Incompressible Case Research Articles

Three-dimensional instability of a flow past a sphere: Mach evolution of the regular and Hopf bifurcations

A fully three-dimensional linear stability analysis is carried out to investigate the unstable bifurcations of a compressible viscous fluid past a sphere. A time-stepper technique is used to compute both equilibrium states and leading eigenmodes. In agreement with previous studies, the numerical results reveal a regular bifurcation under the action of a steady mode and a supercritical Hopf bifurcation that causes the onset of unsteadiness but also illustrate the limitations of previous linear approaches, based on parallel and axisymmetric base flow assumptions, or weakly nonlinear theories. The evolution of the unstable bifurcations is investigated up to low-supersonic speeds. For increasing Mach numbers, the thresholds move towards higher Reynolds numbers. The unsteady fluctuations are weakened and an axisymmetrization of the base flow occurs. For a sufficiently high Reynolds number, the regular bifurcation disappears and the flow directly passes from an unsteady planar-symmetric solution to a stationary axisymmetric stable one when the Mach number is increased. A stability map is drawn by tracking the bifurcation boundaries for different Reynolds and Mach numbers. When supersonic conditions are reached, the flow becomes globally stable and switches to a noise-amplifier system. A continuous Gaussian white noise forcing is applied in front of the shock to examine the convective nature of the flow. A Fourier analysis and a dynamic mode decomposition show a modal response that recalls that of the incompressible unsteady cases. Although transition in the wake does not occur for the chosen Reynolds number and forcing amplitude, this suggests a link between subsonic and supersonic dynamics.

Improved a priori bounds for thermal fluid equations

We consider two hydrodynamic model problems (one incompressible and one compressible) with three dimensional fluid flow on the torus and temperature-dependent viscosity and conductivity. The ambient heat for the fluid is transported by the flow and fed by the local energy dissipation, modeling the transfer of kinetic energy into thermal energy through fluid friction. Both the viscosity and conductivity grow with the local temperature. We prove a strong a priori bound on the enstrophy of the velocity weighed against the temperature for initial data of arbitrary size, requiring only that the conductivity be proportionately larger than the viscosity (and, in the incompressible case, a bound on the temperature as a Muckenhoupt weight).

Unusual extension–torsion–inflation couplings in pressurized thin circular tubes with helical anisotropy

We present a thin tube formulation for coupled extension–torsion–inflation deformation in helically reinforced pressurized circular tubes. Both compressible and incompressible tubes are considered. On applying the thin tube limit, the nonlinear ordinary differential equation to obtain the in-plane radial displacement is converted into a set of two simple algebraic equations for the compressible case and one equation for the incompressible case. This allows us to obtain analytical expressions, in terms of the tube’s intrinsic twist, material constants, and the applied pressure, which can predict whether such tubes would overwind/unwind on being infinitesimally stretched or exhibit positive/negative Poisson’s effect. We further show numerically that such tubes can be tuned to generate initial overwinding followed by rapid unwinding as observed during finite stretching of a torsionally relaxed DNA. Finally, we demonstrate that such tubes can also exhibit usual deflation initially followed by unusual inflation as the tube is finitely stretched.

Nonlinear aeroelastic analysis of high-aspect-ratio wings using indicial aerodynamics

