Conditions for validity of mean flow stability analysis

We study the viscous spatial linear stability characteristics of the time-averaged flow in turbulent subsonic jets issuing from serrated (chevroned) nozzles, and compare them to analogous round jet results. Linear parabolized stability equations (PSE) are used in the calculations to account for the non-parallel base flow. By exploiting the symmetries of the mean flow due to the regular arrangement of serrations, we obtain a series of coupled two-dimensional PSE problems from the original three-dimensional problem. This reduces the solution cost and manifests the symmetries of the stability modes. In the parallel-flow linear stability theory (LST) calculations that are performed near the nozzle to initiate the PSE, we find that the serrated nozzle reduces the growth rates of the most unstable eigenmodes of the jet, but their phase speeds are approximately similar. We obtain encouraging validation of our linear PSE instability wave results vis-à-vis near-field hydrodynamic pressure data acquired on a phased microphone array in experiments, after filtering the latter with proper orthogonal decomposition (POD) to extract the energetically dominant coherent part. Additionally, a large-eddy simulation database of the same serrated jet is investigated, and its POD-filtered pressure field is found to compare favourably with the corresponding PSE solution within the jet plume. We conclude that the coherent hydrodynamic pressure fluctuations of jets from both round and serrated nozzles are reasonably consistent with the linear instability modes of the turbulent mean flow.

In the past stability of boundary layers on sharp-nosed cones with elliptical cross sections using linear stability theory and cross flow correlations was conducted. The calculations relied on mean flow solutions found using a Parabolized Navier-Stokes computer code. In this work, the mean flow of a 2:1 elliptic cone at a freestream Mach number of 7.95 is calculated using a full Navier-Stokes CFD solver. The stability of the boundary layer is then evaluated using Parabolized Stability Equations. Solutions from the current effort are compared to this previous work. In general the mean flows computed with Parabolized Navier-Stokes and the full Navier Stokes solutions compare well. Mean transition correlating parameters such as crossflow percentage and crossflow Reynolds numbers also agreed well between the two calculations. Wall-cooling reduced the crossflow Reynolds number by reducing the boundary layer thickness and percentage crossflow. I. Introduction HE Air Force is interested in the ability to predict the transition of boundary layers over three dimensional vehicles. There have been past studies conducted on such vehicles, but new tools and increases in computational performance of computers warrant the use of more physical models for high fidelity analysis. In this paper, past solutions by a Parabolized Navier-Stokes (PNS) solver combined with linear stability theory are compared to solutions using Full Navier-Stokes (FNS) and Parabolized Stability Equations (PSE). The objective of this work is to examine the issues involved with obtaining high-quality basic state solutions required for stability analysis of 3D bodies and also to back validate the previous PNS simulations. For this comparison an elliptic cone with a 2:1 eccentricity was used. The model half angle in the centerline plane was fixed at 7 degrees, with a model length of 1.016m, Figures 1 and 2 show the wind tunnel model of the geometry. The freestream of the flow around the cone has a Mach number of 7.95 and unit Reynolds number of 3.3x10 6 m -1 . Two wall conditions were considered, adiabatic and a cold wall with Tw/T0 = 0.42 (Tw =303 K). The 2:1 eccentricity sets up a spanwise or circumferential pressure gradient, which induces a flow component perpendicular to the longitudinal axis of a body.

This paper presents an experimental and theoretical investigation of high-Reynolds-number low-density reacting wakes near a hydrodynamic Hopf bifurcation. This configuration is applicable to the wake flows that are commonly used to stabilize flames in high-velocity flows. First, an experimental study is conducted to measure the limit-cycle oscillation of this reacting bluff-body wake. The experiment is repeated while independently varying the bluff-body lip velocity and the density ratio across the flame. In all cases, the wake exhibits a sinuous oscillation. Linear stability analysis is performed on the measured time-averaged velocity and density fields. In the first stage of this analysis, a local spatiotemporal stability analysis is performed on the measured time-averaged velocity and density fields. The stability analysis results are compared to the experimental measurement and demonstrate that the local stability analysis correctly captures the influence of the lip-velocity and density-ratio parameters on the sinuous mode. In the second stage of the analysis, the linear direct and adjoint global modes are estimated by combining the local results. The sensitivity of the eigenvalue to changes in intrinsic feedback mechanisms is found by combining the direct and adjoint global modes. This is referred to as the eigenvalue sensitivity throughout the paper for reasons of brevity. The predicted global mode frequency is consistently within 10 % of the measured value, and the linear global mode shape closely resembles the measured nonlinear oscillations. The adjoint global mode reveals that the oscillation is strongly sensitive to open-loop forcing in the shear layers. The eigenvalue sensitivity identifies a wavemaker in the recirculation zone of the wake. A parametric study shows that these regions change little when the density ratio and lip velocity change. In the third stage of the analysis, the stability analysis is repeated for the varicose hydrodynamic mode. Although not physically observed in this unforced flow, the varicose mode can lock into longitudinal acoustic waves and cause thermoacoustic oscillations to occur. The paper shows that the local stability analysis successfully predicts the global hydrodynamic stability characteristics of this flow and shows that experimental data can be post-processed with this method in order to identify the wavemaker regions and the regions that are most sensitive to external forcing, for example from acoustic waves.

