Incompressible Case Research Articles

Wrinkling of compressible magnetic soft plates

This paper establishes an analytical method for the wrinkling of compressible magnetic soft (MS) plates subject to an in-plane biaxial stretching and an out-of-plane magnetic induction field. The bifurcation analysis is performed with external Maxwell stress considered by combining the surface impedance matrix method and the Stroh formulation in terms of true magnetic field variables. We decouple the resulting bifurcation equations into antisymmetric and symmetric modes and provide the explicit expressions within a neo-Hookean ideal magnetoelastic model. Numerical examples show that the antisymmetric wrinkling usually occurs prior to the symmetric one, unless the permeability of the plates μ is much smaller than that of the surroundings μ′, i.e., the normalized permeability μ/μ′→0. This observation is consistent with the previous studies on incompressible case. However, for nearly incompressible plates with μ/μ′>1, the compressible constitutive relation may impose an additional deformation constraint that noticeably limits the occurrence and extent of wrinkling in the plates. One intriguing observation in particular is that the critical stretches for the thin-plate instability exhibit a nonmonotonic character as the compressibility of plate varies. Release of compressibility plays a positive role on stabilizing the MS plates when 0<μ/μ′<1, yet a negative role when μ/μ′>1. This phenomenon may be attributed to the coupling effect between the compressibility and the normalized permeability μ/μ′, suggesting a potential way to regulate wrinkling behaviors of MS materials by tuning the surrounding permeability. The present work may serve as benchmark solutions for understanding structural failures in various related functional MS-based devices.

Three-dimensional eigensolutions of the unsteady compressible boundary-region equations

A previous paper of the authors (Duck &amp; Stephen, J. Fluid Mech., vol. 917, 2021, A56) considered the effect of three-dimensional, temporally periodic, linear and incompressible disturbances on a Blasius boundary layer, in particular when the disturbance wavelength is both comparable to and longer than the boundary-layer thickness. This previous study revealed that, unlike the two-dimensional counterpart, a mode exists that exhibits regimes of downstream spatial growth. In this paper we extend the analysis to the compressible regime, based on the boundary-region equations methodology. The aforementioned unstable mode is seen to persist into the compressible regime, and is studied using a combination of numerical and asymptotic methods. The paper adopts several approaches. First is a numerical approach in which the spatial development of the disturbances is assessed. This then leads to a consideration of the far-downstream behaviour, using (several) asymptotic limits. Of some note, in addition to unstable modes found in the incompressible case, is the existence of a further class of instability, not found in the incompressible case (which is also analysed asymptotically), corresponding to what amounts to an inviscid instability. The far-downstream analysis enables a (sub-)classification into entropy and non-entropy modes. The former, according to this analysis, are spatially damped, with one caveat, as revealed by our marching procedure, which highlights how spatial development of disturbances can be important.

Atmospheric pressure-induced three-dimensional surface wave propagation in the compressible ocean: Effect of static compression

Under the assumptions of linearized water wave theory, we build a three-dimensional mathematical model that couples atmospheric pressure waves and surface ocean waves, including water compressibility and its static part, to simulate Meteotsunami propagation in the ocean. The solution uses the Laplace–Fourier double transformation technique, emphasizing axisymmetry of the mathematical problem and rigorous treatment of a fairly complicated dispersion relation while using inverse transformations. A novel derivation of the axisymmetric atmospheric pressure front is shown. The impact of water compressibility is shown through a comparative graphical representation against the incompressible case. Faster travel of free-surface waves is observed in the incompressible ocean, followed by the cases with and without static compression of the compressible ocean, respectively. The static compression shifts the phase of the acoustic-gravity modes. The locked wave is hardly influenced by the water compressibility and is entangled with the moving pressure front. The model is validated with the observational pressure data and agrees well with our computed pressure profile. Then, the locked wave profile generated from our model agrees well with the deep-ocean assessment and reporting of tsunami data.

Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework

We consider the validity of Prandtl boundary layer expansion of solutions to the initial boundary value problem for inhomogeneous incompressible magnetohydrodynamics equations in the half-plane when both viscosity and resistivity coefficients tend to zero, where the no-slip boundary condition is imposed on velocity while the perfectly conducting condition is given on magnetic field. Since there exist strong boundary layers, the essential difficulty in establishing the uniform L∞ estimates of the error functions comes from the unboundedness of vorticity of strong boundary layers. Under the assumptions that the viscosity and resistivity coefficients take the same order of a small parameter and the initial tangential component of magnetic field has a positive lower bound near the boundary, we prove the validity of Prandtl boundary layer ansatz in L∞ sense in Sobolev framework. Compared with the homogeneous incompressible case considered in [33], there exists a strong boundary layer of density. Consequently, some suitable functionals should be designed and the elaborated co-normal energy estimates will be involved in analysis due to the variation of density and the interaction between the density and velocity.

Non-universality and dissipative anomaly in compressible magnetohydrodynamic turbulence

We systematically study the dissipative anomaly in compressible magnetohydrodynamic (MHD) turbulence using direct numerical simulations, and show that the total dissipation remains finite as viscosity diminishes. The dimensionless dissipation rate $\mathcal {C}_{\varepsilon }$ fits well with the model $\mathcal {C}_{\varepsilon } = \mathcal {C}_{\varepsilon,\infty } + \mathcal {D}/R_L^-$ for all levels of flow compressibility considered here, where $R_L^-$ is the generalized large-scale Reynolds number. The asymptotic value $\mathcal {C}_{\varepsilon,\infty }$ describes the total energy transfer flux, and decreases with increase of the flow compressibility, indicating non-universality of the dimensionless dissipation rate in compressible MHD turbulence. After introducing an empirically modified dissipation rate, the data from compressible cases collapse to a form similar to the incompressible MHD case depending only on the modified Reynolds number.

Novel pressure-equilibrium and kinetic-energy preserving fluxes for compressible flows based on the harmonic mean

Employing physically-consistent numerical methods is an important step towards attaining robust and accurate numerical simulations. When addressing compressible flows, in addition to preserving kinetic energy at a discrete level, as done in the incompressible case, additional properties are sought after, such as the ability to preserve the equilibrium of pressure that can be found at contact interfaces. This paper investigates the general conditions of the spatial numerical discretizations to achieve the pressure equilibrium preserving property (PEP). Schemes from the literature are analyzed in this respect, and procedures to impart the PEP property to existing discretizations are proposed. Additionally, new PEP numerical schemes are introduced through minor modifications of classical ones. Numerical tests confirmed the theory hereby presented and showed that the modifications, beyond the enforcement of the PEP property, have a generally positive impact on the performances of the original schemes.

Linear stability analysis of compressible boundary layer over an insulated wall using compound matrix method: Existence of multiple unstable modes for Mach number beyond 3

The linear spatial stability of a parallel two-dimensional (2D) compressible boundary layer on an adiabatic plate is investigated by considering both 2D and three-dimensional (3D) disturbances. The compound matrix method is employed here, for the first time, for compressible flows, which, unlike other conventional techniques, can efficiently eliminate the stiffness of the equations governing the spectral amplitudes. The method is first validated with published results in the literature corresponding to spatial and temporal instability of flows ranging from low subsonic to high supersonic Mach numbers (M), which shows a good match depending upon the proper choice of free-stream temperature and the wall dispersion relation. Subsequently, flow compressibility effects and the spanwise variation of disturbances are also investigated for M ranging from low subsonic to high supersonic cases (from M = 0.1 to 6). Mack (AGARD Report No. 709, 1984) reported the existence of two unstable modes for M &amp;gt; 3 from viscous calculations (the so-called “second mode”) that subsequently fuse to create only one unstable zone when M increases. Our calculations show a series of higher-order unstable modes for M &amp;gt; 3 in addition to the findings of Ma and Zhong [“Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions,” J. Fluid Mech. 488, 31–78 (2003)], where such higher-order modes for supersonic boundary layers are all noted to be spatially stable. The number and the frequency extent of the corresponding unstable zones increase with an increase in M beyond 3 while propagating downstream at a higher speed than those corresponding to incompressible, subsonic, and low supersonic (M &amp;lt; 2) cases.

