Three-dimensional Compressible Flow Research Articles

Slip flow characteristics of compressible gaseous in microchannels

In this paper, a numerical model for three-dimensional compressible gaseous slip flow in microchannel was established. Gas viscosity is modified based on Knudsen number using Veijola’s model due to the increased rarefaction effects in microscale. By comparing with experimental data, the model used in the present paper for calculating is credible. The result shows that the gas viscosity needs to be modified in the calculation because it can improve the accuracy of the result. The influences of gas compressibility, rarefaction effect and size effect on flow characteristics are comprehensively described. For compressible flow, a new non-dimensional parameter relative slip length L s /D h is proposed to correlate fRe. The suggested correlation for fRe with L s /D h is validated by comparing with experimental data. This correlation can be used for both slip and non-slip flow, and for both compressible and incompressible flow.

Global stability of swept flow around a parabolic body: connecting attachment-line and crossflow modes

The global linear stability of a three-dimensional compressible flow around a yawed parabolic body of infinite span is investigated using an iterative eigenvalue method in conjunction with direct numerical simulations. The computed global spectrum shows an unstable branch consisting of three-dimensional boundary layer modes whose amplitude distributions exhibit typical characteristics of both attachment-line and crossflow modes. In particular, global eigenfunctions with smaller phase velocities display a more pronounced structure near the stagnation line, reminiscent of attachment-line modes while still featuring strong crossflow vortices further downstream. This analysis establishes a link between the two prevailing instability mechanisms on a swept parabolic body which, so far, have only been studied separately and locally. A parameter study shows maximum modal growth for a spanwise wavenumber of β = 0.213, suggesting a preferred disturbance length scale in the sweep direction.

Time analyticity and backward uniqueness of weak solutions of the Navier–Stokes equations of multidimensional compressible flow

We prove that solutions of the Navier–Stokes equations of three-dimensional compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. As consequences we derive backward uniqueness of solutions as well as sharp rates of smoothing for higher-order Lagrangean time derivatives. The solutions under consideration are in a reasonably broad regularity class corresponding to small-energy initial data with a small degree of regularity, the latter being required for conversion to the Lagrangean coordinate system in which the analysis is carried out.

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R 3 and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.

Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems

This paper is devoted to the study of a transonic shock in three-dimensional steady compressible flow passing a duct with a general section. The flow is described by the steady full Euler system, which is purely hyperbolic in the supersonic region and is of elliptic-hyperbolic type in the subsonic region. The upstream flow at the entrance of the duct is a uniform supersonic one adding a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic-hyperbolic system. The new ingredients of our paper contain the decomposition of the elliptic-hyperbolic system, the determination of the shock front by a pair of partial differential equations coupled with the three-dimensional Euler system, and the regularity analysis of solutions to the boundary value problems introduced in our discussion.

Multi-dimensional limiting process for three-dimensional flow physics analyses

The present paper deals with an efficient and accurate limiting strategy for multi-dimensional compressible flows. The multi-dimensional limiting process (MLP) which was successfully proposed in two-dimensional case [K.H. Kim, C. Kim, Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows. Part II: Multi-dimensional limiting process, J. Comput. Phys. 208 (2) (2005) 570–615] is modified and refined for three-dimensional application. For computational efficiency and easy implementation, the formulation of MLP is newly derived and extended to three-dimensional case without assuming local gradient. Through various test cases and comparisons, it is observed that the newly developed MLP is quite effective in controlling numerical oscillation in multi-dimensional flows including both continuous and discontinuous regions. In addition, compared to conventional TVD approach, MLP combined with improved flux functions does provide remarkable increase in accuracy, convergence and robustness in steady and unsteady three-dimensional compressible flows.