Noise measurements of heated axisymmetric jets at a fixed supersonic acoustic Mach number indicate that the acoustic spectrum reduces when the temperature ratio increases. The “spectral quietening” effect has been observed both experimentally and computationally using large Eddy simulations. It was explained by Afsar, Goldstein, and Fagan [AIAA J. 49, 2522 (2011)] through the cancellation introduced by the enthalpy flux/momentum flux coupling term using the generalized acoustic analogy formulation. But the parallel flow assumption is known to give inaccurate predictions at high jet speeds. In this paper, we therefore extend the nonparallel flow asymptotic theory of Goldstein, Sescu, and Afsar [J. Fluid Mech. 695, 199 (2012)] for the vector Green’s function of the adjoint linearized Euler equations in the analogy. Using the steady Reynolds averaged Navier Stokes calculation for the jet mean flow, we find that the coupling term propagator is positive-definite and asymptotically subdominant at low frequencies corresponding to the peak jet noise when nonparallel flow effects are taken into account and self-consistent approximations for the turbulence structure are made. We then assess the validity of the non-parallel flow-based acoustic analogy model by computing the overall sound pressure level (OASPL) at various observation angles. Interpretation of the latter allows a more rational explanation of the quietening effect. In general, our noise predictions are in very good agreement with acoustic data beyond the peak frequency.

SummaryThe description of transitional boundary layer phenomena is a key issue for aeronautical technology development. For that purpose a certain code robustness and basic flow computational quality is required. The mean flow prediction tools have been thoroughly validated in industry in the past for their use within the local, linear stability theory. The application of nonlocal methods (PSE: Parabolized Stability Equations) is — however — in sight [9]. With such tools streamwise developments of the basic flow, nonlinear effects and receptivity issues can be taken into account; however more mean flow data with sufficient precision are needed.We consider here the effect of systematic streamwise changes of the outer boundary condition on the PSE results for a well defined model flow (2D wedge flow). It is shown that boundary layer and stability computations react very sensitive to small changes in the outer boundary condition.

Stability and structural sensitivity analysis of the turbulent flow in the narrow vaneless diffuser with mean flow method

Wave-packets in a reacting, imperfectly-expanded supersonic jet

Verification and application of a mean flow perturbation method for jet noise

A high lift LPT profile designated L2A is used as a test bed for studying the origin of endwall mixing loss and the role of vortical structures in loss development. It is shown analytically and experimentally that the mixing forces within the endwall wake can be decoupled into either mean flow or turbulent forces, and can be further classified as either reversible or irreversible. Among the irreversible forces, mean flow shear is negligible compared to turbulent shear, suggesting that turbulence dissipation is the dominant cause of loss generation. As a result, the mean flow components of the vortical structures do not generate significant mixing losses. Rather than mixing effects, the mean flow of the vortices cause the suction surface boundary layer to separate inside the passage, thereby generating the large low energy regions typical of endwall flows. Losses are generated as the low energy regions mix out. This vortex separation effect is demonstrated with an experiment using a profile fence and pressure surface modification near the endwall. The findings in this paper suggest that profile modifications near the endwall that suppress suction surface separation may provide loss reductions additive to those that weaken vortical structures, such as endwall contouring.

