Hybrid,numerical solutions,for three-dimensional,compressible Navier-Stokes layer

The author proposes new hybrid solutions for the three-dimensional, compressible Navier-Stokes layer (NSL) over a flying configuration (FC), which use the analytical potential solutions, of the same FC, two times, namely: to reinforce the numerical solutions by multiplying them with these analytical solutions and as outer flow (instead of the parallel flow, used by Prandtl in his boundary layer theory). These hybrid solutions fulfill the last behavior, have correct jumps along the singular lines (like subsonic leading edges, junction lines wing-fuselage, etc.), are split, accurate and rapid convergent. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical study of non-linear convection in a compressible medium

In this paper we consider the problem of non-linear convection in a compressible layer with polytropic structure. After deriving the appropriate forms of the basic conservation equations a single mode expansion is used to obtain a simpler model, within the framework of the anelastic approximation. These equations are integrated, using a band-matrix algorithm, for two characteristic density stratifications. The results are given in the form of graphs and are discussed in detail. An outline of the numerical algorithm is given in an appendix together with a discussion of its effectiveness.     

A discontinuous solution for an evolution compressible stokes system in a bounded domain

An evolution compressible Stokes system is studied in a bounded cylindrical region . The initial datum of pressure is assumed to have a jump at a specified curve C₀ in Ω. As predicted by the Rankine-Hugoniot conditions, the pressure and velocity derivatives have jump discontinuities along the characteristic plane of the curve C₀ directed by an ambient velocity vector. An explicit formula for the jump discontinuity is presented. The jump decays exponentially in time, more rapidly for smaller viscosities. Under suitable conditions of the data, a regularity of the solution is established in a compact subregion of Q away from the jump plane.     

Laminar compressible boundary layer flow at a three-dimensional stagnation point with vectored mass transfer

The effect of surface mass transfer velocities having normal, principal and transverse direction components (‘vectored’ suction and injection) on the steady, laminar, compressible boundary layer at a three-dimensional stagnation point has been investigated both for nodal and saddle points of attachment. The similarity solutions of the boundary layer equations were obtained numerically by the method of parametric differentiation. The principal and transverse direction surface mass transfer velocities significantly affect the skin friction (both in the principal and transverse directions) and the heat transfer. Also the inadequacy of assuming a linear viscosity-temperature relation at low-wall temperatures is shown.     

Numerical study of dynamics of single bubbles and bubble swarms

A systematic computational study of the dynamics of gas bubbles rising in a viscous liquid is presented. Two-dimensional simulations are carried out. Both the dynamics of single bubbles and small groups of bubbles (bubble swarms) are considered. This is a continuation of our previous studies on the two-bubble coalescence and vortex shedding [A. Smolianski, H. Haario, P. Luukka, Vortex shedding behind a rising bubble and two-bubble coalescence: a numerical approach, Appl. Math. Model. 29 (2005) 615–632]. The proposed numerical method allows us to simulate a wide range of flow regimes, accurately capturing the shape of the deforming interface of the bubble and the surface tension effect, while maintaining the mass conservation. The computed time-evolution of bubble’s position and rise velocity shows a good agreement with the available experimental data. At the same time, the results on the dynamics of bubble interface area, which are, up to our knowledge, presented for the first time, show how much the overall mass transfer would be affected by the interface deformation in the case of the bubble dissolution. Another set of experiments that are of interest for chemical engineers modelling bubbly flows concerns the bubble swarms and their behavior in different bubble-shape regimes. The ellipsoidal and spherical shape regimes are considered to represent, respectively, the coalescing and non-coalescing bubble swarms. The average rise velocities of the bubble swarms are computed and analyzed for both regimes.     

Direct numerical simulation of compressible isotropic turbulence

Direct numerical simulation (DNS) of decaying compressible isotropic turbulence at turbulence Mach numbers of M_t = 0.2-0.7 and Taylor Reynolds numbers of 72 and 153 is performed by using the 7th order upwind-biased difference and 8th order center difference schemes. Results show that proper upwind-biased difference schemes can release the limit of“start-up” problem to Mach numbers. Compressibility effects on the statistics of turbulent flow as well as the mechanics of shocklets in compressible turbulence are also studied, and the conclusion is drawn that high Mach number leads to more dissipation. Scaling laws in compressible turbulence are also analyzed. Evidence is obtained that scaling laws and extended self similarity (ESS) hold in the compressible turbulent flow in spite of the presence of shocklets, and compressibility has little effect on scaling exponents.     

Mach configuration in pseudo-stationary compressible flow

This paper is devoted to studying the local structure of Mach reflection, which occurs in the problem of the shock front hitting a ramp. The compressible flow is described by the full unsteady Euler system of gas dynamics. Because of the special geometry, the motion of the fluid can be described by self-similar coordinates, so that the unsteady flow becomes a pseudo-stationary flow in this coordinate system. When the slope of the ramp is less than a critical value, the Mach reflection occurs. The wave configuration in Mach reflection is composed of three shock fronts and a slip line bearing contact discontinuity. The local existence of a flow field with such a configuration under some assumptions is proved in this paper. Our result confirms the reasonableness of the corresponding physical observations and numerical computations in Mach reflection.

