Rapid Distortion Theory is used to perform calculations of unsteadily-forced initially isotropic turbulence so that the physics of such flows can be better understood. The results of these calculations show that there are three distinct regimes of physical behavior for the kind of turbulence that we are considering: (1) turbulence that is forced at a relatively low frequency in which the kinetic energy settles down to a constant value at later times, (2) turbulence that is forced at a slightly higher frequency in which the kinetic energy value oscillates for a time, but then increases dramatically, and (3) turbulence that is forced at a relatively high frequency in which the kinetic energy evolution exhibits a periodic behavior. To better understand the role of the rapid pressure-strain correlation, these results are also compared to Inertial Model results for the same set of forcing frequencies. The results of this comparison show that the rapid pressure plays a key role in determining the stability characteristics of unsteadily-forced turbulence. The evolution equation for kinetic energy is then used to propose a model that describes the behavior approximately in terms of a time lag between applied mean strain and the Reynolds stress. This model suggests that the different responses under the different frequencies of forcing correspond to different stress-strain time lags. Overall, then the results indicate that rapid pressure serves to create a time lag between applied stress and strain, and it is the extent of this time lag that causes turbulence to respond differently under various frequencies of forcing.

Measurements of the turbulent flow field through a highly staggered cascade propulsor configuration in the Virginia Tech cascade wind tunnel have been taken. Predictions of the same flow using Rapid Distortion Theory (RDT) were performed. Measurements and predictions were compared. The comparison was oriented to check the aptness of RDT in describing this kind of flow. Since this study represents the initial steps of a major project, the RDT model was kept simple. The non-penetration condition (blade blocking effect) was not modeled and the viscous effects were roughly accounted for. This work reveals the capabilities of RDT in predicting the development of turbulence convected through a highly staggered cascade propulsor configuration formed by non-symmetrical airfoils. This present study was possible thank to the support from the Office of Naval Research, in particular Candace Wark and Pat Purtell, under grant number NAG 00014-99-1-0230. / Master of Science

Rapid distortion theory (RDT) is applied to compressible ideal-gas turbulence subjected to homogeneous shear flow. The study examines the linear or rapid processes present in turbulence evolution. Specific areas of investigation include:(i) characterization of the multi-stage flow behavior,(ii) changing role of pressure in the three-regime evolution and (iii) influence of thermodynamic fluctuations on the different regimes. Preliminary investigations utilizing the more accurate Favre-averaged RDT approach show promise however, this approach requires careful validation and testing. In this study the Favre-averaged RDT approach is validated against Direct Numerical Simulation (DNS) and Reynolds-averaged RDT results. The three-stage growth of the flow field statistics is first confirmed. The three regime evolution of turbulence is then examined in three different timescales and the physics associated with each regime is discussed in depth. The changing role of pressure in compressible turbulence evolution shows three distinct stages. The physics of each stage is clearly explained. Next, the influence of initial velocity and thermodynamic fluctuations on the flow field are investigated. The evolution of turbulence is shown to be strongly dependent on the initial gradient Mach number and initial temperature fluctuations which tend to delay the onset of the second regime of evolution. The initial turbulent Mach number, which quantifies velocity fluctuations in the flow, influences turbulence evolution only weakly. Comparison of Reynolds-averaged RDT against Favre-averaged RDT for simulations of nonzero initial flow field fluctuations shows the higher fidelity of the latter approach.

The objective of the thesis is to characterize the linear and nonlinear aspects of inertial and pressure effects in turbulent flows. In the first part of the study, computations of Navier-Stokes and 3D Burgers equations are performed in the rapid distortion (RD) limit to analyze the inviscid linear processes in homogeneous turbulence. By contrasting the results of Navier- Stokes RD equations and Burgers RD equations, the effect of pressure can be isolated. The evolution of turbulent kinetic energy and anisotropy components and invariants are examined. In the second part of the thesis, the velocity gradient dynamics in turbulent flows are studied with the help of inviscid 3D Burgers equations and restricted Euler equations. The analytical asymptotic solutions of velocity gradient tensor are obtained for both Burgers and restricted Euler equations. Numerical computations are also performed to identify the stable solutions. The results are compared and contrasted to identify the effect of pressure on nonlinear velocity gradient dynamics. Of particular interest are the sign of the intermediate principle strain-rate and tendency of vorticity to align with the intermediate principle strain-rate. These aspects of velocity gradients provide valuable insight into the role of pressure in the energy cascade process.