A new approach for calculating the unsteady aerodynamic loads based upon the indicial functions concept in combination with a fully third-order nonlinear structural model has been developed to analyze the aeroelastic behavior of high-aspect-ratio wings over the entire range of subsonic flow. The resulting aeroelastic equations including all structural geometric nonlinearities associated with large deformations and mass distributions, along with nonlinear terms due to mass imbalance at wing’s cross section, are then rewritten in the state-space form, introducing an efficient and appropriate approach to use in both eigenvalue and time response analysis. To validate the developed aeroelastic equations, the linear and nonlinear aeroelastic behaviors of a specified wing are compared with those presented for an incompressible aerodynamic case. Quantitative and qualitative agreement between the present results and available ones confirms the unsteady indicial aerodynamics, nonlinear structural modeling, and, consequently, the developed nonlinear aeroelastic model. By changing the wing model and applying the unsteady compressible aerodynamic loads, the nonlinear aeroelastic behavior of Goland wing is then investigated, including the flutter boundary, limit cycle oscillations, pre-flutter, flutter, and post-flutter time responses, phase plane diagrams, and also the effect of flight conditions such as altitude and air speed on the aeroelastic behavior. The results represent the necessity of applying appropriate Mach-dependent aerodynamic loads to provide reasonable description of the aeroelastic analysis in the compressible flight speed regime.

Computation of compressible and incompressible flows with a space–time stabilized finite element method

This paper presents the numerical results of compressible and incompressible flow problems through a unified approach based on a stabilized space–time finite element method. The numerical approach is continuous in space and discontinuous in time. The proposed method starts by the use of a stabilized space–time variational formulation, which allows the use of the same order interpolation functions for all solution variables. The numerical technique is tested through comparison with standard compressible and incompressible flow benchmarks. Compressible flow cases include 1D and 2D shock problems. Incompressible flow cases include lid-driven cavity flow and flow over a backward-facing step computed over a range of Re numbers. The results demonstrate high stability and accuracy of the numerical technique over a wide range of flow regimes, suggesting straightforward extension to many flow cases not yet investigated.

A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction

Abstract The nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in next-to-lowest order Raviart–Thomas spaces is modified in such a way that its anti-symmetric part vanishes in average on each element; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the one for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the incompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations involving different Lamé parameters including the incompressible limit case.

A multiphase level-set approach for all-Mach numbers

In this work, an alternate level-set-based approach is presented that applies uniformly to compressible and incompressible multiphase flows. Fundamental to this work, is the development of analytic transformations from a signed-distance function to species-mass conservation variables. Such transformations can be used to highlight compressible flow difficulties for level set methods, and develop interfacial reinitialization procedures based on different primitive variables. The proposed all-Mach method is based on preserving signed-distance functions within the context of a species-mass conservation equation to evolve the interface, and includes several reinitialization procedures that maintain the spirit of the signed distance function. In addition, we explore hybrid level-set reinitialization procedures that handle sub-grid-scale interfacial breakup. The model is demonstrated on concepts relevant to high-speed marine vehicles based on supercavitation, where a gaseous cavity surrounds a moving vehicle. Results indicate that the present algorithm preserves higher-order numerics, performs well on several incompressible and compressible validation cases, and extends to unsteady, three-dimensional flow.

Energy transfer in compressible magnetohydrodynamic turbulence for isothermal self-gravitating fluids.

Three-dimensional, compressible, magnetohydrodynamic turbulence of an isothermal, self-gravitating fluid is analyzed using two-point statistics in the asymptotic limit of large Reynolds numbers (both kinetic and magnetic). Following an alternative formulation proposed by Banerjee and Galtier [Phys. Rev. E 93, 033120 (2016)2470-004510.1103/PhysRevE.93.033120; J. Phys. A: Math. Theor. 50, 015501 (2017)1751-811310.1088/1751-8113/50/1/015501], an exact relation has been derived for the total energy transfer. This approach results in a simpler relation expressed entirely in terms of mixed second-order structure functions. The kinetic, thermodynamic, magnetic, and gravitational contributions to the energy transfer rate can be easily separated in the present form. By construction, the new formalism includes such additional effects as global rotation, the Hall term in the induction equation, etc. The analysis shows that solid-body rotation cannot alter the energy flux rate of compressible turbulence. However, the contribution of a uniform background magnetic field to the flux is shown to be nontrivial unlike in the incompressible case. Finally, the compressible, turbulent energy flux rate does not vanish completely due to simple alignments, which leads to a zero turbulent energy flux rate in the incompressible case.

