Global Instability Analysis Research Articles

A zero-net-mass-flux wake stabilization method for blunt bodies via global linear instability

A rectangular cylinder, with an aspect ratio of 5, is a widely used bluff body in engineering practice. It undergoes intricate dynamical behavior in response to minute alterations in the flow angle of attack (α). These modifications invariably precipitate the failure of wake control for classical flow control methods with various α values. In this study, global linear instability, adjoint method, and sensitivity analysis are employed to identify the optimal position for flow control. It is found that the sensitive region gradually transitions from the leeward side to the downwind side of the model as α and Reynolds number (Re) increase. So, we set up airflow orifices for flow control in both positions. Jet flow control on the leeward side effectively inhibits vortex shedding (α ≤ 2°). High-order dynamic mode decomposition is employed to reveal the inherent mechanism of control. Suction control on the downside effectively mitigates the shear layer separation phenomenon induced by the altered spatial structure associated with higher α. A novel zero-net-mass-flux wake control, bionics-based breathe-valve control (BVC), is proposed to optimize the control effect. BVC is applicable for various α and Re, with optimal effectiveness achievable through jet velocity adjustments. The prediction-control approach in this investigation provides a targeted method to mitigate flow-induced vibration.

Three-dimensional global instability analysis for high-speed boundary layer flow

Herein, a numerical approach for the three-dimensional global instability analysis of high-speed boundary layer is developed. The approach is based on the Krylov–Schur method for large-scale eigenproblems. A shift-and-invert spectral transformation is employed to accelerate the convergence for eigenvalues of interest. Iterative linear solver is adopted in the transformation to reduce memory usage by avoiding the fill-in problem of direct solver and the explicit construction of matrices. A Robin condition based on BiGlobal instability analysis is imposed at the inflow boundary, and a non-reflecting condition derived from inviscid flow is used for the far-field boundary. The approach is validated in a Blasius boundary layer. According to a comparison between the performances of several algorithms in this case, the conjugate gradient squared method with no preconditioning is chosen as the linear solver. The results provided by three-dimensional global instability analysis consist with those from linear stability theory and BiGlobal instability analysis. A further demonstration of our approach is provided for a three-dimensional roughness-disturbed boundary layer. The fluctuations extracted from direct numerical simulation results illustrate that the symmetric component is the dominant unstable mode in roughness wake, as it grows twice as fast as the antisymmetric component. Two eigenvalues are obtained from the three-dimensional global instability analysis, and these represent the symmetric and antisymmetric unstable modes. The wavelengths and growth rates of both modes delivered by instability analysis agree well with the direct numerical simulation results, and the profiles of disturbance amplitude match the features of the varicose and sinuous modes.

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Nonlinear dynamic modeling and global instability analyses of planetary gear trains considering multi-state engagement and tooth-contact temperature effects

Stability and bifurcation of annular electro-thermo-convection

Abstract We numerically investigated the global linear instability and bifurcations in electro-thermo-convection (ETC) of a dielectric liquid confined in a two-dimensional (2-D) concentric annulus subjected to a strong unipolar injection. Seven kinds of solutions exist in this ETC system due to the complex bifurcations, i.e. saddle-node, subcritical and supercritical Hopf bifurcations. These bifurcation routes constitute at most four solution branches. Global linear instability analysis and energy analysis were conducted to explain the instability mechanism and transition of different solutions and to predict the local instability regions. The linearized lattice Boltzmann method (LLBM) for global linear instability analysis, first proposed by Pérez et al. (Theor. Comput. Fluid Dyn., vol. 31, 2017, pp. 643–664) to analyse incompressible flows, was extended here to solve the whole set of coupled linear equations, including the linear Navier–Stokes equations, the linear energy equation, Poisson's equation and the linear charge conservation equation. A multiscale analysis was also performed to recover the macroscopic linearized Navier–Stokes equations from the four different discrete lattice Boltzmann equations (LBEs). The LLBM was validated by calculating the linear critical value of 2-D natural convection; it has an error of 1.39% compared with the spectral method. Instability with global travelling wave behaviour is a unique behaviour in the annulus configuration electrothermohydrodynamic system, which may be caused by the baroclinity. Finally, the chaotic behaviour was quantitatively analysed through calculation of the fractal dimension and Lyapunov exponent.