Universal deformations and inhomogeneities in isotropic Cauchy elasticity

For a given class of materials, universal deformations are those deformations that can be maintained in the absence of body forces and by applying solely boundary tractions. For inhomogeneous bodies, in addition to the universality constraints that determine the universal deformations, there are extra constraints on the form of the material inhomogeneities—universal inhomogeneity constraints. Those inhomogeneities compatible with the universal inhomogeneity constraints are called universal inhomogeneities. In a Cauchy elastic solid, stress at a given point and at an instance of time is a function of strain at that point and that exact moment in time, without any dependence on prior history. A Cauchy elastic solid does not necessarily have an energy function, i.e. Cauchy elastic solids are, in general, non-hyperelastic (or non-Green elastic). In this paper, we characterize universal deformations in both compressible and incompressible inhomogeneous isotropic Cauchy elasticity. As Cauchy elasticity includes hyperelasticity, one expects the universal deformations of Cauchy elasticity to be a subset of those of hyperelasticity both in compressible and incompressible cases. It is also expected that the universal inhomogeneity constraints to be more stringent than those of hyperelasticity, and hence, the set of universal inhomogeneities to be smaller than that of hyperelasticity. We prove the somewhat unexpected result that the sets of universal deformations of isotropic Cauchy elasticity and isotropic hyperelasticity are identical, in both the compressible and incompressible cases. We also prove that their corresponding universal inhomogeneities are identical as well.

MATRICS: The implicit matrix-free Eulerian hydrodynamics solver

Context. There exists a zoo of different astrophysical fluid dynamics solvers, most of which are based on an explicit formulation and hence stability-limited to small time steps dictated by the Courant number expressing the local speed of sound. With this limitation, the modeling of low-Mach-number flows requires small time steps that introduce significant numerical diffusion, and a large amount of computational resources are needed. On the other hand, implicit methods are often developed to exclusively model the fully incompressible or 1D case since they require the construction and solution of one or more large (non)linear systems per time step, for which direct matrix inversion procedures become unacceptably slow in two or more dimensions. Aims. In this work, we present a globally implicit 3D axisymmetric Eulerian solver for the compressible Navier–Stokes equations including the energy equation using conservative formulation and a fully simultaneous approach. We use the second-order-in-time backward differentiation formula for temporal discretization as well as the κ scheme for spatial discretization. We implement different limiter functions to prohibit the occurrence of spurious oscillations in the vicinity of discontinuities. Our method resembles the well-known monotone upwind scheme for conservation laws (MUSCL). We briefly present efficient solution methods for the arising sparse and nonlinear system of equations. Methods. To deal with the nonlinearity of the Navier–Stokes equations we used a Newton iteration procedure in which the required Jacobian matrix-vector product was reconstructed with a first-order finite difference approximation to machine precision in a matrix-free way. The resulting linear system was solved either completely matrix-free with a combination of a sufficient Krylov solver and an approximate Jacobian preconditioner or semi-matrix-free with the incomplete lower upper factorization technique as a preconditioner. The latter was dependent on a standalone approximation of the Jacobian matrix, which was optionally calculated and needed solely for the purpose of preconditioning. Results. We show our method to be capable of damping sound waves and resolving shocks even at Courant numbers larger than one. Furthermore, we prove the method’s ability to solve boundary value problems like the cylindrical Taylor-Couette flow (TC), including viscosity, and to model transition flows. To show the latter, we recover predicted growth rates for the vertical shear instability, while choosing a time step orders of magnitude larger than the explicit one. Finally, we verify that our method is second order in space by simulating a simplistic, stationary solar wind.

