Accurate numerical solution of compressible,linear stability equations

Summary A compact fourth order accurate finite difference scheme is proposed for solving the eigenvalue problem of the temporal and spatial stability of three-dimensional compressible boundary layers. It is applied to the stability study of the flow on a laminar flow control swept wing. The results indicate that the present method provides an accurate and efficient way of calculating the eigenvalues and eigensolutions of the compressible linear stability equations.

---

Zusammenfassung Ein kompaktes Differenzenschema vierter Ordnung wird vorgeschlagen zur Lösung von Eigenwertproblemen der zeitlichen und räumlichen Stabilität dreidimensionaler zusammendrückbarer Grenzschichten. Es wird zum Stabilitätsstudium der Strömung über einen LFC Flügel angewendet. Die Resultate zeigen, daß es die angewendete Methode erlaubt, genau und effizient die Eigenwerte und Eigenfunktionen der kompressiblen linearen Stabilitätsgleichungen zu berechnen.

---

---

Research for this author was supported by NASA Contract No. NAS 1-16572.

---

Research for this author was supported by NASA Contract No. NCC 1-14.

---

Research for this author was supported by NASA Contracts No. NAS 1-14472 and NAS 1-15810 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, Va. 23665.     

Wall-cooling effect on linear stability of compressible streamwise corner-flow

The influence of constant wall temperatures on the compressible flow along a right-angled corner and its stability behaviour is investigated by temporal local linear stability analysis. Wall-cooling up to 90 % of the adiabatic wall-temperature for a subsonic flow (Ma = 0.95) is considered. The maximum cross-flow velocity along the corner bi-sector for the base flow with adiabatic wall boundary condition is higher than for the base flow with constant adabatic wall-temperature. For increasing wall temperature, the amplification rate of the viscous modes decreases to a lesser extent than that of the corner mode, therefore the critical Reynolds-number is further on defined by the first viscous mode. The spatial behaviour is displayed using N-factors computed from Gaster-transformed modes. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Assessment of high-order discretization schemes for the DNS of compressible swirling mixing layers

We report on results of highly accurate Direct Numerical Simulations (DNS) solving the Navier-Stokes equations in cylindrical coordinates. The DNS code computes a compressible swirling mixing layer at Mach number Ma = 0.8. We present two simulations differing in the spatial discretization schemes for the convective terms. On the same grid, comparisons of flow simulations using different discretization schemes for otherwise identical conditions can be performed quantitatively and improve the understanding of the effects of numerical errors and in particular numerical dissipation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the incompressible limit of inviscid compressible fluids

We consider the Euler equations of barotropic inviscid compressible fluids in a bounded domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. Here we discuss, for the boundary case, the different kinds of convergence under various assumptions on the data, in particular the weak convergence in the case of uniformly bounded initial data and the strong convergence in the norm of the data space.

---

Sunto Nel presente lavoro consideriamo le equazioni di Eulero per un fluido inviscido comprimibile barotropico in un dominio limitato. è ben noto che, quando il numero di Mach tende a zero, i moti comprimibili approssimano le soluzioni delle equazioni del moto relative ad un fluido inviscido incomprimibile. Qui discutiamo, per il problema al contorno, i diversi tipi di convergenza sotto differenti ipotesi sui dati, in particolare la convergenza debole nel caso di dati iniziali uniformemente limitati e la convergenza forte nella norma dello spazio dei dati.

     

The linear stability of inviscid shear flow of a vibrationally excited diatomic gas

The problem of the linear stability of plane-parallel shear flows of a vibrationally excited compressible diatomic gas is investigated using a two-temperature gas dynamics model. The necessary and sufficient conditions for stability of the flows considered are obtained using the energy integrals of the corresponding linearized system for the perturbations. It is proved that thermal relaxation produces an additional dissipation factor, which enhances the flow stability. A region of eigenvalues of unstable perturbations is distinguished in the upper complex half-plane. Numerical calculations of the eigenvalues and eigenfunctions of the unstable inviscid modes are carried out. The dependence on the Mach number of the carrier stream, the vibrational relaxation time τ and the degree of non-equilibrium of the vibrational mode is analysed. The most unstable modes with maximum growth rate are obtained. It is shown that in the limit there is a continuous transition to well-known results for an ideal fluid as the Mach number and τ approach zero and for an ideal gas when τ → 0.     

The inviscid limit for an inflow problem of compressible viscous gas in presence of both shocks and boundary layers

In this paper, we study the inviscid limit problem for the Navier-Stokes equations of one-dimensional compressible viscous gas on half plane. We prove that if the solution of the inviscid Euler system on half plane is piecewise smooth with a single shock satisfying the entropy condition, then there exist solutions to Navier-Stokes equations which converge to the inviscid solution away from the shock discontinuity and the boundary at an optimal rate of ε¹ as the viscosity ε tends to zero.     

A novel type of wave behaviour in a compressible inviscid dipolar fluid and stability characteristics of generalized fluids

Summary A new type of wave behaviour is found for third order waves in a compressible inviscid dipolar fluid. Several stability-like results are presented for the theories of a viscous incompressible dipolar fluid and a mixture of two viscous incompressible fluids.     

Method for enhancing the mixing of a compressible mixing layer

Numerical simulations have been done for a compressible mixing layer, in which the inflow speed on the low speed side was made to have periodic undulations, so as to see if this method could enhance the mixing effect of the layer. Systematic computations for a 2-D compressible mixing layer with Mach numberM _e = 0.6 have been done, and the results showed that the proposed method was indeed effective in enhancing the mixing.     

