Indiana University-Purdue University Indianapolis (IUPUI) / We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_)$ for all values of the real parameter $\gamma$.

Indiana University-Purdue University Indianapolis (IUPUI) / We numerically investigate the underlying physics of two peculiar phenomena, which are axis-switching and saddle-back velocity profile, in both laminar and turbulent rectangular jets using lattice Boltzmann method (LBM). Previously developed computation protocols based on single-relaxation-time (SRT) and multiple-relaxation-time (MRT) lattice Boltzmann equations are utilized to perform direct numerical simulation (DNS) and large eddy simulation (LES) respectively. In the first study, we systematically study the axis-switching behavior in low aspect-ratio (AR), defined as the ratio of width over height, laminar rectangular jets with <italic>AR=1</italic> (square jet), 1.5, 2, 2.5, and 3. Focuses are on various flow properties on transverse planes downstream to investigate the correlation between the streamwise velocity and secondary flow. Three distinct regions of jet development are identified in all the five jets. The <italic>45&deg</italic> and <italic>90&deg</italic> axis-switching occur in characteristic decay (CD) region consecutively at the early and late stage. The half-width contour (HWC) reveals that <italic>45&deg</italic> axis-switching is mainly contributed by the corner effect, whereas the aspect-ratio (elliptic) feature affects the shape of the jet when <italic>45&deg</italic> axis-switching occurs. The close examinations of flow pattern and vorticity contour, as well as the correlation between streamwise velocity and vorticity, indicate that <italic>90&deg</italic> axis-switching results from boundary effect. Specific flow patterns for <italic>45&deg</italic> and <italic>90&deg</italic> axis-switching reveal the mechanism of the two types of axis-switching respectively. In the second study we develop an algorithm to generate a turbulent velocity field for the boundary condition at jet inlet. The turbulent velocity field satisfies incompressible continuity equation with prescribed energy spectrum in wave space. Application study of the turbulent velocity profile is on two turbulent jets with <italic>Re=25900</italic>. In the jets with <italic>AR=1.5</italic>, axis-switching phenomenon driven by the turbulent inlet velocity is more profound and in better agreement with experimental examination over the laminar counterpart. Characteristic jet development driven by both laminar and turbulent inlet velocity profile in square jet (<italic>AR=1</italic>) is also examined. Overall agreement of selected jet features is good, while quantitative match for the turbulence intensity profiles is yet to be obtained in future study. In the third study, we analyze the saddle-back velocity profile phenomenon in turbulent rectangular jets with AR ranging from 2 to 6 driven by the developed turbulent inlet velocity profiles with different turbulence intensity (<italic>I</italic>). Saddle-back velocity profile is observed in all jets. It has been noted that the saddle-back's peak velocities are resulted from the local minimum mixing intensity. Peak-center difference <italic>&Delta_pc</italic> and profound saddle-back (PSB) range are defined to quantify the saddle-back level and the effects of AR and <italic>I</italic> on saddle-back profile. It is found that saddle-back is more profound with larger AR or slimmer rectangular jets, while its relation with <italic>I</italic> is to be further determined.

Indiana University-Purdue University Indianapolis (IUPUI) / This study seeks to improve the understanding of inlet conditions of a large rotor-stator cavity in a turbofan engine, often referred to as the drive cone cavity (DCC). The inlet flow is better understood through a higher fidelity computational fluid dynamics (CFD) modeling of the inlet to the cavity, and a coupled finite element (FE) thermal to CFD fluid analysis of the cavity in order to accurately predict engine component temperatures. Accurately predicting temperature distribution in the cavity is important because temperatures directly affect the material properties including Young's modulus, yield strength, fatigue strength, creep properties. All of these properties directly affect the life of critical engine components. In addition, temperatures cause thermal expansion which changes clearances and in turn affects engine efficiency. The DCC is fed from the last stage of the high pressure compressor. One of its primary functions is to purge the air over the rotor wall to prevent it from overheating. Aero-thermal conditions within the DCC cavity are particularly challenging to predict due to the complex air flow and high heat transfer in the rotating component. Thus, in order to accurately predict metal temperatures a two-way coupled CFD-FE analysis is needed. Historically, when the cavity airflow is modeled for engine design purposes, the inlet condition has been over-simplified for the CFD analysis which impacts the results, particularly in the region around the compressor disc rim. The inlet is typically simplified by circumferentially averaging the velocity field at the inlet to the cavity which removes the effect of pressure wakes from the upstream rotor blades. The way in which these non-axisymmetric flow characteristics affect metal temperatures is not well understood. In addition, a constant air temperature scaled from a previous analysis is used as the simplified cavity inlet air temperature. Therefore, the objectives of this study are: (a) model the DCC cavity with a more physically representative inlet condition while coupling the solid thermal analysis and compressible air flow analysis that includes the fluid velocity, pressure, and temperature fields; (b) run a coupled analysis whose boundary conditions come from computational models, rather than thermocouple data; (c) validate the model using available experimental data; and (d) based on the validation, determine if the model can be used to predict air inlet and metal temperatures for new engine geometries. Verification with experimental results showed that the coupled analysis with the 3D no-bolt CFD model with predictive boundary conditions, over-predicted the HP6 offtake temperature by 16k. The maximum error was an over-prediction of 50k while the average error was 17k. The predictive model with 3D bolts also predicted cavity temperatures with an average error of 17k. For the two CFD models with predicted boundary conditions, the case without bolts performed better than the case with bolts. This is due to the flow errors caused by placing stationary bolts in a rotating reference frame. Therefore it is recommended that this type of analysis only be attempted for drive cone cavities with no bolts or shielded bolts.