A locking-free stabilized mixed finite element method for linear elasticity: the high order case

In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k-order polynomial space is used for the stress and the discontinuous piecewise $$(k-1)$$ -order polynomial space for the displacement, where we require that $$k\ge 3$$ in the two dimensions and $$k\ge 4$$ in the three dimensions. The method is proved to be stable and k-order convergent for the stress in $$H(\mathrm {div})$$ -norm and for the displacement in $$L^2$$ -norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method.

Discrete Functional Analysis Tools for Some Evolution Equations

Abstract We present some discrete functional analysis tools for the proof of convergence of numerical schemes, mainly for equations including diffusion terms such as the Stefan problem or the Navier–Stokes equations in the incompressible and compressible cases. Some of the results covered here have been proved in previous works, coauthored with several coworkers.

A Priori Estimates for the Compressible Euler Equations for a Liquid with Free Surface Boundary and the Incompressible Limit

In this paper, we prove a new type of energy estimate for the compressible Euler equations with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed k. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin to when you have a free boundary. In the incompressible case our energies reduce to those in Christodoulou and Lindblad, and our proof in particular gives a simplified proof of their estimates with improved error estimates. Since for an incompressible irrotational liquid with free surface there are small data global existence results, our result leaves open the possibility of long‐time existence also for slightly compressible liquids with a free surface.© 2017 Wiley Periodicals, Inc.

Turbulence kinetic energy exchanges in flows with highly variable fluid properties

This paper investigates the energy exchanges associated with the half-trace of the velocity fluctuation correlation tensor in a strongly anisothermal low Mach fully developed turbulent channel flow. The study is based on direct numerical simulations of the channel within the low Mach number hypothesis and without gravity. The overall flow behaviour is governed by the variable fluid properties. The temperature of the two channel walls are imposed at 293 K and 586 K to generate the temperature gradient. The mean friction Reynolds number of the simulation is 180. The analysis is carried out in the spatial and spectral domains. The spatial and spectral studies use the same decomposition of the terms of the evolution equation of the half-trace of the velocity fluctuation correlation tensor. The importance of each term of the decomposition in the energy exchanges is assessed. This lets us identify the terms associated with variations or fluctuations of the fluid properties that are not negligible. Then, the behaviour of the terms is investigated. The spectral energy exchanges are first discussed in the incompressible case since the analysis is not present in the literature with the decomposition used in this study. The modification of the energy exchanges by the temperature gradient is then investigated in the spatial and spectral domains. The temperature gradient generates an asymmetry between the two sides of the channel. The asymmetry can in a large part be explained by the combined effect of the mean local variations of the fluid properties, combined with a Reynolds number effect.

A Discrete Adjoint Formulation for Inviscid Flow Nozzle Optimization

A discrete adjoint formulation with an ad hoc flow solver has been recently developed and tested on transonic inviscid flow optimization problems. In the present paper the formulation is extended to compressible as well as incompressible flow solvers. First, the adjoint equations are coupled with an accurate in-house flow solver to test the approach on some inverse design problems involving two- and three-dimensional transonic and subsonic flows. Then, the previous design test cases are re-computed coupling the extended adjoint formulation with commercial and open source flow solvers, without noticing any relevant difference in the optimization convergence histories. Finally, incompressible design test cases are successfully computed by means of commercial solver for incompressible flows. The extended compressible adjoint formulation appears to have a wide application, insofar as it allows to perform accurate and efficient design optimization using different flow conditions, different flow solvers and even a solver for incompressible flows.

