3D Non-Reflecting Boundary Condition
A non-reflecting boundary condition should allow all outgoing unsteady waves to exit the flow domain at the far-field boundary (inlet or outlet) without reflection. Unwanted, non-physical reflections of unsteady waves at the far-field can affect the accuracy of the simulation. The non-reflecting boundary condition is applied by decomposing the unsteady flow perturbations at the far-field into independent waves (unsteady aerodynamic modes). The velocity of each wave is determined and the waves are labeled as incoming or outgoing. The amplitude of the incoming waves are set to zero, except for those that may have been prescribed at the boundary. The outgoing waves are extrapolated out of the domain. It is important that each wave is correctly extrapolated out of the domain to ensure that reflections do not occur.
Unsteady Aerodynamic Modes at Far-field
The unsteady aerodynamic modes at the far-field can be determined analytically for some special cases: uniform 1D flow, and uniform 2D flow, and uniform 3D axial flow. However, this is not possible for non-uniform 3D swirling flows. The unsteady aerodynamic modes at the far-field for non-uniform 3D swirling flows can be determined numerically by performing an eigen analysis on the semi-discretized flow equations at the far-field.
The 3D unsteady Euler equations for a stationary frame of reference are used to model the unsteady flow at the far-field and can be written
where U is the flow solution and F, G, H are non-linear flux vectors. The far-field boundary is assumed to be normal to the x-axis and the spatial derivative in the x direction is linearized. The spatial derivatives in the y and z directions are discretized on a 2D mesh of the far-field boundary and then the discretization is linearized. The governing flow equation now reads
where A and D are constant coefficient matrices, dependent on the steady-state flow and U is the flow solution for the 2D far-field mesh. If the unsteady flow in the far-field is assumed to have the following form
where Us is the time-average flow and Um is the amplitude of the unsteady harmonic flow perturbation, then the unsteady flow equation can be rewritten
The flow equation is now in the form of an eigen problem and the unsteady aerodynamic modes can be determined.
An example of the unsteady aerodynamic modes as calculated by RPMTurbo's 3D non-reflecting boundary condition are shown below. The figures below show the unsteady pressure of acoustic modes in a 3D annulus of various radial orders with an interblade phase angle of 90 degrees.
Test Case: Standard Configuration 10
The figure below shows the aerodynamic damping versus inter-blade phase angle for Standard Configuration 10 due to the pitching mode at the subsonic flow condition. The two solutions employed different methods to determine the unsteady aerodynamic modes needed to apply the non-reflecting boundary condition. The red curve represents a solution calculated using the method of Giles to analytically determine the modes and the green curve represents a solution calculated using RPMTurbo's non-reflecting boundary condition which numerically determines the modes. There is an excellent agreement between the two solutions, this demonstrates that RPMTurbo's non-reflecting boundary condition is correctly determining the unsteady aerodynamic modes.
The figure below shows the axial wave numbers in the complex plane of the unsteady aerodynamic modes at the inlet for the subsonic condition for Standard Configuration 10 for an inter-blade phase angle 90 degrees. The flow Mach number is 0.7 and flow angle 55 degrees. The wave numbers were calculated using the method of Giles (analytical) and RPMTurbo's 3D non-reflecting boundary condition (numerical). For this case, all the acoustic modes are cut-off, that is, their wave numbers are complex. The wave numbers for the -2 and +2 harmonic orders in the pitch-wise direction are shown. The label "A" is for acoustic waves and label "V" is for entropy and vorticity waves. It can be seen that the numerical method is accurately resolving the main unsteady aerodynamic modes.