A newly developed method called parabolized stability equations (PSE) is used to analyze the streamwise evolution of instability waves in flows that are highly unidirectional, such as boundary layers and free shear layers. That the PSE analysis is able to fully account for the nature of diverging mean flow and to resolve the evolving instabilities without numerical difficulties as seen in linear stability theory has made PSE an alternative tool in dealing with the jet instabilities at relatively low computational cost compared with schemes based on direct numerical simulation or large eddy simulation

The dynamics of cervical cancer cells and immune cell interaction are modeled into a three-dimensional nonlinear differential equation system. It is a representation of the interaction model between one prey and two predators, with cervical cancer cells as prey and two types of immune cells as predators. The nature between these predators is a supportive relationship. Based on the mathematical model is obtained that there are six steady states in which three of them exist and the others do not exist for biological cases. Based on the analysis of stability near these steady states there has a bifurcation point about the steady states. By modifying some parameters value, there is a possibility to heal the cervical cancer sufferers.

The present investigation is mainly concerned with a stability analysis for the linearized Navier-Stokes equations for parallel and nonparallel mean flows over an infinite flat plate. The system of equations for parallel flows is presented. The system is viewed as a generalized Orr-Sommerfeld equation. Attention is given to an explicit criterion characterizing the case when the stability of all physically reasonable solutions is determined by the eigenvalues. The proof given in the investigation is applicable to both the generalized Orr-Somerfeld equations and the modified equations for nonparallel flow. The fact that the criterion is independent of the completeness or incompleteness of eigenfunctions is contrary to some expectations.

Stall cells are transverse cellular patterns that often appear on the suction side of airfoils near stalling conditions. Wind-tunnel experiments on a NACA4412 airfoil at Reynolds number ${Re}=3.5 \times 10^5$ show that they appear for angles of attack larger than $\alpha = 11.5^{\circ }\ (\pm 0.5^{\circ })$ . Their onset is further investigated based on global stability analyses of turbulent mean flows computed with the Reynolds-averaged Navier–Stokes (RANS) equations. Using the classical Spalart–Allmaras turbulence model and following Plante et al. (J. Fluid Mech., vol. 908, 2021, A16), we first show that a three-dimensional stationary mode becomes unstable for a critical angle of attack $\alpha = 15.5^{\circ }$ which is much larger than in the experiments. A data-consistent RANS model is then proposed to reinvestigate the onset of these stall cells. Through an adjoint-based data-assimilation approach, several corrections in the turbulence model equation are identified to minimize the differences between assimilated and reference mean-velocity fields, the latter reference field being extracted from direct numerical simulations. Linear stability analysis around the assimilated mean flow obtained with the best correction is performed first using a perturbed eddy-viscosity approach which requires the linearization of both RANS and turbulence model equations. The three-dimensional stationary mode becomes unstable for angle $\alpha = 11^{\circ }$ which is in significantly better agreement with the experimental results. The interest of this perturbed eddy-viscosity approach is demonstrated by comparing with results of two frozen eddy-viscosity approaches that neglect the perturbation of the eddy viscosity. Both approaches predict the primary destabilization of a higher-wavenumber mode which is not experimentally observed. Uncertainties in the stability results are quantified through a sensitivity analysis of the stall cell mode's eigenvalue with respect to residual mean-flow velocity errors. The impact of the correction field on the results of stability analysis is finally assessed.

: The HIFiRE Flight 5 payload was assessed in the NASA LaRC 20-Inch Mach 6 Air Tunnel. The primary objectives were to determine the boundary layer transition characteristics and the effectiveness trips. Boundary layer profiles were extracted from Navier-Stokes calculations of the flow field, and were used to assess trip heights in terms of typical trip correlating parameters. Leading-edge roughness was well-correlated with the R-bar-star parameter. This correlation was used to extrapolate wind tunnel data to full-scale flight conditions. Boundary layer transition was also studied using Parabolized Stability Equation (PSE) analysis. The stability analysis was conducted on the two symmetry planes of the vehicle. The leading edge planes are expected to follow second mode instability growth. The centerline cases experienced a more complicated growth that was mainly driven by second mode, but also showed higher modes. In particular for the lowest altitude case, the higher modes showed to contribute to the maximum N factor growth. This was caused by the complex boundary layer profile that was induced by a pair of vortices along the centerline of the vehicle.

The paper presents the local linear stability analysis of the double annular jets. The calculations show that the first absolutely unstable helical mode can be generated in the non-swirling annular jets by the back-flow in the central recirculation zone or sufficiently strong back-flow in the external recirculation zone. The influence of the back-flow magnitude on the frequency, growth rate and eigenfunctions of the first helical mode is discussed. The calculations are completed with an analysis of the influence of the swirl intensity in the internal and external jets on the characteristics of the first absolutely unstable helical mode.