Sensitivity-aided active control of flow past twin cylinders

The flow-induced vibration (FIV) of twin structures always experiences significantly larger vibration amplitude than that of a single one, which causes severe structural safety problems. However, FIV control methods specifically for twin structures are still lacking. This study investigates the FIV and fluid forces of twin square cylinders under a jet flow control for a laminar flow. The control effects are studied for stationary and free-oscillating cylinders with a gap distance of L = 6.0D in co-shedding regimes (D is the side length of the cylinder). Global linear instability, adjoint method, and sensitive analysis are conducted to find a position where the jet flow should be placed. It was found that the most sensitive region is located in the near-wake region, and we thus set at jet flow at the rear of the cylinders. The control effects show that for stationary cylinders, UC achieves its maximum mean drag (CD) reduction of 11% when Uj > 0.3, meanwhile the CD of the downstream cylinder (DC) and fluctuating lift (CL') ≈ 0. UC plays a primary role in determining the control effects on fluid forces compared to DC, which coincides with the sensitivity analysis. For the free-oscillating case, much larger control energy is required to completely suppress vortex-induced vibration (VIV) with Uj > 2.4. In addition to the splitting effects of the jet flow, the neutralization effect of jet-induced vorticity was found plays a significant role in suppressing VIV, which can neutralize the vorticity generated from the side edge of the cylinders. High-order dynamic mode decomposition (HODMD) is also applied to uncover the underlying control mechanism. It demonstrates that the global energy of the dynamic mode energy is significantly decreased, meanwhile, the primary dynamic mode (i.e., the mode shape introducing VIV) is significantly weakened by the jet flow due to splitting and neutralizing effects.

Electro-thermo-convection in a high Prandtl number fluid: Flow transition and heat transfer

This paper aims to study the electro-thermo-convection of a high Prandtl number fluid (silicone oil) in a rectangular cavity. A high accuracy numerical solver is undertaken to solve the problem. These numerical simulations are performed for a range of Rayleigh and electric Rayleigh parameters, Ra=0,104∼106 and T=0∼1000, which correspond to regimes from laminar to chaotic flow for different cases. Under the condition of weightlessness Ra=0, a strong mixing between heat and cold liquid caused by the electric field is observed, and there is a flow motion transition from large vortices to plume oscillation with the increased value of T. The large vortices motion has a better effencify on heat transfer enhancement. A dual solution at low and moderate Ra numbers is observed. The beingness of these two solutions is validated by a global linear instability analysis based on the linearized lattice Boltzmann method. With the help of some nonlinear analysis techniques, we find that the presence of the electric field may stabilize the thermal convection system. Furthermore, a full map of the heat transfer rate with respect to electric and thermal driving parameters is provided as a reference for applications. Heat transfer enhancement appears only when the electric field force is dominant. A neutral force curve about Ra−T is presented that can characterize the threshold of electric field force to achieve heat transfer enhancement for thermal convection at different Ra numbers.

Optimal computational parameters for maximum accuracy and minimum cost of Arnoldi-based time-stepping methods for flow global stability analysis

Global instability analysis of flows is often performed via time-stepping methods, based on the Arnoldi algorithm. When setting up these methods, several computational parameters must be chosen, which affect intrinsic errors of the procedure, such as the truncation errors, the discretization error of the flow solver, the error associated with the nonlinear terms of the Navier-Stokes equations and the error associated with the limited size of the approximation of the Jacobian matrix. This paper develops theoretical equations for the estimation of optimal balance between accuracy and cost for each case. The 2D open cavity flow is used both for explaining the effect of the parameters on the accuracy and the cost of the solution, and for verifying the quality of the predictions. The equations demonstrate the impact of each parameter on the quality of the solution. For example, if higher order methods are used for approaching a Fr\'echet derivative in the procedure, it is shown that the solution deteriorates more rapidly for larger grids or less accurate flow solvers. On the other hand, lower order approximations are more sensitive to the initial disturbance magnitude. Nevertheless, for accurate flow solvers and moderate grid dimensions, first order Fr\'echet derivative approximation with optimal computational parameters can provide 5 decimal place accurate eigenvalues. It is further shown that optimal parameters based on accuracy tend to also lead to the most cost-effective solution. The predictive equations, guidelines and conclusions are general, and, in principle, applicable to any flow, including 3D ones.

