This slide illustrates the two approaches. Notice that in the moving reference frame approach, the axes are attached to the moving domain and are rotate with it. Therefore the axes always remain in the same position with respect to the moving domain. Again, because the frame is accelerating, additional terms are added to the equations of motion of the fluid. In contrast, the moving/deforming domain approach permits a time-dependent deformation of the computational domain, which requires tracking the mesh with respect to a fixed frame of reference.
This slide illustrates the difference between a single blade passage case on the left, which can be solved using the SRF approach by assuming rotational periodic boundary conditions, versus a multiple domain model on the right, which consists of a blower wheel and casing. Note the interface which separates the two zones. The interface can be treated in a number of ways, each with its own inherent assumptions and limitations.
Let us now briefly examine the forms of the Navier-Stokes equations appropriate for moving reference frames. Basically, the transformation from the stationary frame to the moving frame can be done mathematically in two different ways. In the first form, the momentum equations are expressed in terms of the velocity relative to the moving frame. This form is known as the Relative Velocity Formulation. The second form utilizes the velocities referred to the absolute frame of reference in the momentum equations, and is known as the Absolute Velocity Formulation. In both cases, additional acceleration terms result from the transformations.
It is important to understand the relationship between the velocity viewed from the stationary and relative frames. This relationship is often called the “velocity triangle,” and can be expressed by the simple vector equation: Here, V is the velocity in the stationary or absolute frame ,W is the velocity viewed from the moving frame, and U is the frame velocity. For a rotating reference frame, U is simply Notice from the picture of the moving turbine blade that a flow approaching the blade in the x direction appears to be moving with a positive angle due to the downward motion of the blade.
Let’s now compare the two formulations. To do this we will simply consider the x momentum equation. For the relative velocity formulation, we observe that the x component of the relative velocity is being solver for, and that the convecting velocity is the relative velocity. It should be noted here that in a moving frame, scalars are convected along relative streamlines and we will employ a similar kind of convection term for such scalar equations as turbulence kinetic energy and dissipation rate, and species transport. Also note the presence of additional acceleration terms, which have been placed on the right hand side. The first acceleration is called the Coriolis acceleration (2 x W), and has the property of acting perpendicular to the direction of motion of the fluid and to the axis of rotation. The second term is called the Centripetal acceleration, which acts in the radial direction. Both of these accelerations act as momentum source terms in the momentum equations, and can lead to difficulties if the rotational speed is very large. For the absolute velocity formulation, we now solve for the x component of the absolute velocity. Note that the relative velocity is still the convecting velocity, and that the accerlation terms can be collapsed into a single term ( x V), due to the fact that the Corilolis acceleration becomes ( x W). It should be stressed that these equations are equivalent representations of the flow in the moving frame, and that either form can be used for CFD. However, since the numerical errors are different for each form, on a finite grid there may be an advantage in using one form over another. We will return to this point shortly.