Three-dimensional compressible–incompressible turbulent flow simulation using a pressure-based algorithm

In this work, we extend a finite-volume pressure-based incompressible algorithm to solve three-dimensional compressible and incompressible turbulent flow regimes. To achieve a hybrid algorithm capable of solving either compressible or incompressible flows, the mass flux components instead of the primitive velocity components are chosen as the primary dependent variables in a SIMPLE-based algorithm. This choice warrants to reduce the nonlinearities arose in treating the system of conservative equations. The use of a new Favre-averaging like technique plays a key role to render this benefit. The developed formulations indicate that there is less demand to interpolate the fluxes at the cell faces, which is definitely a merit. To impose the hyperbolic behavior in compressible flow regimes, we introduce an artificial hyperbolicity in pressure correction equation. We choose k– ω turbulence model and incorporate the compressibility effect as a correction. It is shown that the above considerations grant to achieve a robust algorithm with great capabilities in solving both flow regimes with a reasonable range of Mach number applications. To evaluate the ability of the new pressure-based algorithm, three test cases are targeted. They are incompressible backward-facing step problem, compressible flow over a wide range of open to closed cavities, and compressible turbulent flow in a square duct. The current results indicate that there are reliable agreements with those of experiments and other numerical solutions in the entire range of investigation.

In-cylinder charge stratification and fuel-air mixing in a new, low-emission two-stroke engine

A new air-assisted injection system-based two-stroke engine has been developed at the Indian Institute of Science, Bangalore over the past few years and shows the potential to meet emission norms such as Euro II and Euro III and yet deliver satisfactory performance. The 70 cc engine works on the direct mixture injection principle with scavenging performed by air alone instead of an air-fuel mixture. A small 20 cc pump driven off the engine is used for mixture preparation prior to in-cylinder injection of the mixture. In the present study, the mixture injection and subsequent charge stratification process inside the engine cylinder is modelled. A three-dimensional compressible flow code with a standard k-ε turbulence model with wall functions is developed and used for this modelling. To account for the moving boundary or piston in the engine cylinder domain, a non-stationary and deforming grid is used in this region with stationary cells in the ports and connecting ducts. A flux conservation scheme is used in the domain interface to allow the in-cylinder moving mesh to slide past the fixed port meshes. The initial conditions for flow parameters are taken from the output of a three-dimensional scavenging simulation. The state of the inlet charge is obtained from a separate modelling of the air-assisted injection system of this engine. The simulation results show that a large, near-stoichiometric region is present at most operating conditions in the cylinder head plane. The state of the in-cylinder charge at the onset of ignition is studied, leading to a good understanding of the mixing process. In addition, the sensitivity of two critical parameters in the mixing and stratification is investigated. The suggested parameters substantially enhance the flammable proportion at the onset of combustion. The predicted pressure-crank angle history from a combustion simulation supports this recommendation.

Towards stable coupling methods for high-order discretization of fluid–structure interaction: Algorithms and observations

Weak coupling of an explicit spectral/hp finite element ALE fluid solver with a Chebyshev collocation structural solver for fluid–structure interaction problems is addressed. When attempting to couple high-order spatial discretizations of both fluid and structural phenomena, we are required to consider first principles in order to answer two important questions: (1) What information (e.g. forces, velocities, displacements) does one transfer and when? and (2) How does one transfer information? Our goal is to address these questions by considering a non-conventional transfer of velocities of the wet-surface from the structure to the fluid, and the usual transfer of pressures on the wet-surface from the fluid to the structure. A subsonic three-dimensional compressible flow over an elastic non-linear plate model is considered as a representative example problem and we explore the various function spaces in which data (such as pressures, velocities and displacements) lie. We provide arguments as to what projection algorithms to use in combination with a time-staggering scheme to achieve stable and accurate results and at the same time to retain a temporal second-order scheme. We demonstrate the proposed algorithms by numerical examples considering long time integration.

Algebraic and exponential instability of inviscid swirling flow

In this paper we consider the spectrum and stability properties of small-amplitude waves in three-dimensional inviscid compressible swirling flow with non-zero mean vorticity, contained in an infinitely long annular circular cylinder. The mean flow has swirl and sheared axial components which are general functions of radius. We describe the form of the spectrum, in particular the three distinct types of disturbance: sonic (or acoustic) modes; nearly-convected modes; and the non-modal continuous spectrum. The phenomenon of accumulation of infinitely many eigenvalues of the nearly-convected type in the complex wavenumber-plane is classified carefully: we find two different regimes of accumulating neutral modes and one regime of accumulating instability modes, and analytic conditions for the occurrence of each type of behaviour are given. We also discuss the Green's function for the unsteady field, and in particular the contribution made by the continuous spectrum. We show that this contribution can grow algebraically downstream, and is responsible for a new type of convective instability. The algebraic growth rate of this instability is a complicated function of the mean flow parameters, and can be arbitrarily large as a function of radius in cases in which the local convected wavenumber has a local extremum. The algebraic instability we describe is additional to any conventional modal instability which may be present, and indeed we exhibit cases which are convectively stable to modes, but which nevertheless grow algebraically downstream.