In order to prove the result, we formulate the problem to a free boundary value problem of a pseudo-stationary Euler system. In this problem two unknown shock fronts are the free boundary, and the slip line is also an unknown curve inside the flow field. The proof contains some crucial ingredients. The slip line will be transformed to a fixed straight line by a generalized Lagrange transformation. The whole free boundary value problem will be decomposed to a fixed boundary value problem of the Euler system and a problem to updating the location of the shock front. The Euler system in the subsonic region is an elliptic-hyperbolic composite system, which will be decoupled to the elliptic part and the hyperbolic part at the level of principal parts. Then some sophisticated estimates and a suitable iterative scheme are established. The proof leads to the existence and stability of the local structure of Mach reflection.

     

Weak oscillations of a gas bubble in a spherical volume of compressible liquid

The following spherically symmetric problem is considered: a single gas bubble at the centre of a spherical flask filled with a compressible liquid is oscillating in response to forced radial excitation of the flask walls. In the long-wave approximation at low Mach numbers, one obtains a system of differential-difference equations generalizing the Rayleigh-Lamb-Plesseth equation. This system takes into account the compressibility of the liquid and is suitable for describing both free and forced oscillations of the bubble. It includes an ordinary differential equation analogous to the Herring-Flinn-Gilmore equation describing the evolution of the bubble radius, and a delay equation relating the pressure at the flask walls to the variation of the bubble radius. The solutions of this system of differential-difference equations are analysed in the linear approximation and numerical analysis is used to study various modes of weak but non-linear oscillations of the bubble, for different laws governing the variation of the pressure or velocity of the liquid at the flask wall. These solutions are compared with numerical solutions of the complete system of partial differential equations for the radial motion of the compressible liquid around the bubble.     

Steady compressible Navier-Stokes flow in a square

We investigate a steady flow of compressible fluid with inflow boundary condition on the density and slip boundary conditions on the velocity in a square domain Q∈R². We show existence if a solution that is a small perturbation of a constant flow (, ). We also show that this solution is unique in a class of small perturbations of the constant flow . In order to show the existence of the solution we adapt the techniques known from the theory of weak solutions. We apply the method of elliptic regularization and a fixed point argument.     

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.

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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.

     

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.

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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.

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

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

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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.     

Blasius flow of thixotropic fluids: A numerical study

In the present work, laminar, two-dimensional flow of an incompressible thixotropic fluid obeying Harris rheological model is investigated above a fixed semi-infinite plate- the so-called Blasius flow. Assuming that the flow is occurring at high Reynolds number, use will be made of the boundary layer theory to simplify the equations of motion. The equations so obtained are then reduced to a single fourth-order ODE using a suitable similarity variable. It is shown that Harris fluids do not render themselves to a self-similar solution in Blasius flow. A local similarity solution is found which enabled investigating the effects of the model parameters on the velocity profile and wall shear stress at a given location above the plate. Numerical results show that for the Harris model to represent thixotropic fluids, the sign and magnitude of the material parameters appearing in this fluid model cannot be arbitrary.     

Direct numerical simulation of transition and turbulence in compressible mixing layer

The three-dimensional compressible Navier-Stokes equations are approximated by a fifth order upwind compact and a sixth order symmetrical compact difference relations combined with three-stage Ronge-Kutta method. The computed results are presented for convective Mach numberMc = 0.8 andRe = 200 with initial data which have equal and opposite oblique waves. From the computed results we can see the variation of coherent structures with time integration and full process of instability, formation of A -vortices, double horseshoe vortices and mushroom structures. The large structures break into small and smaller vortex structures. Finally, the movement of small structure becomes dominant, and flow field turns into turbulence. It is noted that production of small vortex structures is combined with turning of symmetrical structures to unsymmetrical ones. It is shown in the present computation that the flow field turns into turbulence directly from initial instability and there is not vortex pairing in process of transition. It means that for large convective Mach number the transition mechanism for compressible mixing layer differs from that in incompressible mixing layer.     

Numerical study of the flow in a three-dimensional thermally driven cavity

Solutions for the fully compressible Navier–Stokes equations are presented for the flow and temperature fields in a cubic cavity with large horizontal temperature differences. The ideal-gas approximation for air is assumed and viscosity is computed using Sutherland's law. The three-dimensional case forms an extension of previous studies performed on a two-dimensional square cavity. The influence of imposed boundary conditions in the third dimension is investigated as a numerical experiment. Comparison is made between convergence rates in case of periodic and free-slip boundary conditions. Results with no-slip boundary conditions are presented as well. The effect of the Rayleigh number is studied.     

Statistical modeling of compressible turbulent channel flow

Several questions that are relevant to turbulence modeling are addressed on the basis of recently obtained direct numerical simulation results of turbulent supersonic channel flow. In particular, this concerns the turbulence frequency production mechanism, wall damping effects on turbulence model parameters, and the relevance of compressibility effects. Limited support is found for usually applied models for the turbulence frequency production and wall damping effects. In contrast to that it is shown that turbulence frequency production mechanisms and wall damping effects may be explained very well on the basis of a frequency scaling that characterizes mean flow changes. The influence of compressibility is found to be relevant. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A singular multi-dimensional piston problem in compressible flow

This paper concerns the multi-dimensional piston problem, which is a special initial boundary value problem of multi-dimensional unsteady potential flow equation. The problem is defined in a domain bounded by two conical surfaces, one of them is shock, whose location is also to be determined. By introducing self-similar coordinates, the problem can be reduced to a free boundary value problem of an elliptic equation. The existence of the problem is proved by using partial hodograph transformation and nonlinear alternating iteration. The result also shows the stability of the structure of shock front in symmetric case under small perturbation.