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10.2.2 Equations for a Rotating Reference Frame

Consider a coordinate system which is rotating steadily with angular velocity $\vec{\omega}$ relative to a stationary (inertial) reference frame, as illustrated in Figure  10.2.1. The origin of the rotating system is located by a position vector $\vec{r_0}$.

Figure 10.2.1: Stationary and Rotating Reference Frames
figure

The axis of rotation is defined by a unit direction vector $\hat{a}$ such that


 \vec{\omega} = \omega \hat{a} (10.2-1)

The computational domain for the CFD problem is defined with respect to the rotating frame such that an arbitrary point in the CFD domain is located by a position vector $\vec{r}$ from the origin of the rotating frame.

The fluid velocities can be transformed from the stationary frame to the rotating frame using the following relation:


 \vec{v}_r = \vec{v} - \vec{u}_r (10.2-2)

where

 \vec{u}_r = \vec{\omega} \times \vec{r} (10.2-3)

In the above, $\vec{v}_r$ is the relative velocity (the velocity viewed from the rotating frame), $\vec{v}$ is the absolute velocity (the velocity viewed from the stationary frame), and $\vec{u}_r$ is the "whirl" velocity (the velocity due to the moving frame).

When the equations of motion are solved in the rotating reference frame, the acceleration of the fluid is augmented by additional terms that appear in the momentum equations [ 24]. Moreover, the equations can be formulated in two different ways:

The exact forms of the governing equations for these two formulations will be provided in the sections below. It can be noted here that FLUENT's pressure-based solvers provide the option to use either of these two formulations, whereas the density-based solvers always use the absolute velocity formulation. The advantages of each velocity formulation are discussed in Section  10.7.1.



Relative Velocity Formulation


For the relative velocity formulation, the governing equations of fluid flow for a steadily rotating frame can be written as follows:

Conservation of mass:


 \frac{\partial \rho}{\partial t} + \nabla \cdot \rho {\vec v}_r = 0 (10.2-4)

Conservation of momentum:

 \frac{\partial}{\partial t} (\rho {\vec v}_r) + \nabla \cdot... ...}) = - \nabla p + \nabla \overline{\overline{\tau}}_r+ \vec{F} (10.2-5)

Conservation of energy:

 \frac{\partial}{\partial t} (\rho {E}_r) + \nabla \cdot (\rh... ...nabla T + \overline{\overline{\tau}}_r \cdot {\vec v}_r) + S_h (10.2-6)

The momentum equation contains two additional acceleration terms: the Coriolis acceleration ( $2 {\vec \omega} \times {\vec v}_r$), and the centripetal acceleration ( ${\vec \omega} \times {\vec \omega} \times {\vec r}$). In addition, the viscous stress ( $\overline{\overline{\tau}}_r$) is identical to Equation  9.2-4 except that relative velocity derivatives are used. The energy equation is written in terms of the relative internal energy ( $E_r$) and the relative total enthalpy ( $H_r$), also known as the rothalpy. These variables are defined as:


 E_r = h - \frac{p}{\rho} + \frac{1}{2}({v_r}^2 - {u_r}^2) (10.2-7)


 H_r = E_r + \frac{p}{\rho} (10.2-8)



Absolute Velocity Formulation


For the absolute velocity formulation, the governing equations of fluid flow for a steadily rotating frame can be written as follows:

Conservation of mass:


 \frac{\partial \rho}{\partial t} + \nabla \cdot \rho {\vec v}_r = 0 (10.2-9)

Conservation of momentum:


 \frac{\partial}{\partial t} \rho {\vec v} + \nabla \cdot (\r... ...v}) = - \nabla p + \nabla \overline{\overline{\tau}} + \vec{F} (10.2-10)

Conservation of energy:


 \frac{\partial}{\partial t} \rho E + \nabla \cdot (\rho {\ve... ...(k \nabla T + \overline{\overline{\tau}} \cdot {\vec v}) + S_h (10.2-11)

In this formulation, the Coriolis and centripetal accelerations can be collapsed into a single term ( ${\vec \omega} \times {\vec v}$).


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