 |
| Re | = | Reynolds number [2bρV∞/μB] (–) |
| Re* | = | modified Reynolds number (–) |
| S | = | surface area of the cylinder (m2) |
| V | = | velocity vector (–) |
| V∞ | = | free stream velocity ( m s−1) |
| = | rate of strain tensor (–) | |
| ζ | = | vorticity (–) |
| θ | = | angular position on the surface of cylinder, degree |
| λ | = | angle of inclination (degree) |
| μB | = | plastic viscosity (Pa s) |
| μy | = | yielding viscosity (Pa s) |
| ξ | = | temperature of fluid [≡(T-T∞)/(Tw-T∞) for CWT or k(T-T∞)/(2bqw) for CHF] (–) |
| ρ | = | density of the fluid (kg m−3) |
| τ | = | extra stress tensor (–) |
| τo | = | yield stress (Pa) |
| Subscripts | = | |
| i, j, x, y | = | Cartesian coordinates |
Nomenclature
| a | = | semiminor axis of the elliptical cylinder (m) |
| Ap | = | projected area of the elliptical cylinder per unit length (m2/m) |
| b | = | semimajor axis of the elliptical cylinder (m) |
| Bn | = | Bingham number [2bτo/ (μBV∞)] (–) |
| Bn* | = | modified Bingham number (–) |
| C | = | specific heat of fluid (J/kg K) |
| CD | = | total drag coefficient (–) |
| CL | = | lift coefficient (–) |
| CDF | = | frictional drag coefficient (–) |
| CDP | = | pressure drag coefficient (–) |
| CP | = | pressure coefficient (–) |
| D∞ | = | diameter of the computational domain (m) |
| E | = | aspect ratio of the elliptical cylinder, [=a/b] (–) |
| FD | = | drag force per unit length of the cylinder (N m−1) |
| FDF | = | frictional component of drag force per unit length of the cylinder (N m−1) |
| FDP | = | pressure component of drag force per unit length of the cylinder (N m−1) |
| FL | = | lift force per unit length of the cylinder (N m−1) |
| h | = | local heat transfer coefficient (W/m2 K) |
| k | = | thermal conductivity of fluid (W/m K) |
| m | = | growth rate parameter (s) |
| n | = | unit vector normal to the surface of the cylinder (–) |
| Nu | = | local Nusselt number (–) |
| Nuavg | = | average Nusselt number (–) |
| p | = | pressure (–) |
| ps | = | local pressure on the surface of the cylinder (Pa) |
| p∞ | = | reference pressure far away from the cylinder (Pa) |
| Pe | = | Peclet number [Re × Pr] (–) |
| Pr | = | Prandtl number [CμB/k] (–) |
| Pr* | = | modified Prandtl number (–) |
| qw | = | surface heat flux, W/m2 |
| Re | = | Reynolds number [2bρV∞/μB] (–) |
| Re* | = | modified Reynolds number (–) |
| S | = | surface area of the cylinder (m2) |
| V | = | velocity vector (–) |
| V∞ | = | free stream velocity ( m s−1) |
| = | rate of strain tensor (–) | |
| ζ | = | vorticity (–) |
| θ | = | angular position on the surface of cylinder, degree |
| λ | = | angle of inclination (degree) |
| μB | = | plastic viscosity (Pa s) |
| μy | = | yielding viscosity (Pa s) |
| ξ | = | temperature of fluid [≡(T-T∞)/(Tw-T∞) for CWT or k(T-T∞)/(2bqw) for CHF] (–) |
| ρ | = | density of the fluid (kg m−3) |
| τ | = | extra stress tensor (–) |
| τo | = | yield stress (Pa) |
| Subscripts | = | |
| i, j, x, y | = | Cartesian coordinates |