A third-order compact finite volume scheme on unstructured grid for fluid flows

The large reconstruction stencil is one of the biggest obstacles in design and implementation of high-order finite volume schemes on unstructured grid. To tackle this problem, a third-order compact reconstruction scheme on unstructured grid is developed on the basis of compact least-squares (CLS) finite volume method. In building the reconstruction polynomial, the general boundary condition is adopted to ensure the unified third-order accuracy in both the internal and boundary cells. The reconstruction polynomial is solved in a semi-implicit sense to ensure high computational efficiency without solving the global linear algebraic equations. The theoretical accuracy order and stability limit are firstly verified by the Fourier analysis. The numerical accuracy and efficiency are then verified and validated with several incompressible and compressible flow cases. This work provides a reference to design and analysis of high-order compact finite volume schemes on unstructured grid with unified accuracy.

Localization of energies in Navier–Stokes models with reaction terms

We analyze a general class of coupled systems of stationary Navier–Stokes type equations with variable coefficients and non-homogeneous terms of reaction type in the incompressible case. Existence of solutions satisfying the homogeneous Dirichlet condition in a bounded domain in [Formula: see text], [Formula: see text], and localization results for the corresponding kinetic energy and enstrophy are obtained by using a variational approach and the fixed point index theory.

Preserving non-negative porosity values in a bi-phase elasto-plastic material under Terzaghi’s effective stress principle

Poromechanics is a well-established field of continuum mechanics which seeks to model materials with multiple phases, usually a stiff solid phase and fluid phases of liquids or gases. Applications are widespread particularly in geomechanics where Terzaghi’s effective stress is widely used to solve engineering soil mechanics problems. This approach assumes that the solid phase is incompressible, an assumption that leads to many advantages and simplifications without major loss of fidelity to the real world. Under the assumption of finite (as opposed to infinitesimal) strains, the poromechanics of two- or bi-phase materials gains complexity and while the compressible solid phase case has received attention from researchers, the incompressible case has received less. For the finite strain - incompressible solid phase case there is a fundamental issue with standard material models, in that for some loadings solid skeleton mass conservation is violated and negative Eulerian porosities are predicted. While, to the authors’ best knowledge, acknowledgement of this essential problem has been disregarded in the literature, an elegant solution is presented here, where the constraint on Eulerian porosity can be incorporated into the free energy function for a material. The formulation is explained in detail, soundly grounded in the laws of thermodynamics and validated on a number of illustrative examples.

Global existence and decay estimates of solutions for the compressible Prandtl type equations with small analytic data

This paper aims to address the issues of the global existence and the large-time decay estimates of strong solutions for the two-dimensional compressible Prandtl equations with small initial data, which is analytical in the tangential variable. We investigate a more complicated system, which contains more physics than the incompressible system. Not only the loss of the horizontal derivative in the estimate of nonlinear terms, but also the failure to satisfy the divergence-free condition, may create significant difficulties when closing energy estimates. Motivated by Paicu and Zhang (2021) [36] for the incompressible case, we find a new quantity G=defu+y2〈t〉φ and derive a sufficiently fast decay-in-time estimate of some weighted analytic norm to G. Together with a linear combination of the tangential velocity u with its primitive one φ, which controls the evolution of the analytic radius to solutions ultimately. Another interesting feature of this paper is to take into account the influence of the density, however, the density is independent of the y variable. In order to avoid these obstacles, the approaches we take are some delicate treatments, including the special cut-off function and new inequalities in the process of energy estimates. This paper can be considered as a first global-in-time Cauchy–Kowalevsakya result for the compressible Prandtl equations with small analytic data.