Compressible ideal-gas turbulence subjected to homogeneous shear is investigated at the rapid distortion limit. Specific issues addressed are (i) the interaction between kinetic and internal energies and role of pressure-dilatation; (ii) the modifications to pressure-strain correlation and Reynolds stress anisotropy and (iii) the effect of the composition of velocity fluctuations (solenoidal vs. dilatational). Turbulence evolution is found to be strongly influenced by gradient Mach number, the initial solenoidal-to-dilatational ratio of the velocity field and the initial intensity of the thermodynamic fluctuations. The balance between the initial fluctuations in velocity and thermodynamic variables is also found to be very important. Any imbalance in the two fluctuating fields leads to high levels of pressure-dilatation and intense exchange. For a given initial condition, it is found that the interaction via the pressuredilatation term between the momentum and energy equations reaches a peak at an intermediate gradient Mach number. The energy exchange between internal and kinetic modes is negligible at very high or very low Mach number values due to lack of pressure dilatation. When present, the exchange exhibits oscillations even as the sum of the two energies evolves smoothly. The interaction between shear and solenoidal initial velocity field generates dilatational fluctuations; for some intermediate levels of shear Mach number dilatational fluctuations account for 20% of the total fluctuations. Similarly, the interaction between shear and initial dilatation produces solenoidal oscillations. Somewhat surprisingly, the generation of solenoidal fluctuations increases with gradient Mach number. Larger levels of pressure-strain correlation are seen with dilatational rather than solenoidal initial conditions. Anisotropies of solenoidal and dilatational components are investigated individually. The most interesting observation is that solenoidal and dilatational turbulence tend toward a one componential state but the energetic component is different in each case. As in incompressible shear flows, with solenoidal fluctuations, the streamwise (1,1) component of Reynolds stress is dominant. With dilatational fluctuations, the stream-normal (2,2) component is the strongest. Overall, the study yields valuable insight into the linear processes in high Mach number shear flows and identifies important closure modeling issues.

In this study, the behavior of pressure in the Rapid Distortion Limit, along with its concomitant modeling, are addressed. In the first part of the work, the role of pressure in the initiation, propagation and suppression of flow instabilities for quadratic flows is analyzed. The paradigm of analysis considers the Reynolds stress transport equations to govern the evolution of a dynamical system, in a state space composed of the Reynolds stress tensor components. This dynamical system is scrutinized via the identification of the invariant sets and the bifurcation analysis. The changing role of pressure in quadratic flows, viz. hyperbolic, shear and elliptic, is established mathematically and the underlying physics is explained. Along the maxim of "understanding before prediction", this allows for a deeper insight into the behavior of pressure, thus aiding in its modeling. The second part of this work deals with Rapid Pressure Strain Correlation modeling in earnest. Based on the comprehension developed in the preceding section, the classical pressure strain correlation modeling approaches are revisited. Their shortcomings, along with their successes, are articulated and explained, mathematically and from the viewpoint of the governing physics. Some of the salient issues addressed include, but are not limited to, the requisite nature of the model, viz. a linear or a nonlinear structure, the success of the extant models for hyperbolic flows, their inability to capture elliptic flows and the use of RDT simulations to validate models. Through this analysis, the schism between mathematical and physical guidelines and the engineering approach, at present, is substantiated. Subsequently, a model is developed that adheres to the classical modeling framework and shows excellent agreement with the RDT simulations. The performance of this model is compared to that of other nominations prevalent in engineering simulations. The work concludes with a summary, pertinent observations and recommendations for future research in the germane field.

Three different topics of turbulence research that cover modeling, theory and model computation categories are selected and studied in depth. In the first topic, "velocity gradient dynamics in turbulence" (modeling), the Lagrangian linear diffusion model that accounts for the viscous-effect is proposed to make the existing restricted-Euler velocity gradient dynamics model quantitatively useful. Results show good agreement with DNS data. In the second topic, "pressure-strain correlation in homogeneous anisotropic turbulence subject to rapid strain-dominated distortion" (theory), extensive rapid distortion calculation is performed for various anisotropic initial turbulence conditions in strain-dominated mean flows. The behavior of the rapid pressure-strain correlation is investigated and constraining criteria for the rapid pressure-strain correlation models are developed. In the last topic, "unsteady computation of turbulent flow past a square cylinder using partially-averaged Navier-Stokes method" (model computation), the basic philosophy of the PANS method is reviewed and a practical problem of flow past a square cylinder is computed for various levels of physical resolution. It is revealed that the PANS method can capture many important unsteady flow features at an affordable computational effort.

Three different topics of turbulence research that cover modeling, theory and model computation categories are selected and studied in depth. In the first topic, "velocity gradient dynamics in turbulence" (modeling), the Lagrangian linear diffusion model that accounts for the viscous-effect is proposed to make the existing restricted-Euler velocity gradient dynamics model quantitatively useful. Results show good agreement with DNS data. In the second topic, "pressure-strain correlation in homogeneous anisotropic turbulence subject to rapid strain-dominated distortion" (theory), extensive rapid distortion calculation is performed for various anisotropic initial turbulence conditions in strain-dominated mean flows. The behavior of the rapid pressure-strain correlation is investigated and constraining criteria for the rapid pressure-strain correlation models are developed. In the last topic, "unsteady computation of turbulent flow past a square cylinder using partially-averaged Navier-Stokes method" (model computation), the basic philosophy of the PANS method is reviewed and a practical problem of flow past a square cylinder is computed for various levels of physical resolution. It is revealed that the PANS method can capture many important unsteady flow features at an affordable computational effort.