Numerical measurements of scaling relations in two-dimensional conformal fluid turbulence

We present measurements of relativistic scaling relations in (2+1)-dimensional conformal fluid turbulence from direct numerical simulations, in the weakly compressible regime. These relations were analytically derived previously in [1] for a relativistic fluid; this work is a continuation of that study, providing further analytical insights together with numerical experiments to test the scaling relations and extract other important features characterizing the turbulent behavior. We first explicitly demonstrate that the non-relativistic limit of these scaling relations reduce to known results from the statistical theory of incompressible Navier-Stokes turbulence. In simulations of the inverse-cascade range, we find the relevant relativistic scaling relation is satisfied to a high degree of ac-curacy. We observe that the non-relativistic versions of this scaling relation underperform the relativistic one in both an absolute and relative sense, with a progressive degradation as the rms Mach number increases from 0.14 to 0.19. In the direct-cascade range, the two relevant relativistic scaling relations are satisfied with a lower degree of accuracy in a simulation with rms Mach number 0.11. We elucidate the poorer agreement with further simulations of an incompressible Navier-Stokes fluid. Finally, as has been observed in the incompressible Navier-Stokes case, we show that the energy spectrum in the inverse-cascade of the conformal fluid exhibits k−2 scaling rather than the Kolmogorov/Kraichnan expectation of k−5/3, and that it is not necessarily associated with compressive effects. We comment on the implications for a recent calculation of the fractal dimension of a turbulent (3 + 1)-dimensional AdS black brane.

Visualization of Görtler vortices in supersonic concave boundary layer

By employing a nanoparticle-based planar laser scattering technique, with inflow Mach number of 2.95, boundary layers formed over two concave walls with respective curvature radii of 113 and 908 mm are visualized. The formation and the breakup process of the Gortler vortices have been clearly captured, which has seldom been done before under the supersonic condition. Different from the incompressible case, the breakup of the Gortler vortices in the supersonic boundary layer is found to be mainly attributed to the varicose mode instability. The Gortler vortices are not stationary, instead they are moving in the spanwise direction.

Compressible Turbulence Model Consistency for Separated High-Speed Flow Regimes

Fluid compressibility can remarkably modify the behavior of turbulent flow with respect to the incompressible case. The presence of strong pressure-induced density changes adds to the turbulence phenomenology new specific processes and mechanisms, such as new pathways for energy exchanges and a strong coupling between momentum and energy exchanges, significantly altering the flow behavior. Many research groups are working on methods to include these compressibility effects in conventional turbulence models to predict the flowfield for compressible conditions more realistically. This work aims at the assessment and validation of several compressible corrections to an in-house hypersonic multigrid reactive Navier–Stokes code within the “baseline” Spalart–Allmaras turbulence model. The base pressure enhancement for supersonic afterbody flow computations is first addressed. Then, the shock/turbulent boundary-layer interaction at the junction of a cone/flare configuration is dealt with. The analysis of the numerical results and their comparison with the experimental data show that, in most cases, the proposed corrections produce a significant simulation improvement. Finally, an in-house hypersonic wind-tunnel test campaign is performed on generic blunt-cone/flare configurations. The pressure distributions along the model as well as the forces and moments are used to calibrate the numerical calculations in the range of Mach numbers of 4.8 to 6.2.

Simultaneous influence of helicity and compressibility on anomalous scaling of the magnetic field in the Kazantsev-Kraichnan model.