Boundary-Layer Instabilities over a Cone–Cylinder–Flare Model at Mach 6

Computations are performed to investigate the convective and global boundary-layer instabilities over a sharp cone–cylinder–flare model at zero-degree angle of attack. The model geometry and the flow conditions are selected to match the experiments conducted in the Boeing/AFOSR Mach 6 Quiet Tunnel at Purdue University. The geometry consists of a nominally sharp 5 deg half-angle cone, followed by a cylindrical segment, and then a 10 deg flare. Additionally, flare half angles equal to 8 and 12 deg and nosetip radii equal to 0.1, 1, and 5 mm are studied. An axisymmetric separation bubble is generated as a result of the laminar shock–boundary-layer interaction in the cylinder–flare region. The comparison of the laminar flow solution and the schlieren images shows a remarkable agreement between the respective locations of both the boundary-layer edge and the reattachment shock. The linear amplification of first and second Mack mode instabilities that begin to amplify in the cone region are computed with a combination of the parabolized stability equations and the harmonic linearized Navier–Stokes equations. The predicted frequency spectra of the surface pressure fluctuations associated with both planar and oblique instability waves capture the distinct lobes within the disturbance amplification spectra measured with PCB® and Kulite sensors. To our knowledge, this represents the first successful comparison between convective instability analysis and measured surface pressure fluctuations for a hypersonic configuration with a separation bubble. Finally, the global instability analysis shows that the laminar flow becomes supercritical for flare half angles larger than 8 deg.

Global linear instability analysis of thermal convective flow using the linearized lattice Boltzmann method

Modal global linear stability analysis of thermal convection is performed with the linearized lattice Boltzmann method (LLBM). The onset of Rayleigh–Bénard convection in rectangular cavities with conducting and adiabatic sidewalls and the instability of two-dimensional (2-D) and three-dimensional (3-D) natural convection in cavities are studied. The method of linearizing the local equilibrium probability distribution function that was first proposed by Pérez et al. (Theor. Comp. Fluid Dyn., vol. 31, 2017, pp. 643–664) is extended to solve the coupled linear Navier–Stokes equations together with the linear energy equation in this work. A multiscale analysis is also performed to recover the macroscopic linear Navier–Stokes equations from the discrete lattice Boltzmann equations for both the single and multiple relaxation time models. The present LLBM is implemented in the framework of the Palabos library. It is validated by calculating the linear critical value of 2-D natural convection that the LLBM with the multiple relaxation time model has an error less than 1 % compared with the spectral method. The instability mechanism of the flow is explained by kinetic energy transfer analysis. It is shown that the buoyancy mechanism and inertial mechanism tend to stabilize the Hopf bifurcation of the 2-D natural convection at Pr < 0.08 and Pr > 1, respectively. For 3-D natural convection, subcritical bifurcation of the Hopf type is found for low-Prandtl-number fluids (Pr < 0.1).

Effects of Extended Laminar Flow on Wing Buffet-Onset Characteristics

Global instability analysis is used to investigate the effects of extended regions of laminar flow on both unswept and swept infinite-span wings. The formulation is based on the Reynolds-averaged Navier–Stokes equations and differs from earlier studies on fully turbulent flows in the activation of the trip term in the Spalart–Allmaras eddy-viscosity transport equation. The trip term ensures a rapid transition to turbulence at an arbitrary-specified location on the wing surface, analogous a trip strip in an experiment. A parametric study is conducted for the ONERA OAT15A extruded airfoil, as an example. Results show that an extended region of laminar flow leads to a reduction in the critical angle of attack for the buffet onset, as compared with the fully turbulent conditions. All modes of instability show enhanced growth as a result of the laminar flow. However, increased lift (at fixed angle of attack) associated with an extended region of laminar flow results in a higher lift coefficient at buffet onset as compared with the fully turbulent case. Results show that the laminar flow effects on the buffet onset are linked to the suction-side laminar flow and are largely independent of the boundary-layer state on the pressure side of the wing.

A CFD-Aided Galerkin Method for Global Linear Instability Analysis

A CFD-Aided Galerkin Method for Global Linear Instability Analysis

Stability analysis of cross buoyancy flow past a circular cylinder using OpenFOAM

A global instability analysis has been performed to study the influence of buoyancy on flow past a circular cylinder placed near a moving wall for −1 ≤ Ri ≤ 1 at a fixed value of G/D = 0.3 and a constant Re = 105. An open source finite-volume code OpenFOAM has been used. Selective frequency damping (SFD) method is implemented and used for calculations of the base flow. For the solution of the perturbation quantities linearized Navier-Stokes equations have been implemented in OpenFOAM. The growth rate and frequency have been obtained using the power iteration method. It has been observed that the flow is steady for Ri = 0 while it becomes unsteady for both positive and negative buoyancies. The most unstable modes appear as the traveling modes. These modes are similar to the Kelvin-Helmholtz (KH) modes with some distortion due to the presence of the moving wall.

