# A turbulence model for the heat transfer near stagnation point of a circular cylinder

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## Abstract

A one-equation low-Reynolds number turbulence model has been applied successfully to the flow and heat transfer over a circular cylinder in turbulent cross flow. The turbulence length-scale was found to be equal 3.7*y* up to a distance 0.05δ and then constant equal to 0.185δ up to the edge of the boundary layer (where*y* is the distance from the surface and δ is the boundary layer thickness).

The model predictions for heat transfer coefficient, skin friction factor, velocity and kinetic energy profiles were in good agreement with the data. The model was applied for Re ≤250,000 and Tu_{∞}≤0.07.

### Keywords

Heat Transfer Boundary Layer Transfer Coefficient Heat Transfer Coefficient Turbulence Model### Nomenclature

*µ*,*C*_{D}Constants in the turbulence kinetic energy equation

*C*_{1},*C*_{2}Constants in the turbulence length-scale equation

- \(C_{f_x } \)
Skin friction coefficient\(\frac{{\tau _w }}{{\tfrac{1}{2}\varrho u_\infty ^2 }}\) at

*x**D*Cylinder diameter

*F*Dimensionless flow streamwise velocity

*u/u*_{ e }*k*Turbulence kinetic energy =1/2 the sum of the squared three fluctuating velocities

*K*Dimensionless turbulence kinetic energy

*k/u*_{ e }^{/2}*I*Dimensionless temperature (

*T−T*_{ w })/(*T*_{∞}−*T*_{ w })*l*Turbulence length-scale

*l*_{e}Turbulence length-scale at outer region

- Nu
_{D} Nusselt number

*p*Pressure

- Pr
Prandtl number

- Pr
_{t} Turbulent Prandtl number

- Pr
_{k} Constant in the turbulence kinetic energy equation

*R*Cylinder radius

- Re
_{D} Reynolds number ϱ

_{∞}*u*_{∞}*D*/µ_{∞}- Re
_{x} Reynolds number ϱ

_{∞}*u*_{∞}*x*/µ_{∞}- R
_{K} Reynolds number of turbulence

*T*Mean temperature

*T*_{∞}Mean temperature at ambient

*T*_{s}Mean temperature at surface

- Tu
_{∞} Cross flow turbulence intensity,\(\sqrt {u'^2 /u_\infty } \)

*u*Mean flow streamwise velocity

*u*′Fluctuating streamwise velocity

*u*_{e}Mean flow velocity at far field distance

*u*_{∞}Mean flow velocity at ambient

*u**Friction velocity\( = \sqrt {\tau _w /\varrho } \)

*v*Mean velocity normal to surface

*V*Dimensionless mean velocity normal to surface

*x*,*x*_{1}Distance along the surface

*y*Distance normal to surface

*β*Dimensionless pressure gradient parameter\( = - \frac{{x_1 }}{{\varrho u_e^2 }}\left( {\frac{{dp}}{{dx_1 }}} \right)\)

*δ*Boundary layer thickness at

*u*=0.9995*u*_{ e }*η*Transformed coordinate in

*y*direction*μ*Fluid molecular viscosity

*μ*_{t}Turbulent viscosity

*μ*_{eff}*μ+μ*_{ t }- µ
_{∞} Fluid molecular viscosity at ambient

*ν*Kinematic viscosity

*μ/ϱ**ϱ*Density

- ϱ
_{∞} Density at ambient

*τ*_{w}Wall shear stress

*τ*_{w,0}Wall shear stress at zero free stream turbulence

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