Direct Spatial Resonance in the Compressible Boundary Layer on a Rotating-disk

This paper focuses on a resonance mechanism that can lead to significant disturbance amplification at conditions which are sub-critical to nonlinear instabilities. Particularly, direct spatial resonance instability is investigated, which is present in the basic three-dimensional viscous compressible boundary-layer flow due to a rotating-disk. Within this purpose, the linearized system of stability equations is treated numerically making use of a spectral Chebyshev collocation method. The analysis provides critical resonant Reynolds numbers above which growth occurs. Amplitudes of the response of the degeneracies decaying rapidly due to their high damping rates are shown to exist for small enough Reynolds numbers while the flow is still in the laminar state. If the flow is restricted to the incompressible case, the results of Turkyilmazoglu and Gajjar (in Sadhana Acad P Engs 25:601–617, 2000) are completely reproduced. The influences of compressibility are then explored by means of varying the Mach and Prandtl numbers in the cases of heating/cooling the wall as well as the isothermal wall. In general, compressibility effects are found strongly in favor of stabilizing as the Mach number increases, while a strong destabilization is observed by lowering the critical values of Reynolds numbers in the cases of wall heating and insulation. The modal interaction and coalescence of the eigenmodes calculated here create local algebraic growth by rapid development of relatively large amplitudes which might then provide the onset of nonlinear effects followed by transition.

Numerical modeling of three-dimensional compressible gas flow in microchannnels

Nitrogen gas flow in long microchannels with square cross-sections was simulated numerically with a three-dimensional continuum model with slip and no-slip boundary conditions. The governing equations of the model were solved by a control volume method. The numerical model was validated with the available experimental and numerical results. For incompressible flow, it was found that when Dh was less than 60 µm, a slip boundary condition must be applied. An analytical expression for normalized friction coefficients, C*IC, i.e. the ratio of f Re (slip) to f Re (no-slip), was developed on the basis of incompressible flow behavior. For compressible flow, a parametric study was conducted for Dh = 1 µm, L/Dh = 200 and with varying pressure ratios (PR = 1.5–5.0). It was found that as the pressure ratio increased from 1.5 to 5.0, compressibility effects increased while the rarefaction effects started diminishing. Slip effects also played an important role in the friction characteristics of microchannel flows. An analytical expression for normalized friction coefficients, C*C, i.e. the ratio of f Re (compressible) to f Re (incompressible), was developed on the basis of flow behavior for compressible flow. A comparative study of two-dimensional and three-dimensional flows was also conducted, and it was shown that the two-dimensional assumption for the compressible flow was not valid since it predicted 15–45% higher flow velocities, and 7–12% lower friction factors than those predicted by the 3D models.

Computational turbulent stress closure for large-eddy simulation of compressible flow

Large-eddy simulation (LES) focuses on the primary features of a turbulent flow with length scales larger than an externally specified filter width Δ. Scales smaller than Δ are removed from the simulation by spatial filtering which allows the use of a grid spacing h on the order of Δ. In most LES, the subgrid resolution r = Δ/h is rather coarse (r≈ 1–2) and an additional filtering is induced by the spatial discretization method. This modifies the dynamics of the smaller resolved scales. The computational turbulent stress tensor ξ ij expresses this modification and incorporates the regular turbulent stress as well as numerical high-pass filter contributions. We consider the computational turbulent stress for three-dimensional compressible flow. Compressibility motivates a decomposition of ξ ij in a part that is primarily associated with velocity variations and a part associated with density variations. The density variations depend strongly on the Mach number while the velocity variations remain much closer to their incompressible level, even in low supersonic flow. An a priori investigation of a turbulent mixing layer quantifies the significance of the density variations at sufficiently high Mach number with a magnitude of up to 20% of ξ ij . An explicit modelling of density variations in terms of approximate deconvolution is proposed and shown to accurately follow the Mach number dependence.