Study of the linear models in estimating coherent velocity and temperature structures for compressible turbulent channel flows

Linear models, based on stochastically forced linearized equations, are deployed for spectral linear stochastic estimation (SLSE) of the velocity and temperature fluctuations in compressible turbulent channel flows with a bulk Mach number of 1.5. Through comparison with the direct numerical simulation (DNS) data, an eddy-viscosity-enhanced model (eLNS) outperforms the one not enhanced (LNS) in computing the coherence and amplitude ratio of streamwise velocity at different wall-normal heights, but they both largely deviate from DNS regarding the temperature prediction. For further investigation, the eigenspectra and pseudospectra of the linear operators are scrutinized. The eddy viscosity is shown to stabilize the eigenmodes and decrease the non-normality of the vortical modes. Consequently, the relative importance of acoustic and entropy modes increases, and they can contribute 20 % to 55 % of the response growth, which is not supported by DNS. Hence, it is an intrinsic defect of the eLNS model introduced by turbulence modelling. After a procedure of cospectrum decomposition, the contributions of acoustic and entropy components are filtered out. The resulting SLSE quantities for velocity, temperature and their coupling are basically agreeable with DNS, demonstrating that the coherent temperature fluctuation is dominated by advection and other vortical motions, instead of the compressibility effects. Moreover, a parameter study of Reynolds and Mach numbers (from 0.3 to 4) is conducted. The semi-local units are shown to well collapse the velocity SLSE quantities to the incompressible case for streamwise-elongated structures of high coherence.

Novel algorithm for compressible Newtonian axisymmetric thermal flow

In this paper, we mainly focus on the development of the Taylor-Galerkin/Pressure-Correction method (TG/PC) method to solve the flow of a compressible Newtonian fluid under nonisothermal conditions. The process of development needs applying the two-step Lax–Wendroff method on the energy equation, after which two new steps for energy equations will be obtained within the TG/PC algorithm. In addition, for the density component, the Tait equation of state is adopted to relate density to pressure, which also yields a new step within the novel method to give the correction formula of compressible pressure. To perform the analysis of the new algorithm, Poiseuille flow through axisymmetric rectangular channel for the Newtonian thermal flow is utilized as a simple problem test. The effect of nondimensional factors such as Reynolds number (Re) and Prandtl number (Pr) is discussed in this study. The influence of thermal conductivity [Formula: see text] and viscosity [Formula: see text] on the solution components is presented as well. Also, comparison for both compressible and incompressible flows is provided in addition to a comparison between both cases’ convergence, as it turns out that the convergence of the incompressible case is better than the compressible case. All the results of the effects presented in this study agree well with the approved physical trend.

Application extension of the meshless local Petrov-Galerkin method: Non-Newtonian fluid flow implementations

A computer code is developed to enhance the meshless local Petrov-Galerkin method capability to solve non-Newtonian fluid flow problems. In particular, the mesh-free method here, is to be empowered to simulate and solve the two-dimensional laminar incompressible power-law cases for the first time. The newly developed computer code expresses the appropriate governing equations in terms of the vorticity-stream function formulation and sustains a weighting function of unity. The local quadrature domain integrated by parts over the appropriate control volumes provides the local weak form of the considered governing equations. The extended scheme implements the moving least square interpolation technique to approximate the obtained field variables. The ability of the developed code to handle the Ostwald-de Waele (also known as the power-law) fluid flow occurrences were assessed through comparing the obtained results of the extended method to those of the conventional mesh-based methods for some benchmarking cases available in the paper. Based on this comparison, the extended code demonstrates a good capability to address the power-law fluid flow problems for different indices for one of the most popular non-Newtonian fluid flow models.

