A three-dimensional numerical study of flow characteristics in strongly curved channel bends with different side slopes

Abstract

The influence of channel side slope on flow in strongly curved channel bends is studied numerically. The performances of five different turbulence models are investigated. Comparison to experimental measurements demonstrates that the fully 3D numerical model can reliably simulate a channel bend flow field. Among the tested turbulence models, the realizable k − ε model performed best. The present study also demonstrates that the realizable k − ε model can satisfactorily predict smaller flow features in bend flow, such as the outer-bank circulation cell. The validated model is employed to carry out additional computations for channel bends with different side slopes. It is found that the number, position, and strength of secondary flow cells varies with the channel side slope, with corresponding influence on flow distribution and flow vorticity.

Appendix: turbulence models

The standard k − ε model:

$$\frac{\partial k}{\partial t} + \frac{{\partial ku_{i} }}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left( {D_{keff} \frac{\partial k}{{\partial x_{i} }}} \right) = G - \varepsilon$$

(23)

$$\frac{\partial \varepsilon }{\partial t} + \frac{{\partial \varepsilon u_{i} }}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left( {D_{\varepsilon eff} \frac{\partial \varepsilon }{{\partial x_{i} }}} \right) = c_{1\varepsilon } \frac{\varepsilon }{k}G - c_{2\varepsilon } \frac{{\varepsilon^{2} }}{k}$$

(24)

$$D_{keff} = \nu_{t} + \nu$$

(25)

$$D_{\varepsilon eff} = \frac{{\nu_{t} }}{{\sigma_{\varepsilon } }} + \nu$$

(26)

$$\nu_{t} = c_{\mu } \frac{{k^{2} }}{\varepsilon }$$

(27)

$$G = 2\nu_{t} S_{ij} S_{ij}$$

(28)

$$S_{ij} = \frac{1}{2}\left( {\frac{{\partial u_{j} }}{{\partial x_{i} }} + \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right)$$

(29)

where σ_ε, c_1ε, c_2ε, c_μ are model constants equal to 1.3, 1.44, 1.92, and 0.09, respectively.

The RNG k − ε model:

$$\frac{\partial k}{\partial t} + \frac{{\partial ku_{i} }}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left( {D_{keff} \frac{\partial k}{{\partial x_{i} }}} \right) = G - \varepsilon$$

(30)

$$\frac{\partial \varepsilon }{\partial t} + \frac{{\partial \varepsilon u_{i} }}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left( {D_{\varepsilon eff} \frac{\partial \varepsilon }{{\partial x_{i} }}} \right) = \left( {c_{1\varepsilon } - R_{\varepsilon } } \right)\frac{\varepsilon }{k}G - c_{2\varepsilon } \frac{{\varepsilon^{2} }}{k}$$

(31)

$$D_{keff} = \frac{{\nu_{t} }}{{\sigma_{k} }} + \nu$$

(32)

$$D_{\varepsilon eff} = \frac{{\nu_{t} }}{{\sigma_{\varepsilon } }} + \nu$$

(33)

$$\nu_{t} = c_{\mu } \frac{{k^{2} }}{\varepsilon }$$

(34)

$$G = \nu_{t} S_{2}$$

(35)

$$S_{2} = 2S_{ij} S_{ij}$$

(36)

$$S_{ij} = \frac{1}{2}\left( {\frac{{\partial u_{j} }}{{\partial x_{i} }} + \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right)$$

(37)

$$R_{\varepsilon } = \frac{{\eta \left( {1 - {\eta \mathord{\left/ {\vphantom {\eta {\eta_{0} }}} \right. \kern-0pt} {\eta_{0} }}} \right)}}{{1 + \beta \eta^{3} }}$$

(38)

$$\eta = \sqrt {S_{2} } \frac{k}{\varepsilon }$$

(39)

where σ_k, σ_ε, c_1ε, c_2ε, c_μ, η₀, and β are model constants equal to 0.71942, 0.71942, 1.42, 1.68, 0.0845, and 0.012 respectively.

The k − ω model:

$$\frac{\partial k}{\partial t} + \frac{{\partial ku_{i} }}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left( {D_{keff} \frac{\partial k}{{\partial x_{i} }}} \right) = G - c_{\mu } k\omega$$

(40)

$$\frac{\partial \omega }{\partial t} + \frac{{\partial \omega u_{i} }}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left( {D_{\omega eff} \frac{\partial \omega }{{\partial x_{i} }}} \right) = \alpha_{k\omega } G\frac{\omega }{k} - \beta \omega^{2}$$

(41)

$$D_{keff} = \alpha_{k} \nu_{t} + \nu$$

(42)

$$D_{\omega eff} = \alpha_{\omega } \nu_{t} + \nu$$

(43)

$$\nu_{t} = \frac{k}{\omega }$$

(44)

$$G = 2\nu_{t} S_{ij} S_{ij}$$

(45)

$$S_{ij} = \frac{1}{2}\left( {\frac{{\partial u_{j} }}{{\partial x_{i} }} + \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right)$$

(46)

where ω is the specific dissipation rate; c_μ, α_k, α_ω, α_kω, β are model constants equal to 0.09, 0.5, 0.5, 0.52 and 0.072 respectively.

The k − ω SST model:

$$\frac{\partial k}{\partial t} + \frac{{\partial ku_{i} }}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left( {D_{keffi} \frac{\partial k}{{\partial x_{i} }}} \right) = \hbox{min} \left( {G,c_{1} \beta^{*} k\omega } \right) - \beta^{*} k\omega$$

(47)

$$\frac{\partial \omega }{\partial t} + \frac{{\partial \omega u_{i} }}{{\partial x_{i} }} - \frac{\partial }{{\partial x_{i} }}\left( {D_{\omega eff} \frac{\partial \omega }{{\partial x_{i} }}} \right) = \gamma_{i} \hbox{min} \left[ {S_{2} ,\frac{{c_{1} }}{{a_{1} }}\beta^{*} \omega \hbox{max} \left( {a_{1} \omega ,b_{1} F_{23} \sqrt {S_{2} } } \right)} \right] - \beta \omega^{2} - \left( {F_{1} - 1} \right)CD_{k\omega }$$

(48)

$$D_{keff} = \alpha_{ki} \nu_{t} + \nu$$

(49)

$$D_{\omega effi} = \alpha_{\omega i} \nu_{t} + \nu$$

(50)

$$\nu_{t} = \frac{{a_{1} k}}{{\hbox{max} \left( {a_{1} \omega ,b_{1} F_{23} \sqrt {S_{2} } } \right)}}$$

(51)

$$S_{2} = 2S_{ij} S_{ij}$$

(52)

$$S_{ij} = \frac{1}{2}\left( {\frac{{\partial u_{j} }}{{\partial x_{i} }} + \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right)$$

(53)

$$G = 2\nu_{t} S_{ij} S_{ij}$$

(54)

$$CD_{k\omega } = 2\alpha_{\omega 2} \frac{\partial k}{{\partial x_{i} }}\frac{\partial \omega }{{\partial x_{i} }}\frac{1}{\omega }$$

(55)

where the subscript i can be either 1 or 2, depending on the blending functions in the model. α_k1, α_k2, α_ω1, α_ω2, α₁, b₁, c₁, β*, γ₁, γ₂, are model constants equal to 0.85, 1, 0.5, 0.856, 0.31, 1, 10, 0.09, 5/9, 0.44. F₁ and F₂₃ are blending functions.