Linear stability analysis of oscillating Ekman boundary layers

The analysed Ekman layer is generated in a fluid layer rotating around an axis normal to its two bounding rigid plates. One of the plates is stationary, the other moving at certain Reynolds numbers. An additional oscillation is added to the moving plate at different amplitudes and frequencies. The linear stability of this system is determined via a Floquet analysis and a Galerkin-approximation of the corresponding Navier-Stokes-Equations. If the frequencies of the oscillations are small the critical Reynolds numbers of the Type I and Type II instabilities do not differ much from steady Ekman layers. Also for a purely oscillating system the critical values of the instabilities are almost consistent with those for a steady system. Interestingly, for higher frequencies the Type II instability does not appear any more. Instead the boundary layer becomes unstable only in terms of a Type I instability. In comparison with findings of other authors these results seem to be quite reasonable. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of transonic and subsonic flow of compressible inviscid and viscous fluid

The work presents two numerical solutions of compressible flows problems with high and very low Mach numbers. Both problems are numerically solved by finite volume method and the explicit MacCormack scheme using a grid of quadrilateral cells. Moved grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian–Eulerian method. In the first case, inviscid transonic flow through cascade DCA 8% is presented and the numerical results are compared to experimental data. The second case, numerical solution of unsteady viscous flow in the channel for upstream Mach number M_∞=0.012 and frequency of the wall motions 100 Hz is presented. The unsteady case can represent a simplified model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The inviscid stability of a trailing line vortex

Summary A finite difference method has been developed to study the inviscid stability of swirling flows to small non-axisymmetric disturbances. We apply the method to Batchelor's trailing line vortex solution [3]. The method appears to be more efficient, and simpler to implement for this class of problem, than previously reported methods.

---

Zusammenfassung Zur Untersuchung der Stabilität von Wirbelströmungen bei nichtaxisymmetrischen kleinen Störungen ist eine Differenzemethode entwickelt worden. Sie wird auf Batchelor's Lösung für die abgehende Wirbellinie angewendet. Die Methode erscheint für diese Art von Problemen gut geeignet und einfach in der Anwendung.

     

Implementing a discontinuous Galerkin method for the compressible,inviscid Euler equations in the DUNE framework

A discontinuous Galerkin scheme was implemented in the DUNE framework to solve the compressible, inviscid Euler equations in a multi-dimensional Cartesian grid. It uses a HLLC Riemann solver for the numerical fluxes in the interfaces, a total variation bounded limiter to handle discontinuities, a positivity preserving limiter for near vacuum conditions, and adaptive mesh refinement (AMR). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Verification of compressible closure models for turbulent multifluid mixing

For compressible multiphase flow models the proper definition of averages of nonlinear terms is an essential problem which is called the closure problem. The purpose of the present work is to verify the spatial homogeneity closure, based on analysis of the simulation data. Various proposed closures are compared in a verification study to spatial averages of numerical simulations.     

Spatial problems of stability theory of stratified,compressible, composite materials

We consider spatial, non-axially-symmetric problems of stability theory of stratified, compressible, composite materials for biaxial uniform compression with the use of three-dimensional, linearized stability theory of deformed bodies. It is shown in general form for arbitrary models of compressible elastic and elastoplastic bodies that the characteristic equation corresponding to spatial non-axially-symmetric problems reduces to the characteristic equation corresponding to planar problems of single-axis compression. In this context all numerical results obtained earlier for the planar problems mentioned are also equally valid for spatial, non-axially-symmetric problems with the corresponding notations for the wave formation parameters.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 24–32, 1988.     

Self-similar solution of unsteady compressible three-dimensional stagnation-point boundary layers

Summary The self-similar solution of the unsteady laminar compressible boundary-layer flow with variable properties at a three-dimensional stagnation point with mass transfer has been obtained when the free-stream velocity varies inversely as a linear function of time. The resulting ordinary differential equations have been solved numerically using an implicit finite-difference scheme. The results are found to be strongly dependent on the parameter characterizing the unsteadiness in the free-stream velocity. The velocity profiles show some features not encountered in steady flows.

---

Zusammenfassung Ähnliche Lösungen der nichtstationären laminaren kompressiblen Grenzschichtströmung mit veränderlichen Stoffgrößen ist für einen dreidimensionalen Staupunkt mit Massen-Übertragung erhalten worden, für den Fall, daß die Anströmung invers zu einer linearen Funktion der Zeit variiert. Es entsteht eine gewöhnliche Differentialgleichung, die numerisch mit einer impliziten Differenzenmethode gelöst wurde. Die Ergebnisse hängen stark von dem Parameter ab, der die Nicht-Stationärität der Anströmung kennzeichnet. Die Geschwindigkeitsprofile zeigen einige Eigenschaften die in stationärer Strömung nicht auftreten.

     

Dynamical stability of compressible drops and stars

Interest is directed to linearized free boundary motion of a compressible liquid subject to surface tension and self‐gravitation respectively. Linearization relative to an a‐priori given solution to the non‐linear equations leads to a non‐local second order evolution problem to be posed in a space‐time cylinder with variable cross section subject to Fréchet boundary conditions along the lateral boundary part. Well‐posedness of the corresponding initial value problem in a natural weak formulation is proved. Copyright © 2005 John Wiley & Sons, Ltd.     

Transient inviscid flow in a passive linear model of the cochlea

Detailed knowledge of the fluid dynamics within the cochlea is important for a complete understanding of the physiology of this complex organ. We present a numerical simulation in the time domain for the inviscid two-dimensional flow in a passive cochlea. Numerical results are presented for a click stimulation and an impulsively started tone. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)