A procedure that generates steady-states with accurate far-field entrainment is described here. It can be applied to any existing code originally designed for unsteady simulations. Such steady-states are obtained with physical-time damping, introduced through a dual time stepping methodology to effectively eliminate all transient fluctuations without affecting spatial resolution. Far-field entrainment is guaranteed by using properly selected local boundary conditions. Hence, reference profiles with accurate far-field entrainment are generated from the very same code that will eventually use them in unsteady simulations. This new procedure is tested on an unsteady code designed for compressible planar mixing-layer simulations at arbitrary Mach numbers. It is numerically stable for a wide range of Mach numbers and velocity ratios.

Modelling far-field entrainment in compressible flows

On the controversy regarding the effect of flow shear on the Strouhal number of cylinder vortex shedding

This paper presents a family of finite difference schemes for the first and second derivatives of smooth functions. The schemes are Hermitian and symmetric and may be considered a more general version of the standard compact (Pade) schemes discussed by Lele. They are different from the standard Pade schemes, in that the first and second derivatives are evaluated simultaneously. For the same stencil width, the proposed schemes are two orders higher in accuracy, and have significantly better spectral representation. Eigenvalue analysis, and numerical solutions of the one-dimensional advection equation are used to demonstrate the numerical stability of the schemes. The computational cost of computing both derivatives is assessed and shown to be essentially the same as the standard Pade schemes. The proposed schemes appear to be attractive alternatives to the standard Pade schemes for computations of the Navier?Stokes equations.

High order finite difference schemes with good spectral resolution

Reference solutions are important in several applications. They are used as base states in linear stability analyses as well as initial conditions and reference states for sponge zones in numerical simulations, just to name a few examples. Their accuracy is also paramount in both fields, leading to more reliable analyses and efficient simulations, respectively. Hence, steady-states usually make the best reference solutions. Unfortunately, standard marching schemes utilized for accurate unsteady simulations almost never reach steady-states of unstable flows. Steady governing equations could be solved instead, by employing Newton-type methods often coupled with continuation techniques. However, such iterative approaches do require large computational resources and very good initial guesses to converge. These difficulties motivated the development of a technique known as selective frequency damping (SFD) (Akervik et al. in Phys Fluids 18(6):068102, 2006). It adds a source term to the unsteady governing equations that filters out the unstable frequencies, allowing a steady-state to be reached. This approach does not require a good initial condition and works well for self-excited flows, where a single nonzero excitation frequency is selected by either absolute or global instability mechanisms. On the other hand, it seems unable to damp stationary disturbances. Furthermore, flows with a broad unstable frequency spectrum might require the use of multiple filters, which delays convergence significantly. Both scenarios appear in convectively, absolutely or globally unstable flows. An alternative approach is proposed in the present paper. It modifies the coefficients of a marching scheme in such a way that makes the absolute value of its linear gain smaller than one within the required unstable frequency spectra, allowing the respective disturbance amplitudes to decay given enough time. These ideas are applied here to implicit multi-step schemes. A few chosen test cases shows that they enable convergence toward solutions that are unstable to stationary and oscillatory disturbances, with either a single or multiple frequency content. Finally, comparisons with SFD are also performed, showing significant reduction in computer cost for complex flows by using the implicit multi-step MGM schemes.

Minimal gain marching schemes: searching for unstable steady-states with unsteady solvers

The Rossiter modes of an open cavity were studied using bi-global linear analysis, local instability analysis and nonlinear numerical simulations. Rossiter modes are normally seen only for short cavities; hence, in the study, the length over depth ratio was two. We focus on the critical region; hence, the Reynolds numbers based on cavity depth were close to 1000. We investigated the effect of the ratio boundary layer thickness to cavity depth, a parameter often overlooked in the literature. Increasing this ratio is destabilizing and increases the number of unstable Rossiter modes. Local instability analysis revealed that the hierarchy of unstable modes was governed by the mixing in the cavity opening. The effect of Mach number was also studied for thin and thick boundary layers. Compressibility had a very destabilizing effect at low Mach numbers. Analysis of the Rossiter mode eigenfunctions indicated that the acoustic feedback scaled to $$\mathrm{Ma}^3$$
 and explained the strong destabilizing effect of compressibility at low Mach numbers. At moderate Mach numbers, the instability either saturated with Mach number or had an irregular dependence on it. This was associated with resonances between Rossiter modes and acoustic cavity modes. The analysis explained why this irregular dependence occurred only for higher-order Rossiter modes. In this parameter region, three-dimensional modes are either stable or marginally unstable. Two-dimensional simulations were performed to evaluate how much of the nonlinear regime could be captured by the linear stability results. The instability was triggered by the $$10^{-13}$$
 flow solver noise floor. The simulations initially agreed with linear theory and later became nonlinearly saturated. The simulations showed that, as the flow becomes more unstable, an increasingly more complex final stage is reached. Yet, the spectra present distinct tones that are not far from linear predictions, with the thin boundary layer cases being closer to empirical predictions. The final stage, in general, was dominated by first Rossiter mode, even though the second one was the most unstable linearly. It seems this may be associated with nonlinear boundary layer thickening, which favors lower frequency in the mixing layer, or vortex pairing of the second Rossiter mode. The spectra in the final stages are well described by the mode R1 and a cascade of nonlinearly generated harmonics, with little reminiscence of the linear instability.