Particle dynamics in compressible turbulent vertical channel flows

In this work, we carry out direct numerical simulations of particle suspensions in the compressible turbulent vertical channel (TVC) flows with Mach number Ma = 1.5 and particle Stokes number St = 1–100. The compressibility effect is considered in the particle dynamic model for the first time in the study of compressible particle-laden wall turbulence. We find that in both incompressible and compressible flow, gravity weakens the wall-normal and spanwise fluctuations of particle velocities as the Stokes number increases. However, compared to the incompressible flow case, the compressible effect amplifies the mean velocity, fluctuations of velocity, and slip velocity of particle in the streamwise direction. The wall-normal and spanwise fluctuations of particle velocities are augmented by the compressible effect in the channel core region. Moreover, in the core region, the effect of fluid compressibility on the wall-normal and spanwise fluctuations of particle velocities attenuates as the Stokes number increases, indicating a competition between the compressible effect and the particle inertia effect. We, furthermore, conduct the quadrant analysis of the local fluctuation velocities of fluid at particle positions and observe preferential distributions in the second and the fourth quadrants at y+ = 12.5–13.5. For compressible TVC flows, the pattern of probability distributions is more elongated, and the percentage is slightly higher in the second and fourth quadrants than that of incompressible flows. This observation implies that more particles locate in the ejection and sweep events in compressible flows than that in incompressible flows, which is anticipated to influence the particle wall-normal transport.

Analysis of weakly symmetric mixed finite elements for elasticity

We consider mixed finite element methods for linear elasticity where the symmetry of the stress tensor is weakly enforced. Both an a priori and a posteriori error analysis are given for several known families of methods that are uniformly valid in the incompressible limit. A posteriori estimates are derived for both the compressible and incompressible cases. The results are verified by numerical examples.

Which Measure-Valued Solutions of the Monoatomic Gas Equations are Generated by Weak Solutions?

Contrary to the incompressible case, not every measure-valued solution of the compressible Euler equations can be generated by weak solutions or a vanishing viscosity sequence. In the present paper we give sufficient conditions on an admissible measure-valued solution of the isentropic Euler system to be generated by weak solutions. As one of the crucial steps we prove a characterization result for generating {mathcal {A}}-free Young measures in terms of potential operators including uniform L^{infty }-bounds. More concrete versions of our results are presented in the case of a solution consisting of two Dirac measures. We conclude by discussing that are also necessary conditions for generating a measure-valued solution by weak solutions or a vanishing viscosity sequence and will point out that the resulting gap mainly results from obtaining only uniform L^p-bounds for 1<p<infty instead of p=infty .

Study of compressible turbulent plane Couette flows via direct numerical simulation

Compressible turbulent plane Couette flows are studied via direct numerical simulation for wall Reynolds numbers up to $Re_w=10\ 000$ and wall Mach numbers up to $M_w=5$ . Various turbulence statistics are compared with their incompressible counterparts at comparable semilocal Reynolds numbers $Re^*_{\tau,c}$ . The skin friction coefficient $C_f$ , which decreases with $Re_w$ , only weakly depends on $M_w$ . On the other hand, the thermodynamic properties (mean temperature, density and others) strongly vary with $M_w$ . Under proper scaling transformations, the mean velocity profiles for the compressible and incompressible cases collapse well and show a logarithmic region with the Kárman constant $\kappa =0.41$ . Compared with wall units, the semilocal units yield a better collapse for the profiles of the Reynolds stresses. While the wall-normal and spanwise Reynolds stress components slightly decrease in the near-wall region, the inner peak of the streamwise component notably increases with increasing $M_w$ – indicating that flow becomes more anisotropic when compressible. In addition, the near-wall turbulence production decreases as $M_w$ increases – due to rapid wall-normal changes of viscosity caused by viscous heating. The streamwise and spanwise energy spectra show that the length scale of near-wall coherent structures does not vary with $M_w$ in semilocal units. Consistent with those in incompressible flows, the superstructures (the large-scale streamwise rollers) with a typical spanwise scale of $\lambda _z/h\approx 1.5{\rm \pi}$ become stronger with increasing $Re_w$ . For the highest $Re_w$ studied, they contribute about $40\,\%$ of the Reynolds shear stress at the channel centre. Interestingly, flow visualization and correlation analysis show that the streamwise coherence of these structures degrades with increasing $M_w$ . In addition, at comparable $Re^*_{\tau,c}$ , the amplitude modulation of these structures on the near-wall small scales is quite similar between incompressible and compressible cases – but much stronger than that in plane Poiseuille flows.