Cfd modelling of h-darrieus vertical axis wind turbine

 
Eq (3.18) 
where φ is a fluid property and Γ is the diffusion coefficient. 
By integration of equation 3.18 over a control volume, the equation becomes:
∫
𝜕(𝜌𝜑)
𝜕𝑡
𝑑𝑉 +
𝐶𝑉
∫ 𝑑𝑖𝑣(𝜌𝜑𝒖)𝑑𝑉
𝐶𝑉
= ∫ 𝑑𝑖𝑣(Γ grad φ)𝑑𝑣
𝐶𝑉
+ ∫ 𝑆
𝜑
𝑑𝑣
𝐶𝑉
 
Eq (3.19) 
Figure 9: A typical 2D finite volume grid (rectangular grid) [88] 

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25 
By applying Gauss' divergence theorem, equation 3.19 can be written as follows:
𝜕
𝜕𝑡
(∫ 𝜌𝜑𝑑𝑣
𝐶𝑉
) + ∫ 𝒏 ∙ (ρφ𝐮)𝑑𝐴 =
𝐴
∫ 𝒏 ∙ (Γ grad φ)𝑑𝐴
𝐴
+ ∫ 𝑆
𝜑
𝑑𝑣
𝐶𝑉
 
Eq (3.20) 
The change rating term of 3.19 for the steady state problems is equal to zero, therefore, 
∫ 𝒏 ∙ (ρφ𝐮)𝑑𝐴 =
𝐴
∫ 𝒏 ∙ (Γ grad φ)𝑑𝐴
𝐴
+ ∫ 𝑆
𝜑
𝑑𝑣
𝐶𝑉
 
Eq (3.21) 
And for the transient problems, the equation will be:
∫
𝜕
𝜕𝑥
(∫ (𝜌𝜑)𝑑𝑣
𝐶𝑉
) 𝑑𝑡 + ∫ ∫ 𝒏 ∙ (ρφ𝐮)𝑑𝐴
𝐴
𝑑𝑡
𝛥𝑡
𝛥𝑡
= ∫ ∫ 𝒏 ∙ ( Γ
𝜑
grad φ)𝑑𝐴
𝐴
𝑑𝑡 + ∫ 𝑆
𝜑
𝑑𝑣𝑑𝑡
𝛥𝑡
𝛥𝑡
 
Eq (3.22) 
 
3.2.3 Turbulence Modelling 
The turbulence flow is a highly erratic flow of fluid (gas or liquid) where the various properties 
like velocity continuously fluctuates in different directions. The equations to describe such kind 
of flow is very complex, and they are nonlinear, and time-dependent. It is very complicated to 
use the Navier-Stokes equations for these flows where the equations are three dimensional. As a 
result, for calculating industrial flows, most commonly, the Reynolds Averaged Navier-Stokes 
Equation (RANS) is used. The RANS equations are obtained by taking the mean of different 
properties like mean pressure, mean velocities, mean stresses etc. and plugging them into the 
Navier-stokes equations. Hence, the equations 3.15 to 3.17 become 
𝜕(𝜌𝑈)
𝜕𝑡
+ div(ρU𝐔)
= −
𝜕𝑃
𝜕𝑥
+ 𝑑𝑖𝑣 (𝜇 𝑔𝑟𝑎𝑑 𝑈)
+ [−
𝜕(𝜌𝑢
ʹ2
)
̅̅̅̅̅
𝜕𝑥
−
𝜕(𝜌𝑢
ʹ
𝑣
ʹ
)
̅̅̅̅̅̅
𝜕𝑦
−
𝜕(𝜌𝑢
ʹ
𝑤
ʹ
)
̅̅̅̅̅̅̅
𝜕𝑧
] +𝑆
𝑀𝑥