Date of Award

Spring 2014

Document Type


Degree Name

Doctor of Philosophy (PhD)


Mechanical & Aerospace Engineering


Aerospace Engineering

Committee Director

Robert L. Ash

Committee Member

Miltiadis Kotinis

Committee Member

Li-Shi Luo

Committee Member

Shizhi Qian


This research has utilized a state variable model for unsteady two dimensional axial vortex flows experiencing non-equilibrium pressure gradient forces. The model was developed successfully using perturbed radial and azimuthal momentum equations and a pressure Poisson's equations. Three main regions of the axial vortex flow were highlighted in this study including: a laminar core region, a non-equilibrium pressure envelope, and an outer potential vortex.

Linear stability theory was utilized to formulate the model and the perturbation functions were assumed to be of the Fourier type. The flow parameters considered were the Reynolds numbers, ranging between 6,000 and 14,000, and a new non-equilibrium swirl parameter, Np determining the area of significant non-equilibrium pressure forces. Two other state variable parameters were imposed-complex frequency and associated azimuthal mode number. Perturbation outputs included primary Reynolds stress, radial and azimuthal velocity amplitudes, and radial pressure gradient amplitudes.

Maximum perturbation growth occurred inside the non-equilibrium pressure zone between one and five core radii from the rotational axis, while the inner core remained laminar. The maximum amplitudes and critical radii depended on the four physical and state variable parameters. Increases in Np resulted in lower perturbation pressure gradient amplitudes, moving the critical radius closer to the vortex core, and expanding the non-equilibrium pressure zone. Increasing the frequency resulted in steady increases in the perturbation amplitudes until a particular dimensionless frequency was reached. Beyond that frequency, additional perturbation growth was insignificant or the amplitude decayed because of a high damping factor. Two types of azimuthal modes were unstable, the ±½ modes inside the non-equilibrium pressure zone, causing the pressure gradient amplitudes to peak even though the azimuthal velocity profile remained stable, and ± 1 helical modes associated with growing pressure gradient amplitudes in the outer potential region. The symmetrical azimuthal modes were globally stable.

The state variable model was stable numerically inside the non-equilibrium pressure zone, even though the perturbation amplitudes exhibited instability. Inside that region, unstable pressure eigenmodes were detected in the form of relaxation Reynolds stresses in response to perturbations in the flow. The width of the non-equilibrium pressure zone was again determined using eigenmode plots for different Np. The positive real parts of the unstable modes were slightly larger in the outer potential region causing slow growth profiles.

The current vortex state variable model can be utilized to explore the development of small perturbations in the non-equilibrium zone as the flow becomes turbulent, via a bifurcation cycle study where coherent structures can be identified. Experimental verification using hot-wire probes is needed to validate the theory and adjust the state variable model parameters. A side effect of the non-equilibrium pressure model for this vortical flow is the likely sound propagation causing small density perturbations that are balanced by the contracted pressure gradient-velocity tensor terms in the pressure relaxation equations. This non-equilibrium balance process appears to vanish in the outer potential vortex region.