Using the field theoretic renormalization group technique and the operator product expansion, the systematic investigation of the influence of the spatial parity violation on the anomalous scaling behavior of correlation functions of the weak passive magnetic field in the framework of the compressible Kazantsev-Kraichnan model with the presence of a large-scale anisotropy is performed up to the second order of the perturbation theory (two-loop approximation). The renormalization group analysis of the model is done and the two-loop explicit expressions for the anomalous and critical dimensions of the leading composite operators are found as functions of the helicity and compressibility parameters and their anisotropic hierarchies are discussed. It is shown that for arbitrary values of the helicity parameter and for physically acceptable (small enough) values of the compressibility parameter, the main role is played by the composite operators near the isotropic shell in accordance with the Kolmogorov's local isotropy restoration hypothesis. The anomalous dimensions of the relevant composite operators are then compared with the anomalous dimensions of the corresponding leading composite operators in the Kraichnan model of passively advected scalar field. The significant difference between these two sets of anomalous dimensions is discussed. The two-loop inertial-range scaling exponents of the single-time two-point correlation functions of the magnetic field are found and their dependence on the helicity and compressibility parameters is studied in detail. It is shown that while the presence of the helicity leads to more pronounced anomalous scaling for correlation functions of arbitrary order, the compressibility, in general, makes the anomalous scaling more pronounced in comparison to the incompressible case only for low-order correlation functions. The persistence of the anisotropy deep inside the inertial interval is investigated using the appropriate odd ratios of the correlation functions. It is shown that, in general, the persistence of the anisotropy is much more pronounced in the helical systems, while in the compressible turbulent environments this is true only for low-order odd ratios of the correlation functions.

Compressible starting jet: pinch-off and vortex ring–trailing jet interaction

The dominant feature of the compressible starting jet is the interaction between the emerging vortex ring and the trailing jet. There are two types of interaction: the shock–shear layer–vortex interaction and the shear layer–vortex interaction. The former is clearly not present in the incompressible case, since there are no shocks. The shear layer–vortex interaction has been reported in the literature in the incompressible case and it was found that compressibility reduces the critical Reynolds number for the interaction. Four governing parameters describe the compressible starting jet: the non-dimensional mass supply, the Reynolds number, the reservoir to unbounded chamber temperature ratio and the reservoir to unbounded chamber pressure ratio. The latter parameter does not exist in the incompressible case. For large Reynolds numbers, the vortex pinch-off takes place in a multiple way. We studied the compressible starting jet numerically and found that the interaction strongly links the vortex ring and the trailing jet. The shear layer–vortex interaction leads to a rapid breakdown of the head vortex ring when the flow impacted by the Kelvin–Helmholtz instabilities is ingested into the head vortex ring. The shock–shear layer–vortex interaction is similar to the noise generation mechanism of broadband shock noise in a continuously blowing jet and results in similar sound pressure amplitudes in the far field.

Global Existence and Asymptotic Behavior of Affine Motion of 3D Ideal Fluids Surrounded by Vacuum

The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $\rm{GL}^+(3,\mathbb R)$. The evolution of the fluid domain is described by a family ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank $r=1$, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along $3-r$ of its principal axes. In the compressible case, the asymptotic limit has rank $r=3$, and asymptotic completeness holds, when the adiabatic index $\gamma$ satisfies $4/3<\gamma<2$. The number of possible degeneracies, $3-r$, increases with the value of the adiabatic index $\gamma$. In the incompressible case, affine motion reduces to geodesic flow in $\rm{SL}(3,\mathbb R)$ with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes.

Compressibility effect on coherent structures, energy transfer, and scaling in magnetohydrodynamic turbulence

Compressible magnetohydrodynamic (MHD) turbulence, a model often used to study space and astrophysical plasmas, differs from incompressible magnetohydrodynamic and hydrodynamic (HD) turbulence in many ways. Here direct numerical simulations of mechanically forced compressible MHD turbulence are used to study the degree to which some turbulence theories proposed in the incompressible case remain applicable in the compressible one. Several aspects of compressible turbulence are studied: (i) Intermittency in the compressible case is studied by addressing flows driven with varying forcing mechanisms; these display different features, such as compression and coherent structures. The more compressive simulation is characterized by sheet-like current density structures and shocks, which lead to saturated scaling exponents of high order structure functions of density and compressive velocity. (ii) Further investigations employing conditional averages of different energy transfer fluxes reveal that the influence of shocks (or compressions) makes differences in energy cascade and magnetic amplification. (iii) Cascade is also studied by examining the validity of Yaglom-type relations that verify the scale invariant property within the inertial range, where the transfer is relatively free of the effect of pressure dilation.