Analysis of nonequilibrium effects characterizes shock wave-boundary layer interactions in hypersonic flow

The study combines direct simulation Monte Carlo calculations, linear global instability analysis and momentum potential theory to characterize nonequilibrium effects.

On the unsteadiness of shock–laminar boundary layer interactions of hypersonic flows over a double cone

Unsteadiness of axisymmetric shock-dominated hypersonic laminar separated flow over a double cone is studied for the first time using a combination of time accurate Direct Simulation Monte Carlo (DSMC) calculations, linear global instability analysis, and momentum potential theory (MPT). Close to steady state linear analysis reveals the spatial structure of the underlying temporally stable global modes. At all Reynolds numbers examined, the amplitude functions demonstrate the strong coupling between the separated flow region at the cone junction with the entire shock system, including pressure and temperature waves generated behind the shock and spatially amplified Kelvin-Helmholtz waves. In addition, as the Reynolds number is increased, temporally damped harmonic shock oscillations and multiple-reflected λ-shock patterns emerge in the eigenfunctions. Application of the MPT (valid for both linear and nonlinear signals) to the highest Reynolds number DSMC results shows that large acoustic and thermal potential variations exist in the vicinity of the separation shock, the λ-shock patterns, and the shear layers. It is further shown that the motion of the bow shock system is highly affected by non-uniformities in the acoustic field. At the highest Reynolds number considered here, the unsteadiness is characterized by Strouhal numbers in the shear layer and bow-shock regions and is found to be in qualitative agreement with earlier experimental and numerical work.

Stall cell formation over a post-stall airfoil: effects of active perturbations using plasma actuators

Stall cells have been observed over stalled airfoils by many researchers and there have been extensive discussions in the literature on their formation mechanism. They have been known to be unsteady and sensitive to even the smallest upstream perturbations, and therefore, can be made spatially steady by fixing the upstream perturbation location. In the current experimental work, flow with a Reynolds number of 5·105 over a stalled VR-7 airfoil with an aspect ratio 3 was perturbed using a single nanosecond dielectric barrier discharge (NS-DBD) plasma actuator positioned near the leading edge and covering the span of the airfoil. Surface oil flow visualization and stereo particle image velocimetry were employed to investigate spanwise non-uniformities on the airfoil surface as well as in the flow. The results show a gradual appearance of stall cells by increasing the perturbation Strouhal number to approximately 3 times the natural shedding Strouhal number of Stn = 0.6 (Ste ~ 3Stn). The stall cells become well-defined and eventually saturated with no further changes with increasing perturbation Strouhal number beyond 10Stn. The instability responsible for the stall cells is self-sustained as the stall cells persist after the perturbations are stopped. The results of the current work are discussed and interpreted using recent modal and non-modal linear global instability analyses, even though they are carried out in much lower Reynolds number laminar stalled flows over airfoils.

Global instability analysis and experiments on buoyant plumes

The present work investigates the puffing instability of circular buoyant plumes by performing global linear stability analysis and experiments. In the non-dimensional parameter space investigated, plumes exhibit global instability only for axisymmetric perturbations with two unstable modes, which are of oscillatory type. The frequencies of these two unstable global modes agree well with the experiments which suggest that puffing occurs in buoyant plumes as a result of linear global instability. A comprehensive investigation on the effect of various non-dimensional parameters and inlet velocity profiles on frequency and growth rates of the global modes is carried out. The results are used to delineate the stability boundaries for these global modes and to obtain scaling laws for the associated oscillation frequencies. The analysis demonstrates that the two buoyancy parameters, Froude number and source-to-ambient density ratio, play dominant roles in impacting plume transition and oscillation frequencies. Results from global linear stability analysis and earlier experiments have majorly differed in two aspects. The earlier experiments reported a switch in puffing frequency scaling in Richardson number range 100–500, while the instability analysis predicts this switch at around 6000. Also, the instability analysis predicts the occurrence of puffing at density ratios higher than the critical value 0.5–0.6 reported in earlier experiments. To address these differences and validate the results obtained from global linear stability analysis, experiments are performed in a set-up that has been carefully designed to minimize the settling chamber disturbances. The present experiments corroborate the findings of global linear stability analysis. The mechanisms responsible for global instability in plumes have been identified using perturbation vorticity transport equation.