The effect of incoming boundary layer thickness and Mach number on linear and nonlinear Rossiter modes in open cavity flows

Mixing layers are present in very different types of physical situations such as atmospheric flows, aerodynamics and combustion. It is, therefore, a well researched subject, but there are aspects that require further studies. Here the instability of two-and three-dimensional perturbations in the compressible mixing layer was investigated by numerical simulations. In the numerical code, the derivatives were discretized using high-order compact finite-difference schemes. A stretching in the normal direction was implemented with both the objective of reducing the sound waves generated by the shear region and improving the resolution near the center. The compact schemes were modified to work with non-uniform grids. Numerical tests started with an analysis of the growth rate in the linear regime to verify the code implementation. Tests were also performed in the non-linear regime and it was possible to reproduce the vortex roll-up and pairing, both in two-and three-dimensional situations. Amplification rate analysis was also performed for the secondary instability of this flow. It was found that, for essentially incompressible flow, maximum growth rates occurred for a spanwise wavelength of approximately 2/3 of the streamwise spacing of the vortices. The result demonstrated the applicability of the theory developed by Pierrehumbet and Widnall. Compressibility effects were then considered and the maximum growth rates obtained for relatively high Mach numbers (typically under 0.8) were also presented.

/pdf/numerical-investigation-of-the-three-dimensional-secondary-48pe6tja3m.pdf

Numerical Investigation of the Three- Dimensional Secondary Instabilities in the Time-developing Compressible Mixing Layer

The engineering research and design requirements of today pose great challenges in computer simulation to engineers and scientists who are called on to analyze phenomena in continuum mechanics. The study of hydrodynamic instability is a subject well researched, seeking among other objectives, the development of propulsion systems based on supersonic combustion. In the current work, the instability of the compressible free shear layer at relatively low Reynolds number was investigated. In the aerospace context, important applications involve compressible flows at relatively low Reynolds number. Among them, the flow on gas turbine blades and the flow on high lift devices such as slats and flaps at high angle of attack are particularly important. In aerodynamic applications in low Reynolds number, often a substantial portion of the flow is in the transition regime, or in the initial stages of a turbulent flow. Despite the extensive research carried out in the field, there are various aspects of the transition process that require further studies. The transition in compressible flow is an example. In this work the simulation of compressible Navier-Stokes equations were performed. The spatial derivatives of these equations were discretized through of the high-order finite difference method. To solve the temporal derivatives the high-order Runge-Kutta method was adopted. Firstly, an analysis of the amplification rate in linear regime was performed in order to verify the numerical code. Tests were also performed in the nonlinear regime of two-dimensional waves and it was possible to reproduce the vortex roll up and pairing. Finally the initial results with the three-dimensional equations were presented.

https://www.abcm.org.br/anais/cobem/2005/PDF/COBEM2005-1479.pdf

Development of a code for a direct numerical simulation of compressible shear flow instabilities

The governing equations of the acoustic problem are the compressible Euler equations. The discretization of these equations has to ensure that the acoustic waves are transported with non-dispersive and non-dissipative characteristics. In the present study numerical simulations of a standing acoustic wave are performed. Four different space discretization schemes are tested, namely, a second order finite-differences, a fourth order finitedifferences, a fourth order finite-differences compact scheme and a sixth order finite-differences compact scheme. The time integration is done with a fourth order Runge-Kutta scheme. The results obtained are compared with linearized analytical solutions. The influence of the dispersion on the simulation of a standing wave is analyzed. The results confirm that high order accuracy schemes can be more efficient for simulation of acoustic waves, especially the waves with high frequency.

/pdf/analysis-of-dispersion-errors-in-acoustic-wave-simulations-4gw61jq3x4.pdf

Analysis of dispersion errors in acoustic wave simulations

We present numerical and theoretical results on the natural transition in a plane Poiseuille flow. The natural transition is often studied in boundary layers, but the analysis in a parallel flow allows the use of the Reynolds number as a control parameter. The results indicated that the classical scenarios described in the literature play a role in natural transition and can lead to turbulent spots. In natural transition the wave systems are modulated in streamwise and spanwise directions and cause premature nonlinearity [1]. Spikes and turbulent spots are often present in natural transition, but the classical routes, namely, K-type and H-type instabilities and the oblique transition, have not been linked to these phenomena. Therefore, many questions remain regarding the relevance of these studies to natural transition. For instance, it is unknown whether such classical routes actually occur in natural transition, whether they occur simultaneously or interact, whether there is a dominant mechanism under some circumstances, etc. Such questions motivated the present work. Numerical simulation of the incompressible three-dimensional Navier-Stokes equations in a vorticity velocity formulation was performed using high accurate numerical schemes. Owing to the complexity of the natural transition, the streamwise modulation was initially not considered. Therefore, only the three dimensionality was studied yielding spanwise modulated wavetrains. Two cases were judiciously selected at Re = 8000. One case was located near the first branch of the two-dimensional stability diagram where the linear process is dominated by three-dimensional Tollmien-Schlichting (TS) waves. The other case was located far from the first branch where the linear process is dominated by a two-dimensional TS waves. Preliminary results can be found in [2]. Similar studies, but for boundary layers, can be found in [3].

Natural Transition in Plane Poiseuille Flow

This abstract presents numerical results and analysis related to natural transition in a plane Poiseuille flow. The natural transition scenario was modeled by three–dimentional wavepackets. Natural transition is often studied in boundary layers, but the analysis in a parallel flow allows the use of the Reynolds number as a control parameter. The results indicated that the interactions between wavepackets lead to oblique transition which is already often linked to natural transition.

Ricardo Alberto Coppola Germanos

Papers

Numerical Investigation of the Three- Dimensional Secondary Instabilities in the Time-developing Compressible Mixing Layer

Development of a code for a direct numerical simulation of compressible shear flow instabilities

Analysis of dispersion errors in acoustic wave simulations

Natural Transition in Plane Poiseuille Flow

Nonlinear Interaction Between Wavepackets in Plane Poiseuille Flow