An efficient parallel/unstructured-multigrid preconditioned implicit method for simulating 3D unsteady compressible flows with moving objects

This paper presents the development and validation of a new parallel unstructured multi-grid preconditioned implicit method for the simulation of three-dimensional (3D) unsteady compressible flows using a so-called immersed membrane method (IMM) [G.H. Xia, Y. Zhao, J.H. Yeo, An immersed membrane method for simulation of fluid–structure interaction in bio-fluid flows, in: 1st International BioEngineering Conference (IBEC 2004), 08–10 September 2004]. The novel feature of the method presented is that it employs a unique combination of a parallel multi-grid scheme with low-Mach-number preconditioning and the IMM so that 3D unsteady low-Mach-number flows with arbitrarily moving objects can be efficiently simulated.

Validation Study of a Multidomain Spectral Code for Simulation of Turbulent Flows

A Chebyshev multidomain staggered-grid method (CMSM) presented by Kopriva for numerical simulation of three-dimensional compressible turbulent flows, is validated. Computations with the CMSM of an isotropic turbulence and a fully developed turbulent channel flow are compared with previously published results. Turbulence initiation of the isotropic turbulence on the Chebyshev grid is discussed. Low-resolution simulations ensure transition of an initial laminar channel flow to a turbulent channel flow. The turbulent kinetic energy spectrum has sufficient dropoff (indicating a resolved solution) at a minimum of 3 to 6 points per wave number for a polynomial approximation of 13 to 10, respectively

Simulation des Grandes Échelles d'écoulements turbulents compressibles dans des conduits courbes : étude des transferts thermiques

Large Eddy Simulations of three-dimensional turbulent compressible flows in curved duct of rectangular cross-section are presented. Strong secondary flows and turbulent structures are the typical features of this type of flow: these are responsible for most of the heat and mass transfer within the duct. Our goal is here to investigate the influence of the duct aspect ratio on the flow development. The results show that this parameter modifies the flow structures in terms of intensity and localisation. Thus it significantly influences the heat transfers when a higher temperature is imposed on the duct convex wall.

Multiple spatial scales in engineering and atmospheric low Mach number flows

The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a single spatial scale reproduce automatically the zero Mach number variable density flow equations for the small scales, and the linear acoustic equations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number of well-known simplified equations of theoretical meteorology can be derived in a unified fashion directly from the three-dimensional compressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are. however, characterized by several singular perturbation parameters that appear in addition to the Mach number; and that are defined independently of any particular length or time scale associated with some specific flow phenomenon. These are the ratio of the centripetal acceleration due to the earth's rotation vs. the acceleration of gravity, and the ratio of the sound speed vs. the rotational velocity of points on the equator. To systematically incorporate these parameters in an asymptotic approach, we couple them with the square root of the Mach number in a particular distinguished so that we are left with a single small asymptotic expansion parameter, e. Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of £ as well. Next we consider a very general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales. Various restrictions of the general ansatz to only one horizontal, one vertical. and one time scale lead directly to the family of simplified model equations mentioned above. Of course, the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family. In this context we will summarize a recent systematic development of multiscale models for the tropics (with Majda).

High-Reynolds Number Viscous Flow Computations Using an Unstructured-Grid Method

An unstructured grid method is presented to compute three-dimensional compressible turbulent flows for complex geometries. The Navier-Stokes equations, along with the one-equation turbulence model of Spalart-Allmaras are solved by the use of a parallel, matrix-free implicit method on unstructured tetrahedral grids. The developed method has been used to predict drags in the transonic regime for both DLR-F4 and DLR-F6 configurations to assess the accuracy and efficiency of the method

Prediction of the compression pressure wave generated by a high-speed train entering a tunnel

The pressure history of the compression acoustic wave generated by a high-speed train tunnel entry is calculated in a finite element flow analysis. The generation of the pressure wave and its propagation into the tunnel is numerically simulated. To obtain an accurate prediction of the pressure gradient, a three-dimensional compressible transient finite element flow analysis with arbitrary Lagrangian–Eulerian (ALE) formulation is performed on a multi-domain parallel iterative solver. Also an alternative approximation method of predicting the pressure wave shape based on “v-wall” method is developed. The result of the numerical simulation is compared with that of the 1/50th-scale experimental test.

Global Helically Symmetric Solutions for the Stokes Approximation Equations for Three-Dimensional Compressible Viscous Flows

Global Helically Symmetric Solutions for the Stokes Approximation Equations for Three-Dimensional Compressible Viscous Flows