The linearized pressure Poisson equation for global instability analysis of incompressible flows

The linearized pressure Poisson equation (LPPE) is used in two and three spatial dimensions in the respective matrix-forming solution of the BiGlobal and TriGlobal eigenvalue problem in primitive variables on collocated grids. It provides a disturbance pressure boundary condition which is compatible with the recovery of perturbation velocity components that satisfy exactly the linearized continuity equation. The LPPE is employed to analyze instability in wall-bounded flows and in the prototype open Blasius boundary layer flow. In the closed flows, excellent agreement is shown between results of the LPPE and those of global linear instability analyses based on the time-stepping nektar++, Semtex and nek5000 codes, as well as with those obtained from the FreeFEM++ matrix-forming code. In the flat plate boundary layer, solutions extracted from the two-dimensional LPPE eigenvector at constant streamwise locations are found to be in very good agreement with profiles delivered by the NOLOT/PSE space marching code. Benchmark eigenvalue data are provided in all flows analyzed. The performance of the LPPE is seen to be superior to that of the commonly used pressure compatibility (PC) boundary condition: at any given resolution, the discrete part of the LPPE eigenspectrum contains converged and not converged, but physically correct, eigenvalues. By contrast, the PC boundary closure delivers some of the LPPE eigenvalues and, in addition, physically wrong eigenmodes. It is concluded that the LPPE should be used in place of the PC pressure boundary closure, when BiGlobal or TriGlobal eigenvalue problems are solved in primitive variables by the matrix-forming approach on collocated grids.

Lattice Boltzmann methods for global linear instability analysis

Modal global linear instability analysis is performed using, for the first time ever, the lattice Boltzmann method (LBM) to analyze incompressible flows with two and three inhomogeneous spatial directions. Four linearization models have been implemented in order to recover the linearized Navier–Stokes equations in the incompressible limit. Two of those models employ the single relaxation time and have been proposed previously in the literature as linearization of the collision operator of the lattice Boltzmann equation. Two additional models are derived herein for the first time by linearizing the local equilibrium probability distribution function. Instability analysis results are obtained in three benchmark problems, two in closed geometries and one in open flow, namely the square and cubic lid-driven cavity flow and flow in the wake of the circular cylinder. Comparisons with results delivered by classic spectral element methods verify the accuracy of the proposed new methodologies and point potential limitations particular to the LBM approach. The known issue of appearance of numerical instabilities when the SRT model is used in direct numerical simulations employing the LBM is shown to be reflected in a spurious global eigenmode when the SRT model is used in the instability analysis. Although this mode is absent in the multiple relaxation times model, other spurious instabilities can also arise and are documented herein. Areas of potential improvements in order to make the proposed methodology competitive with established approaches for global instability analysis are discussed.

Morphological optimization of tensegrity-type metamaterials

We analyze the problem of mass and morphological optimization for metamaterials with tensegrity-type cells. Our approach is based on both local and global instability analyses. As a meaningful prototypical example we tassellate a compressed slab into periodic patterns of triangular, square and hexagonal cells.

Linear spatial instability analysis in 3D boundary layers using plane-marching 3D-LPSE

It is widely accepted that a robust and efficient method to compute the linear spatial amplified rate ought to be developed in three-dimensional (3D) boundary layers to predict the transition with the e N method, especially when the boundary layer varies significantly in the spanwise direction. The 3D-linear parabolized stability equation (3D-LPSE) approach, a 3D extension of the two-dimensional LPSE (2D-LPSE), is developed with a plane-marching procedure for investigating the instability of a 3D boundary layer with a significant spanwise variation. The method is suitable for a full Mach number region, and is validated by computing the unstable modes in 2D and 3D boundary layers, in both global and local instability problems. The predictions are in better agreement with the ones of the direct numerical simulation (DNS) rather than a 2D-eigenvalue problem (EVP) procedure. These results suggest that the plane-marching 3D-LPSE approach is a robust, efficient, and accurate choice for the local and global instability analysis in 2D and 3D boundary layers for all free-stream Mach numbers.