An aircraft is cruising at a constant velocity and we are interested in
resolving the physics on the aircraft. The aircraft here is our object of interest.
In fluid dynamics, we encounter engineering applications where
the object of interest is translating through a fluid at a constant velocity. To resolve
the flow physics, we solve the flow field by changing the frame of reference from the fluid
to the moving object. In other words, we invert the problem and the object of interest becomes
stationary and the fluid is moving over this stationary body at a constant freestream velocity.
However, there are specific scenarios where moving objects are not undergoing pure translation at
a constant velocity. There might be scenarios where the object of interest is accelerating with
respect to the fluid. A good example is rotating machinery such as turbines and compressors where
the fluid is undergoing rotation. Another engineering scenario involves multiple bodies
each moving at different speeds. The motion of a drone is an excellent example.
While the rotors of the drone are rotating about their respective axes,
the drone itself can move in any direction. Under these conditions, the reference frame
transformation is complicated and in certain cases, next to impossible. In this lesson,
we will learn about these classes of problems and some methods employed to solve them.
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There are two ways to solve such a problem.
The first way is to solve the governing equations of fluid motion in the stationary reference frame
of the fluid and include the motion of the moving object into the analysis. The second way is to
recast the governing equations in the reference frame of the accelerating object. It is not always
possible to recast the governing equations in an accelerating frame of reference.
In case of using a stationary reference frame,
the regular generalized Navier-Stokes Equations are used to define the motion of the fluid.
Here, the no-slip boundary condition is applied to the solid-fluid interface.
Because the object of interest either accelerates or exhibits a complex motion,
a transient formulation, that is one varying in time, is applied in these scenarios.
The motion of the solid object is prescribed
as a rigid body motion and includes both rotational and translational components.
These equations have no analytical solutions and are solved using computational fluid dynamics.
In the second approach, we change this reference frame from stationary to that of the moving
object. This type of analysis is typically called moving reference frame analysis or MRF in short.
The biggest advantage of this approach is that
certain type of problems that are inherently transient in a stationary reference frame
show steady behavior when the reference frame is changed to the moving object.
For example, if the fluid velocity were measured at a point near the propeller of a plane,
we would obtain an unsteady velocity profile.
But when the same measurement point is on the propeller blade, the fluid velocity becomes
steady. Because steady problems are easier to solve and this approach does not require
any rigid body motion of parts, it is less tedious compared to the previous approach.
Although MRF analysis can be applied to both translational and rotational motions,
rotational problems involving turbomachinery are most suitable candidates for this.
When we model accelerating objects, additional force terms, such as the gravitational force,
show-up in the momentum equation of the Navier-Stokes formulation. These additional forces
show-up even in simple cases where the reference frame is rotating at a constant angular velocity.
The fluid feels like it is being pulled in different directions
during this rotation. Imagine what you feed while you are sitting inside a roller coaster.
Your body feels pulled in different directions based on the movement of the roller coaster.
When the roller coaster makes a sudden right turn, your body feels a pulled to the left.
This is similar to the additional forces that the fluid would be experiencing as well.
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In MRF approach, the fluid velocity can be defined either as absolute or relative velocity. While the
absolute velocity is defined with respect to the stationary or absolute frame of reference,
the relative velocity is measured with respect to the moving reference frame. These two variables
are related to the frame velocity using the simple velocity triangle rule for vector addition. In the
MRF approach, the governing equations are first written out in a stationary frame of reference.
The relationship between the stationary and moving frames is used to recast these equations into the
moving reference frame. Therefore, depending on the type of analysis, we have two different sets
of governing equations – the relative velocity formulation and the absolute velocity formulation.
The Navier-Stokes equations based on a stationary reference frame are transformed to a moving or
rotating reference frame to obtain the relative velocity formulation. The equations for the
absolute velocity formulation are obtained from the relative velocity formulation.
The momentum equation in both formulations have additional Coriolis and Centrifugal
acceleration terms as well. Moreover, the force balance in the relative velocity formulation
contains an additional term: the frame acceleration term. In totality, the equations
for the relative velocity formulation has both rotational as well as translational accelerations.
Even though these equations look different and contain different terms,
they are equivalent, and we can obtain identical solutions when equivalent boundary conditions
are applied to solve a flow problem. Additional scalar transport equations might be necessary
for certain problems and have the same form in both velocity formulations.
MRF is most suitable for solving the flow field in rotating frame applications such as turbines,
compressor, fans, propellers, and pumps. Pumps and turbines seem similar in their
engineering construction but are fundamentally different.
Pumps are used to inject extra work to the fluid thereby increasing its pressure,
while turbines use the energy from the fluid for power generation. In addition,
these modeling techniques can be employed to design mixing tanks where it is required to
figure out how to efficiently mix either two or more fluids or a fluid and some solids.
Other engineering applications where this type of rotating frame of reference analysis is
critical are rotating cavities, disk drives, and electric motors for developing electric vehicles.