Aerodynamics and Heat transfer computation in Turbo-machinery

In most cases,the flow field across a turbine blade consists of three domains. These are the Laminar, the Transitional and the Turbulent region. Refer to Fig.1.

The Laminar flow can be solved on the basis of the Navier-Stokes equation.

The Transition zone is very hard to solve for but is usually solved via non-dimensionalizing and statistical models.

The Turbulent zone can be solved for with the help of boundary layer equations combined with the Navier-Stokes Equation.

The major problem is when one had to define the boundary conditions and formulate the closure equations. These are to, often solved numerically as they have a lot of boundary conditions, a lot of irregularities and are highly non-linear.

There are certain things which are hence essential to the prediction of the Aerodynamics fields and heat transfer losses. These include:

Pressure Difference computations

Prediction of the boundary layers and their growth

Prediction of transition length and heat transfer

Prediction of temperature gradients within the blade

The prediction methods take into account:

Heat flux models- To carry out the computation of viscous layer and heat transfer, there is a need to develop constitutive equations for the temperature and velocity correlations. These equations are developed under heat flux models
Governing equations and boundary conditions- A brief review of the computational efforts for heat transfer prediction is covered in this section. These efforts constitute the application of boundary conditions and the selection of appropriate governing equations.

Using these models we can then discuss Computation for uncooled/convectively cooled blades Computation for Film-blades

Governing Equations & Boundary Conditions

Flow Boundary Conditions

The boundary conditions in the injection region, while considering film cooling are as follows:

$u(x,0)=u_{\rm {f}},\;\;v(x,0)=v_{\rm {f}},\;\;w(x,0)=w_{\rm {f}}$

$u(x,h)=u_{\rm {e}},\;\;v(x,h)=v_{\rm {e}},\;\;w(x,h)=w_{\rm {e}},\;\;h>\delta$

where u_f, v_f and w_f are streamwise, normal, and radial components of coolant velocity, respectively, at the location of any openings or holes for film-cooled or transportation-cooled blades. At any other location, it is considered to be zero.

Thermal Boundary Conditions

The thermal wall boundary conditions are as follows:

$c_{\rm {p}}{\frac {\partial T(x,0)}{\partial x}}=-q_{\rm {w}}(x){PR \over \mu }$

$T(x,0)=T_{\rm {w}}(x)$

The inlet boundary conditions will be:

$u(0,y)=u(y),\;\;v(0,y)=v(y),\;\;w(0,y)=w(y),\;\;T(0,y)=T(y)$

The k-ε model has been successfully used If the model used is k-ε, the free-stream turbulence intensity will be specified, and low-Reynolds number k-ε models will be used to capture the transition location.

Boundary conditions for k-ε equations :

$x=0,\;\;y=0,\;\;k_{\rm {w}}={\frac {\partial \epsilon }{\partial y}}_{\rm {w}}=0$

$x=x,\;\;y>\delta ,\;\;u_{\rm {e}}{\frac {\partial k_{\rm {e}}}{\partial x}}=-\epsilon _{\rm {e}};\;\;u_{\rm {e}}{\frac {\partial \epsilon _{\rm {e}}}{\partial x}}=-C_{\rm {\epsilon e}}{\frac {\epsilon _{\rm {e}}^{2}}{k_{\rm {e}}}}$

Initial condition should also be prescribed to start the solution, including free-stream turbulence intensity. In that case, u, k, & ε necessarily needs to be prescribed. The velocity profile is prescribed either from experimental data or from a known analytical solution.
While computing the flow field in the vicinity of blade leading edge, it should be noted that the free stream turbulence intensity changes substantially in the vicinity of the stagnation point. This is because of the large velocity gradient due to flow deceleration which is an inviscid effect. The k-ε model is not capable of capturing turbulence production controlled by normal components of Reynolds stress. When both the full Reynolds stress equations and the k-ε equations was used, the prediction of k profile from the k-ε model was poor but the predictions from the full Reynolds stress equations was in good agreement with the data from a cylinder. While on the other hand, downstream of the leading edge and in the boundary layer region, the k-ε model performed well.